L _ Maw 49408 fl.-_.u ABSTRACT EMPIRICAL TESTING OF THE FRIEDMAN— MEISELMAN HYPOTHESIS By Harland William Whitmore, Jr. The purpose of this paper is to test empirically the Friedman-Meiselman hypothesis that the money supply is more important than government expenditures in deter- mining changes in total spending. An apparent implica— tion of this hypothesis is that monetary policy is more powerful than fiscal policy in bringing about desired changes in aggregate income. The significance of this issue is readily perceived. If monetary policy is in fact more effective, the monetary authority should assume the greater share of the burden in implementing economic stabilization policy. The goal of this paper is to provide further evi- dence that might aid in making a decision as to the validity of the Friedman-Meiselman hypothesis. This extension of evidence is three-fold. First, we retain Friedman and Meiselman's equations and their statiStical definitions of the variables, and we introduce revised data which are presumably better than theirs. Second, we discuss possible alternative statistical definitions of autonomous expenditures that were offered by Harland William Whitmore, Jr. Friedman and Meiselman's critics and use these alternative measurements to find new correlation coefficients for the same equations. Third, the simplified versions of Fried- man and Meiselman's equations of income determination are replaced by more extensive dynamic econometric models of the U. S. economy. The third extension comprises the major part of our investigation and involves an examina- tion of the dynamic properties of Klein Models II and III. In this aspect of the study we base our test of the Friedman-Meiselman hypothesis on a comparison of the dynamic and long run multipliers for the money supply and government expenditures and on an analysis of causes of changes in real net national product over the sample period. The simple single equation models we tested ini- tially did not contradict Friedman and Meiselman's find- ings that the correlation between income and the money supply is greater than that between income and autono- mous expenditures. Estimating Klein Model IIJwe found the long run and impact government expenditure multipliers to be greater than those corresponding to the money stock. Hence these estimates support the other side of the issue. An analysis of the dynamic properties of Klein Model III yielded policy implications that are less apparent than those suggested by the simpler models. Harland William Whitmore, Jr. Our findings indicate that the relative efficacy of the money supply and government expenditures depends signifi- cantly on the particular definition of money we adopt and on whether a comparison is based on concurrent or cumulative effects of the policy instruments. In short and contrary to the claims made by Friedman and Meiselman, we did not find a clear answer to the relative effective- ness of monetary and fiscal policy. EMPIRICAL TESTING OF THE FRIEDMAN- MEISELMAN HYPOTHESIS By Harland William Whitmore, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1969 ACKNOWLEDGMENTS I wish to thank Robert L. Gustafson and Thomas R. Saving for their helpful comments. I am also grateful to James B. Ramsey for his numerous suggestions and use- ful criticisms. Jan Kmenta suggested the topic to me. Also, as chairman of the dissertation committee, he served as an unfailing source of encouragement and guidance. I owe a special debt to him for the help he gave me on method- ology, content, and organization. Finally, I wish to thank my wife, Debby, who typed all but the final draft of this paper, and provided suggestions on writing style. Her patience and continu- ous understanding have made it all worthwhile. ii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES. LIST OF FIGURES . . . . . . . Chapter I. STATEMENT OF THE PROBLEM. II. THE FRIEDMAN-MEISELMAN ANALYSIS III. IV. V. Introduction. . . . . . Purpose and Procedure of the FM Analysis Definition of the Variables. . . . Results of the Study . . Conclusion . THE CRITICS OF FM . . . . . . . Introduction. . . . . . . . Hester's Analysis . . . . . . . AM's Analysis . . . . . . . . . . DM's Analysis . . . . . . . . . . Further Comments . . . . . . . . . Conclusion Appendix A . . . . . . . . . . . Appendix B . . . . . . . . . . . TESTS OF ALTERNATIVE MEASUREMENTS. . Introduction. . Re—estimation of FM Equations Using Revised Data and an Alternative Definition of M . . Re-estimation of Alternative Definitions of A to Conform to the Periods Tested by FM. Conclusion . . . . . . . . . . . RE-ESTIMATION OF KLEIN MODELS II AND III Introduction. . . . . . Klein Model II . . . . . . . . Klein Model III. . . . . . . . Results . . . . . . . . . Conclusion . . . . . . . . . . iii Page ii vi 10 19 52 56 63 63 64 74 97 104 107 109 115 128 128 130 137 142 144 144 146 157 187 193 Chapter VI. DYNAMIC PROPERTIES OF THE KLEIN MODEL II An Introduction to Dynamic Analysis Dynamic Analysis of Klein Model II VII. DYNAMIC ANALYSIS OF KLEIN MODEL III . . Introduction. . . . Fundamental Dynamic Equation for Y + T Dynamic Properties of the Adopted Fundamental Dynamic Equation for Net National Product. . Search for Crucial Coefficients that Determine Whether System is Stable and Whether Long Run Multipliers of M and G are Positive . . . . . . . Appendix . . . . . . . . . . VIII. CONCLUSION Possible Shortcomings of Klein Model III and Suggestions for Further Research . BIBLIOGRAPHY iv Page 196 196 212 215 215 216 232 247 259 264 266 270 Table 10. ll. 12. LIST OF TABLES Correlations between alternative definitions of money and income . . . . . . . Correlations between synchronous variables in nominal terms Correlations between synchronous variables in real terms Correlations between consumption and the money supply and various measures of autonomous expenditure Multiple correlation coefficients and unexplained variances of equations 8, 9, 11, and 12 with c and cf as dependent variables . . . . . . . Multiple correlation coefficients and residual variances for AM's various regression equations on the money supply, 1929-1958, excluding 1942-1946 Coefficients of determinations for various periods for FM equations using revised data and alternative definitions of the money supply . . . . . . . . . Coefficients of determination for alterna— tive definitions of autonomous expenditures Dynamic multipliers for G on the variable Y for three equation model . . . . . Dynamic multipliers for the time path of real NNP. . . . . . . . . Analysis of causes of annual changes in Annual changes in real NNP cue to current and cumulative changes in exogenous variables (in constant 1958 dollars) Page 23 55 56 73 83 90 136 141 208 236 242 245 LIST OF FIGURES Figure Page 1. Variables participation matrix for Klein Model III . . . . . . . . . . . 183 2. Dynamic multipliers of M, G, and T for current period and first eight lags. . . 237 vi CHAPTER I STATEMENT OF THE PROBLEM The purpose of this paper is to test empirically the Friedman-Meiselman hypothesis that the money supply is a more important determinant of changes in total spending than are government expenditures. An apparent implication of this hypothesis is that monetary policy is a more powerful tool to bring about desired changes in aggregate income than is fiscal policy. The significance of this issue is readily perceived. If monetary policy is in fact more effective, the monetary authority Should assume the greater share of the burden in implementing economic stabilization policy. As will be shown in detail in Chapter II, Friedman and Meiselman tested their hypothesis using estimates of alternative single-equations of income determination. One such equation Friedman and Meiselman call "the quantity "1 This expresses the level of income as theory equation. a linear function of the size of the money stock; accord- ing to Friedman and Meiselman it also represents the view that "money matters."2 Friedman and Meiselman choose as an alternative to "the quantity theory equation" a linear equation relating income to autonomous expenditures. They l name this expression "the income—expenditure theory equa— tion."3 It allegedly represents the view that "money does not matter."Ll These equations were then fitted to the data for selected sub—periods between 1897 and 1958. Correlation coefficients for the fitted equations were then compared to ascertain whether the money supply or autonomous expenditure is more important in determining aggregate income. The Justification Friedman and Meiselman offer for proceeding in this manner is that in an initial investi- gation into the question of the relative effectiveness of monetary and fiscal policy, it is preferable "to rely on a wide range of evidence interpreted on a rather simple level than on the more indirect and longer chain of con- nections inevitable in a sophisticated analysis resting on a narrower base."5 The authors freely acknowledge that since their approach is on a "simple level," their "results cannot be decisive."6 The goal of this paper is to provide further evi- dence that might aid in making a decision as to the validity of the Friedman-Meiselman hypothesis. We will extend the Friedman—Meiselman analysis. This extension will be three-fold. First, retaining Friedman and Meiselman's equations and their statistical definitions of autonomous expenditures and the money supply, we shall introduce revised data which are presumably better than theirs. These revised data cover a portion of the 1897~H958 period. Second, we shall discuss possible alternative statistical definitions of autonomous expen- ditures that were offered by Friedman and Meiselman's critics and use these alternative measurements to find new correlation coefficients for the same equations. Third, the simplified versions of Friedman and Meiselman's equations of income determination will be replaced by a more extensive dynamic econometric model of the U. S. economy. These extensions will help determine whether the conclusions reached by Friedman and Meiselman also hold using (a) revised data, (b) alternative statistical definitions of the variables, and (c) more sophisticated models of income determination. The third extension will comprise the major part of our investigation into the Friedman-Meiselman hypo- thesis. A dynamic model allows inquiry into the stabil- ity of the time paths of the endogenous variables. It also permits an analysis of the changes which occur in the endogenous variables over the sample period. The change in an endogenous variable in a given period may be traced (a) to changes in the exogenous variables during that period, (b) to changes in the exogenous variables in each preceding period, and (c) to the initial conditions which prevailed at the beginning of the sample period. Dynamic analysis of linear models also permits a derivation of so—called dynamic multipliers for the policy variables. These dynamic multipliers indicate the relative effects of, for instance, a one billion dollar change in the stock of money and of a (sustained) one billion dollar change in government eXpenditures on aggregate income in the current period as well as in each succeeding period. From the dynamic multipliers it is also possible to determine the respective long run multipliers of the policy variables on income. The central aspect of the Friedman-Meiselman tech— nique is a comparison of correlation coefficients between the stock of money and the level of income on the one hand, and between the level of government expenditures and income on the other. It is clear that a full dynamic analysis of a general equilibrium econometric model pro- vides a firmer basis upon which to Judge the relative effectiveness of monetary and fiscal policy to influence income. The plan of this study is as follows. Chapter II contains a discussion of the purpose and procedure of the Friedman-Meiselman analysis as well as of their results and conclusions. In Chapter III we review the criticisms of the Friedman-Meiselman analysis that are offered by Donald Hester, Ando and Modigliani, and DePrano and Mayer. We also include the results of the tests per— formed by the critics and the Friedman-Meiselman rebuttals. This chapter also includes our comments on points raised by all parties involved in the exchange. Chapter IV consists of two parts. In the first part we test the Friedman-Meiselman equations using revised data for the policy variables, and in the second part we use the alternative definitions of autonomous expenditures that were offered by Friedman and Meiselman's critics. This provides an investigation of the resili- ency of Friedman and Meiselman's conclusions to changes in statistical series and to alterations in definitions of the variables. In Chapter V we look for alternative analytical frameworks within which to place our examination of the relative effectiveness of monetary and fiscal policy. After a brief discussion of econometric models that are not suited to this investigation, we re-estimate Klein Models II and III to cover longer sample periods than that for which they were originally constructed. Klein Model II is re-estimated for the period 1922-1941 and 1946—1965. Klein Model III is re-estimated for the period 1923-1941 and 1947-1965. In both models the most recently revised data are used. Also, some of the equa- tions in Klein Model III have to be altered to allow a more direct comparison of the money supply and govern- ment expenditures as policy instruments. In Chapter VI we begin with a general discussion of the elements involved in dynamic analysis and examine the dynamic properties of the relatively simple Klein Model II. In Chapter VII we undertake the major aspect of our study with an examination of the dynamic prOperties of Klein Model III. We begin this chapter by deriving the fundamental dynamic equation for real net national pro- duct. Next we delineate the difficulties encountered in finding a stable fundamental dynamic equation displaying non—negative equilibrium policy multipliers. Proceeding, we examine the dynamic multipliers for the money supply and government expenditures. We then turn to an analysis of causes of changes in net national product over the sample period. This provides the basis upon which we test the Friedman-Meiselman hypothesis. We end the chapter with a general discussion of the coefficients in dynamic models which are important for stability and non- negative long run multipliers. In Chapter VIII we offer our conclusions and the policy implications of the study. FOOTNOTES--CHAPTER I 1Milton Friedman and David Meiselman, "The Rela- tive Stability of Monetary Velocity and the Investment Multiplier in the United States, 1897-1958," Stabili- zation Policies by E. C. Brown, et al. (A Series of Research Studies Prepared for the Commission on Money and Credit; Englewood Cliffs, N.J.: Prestice-Hall, 1963), pp. 170-171. 2Ibid., p. 166. 3Ibid., pp. 170—171. “Ibid., p. 167. 5Ibid., p. 170. 6Ibid., p. 174. CHAPTER II THE FRIEDMAN-MEISELMAN ANALYSIS Introduction In their study prepared for the Commission on Money 1 Friedman and Meiselman (hereafter referred and Credit, to as FM) examine two stochastic relationships relevant to a comparison of the effectiveness of monetary and fiscal policy. One relationship expresses the level of aggregate income as dependent upon the size of the money stock; the other relates aggregate income to autonomous expenditures. Before we present a detailed account of FM's analysis of these relationships, it might be useful to review briefly some fundamental concepts involved in the estimation of stochastic relationships. All stochastic relationships involving one inde- pendent or explanatory variable assign a conditional probability distribution to the dependent variable for each given value of the explanatory, or conditioning, variable. This means that given a value of the ex- planatory variable, probabilities are assigned to all possible values of the dependent variable. Each condi- tional probability distribution has a mean or expected value, referred to as the mean of the dependent variable. 8 In practice, interest is focused on how the mean of the dependent variable varies with the values assumed by the explanatory variable. The function which describes this variation is called the population regression function.2 Denoting the dependent variable by Y and the explanatory variable by X, we may wish to postulate that the popula- tion regression function is linear. That is, we may assume that the expected value of Y given X, E(Y|X), is of the form (a) B = a + ex where a and B are unknown parameters. In order to estimate these unknown parameters, a sample of pairs of observations of X and Y is taken-- (X1,Y1), (X2,Y2),....(Xn,Yn). Then, a straight line denoted by is fitted to the pairs of observations. This line serves as an estimate of the true regression, equation (a); a and B are the corresponding estimates of a and B. As is well known, the least squares method is commonly employed to obtain equation (b). This method involves finding the pair of values a and B which minimizes the sum of squares of deviations between the observed values, Y1, and the estimated values, §i3. 10 Associated with equation (b) is a statistic, de- noted by R2, called the coefficient of determination. Often this statistic is used as a measure of the "goodness- of—fit" of equation (b) to the sample data.” It is pos— sible to interpret R2 in this manner (if least squares estimation is used) because then R2 is equal to the pro- portion of the sample variation in Y that is explained by the linear influence of X.5 The coefficient of determination plays a crucial role in the FM analysis. FM based their choice between "competing" models primarily on a comparison of values of the R2's associated with these models. Now we turn to a discussion of the FM analysis. We begin with a state- ment of the purpose and procedure of this study. Purpose and Procedure of the FM Analysis Purpose of Study 6 FM proposed to examine the "relative stability" of "a relation between income [Y] and the stock of money [M] suggested by the quantity theory of money . . . [and of] a relation between income [Y] and autonomous expenditures [A] suggested by the income expenditure theory. . . ."7 These so called "relations," FM argue, are respectively the "marginal income velocity"8 of money, V', and the "marginal [autonomous expenditure] multiplier,"9 K', as expressed in the equations10 ll (1) Y a + V'M (2) Y a + K'A. Equations (1) and (2) reflect, according to FM, ". the simplest form of the quantity theory . . . and the simplest form of the income expenditure theory."ll FM readily acknowledge that the deterministic rela- tionships embodied in a macroeconomic model could easily specify income as a function of both the money supply and the level of autonomous expenditures. However, they claim an important difference of opinion exists among those in the profession . . . about which set of relations in the more generalized theoretical system is (a) critical in the sense of being in practice the primary source of change and disturbance and (b) stable in the sense of expressing empirically consistent rela— tions which can be depended on to remain the same from time to time. In other words, the crucial questions are (a) whether investment on [sic] the stock of money can better be regarded as subject to independent change, and to changes that have major effects on other variables, and (b) whether the multiplier (the ratio of the flow of income or consumption to the flow of investment) or velocity (the ratio of the flow of income or con— sumption to the stock of money) is the more stabie.12 FM limit their study to the latter question: "The aim of this paper is to present some evidence bearing on the second of the two crucial issues-—the relative stability of the multiplier and of velocity."13 FM state further that their ". . . main approach to exploring the relative stability of velocity and of the multiplier will be to 12 fit equations such as equations (1) and (2) to data for various periods of time in order to determine which of the two fits the data better."lu It seems desirable to recast FM's above statements in more formal terminology. This involves using the concepts presented in the introduction to this chapter. The purpose of the FM study is to determine whether a more significant functional relationship exists between the money stock and the level of income or between autono- mous expenditures and income. Which of the two gives a better explanation of the level of income will be de- cided by "fitting" equations such as (l) and (2) to U. 8. data for various sub-periods between 1897 and 1956 using the least squares method. The coefficients of determination, the R2's, are calculated for each equa- tion for each sub—period. The equation displaying the higher R2 more often is the one which FM declared to "fit the data best." According to FM, if one relationship provides a higher R2 for more sub—periods than does the other, the first is labeled "more stable." Further, FM argue, the one found to be more "stable" is to be stressed in economic theory and policy. Procedural Considerations FM consider several procedural questions. These pertain to the tasks of specifying alternative equations to be tested, grouping data into subperiods, and selecting 13 the national income account categories to be included in the definitions of the variables. A detailed discussion of these procedural considerations follows. Equations To Be Tested Besides testing equations (1) and (2), FM consider four others. Two include both the money supply and autonomous expenditures as exogenous with one of these equations also including the price level as an inde- pendent variable. The remaining two equations have the price level added as an explanatory variable to equations (1) and (2). These additional equations are given below along with equations (1) and (2): (1) Y = d1 + d2M (2) Y = 81 + 82A (3) Y = a3 + duM + dSP (4) Y = 83 + 84A + BSP (5) Y = Y1 + Y2M + Y3A (6) Y Y4 + YSM + Y6A + Y7P where Y = level of income, M = stock of money, P = an index of prices, and A = autonomous eXpenditures.15 It should be noted, though, that the dependent variable actually used by FM in testing each of the above equations is not the level of income, Y, but rather the level of "induced" expenditures, U. FM argue that the level of income can be defined as the sum of autonomous l4 and induced expenditures, Y = A + U.16 Therefore if, for instance, equation (2) were tested as presented above, the resulting "fit" would involve fitting Y "with part of itself."17 This, FM claim, gives an unfair advantage to A over M in determining the level of Y since the R2 for equation (2), for example, would tell "nothing about the stability of any economic relation."18 According to FM, "the independent contribution of the multiplier analysis is to predict the other component of income, consumption [i.e. induced expenditures, U] from the known (or pre- 19 FM argue dicted)autonomous expenditure component." further: "When the data are synchronous, that is, when no lagged responses are introduced, it therefore is pre- ferable to replace equation (2) by one obtained by sub- ."20 FM tracting A from both sides of equation (2). then argue that since U is the prOper dependent variable in equations involving A, U must also replace Y as the dependent variable in equations containing the independ- ent variable M. This is necessary because the purpose of the study is to compare the ability of M and A to influ- ence a common dependent variable. We therefore re-write equations (1) through (6) as: (1') U = d* + d*M l 2 x x (2') U = 81 + 32A I x x x (3 ) U = d3 + duM + dSP 15 x x x (4 ) U = 83 + BuA + ESP * x _ x (5') U = Y1 + 72M + 73A * * x * (6') U = Y4 + YSM + Y6A + y7P The reason FM give for testing equation (5') is that it allows them "to obtain a valid statistical test whether the correlation between C [our U] and M is significantly different from the correlation between C and A."2l’ 22 FM argue that the partial correlations on each of the vari— ables, M and A, keeping the other constant, indi— cate the net contribution of each to the explana- tion of C. . . . The simple correlations and the partial correlations necessarily differ in the same direction, so that the partial correlations add to our understanding of magnitude of effect but cannot reverse a conclusion [given by compari- son of the sample correlations] about which variable is more highly correlated with consump- tion. If M and A were entirely independent of one another, in the sense that there was no statistical correlation between them, then, on the average, the partial correlations would equal the simple correlations. . . . However, in prac- tice M and A are positively correlated and that is to be expected under either of the theories under consideration [emphasis mine]? . . . A positive simple correlation between A and C may simply be a disguised reflection of the effect of M on C; alternatively, a positive simple correla- tion between M and C may simply by a disguised reflection of the effect of A on C. . . . Pre- sumably the disguised effect will be smaller and less consistent than the direct effect which is why a comparison of the simple correlations is relevant and will yield the same result with re— spect to direction as a comparison of the partial correlations. But only the partial correlation can indicate how much of either simple correla— tion is produced by the disguised effect of the other variable.23 16 Hence, FM argue that the simple correlation coefficients (the positive square root of the coefficient of determi- nation) associated with equations (l') and (2') will pro- vide measures of the total effect of M and A, respectively, on U. The partial correlation coefficients, between M and C on the one hand and between A and C on the other, of equation (5') provide measures of the "undisguised"por- tion of the total effects on U. The price level, P, is added as a new variable to equations (1'), (2') and (5') to form equations (3'), (4') and (6'). This is done to check the degree to which the changes in money values of A and C and nominal values of M can be eXplained by variations in the price level.2u Parenthetically, FM decide to add P as a new variable rather than replace A, C, and M by their "real" values because "dividing variables initially expressed in money terms by some index of prices introduces spurious corre- lation, since errors of measurement in the price index are introduced alike into both sides of the equation."25 FM's procedure involves making the following four comparisons. (i) the correlation coefficient between U and M (rUM) of equation (1') with that between U and A (rUA) of equation (2') 17 (ii) the partial correlation coefficient between U and M (r ) of equation (3') with that UM-P between U and A (r ) of equation (4') UA-P (iii) the partial correlation coefficients between U and M (r ) and between U and A (r UM-A UA-M) of equation (5') (iv) the partial correlation coefficients between U and M (r and between U and A UM-AP) (rUA-MP) of equation (6').26 Time Periods FM propose to divide the period 1897 to 1958 into various subperiods and fit equations (1') through (6') to each subperiod rather than to the entire period. The reason FM give for this procedure is that they are pri- marily concerned with measuring "short-run stability of d."27 FM reason further than the realtions being compare "since the relations may differ at different phases of the cycle, it seems desirable that any one comparison should ”28 Since only cover one or more complete cycles annual data are available for the pre-World War II period and because the cycles during this period were too short to provide a large enough "number of observations to yield statistically meaningful results,"29 FM settle on a com— promise. They decide 18 to divide the period [1897—1958] for which data are available into two sets of overlapping segments, one set marked out by the troughs of the major depressions during the period (1896, 1907, 1921, 1933, 1938) except for the post- World War II period, which we have marked off simply by the end of the war; a second set, by peaks intermediate between the troughs of major depressions, except again for dates separating out World War II. . . . The dates we have used are 1903, 1913 (to get a period excluding World War I), 1920, 1929, 1939 (to get a period ex- eluding World War II), 1948, and 1957.30 In addition, the equations are tested for the periods 1938-53, 1929-58, and for the total period 1897-1958. The period 1929-58 is a "peak-to-peak" cycle of the last three sub-periods quoted above (1929-39, 1939-48, and 1948—57); 1938-53 is a "trough-to—trough" or depression cycle which completes the sequence of depression cycles covering the whole 1897-1958 period. Equations (1') through (6') are also tested with quarterly post-World War II data for period 1945111-1958IV and 19461- 1958IV.3l PM are primarily concerned with testing equations using the levels of the variables U, A, and M; but they supplement their study by testing equations containing first differences (i.e. year—to-year and quarter-to- quarter changes) of U, A, and M. The periods used in the tests of first differences differ only slightly from those used in testing equations (1') through (6'). FM also test equations with lagged explanatory variables 'I-mznnf ,j 19 of up to five periods for quarterly data 1945111- 1958Iv-32 Definition of the Variables Before the comparisons (listed on pp. 16-17) are made, FM decide which national income and product account categories to include in the definitions of autonomous expenditures and the money supply. Induced expenditures comprise those residual items of the income concept not included in the autonomous category. Because FM feel a decision cannot be made a priori, they construct certain "conditions" which,when satisfied, determine the account— ing items to be contained in the definitions. We will now discuss this construction in detail, since much of the criticism leveled at the FM analysis concerns their choice of items to be included. Money Supply In order to decide which accounting items the defi- nition of the stock of money should contain, FM pro— visionally define the money supply as currency in circu- lation plus adjusted demand deposits. Because there is a degree of substitutability between the money supply as tentatively defined and time deposits at commercial banks, FM consider time deposits for inclusion as well. If time deposits, through investigation, are found to be "close substitutes for the other monetary items . . . it 20 is preferable to treat them as if they were perfect sub- stitutes than to omit them."33 If time deposits are per- fect substitutes for money, shifting a dollar's worth of time deposits to a dollar as money would have no effect on aggregate money income or induced expendi- tures.3u Thus, FM argue that, "this suggests than an appropriate criterion whether time deposits are suffi- ciently close substitutes for other items [i.e. currency in circulation plus adjusted demand deposits] is whether income is more highly correlated with their sum than 1135 with each component separately. In other words, FM regress money on income, time deposits, T, on income, and money plus time deposits, M+T, on income. Time deposits are accepted as a perfect substitute for money if the following "condition" holds:36 ' r YM rY(M+T) > J and k rYT where rY(M+T) = simple correlation coefficient between income and the money supply plus time deposits rYM = simple correlation coefficient between income and the money supply and rYT = simple correlation coefficient between income and time deposits. 21 In effect, FM are saying that if M and T are perfect sub- stitutes (and thus both can be considered as money) then the above condition holds; therefore, we should test to see if the condition holds and if it does, it can be concluded that M and T and perfect substitutes. If this condition does not hold, then M and T are not perfect substitutes and T cannot be accepted as part of the money supply. Another alternative definition of money is con- sidered, namely, currency in circulation plus adjusted demand deposits plus time deposits plus mutual savings bank deposits plus postal savings accounts plus savings and loan shares. Mutual savings bank deposits, postal savings accounts, and savings and loan shares are to be accepted as perfect substitutes for money plus time deposits if: , rYM2 > J and r Y(M3-M2) k currency in circulation plus adjusted demand deposits plus time deposits (same as M+T) where M2 M3 = M2 plus mutual savings bank deposits plus postal savings accounts plus savings and loan shares. FM use two alternative definitions of income in their tests. These are: Y1 = personal disposable income 22 plus statistical discrepancy, and Y2 = Yl plus corporate retained earnings plus corporate inventory valuation t.37 adjustmen Ten correlation coefficients are computed for each period taking simple linear regressions of M1 (equals currency in circulation plus adjusted demand deposits), M (equals M and time deposits at commercial 2 1 banks), M3, (M2-Ml), and (M3-M2) on each alternative definition of income. Six periods were tested, three of which (1929-1939, 1940-1952, and 1929-1952) were tested by fitting the regressions to annual data. The remaining three (1946-1958, 1946—1950, and 1951-1958) were tested with quarterly data.38 For all periods in which annual data were used and for the 1946-1958 period as well, FM found M2 to be more 1 or M3 with both Y1 The correlations between M2 and income are also highly correlated than either M and Y2. higher than that of M of income.”0 1 or (M2-Ml) with each definition For the periods 1946-1950 and 1951-1958 the re- sults are mixed. The correlation between M2 and income is less than that between M and income for both Y1 and 3 Y2 in both periods. In period 1946—1950 the correlation of M2 is greater than that for either of its components, M and (M2-Ml), for definition Y but less than the l l correlation of (M2—Ml) for definition Y2. In period 1951-1958 the correlation of M is greater than that 2 23 TABLE l.—-Correlations between alternative definitions of money and income. M1 M2 M2—Ml M3 M3—M2 I Annual data, 1929—1939 Y1 .498 .849 .592 .828 (-).351 Y2 .512 .835 .548 .813 (-).361 II Annual data, 1940-1952 Yl .890 .891 !.844 .869 .506 Y2 .882 .886 .845 .858 .484 III Annual data, 1929-1952 Yl .958 .961 .882 .952 .765 Y2 .955 .958 .880 .947 .752 IV Quarterly data, 1946-1958 Yl .961 .967 .888 .962 .941 Y2 4956 .957 .873 .950 .928 V Quarterly data, 1946-1950 Yl .661 .758 .749 .893 .925 Y2 .654 .779 .807 .904 .922 VI Quarterly data, 1951—1958 Yl .896 .959 .933 .977 .982 Y2 .899 .950 .916 .967 .971 24 for either of its components for both definitions of income.ul Since the correlation of M2 was greater than both of its components for definition Yl in all six periods and greater than both of its components for definition Y2 in all but one period, and since the correlation of M2 was greater than that of M3 for both definitions of income in four periods; M2 was chosen as the best empirical definition of money to use in the equations testing the relative stability of the velocity of money and the investment multiplier.“2 Autonomous Expenditures An approach similar to the one followed in decid- ing upon what items to include in the definition of the money supply is used to settle upon a definition of autonomous expenditures. However, when dealing with this particular topic, PM do a considerable amount of juggling of the national income and product account categories. Income and Product Accounts Because of the detailed way in which FM manipulate the income and product accounts when they discuss the various candidates for inclusion into "autonomous expendi- tures," it is necessary to insert a short presentation of these accounts. This way, we will be able to fit the FM analysis into a compact framework and give an orderly v-d __... ._ "'1 -., 4 ‘Yr id 25 presentation of their tests and accompanying results. We begin with the national income account known as the "foreign transaction account." From the "foreign transactions account": (7) E = I + T + F f where E = exports I = imports Tf = transfer payments to foreigners and F = net foreign balance. The "gross savings and investment account" shows: S + W + R + D + GS + H (8) GDPI + F IPA or K + F — S + W + R + GSIPA + H where K = GDPI-D GDPI = gross private domestic investment 8 = personal saving W = excess of wages accruals over disbursement R = corporate retained earnings after taxes plus inventory valuation adjustment D = capital consumption allowance GSIPA = government surplus on income and product accounts and H = statistical discrepancy. From (7) and (8) personal saving, S, is equal to: (9) s = K + (E-I) - Tf - w - R - H - GSIPA. From the "government receipts and expenditures account": + Q + GS = T + T (10) G + T IPA p c + T + T g f i + Tb + T8 .7 26 where G = total government purchases of goods and services T8 = government transfer payments T1 = net interest paid by government Q = subsidies less current surplus of government enterprises Tp = personal tax and non-tax payments To = corporate profit tax accruals Tb = indirect business tax TS = contributions to social insurance. Substituting GS from identity (10) into identity (9), IPA S is equal to: (11) S = K + (E-I) — T - W - R — H + G + T f + Tg + T1 + Q — Tp - Tc - Tb K + (E-I) - w - R - H + G + Tg + T f _T = S i + Q - Tp - TC - Tb — Ts. Most of the alternative definitions of autonomous expenditures which FM consider can be taken from identity (11). However, when FM conduct experiments on specific items of government expenditures and receipts, an impli- cit reference is made to the "Social Insurance Funds" account. Therefore, we need a further breakdown of some items found in identity (11). From the "Social Insur- ance Funds" account: (12) TS = ng + Tg1 + Tg3 + GSSIF — T12 where ng transfers to general government Tgl federal government benefits from social insurance funds Tg3 = state and local benefits from social insur- ance funds GSSIF = surplus of social insurance funds and T12 investment income. 27 Our task now becomes one of incorporating the entries of the social insurance funds account into the government receipts and expenditures account. That is, we must break down government receipts and expenditures to show explicitly the items involved in the social insurance funds. Contributions to social insurance, TS, is the only item in identity (12) which enters explicitly into identity (10). Transfers to general government, T is not capable of explicit treatment in equation ss’ (10) since these transfers to general government are intra-governmental. Thus, they ultimately become expen- ditures of either (a) purchase of goods and services (i.e. a part of G), (b) transfer payments (i.e. a part of Tf or T8)’ (c) net interest paid (i.e. a part of Ti)’ or (d) a subset of subsidies less current surplus of government, Q. Using the following identity, items Tgl and Tg3 can be integrated into equation (10): (13) Tg = T + T + T + + s1 22 s3 T24 T g5 2 federal transfer payments to persons other 8 than federal government benefits from social insurance funds 4 = direct relief payments by state and local g governments T 5 = state and local government transfer payments g to persons other than T83 and T8“. Since FM consider the item "direct relief payments by 43 H where T T state and local governments we are explicitly enter- ing it into our accounting framework. 28 The surplus of social insurance funds could be shown by dividing the government surplus on the income and product account into two categories: (a) surplus of social insurance funds and (b) other government sur- plus; however, this division is not necessary for the FM analysis. By means of the following identity Ti2 can be explicitly entered into (10): (1”) T1 = T11 ’ T12 ‘ T13 where T1 = net interest paid Til = total interest paid T12 = investment income on social insurance funds and T13 = interest received other than investment income on social insuranbe funds. Making the above adjustments, identity (10) be— comes: + + + + + + + (15) G Tf Tg1 Tg2 _ Tg3 Tgu Tg5 Til +Q+GS =Tp+I‘c+T ‘ T ' T IPA i2 + T . s 13 b Substituting GS from (15) into (9) personal saving, S, IPA becomes equal to: (16) S = K + (E-I) — w — R - H + G + Tg1 + Tg5 + Til - Ti2 - T . s +T +T +T 23 g“ + Q — Tp - Tc - T g2 ‘ T13 b 29 Define T* and G* respectively as: v x = (16 ) T Tp + TC + Tb + TS " * = + + + + + (16 ) G G Tf Tg1 Tg2 Tg3 Tg4 + Tg5 + Til - Ti2 - Ti3 + Q. (G*-T*) is equal to minus the government surplus on the income and product account (see identity (15)). If the government surplus is itself negative, (G*-T*) equals the government deficit and personal saving is equal to net private domestic investment plus exports minus imports plus the government deficit minus (1) excess of wage accruals over disbursements, (2) inventory valuation adjustment and corporate retained earnings, (3) the statistical discrepancy, and (4) transfer payments to foreigners. Criteria Used in Experiments Having outlined the accounting framework necessary for an orderly presentation of the FM quest for an appro- priate definition of autonomous expenditures, we discuss, first, the criteria upon which they base their experi- ments. Next, we will provide a detailed account of FM's consideration of each of the alternative defini— tions. The definition which best satisfies the criteria is the one chosen as the "autonomous expenditures" 30 variable used in equations (1') through (6'), which were discussed previously. Two criteria are used to decide which items to include in the definition of autonomous expenditures. The first is similar to the one used in determining the appropriate definition of the money supply. Recall, first, that the criterion used for determining the appropriate definition of the money supply involved cor— relating income with (a) a tentatively accepted defini- tion of the money supply plus (b) an item which is being considered for inclusion in the definition of the money supply. If this correlation coefficient is greater than the correlation coefficients between income and each of the items (a) and (b) separately, the conclusion is that the proper definition of the money supply is (a) plus (b). This criterion needs little alteration to be applied to the definition of autonomous expenditures: The application of this . . . approach to the definition of autonomous expenditures can be illustrated by considering the question whether durable consumer goods should be included in consumption or in autonomous expenditures. Let D stand for consumption expenditures or durable goods, N for non—durable goods, 0 for their total, and A for autonomous, according to some tentative definition that excludes durable consumer goods but settles other doubtful items. The question to be decided is whether D + A or A alone is a preferable definition for autonomous expenditures. If D and A were perfect substitutes as autonomous or income-generating eXpenditures, then a shift of $1 from D to A or from A to D would have no effect on N. Hence N would tend to have a lower ‘v F 4.. A\V «9" Ann Q» 31 correlation with either D or A alone than with their sum. Consequently, this approach implies that a necessary condition for the inclusion of D in autonomous is that: [i] rN(D+A) > J and FM add a new twist, however, and develop a second cri- terion as well: The requirement that the sum of autonomous and induced expenditures equal income gives rise to a similar test in the other direction, a possi- bility that did not arise for the simpler example of time deposits. Suppose [i] is not satisfied. If this occured because D was a part of induced expenditures along with N, one might expect shifts between D and N to be independent of changes in A. Changes in A would affect only their sum. But this would imply that I'AD [ii] r'A(D+N) > A and t I'AN This approach therefore yields the following criterion: Possibility Condition [i] Condition [ii] Conclusion (a) Satisfied Not satisfied D autonomous (b) Not satisfied Satisfied D induced (c) Satisfied Satisfied Ambiguous 5 (d) Not satisfied Not satisfied Ambiguous These passages are quoted in full to show that even though FM argue--that if D and A are perfect substitutes as autonomous expenditures, then condition (1) will be satisfied and that if D is induced, then condition (ii) 32 will be satisfied——they are, in fact, testing conditions (i) and (ii) to decide whether D is autonomous or in- duced. For instance, if they find condition (1) to be satisfied they will then infer that D and A are autono- mous. Thus, they are treating a conditional statement and its converse as equivalent. Comparisons of Alternative Concepts The alternative definitions of autonomous expendi- 46 tures tested are: A Net private domestic investment plus net foreign balance plus government deficit on income and pro- duct account A1 = A plus consumer durable expenditures A2 = A plus imports [FM call (E—I) "net foreign balance." Therefore, they say (A+I) "is equivalent to treat- ing exports as autonomous . . . and imports as induced."u7] A3 = A plus "some part of government receipts which is equivalent to treating most government expendi- tures as autonomous and all taxes and some govern- ment expenditures as induced.”8 Before presenting the FM results from testing these alternative definitions, we tie definition A to the accounting framework we developed earlier. Identity (9) is: y. 0,} A. u- {/1 33 s = K + (E-I) - w — R — H - GsIPA - Tf. If the government surplus is negative, (9) becomes: (9) s = K + (E-I) — w - R - H + GD — T IPA f where GDIPA = government deficit on income and product account. Therefore, FM's definition of A is: (17) A = s + w + R + H + Tf. From the "Personal Income and Outlay" account: (18) Y = T + C + S p p where C personal consumption expenditures and Yp = personal income. Re-writing (18) we have: (18') C + s = Yd where Yd = Yp - Tp = personal disposable income. Therefore, (19) C + A = Yd + w + R + H + Tf = Y. Thus, if A is the autonomous concept, Y is the correspond- ing income concept. Our journey through the details of the income and product accounts has yielded the implicit and unique income concept FM are assuming in their study. Definition A vs A (consideration of consumers durables 1 as autonomous) 34 The test undertaken in comparing definition A l with definition A involves a consideration as to whether consumer durable expenditures is better looked upon as being autonomous or induced. Looking at (19) we see that durable consumer expenditures, D, is included in total personal consumption, 0, which is the induced com- ponent. However if A is the appropriate autonomous item, 1 then the induced item is C-D = N and we have N+A1 = Y. Criteria.-—The criteria for this test are exactly those quoted above, i.e. if rNAl > rND and rNA then D is autonomous provided condition (ii), rAC AC and rAN, is not satisfied. The experiments were run with annual data >1“ for periods 1929—39, 1940-52, and 1929-52 and with quarterly data for 1946-58. Instead of using A for the periods using annual data, however, FM use A* = A plus surplus of government social insurance funds, GSSIF’ less excess of wage accruals over disbursements, W.“9 Thus d * = C + A Y + R + H + Tf + GSSIF' This is the income concept FM are assuming. Al becomes A* + D = A3. There- fore, for the annual data, conditions (i) and (ii) are: r r rND rA*D (i) rNAi > Jand (ii) r“C > yand rNA* rA*N. Results of experiments.—-Since rND was found to be greater than rNA* for all periods in which annual data 1 35 were used and since r was also reater than r for ND 8 NAl the period involving quarterly data, condition (1) was 50 not satisfied in any of the periods. As for condition (ii), rA*D was not computed in any of the annual experi— ments; r!”N was less than but close to r“C for periods 1929-39 and 1940-52 and greater than but close to n“C for 1929-52.51 In the quarterly experiment, r1”,D was less than r and r was greater than rA*C.52 Thus, A*C A*N in a strict sense, condition (ii) was not satisfied for the quarterly data and for one annual period (1929-52). It was satisfied in part for the remaining two annual periods. Because the results were mixed for condition (ii) and rejected condition (1) in every case, FM decide that D is best considered as induced. Therefore, FM's tentative definition of autonomous expenditures remains as A. Definition A2 vs A (consideration of "net foreign invest- ment," (E-I), as autonomous) According to FM (E-I) might be autonomous either (a) because each item is or (b) "on its own account"53 with imports induced and exports mixed (i.e. exports partly induced and partly autonomous).514 They consider three possibilities. Three possibilities.-—(1) If imports, I, are autono— mous (and exports, E, are assumed to be autonomous) the appropriate definition of autonomous expenditures would 36 be A and the induced component would be C. Condition (1) states that the correlation between the induced item 0 and the more inclusive autonomous item, A, must be greater than the correlations between C and each of the components of A, namely A and I. It is to be noted that A is the 2 more inclusive autonomous concept since I is included in the definition, though as a negative entity. This is different from the case we considered in the previous set of experiments since durable consumption is not included in A. From this we see that condition (i) would be: 01 r > and rCA \ 2. That is, I is autonomous (and exports autonomous) if con- dition (i) is satisfied and condition (ii) is not satis- fied. FM argue, however, that since imports are a part of personal consumption, condition (1) involves correlat— ing consumption with part of itself which leads to "spurious correlation." To remedy this they re-write condition (i) as: r(C-I)I r 55 (C—I)A2. L 37 If I is induced (and exports autonomous), the appro- priate definition of autonomous expenditures is A2. Since induced expenditures plus autonomous expenditures must equal Y, and since C + A = Y and A2 = A + I, the appro- priate induced component becomes C-I. And condition (ii) becomes: A20 rA2(C-I) > ,and FM neither test nor even state condition (ii), however. They only say that if I is autonomous, then r(C-I)A must be greater than r(C-I)A2‘ Since, in general, this was not found to be the case and since, also, r(C-I)I was found to be quite large, they decide imports are best treated as induced.56 (2) Next FM consider the possibility that (E-I) is "mixed," i.e. that E is autonomous but I is induced.57 If E were autonomous, the appropriate definition of autonomous expenditures is A2 and the appropriate in- duced concept is 0-1. The components of A2 are K + (G* - T*) and E. Letting K + (G* - T*) be equal to Do’ condition (1) is: 38 ( r(C—I)DO r'(C_I)A2 > land r(C-I)E' \ If E were induced, the appropriate autonomous item would be DO and the appropriate induced item would be (C-I) + E. From this, condition (ii) is: r rDom-I) rDO[(C-I)+E] > (and Again, FM consider only condition (i). FM did not men- tion condition (ii). Nevertheless, we include what we believe to be its correct formulation. In the three periods tested FM found r(C-I)A2 to be less than r(C—I)E which they say contradicts the hypothesis that E'is autonomous.58 (3) The third and final possibility FM consider is the case in which (E—I) is "autonomous directly with imports induced and exports mixed."59 If (E-I) is autono- mous, then total autonomous expenditures is A and from identity (19), C is the corresponding induced expenditures. The components of A are K + (G*-T*) and E—I. Again let- ting K + (G*-T*) = Do’ condition (i) is: 39 . rCDO rCA > land rC(E-I)' N If (E—I) is induced, the total autonomous item is DO and induced expenditures are C + (E-I). Condition (ii) be- comes: rr DOC rDOEC+(E-I)J > (and r . \ DO(E-I) Once again FM do not consider condition (ii).6O Their full statement of what should be true if (E-I) is autono- mous directly is: . . then (C-m) [m= I] should be more highly cor- related with (DO+F) [F= (E- I) and (DO+F) = A] than with DC alone or F alone. These were, in fact, the results for all three annual periods, which were consistent with our decision to classify F as autonomous. In addition, if F is autonomous, 0 should be more highly correlated with D0 + F than with DO alone or F alone. However, we have not made these calculations. Supposedly (but PM do not state this), the reason condi- tion (1) was altered by correlating Do’ E-I, and A with C rather than with 0-1 is that I is induced and "spurious" correlation results from correlating C with "a part of itself." Definition A3 vs A (consideration of various government budget items as autonomous) 40 Division of government budget into G' and T'.-—In order to make a clear presentation of this set of FM's experiments, we bring forward identity (16) which we developed earlier.62 (16) S = K + (E—I) — W — R — H + G + Tgl + T + T - T + T + T g5 11 s2 + T s3 24 i2 - Ti3 + Q - Tp - TC - Tb - Ts. The autonomous item FM select for the first set of experi- 63 ments of this section if A = S + H + R + GS . In 3 SIF terms of identity (16): 2 = + — — + + + ( 0) A3 K (E I) W G Tgl T +Tg4+Tg5+Til-Ti2'T + T E2 E3 13 + Q ' Tp - T - T - T + GS c s b SIF 01" (20') A = K + (E-I) - W - T + (G*-T*) + GS 3 f SIF Ol” (20") A = K + F - W + (G*-T*) + GS 3 SIF where G* and T* are defined as in (16') and (16") above, and brought forward here: 41 " * = + T + + T + + (16 ) G G f Tg1 g2 Tg3 Tgu + Tg5 + Til — Ti2 — Ti3 + Q 1 «x- = (16 ) T Tp + TC + Tb + TS However, instead of breaking down the government budget in terms of receipts and expenditures, FM decide to categorize all items (save K, F, and W) on the right hand side of (20") into an induced (by assumption) com- ponent and a residual component. FM submit that the induced items of the budget are: state and local relief payments, Tg45 personal tax and non-tax payments, Tp; corporate profit tax accruals, Tc; 64 and indirect business taxes, Tb' Re-writing (20") we have: (21) A3 = K + F - W + [G + Tf + Tg1 + Tg2 +T +T +T.-T.- 23 g5 11 12 T13 + Q + GS SIF - TS] - [Tp + TO + Tb - Tg4] where the items inside the first bracket are the residual components of the government budget; inside the second bracket are the items FM assume to be induced. Substi- tuting for T8 from (12) where: (12) T3 = ng + Tg1 + Tg3 + GSSIF - Ti2 42 into the first bracket of (21) we have: (22) A = K + F — w + [G + T + T + T 3 f E2 25 + — + — - T11 T13 Q TEE] [Tp + TC + Tb - Tgu]. In terms of G* and T* the sums of the items in each bracket are respectively equal to: (23) G* — T — T g1 _Tg4+T —T s3 12 as (24) T* - TS - Tgu. 65 FM call item (23) G' and item (24) T'. Identity (21) then becomes: (25) A3 K + F — w + (G'-T'). FM, however, omit W from consideration. They also in- correctly find the residual, G', to be G* - Tg1 - Tg3 66 .. V v + Ti2 ng. That is, their G is equal to our G + Thus, their definition of A T is actually: g4' 3 (26) A3 = K + F + (G'—T') + Tg4' Because of this, A' = s + w + R + H + GS + T 3 SIF g4' d : G Since C + S = Y , we have C + A3 = Y + W + H + R + GDSIF + Tg4 are implicitly considering in this set of experiments. Y. This is the income concept which FM 43 Since FM's entire discussion of the definitions of G' and T' was limited to one short paragraph, it was necessary to develop the detailed income account relationships above so that we could keep track of and check their sketchy treatment.67 Having thus disposed of the means by which the definitions of G' and T' are derived, we now attempt to unravel the experiments which FM made using these vari- ables. Criteria for tests on G' and T'.--FM do not specify what conditions are to be satisfied in the experiments on G' and T'. However, from the above discussion of the derivation of G' and T' in which they assumed T' to be induced, and from the following paragraph we have some indication of the approach they used in this section. G' would be the appropriate government autonomous concept if T' were induced and total autonomous would be (A*+T') or A' [A* a A; of (26)]. How- ever, if T' were autonomous,A would be the appropriate total autonomous. C was correlated with A* and the components of A*, namely T', G', IN [INP = K + F]. (C—T'g8 was also cor- relate with A*, INP’ and G' Given that A3 = K + F + (G'—T') = INP + G' - T' and C + A3 = Y, [i.e. using FM's definition of A; while drOpping Tg4 which was no doubt an oversight on their part in the first place] we submit FM first test for G' being autonomous, while assuming that T' is induced. In this case, the appropriate total autonomous item is 44 INP + G' = A3 autonomous expenditures must add to Y (= C + A3), the + T' = A'. Since induced expenditures plus corresponding induced item is C - T'. Condition (1), to be consistent with previous experiments, must be: fr ' r(C-T')A' > land r(C-T')G'° \ This, however, causes one to wonder if the last line of the last quote contains a misprint. (C-T') should be correlated with A' and not A* [our A3]. Further confu- sion of FM's intentions is added by the fact that appendix Table II—A-4 has (C—T') correlated with A (and not with A* as the FM quote suggests). We grant FM the benefit of the doubt and assume they meant to say (C—T') is correlated with A'. Results.—-For periods 1940-52 and 1929-52 condition (1) is not satisfied. It is satisfied for 1929-39, how- .632, r ever. For 1946—53 r .040, (C-T')A' (C—T')INp = and r .645. Therefore, condition (i) is not (C-T')G' satisfied. Condition (ii) was not tested.69 However, to be consistent with their previous experiments, condition > and r (ii) must be rINP(C-T' + G') rINP(C-T') INPG'° 45 Further experiments on G' and T'.——Next, we submit FM test to see if G' and T' are each autonomous. If this is the case, condition (1) should be: r I,CINp and rCA > JrCG' and \rCT" This follows from the fact that if G' and T' are both autonomous, the appropriate total autonomous item is A3 and the corresponding induced item is C. Also, as FM stated when they initially outlined the condition (1) against which all the alternative definitions of autono- mous expenditures were to be tested, the correlation be- tween the induced item and the total autonomous item must be greater than the correlations between the induced item and each component of the autonomous item. However, the condition FM test violates their original statement that only one component is to be tested at one time. It seems they forgot this when considering the government budget even though they had remembered it in the previous experi- ments. Again, as has been the case in all experiments save the ones considering durable expenditures as autono- mous, condition (ii) was not tested. Condition (1) is not satisfied for periods 1940-52, 1929-52, and 1946-53. In period 1929-39 rCA = .852 and 3 46 CG' CT' while rCINP = .910. Thus, strictly speaking, the condition is not satisfied is greater than both r and r for 1929-39, either. Thus, they conclude G' and T' are not separately autonomous.7O The final test for this set of experiments in this section is the one which considers (G'—T') to be autono- mous directly. If this is the case, the apprOpriate total autonomous item is once again A and condition 3 (i) is: rCINP r > and rC(A 3‘INP) \ where A3 - INP is equal to (G'—T'). For the periods 1929-52 and 1940-52 this condition is not satisfied. However, it is satisfied for 1946-53. For the 1929-39 ); period r equals .852 and is greater than r CA3 is equal to .910. FM state that r C(A3-INP H CINP CA3 differs "71 little from the correlation between C and INP' Let us summarize the results for the test of l” "(G'-T') autonomous directly" and the test of "G' autono— mous and T' induced" in juxtaposition. In both cases, condition (1) is not satisfied for periods 1940-52 and 1929—52. For period 1946-53 condition (i) is satisfied for (G'-T') directly autonomous. Condition (1), strictly 47 speaking, is not satisfied for 1946-53 period, for G' autonomous and T' induced since r(C-T')A' = .632 and r(C-T')G' = .645 for this period. In the 1929-39 period condition (1) for G' autonomous and T' induced is satisfied. For this period condition (i), strictly speaking, is not satisfied for (G'-T') directly autono- CINP that rCA3 is greater than r(C-T')A' for the periods mous since rCA3 = .852 and r = .910. FM point out 1929-39 and 1946-53, implying that this may be enough to swing the scales in favor of the definition of (G'-T') as directly autonomous. However, for periods 1940-52 and 1929-52, r(C-T')A' is greater than rCA3' Thus, neither 72 definition seems to be preferable to the other. Alternative division ofggovernment budget.-—Possibly because of the inconclusive results obtained by the above set of experiments, FM devise a second series based upon a different division of the government budget into in- duced and autonomous components. This set of experiments covers annual and quarterly data for the period 1946-58.73 The tentative (i.e. assumed) autonomous concept is equal to A4 = S + H + R + w = K + F + G* - T*. All government expenditures, G*, except state unemployment benefits, UB , were taken as the autonomous portion of S the government budget. Defining G" = G* - UBS as the autonomous portion of the government budget, we have: .48 K + F + G" + UBS - T* (27) Au K + F + G" - (T* — UBS). Setting the induced part of the budget, (T* - UB3), equal to T", A“ becomes: (28) A4 = K + F + G" - T". Discussing the set of experiments they ran using definition A4 FM claim: A* [Au] would be autonomous if T" were autono— mous. If T" were autonomous then we should find that r(C—T")A* r(A*+T")(C—T") > (and r(C-T")T"' L However, r(A*+T")(C-T") is less than r(C-T")T" which suggests that T" is not autonomous. On the other hand, if T" were induced, the cor- relation between A* [A4] and C should be higher than the correlation between A* and T" alone, or A* and C-T" alone. However, this is not so for the 1946—58 period. A [A*] is more highly cor- related with T" alone than with C. The results are therefore inconsistent and ambiguous.7 The above statement is highly confusing in that the conditions are inconsistent with all previous statements regarding condition (1). Also, nothing is said about G". A4 would be the appropriate definition of total autono- mous expenditures if both G" and T" were autonomous. If we are testing whether T" is autonomous, assuming 49 G" to be autonomous, condition (i), in order to be con— sistent with what has been done previously, would be: rC(Au+T") rCAu > land rCT" where A4 + T" = K + F + G" and T" are the settled and unsettled components of the definition of autonomous expenditures. Similarly, if T" were induced, assuming G" to be autonomous, the appropriate total autonomous item would be A4 + T" and the corresponding induced component would be C - T". Condition (ii) would be: r(Au+T")C r(Au+T':> > land r< 11 '1' Au+T )T t It is to be noted that FM's above quoted claim can- not be referring to tests of whether T" is autonomous while assuming G" to be induced. The reason is that such an assumption precludes using A“ (FM's A*) as the initial tentative definition of autonomous expenditures. However, if we are to assume G" to be autonomous in interpreting the above quotation, we see that the posi- tions FM gave correlations r(C-T")A* and 50 r(A* + T")(C—T") should be reversed. Secondly, all three correlations should involve C and not C - T". We also see that the condition which FM submit must hold if T" is induced is incorrect. They should have used A4 + T" [i.e. A*+T"] throughout instead of A4 [A*]. Also, the relationships between the correlation coefficients are stated incorrectly. The inconsistency of this section of the study can be verified by referring to the section "Definition of A2 v§_A" where the first possibility is discussed. If the reader substitutes T" for I and abstracts from the pro- position that I is to be considered as part of consumption, the inconsistency will be blatant. Also, the reader may compare this section with the original FM statement of the proper conditions to be tested. We turn, now, to the conclusions reached by PM as to what part of the government budget should be considered as autonomous. Under the circumstances, we decided to adopt the usual treatment of the deficit alone as the autono- mous contribution of the government, though we cannot demonstrate that this is the best treatment. The alternative would be to treat government expendi- tures alone as autonomous. This, too, is not sup- ported by the data. Further, it would have led to the designation of C-T" as the induced concept. The sum of consumer expenditures and essentially tax payments would make for a rather novel consump- tion function [emphasis mine]. In addition, it does not conform to the sense of the literature on money-income relations. The procuedure we followed therefore seemed the least bad. 51 This paragraph points to the item which swung the pendulum in favor of treating the government deficit as the appropriate autonomous item as opposed to either government expenditures or receipts alone. For, taking the results of all experiments prior to this one into account, the treatment of the government deficit as autonomous leaves C as the only induced item and equa- tions (1') through (6') represent consumption functions. FM's referral to a "novel consumption function" shows that they have become confused, at this point. They have become so accustomed to using the letter C and the term "consumption" in their study that they have forgotten they originally meant "consumption" to denote "induced expenditures" and not necessarily just personal consump- tion expenditures. Also, until now we had been implicitly assuming that FM were viewing the equation U = a3 + auA as a reduced form equation. But their above comment seems to indicate that they now view it as a "consump- tion function." Before we outline the results of the tests FM made on equations (1') through (6'), it should be explained why the experiments which FM performed to arrive at "empirical definitions" of autonomous expenditures and the money supply were presented in such detail. Mainly, this was done to illustrate the degree to which their analysis deviates from the utilization of the conditions 52 as originally stated. Secondly, since a significant source of criticism in the literature concerns the treat- ment of the government budget in the definition of autono- mous expenditures, we presented the complete analysis of the alternative definitions considered so that a compari- son could be made between the method used in considering the government budget with that used in considering the other items for inclusion in the definition of autonomous expenditures. It has been shown that, even if we set aside the fact that conditions (i) and (ii) do not imply what FM contend they do, a) FM did not, except in one case, test condition (ii), b) their consideration of the various components of the government budget was inconsistent with their consideration of other items, 0) the results were not clear-cut especially for the consideration of the government budget, and d) the concept tested for govern— ment budget was not expenditures minus receipts but rather autonomous budget items minus induced budget items. Results of the Study Having decided upon the following "empirical defi- nitions" of the variables: M = currency in circulation plus adjusted demand deposits plus time deposits in commercial banks 53 A = net private domestic investment plus the government deficit on income and product account plus exports minus imports U = personal consumption expenditures (C) P = index of consumer prices, 1954 = 100 equations (1') through (6') and their counterparts in terms of first differences are now tested. The results, FM argue, clearly indicate that "the income velocity of circulation of money in consistently and decidedly stabler n76 [sic] than the investment multiplier Comparing equations (1') and (2'), FM found r to be greater than CM r for both quarterly periods and for all annual periods CA except 1929-39. They say, however, that 1929 was an exceptional year and that if the 1929-39 period is altered to include only the years 1930—39, the correlation be- tween consumption and money is greater than the correla- tion between consumption and autonomous expenditures for this period as well.77 The same results are obtained when equation (5') is considered and a comparison is made be- tween rCA-N and rCM-A’ the partial correlation coeffi— cients between consumption and autonomous expenditures and between consumption and money, respectively.78 Since FM find the correlations between money and autonomous expenditures to be positive for all periods, quarterly and annual, they say that one of the simple correlations between consumption and money and between 54 consumption and autonomous expenditures is probably the disguised effect of the other variable on consumption.79 Since the partial correlation coefficients between con- sumption and money is greater than those between consump- tion and autonomous expenditures, FM argue that the effect of money on consumption is disguised in the simple corre- lation between consumption and autonomous expenditures rather than the other way around. In their comparison of equations (3') and (4') FM found the simple correlations between consumption and money to be greater than those between consumption and autonomous expenditures in all periods except for the period 1938—53 "for which both correlations are nega- tive."81 Testing equation (6') FM found the partial correlations r to be consistently greater than 82 to be greater than rCA-MP' CM-P rCA-P and, also, rCM-AP These findings, FM argue, "clearly indicate that the results of the comparisons are even more one-sided when the statistical effects of the price level are held constant."83 The results of tests using first differences in equations (1') through (6') are too sketchy to merit any discussion. FM consider finally lagged values of M and A and regress them on C for the quarterly time period 194SIII-l958lv. We shall not discuss the results of this part of their study, either, since it seems to 55 be an after thought and violates credance of their treat- ment of equations (1') through (6') as reduced forms. The following tables show the results FM obtained from making the four comparisons of correlation coefficients listed in pp. 16-17. TABLE 2.——Correlations between synchflonous variables in nominal terms.8 Income Expenditure Quantity Theory Theory Period rCA rCA-M rCM I'CM-A rYM rAM 1897-1958 .756 —.222 .985 .967 .988 .791 1897-1908 .587 -.496 .996 .996 .991 .622 1903-1913 .485 —.l27 .997 .996 .987 .495 1908-1921 .672 .400 .995 .993 .975 .646 1913—1920 .791 .423 .991 .980 .975 .761 1920—1929 .569 .288 .968 .956 .933 .524 1921—1933 .843 .884 .897 .923 .810 .586 1929-1939 .937 .688 .912 .529 .915 .880 1933-1938 .935 .414 .991 .938 .985 .921 1938—1953 .397 -.328 .958 .955 .966 .500 1939-1948 .173 —.562 .963 .974 .967 .327 1948-1957 .747 .361 .990 .980 .986 .719 1929—1958 .705 -.424 .974 .957 .983 .784 56 TABLE 3.——Corre1ations between synghronous variables in real terms. 5 Period rCA-P rCA-MP rCM-P I’CM-AP rYM-P 1897-1958 .157 .314 .878 .888 .901 1897-1908 .290 -.570 .911 .935 .910 1903—1913 .126 -.113 .918 .917 .757 1908-1921 -.673 -.443 .919 .880 .137 1913—1920 -.701 —.662 .863 .848 .059 1920-1929 .611 .190 .970 .954 .944 1921-1933 .611 .387 .956 .940 .917 1929-1939 .909 .807 .946 .887 .912 1933-1938 .442 .097 .952 .940 .896 1938-1953 -.513 -.472 -.342 -.261 -.010 1939-1948 -.904 -.929 .083 .505 .287 1948-1957 -.606 .203 .856 .771 .781 1929-1958 -.207 -.352 .222 .360 .485 Conclusion The conclusions which FM draw from their results show they had no trouble selecting the "more stable rela— tionship." This is evidenced by their following comments: The major implications of our findings are so obvious as to require little elaboration. For scientific analysis, they indicate that the quantity-theory approach to income change is likely to be more fruitful than the income- expenditure theory approach; that the first 57 corresponds to empirical relations that are far more stable over the course of business cycles than the second. . . . For economic policy, our findings indicate that control over the stock of money is a far more useful tool for affecting the level of aggregate money demand than control over autonomous expenditures . . . 86 FM conclude that ". . . it is what monetary policy does to the stock of money rather than what it does to interest rates that matters most."87 Also, "changes in the stock of money have an effect on a much broader range of capital assets and correspondingly broader range of associated 88 ,1 E 1 i i expenditure" than is recognized by the income-expenditure approach. Much of the procedure FM followed in this study has been subjected to criticism in the literature. Chapter III will fully present the objections others have found to this study as well as some criticisms not yet appear- ing in print and, so far as is known, are original with this author. FOOTNOTES--CHAPTER II 1Milton Friedman and David Meiselman, "The Rela— tive Stability of Monetary Velocity and the Investment Multiplier in the United States, 1897-1959," Stabili— zation Policies by E. C. Brown, et al. (A Series of Research Studies Prepared for the Commission on Money and Credit; Englewood Cliffs, N.J.: Prentice-Hall, 1963), pp. 165-268. 2Arthur S. Goldberger, Topics in Regression Analy- sis (New York: The MacMillan Co., 1968), p. 3. 3J. Johnston, Econometric Methods (New York: MCGraw—Hill Co., Inc., 1963), pp. 9—11. ,uGoldberger, p. 40. 5Johnston, p. 31. 6Friedman andffleiselman, p. 169. 7Ibid., p. 165. 8Ibid., p. 170. 9Ibid. 10Ihid. 11Ibid., p. 171. l2Ibid., pp. 168-619. 13Ibid., p. 169. lL‘Ibid., p. 174. 15Ibid., pp. 175-178. 16Ibid., p. 175. l7Ihid. 18Ibid. 58 59 19Ibid. 2OIbid. 21Ibid., p. 177. 22FM consistently, when describing the equations to be tested, refer to induced expenditures as "consumption" and label this item with the letter C. This has caused a great deal of confusion as evidenced by the comments of the critics of PM. PM later acknowledged that the confu- sion was caused by their unfortunate choice of a substi— tute term for induced expenditures. In fact, the termi— nology even caused FM to stumble as we will demonstrate later in this chapter. 23 24 Friedman and Meiselman, pp. 177-178. Ibid., p. 178. 251bid. 26Ibid., pp. 186-209. 27Ibid., p. 174. 28Ibid. 29Ihid., p. 175. 3OIbid. 31Ibid., p. 190. 32Ibid., pp. 234-241. 33Ibid., p. 182. 3thid. 35Ibid. 36Ibid. 37Ibid., p. 182. See, also, p. 242. 38Ibid., p. 243. 60 39Each entry is the coefficient of determination. The signs in parentheses are the signs of the correla- tion coefficients. This table is reproduced from Friedman and Meiselman's Appendix Table II-Al, Experiments with Alternative Concepts of Money and Income, p. 244. 40 Ibid., pp. 243-246. “lIhid. u2Ibid., p. 246. 73Ibid., p. 243. See, also, p. 238. } 05"“ [ Ibid., pp. 182-183. uSIbid., p. 183. 46 Ibid., pp. 246-247. u7Ibid., p. 247. 78Ibid. ugIbid., p. 249. Solbid. 51Ibid. 52Ibid. 53Ibid., p. 251. BuIbid. 55Ibid. 56Ibid. 57Ibid. 58Ibid., p. 252. 59Ibid 60After the first draft of this paper was written, M. K. Lewis in a comment "Friedman and Meiselman and Autonomous Expenditures" in the American Economic Review, LVII (June, 1967), also points out the inconsistencies in the FM procedure. Lewis claims that "a correct application 61 of their [FM] criteria for selection provides support for the definitions they used" (p. 542). Since in the next chapter we are going to show the FM criteria are totally invalid in the first place, we will not concern ourselves with Lewis's detailed criticism of FM's application of these criteria. 61Friedman and Meiselman, p. 252. 62 63 See p. 28 above. Friedman and Meiselman, p. 254.. 6“Friedman and Meiselman, p. 254. 651219- 6622;9- 67221g. 681212- 691pig., pp. 254-255. 70Ibid., p. 257. 7llbid., p. 255. 72;p;q., p. 257. 73;bid., p. 255. 771212-, p. 256. 751212° 76 Ibid., p. 186. 77Ibid., p. 189. 78 . Ibid., pp. 204—205. 791bid., p. 204. 80Ibid. 81 Ibid., p. 207. 82Ibid. 83Ibid. 62 4 8 This table is reproduced from FM's Table II-l on p. 190. 8 5This table is reproduced from FM's Table 11-3 on p. 228. 6Friedman and Meiselman, p. 213. 87 881bid., p. 217. Ibid., p. 216. (l) (J ‘11 .1' (I) fin Va CHAPTER III THE CRITICS OF FM Introduction In this chapter we shall review the criticisms of the FM analysis contributed by Donald Hester,l Albert Ando and Franco Modigliani2 (hereafter referred to as AM), and Michael DePrano and Thomas Mayer3 (hereafter re- ferred to as DM). The three sets of authors frequently offer similar criticisms. We shall try to avoid repeti- tion by including in our review of AM only those criti- cisms not fully discussed by Hester and in our review of DM only those criticisms presented by neither Hester nor AM. FM have responded to the objections raised by their critics;u the critics, in turn, have issued rejoinders.5 We shall include the pertinent remarks from these ex- changes as we proceed, rather than place them in a sepa- rate section. Along the way, we shall also interject our own criticisms, comments, and interpretations. 63 64 Hester's Analysis Objections to the FM Analysis Hester limits his comments to FM's equations (1') and (2')6 and, therefore, to only the first of the four comparisons listed above.7 He objects to the national income and product account categories FM included in their concept of autonomous expenditures and to the "criteria" (i.e. conditions (i) and (ii)8) FM used to choose these categories. Definition of Autonomous Expenditures Hester submits that T and I "are not likely to be exogenous"9 and that, therefore, A [= K + (G-T) + (E-I)] contains endogenous elements. Setting aside the fact that imports may be endogenous, Hester argues that if T is induced, the proper autonomous concept is L where L = K + G + (E-I). He then proceeds to show that it is pos- ,10 CL without a 11 correspondingly high value being assigned to rCA' Thus, sible for r to take on a "high value' if the proper autonomous item is L and not A we could arrive at a high value of r which would support the CL "Keynesian" income-expenditure theory and a low value of r which would reject its importance. Since FM used A CA as their measure of autonomous expenditure and since L is a more reasonable definition, Hester states that: 65 "Friedman and Meiselman have stacked the cards against the Keynesian model in their comparisons by ignoring the fact that taxes are a function of income."12 Hester considers next the dependency of imports on income and says that "by an argument completely analogous to that for taxes, failure to eliminate imports from L will serve to misrepresent the autonomous expenditure model."13 Thus another possible and more "appealing" measure of autonomous expenditure, Hester feels, is L' where L' = G + GPDI + E = L + I + D where D is capital consumption allowances. Hester adds D to K because he feels that gross private domestic investment is more accurately measured by the national income statisticians than is K. This is because K is computed by subtracting D from gross investment and D is only an "imperfect "14 approximation for actual depreciation. Later he states that while in principle net investment is the ideal concept, it is well known that depreciation mea- sures are highly imperfect and that measurement errors bias correlation coefficients toward zero. The fact that we don't know how to measure depre- ciation is not grounds for rejecting an autono- mous expenditure theory.15 Hester proposes two more revisions in the autono- mous concept. First, "spurious correlation exists be- tween M [our I] and C"16 since "part of imports is in- 17 cluded in consumption." Therefore, Hester subtracts imports from L' which results in a new concept, L", 66 equal to G + GPDI + E - I. Since L" = L' - I and since L' = L + I + D, Hester defines L" = L + D. Therefore, Hester claims the "spurious correlation" of imports with consumption will be removed by correlating C with L".18 This, of course, is nonsense! L' does not contain the variable I; subtracting I from L' re—introduces imports into the autonomous concept and simultaneously, therefore, re-introduces the so-called "spurious correlation" with C. In this case, Hester has done precisely the opposite of what he claims to have done. The second revision Hester proposes is L"'. L"' is defined as L" minus 19 This adjustment is made because inventory investment. Hester feels it is proper to assume that inventory in- vestment (which is included in GPDI) is endogenous; that is, "variations in consumption may cause negative varia— tions in inventories."20 FM's reply.--In response to Hester, FM point out that their initial treatment does allow both T and I to be endogenously determined. FM contend that (G-T) and (E-I) are exogenous on their own account, with T and I each being endogenous (both are functions of income) and G and E each being "mixed" categories, i.e. "'the 21 sum of an induced and autonomous item.'" FM feel that "'foreign countries . . . spend on U. S. goods . . . [a certain amount which is determined exogenously] . . . plus 22 I" what they earn for (U. S.) imports. As for government n) I In 'vh ‘4. 1h ‘A ‘V “TIN. 67 expenditures, FM say "we can regard the government as deciding that total expenditures shall equal what is raised by taxes plus (or minus) a specified sum to be financed by borrowing (or used to repay debt)."23 If we apply FM's rationale for defining a variable as falling into a "mixed category" to, for instance, consumption, we easily discover the absurdity of such a definition. The consumption function, C = d + BY, is the sum of an "autonomous" amount, a, and an induced amount, BY. Therefore, to be consistent with their above treat- ment of exports and government expenditures as "mixed," FM would have to regard consumption as "mixed" as well. Consumption could be considered as endogenous, according to their reasoning, only if the consumption function were C = BY. Clearly, if a variable is influenced by other variables specified in the set of structural equations, it must be considered as an endogenous variable. If it is not endogenous, it must be exogenous since it is then completely determined outside of the model. All this is not to say that a model could not be constructed which would treat, for instance, the government deficit as exogenous while simultaneously treating G and T each as endogenous variables; but it seems it would be necessary to include in the model some type of "decision" or "reac- tion" functions which would explain how the government «1‘ UL. fif' U\- r I. ll. r- 6 V1; 6». 68 will behave in order to keep the deficit at its exoge- nously determined level. FM attempt to show that their model is consistent with treating T as endogenous. This is done by defining the consumption function as C = a + bY and defining in- come as Y = C+A where A = K + (G-T) + (E-I). These two equations form a "complete model";2u the reduced forms for C and Y are C=—§—+——AandY=—§—+LA l-b l-b l-b 1-b ° FM then argue that "the value of T is not required for 25 the solution"; but, setting T = f + gY, T can be derived as According to FM, "these equations demonstrate that, con- trary to Hester's assertions, there is no inconsistency between our model and the treatment of taxes as in- duced."26 This argument does not seem to be correct, since T enters the variable A. A is then equal to K + G - f - gY + E-I. Thus, A is induced and the FM reduced forms are not reduced forms at all. The model FM present is inconsistent with taxes being endogenously determined and A exogenous. 69 PM respond to Hester's demonstration that the corre— lation coefficient between C and L will be at least as great as the correlation coefficient between C and A with the assertion that "since L = A + T, it is easy to see that rCL is the correlation of C with A plus part of itself and hence will be larger than rCA if rCA < 1."27 This is disturbing on two counts. First, this statement is inconcsistent with FM's remark that "the value of T is not required for the solution" to the system of equations discussed in the above paragraph because they are now arguing that A, in fact, includes T. This verifies that our argument in the above paragraph is a proper one. Secondly, since L = A + T = K + G + E - I, it is not true that L is the sum of any item "plus a part of itself." L, in fact, only removes a variable previously included in the definition of autonomous expenditures. It seems FM refuse to admit the obvious fact that the government deficit plus taxes must equal government expenditures. FM state that "we examined explicitly all but one of the alternative definitions he [Hester] proposes (we did not consider L"', which excludes inventory invest- ment); . . . we presented statistical evidence on each; and . . . we explained why the evidence seemed to favor the concept we finally used."28 Since FM based their choice of the definition of autonomous expenditure on 70 the criteria presented above, we proceed to Hester's criticism of these conditions. Criteria for Defining Autonomous Expenditures Regarding the criteria (conditions (i) and (11)) FM used to decide which national income account cate- gories to include in the concept of autonomous expendi- tures, Hester writes: Suppose there exist two doubtful components of autonomous expenditures, G and H. Somehow I is known to be autonomous. Then Friedman and Meiselman argue that a necessary condition for G to be autonomous is that PC(I+G) > rCI and fog. Suppose in fact G is autonomous. Assume H is also autonomous and negatively correlated with G, but independent of I. In this case, rCI may exceed PC(I+G) and G will be erroneously rejected as autonomous. Their test is sensitive to the variances and covariances of I, G, and H. The Friedman—Meiselman test is ill-suited for its task;components of autonomous expenditure will not be reliably selected by their procedure. Theory or "intuition" is necessary to specify components of autonomous expenditure. 9 Some remakrs on Hester's criticism.—-Hester is correct in his assertion that FM's criteria are invalid. This holds true not just in the case of finding a defi- nition of autonomous expenditures, but also in the FM attempt to define the money supply. We have shown above30 that when FM consider a defi- nition for money they state that if M and T are perfect substitutes then the condition, rY(M+T) > rYM and rYT’ holds. They argue that they should, therefore, test to 71 see whether the condition holds. If it does they will claim M and T are perfect substitutes (implying both items can be considered as money). If the condition does not hold, then M and T are not perfect substitutes and T cannot be accepted as part of the money supply. This method is invalid. If the condition is found to be satisfied, we cannot say anything about the sub— stitutive nature of M and T. The condition, rY(M+T) > is not a sufficient condition for M and T r and r YM YT’ to be perfect substitutes. Therefore, even if the condi- tion is satisfied, M and T may or may not be perfect substitutes. We will call the statement "M and T are perfect substitutes" p; and call the statement "the con- dition holds" q. The statement "M and T are not perfect substitutes" will be denoted by ~ p, and "the condition does not hold" we will call ~ q. The conditional state- ment "if p then q" is not equivalent to the converse "if q then p." The contrapositive "if ~ q then ~ p" is equivalent to "if p then q." Therefore, if the condition rY(M+T) > rYM and rYT is not satisfied we can say that M and T are not perfect substitutes; but we can say this only if the original conditional statement is true. However, PM are completely incorrect in treating the conditional statement as equivalent to the converse. We can now interpret Hester's criticism of FM's condition (1) concerning the definition of autonomous 72 expenditures. (Condition (1) is directly analogous to the single condition for money.) Hester is saying that since the element we are testing can be autonomous even though condition (i) is not satisfied, the contrapositive and, therefore, the original conditional statement is not true. That is, he argues that G being autonomous is not sufficient for condition (i) to hold; or, in other words, it is not true that a necessary condition for G to be autonomous is that rC(I+G) > rCI and rCG' Therefore, he concludes that the FM experiments are based on an invalid statement causing an invalid statistical definition of autonomous expenditures. Hester only argues that the conditional statement is not true. In Appendix A we offer a formal proof that Al and A2 being perfect substitutes as autonomous ex- penditures does not imply that condition (1) holds. Obviously, this proof will also show, simultaneously, that the conditional statement for M and T is not true either. Hester's Tests and Results Hester's battery of tests is not very elaborate. It involves computing (using annual data) the correlation coefficients between C and each of the four definitions of autonomous expenditures which he feels are improve- ments upon FM's A. We repeat them here in order to aid the continuity of presentation. They are: 73 a) L = G + K + E - I b) L" = G + GPDI + E c) L" = G + GPDI + E — I d) L"' = G + GPDI + E - V where V is change in inventories. The correlation coefficients were computed for various subperiods between 1929 and 1958. We reproduce Hester's table whicn presents the results of his tests. TABLE 4.--Correlations between consumption and the money supply and various measures of autonomous expenditure.3l Years rCM I'CA I'CL r'CL' rCL" I’CL'H 1929-1939 .912 .937 .903 .957 .933 .976 1933—1938 .991 .935 .995 .992 .997 .997 1938-1953 .958 .397 .755 .837 .809 .817 1939-1948 .964 .173 .471 .566 .519 .527 1948-1957 .990 .756 .925 .964 .961 .969 1929-1958 .974 .706 .915 .953 .943 .949 Hester finds, in general, that his four definitions performed better than the FM definition in the sense that "with the exception of the 1929—1939 period, the correla— tion between consumption and every proposed measure of autonomous expenditure exceeds r A as expected."32 C Hester concludes his paper with an evaluation of the scope of the FM analysis: ". . . as both autonomous expenditure and quantity models predict a high correlation 74 between Y and M (and hence C and M), high correlation be— tween the money supply and consumption are of little value in discriminating between the models."33 FM's Reply It is FM's contention that "the appearance of sub- stantial difference between his [Hester's] results and ours derives primarily from the Shorter period his calcu- "37 FM point out that Hester's alternative lations cover. measures of autonomous expenditures display higher correla— tion coefficients than does the money supply only for periods 1929-1939 and 1933-1938.35 AM's Analysis Critique of FM's Study FM's Treatment of the Autonomous Expenditures Model AM assert that the FM results from testing the cor- relation coefficients between autonomous eXpenditures and "36 This irrelevancy, AM consumption are "irrelevant. submit, arises from a misspecification of the consumption function; the inclusion of the war years, 1942-1946, in three of the six subperiods after 1929; the inclusion of induced components in the explanatory variable which re- sults in a least squares bias; and the combination of exogenous variables to form a single variable. 75 Misspecification of the consumption function.-—AM claim the misspecification of the FM consumption function is due to the structural equation (a) C = a + bY + 0 where 6 = random disturbance. We recall that equation (a) along with (b) Y = C + A leads to the reduced form equation for C (c) C = —E— + Ig6 A + l 37 l-b ——— 6. 1-b AM object to equation (a) and say it should be replaced by the more "conventional" consumption function (a') C = a + bYd + 0 where Yd = disposable income. So far, this is the same argument as Hester's. However AM also replace (b) with (b') Yd = C + S. Equations (a') and (b') yield a new reduced form equation for consumption _a_+ ' = —— (C ) C l—b l-b l—b 76 AM then argue that since S = A — R — H — W - T the FM f, reduced form is valid only if Yd is replaced by Yd + R + H + W + Tf in equation (a'). "But this surely in- volves a grievous misspecification of the consumption function "39 The war years.--AM claim that during the war years . consumers may have been persuaded to consume abnormally small proportions of their income for patriotic reasons, and/or they may have changed their consumption habits in response to rationing and to unavailability of some goods. Hence any test including these years is worthless unless it has been shown that the results are largely invarianE whether these years are included or omitted. AM tested four regression equations to discover whether this invariance existed. One equation regressed A on C for the 1929-1958 period; the second regressed A on C for 1929-1958 exclusive of the years 1942-1946; the third regressed M on C for 1929-1958; and the fourth regressed M on C for 1929—1958 exclusive of 1942—1946. The coefficient of determination between A and C rose from 0.49 to 0.92 when the war years were excluded; the R2 between M and C changed from 0.94 to 0.98 when the same years were not included.“1 Because of the differ- ences in the R2's for the non—war period and the total period, AM claim "the omission of these years [1942—1946] makes an overwhelming difference.”2 FM discount AM's results Since FM based their con— clusions on the results for Shorter subperiods rather 77 than on the period as a whole.“3 However, the lowest correlation coefficients FM obtained were for the periods 1938-1953 (rCA = .397) and 1939-1948 (rCA = .173) both of which include war years.uu Induced components in the explanatory variable.-- The third objection AM raise concerns the explanatory variables A in equation (c) and S in equation (c'). Dropping H and W from the definition of personal saving and re-writing S as K + G + E - [T+I+R], AM argue that S (and, also, A) cannot be considered autonomous because the variables T, I, and R are not ". . . uncorrelated with the residual error of the consumption function "“5 AM decide to call "autonomous" those variables that are expected to be uncorrelated with the error term of the test equation under consideration, and call "induced" all other variables. Autonomous variables in this sense are not necessarily "exogenous" in the usual sense of being deter— mined entirely outside the economic system and therefore uncorrelated with the error term of any structural equation. Thus exogenous variables are autonomous, but not al autonomous variables are necessarily exogenous. Thus, AM conclude: The three components in the square brackets could not possibly be regarded as autonomous in the sense defined above. The movements of each of these three components are closely related to that of consumption (either directly as in the case of imports or through income as in the case of taxes) which in turn is clearly related with the error term 2 [our 6] of the consumption function. Since S thus includes items correlated with e [emphasis mine], it will in general be 78 itself correlated with s. It is well known that under these conditions direct regression of C on S will yield biased estimates of the coeffi- cients as well as of the variance of the error term. Two comments are necessary before proceeding. First, by defining as autonomous "those variable ex- pected to be uncorrelated with the error term of the test equation under consideration," AM are allowing lag- ged endogenous variables with no autocorrelation to be defined as autonomous. Secondly, AM's discussion of items being related to C and therefore with e is valid only if "related" is synonomous with "dependency." The term, related, cannot mean merely that the components are correlated with consumption. For instance, we could assume G to be exogenous and find that G is also highly Correlated with C; but the correlation with 0 would not imply that G was also correlated with 5. AM argue, further, that since 5 will be positively correlated with I and T, fluctuations in a will tend to be negatively correlated with fluctuations in S (and also A). Thus, the coefficients of the regression equa- tions associated with (c) and (0') will be biased down— ward. This may cause "the regression coefficient of S (or A) on C and hence also the correlation coeffi- "78 The cient . . . [to be] zero or even negative relationship between R2 and the coefficient of S (or A) in the regression equation is 79 2(Si-S)2 2(01-5)2 where Si and C1 are the observation on S and C respec- tively and S and C are the correspondingb means of the sample values.“9 Clearly the smaller (% b) is in absolute value, other things remaining unchanged, the smaller R2 is. However, the SLS estimate of (IgE) is 2(S1 -S)(C i-C) equal to ; it is extremely difficult to 2(Si -S)2 consider a smaller (% b) without simultaneously allowing 2(Si -s)2 ———————§ to vary as well. Therefore, AM's last state- ment is not obviously true. Combining exogenous variables.--AM point out that the various exogenous variables FM consider as components of A, namely (G-T), K, and (E-I) can be combined to form A only if the coefficients of each of these variables explaining induced expenditures are equal to each other.50 The veracity of this statement can be illustrated by considering (1) Ut = 80 + BlAtl + B2At2 + E to be the true model, where Atl and At2 are two exogenous items which FM include in their one variable, A. Assume the estimated equation is 80 = * The estimates of bl, 81, is Z — - (3) 8 — - l m 2 2 (1+2)(1+2) t[(Atl+At2)-(A1+A2)] Since the number of observations of Al equals the number A + A = A + A of observations of A2, 1 2 l 2. Thus, Z I 1 v A t rYAl and rYA2; however FM also argue that all correlation coefficients with Y should substitutes implies RY(A1+A2) be positive so that our condition is equivalent to theirs. Using the definitions of r and rYA and (3) we YAl 2 see that: RY(A +A ) m2 m + m2 m + 2m m m (5) 1 2 _ Y1 11 Y2 11 Y1 Y2 11 2 - 2 I'YA (mll + m22 + 2ml2)mYl 1 '4' C‘ . a 112 R 2 2 (6) Y(A1+A2) = in m22 + mY2 m22 + 2mY1 mY2 m22 2 2 rYA (mll + m22 + 2m12)mY2 We see that since the denominators in (5) and (6) must be positive, R2 Y(A1+A2) 2 is greater than 1 if and only if r YAl 2 ? 2 (7) mY2 m11 + 2mY1 mY2 m11 > (m22 + 2m12) mY1 2 RY(A1+A2) while 2 is greater than 1 if and only if r’YA 2 2 ? 2 (8) in m22 + 2mY1 mY2 m22 > (mll + 2m12) mY2' We now substitute (4) into (7) and (8) respectively. If Al and A2 being perfect substitutes does imply 2 2 2 . RY(A1+A2) > r and rYA then this substitution Will YAl 2 yield results which are always true. Substituting (4) into (7) we have ? 2 2 ' - (7 ) mY2 m11 mY1 m22 > 2m2 mY2 m11 ‘ in m22 _ 2m m m Y1 Y1 Y2 ll ' 113 Assuming mYl > mY2, this yields 2 3 3 ' - - (7' ) mY2 m11 mY1 mY2 m11 mYl m22 2 ? 2 3 + mY1 m22 mY2 > 2mY1 mY2 mll ‘ 2mY1 m22 2 2 ' 2mY1 mY2 m11 + 2mYl mY2 m11 which reduces to (7"') m3 m + m2 m m 3 m3 m Y1 '22 Y1 22 Y2 Y2 11 + m m2 m Y1 Y2 11° Inequality (7"') becomes 2 ? 2 it (7 ) m22 mYl (mYl + mY2) > m11 mY2 (mY2 + mYl)° The direction of the inequality is not reversed if mY2 + * mY1 > 0. Assuming mY1 + mY2 > 0, (7 ) becomes (7**) m m2 3 m m2 22 Y1 11 Y2 “which is not always true. Substituting (4) into (8) we have (assuming mYl > mY2) 114 2 3 3 (8') mY2 mY1 m11 ' mY2 m11 ‘ mY1 m22 & —2m +m2 m m m Yl 22 Y2 Y2 11 + 2m2 + 2m Y2 m m m Yl 22 Yl mY2 22 - 2m Y1 Y2 22 ’ which reduces to 2 O - 2 (8") mY2 m11 (mY2 + mYl) < mY1 m22 (mYl + mY2)‘ Assuming mY2 + mYl > 0, (8") becomes 2 2 99* (8 ) mY2 m11 < mY1 m22 which is precisely the inequality (7**). However, as was pointed out (7**) is not obviously true. If mYl > 8**) would not be satisfied if m were large mY2’ ( 11 enough relative to m22. Therefore, FM's conditional statement that if Al and A2 are perfect substitues then RY(A +A ) > rYA d 2 i t t l 2 1 an rYA2’ S no rue. Appendix B AS stated on p. 81, we are interested in the com— 2's in equations (1) and (2). 2 2, parative Sizes of the R As shown in Appendix A, the R2 of equation (2), R is 2 2 2 Y2) _ mY1 + mY2 + 2"Y1 "Y2, ("11 + "22 + 2ml2meY ("11 + "22 + 2"127"YY (m + m 2 _ Y1 2 l’ is The coefficient of determination of equation (1), R 2 YAl-A2 le + YA2-Al mY2 (10) R1 "YY 2 "ll "Y2 ‘ 2" 2 ("11 "22 ‘ "l2 2 "22 "Y1 + 12 "Y1 "Y2 ) "YY where bYA -A and b are defined in Appendix A. l 2 2 1 We set out, now to determine whether R3 is greater than Hi. The ratio Rg/Ri is greater than unity if and only if YA -A 2 2 2 2 ("11 "22 ’ "12) "Y1 + (mll "22 ' "12) "Y2 + 2m (11) 2 "22("11 + "22 12) "Y1 + "11("11 + "22 2 + 2(mll "22 ' "12) "Y1 "Y2 ‘ 2"12("11 + "22 + 2"12) "Y1 "Y2 2 + 2ml2)mY2 115 116 or (11') [m m - m 2 — m (m + m + 2m )] m2 11 22 12 22 11 22 12 Y1 + [m m - m 2 - m (m + m + 2m )] m2 11 22 12 11 11 22 12 Y2 2 + 2["11 "12 ‘ "12 + "12<"11 + "22 + 2"12)J ? mY1 mY2 > 0. The coefficient of mil is > 0 when m m - m 2 - m (m + m + 2m ) > 0 11 22 12 22 11 22 12 or m m - m 2 — m m — m 2 — 2m m > 0 11 22 12 11 22 22 12 22 2 . . or (ml2 + m22) < 0 which is impossible. The coefficient of m§2 is > 0 when m m — m 2 — m (m + m + 2m ) > 0 11 22 12 11 11 22 12 or m m — m 2 - m 2 - m m — 2m m > 0 11 22 12 11 11 22 11 12 2 or (m12 + mll) < 0 which is impossible. Since the coefficients of mYl and m§2 are always 1 0 and since mil and m§2 are always 1 0, the left hand Side of (11') is > 0 if and only if 117 2 + 2[ 22 ' m12 "12 ("11 + m22 + 2"12)J "Y1 "Y2 mll m 2 2 (12) > ' ':"11 "22 ”"12 ' "22 ("11 + "22 + 2"12)J "Y1 2 2 "22 ' "12 ‘ "ll ("11 + " + 2"12)J "Y ["11 22 2 or ? (l3) 2(mll + "12)("12 + "22) "Y1 "Y2 > 2 2 2 ("12 + "22) "Y1 ("11 ) + + "12 Y2 m + m (13) becomes Setting a = ml2 + m22 and b 11 12’ (14) 2(a mYl)(b mY2) 2 (a mY1)2 + (b mY2)2 01" 2 2 (a m V90 (15) 0 = 2(a mY1)(b mY2) + (b mY2) Y1) 2 . or 0 > (a mY1 - b mY2) which is impossible. Therefore, inequality (11) cannot hold and we see that R3 2 _ is less than Rl except when (ml2 + m22) mYl — (mll + 2 = 2 ml2) mY2. In this latter case R2 R1. But this equality is precisely the same as mY1 m22 - mY2 ml2 = mY2 mll — mYl mll which holds if and only if bYAl:A2 = bYA2-Al' Thus, if 21 fi 82, R2 from the estimated equation (2) is always less than the R2 of the true model (1), unless the sample just happened to generate bYA .A = bYA 1 2 'A 2 1 118 Since FM also combine the components money, M1’ and 2 time deposits, M2, into a single variable, the R obtained from the estimated equation: x (20) Y a0 + a1 (Ml + M2) + e a + a M + 6* or Y 0 1 will always be less than or equal to the R2 obtained from the true model: (21) Y = do + dlMl + d2M2 + e. The PM study compares the R2 of equation (20) with that of equation (2). The question now becomes whether 2 2 the R of (2) is less than the R of (l) by a greater or 2 of smaller amount than the R2 of (20) is less than the R (21). That is, does combining autonomous expenditures cause the R2 to be lower than that of model (1) by more causes the R2 to be or less than the combining M and M 1 2 lower than that of model (21)? The key to this question rests on the fact that the A used by FM in equation (2) is composed of the elements K + G — T + E - I while the definition of M used in (20) is M1 + M2. We would expect K, G, T, E, and I to all be positively correlated with the dependent variable of equa- tion (2). We would also expect M1 and M2 to be positively correlated with the dependent variable of equation (20). 119 And, we would expect -T and -I to be negatively correlated with the dependent variable of (2) as well as with K, G, and E. Therefore, we examine R2 and R3 to see whether they 1 are affected by the fact that one of the components, A2, is negatively correlated with the dependent variable and With A1. We note that m m2 + 2 2m m m 2 22 Y1 "11 "Y2 ' 12 Y1 Y2 2 (mll "22 ‘ "12) "YY We are assuming rl2 and rY2 to be negative. m Since r = 12 12 7m /m 11 22 m and r = Y2 Y2 «m /m 22 YY and since m22 and m are both positive (both being sums YY of squares), the assumption that rl2 and rY2 are negative leads to the condition that m12 and m2Y are negative. Look— ing at R2, we see that it is unaffected by negative values 12 and m2y. When ml2 term of the right hand side, they appear as squares. The of m and m2Y appear separately in a size of R3 is dependent on whether ml2 and m2Y are negative or positive, however. 120 We now check to see whether adding the two com- ponents Al and A2 when r12 < 0, mY2 < 0, and mY1 > 0, lowers R3 more than adding A and A2 when r > 0, 1 l2 mY2 > 0, and mYl > 0. We call the R2 connected with the first case R§(_) and the R2 associated with the second 2 2 2 R2<+). Since Rl(+) equals R1(_) (that is, since the coefficient of determination of the true model is the same whether r12 < 0 and m < 0 or whether r > 0 and Y2 12 mY2 > 0), the relative damage to the true model from adding components Al and A2 when r12 < 0 and mY2 < 0 versus when rl2 > 0 and m > 0 will be measured by Y2 2 2 R2<+> - R2(_). If this difference is positive, the addition of negatively correlated components causes the computed R2 to be lower than the true R2 (i.e. R2) by more than does the addition of positively correlated components. We define 2 2 R2 _ "Y1 + "Y2 + 2"Y1 "Y2 2(+) - ("11 + "22 ‘ 2"l2) "YY 2 2 2 _ "Y1 + "Y2 ’ 2"Y1 "Y2 R2<-> ‘ ("11 + "22 ' 2"12) "YY where all m's are positive. 121 Setting 2 2 _ _ "Y1 + "Y2 ' a "11 + "22 ‘ ° 2mY1 mY2 -= b 2ml2 = d the difference a + b a - b R2 — R2 = - 2(+) 2(-) (c - d)mYY (c + d) m YY This difference is greater than zero if and only if (a+b)(c-d)-(a-b)(c+d):0 ? or be > ad or m m ( + ) 3 (m2 + m2 ) m Y1 Y2 "ll "22 Y1 Y2 12' It is not clear that combining negatively correlated items will do greater damage to the R2 than does combin- ing positively correlated components. FOOTNOTES—-CHAPTER III 1Donald D. Hester, "Keynes and the Quantity Theory: A Comment on the Friendman-Meiselman CMC Paper," Review of Economics and Statistics, XLVI (November, 1964), 364— 368. 2Albert Ando and Franco Modigliani, "The Relative Stability of Monetary Velocity and the Investment Mul— tiplier," American Economic Review, LV (September, 1965), 693-728. 3Michael DePrano and Thomas Mayer, "Tests of the Relative Importance of Autonomous EXpenditures and Money," American Economic Review, LV (September, 1965), 729-752. ”Milton Friedman and David Meiselman, "Reply to Donald Hester," Review of Economics and Statistics, XLVI (November, 1964), 369-376; and "Reply to Ando and Modigliani and to DePrano and Mayer," American Economic Review, LV (September, 1965), 753-785. 5Donald D. Hester, "Rejoinder," Review of Economics and Statistics, XLVI (November, 1964), 376:377; Albert Ando and Franco Modigliani, "Rejoinder," American Economic Review, LV (September, 1965), 786-790; Michael DePrano and Thomas Mayer, "Rejoinder," American Economic Review, LV (September, 1965), 791—792. 6Above, p. 14. 7Above, p. 16. 8Above, pp. 30—31. 9Hester, "Keynes and the . . . ," p. 365. lolplg., p. 366. lllglgn 121219- 131219: 122 123 luIbid. 15Ibid., p. 368. 16Ibid., p. 366. l7Ibid. 18Ibid., pp. 366-367. 19ibid., p. 367. 2OIbid. 21 p. 374. 22Ibid. 23Ibid., p. 375. 2"Ibid., p. 372. 25Ibid. 26 Friedman and Meiselman, "Reply to . . . Hester," Ibid. 27Ibid., p. 373. 28Ibid., p. 370. 29Hester, "Rejoinder," p. 377. 30Above, p. 20. 31Hester, "Keynes and the . . . ," p. 367. 32Ibid. 33Ibid., p. 368. 3“Friedman and Meiselman, "Reply to . . . Hester," p. 369. 35Ibid. 36Ando and Modigliani, "The Relative Stability H , p. 696. 37Above, p. 68. 124 38Ando and Modigliani, "The Relative Stability ," p. 696. 39Ibid. 40 Ibid., p. 697. 41 42 Ibid., pp. 697—698. Ibid., p. 697. 23Friedman and Meiselman, "Reply to Ando . . . ," pp' 756—7570 22Friedman and Meiselman, "The Relative Stability ," p. 190. 25Ando and Modigliani, "The Relative Stability ," p. 697. 46 p. 3212-: pp. 697-698. “7gpgq , p. 698. 281212-2 p. 699. 29Johnston, p. 31. 50Ando and Modigliani, "The Relative Stability ," p. 702. 51Ibid., p. 696. 52Ibid., p. 701. 53ibid., pp. 695-696. 514Ibid., p. 705. 55Ibid., pp. 704, 695-696. 56Ibid., p. 706. 57Ibid. 58These values were taken from AM's Table 3; Ibid. 704. 59Ibid., p. 703. 60These values were taken from AM's Table 2; Ibid. p. 698. 61 62 Ibid., p. 63Ibid. 62"Friedman p. 756. 65Ibid., p. 66 Ibid., p. 67Ibid., p. 68 Ibid., p. 69Ando and ," p. 708. 7OIbid. 71Ibid. 721bid., p. 73Ibid., p. 72Ibid. 7SIbid., 76Ibid., 78Ibid., 79Friedman p. 780. 80Ando and ," p. 695. 81 82 Ibid., p. p. 712. p p 77Ibid., p. p Ibid., p. 125 Ibid., pp. 703-704. 714. and Meiselman, "Reply to Ando . . . ," 762. 766. 756. f“ 767. : Modigliani, "The Relative Stability 709. 710. 711. 713. 695. . 713. and Meiselman, "Reply to Ando . . . ," Modigliani, "The Relative Stability 711. 713-714. 83These values were taken from AM's Table 4; Ibid., 126 821bid., p. 719. 85Ibid., p. 717. 86 87 88 Ibid., p. 718. Ibid. Ibid., p. 716. 89Ibid., p. 719 90Ibid. 91Ibid. 92Ibid. 93Ibid., p. 718. 9“Friedman and Meiselman, "Reply to Ando p‘ 7830 95Ibid. 96Ibid. 97Ibid. 98DePrano and Mayer, "Tests of the Relative pp. 747—748. 99Ibid., p. 748. 100Ibid. Ibid., pp. 732-734. Above, pp. 77-79. 101 102 103DePrano and Mayer, "Tests of the Relative p. 732- 102Above, pp. 79-81- 105DePrano and Mayer, "Tests of the Relative p. 749. 106Ibid., p. 732. 127 lO7DePI'anO and Mayer, "Rejoinder," p. 792. 108Ibid. 109Friedman and Meiselman, "Reply to Ando . . . ," ID— 766. llOIbid. 111Ibid. 112 H H DePrano and Mayer, Tests of the Relative . . . , p0 73“. F:-’l ll 3 31bid-. pp. 735-737. 114 Ibid., p. 737. . 11 2Ibid., p. 739. 116 Ibid., pp. 741-742. 117Ibid., p. 745. 118Goldberger, Econometric Theory, p. 201. 1 l908.1"1 Christ, Econometric Models and Method (New 'York: John Wiley & Sons, 1966), p. 169. 120Friedman and Meiselman, "Reply to . . . Hester," ID. 373. 121Friedman and Meiselman, "Reply to Ando ' ° ' ’n r). 784. 122 Ibid., p. 754. 123Hester, "Rejoinder," p. 377. 122Friedman and Meiselman, "The Relative Stability . . ," p. 182. CHAPTER IV TESTS OF ALTERNATIVE MEASUREMENTS Introduction Our discussion of FM's critics has uncovered several alternative definitions of autonomous expenditures. Hester prOposed the concepts L, L', L", and L"'; AM a; and DM offered a "gross"concept, A* = preferred 22 + X L"' minus state and local government purchases, and a "net" concept, A** = A* - D. When the critics fitted their definitions to the data they invariably found a liigher correlation coefficient or coefficient of determi- Iiation between their concepts and consumption, C, than IPM found between A and 0. FM claim these results are suspect, first, because ‘their critics should have used the same periods FM did in tsheir regression equations and, second, because the re- ggressions used by their critics included the wrong de— IDendent variable. On this latter point, FM argue that ‘their use of C was derived from the fact that it was the (only induced item in their concept of income; hence (in <3rder to be consistent with the FM procedure) when their critics altered the items included in autonomous spending, 128 129 ‘bhey should have simultaneously altered the dependent \Iariable to include the new endogenous items. In light of FM's comments, one of the objectives <>f this chapter is to re-estimate the coefficients of cietermination of regression equations of C on the vari- ous concepts of autonomous expenditures for the same periods FM used and compare their values. Since some of the proposed alternative concepts of A involve a detailed national income accounting framework, we will compare R2's only for time periods including the years 1929-1965. We have not heeded FM's second objection and their implicit advice to calculate another set of coefficients of determination for equations involving alternative concepts, AA, as the explanatory variable and CA (Y - A, where AA is the corresponding autonomous concept) as .A ‘the dependent variable. Partially, the reason we have not clone so is that FM's equations suffer from the greater nnalady of induced components contained in their so called éiutonomous concept. Also, since FM did not present a Iformal structural model of income determination in the IIirst place and since they merely stumbled upon their (zoncept of income (Y was found by summing U and A which \Nere determined from statistical "experiments"), it seems Vve can do little more than compare what appear to be tests on alternative reduced form equations for consump— tion. The results of the refined estimation procedures 130 and dynamic analyses applied to more SOphisticated models in the chapters to follow are bound to overshadow any significance we could discover from manipulating the naive models we have reviewed so far. While the major purpose of this chapter is to re- estimate equations offered by FM's critics using the same time periods FM considered, we have decided to test again FM's equations (1') through (6')1 with revised data for M, A, C, and P. Secondly, since we object to FM's inclusion of time deposits in the definition of M, we will re-test equations (1'), (3'), (5'), and (6') using currency in circulation plus demand deposits as the definition of M. Also, we add the subperiods 1957-1965 and 1929-1965 to those suggested by FM. Re-estimation of FM Equations Using Revised Data and an Alternative Definition of M Equations Tested The FM equations (1') through (6') were re-estimated with revised data. Equations (1'), (3'), (5'), and (6') were also tested with an alternative definition of the money supply. These equations are M + = 2 + D+T El (1")013 1 0‘2 7' (2 ) CB G3 + 24MD + £2 (3") CM = 81 + 82A + £3 131 (4") C = B + 842 + 85p + £4 (5") C = d + 06M + d7p + £5 D+T (6") C = G8 + dgMD + d8p + 26 / v: _ '7 ) CB ' Yl + YZAB + Y3MD+T + 67 v: = (8 ) CB Y4 + YSAB + Y6MD + 28 (9") CB = Y7 + Y8A8 + Y9MD+T + Y10p + 89 (10") C = B Y11 + Y12AB + Y13MD + Y14p I 610 Equation (1") was tested for subperiods between 1897 and 1965, equation (2") was fitted to subperiods between 1915 and 1965, and equations (3") through (10") were tested for subperiods with years between 1929 and 1965. Definition of Variables CB = Personal consumption in billions of current dollars. For 1929-1965 CB = CM/1000. For 1897-1928 CB was computed using the regression equation: COBE = 0.5821 + 0.9508 CK which was computed by regressing COBE on CK for the period 1929-1941. C r C58 x P58 where OBE - OBE C PC: Implicit price deflator for per- sonal consumption expenditures (1958 = 100) for years 1929-65, 132 U. S. Department of Commerce, Bureau of the Census, Long Term Economic Growth, 1970—1965, Superintendent of Documents, U. S. Government Printing Office, Washington, D.C., October, 1966, pp. 200-201, Series B66. Personal consumption expenditures in re billions of 1958 dollars, Ibid., pp. 170-171, Series A24. And where 29 x PK Implicit price deflator for personal consumption expenditures (1929 = 100) for years 1899 to 1929, Ipig,, pp. 200— 201, Series B65. This price deflator was extended to the years 1930-41 by the formula: P29 = P29 x 100/55.3 t >1929 Kt OBE where 100 is the index of Pig for the year 1929 and 55.3 is the index of ngE for the year 1929. Personal consumption expenditures in millions of 1929 dollars, Ipid., pp. 170-171, Series A23. This series was Changed to billions of 1929 dollars prior to running the regression. 133 Personal consumption expenditures in millions of current dollars. Data was taken from U. S. Department of Commerce, Office of Business Economics, "The National Income and Product Accounts of the United States, 1929-1965; (Statis- tical Tables)," A Supplement to the Survey of Current Business (Washington: Government Printing tn Office, .966), Table 1.1, line 2, pp. 2-3. ‘ D+T = Money supply plus time deposits (currency in cir— culation, demand deposits at commercial banks other than those due to commercial banks and U. S. government, less cash items in process of collec- tion and Federal Reserve float, and time deposits at commercial banks) in billions of dollars. For 1897-1946 data is taken from U. S. Department of Commerce, Bureau of the Census, Long Term Economic Growth, 1860-1965, Series Blll, pp. 208-209 (data is twelve month centered means of seasonally ad- justed data). For 1947-1965 data is twelve month centered means of seasonally adjusted monthly data of total money supply plus time deposits at com- mercial banks taken from data presented in Board of Governors, Federal Reserve Bulletin, Table-- Money Supply and Related Data (in billions of dollars) June, 1964, pp. 682—692; June, 1965, 134 p. 978; May, 1966, p. 678; and April, 1967, p. 608. Money supply in billions of dollars. For 1915— 1946 data taken from U. S. Department of Com- merce, Bureau of Census, Long term . . . , Series B109, pp. 208-209. (Data is centered means of seasonally adjusted data.) For l9H7-l965 data is twelve month centered means of seasonally adjusted monthly data of total money supply taken from Board of Governors, Federal Research Bulletin, Table--Money Supply and Related Data (in billions of dollars), same issues and pages as for MD+T' FM's autonomous expenditure concept. F SL IPA ’ GSIPA Data is for 1929-1965 period where: GPDI - CCA + E - I — GS GPDI = Gross private domestic investment in millions of current dollars, U. S. Department of Commerce, Office of Business Economics, "The National Income . . . ," Supplement to the Survey of Current Business, Table 1.1, line 6, pp. 2-3. CCA = Capital consumption allowance in millions of current dollars, U. S. Department of Commerce, OBE, Table 1.9, line 2, pp. 12-13. 135 E = Exports in millions of current dollars, U. S. Department of Commerce, OBE, Table 1.1, line 18. I = Imports in millions of current dollars, U. S. Department of Commerce, OBE, Table 1.1, line 19. GSIPA = Federal government surplus on income and product account in millions of current dollars, U. S. Department of Commerce, OBE, Table 3.1, line 31, pp. 52-53. GsfigA — State and local government surplus on income and product account in millions of current dollars, U. S. Department of Commerce, OBE, Table 3.3, line 33, pp. Bu-SS. 3> ll A/lOOO Implicit price deflator for personal consumption 'U ll expenditures (1958 = 100), U. S. Department of Commerce, OBE, Table 8.1, line 2, pp. 158—159. Results The results of the series of tests are given in Table 1 below. These results for the single variable equations show that the coefficients of determination are generally lowest when A is the explanatory variable and generally highest when M is the single explanatory D+T I. 1 I . It i . uvas..w>a .M .\ y. . a u. . . u x n . . v 1 I . A v s—h— -— . _._ .— —-.~J- .- -¢m r~ sh u~.Av~aNf.-.‘N ~—-\u ~d > _av~ ——.vfiv u ah -§~ v s as . ~ V n k N: K ~.I¢.y .ys.\.\o\.».clnl...\ ...N..N.\—\\ 136 000.0 000.0 000.0 000 0 000.0 000.0 000.0 A..000 000.0 000.0 000.0 000.0 000.0 000 0 000.0 A..00 000.0 000.0 000.0 000.0 000.0 000.0 000 0 A..0V 000.0 000.0 000.0 000 0 000.0 000.0 000.0 A..00 000.0 000.0 000.0 000.0 000.0 000.0 000.0 A..0V 000.0 000.0 000.0 000.0 000 0 000.0 000 0 A..00 000.0 000.0 000.0 000.0 000.0 000.0 000.0 A..0V 000.0 000.0 000.0 000.0 000.0 000.0 000.0 A..00 000.0 000.0 000.0 000.0 000.0 000 0 000.0 000.0 000.0 000.0 A..00 .000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 A..0V 0000 0000 0000 0:00 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 -0000 000 0 0pm: 0 mU00%om .00Qd50 00:08 on» mo wc00u0C0wmo m>0pmcmmp0m paw mumo vmm0>mp 0:00: 0:00pmsvm Em pom m000pmd 030000> you :00pmz0Emmpmp mo mucm0o0mgmoo-I.0 m0m<9 137 variable. When the price level and one other variable are considered as independent, the lowest coefficients of determination arise when A is that variable and high- est when the second independent variable is MD+T' Equa- tions with A and M as explanatory variables display higher R2's when M + M than when M = M . And, finally, when D+T D A, P, and M are the explanatory variables M = M D+T yields higher coefficients of determination than when M = MD. Naturally, the R2's increase as the number of variables is increased from one to two and from two to three. Re-estimation of Alternative Definitions of A to Conform to the Periods Tested by FM Answering FM's charge that their critics should have considered the same sample periods FM did when testing their equations, we fit the following equations to the data for various subperiods since 1929. We also add the two periods 1957-1965 and 1929-1965. Equations Tested The following seven equations proposed by FM's critics are taken as alternative reduced forms for con- sumption. (11") CM YO + YlL + 811 (12") CM = Y2 + Y3L' + $12 138 (13") CM = Yu + YSL" + 613 (1M**) 0 = Y6 + y7L"' + ralLl (15") CM = Y8 + Y9A* + 615 H = 96* (16 ) CM Y10 + Y11A + 816 A H = (17 ) CM Yl2 + 713A + :17 Definition of the Variables CM L? L?! Personal consumption expenditures in millions of current dollars. F SL A + GSIPA + GSIPA + GF + GSL K + E - I + G = Hester's L where A, GSF , and GSSL are defined above and IPA IPA GF = Federal government expenditures in millions of current dollars, U. S. Department of Commerce, OBE, Table 1.1, line 21, pp. 2—3. GSL = State and local government expenditure in millions of current dollars, U. S. Depart— ment of Commerce, OBE, Table 1.1, line 2U, pp. 2-3. L + CCA + I GPDI + E + G = Hester's L' where data sources for CCA and I are shown in the above section. L + CCA (Hester's L") vav = INV = A* = AA = Tg2 Tb2 TiF TiS Tg2 139 L - INV (Hester's L"') where Net changes in inventory in millions of current dollars, U. S. Department of Commerce, OBE, Table 1.1, line 1h, pp. 2-3. GPDI - CCA — INV + E + G + G + T GPDI + E + G - INV (DM'S "gross" concept) F A* - CCA (DM'S "net" concept) + T + T + F SL b2 1F iS + Q — H - w (AM's za + xa) where Property tax portion of indirect business taxes in millions of current dollars, U. S. Department of Commerce, OBE, Table 3.3, line 19, pp. 54-55. Federal net interest paid in millions of current dollars, U. S. Department of Commerce, OBE, Table 3.1, line 29, pp. 52-53. State and local net interest paid in millions of current dollars, U. S. Depart— ment of Commerce, OBE, Table 3.3, line 31, pp. SA-SS. T - T where s s1 T = T + T and T = SB + RB where s sF g8 g1 TgF = Federal government transfer payments to persons in millions of current dollars, U. S. 1N0 Department of Commerce, OBE, Table 3.1, line 26, pp. 52-53. TgS = State and local government transfer payments to persons in millions of current dollars, U. S. Department of Commerce, OBE, Table 3.3, line 30, pp. SH-SS. SB = State unemployment benefits in :fig millions of current dollars, U. S. p _ Department of Commerce, OBE, Table l 3.9, line 5, pp. 58-59. RB = Railroad unemployment benefits in millions of current dollars, U. S. Department of Commerce, OBE, Table 3.9, line 7, pp. 58-59. Q = QF + QSL where QF Subsidies less current surplus of federal government enterprises in millions of current dollars, U. S. Department of Com- merce, OBE, Table 3.1, line 30, pp. 52—53. Subsidies less current surplus of state and local government enterprises in millions of current dollars, U. S. Department of Commerce, OBE, Table 3.3, line 32, pp. SN-SS. 141 = Statistical discrepancy in millions of current dollars, U. S. Department of Commerce, OBE, Table 5.1, line 14, pp. 78—79. = Wage accruals less disbursements in millions of current dollars, U. S. Department of Commerce, OBE, Table 5.1, line 7, pp. 78—79. Results Table 8 presents the coefficients of determination of these regressions for each of seven subperiods between 1929 and 1965. TABLE 8.--Coefficients of determination for alternative definitions of autonomous expenditures. Periods Equations 1929- 1933— 1938- 1939- 1948— 1957- 1929- 1939 1938 1953 1948 1957 1965 1965 (11' (12' (13' (14' (15' (16' (17' ') 0.834 0.986 0.610 0.264 0.865 0.956 0.925 ') 0.919 0.980 0.729 0.355 0.936 0.961 0.964 ') 0.882 0.990 0.688 0.309 0.932 0.962 0.957 ') 0.959 0.993 0.699 0.312 0.949 0.965 0.961 ') 0.948 0.990 0.594 0.241 0.906 0.944 0.924 ') 0.913 0.993 0.492 0.199 0.806 0.915 0.851 ') 0.901 0.997 0.752 0.440 0.966 0.978 0.969 This table shows that, in general, all alternative concepts of autonomous expenditures yield higher 142 coefficients of determination than does A. Except for the time periods 1938—1953 and 1939-1948, when all R2's are rather low, the coefficients of determination are of "roughly the same magnitude" as DM have suggested. Conclusion. Little insight into the relative effectiveness of monetary and fiscal policy has been offered by the re— finements carried out in this chapter or by the entire exchange among authors we have reviewed to this point. As we have shown, FM's methodology is naive and not altogether sound.2 Consequently their crowning the "quantity theory" with laurel seems somewhat persumptuous. Fortunately, there is a better method at our dis- posal than that utilized by FM, namely, beginning at the beginning with a specific, more sophisticated, yet manage- able, theoretical model of income determination.3 This model is then tested statistically using econometric methods which account for the difficulties which arise in measuring economic relationships. Finally, but certainly not the least significantly, there is a large body of knowledge which can be brought to bear in analyzing the model (especially a dynamic one) after it has been esti- mated. It is this methodology which will guide us through the next three chapters. FOOTNOTES-—CHAPTER IV 1See above, pp. 2Not only is it unsound; it is contagious. See for instance R. H. Timberlake, Jr. and James Forston, "Time Deposits in the Definition of Money," American Economic Review, LVII (March, 1967), 190-194. 3This is in fact what FM suggested should be done. See Friedman and Meiselman, "Reply to Ando . . . ," p. 753 and Ando and Modigliani, "The Relative Stability . . . ," p. 716. 143 CHAPTER V RE—ESTIMATION OF KLEIN MODELS II AND III Introduction Having demonstrated that the studies reviewed in Chapters II and III are likely to be inadequate ex— plorations of the effects of money and autonomous expendi— tures on the level of income, we try to construct a more comprehensive framework to analyze the relative effective- ness of monetary and fiscal policy. Building a general equilibrium econometric model involves a myriad of problems and uncertainties as to performance. Therefore, it was decided to adopt and, if necessary, revise existing econometric models. The lei- sure consumed as a result of this decision and the free- dom of responsibility for defects in the models used were felt to more than offset the accompanying costs. These costs include damages to the predictability of the models which may result from manipulation of the time periods, alteration of the status of variables as to their determi— nation within or outside the model, and the deletion or addition of equations. We considered several general equilibrium econometric models for adoption. The first condition we set down was 144 145 that the model by dynamic and use annual data. Quarterly models cannot include the years prior to World War II since quarterly data are unavailable for the pre-war period. Secondly, the model chosen must have treated government expenditures and the money stock as exogenously determined. Thirdly, we were interested in finding a model that was originally tested for as many of the years 1897-1958 as possible--the interval tested by FM. Klein's models I, II, and III and the Klein- Goldberger, Valavanis—Vali, and Morishima-Saito models comprise the field of candidates from which our selection 1 Klein model I lacks a monetary sector, the was made. Valavanis—Vail model lumps government spending with con- sumption into one endogenous variable, and the Morishima- Saito model specifies an endogenous variable which com- bines government spending with net private domestic invest— ment. Therefore, these candidates do not contain the requisite exogenous categories for a comparison of the effectiveness of monetary and fiscal policy. While the Klein-Goldberger model contains a monetary sector, it exercises no influence over the real sector. By the process of elimination, we decided to base our analysis on Klein's models II and III. Both treat the money supply and government spending as exogenous variables. However, both suffer from being previously tested for pre-World War II years only. Also, Klein 146 model III contains non-linear equations. Therefore, in order to carry out the dynamic analysis on Klein model III,other than by simulation methods, we must be satisfied with using linear approximations of the equations. Though significant, these deficiencies do not preclude employing Klein models II and III in a more sophisticated compari- son of the relative effectiveness of monetary and fiscal policy than has been offered by the studies reviewed earlier. Klein Model II Klein Model II is a simple three equation model of income determination and is only a modest step toward a better analytical framework within which to analyze the issue at hand. This model comprises the following three equations: (l)C/pN=a+aY+o-Y +dfl_ 0 lsfi ZCEfi)-l 3(pN)—l +1.1 (2) GNP = C + I' + G (3)GNP=Y+T where C = consumption in current dollars Y = disposable income in current dollars M = money supply in current dollars I' = gross investment in current dollars G = government expenditure plus foreign balance in current dollars GNP = gross national product in current dollars p = cost-of-living index N = population in United States 147 T = government receipts plus corporate savings plus business reserves minus transfer payments minus inventory profits, all measured in current dollars. The endogenous variables are C, Y, GNP. The exogenous 2 1' variables are I'/pN, G/pN, T/pN and M/pN_ Data Revisions Klein Model II was extended to cover the pre-war period 1922-1941 and the post—war period 1946-1965. The following presents the data and its revisions for the period 1920-41 and 1945-65. C: Personal consumption expenditures in billions of current dollars. For 1929-65 data was taken from U. S. Department of Commerce, Office of Business Economics, "The National Income and Product Accounts of the United States, 1929-1965; (Statistical Tables)," A Supplement to the Survey_of Current Business (Washington, D.C., 1966), Table 1.1, pp. 2-3, line 2. For 1920-28 data for personal consumption expenditures in billions of current dollars was computed as follows: C = 0.5821 + 0.9508 CK was computed using the OBE years 1929-1941 and then 0 BE for years 0 1920 to 1928 was estimated from the regression where C OBE 58 OBE P58: 0 = C 148 8 x P2 where Implicit price deflator for per- sonal consumption expenditures (1958 = 100) for years 1929-65, U. S. Department of Commerce, Bureau of the Census, Long Term Economic Growth, 1860-1965, Superintendent of Documents, U. S. Government Printing Office, Washington, D.C., October, 1966, pp. 200—201, Series B66. Personal consumption expenditures in billions of 1958 dollars, Ibid., pp. 170-171, Series A24. And where Implicit price deflator for personal consumption expenditures (1929 = 100) for years 1899 to 1929, U. S. Department of Commerce, Bureau of the Census, Lonngerm . . . , pp. 200-201, Series B65. This price deflator was extended to the years 1930-41 by the formula: P29 100 t > 1929 where 100 OBE x 5923 is the index of Pig for the year 149 1929 and 55.3 is the index of 233E for the year 1929. Personal consumption expenditures in millions of 1929 dollars, 1610,, pp. 170-171, Series A23. This series was changed to billions of 1929 dollars prior to running the regression. G: Government expenditures in billions of current doll ars . GF: SL: F + GSL where Federal government expenditures, U. S. For 1929-1965, 0 = G Department of Commerce, OBE, "The National Income . . . ," Table 3.1, pp. 52-53, line 19. State and local government expenditures, Ibid., Table 3.3, pp. 54-55, line 24. For 1920-28 estimates of 0 were obtained using the regression of GOBE on GK for years 1929-41: GOBE = 2.92 + 1.104 G where G G OBE: GNP: _ 58 58 K OBE ‘ GOBE x PGNP' Government purchases of goods and services in billions of 1958 dollars, U. S. Depart- ment of Commerce, Bureau of Census, Long Term . . . , pp. 172-173, Series A34. Implicit price deflator for GNP (1958 = 100), Ibid., pp. 200-201, Series B62. 150 _ 29 29 and whereGK - GK x pGNP Gig: Government purchases of goods and services in billions of 1929 dollars, Ibid., pp. 170-171, Series A33. pggpz Implicit price deflator for GNP (1929 100), Ibid., pp. 200-201, Series B61. Series extended using expression 933p = 58 100 pGNP x 5573 Where 50.6 is the value of 58 pGNP for 1929. Population in billions for 1920-65 (converted from data in thousands), The U. S. Book of Facts, Statis- tics,and Information,(New York: Washington Square Press, Inc., 1966), Table No. 2--Estimated Popula- tion 1900 to 1966 (original source: U. S. Depart- ment of Commerce, Bureau of the Census, Current Population Reports, Series P—25, Nos. 331 and 340). Disposable personal income in billions of current dollars (converted from millions). For 1929-1965 'data from U. S. Department of Commerce, OBE, "The National Income . . . ," Table 2.1, pp. 32-33, line 22. For 1920-1928 Y is regressed on Y for K years 1929-41: Y = 1.042 Y where Y are Klein's K K data for disposable income. Gross private domestic investment in billions of dollars (converted from millions of dollars). 151 1929-65 data from Ibid., Table 1.1, pp. 2-3, line 6. For 1920-1928 estimates of I were obtained using the regression of IOBE on 1K for years 1929-41: IOBE = 1.238 + 0.8204 IK where IOBE I Iggz x p18 IggE: Gross private domestic investment in billions of 1958 dollars, U. S. Department of Commerce, Bureau of Census, Long Term . . . , pp. 170- 171, Series A28. 58 pI : Implicit price deflator for fixed investment, (1958 = 100), Ibid., pp. 200-201, Series B68. _ 29 29 IK - IK x pI IE9: Gross private domestic investment in billions of 1929 dollars (con- verted from millions of dollars), Ibig,, pp. 170-171, Series A27. P29: Implicit price deflator for fixed investment, 1929 = 100, Ibid., pp. 200-201, Series B67. For 1930-1941 series extended by use of following expression: 29 _ 58 100 P ‘ p X It It 39. t > 1929 152 where 39.4 = the value of 3% for the year 1929. Implicit price deflator for GNP (1958 = 100). For 1929-1965 data from U. S. Department of Commerce, OBE, "The National Income . . . ," Table 8.1, pp. _ 29 50.6 158-59, line 1. For 1920-1928 pl - pK x _T00 where 50.6 is value of pl for 1929 and pig and 29 pGNP Implicit price deflator for personal consumption are as defined above. expenditures (1958 = 100). For 1929-65 data, Ibid., Table 8.1, pp. 158—59, line 2. For 1921-1928, - 29 55.3 p2 - pC x -I00' where p39: Implicit price deflator for personal con- sumption expenditures (1929 = 100), U. S. Department of Commerce, Bureau of Census, Long Term . . . , pp. 200-201, Series B65; 55.3 is the value of p2 for 1929. Money supply billions of dollars,cf. ME under Klein Model III. Money supply plus time deposite in billions of current dollars, cf. MS under Klein Model III. Estimation Procedure The stochastic equation of Klein Model II is (l) C/pN = a + 01(Y/pN) + 02(Y/pN)_l + 03(M/pN)_l + u. 0 153 It was estimated several times, both for the period 1922- 1941 and the extended period 1922-1941 and 1946-1965 using different measures of the implicit price deflator and the money supply. 0 and Y are endogenous variables in the system of equations comprising Klein Model II. We assume u is normally distributed with mean zero and variance t o2 for all t, independent of past and future distrubance terms, and independent of the predetermined variables in the system. Applying simple least squares to equation (1) would yield the undesirable result of obtaining biased and inconsistent estimates of the 0's. This arises from the fact that Y/pN is correlated with u Therefore, we t. use two alternative methods to estimate equation (1) which take account of this problem. These methods are two stage least squares (2SLS) and limited information maximum likelihood (LI). Results of Re-estimating Klein Model II We now turn to the estimates of the coefficients we obtained using both the 2SLS and LI estimation methods for the years 1922-1941 and for the extended period 1922- 1941 and 1946-1965. The figures in parentheses are the standard errors of the coefficients; those in brackets are the ratios of the coefficients to their standard errors . 154 Using p = p1, M = M2 (1922—1941) Two Stage Least Squares C/pN = 183.66911 + 0.56001 Y/pN + 0. 02577 (Y/pN) —1 (45.39866) (0. 06344) (0.07451) [4.04569] [8 827721 [0. 345911 + 0.28193 (Mg/pN)_ 1 (0.08416) [3.34981] All explanatory variables except (Y/pN)_l are highly significant at the 95 per cent level of confidence. Limited Information C/pN = 189.79843 + 0. 51625 Y/pN + 0. 05378 (Y/pN) (47.28288) (0. 06607) (0.07761) 1 [4.01411] [7. 81362] [0. 69298] + 0. 29946 (M 2/pN) —1 (0.08766) [3. 41628] Once again (Y/pN)_l is not statistically significant at the 95 per cent level of confidence. Using p = pl, M = M2 (1922—1941 and 1946-1965) Two Stage Least Squares C/pN = 40. 22769 + 0.69077 (Y/pN) + 0.18972 (Y/pN) (14.07767) (0.06063) (0.06942) 1 [2. 857551[11.39288] [2. 732891 + 0.03027 (M 2/pN) -1 (0.03683) [0. 82195] Limited Information C/pN = 41.18210 + 0. 63995 (Y/pN) + 0. 24329 (Y/pN) -1 (14.55739) (0.06270) (0.07179) [2. 82895][10. 20692] [3. 38911] + 0.02657 (M 2/pN) (0. 03809) [0. 69751] -1 155 Extending the sample period causes variable (M2/pN)-l to become not statistically significant and the variable Y/pN)_l to become significant. Using p = p2, M = M2 (1922—1941) Two Stage Least Squares II l\) H O C/pN .32619 + 0.52356 (Y/pN) + 0.06326 (Y/pN) (51.45664) (0. 06758) (0. 07813) 1 [4.087451 [7. 74724] [0.80972] + 0.26442 (M2/pN)_ 1 (0.08126) [3.25406] Limited Information C/pN = 219.12311 + 0. 47434 (Y/pN) + 0. 09693 (Y/pN) _1 (53.76023) (0.07061) (0.08163) [4.07593] [6. 71825] [1.18745] + 0.27851 (M 2_1/pN) (0.08490) [3.28062] Using p = p2, M = M2 (1922-41 and 1946-65) Two Stage Least Squares C/pN = 48. 58589 + 0. 63614 (Y/pN) + 0. 23401 (Y/pN) —1 (16 57955) (0.06807) (0.07514) [2 93047] [9.34564] [3.11459] + 0.04115 (M2/pN)_ 1 (0.03799) [1.08328] Limited Information C/pN = 49. 60436 + 0.57583 (Y/pN) + 0.29565 (Y/pN) _l (17.28810) (0.07098) (0.07835) [2. 86928] [8.11286] [3 77367] + 0.03995 (M C/pN) -1 (0. 03961) [1.00849] 156 This sub—group using p = p2 and M = M2 is the one which corresponds most closely to the definitions of the vari- ables used by Klein. The results of extending the period are precisely those we found when p = p1 and M = M2. We now redefine M to be the money supply (rather than money supply plus time deposits). Using p = p1 and M = M (1922-1941) 1 Two Stage Least Squares C/pN = 215.27562 + 0. 48434 (Y/pN) + 0.12025 (Y/pN) —1 (46.97485) (0.07363) (0.06969) [4.58278] [6.57828] [1. 72551] + 0.32398 (M l/pN) —1 (0.09484) [3.41621] Limited Information cz/pN = 220. 53146 + 0. 45313 (Y/pN) + 0.13857 (Y/pN) -1 (48.35036) (0. 07734) (0.07223) [4.56111] [5.85896] [1.91845] + C 34580 (M l/pN) —1 (0.09810) [3.52489] Elsing_p = p1 and M = M:L (1922-41 and 1946-65) Two Stage Least Squares C/pN = 38 55807 + 0. 68786 (Y/pN) + 0. 21513 (Y/pN) -1 (14. 37861) (0. 06080) (0. 06536) [2 68l631£ll. 313341 [3 291351 - 0 00305 (Ml/pN)_ 1 (0. 03148) [— 0. 09688] 157 Limited Information C/pN = 39.18944 + 0.65242 (Y/pN) + 0.25163 (Y/pN)_l (14.67310) (0.06514) (0.06976) [2 67084][10 01547] [3.60726] — 0.00486 (Ml/pN) 1 (0.03213) _ [—0.15132] From these sets of regressions we see that consider- able damage is done to the significance of money in Klein IWodel II when the sampling period is extended. The money supply variable loses its significance for this model in csach case. This gives us further evidence that different Especifications of models may be needed to explain pre— VJorld War II and post-World War II macroeconomic behavior :for the United States. We shall examine the dynamic g>roperties of Klein Model II in Chapter VI. This model Ioroves to be a very simple one to subject to dynamic Einalysis and will, therefore, provide a demonstration of tzhe principles to be applied to the more sophisticated I 2 (definition of rent payments).3 Adjustments in Structural Equations Klein tested the above equations for the period 1922- 1941. Before this model was tested for the extended period of 1922-41 and 1947-65, several minor adjustments were made in the structural equations. First, equation (16) was drOpped and the status of R1 was changed to that of an exogenous variable because of the degree to which the equation is non-linear. Second, the interest adjust- ment equation (11) was replaced by the equilibrium condi- tion: , = D D (11 ) Ms M1 + M2. This adjustment allows a comparison to be made between the money supply and autonomous expenditure and is in the spirit of the FM analysis. Third, the variables p“l and t were dropped from equation (3) and t was also dropped as an explanatory variable from equation (2). This third revision follows that of Klein who found these variables not to be statistically significant. Data Revisions Klein's raw data are for the period 1920-41. Several revisions and estimations were needed to extend the 163 sampling period to include the years 1945—1965 as well. Klein expressed some variables in constant 1934 dollars. In these instances we have changed the base year to 1958. The following are the data and sources used in the re- vised model for years 1929-1965. C: Personal consumption expenditures (billions of 1958 dollars). For years 1929-1965 data taken from U. S. Department of Commerce, Bureau of Census Long Term . . . , Series A-24, pp. 170-171. To generate data for 1920-1928 consistent with that for 1929- 1965 we used a regression of COBE on CK for 1929-41 and estimated C from this regression for 1920-28: OBE C = 0.5821 + 0.9508 CK. See discussion of C OBE under data revisions for Klein Model II. : Net fixed business investment in billions of 1958 dollars. For 1929—65: I' - CCAI I where péB: Implicit price deflator for non-residential fixed investment, (1958 = 100), U. S. De- partment of Commerce, OBE, "The National Income . . . ," Table 8.1, pp. 158-159, line 8. I': Gross fixed business investment in billions of current dollars, Ibid., Table 1.1, pp. 2—3, line 8. AH: 164 CCAI = CCA — CCAD: where CCA: Capital consumption allowance in billions of current dollars, Ibig,, Table 1.9, pp. 12—13, line 2. CCA : Depreciation on all residences in billions of current dollars. This variable was calculated as shown below. For 1920-1928 we regressed our I on Klein's I for 1929-41 and estimated I for 1920-28 from I 8 = 1.031 + 3.702 I 5 34' Price index of fixed investment, (1958 = 100), ggig., Table 8.1, pp. 158-159, line 7. For 1920—28 we regressed our q58 on Klein's q for 1929-41 and esti- mated Q58 for 1920-28 from q58 = .254 + .345 q3u. Net change in inventories (billions of 1958 dollars), Ibid., Table 1.2, pp. 4-5, line 14. For 1920-28 we used regression AH38 = 0.838 + 2.506 AH3u : Gross expenditure on construction of owner-occupied, single family, non-farm residences, billions of 1958 dollars. I .‘ 2.. D1 - ql where D': Estimated construction cost of single family, non-farm starts in millions of current dollars converted to billions, 1945-1965, D": 165 Department of Housing and Urban Development, Housing Statistics-~Annual Data (Washington, D.C., v61, XIX, No. 5, May, 1966), Table A-5, p. 4, column 5 (old series was extended to the years 1960 through 1965 by taking ratio of old series to new series figures for 1959 (0.890) and multiplying new series figures for 1960-65 by this ratio). For 1920-1941 Klein's D1 (1934 dollars) was multiplied by Klein's ql (1934 = 100) and then divided by 01 (1958 = 100). Gross expenditure on construction of rented, non-farm residences, measured in billions of 1958 dollars. D2 = D - Dl D = non-farm building--new construction, U. S. Department of Commerce, OBE, "The National Income . . . ," Table 5.3, pp. 82-83, line 5. For 1920-28 we used D38 = .700 + 3.754 D3“. ° Gross private domestic investment on residential structures--farm (billions of 1958 dollars), Table 1.2, line 13, pp. 4-5, Ibid. Depreciation of all residences (farm and non-farm) in billions of 1958 dollars. For years 1929-1965: CCA' D" = fiT_-' where D58 p' D58 CCA': Implicit 166 price deflator for construction eXpenditures on all residences (1958 = 100), Ibid., Table 8.1, pp. 58—59, line 11. (a) For 1934: CCA' = (67.6) (0.03) + D (D34) (.015) where 67.6 D': 0.015 is the value in billions of dollars, Jan. 1, 1934, of the stock of residential dwellings in the United States, L. R. Klein, Economic Fluctuations . . . , p. 148. represents 3% depreciation over the year of that housing in exist- ence at beginning of year. Gross expenditures on construction of residential structures, billions of current dollars, U. S. Depart- ment of Commerce, OBE, "The National Income . . . ," Table 1.1, pp. 2-3, line 11. represents 1.5% depreciation (on the average) of housing build dur- ing the year. The implicit assump- tion is that housing is constructed uniformly over the year. 167 (b) For years 1935—1965: CCAfi = DA + t t D + B + D* where Bt Ct t 0A = (67.6) (0.97)t‘193“ (0.03). t This is the depreciation during the year on that fraction still in existence on January 1 of the year t of housing in existence on January 1, 1934- D = (D ) (0.015) Bt 1t 0% = (Dé_l) (0.985) (0.03). 98.5% of last years construction remains on Jan. 1 of year t. During year t it depreciates at 3% rate. C. t D = .97D* but D* = 0 C1935 1934 1934 1935 .97D§935;ror t > 1936 pct = .97 (D + D* ) ct_l t-l (c) For years 1929-33--Part of the value of 67.6 which was in existence on January 1, 1934 was not built by January 1, 1933. Klein begins by taking 67.6 and multiplying this value by (0.97)‘l, the 168 value which would have existed on January 1, 1933 if all of the 67.6 value were due to housing in existence prior to January 1, 1933. This product is then multiplied by 0.03 to find the 3% depreciation. However, (0.97)"1 D' of the (67.6) (0.97)'1 value was created during 1933 and should not have been depreciated by 3%. Assuming housing was added uniformly over 1933, the deprecia- tion on Di933 is only (Di933) (0.015) (0.97)"1 and therefore the actual de— D' = 67.6 (0.97)‘1 (0.03) - (01933) (0.015) (0.97)‘1. 1. For 1932: The value which preciation for 1933 is CCA existed on Jan. 1, 1933 is equal to the value existing on Jan. 1, 1934, 67.6, less the net value added during 1933, D' 33 Therefore, if all the value - CCA , . D 33 existing on Jan. 1, 1933 were also in existence on Jan. 1, 1932, the depreciation during 1932 would be: [67.6 - (D3,)3 - CCAD, )1 (0.97)'1 (0.03). 33 169 However, the amount Dé2 was not in existence on Jan. 1, 1932 and therefore the depreciation for 1932 is not [67.6 - (Dé3 - CCAD,33)] 0.97'1 (0.03), but rather [67.6 - (Dé3 — CCAD,33)] (0.97)‘1 (0.03) - 0&2 (0.97)-1 (0.015). In general, for years prior to 1933: [6 6 1933 ( CCA , = 7. - z 0' - D1 i=t+l 1 0.03 0.015 . CCADi)]'0T97 ‘ Dt(‘0797 ° This formula is the same as Klein's (p. 148) except that his in- volves D1, D2, and D3 in 1934 dollars. Parenthetically, Klein's formula for years later than 1934 is incorrect. Accord- ing to his expression, D" for 1935 would be: (1) Déé = (67.6) (0.97) (0.03) + (01 + 02 + D3>1934 (0.985) (0.97)‘1 (0.03) + (D1 + D2 + D3)1935 (0.015). Substituting our notation for ' V 'l Klein 5 D and (D1 + D2 + D3)i 170 equation (1) becomes CCAD, = 35 (67.6) (0.97) (0.03) + D34 (0 985) (0.97)‘1 (0.03) + D35 power of (0.97) in the second (0.015). However, the term should be zero and not minus 1. (d) For years 1920-1929: Since the index of construction prices which we have been using as a deflator for CCA has D been computed back to only 1929, a different approach must be followed to find D" in 1958 dollars for the years 1920-1929. This is accomplished by regressing D" in 1958 dollars for the years 1929-41 on D" in 1934 dol— lars as computed by Klein and then using the resulting equation to esti- mate the earlier D" (1958 dollars) figures using Klein's calculations. The equation used was Déé = 3.238 + 1.449 Défi. Government expenditures on goods and services and net exports (billions of 1958 dollars). For 1929— 1965, U. S. Department of Commerce, Bureau of the Census, Long Term . . . , Series A32 and A34, 171 pp- 170-173. For 1920-1928 used regression G58 = 2.00 + 2.426 G3“. Y + T: Net national product (billions of 1934 dollars). = _ '1 Y + T C + I + H + D1 + D2 + D3 D + G. Y: Disposable income, measured in billions of 1958 dollars, Ipig,, Series A41, pp. 172-173. For 1920- 28 used regression Y58 = 13.292 + 2.143 Y3“. p: Price index of GNP (1958 = 100), U. S. Department of Commerce, OBE, "The National Income . . . ," Table 8.1, pp. 158-159, line 1; p58 for 1920-28 was generated using p58 = 8.677 + 0.3440 p3“. Private wage-salary bill in billions of current dollars, Ipid., Table 6.2, pp. 94—95, line 87, and pp. 96-97, line 85. For 1920-28 W1 8 was found using 5 the regression Wl = 1.415 + 0.9139 W1 58 34 W2: Government and government enterprises wages and salaries (millions of dollars converted to billions of dollars), Ibid., Table 6.2, pp. 94-95, line 73, and pp. 96-97, line 71. For 1920—28 used W2 8 = 5 —0.480 + 1.089 W2 . 34 R : Non-farm rentals, paid and imputed, for residential dwellings (billions of dollars). _ R58 58 , R58 58 - x p x p where 1 1,0 R1,0 1,T R1,T R 172 RI?0: Space rental value of owner—occupied non- farm dwellings (billions of 1958 dollars), Ipid., Table 2.6, pp. 48-49, line 36. RI?T: Space rental value of tenant-occupied non- farm dwellings (billions of 1958 dollars), Ipid., Table 2.6, pp. 48-49, line 37. pg: 0: Implicit price deflator for space rental ’ value of owner-occupied non-farm dwellings (1958 = 100), Ip;g., Table 8.6, pp. 162- 163, line 36. pg: T: Implicit price deflator for space rental ’ value of tenant-occupied non-farm dwell- ings (1958 = 100), Ipig., Table 8.6, pp. 162-163, line 37. For 1920-28 used R1 = 0.352 + 1.037 R1 58 34 R2: Farm rentals (billions of dollars). R2 = R28 x pg: where R28: Rental value of farm houses (billions of 1958 dollars), IQ;§,, Table 2.6, pp. 48—49, line 38. p22: Implicit price deflator for rental value of farm houses (1958 = 100), Ibid., Table 8.6, pp, 162—163, line 38. For 1920—28 used R2 8 = 5 0.100 + 0.939 R 234. 173 Index of rents (1958 = 100), Ibid., Table 8.6, pp. 162—163, line 35. For 1920-28 used r = 3.423 + 58 0.523 r3“. Excise taxes in billions of current dollars. E = ESL + EF where ESL: State and local sales taxes (millions of current dollars converted to billions of current dollars), Ipig,, Table 3.3, pp. 54-55, line 11. E : Federal excise taxes (millions of dollars converted to billions of dollars), Ipid,, Table 3.1, pp. 52—53, line 11, For 1920- 28 used E = -0.02 + 0.9683 E 58 34' Index of construction costs (1958 = 100), Ibid., Table 8.7, pp. 165-165, line 5. For 1920-28 used regression q1 = 1.200 + 0.282 ql . 58 34 ° Average corporate bond yield, Board of Governors, Federal Reserve Bulletin, Table--"Bond and Stock Yields," column entitled "average total corporate bonds yields." End—of—year stock of fixed business investment, billions of 1958 dollars. Kt = l%%f%-x 100 = 311.6 t = 1934 where 107.8 = Klein's value for K for 1934 in current (i.e. 1934) dollars. 34.9 = Kt = 311.6 + Kt = 311 6 — : End-of-year dollars. _ 21.8 Ht ‘ 40.8 x 21.8 = 40.8 = Ht = 53.4 - 174 Price deflator for the year 1934 for non- residential fixed investment (1958 = 100), U. S. Department of Commerce, OBE, "The National Income . . . ," Table 8.1, pp. 158-159, line 8. t 2 I1 t > 1934 i=l935 1934 2 I1 t < 1934 i=t+1 stock of inventories, billions of 1958 100 = 53.4 t = 1934 Klein's value for H for end of 1934 in current (i.e. 1934) dollars. Derived from wholesale price index, all commodities, BLS (1957-1959 = 100) as shown in series B69, U. S. Department of Commerce, Bureau of Census, Long Term , pp. 201-202. Series shows 1958 index to be 100.4 and 1934 index to be 41.0. Index for 1934 to base 1958 was computed as I%%;% x 100 = 40.8. 1934 2 (AH) i=t+l 1 t < 1934 175 t H = 53.4 + 2 (AH) t t > 1935 i=l935 1 AF: Thousands of new non-farm families. AF = F - F where t t t-l F: Number of non-farm families (converted to thousands), U. S. Department of Commerce, Bureau of the Census, Current Population Reports, Series P 20, "Population Character- istics," Households and Families by Type, various March issues, Table--Households by Type by Color of Head and Residence f6r the United States for 1945-1965. AFt for 1920- 41 were taken from Klein's data, p. 144. : Millions of available non—farm dwelling units at the end of the year. NS = Klein's data 1920—1928 t N: = 24.6 + 2 Si where i=1929 24.6 = Klein's value for end of 1928. S1 = New non-farm housing units started in thousands (converted to millions), U. S. Department of Commerce, Bureau of the Census, Construction Reports, Series C 20, various issues. 176 Percentage of non-farm housing units occupied at the end of the year. For 1920-41 Klein's data was used. For 1956-1965 data from "Vacant Housing Units in the United States: 1956 to 1965," Current Housing Reports--Housing Vacancies, Series H-lll, No. 43, June, 1966, U. S. Department of Commerce, Bureau of the Census (Washington, D.C.), Table 1, Annual Average Vacancy Rates by Condition and Type of Vacancy for the United States, Inside and Outside Standard Metropolitan Statistical Areas and Regions: 1956 to 1965, p. 22, line 22. For 1941 to 1956: "v" was looked upon as the ratio of the number of non-farm housing units occupied to the variable NS. The numerator was calculated for the years 1956-1965 by vt - N: = Ot' Ot was then regressed on Ft for 1956-1965 and, then, Ot was estimated for the period 1941 to 1956. The estimates of 0 along with the values of NS were then substi- t tuted into vt . N: = 0t to obtain estimates of vt for the years, 1941 to 1956. Money supply billions of dollars. For 1920-1946, U. S. Department of Commerce, Bureau of the Census, Long Term . . . ,.pp. 208-209, Series B109. For 1947-1965 centered means of seasonally adjusted monthly data of total money supply, Board of 177 Governors, Federal Reserve Bulletin, June, 1964, pp. 682—692; June, 1965, p. 978; May, 1966, p. 678; and April, 1967, p. 608; Table--Money Supply and Related Data (in billions of dollars). M : Money supply plus time deposits in billions of dollars. For 1920-1946 U. S. Department of Com— merce, Bureau of Census, Long Term . . . , pp. 208- 209, Series B111. For 1947—1965 centered means of seasonally adjusted monthly data of total money supply plus time deposits, Board of Governors, Federal Reserve Bulletin, June, 1964, pp. 682-692; June, 1965, p. 978; May, 1966, p. 678; and April, 1967, p. 608. Table-—Money Supply and Related Data (billions of dollars). Estimation Procedure Let us consider the problem of estimating the structural coefficients by the method of 2SLS. For this we need the reduced form equations for Y, X, p, AH, i, and r. Beginning with a linear approximation of equation (14), we have (17) x = (Y+T) + ¢0 + ¢lp + ¢2(w2 + R1 + R2). From (12) and (3), setting Y3 = 0, we find 2. 7 (18) p w' + n'p_l + né(Y+T) + 03(W2 + R 0 1 + R2) + l 178 Also from (8), setting l/r__l = z (19) r = 06 + 0 Substituting for M? from (9) into (11') (after taking a linear approximation of (9)), then substituting for M? from (11') into (10), and then solving (10) for i we have = t v v v v (20) i “0 + leS + u2p + p3(Y+T) + put D ' Y Y Finally, from (3) setting Y3 = 0, we see that = v v v v v (21) AH y 0 + le + y2p + y3H_l + yut. Viewing equations (17) - (21) it is now clear that finding the reduced form equation for Y will immediately yield the reduced forms of r and p. The reduced form for Y along with that for p will then give the reduced forms for X and 1. Finally, substituting the reduced forms for X and p into (21) will result in the reduced form equation for AH. All that remains is the derivation of the reduced form for Y. Beginning with equation (13) (13) Y + T = I + AH + C + D + D _ v v 1 2 + (D3 D + G) and substituting for C and 1, yields 179 (22) Y + T = do + dlp + a2X + d3E + duq + a5X_l + d6K_l + d7t + 08H + 09H_l + leDl _ H + allD2 + 0112(D3 D + G) + 013T + o‘149—1 + O‘15E—1 + O‘16‘1--1° Substituting for H: (23) Y + T = 80 + 81p + 82X + 83E + 349 + BSX-l + B6K_l + B7t + 88H_l + 89p_l + 61001 + D - D" + G) + B 811 2 + B12(D3 l3T + B14E—l + 81581—1 and then substituting for D D and r into (23) yields 1’ 2’ (24) Y + T = 60 + 61p + 62X + 53E + duq + 6 5X-1 + 6 9p-1 10V-l + 6 + 3 Y + 613Y_2 + SluAF llql 12 —1 + 515r-1'+516(q1)—1 + 617(q1)-2 + 5181 + 6 T + 521(D3 - D" + G) 619AF-1 20 + 0 + 228-1 523E-1° Substituting for Mg 1 from (10) we have from (11') and then substituting for 180 (25) Y + T = no + 01p + n2X + n3E + nuq + n X 7 + Y n10V—1 + nllql + n12 -1 + n13Y—2 + n14AF + n15r—1 + n16(q1)-1 I + n17(ql)—2 + n181-1 + n19Ms D D 1 + n21 2)-l + n22AF M (M n20 —1 + n23T + “24(D3 - D" + G) + n25E_1 ”26q—l° Substituting for M1: = v t v v v v (26) Y + T ”0 + nlp + n2X + n3E + nuq + ”BX-1 + néK_l + nit + néH_l + n§p_l V V ' + “10V—l + n'1181 + n12Y-1 + n13Y—2 ' 7 7 ' + nl4AF + n15111 + ”16(q1)-1 D 3 v ' + n17(q1)-2 + ”18* + n19Ms + n20(M2)-1 + nélAF_1 + UégT + né3(D3 - D" + G) 1 1 + n24E—1 + n25q_l. Substituting for X: (27) Y + T = p0 + plp + p2(w2 + R1 + R2) + p3E + p9p-1 + p10V-1 + pllql + p12Y—l 181 Y + pluAF p13 -2 + p 15r-1 + p16(ql)—1 + p17(ql)—2 + 9181' (MD 2)-1 + p21AF T p19Ms + 020 —1 + 022 + 023(D3 - 0" + 0) + p2uE_l + p25q_l Substituting for X into (12), then substituting for p into (27): (28) Y + T = w + le_l + w2H_2 + w3p_2 + + wu(w2 + R + R2) + w E + w6q + w E_ l 1 5 7 + w8K-1 + wgt + wlOH-l + wllp-l + w Y + + w 14 —1 “l5Y-2 v + w 12 -1 13q1 17r-1 + “18(ql)-1 19(q1)-2 i + w M + w MD) 820 —l 21 s 22( 2 + w -1 23AF-1 __ H + N24T + 0025(D3 D + G) + w26q_l. Since equations (17) - (21) do not contain any more predetermined variables than those given by the reduced form for Y + T (and therefore for Y), the reduced form equations for X, AH, p, i, and r will contain precisely the predetermined variables given by equation (28). Equa— tion (28) involves 26 predetermined variables. Because we wish to re-estimate Klein Model III for the period 1922-1941 and inspect the damage to Klein's structural parameters resulting from our manipulation of the variables, 182 we cannot estimate the reduced form coefficients. We have data for only the 22 years from 1920—1941, thus there are too few observations to calculate all 27 coefficients of the reduced form. Also, two observations must be dropped to allow for the lagged variables. Variables Participation Matrix (VPM) What is needed, then, is a method to decide the sub- sets of predetermined variables in the original reduced forms to include in the moidfied reduced form equations of the endogenous variables. In the spirit of Franklin FisherLl we first set up a variables participation matrix. This matrix is a transform of the coefficients matrix, B, of the structural equations: BY + FZt = u t t' We transform B into the VPM by replacing the bij's such that if b = 0 we write 0, and if b # 0 we write "i". 13 i.) The variables participation matrix provides a priori structural information which can be used to decide which predetermined variables to include in the first stage of the 2SLS method. As can be seen from the VPM below, the Klein Model III is an integrated structure. An integrated structure is one for which the coefficients matrix, B, is neither block-diagonal nor block-triangular. I 183 FIGURE l.—-Variab1es participation matrix for Klein Model III Y I Wl p X H K +_ +_ 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 +. 0 0 0 0 0 0 0 0 +_ 0 0 +. +_ +_ +_ 0 +_ 0 0 +_ +_ 0 0 0 +_ 0 0 0 0 0 0 0 0 0 +_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +_ 0 +_ 0 0 +_ +_ +_ +_ 0 0 +_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +_ 0 0 0 0 0 2 nil DK2 D i M N 184 If the coefficients matrix were "block—diagonal" the structure would be defined as "non-integrated"; if the matrix were "block-triangular" the structure would be defined as "causal" or "block recursive." These classi- fications pertain to linear structures. The Klein Model III is not linear. However, this situation can be remedied by taking a linear approximation of the structural equa- tions in the system for the purpose of deriving the VPM. While the VPM for Klein Model III falls under the integrated category, it is quite close to being block- m‘. n.‘ .a.‘ T A triangular and therefore approximates the causal variety. In fact, it is nearly block-diagonal. Because of this the model may be viewed as composed of three sectors and this a priori information used to decide the variables to in— clude as the predetermined variables in the first stage of the 2SLS method. Predetermined variables included in the first spage.—-The current endogenous variables p, AH, X, Y, r, and 1 appear as explanatory variables in the structural equations. We decide on the predetermined variables to include in estimating the reduced form equations of these current endogenous variables by noticing, first, the sector into which each variable falls in the VPM. Since the equations for H, p, and X fall in the first sector (numbering the sectors along the principal diagonal from left to right), we decide to include in the "modified 185 reduced forms" for H, p, and X those predetermined vari- ables appearing in the structural equations of sector one. Clearly q, t, E (W2 + R1 + R2), K-l’ E-k’ q_l, 8-1’ and X_l belong to this set as can be seen from an examination of structural equations (1), (2), (3), (12), (14), and (15). Since (12) includes the term (u3)_l, we also include H_2 and p_2 (this time we allow Y3 to be non-zero). Since X is directly affected by Y + T which in turn is directly affected by (D3 - D" + G), we include (D3 - D" + G) as well in these modified reduced forms. H_l was included in the modified reduced form for p, but was excluded from the first stage estimation of AH and X. The reason it was excluded from participating in the esti- mation of AH is obvious. AH is by definition H - H—l; hence we would expect to find a very high dependency be- tween H_l and the distrubance term in the equation esti- mating AH. The reason H_1 was excluded from the reduced form for X is procedural rather than theoretical-—as it happened, the reduced form estimates for AH and X were run simultaneously on the computer and it was easier to leave H_l out altogether. Sector two involves the structural equations for Y, C, D1, r, and v as shown by the VPM. Even though we originally joined D2 with the variables 1, ME, and MB to form sector three, D2 is a part of the housing market and therefore is closely mated to the variables Dl’ r, and v. 186 Also, the nature of the VPM would not be altered signifi- cantly by constructing sector two to include D2. All that happens is that the "i" in cell (Y, D2) now becomes a part of the second diagonal matrix and the "i" in cell (D2, 1) falls above the third diagonal matrix. In either case, the influence of the monetary sector on the other sectors of the model will operate through only one variable. As L the VPM was originally constructed, the monetary sector ‘1 influenced Y through its effect on D2. When sector two is expanded to include D changes in the monetary sector my“ igqibl .m I . . 25 influence sector two via the interest rate. Nevertheless, the truncated reduced forms for r and Y were estimated using the predetermined variables appear- ing in the structural equations for Y, C, D1, r, and D2. Thus, r and Y were estimated using the variables: ql, K—l’ t, AF, NS, (03 - 0H + G), T, H_l, Y_l, Y_2, r_l, (ql)_l, (q1)_2, AF_1, and V-l' The variable K_l crept into these equations by substituting for I in equation (13) its identical counterpart, K - K-l' The truncated reduced form for i was estimated using , D the variables. MS, (M2)_l, i_l, r_l, (ql)_1, (q1)_2, l' The first eight variables enter the equations of sector three--(6), (9), AF_l, t, (03 — 0" + 0), K_1, and H_ (10), and (ll')--direct1y. The remaining variables entered the estimated equation through their direct in- influence on Y + T. 187 Second Stage of 2SLS After the estimates of the current endogenous vari- ables, r, p, Y, AH, X, and i, were found, they were sub- stituted into the structural equations as explanatory variables. Then the second stage of 2SLS was taken by performing ordinary least squares with the resulting set of variables. The equations were first estimated for the period 1922-1941 in order to find some indication of the damage to Model III caused by the revisions discussed j earlier. 1 Results Klein's Results The following are the estimates which Klein obtained using the "method of reduced forms" (now commonly referred to as limited information maximum likelihood). w = 5.04 + 0.41 (pX-E) + 0.17 (pX-E)_ 1 1 + 0.17 (t-l931) I = 2.59 + 0.12 (pg-E) + 0.04 (pg-E)_l - 0.10 K_l H = 1.17 + 4.60 p + 0.12 (X-AH) + 0.50 H_ 1 c = 11.87 + 0.73 Y + 0.04 (t-l931) D1 = -9.03 + 3.74 (§_J + 0.02 (Y + Y_l + Y_2) q1 + 0.0043 AF D2 = —2.14 + 2.81 r_l + 0.02 (q1)_l - 0.44 (q1)_2 + 0.0016 AF_ - 0.18 1 1 188 v = 178.01 + 0.29 Y - 2.62 r + 1.42 (t-l93l) — 3.76 NS Ar = -2.15 + 0.02 v_l + 0.00071 Y + 0.17 _i_ r -1 ME = 8.45 + 0.24 p(Y+T) + 0.03 p(Y+T)(t-1931) — 1.43 (t-1931) D _ D M2 - 15.37 + 0.28 i — 1.90 i_l + 0.74 (M2)-l — 0.18 (t—l931) AX = 2.55 — 4.46 (u3)_1 + 82.76 Ap Reestimation of Klein Lg Model III for Period 1922-1941 The following are the 2SLS estimates of Klein Model III derived from the use of revised data, the replacement of Klein's equation (11) with equation (11'), and the treatment of R1 as exogenously determined. All equations except the output adjustment equation (equation (12)) were estimated for 1922-1941. Equation (12) was estimated for 1923-1941 since it contains as an explanatory variable the residual of equation (3) lagged one period. (1*) W1 = 7.26768 + 0.40159 (pX-E) + 0.09670 (px-E)_l (0.03373) (0-03886) + 0.02953 t, R2 = 0.9713 (0.05152) (2*) 1 = 7.03134 + 0.12187 px—E + 0.06759 pX—E (0.02284)( q ) (0.02435)( q )'l - 0.10617 K‘l, R2 = 0.9343 (0.01817) 189 (3*) H = -3.25596 + 18.76474 p + 0.14090 (X—AH) (7.54425) (0.01601) + 0.63565 H 1’ R2 = 0.9500 (0.06659) ' (4*) C = 28.78560 + 0.63131 Y + 0.98523 t, (0.04643) (0.14710) R2 = 0.9685 (5*) D1 = —24.7604l + 5.20749 (r/ql) (0.87043) + 0.03038 (Y+Y +Y ) + 0.01106 AF, -2 (0.00488) ’1 (0.0157) R2 = 0.7913 (6*) D2 = —5.36336 + 22.81729 r_l - 6.89784 (ql)_l (5.01210) (14.96885) — 1.63128 (q ) 2 + 0.00507 AF_l (7.91248) " (0.00224) — 1.30531 1, R2 = 0.8731 (0.52135) (7*) v = 158.45360 + 0.09550 Y + 2.71645 r (0.02441) (6.84348) + 1.69510 t — 3.86616 NS, R2 = 0.9179 (0.35635) (0.44540) (8*) Ar = -1.16478 + 0.01079 v_l + 0.00036 Y (0.00116) (0.00017) + 0.05138 _1_ . 82 = 0.8701 1" —1 (9*) ME = 19.92253 + 0.00575 p(Y+T) (0.05200) + 0.01715 p(Y+T)t — 0.79942 t, R2 = 0.9781 (0.00321) (0.26779) (10*) M2 = 18.01618 — 1.39971 1 - 0.75883 1_l (0.36626) (0.35205) + 0.81421 (Mg)_l — 0.32694 t, R2 = 0.9715 (0.05848) (0.03272) 190 _ D D (11*) MS - Ml + M2 (12*) AX = 6.28263 — 2.50434 (u3)_l (1.81979) + 635 70364 Ap, R2 = 0.6340 (132.32889) (13*) Y + T = I + AH + C + D1 + D2 + D3 - D" + G p(Y+T) - W2 - Rl — R2 (14*) X ll --T-p— I p (15*) AK I Eleven coefficients appearing in the revised set of 4 sample regressions appear to differ significantly from 4) those Klein obtained. Aside from the coefficient tied to t in equation (4*) and the one associated with p(Y+T) in equation (9*), all coefficients which blatantly deviate from Klein's are associated with indexes of prices or interest rates. In equation (6*) the coefficients of r_l, (ql)_l, (ql)_2, and i are not of roughly the same magnitude as those previously found. Other sizeable dif- ferences between Klein's estimates and those presented here are connected to p of equation (3*), Ap of equation (12*), r of equation (7*), Fl; of equation (8*), and i of equation (10*). - While eleven coefficients contained in the eleven Stochastic equations differ from the order of magnitude computed by Klein, the picture is brighter in terms of the number of variables which are no longer statistically 191 significant at the 95 p8r cent level of confidence. The adjustments we have made in Klein Model III have caused only the coefficients of p(Y+T) in equation (9*), (ql)_l and (ql).__2 in equation (6*), r in equation (7*), and (1.13)_l in equation (12*) to become not significantly different from zero. Because the damage inflicted seems to be relatively minor, we proceed to estimate Model III for the extended period 1922-41 and 1947-1965. Results for the Extended Period For the extended period, the sample regression equa- tions are: (1**) W1 = -1.10413 + 0.41425 (pX-E) (0.05223) + 0. 22280 (pX— E) _l + 0.01825 t, R2 = 0.9985 (8 05400) (0.11805) (2**) I = 10.85164 + 0.10246 ( x- -E) (8 02101) + 0. 07957 ( x— -E)_l - 0.11434 K_l, R2 = (0. 02211) (0.00822) 0.9342 (3**) H = 11.65666 + 0.15125 (X-AH) — 9.04019 p (0.02406) (8.17592) + 0.57622 H 1’ R2 = 0.9939 (0.07286) _ (44*) c = 4. 96737 + 0. 83644 Y + 0. 57315 t, R2 = (0. 02826) (0.19864) 0.9971 (5**) D1 = 0. 69715 - 1.11282 (r/ql ) (1.03702) + 8 01111 (Y+Y wl+Y ) + 0. 00161 AF, (0. 00203) (0.00091) R2 = 0.8485 (6**) (7**) (8**) (9**) (lO**) (ll**) (l2**) (l3**) (14**) (15**) Ar AX Y AK 4. —2.94203 + 23.0“778 r -0.80098 + 0.00738 V + 0 2.20494 - 3.21538 1 + 2.42517 1 + l. (0. D M1 “.57200 - 1.HU730 (u R2 .01133 p(Y+T)t + 1.85179 t, R .00096) (0.20065) + M I + AH + C + D 192 + 3.88784 (q ) (3.31570) ‘1 (6.16504) ‘1 .90503 (ql)_2 + 0.00117 AF_l .02974) (0.00058) 2 .82335 i, R = 0.9047 .38517) .44945 + 0.09504 Y + 19.87004 r (6,5“262) 2 (0.03111) .30607 t — 2.15627 NS, R = 0.6292 .23421) (0.30316) + 0.00025 Y ‘1 (0.00003) (0.00081) 2 (0.03519 1 , R = 0.7739 01166)F:I .43881 + 0.69677 p(Y+T) (0.04883) 2 (1.19702) (1.00185) ’1 2 2U003 (M 03141) D 2 — 0.18928 t, R D) 2 '1 (0.07478) 3)-1 (0.84718) (101.88610) 0.3778 + D - 0" + G + D 3 1 2 p(Y+T) - W2 - R - R l 2 p 0.9896 0.9971 + 378.U5320 Ap, REE? 193 Equation (l2**) was estimated for the years 1923- 1941 and l9u8-l965. All coefficients except those for t in equation (1**), (”3)—1 in equation (12**), g: and AF in equation (5**), and (ql)—l and (ql)_2 in equation (6**) are significantly different from zero at the 95 per cent confidence level. Conclusion 8 This chapter has outlined the data revisions and other alterations we made in Klein Models II and III. ' “‘4“ ha: J. . Klein Model II was estimated by the 2SLS and the LI methods for alternative definitions of the price index and the money supply. Klein Model III was estimated using BSLS. We outlined the difficulties encountered as a result of the large number of variables present in the reduced form equations and our procedural remedy. Since the F values computed for all equations for the extended period show that a highly significant relationship exists between the variables in each equation, we retain Model III. The building and estimation of the structural equa- tions of a dynamic model, while considered by many to be the end product (because it is more or less all that is needed for forecasting), are only a start toward a thorough analysis of causes of change in the endogenous variables. In Chapter VI the fundamentals of dynamic 194 analysis are presented and then applied to the naive Klein Model II. In Chapter VII the complexity of analysis in- creases somewhat with an exploration of the dynamic pro- perties of Model III. FOOTNOTES—-CHAPTER V lKlein's Models I, II, and III are found in L. R. Klein, Economic Fluctuations in the United States, 1921- 1941 (Cowles Commission Monograph No. 11; New York: John Wiley and Sons, 1950), 168 pp.; L. R. Klein and A. S. Goldberger, An Econometric Model of the United States 1929-1952 (Amsterdam: North Holland, 1955), 165 pp.; Stefan Valavanis-Vail, "An Econometric Model of Growth, U. S. A. 1869-1953," American Economic Review, XXXXV (May, 1955), 208-221; Michio Morishima and Mitsuo Saito, "A Dynamic Analysis of the American Economy 1902-1952," International Economic Review, V (May, 196M), 125-164. 2 Klein, Economic Fluctuations . . . , p. 80. 3Ibid., pp. 102—105. “Franklin Fisher, "Dynamic Structure and Estimation in Economy-wide Econometric Models," in The Brookings Quarterly Econometric Model of the gnited States, ed. by J. S. Duesenberry, G. Froom, L. R. Klein, and E. Kuh (Chicago: Rand McNally, 1965), pp. 589-635. 195 CHAPTER VI DYNAMIC PROPERTIES OF THE KLEIN MODEL II Our objective is to collect further evidence per- taining to the effectiveness of monetary and fiscal policy. The policy variables we select for examination by virtue of our choosing Klein Models II and III are government expenditures, personal taxes, and the money supply. In this chapter we subject Klein Model II to dy— namic analysis. Since the model is a naive one, the re- sults we obtain may be suspect. However, Kelin Model II will provide the training ground for applying similar maneuvers to Klein Model III. Before proceeding with an examination of Model II, we pause to explain the various facets pertaining to dynamic analysis per se.’ This di- gression will point out the several bases that dynamic analysis provides to compare the effectiveness of policy variables. An Introduction to Dynamic Analysis In our review of the studies of FM, DM, AM, and Hester, we found that these studies went no further than 196 w.-.L‘ ‘- #- 197 to compare the correlation coefficients associated with alternative reduced form equations for consumption. Any decisions as to the relative effectiveness of monetary and fiscal policy were based primarily on these compari— sons. The major disputes arose over the definition of the variables included in the equations and the specifi- cation of this system. Much more can, in fact, be done L by concentrating more fully on forms which are derivable from the structure. I, . ,.IIJLK’MO‘. A full dynamic analysis examines the reduced form {“1 equations of the jointly determined variables as well as the so called fundamental dynamic equations. Proper attention to these equations uncovers a wealth of informa- tion. First, it allows an examination of the stability of the system. Second, it enables us to estimate the impact and dynamic multipliers which in turn may be used to analyze the causes of changes in the endogenous vari- ables. The impact and dynamic multipliers allocate the portions of the changes in the endogenous variables which are caused by (a) changes in the exogenous variables in- cluding the trend variable; (b) the given initial values of the exogenous variables, the so called "starting values"; (c) the given initial values of the endogenous variables, the so called "initial conditions"; (d) changes in the constant term, the changes in exogenous 198 variables not specified by the structure; and (e) the effects of disturbances and errors. Dynamic models are systems of difference equations. As such, they are considered better approximations of macroeconomic phenomena than comparative static models since economic phenomena are likely to affect future events, and to be affected by past events. Comparative l statics can reveal only the ultimate changes in static : equilibrium positions resulting from changes in magni- tudes of structural parameters and exogenous variables. .‘fl-‘V The derivative of an endogenous variable with respect to a policy variable does divulge the long run or "equilib— rium multiplier" of that policy variable on the endogenous variable. However, the multipliers derived in comparative statics do not offer the slightest insight into (a) the length of time it will take to reach the new equilibrium position, (b) the proportion of the ultimate change in the endogenous variable that will take place within a few unit time intervals, (0) the time paths of the endoge- nous variables given the values of exogenous variables and the initial conditions, (d) the stability of the system, and (e) the effects on the time paths of the endogenous variables resulting from a "one-shot" or impulse change in an exogenous variable. Any serious attempt to decide the comparative effectiveness of policy variables must consider these effects. A complete dynamic analysis 199 does allow for consideration of the temporal properties of the Structure. Example It may be easiest to illustrate the efficacy of dynamic analysis by use of an example. Assume that we have the following three equation deterministic structural l model: j.“ . is (l) Ct = 00 + alYt K E; (2) It = 80 + BlYt-l 8' (3) Yt = Ct + It + Gt where Ct = consumption It = investment Gt = government expenditure, and Yt = income. For simplicity we choose a model with only a one period lag and only one exogenous variable. Since the first two equations are stochastic, we are also abstracting from the presence of the disturbance terms in these equations. Reduced Forms First we derive the reduced form equations which express each endogenous variable in terms of the pre- determined variables, Gt and Yt-l' The reduced form equation for Yt is found by substituting for 0t and It in equation (3): Yt = do + dlYt + 80 + BlYt-l + Gt therefore 80 81 1 (A) Yt = 0‘0 + l-d + l—d Yt-l + -d Gt 1 l 1 Substituting for Yt from equation (4) into equation (1) we find the reduced form for C : t B o a . _ 0 . 1 1 1 (5) Ct ‘ 0‘0 + 1—ol I l-dlYt-l + l—let° The reduced form equation for It is the same as its structural equation, i.e. (6) It = 80 + BlYt-l' The coefficients of the predetermined variables in reduced form equations (A), (5), and (6) are expressed in terms of the structural parameters and are called the impact multipliers. They measure the direct (but not necessarily the total) effect on the endogenous variable in period t of a unit change in the value of the pre- determined variable to which the coefficient is attached holding the magnitudes of all other predetermined vari- ables fixed. Therefore, act/act = dl/l-dl measures the direct effect (in this case, the total effect as well) on current consumption expenditures of a unit change of 201 current government expenditures, given the value of the level of income in the previous period. Fundamental Dynamic Equations The reduced form equations show part of the effect on the current levels of the endogenous variables to be due to the values which the endogenous variables took on in the past. These past values, however, were influenced in turn by previous magnitudes assumed by the exogenous variable, G. Hence we attempt to purge the reduced form equations of lagged values of endogenous variables other than the one explained by the reduced form equation. That is, we transform the reduced form equations so that each equation expresses a current endogenous variable only in terms of past values of that variable and current and past values of the exogenous variables. These new ex- pressions are called "fundamental dynamic" equations. In the present example, the reduced form equation for Yt is also the fundamental dynamic equation for Y since the only endogenous variable in this reduced form equation is Y. To find the fundamental dynamic equation for Ct we first substitute equation (2) for I in equation (3). t This gives an expression for Yt in terms of Ct’ Yt-l’ and Gt: 202 (7) Yt = 80 + Ct + BlYt-l + Gt. Next, we eliminate Yt from equation (1) by substituting into this equation the expression for Yt given by (7): (8) Ct = a0 + a1 [80 + Ct + BlYt-l + Gt]. Lagging equation (1) one period and multiplying by -81 we find: (9) -B Ct-l = -BlaO-81Yt_l. Next, we add equations (8) and (9): (10) Ct-BlCt—l = do + also-Blao + alCt + ath or a (1-8 ) + a B B a __ 0 1 10 1 1 (11) Ct ‘ 1—ol + l-aiCt-l + ILEIGt' This procedure of lagging equation (1), multiplying the lagged equation by minus the coefficient of Yt-l in (l), and adding the new expression to equation (8) is called a Koyck transformation. This transformation has elimi- nated Yt-l from equation (8) and has given the fundamental dynamic equation for C. To find the fundamental dynamic equation for It’ we proceed in a manner similar to that followed for C. First, we substitute for C into (3): t 1?: _""’“ _‘. ‘7 _. 'T‘-'_.. 203 Yt = do + alYt + It + Gt or a O 1 1 Y = —— + ——I + ——-G . t l—ol l-dl t l-al t Therefore, “0 1 1 It = 80 + 81 [ T—-ol + __l-01It-l + _Gl-al t-l J °r 8 (1-9 ) + B a B B _ 0 1 1 0 l 1 (12) It - (1-01) + l-alIt-l + l—ath—l' StabilityVConditions We can now examine the system for stability by trans- ferring all terms involving Y to the left hand side of the fundamental dynamic equation for Y and setting the right hand side equal to zero: This gives the auxiliary equation 8 1 (13) A - T:EI - I O Equation (13) is a first order dynamic equation; it is easily seen that the general solution is: 204 B l t 0(1-0 ) (14) Y = Y t 1 The system is, therefore, inherently stable if Bl/l-al is less than one in absolute value. Equation (14) is the general solution to the homogenous equation. The particu- lar solution depends on the time paths of the exogenous r {T‘- o; 40w 1' - .- ‘1th ‘- .,_ variables. But if the magnitudes of the exogenous vari- ables are constant over time, the particular solution is either a constant or is proportional to t. Thus, the 5...». general solution of the non-homogenous equation will be 8 l 1 )t and the second term will dominate. Thus, if Bl/l-al is less than one in absolute value, the non-homogenous system in said to be inherently stable. The inherent time path of Y depends on the values of the parameters, 01 and 81, associated with the endogenous and lagged endogenous variables, Yt and Yt-l' Longerun Multiplier The fundamental dynamic equation for Y can be used to find the long-run or equilibrium multiplier (if it exists) of a particular exogenous variable when the values of the other exogenous variables remain unchanged. To find this multiplier, we replace the subscripted Y's 205 with Y, and the subscripted exogenous variables with 5. We then find dY/da. In our example we have a + 80 8 ' _ O l - l - (.15) Y - W.- + l-a Y WC} 01" l l l B a + B l ‘ 0 0 l ‘ (l - ————) Y = + ————G or 1-01 l-al l-al l 0‘0 + £30 1 l - l - ———— l-al l-al l 6 _ _ l—al (l ) dY/dG = ———————— = l-al This long—run multiplier assumes that we begin at a static equilibrium position and indicates what will be the ultimate change in Y per unit change in G by the time the new static equilibrium position is reached. Since G is a flow variable, the new level of G would necessarily be sustained each period in order for equation (16) to represent the long-run multipliers. Dynamic Multipliers and the "Final Form‘r We can also use the fundamental dynamic equation for a given endogenous variable to derive an expression of the 206 current level of that variable entirely in terms of the exogenous variables. This representation is called the 1 All that needs to be done is to purge "final form." the lagged values of the endogenous variable from the fundamental dynamic equation. The fundamental dynamic equation for the first unit 94,.— time interval of the sample period, i.e. t=l, expresses the value of the endogenous variable for t=l in terms of the initial conditions, the starting values of the exogenous variables (all assumed to be given), and current r“ values of the exogenous variables. The value of the endogenous variable at t=1, Y1, is then substituted into the fundamental dynamic equation for t=2. In our example, from the fundamental dynamic equation for Y, equation (4), we have: a + B B 0 0 1 1 Y1 ' 1-a + 1—a Y0 + lion—GI or 1 l l (17) Y = a + ——l—G l 1 l-al l a + B _ 0 0 1 where 8.1 - -—l:al—- + l-alY0° Hence Yl is determined by the initial condition, YO; the starting value of the exogenous variable, GO; and the 207 <2urrent level of the exogenous variable, G1. Re-writing (17) as (17') Y1 = a1 + Y2Gl 1N8 substitute (17') for Yl in the fundamental dynamic equation for t=2: 0‘0 + 80 B1 _ 1 1 (l8) Y2 ‘ “I:E" + 135‘ [a1 + 125‘61] + TIE-G2 or 1 1 1 1 (18') Y = a + —-l—G + 81 [ 1 JG 2 2 l—a 2 1-u l-a 1 1 1 1 a + B B _ 0 0 1 where 8.2 - T + l-a a1 1 1 'Then: a + B B _ 0 0 1 1 (19) Y3 ‘ l-a + 1—d' [a2 + 1-o'G2 1 1 1 + 81 . ——l—G ] + -—l—G or l-al 1-01 1 l-al 3 _ 2 (19') Y3 — a3 + Y2G3 + le2G2 + 717201. Continuing in this manner, we find: _ 2 (20) Yt - at + Y2Gt + Y1Y2Gt-l + Y1Y2Gt-2 + .... + t-l Y1 Y2G1° 208 The coefficients of the G's in equation (20) are the dynamic multipliers. Each multiplier shows the effect on the time path of Y resulting from an impulse change in G in a particular period.2 The dynamic multipliers are set down in tabular form here with the lag (denoted by k) shown in the first column and the corresponding dynamic multiplier of Gt—k in the second. TABLE 9.--Dynamic multipliers for G on the variable Y for three equation model. Lag (k) Multiplier 0 Y2 = l/l-al 2 l 7172 = Bl/(l-al) 2 yiyz = ei/(l—al)3 In this example, the multiplier associated with a particular lag is Y1 times the dynamic multiplier one period earlier. Since Y1 is equal to Bl/l-al and since Bl/l-al must be less than one in absolute value if the system is stable, the multipliers become smaller in W.l_'— ' "tnd‘ IE fig-«nu . 1.? 209 absolute value the greater the lag from the current period for models that are inherently stable. The long-run multiplier discussed earlier can be computed in an alternative manner by summing all the dynamic multipliers for G. That is, we set all subscripted G's in equation (20) equal to G and sum up the coeffi- cients.3 In our case, the long-run multiplier is equal to: !_ i 1 B B B 1 1 1 2 1 t-l i (21)T[l+iT+(m—) + ...+(1—_—a—) J g l 1 l l ‘5 is B .l t 1 - (l-al) 1 = 31 l-a l l - (l-dl) As t approaches infinity, [Bl/(l—al)]t will approach zero. Hence the long—run multiplier will equal l l—a (22) 3; =1_a1_8 l 1 l l - l-a l which is what we found earlier. The long-run multiplier will be finite if and only if the system is stable. These results show that it is possible to determine how many periods are needed before, say, 90 per cent of 99 per cent of the equilibrium multiplier is attained. 210 Also, because the first few dynamic multipliers are the most important—~se1dom is a given policy variable altered and then sustained at the new level for more than a few periods before being changed again--we can sum the dynamic multipliers for the first few lags and use the ratio of this sum to the long-run multiplier as a measure of compari- son of the effectiveness of policy variables. The final form for Y is useful in finding the inherent time path of Y. In our example: _ 2 Yt ‘ at + Y2Gt + Y1Y2Gt—1 + Y1Y2Gt-2 + .... + YE-lY2Gl. The constant term a captures the movement of Y not due t t to government expenditures. Thus the inherent time path of Y is captured in the values of a The value of a t' t expresses the influence of the initial conditions on the time path of Y holding the levels of the exogenous vari- ables constant. In terms of the initial conditions, the V at S are a1 = Y0 + Y1Y0 a2 = Y0 + Y131 = Y0 + Y1(70 + YlYO) 2 70(1 + 71) + YlYO 211 _ 3 YO + 71a2 - YOEl + 71(1 + 71)] + YlYO - 3 2 4 2 t-2 t t yo[l + 71 + yl + .... + yl (1 + yl)] + leO. m ll Thus, the effect of the level of Y in the initial period, YO, disappears as t approaches infinity; the 11m a n-Hao a constant that is independent of the initial conditions. t equals Actual Movements in Y Until now we have been concerned with hypothetical changes in policy variables and their effects on the level of income. We can also trace the observed changes in income that were caused by actual changes in the exogenous variables. In our example: (23) (Yt ‘ Yt—l) = (at ‘ at-l) + bO(Gt ‘ Gt-l) + G - G + .... + b G b t-l t-2) t 1' 1( If we had more than one exogenous variable in our system and therefore on the right hand side of (23) as well, we could, by substituting in the actual values, find the 212 role that each policy measure played in affecting the observed change in the level of income. Dynamic Analysis of Klein Model II The previous section outlined the nature of a com— plete dynamic analysis for a linear structural system. At this point we apply this analysis to the estimates of p the Klein Model II we obtained in Chapter v. f The reduced form equations for this model were 7 presented in Chapter V; but we shall reproduce the re- f duced from for Y/pN here using the estimated coefficients EV of the stochastic equation with p = p2 and Me= M2 for the period 1921—1941 and 1945-1965 derived by 2SLS. The reduced form equation for per capita real disposable income is 48.58589 0.23401 Y/pN = 1 — 0.63614 + l — 0.63614 (Y/pN)-1 + 0.04115 1 — 0.63614 (M/pN)-l + l l — 0.63614 (I + G ' T/pN or (24) Y/pN = 133.52908 + 0.64313 (Y/pN)_l + + 0.11309 (M/pN)_ + 2.74831 (I + 0 — T/pN). 1 Equation (24) is also the fundamental dynamic equation for Y/pN. Solving the homogeneous auxiliary equation 213 Y/pN - 0.64313 (Y/pN)_l = O we see that the general solu— tion is (Y/pN)t = k(0.64313)t. Therefore, the system is inherently stable. The long—run multipliers for G and M are respectively 2.74831/0.35687 7.70115 aY/aé 37/8M 0.11309/0.35687 0.31689. The results of the estimation of the stochastic structural equation, which show the coefficient of the money supply to be not statistically significant, indicates that the Klein Model II may not provide a good description of the U. S. economy. Therefore, we shall not carry out a dynamic analysis of this model and, instead, turn to Klein Model III. WY ‘ t!) ' .'.-|._‘i}\ (Elm FOOTNOTES——CHAPTER VI 1Arthur S. Goldberger, Econometric Theory (New York: John Wiley and Sons, 1964), p. 374. 2Ibid., p. 375. 3Ibid. 214 CHAPTER VII DYNAMIC ANALYSIS OF KLEIN MODEL III Introduction In this chapter our main purpose is to examine the dynamic properties of Klein Model III. Of particular interest is the relative effectiveness of the policy variables——the money supply, government expenditures, and (to a lesser extent) taxes--to induce changes in net national product over time. Since Klein Model III is more intricate than either Klein Model II or the models prOposed by FM and their critics, it should describe more comprehensively the dependency of the time path of net national product on exogenous forces than the models we discussed heretofore. Thus, this model should place an analysis of the relative effectiveness of monetary and discal policy on firmer ground. Consequently, the analysis of the dynamic prOperties of Klein Model III constitutes the major thrust of our study. Because we are primarily concerned with the exogenous forces influencing the time path of net national product, we begin with a derivation of the fundamental dynamic equation for this variable. After finding the fundamental dynamic equation, we are able to test the inherent 215 216 stability of the system and find the dynamic multipliers associated with the policy variables. Also, the dynamic multipliers along with the actual changes in the policy variables and net national product over the sample period enable us to analyze the relative importance of the policy variables to cause changes in the level of net national product. Finally, we search for the cricial variables that determine the stability or instability of the system and the size of the policy multipliers. Fundamental Dynamic Equation for Y + T Initial Revisions in the Structural Equations The re-estimated structural equations presented in Chapter V furnish the starting point of this dynamic analysis. The equations are reproduced here after drop- ping those variables from the structure which have re- estimated coefficients that were found to be not signifi— cantly different from zero. Including only those vari- ables in the structure with statistically significant coefficients makes our task easier in two respects. First, it reduces the number of exogenous variables included in the fundamental dynamic equation as well as the number of lags for some of those exogenous variables which remain. Secondly, since many of the structural equations are non- linear, dropping the variables whose coefficients are 217 not significantly different from zero reduces the number of linear approximations to be performed. The revised re-estimated Klein Model III becomes: (1') M1 = -1.10413 + 0.41425 (pX-E)t + 0.22280 (px—E)t l t - +1.1 1t = 21:2 X-E (2') It 10.85164 + 0.10246 ( q )t + 0.07957 (Eq———)t_l -0.ll434 Kt—l + u21: (3') Ht = 11.65666 + 0.15125 (x—AR)t + 0.57622 Ht_l +u 3t (4') ct = 4.96737 + 0.83644 Yt + 0.57315 t + u“ t (5') D1t = 0.69715 + 0.0111 (Yt+Yt_l+Yt_2) + u5": (6') D2t = —2.94203 + 23.04778 rt_l + 0 00117 AFt_1 -1.82335 it + u6t (7') v = 107.4495 + 0.09504 Yt + 19.87004 rt + 1.30607 t —2.l5627 N: + u 7t (8') Art = -0.80098 + 0.00738 Vt—1 + 0.00025 Yt + 0.03519 1/rt_l + u81; (9') ME = -37.43881 + 0.69677 pt(Y+T)t t -0.01133 tpt(Y+T)t + 1.85179t + u9 t 218 (10') M2 = 2.20494 - 3.21538 it + 2.42517 it_l t + 1.24003 Mg — 0.18928 t + u10 t—l t (11') MS = M? + M? t t t (12') axt = 4.57200 + 378.453204pt + ullt (13') (Y+T)t = It + AHt + Ct + D1 + D2 + D3 - Dé' + Gt E_ t t t E pt(Y+T)t -W2t -th -R2t (14') xt = 1 p .. t .; 47 (15') AKt = It. We will be concerned only with the deterministic part of the structural equations. That is, we will abstract from the disturbance terms in the above eleven stochastic equa- tions. The Quest for a Suitable Fundamental Dynamic Equation Our fundamental dynamic equation for net national product must exhibit certain properties if it is to be representative of the United States economy as well as consistent with economic theory. Namely, the experience of the United States economy dictates that the fundamental dynamic equation not exhibit explosive prOperties. Also, economic theory imposes the condition that the long run multipliers for government expenditures and the money 219 supply must be non-negative. With these a priori restric— tions in mind we proceed to a derivation of the fundamental dynamic equation for net national product starting with the revised re-estimated Klein Model III given at the beginning of this section. The First Attempt The initial step in the derivation involves substi- tuting for C in (15') its equivalent as given by equation (4'): t + 0.57315 t + I + AH (l) (Y+T)t = 4.96737 + 0.83644 Y t t t + D + D + D - 0" + G . 1t 2t 3t t t Next we eliminate D1 and D2 from (1) in the same manner t t that we eliminated Ct' This yields: (2) 0.15245 (Y+T)t — 0.01111 (Y+T)t-l - 0.01111 (Y+T)t_2 = 2.72249 + 23.04778 rt_l + 0.00117 AFt-l - 1.82335 it - 0.84755 Tt - 0.01111 Tt_l - 0.01111 Tt_2 + 0.57315 t + It + AHt + 031: 'Y + Dt + Gt. Substituting for v from (7') into equation (8'), t finding a linear approximation of the term l/rt-l in (8') by taking the Taylor's expansion around the sample mean 1 “Ki Quilt-o; .1“ m .- 220 of rt-l (= 0.82015), and subsequently solving for rt, we find: (3) rt = 0.06817 + 0.00025 Yt + 0.0007 Yt_ + 1.09433 r l t-l S + 0.00964 t - 0.01591 Nt_l. Performing a Koyck transformation on (2) in order to eliminate rt—l yields: (4) 0.15245 (Y+T)t - 0.18370 (Y+T)t_l - 0.01512 (Y+T)t—2 WW‘“‘“"‘“‘"‘T—' + 0.01216 (Y +T)t_3 = 1.71947 + 0.00117 AFt_l - 0.00128 AFt-2 I I—' .82335 it + 1.99534 it_l — 0.84755 Tt + 0.01216 T + 0.91062 T t_2 t_3 t_l — 0.01512 T + O .16809 t + I - 1 09433 It_ + AH t 1 t - 1.09433 AHt_ + (D3-D"+G)t - 1 s t-2' I [.1 .09433 (D3—D"+G)t_l - 0.36677 N Next It is eliminated from (4) by substituting its definitional counterpart, Kt - Kt-l‘ In order to eliminate the current and lagged K variables from the resulting equation, we substitute Kt - Kt-l for It in equation (2') and solve equation (2') for K In the process of solving t. equation (2') for Kt’ we "linearize" the terms involving (E§:§)t and (2%:§)t—1 by the Taylor's expansion about the 221 sample means of Xt’ qt’ pt’ Et’ Xt-l’ qt-l’ pt-l’ and Et-l' The result is the following expression for Kt: (5) Kt = 8.55846 + 36.53032 pt + 0.11282 x - 38.15468 qt t - 0.16111 Et + 27.99212 pt_l + 0.08831 xt—l - 29.49701 qt_l - 0.12786 Et_ + 0.88566 Kt_ 1 1' Next, K is eliminated from (4) by the same procedure as mentioned above. After K is purged from (4) we set out to eliminate i from the resulting equation. We begin this operation by re-writing (10') as: (6) 3.21538 it 2.20494 — MD — 0 18928 t + 2.42517 1 2t t-l D + 1.24003 M2 t-l From (11') MD = M - MD Therefore, we substitute for 2t St 1t M? into (6). This results in an expression for it in D D t terms of MS , Ml , t, it—l’ M and M1 t t St—l’ t-l Next, linearizing equation (9') gives: (7) ME = —65.60966 + 118.08776 pt + 0.29405 (Y+T)t t - 0.42359 t. To find the expression for i in terms of (a) the t endogenous variables yet to be purged from the expression 222 for (Y+T) and (b) the exogenous variables, we substitute for ME into the revised expression for i t t (8) it = 5.42019 - 0.31101 MS + 0.38566 Ms t t—l _ 0 02725 t + 36.72591 pt — 45.53528 pt_l + 0.09145 (Y+T)t — 0.11340 (Y+T)t_l + 0.75424 it_l. Then it is eliminated from the expression for (Y+T) by performing a Koyck transformation. From (14'), after linearizing, we have: (9) xt = -64.18528 + (Y+T)t - 1.42805 (w2+Rl+R2)t + 91.65968 pt. By direct substitution of (9) into the new expression for (Y+T), the Xt's are purged from the expression leaving only the endogenous variables p and H. To eliminate the p's we recast equation (12') as (10) 378.45320 pt = -4.57200 + Xt - Xt-l + 378.45320 pt-l' Substituting (9) into (10) we have: (11) 378.45320 pt = -4.57200 + (Y+T)t ‘ (Y+T)t-l - 1.42805 (W2+R +R2)t + 1.42805 1 (W2+R1+R2)t_l + 91.65968 pt - 91 65968 pt_l + 378.45320 pt_l. 223 Solving (11) for pt will obviously make the coefficient of pt”1 equal to unity. Therefore, if a Koyck transfor- mation were to be made to purge the p's from the expres— sion for (Y+T), the multiplying constant would be unity causing the long run multipliers of M and G to be in- determinate. This result follows from the fact that the Koyck transformation, first, multiplies all the coeffi- cients in the expression for (Y+T) lagged one period by the coefficient of pt-l in the expression for pt. In this case the coefficient in unity. Then the altered lagged equation for (Y+T) is subtracted from its unlagged version. In this particular case, all coefficients appearing in the altered lagged equation are precisely equal to the coefficients appearing in the unaltered equation one period later. Thus, adding all coefficients connected with either M or G in the expression for (Y+T) (after the Koyck transformation is completed) causes both sums to equal zero. The same sum is reached when the coefficients of the (Y+T)'s are added—~first, because of the Koyck transformation and, second, because the coeffi- cient of(Y+T)t_1 is equal to minus the coefficient of (Y+T)t in the expression for pt. Thus the long run multipliers for M and G are each equal to 0/0. It should be noted that this will be the result no matter what the values of the estimated coefficients. In equation (10) 54.119 man—1 \' J In! 224 the value of the coefficient of pt will be equal to the coefficient of pt-l' Whether (14') is linearized to obtain (9) or kept in its non—linear form will not alter the fact that pt will have the same coefficient as pt—l in equation (11) after X has been purged from equation t (10). The reason is that X will be expressed in terms t of current values of p only and that the coefficient of Xt in equation (10) is minus the coefficient of X Thus, when X t-l' is purged from equation (10) and equation t (11) is solved for pt, the coefficient of pt-l will equal unity. Thus, the Koyck transformation performed to purge p from the expression for Y+T will yield indeterminate long run multipliers for all samples. To circumvent this indeterminacy we could either (a) stop with the elimination of the X's and change p and H to exogenous variables,(b) change p to an exogenous variable and proceed to eliminate H from the expression for (Y+T), or (c) reinstate the variable (u as it 3)t-l appears in (l2**) of Chapter V even though the coefficient of this variable is not statistically significant at the 95 per cent level of confidence. We choose to reinstate (u3)t-1’ the lag of the residual term of equation (3'), in equation (12'). Thus (11) becomes (12) 378.45320 pt = -4.57200 + Xt - Xt-l + 378.45320 pt_1 + 1.44730 (u3)t_l “ryunv ”h:-I£‘j i“ 1 m...:‘; u- I\_' C-.."- 225 where, from (3'), (u3) is: t-l = —ll.65666 - 0.15125 xt_ + 1.15125 Ht- (”3)t—1 l 1 - 0.72747 Ht_2. Substituting for (u3)t-l into (12) and solving for pt yields: (13) pt = -0.02577 5552 + 0.93003 7879 pt_l + 0.00348 6820 (Y+T)t - 0.00425 01104 (Y+T)t_l + 0.00580 97691 H — 0.00367 11685 H t-l t—2 - 0.00497 93606 (w2+Rl+R2)t + 0.00606 93631 (w2+Rl+R2)t_1. When (13) is used to purge pt from the quasi-fundamental dynamic equation for (Y+T), the left hand side of this equation becomes (133) 2.7643 37382 (Y+T)t - 1.42698 05652 (Y+T)t_l + 2.81804 09537 (Y+T)t_2 - 2.63533 96798 (Y+T)t_3 + 1.13355 46647 (Y+T)t_u - 0.15832 50781 (Y+T)t_5 - 0.00755 33216 (Y+T)t-6‘ The right hand side contains only one endogenous variable, namely, H. 226 From equation (3'), after substituting for Xt and ignoring the residual (u3)t, we arrive at (14) Ht = -1.94863 63788 + 0.15125 (Y+T)t - 0.21599 23105 (W2+R + 12.04215 1539 pt 1+R2)t + 0.63189 5765 Ht-l' After eliminating pt from (14), we have (15) Ht = 40.44672 3837 + 1.63189 5764 Ht_l - 0.63189 5764 H t_2 + 0.19323 8815 (Y+T)t - 0.19184 8703 (Y+T)t_l - 0.27595 45254 (w2+Rl+R2)t + 0.27396 92205 (W2+Rl+R2)t_1. Equation (15) could be used to eliminate H from the quasi-fundamental dynamic equation for (Y+T). However, to purge H, it is necessary to perform two Koyck trans- formations. First, we would lag the expression for (Y+T) one period and multiply by 1.63189 5764. Second, we would lag the expression for (Y+T) two periods and multiply by -0.63189 5764. Then each equation would be subtracted from the original expression for (Y+T) after substituting into the original expression for the H's their equivalents as given by (15). However, performing these two Koyck transformations will result in the sum of the coefficients of M and G being equal to zero. This is 227 because we are first multiplying each coefficient by 1.63189 5764 and then by -0.63189 5764; the net effect is multiplication by unity. Then these coefficients are subtracted from the original ones causing the sum of the coefficients for each variable, M and G, to equal zero. In the Appendix to this chapter we show that this result is independent of the sample chosen. The effect on the coefficients of (Y+T), after per— forming the above Koyck transformations, is twofold. The first effect is the same as that on M and G. However, (Y+T)t and (Y+T)t—l enter (15) with the sum of the coeffi- cients not equal to zero which by itself would imply that the sum of the coefficients of (Y+T) in the final expres- sion would be non-zero. However, the H's appear in the quasi-fundamental dynamic equation in the form of first differences (see equation (4)). Therefore, when we re- write the expression for (Y+T) with H's instead of AH's, the coefficients of the H's sum to zero. Thus the par- tial sum of the coefficients of (Y+T) in the final equa— tion arising from (15) will be equal to zero after (15) is substituted for each H. Thus the combined effect is for the sum of all coefficients of (Y+T) in the funda— mental dynamic equation of (Y+T) to equal zero. Con- sequently, the long run multipliers of the policy vari- ables on (Y+T) will be indeterminate since the sums of the coefficients of each of these variables are also 228 equal to zero. The equilibrium point of income is also indeterminate because the sum of the coefficients of (Y+T) is equal to zero. The Second Attempt Since the elimination of H from the expression for (Y+T) results in indeterminate long run multipliers, we did not test the resulting fundamental dynamic equation for stability. Instead, we dropped equation (3') from the model and changed H to an exogenous variable. This made the expression for (Y+T), which was found after eliminating p, the new fundamental dynamic equation. This equation (the left hand side of which is given in (13a)) was tested for stability. Since the largest root of the equation formed by setting (13a) equal to zero is 1.70188, the system is unstable. Parenthetically, the long run money multiplier is also negative. Thus, we have yet to find a suitable fundamental dynamic equation for (Y+T). It is not our intention to describe at this juncture the remaining abortive attempts to find a fundamental dynamic equation displaying stability and positive long run money and government expenditure multipliers. The first two attempts were outlined here to show that making no adjustments in Klein Model III (other than dropping some coefficients that are not statistically significant) before undertaking a derivation of the fundamental dynamic 229 equation for (Y+T) is fruitless. Secondly, making an initial "dry run" allowed us to set down the equations relevant to a discussion of the adjustments in the model which are necessary if the fundamental dynamic equation is to be stable and exhibit suitable policy multipliers. At this point we turn to the fundamental dynamic equation that was finally adopted, delaying until the last section of this chapter a full discussion of other trials. Derivation of the AdOpted Fundamental Dynamic Equation for (Y+T) The derivation of the adOpted equation begins as in the first attempt with the elimination of Ct’ Dl , and D2 t from the definitional equation (15'). A variant of equation (3) is found by dropping rt from equation (7') and Y from (8'). This yields, instead of equation (3): t (3") rt = 0.06817 17151 + 0 00070 13952 Yt—l + 0.94768 42221 r + 0.00963 87966 t -1 - 0.01591 32726 N CPU) ('1‘ H and (4) becomes (setting It = Kt - Kt-l): (4") 0.15245 (Y+T)t - 0.15558 44597 (Y+T)t-l _ 0.01674 58306 (y+T)t_2 + 0.01052 87717 (Y+T)t_3 = 2.03464 82224 + 0.00117 AFt_l - 0.00110 87905 AF - 1.82335 1 t-2 t t : 3,133!“ "'1‘ l .‘3‘. Inf-m t. TWII} I .2 "I 230 + 1.72796 00264 it_l - 0.84755 Tt + 0.79209 97624 Tt_l — 0.00058 12283 Tt_2 + 0.01052 87717 Tt_3 + 0.25213 76516 t + 0.36676 56060 N:_2 + AHt - 0.94768 42221 AHt_l + Gt - 0.94768 42221 Gt_l + 03t — 0 94768 42221 D3t—1 - Dé' + 0.94768 42221 D" + K - 1.94768 42221 K + 0.94768 42221 K t-l t—2' Using equation (5) the K 's are eliminated from (4") t via a Koyck transformation. Before i is eliminated from the resulting equation, the variables Ms and (Y+T)t-l t-l are dropped from equation (8). Then the X's are removed from the expression for (Y+T) by direct substitution using equation (9). Finally, equations (3') and (12') are dropped from the system and p and H converted to exogenous variables.1 The resulting fundamental dynamic equation for (Y+T), after rounding to five decimal places, is: (16) 0.20637 (Y+T)t - 0.49477 (Y+T)t_l + 0-44614 (Y+T)t_2 _ 0.19872 (Y+T)t_3 + 0.03467 (Y+T)t_u + 0.00703 (Y+T)t_5 = 0.08237 + 0.00738 t + Gt - 2.58758 Gt-l 231 + 2.22218 G - 0.63305 Gt_ + AH 3 t — 0.63305 AHt_ t-2 2.58758 AH + 2.22218 AH t—l t-2 3 + D — 2.58758 D + 2.22218 D 3t 3t—l 3t—2 0.63305 0 — 0" + 2.58758 0%; t 1 2.22218 D': + 0.63305 Dél3 + 0.56707 MS t ’7' 3‘ 1.03964 M + 0.47596 M St—l St-2 20.09274 pt + 115 23867 pt_l - 192.65024 pt_2 “in .-"I..‘.ll'n. sh... S + 123.39447 pt_3 — 25.79407 pt_u + 0.36677 Nt_2 <4 - 0.60146 N:_3 + 0.24500 N:_u - 0.84755 Tt + 2.18200 Tt_l - 1.86571 Tt_2 + 0.54061 Tt_3 - 0.01765 Tt_u + 0.00703 Tt_5 — 0 16111 Et + 0.30745 Et-l - 0.04389 Et—2 - 0.19385 Et-3 + - 0.09139 Et_u 38.15468 qt + 73.59406 qt_l 12.50995 qt_2 - 44.01336 qt_3 + 21-08393 9t_u 0 16111 (w2+Rl+R2)t + 0.30921 (W2+R1+R2)t_l 0.04862 (W2+R1+R2)t_2 - 0.18961 (W2+R1+R2)t_3 + 0.09014 (w2+R +R 1 2)t-4 + 0.00117 AF t—l 0.00303 AF + 0.00260 AF - 0.00074 AF t-2 t-3 t—4' 232 Dynamic Properties of the Adgpted Fundamental Dynamic Equation for Net National Product Stability of the System The auxiliary equation associated with (16) is: 0.20637 (Y+T)t - 0.49477 (Y+T)t_l + 0-4451” (Y+T)t-2 0.19872 (Y+T)t_3 + 0.03467 (Y+T)t_4 + 0-00703 (Y+T)t-5 0. The three real roots are 0.96565, 0.86737, and —0.11145; the pair of conjugate complex roots is 0.33796 i 0.50070 1. Since all three real roots and the F .1 ”3".“ .d‘. has. modulus of the conjugate complex roots are less than unity -7 in absolute value, the system is inherently stable.2 The largest root, 0.96565, dominates. Hence the inherent time path of net national product is not highly damped. Also, since there is a pair of complex roots and a nega- tive valued real root, the inherent time path of (Y+T) is oscillatory. Long Run Multipliers Setting (Y+T)t-k = (YIT) for k = 0, ...., 5 and (Ms)t-k = MS for k = 0, l, 2, we find the long run (equilibrium) money multiplier associated with equation (16) by first adding the coefficients for both (YPT) and Ms and then taking the partial derivative of (TIT) in (16) with respect to MS. In this case, 233 _——) K: a( *< +T) _ 3( — 4.615. Similarly 8 _ = 2.101 and 341:11 = 3G 3T 3| 8 -l.74l. All three long run multipliers have signs con- sistent with economic theory. Dynamic Multipliers In order to form the dynamic multipliers, equation (16) is solved for (Y+T)t' Hence our alternative form of the fundamental dynamic equation becomes: (17) (Y+T)t = 2.39742 (Y+T)t_l - 2.16180 (Y+T)t_2 + 0.96290 (Y+T)t_3 - 0.16799 (Y+T)t-4 0.03408 (Y+T)t_5 + 0.39911 + 0.03577 t + 4.84557 G - 12.53833 D 3.06750 Gt_3 + 4.84557 D3t + 10.76775 D ~ 3.06750 D 3t-2 3t-3 + 4.84557 AHt — 12.53833 AHt_l t - 12.53833 Gt—l + 10.76775 Gt- 2 3t—l + 10.76775 AHt_ - 3.06750 AHt_3 — 4.84557 Dé' + 12.53833 DéLl - 10.76775 Délg + 3.06750 Dél3 + 2.74779 MSt - 5.03764 M + 2.30629 M - 97.36082 pt St-l St-2 + 558.39731 pt-l - 933.50067 pt_2 + 597.91680 pt_3 — 124.98704 pt_,4 2 234 s S S + 1.77719 Nt-2 — 2.91441 Nt—3 + 1.18716 Nt-4 - 4.10686 Tt + 10.57303 Tt-l - 9.04043 Tt-2 + 2.61954 Tt-3 — 0.08555 Tt-4 + 0.03408 Tt-5 — 0.78069 E + 1.48979 Et_ t - 0.21267 Et_ 1 2 _ 0.93929 Et_3 + 0.44286 '8thl - 184.88125 qt + 356.60533 qt_l - 60.61788 qt_2 _ 213,26990 qt_3 + 102.16371 qt-4 - 0.78069 (W2+R + 1.49828 (W2+R 1+R2)t _ 0.23559 (W2+R1+R2)t-2 - 0-91879 (W2+R1+R2)t + 0.43679 (W2+R1+R2)t-4 + 0.00567 AFt—l + 0.01260 AF 0.01467 AFt_2 t_3 0.00359 AFt_u. The inherent stability of the time path of (Y+T) discussed above relates to the movement of (Y+T) when all exogenous variables are held constant over time. We found that the movement of (Y+T) was in fact stable. Thus any instability of the time path must be caused by the actual levels of the exogenous variables over the period, the starting values of the exogenous variables, or random disturbances. We, therefore, turn to a study of the effects of changes in the exogenous variables on the time path of l+R2)t-l 3 235 (Y+T). This is facilitated using the fundamental dynamic equation as the point of departure. For it is possible to derive from the fundamental dynamic equation the effect of a unit change in a given exogenous variable at time (t-k) on the expected value of (Y+T)t while holding con- stant the rest of the time path of this exogenous vari- able (that is, the initial change is unsustained), as well as fixing the values of all other exogenous vari- ables. The resultant effects are called the dynamic multipliers. To reveal these multipliers we derive the "final form" of (Y+T) by purging the lagged values of (Y+T) from the fundamental dynamic equation as shown in Chapter VI. The resulting coefficients of the "final form" are the dynamic multipliers. In Table 10 the multipliers are presented for the money supply, government expenditures, and taxes. Several observations are possible as to the nature of these policy multipliers. First, all multipliers show damped oscillatory movement and converge to zero as the time lag increases. Second, government expenditures alternate between being stimulating and depressing for the first eleven lags and then are stimulating for the remainder of the period. Third, taxes alternate between being depressing and stimulating also for eleven lags and from then on are depressing. Fourth, the money supply 236 TABLE 10.--Dynamic multipliers for the time path of real NNP. Coefficient of Lag k T St—k t-k t—k 0 2.74779 4.84557 -4.10686 1 1.54997 -0 92146 0.72716 2 0.08105 —1 91653 1.58109 3 -0.50817 —1 00443 0.88361 4 -0.36481 0 03383 0.00492 5 -0.05105 0 39671 —0.32408 6 0.11034 0 26414 -0.22715 7 0.10617 0 04228 -0.04157 8 0.04547 -0 05913 0.04841 9 0.00673 -0.04660 0.04146 10 0.00328 -0.00109 0.00393 11 0.01550 0.02510 -0.01887 12 0.02530 0.02615 -0.02053 13 0.02761 0.01722 -0.01325 14 0.02566 0.01069 -0.00764 15 0.02347 0.00942 -0.00639 16 0.02260 0.01078 -0.00748 17 0.02265 0.01201 -0.00855 18 0.02280 0.01217 -0.00874 19 0.02264 0.01165 -0.00834 20 0.02219 0.01105 -0.00786 21 0.02165 0.01064 -0.00754 22 0.02112 0.01039 -0.00735 23 0.02063 0.01017 -0.00720 24 0.02013 0.00991 -0.00702 25 0.01962 0.00962 —0.00681 26 0.01910 0.00932 -0.00659 27 0.01857 0.00903 -0.00638 28 0.01803 0.00875 -0.00618 29 0.01750 0.00848 -0.00599 30 0.01698 0.00821 -0.00580 31 0.01646 0.00795 ~0.0056l 32 0.01595 0.00769 -0.00543 33 0.01545 0.00744 -0.00525 34 0.01496 0.00720 -0.00508 35 0.01448 0.00696 —0.00491 36 0.01401 0.00673 -0.00474 37 0.01355 0.00650 —0.00458 38 0.01311 0.00628 -0.00443 39 0.01267 0.00607 -0.00428 40 0.01225 0.00586 -0.00413 lfl‘m-“rD-Imn-A ”if" {L w. .- .51 -2 -4 FIGURE 2.——Dynamic multipliers of M, G, period and first eight lags. 237 1 .__. - Coefficients of "a t-k x x- Coefficients of th ---.- -L 0- Coefficients of Tt-k Ii 4L- Y I I I I «b- 1 I I II V and T for current v-h'.’~A .IA"‘.n A . L .h—_. 238 alternates between being stimulating and depressing for the initial six periods (the first three are stimulating and the next three depressing) and then is stimulating for the remainder of the time period. After twelve years the cumulative money supply multiplier is 3.74327 (81.1% of its long run value) while the cumulative government expenditure multiplier is 1.65839 (78.9% of its equilib- rium multiplier). These compare with cumulative mul- tipliers after six years of 3.45778 (74.9% of the equilibrium multiplier) for money and 1.43369 (68.2% of the long run multiplier) for government expenditures. The dynamic multipliers given in Table 10 illustrate the relative efficacy of a unit change in monetary and fiscal policy variables to influence changes in the level of real net national product. For instance, a comparison of the values listed in Table 10 indicates how effective a one billion dollar increase in the money supply this year will effect real NNP in each succeeding year com- pared to a similar increase in government expenditures. These multipliers are relevant in an analysis of the Friedman-Meiselman hypothesis for they demonstrate the relative effectiveness of monetary and fiscal policy after all cross-temporal and inter—temporal effects are considered. They also allow for a comparison of the speed at which the policy multipliers approach their respective equilibrium or long run values. 239 I Analysis of Causes of Changes in NNP The dynamic multipliers we estimated in the previous section demonstrate the effect on real NNP of hypotheti- cal unit changes in the exogenous variables. These mul- tipliers are an aid in a discussion of the broader issue to which we now turn. To attack the question of the relative effectiveness of monetary and fiscal policy from 1925—41 and 1947-65 we need to consider also the actual changes that occurred in the policy variables over the sample period. The dynamic multipliers and the actual changes in the variables, coupled with the assumption that the policymakers' only objective was to induce de- sirable changes in real NNP over the period, will provide the needed framework to delineate the historical perform- ance of the fiscal and monetary authorities. The "final form" of the fundamental dynamic equa- tion for (Y+T) is the starting point of our probe into the causes of changes in NNP. Lagging this "final form" one period and subtracting the lagged version from the equation for year t yields: /\ /\ (18) (Y+T)t - (Y+T)t_l = (at — at_l) + 60(Mt - Mt_l) + bl(M - M ) + t-l t-2 + bt-lMl + cO(Gt — Gt-l) + c G t-l 1 + d0(Tt ‘ T t-l) Equation (18) expresses the change in estimated NNP, (Y+T), during a given year in terms of (a) the change in the constant term, (b) the actual changes in the exogenous variables—-the coefficients of these changes are dynamic multipliers-—since the beginning of the sample period, and (3) the starting values—-i.e. the levels of the exogenous variables in the first year of the sample period. The change in the constant term represents the effect of the initial conditions (the values of NNP and the exogenous variables prior to the beginning of the sample period) on the change in NNP. The terms containing the changes in the exogenous variables show the effects on the current change in NNP of each exogenous variable in each preceding year. The terms representing the starting values relate the change in NNP to the beginning values of the exogenous variables. Since the coefficients associated with these starting values are dynamic multipliers lagged t—l years and since these multipliers decrease in absolute value as the time lag increases, the effects of the starting values will decrease with the advance of time. Equation (18) provides a basis for a comparison of both the cumulative and concurrent effects of the money supply and autonomous expenditure on the time path of NNP 241 over the sample period. A comparison of the absolute values of the cumulative effects will describe the rela- tive importance the policy instruments played in influenc- ing economic activity. The signed values of the cumula- tive effects and, alternatively, the signed values of the concurrent or impact effects will provide two measures of the relative abilities of the monetary and fiscal authori- ties to realize some "optimal" (arbitrarily determined) rate of growth in real net national product. Hence equa- tion (18) facilitates an analysis of the relative effective- ness of monetary and fiscal policy in several respects. Analysis of Cumulative Effects In Table 11 we present the cumulative effects of (1) the policy variables--money supply "broadly" defined, MS; money supply "narrowly" defined, Ml; government ex- penditures plus net exports, G; and "net" taxes (net national product minue disposable income), T; (2) the other exogenous variables combined (i.e. other than Ml’ G, and T); (3) the starting values; and (4) the initial conditions on current NNP. The cumulative effect of each exogenous variable is obtained by adding all terms on the right hand side of equation (18) which are associated with the concurrent and past changes in that exogenous variable. The residuals presented in column (10) are found by sub- tracting from the actual changes in NNP the estimated TABLE ll.-—Analysis of causes of annual changes in real NNP: 2112 (constant 1958 dollars) 19 26-1965. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Time in Eflagggs Sources of Change Years (Y+T) r ctr r A 9t rti AM HI AG AT EYog.173r. Constant uvzluegg Residuals 1926 10.7 4.396 1.374 1 54 -21.356 7.904 -148.975 200.164 —°9.9 1927 1.1 5.503 0.775 12 807 5.013 — 7.072 -111.115 132.874 —32.3 1928 0.4 6.233 0.865 — 1.1-5 2.254 - 2.996 - 15.39 26.317 - 9.5 1929 17.8 2.307 0.760 0 93) -27.716 24.320 37.738 — 24.102 0.9 1930 -21.0 3.174 _ 2.321 5 902 46.396 _1t.596 37.381 - 18.961 -70.8 1931 -15.6 11.742 — 6.228 — 1 994 47.664 - 9.669 16.955 2.841 -65.2 1932 —26.3 24.988 —11.3P3 ~12 '20 2 .583 -32.525 2.549 15.650 -15.9 1933 - 2.1 21.793 - 7.8’0 - 8.314 -48.540 - 1.053 — 0 363 16.267 47.7 1934 15.4 1.752 4.344 21 ~30 —55.518 8 841 2.339 11.905 2 .5 1935 10.6 2 .752 16 136 — z 191 6 396 13.234 4 822 8.625 —36.4 1936 28.0 25.236 15 113 1 494 -27 495 37 976 E 069 7.869 —30 9 19,7 11.7 14.125 9.1 1 - 9 044 — E 799 19.177 3.972 8.422 - 2.1 1938 -10.3 0.827 — 2.122 19 2:4 27 137 -43.169 2.841 8.929 —2 .1 1939 16.4 4 5'9 5.(16 — 4.(33 0 019 22.5‘9 2.'04 8 923 -17.7 1940 17.8 17 874 17 (MO - 0 570 -“5.165 33 102 1.933 8.582 -17.8 1941 36.1 21.867 27.595 C1.90 -42 651 21.429 1.314 5 201 —h2.6 1942 35.430 35.05C 262 353 —19 435 -4 27; 1.61' 7 911 1943 62 670 57 ~44 12 4 3 -92 ("r - 2 13 1 41% 7.795 1944 "0.823 55.076 --. 1") -1“ 506 40 7 1 ‘ r " 61; 1945 75 “95 5” 36f —’t' 14“ 9: {9: I? 749 1 '1" 7 _-5 1946 5? ”73 “9 = : —'1 :. 1 n 27‘ 1: 3 . 8'~ 7 3 1947 27 010 1« ..z 1.- L61 - r ~74 _«( 124 r 7f) c :60 1948 2 4 4.700 1.910 246 "9 —114.1F' '0 N35 0.r ~ 6.642 -134 7 1949 - 2.0 8.094 _ 7.336 159 411 —20 277 —24 '99 0 6. u 431 -106 1 1950 29.4 2.82— 3.193 —32.10( —35.746 37.9' 0.487 6..“6 10 2 1951 26.5 20 349 19.11r 6‘.1~’ -t" 77. 9 0€X 9 ~17 6 -26 -13 3 1952 9.8 35 967 28 0 3 16 747 31 251 — 4 3»‘ 0.3‘5 L 839 —'3 0 1953 15.3 33.610 20.t~7 —51 59r “9 0r9 31.237 0. 0» 5.137 —30.1 1954 - 8.6 25.794 9.244 -(9.352 Ee.9:_ 41.214 0.2(‘ 5 4%0 -4T 3 1955 28.1 23.472 11 755 -25 2’2 —47.3L€ 79 215 E 224 5.2:~ 4.8 1956 7.3 15.290 8.9 35 469 3.801 5.0 0 9.1»n . 091 —51 2 1957 5.2 16.445 5. 9 49 ’42 1 15‘ - 9 885 0.1ti 4 92- —46 n 1958 - 6.3 30.921 5.508 4 8‘8 35 r40 ‘3 501 0.137 4. '3 —90.7 1959 27.0 38.897 15.877 -22 704 -62.392 EC 540 0.115 4.r.. 4.9 1960 10.2 15.620 3.738 9 900 -19 9-3 45 186 0.097 4.435 —34.2 1961 7.8 29 641 4.994 28.190 25 047 24 A94 0.981 4.323 -7u.5 1962 28.5 56.129 11 514 22.1)“ —,4 952 78.397 0.067 4.13” —52 4 1963 19.0 76.566 18 641 — 3 284 -10 848 72.871 0.055 3.994 —62.4 1964 26.7 82.445 24.307 1 853 11 22 63 560 0.045 3 3:7 -7E.1 1965 31.9 91.28 27.111 —12 108 -15 907 if“ 93L 0.036 3.722 -71.2 243 1‘ changes in NNP caused by the exogenous variables, the starting values, and the initial conditions. The years 1942-47 are not included for comparison since we elimi- nated 1942-46 in our estimation of Klein Model III. The first year estimated in the second half of the time period is 1947 and the change between 1947 and 1948 is given in the row corresponding to 1948. Absolute values of the cumulative effects.--Accord- ing to Table 11, the cumulative effect of net taxes was greater than that associated with the broad definition of money in 20 out of 3” cases. The item net taxes was also a more important influence on real NNP than the money supply "narrowly" defined in 26 of the 34 years. Govern- ment expenditures plus net exports had a greater absolute cumulative effect on income than did the money supply "broadly" defined in only 14 cases while "autonomous" expenditures were more important in influencing income than currency plus adjusted demand deposits in 24 cases. Since T is no doubt misspecified in Klein Model III as an exogenous variable we limit our remaining discussion to the effects of the money supply and autonomous expendi- tures on NNP. Signed values of the cumulative effects.-—We now impose an arbitrary "goal function"3 on the time path of NNP by assuming that a desirable rate of change in real 3'1” I'lllll‘l‘ 244 NNP over the period was $10 billion (in constant 1958 dollars). The residuals as given in column (10) repre- sent the changes that would have occurred in NNP had there been no concurrent or past changes in any of the exogenous variables. If the residual in any given year was greater than $10 billion the policy makers should have, according to the "goal function," attempted to de- crease NNP; if the residual was less than $10 billion the cumulative effects of each of M and G should have been expansionary. Applying the above criterion to the residuals of column (10), we find that changes in the money supply (defined either way) affected the level of NNP in the wrong direction only 7 out of 34 times while the changes in autonomous expenditures produced the wrong effect 13 times. When the criterion is changed to a $15 billion (in constant 1958 dollars) increase in real NNP, column (10) shows that the effects on NNP were in the wrong direction 6 times in the case of money and 14 times in the case of autonomous expenditures. Analysis of Impact Effects In Table 12 we present the estimated changes of the concurrent effects of the money supply and autono- mous expenditures on NNP. In columns (2), (3), and (4) the concurrent effects on NNP are listed for the broadly 245 TABLE l2.--Annual changes in real NNP cue to current and cumulative changes in exogenous variables (in constant 1958 dollars). (l) (2) (3) (4) (S) (0) (7) Time in Change in Sources of Change Years Y+T Current Change Current Change Current Change Residual Residual in MS in M1 in G (Ms, G) (M1, 0) 1926 10.7 4.396 1.374 1.454 4.8 7 9 1927 1.1 3.023 0.000 13.083 - 15.0 — 12 0 1928 0.4 4.396 0.824 1.938 — 5.9 — 2.4 1929 17.8 0.550 0.550 6.784 10.5 10.5 1930 —21.0 - 2.473 - 2.473 10.660 - 29.2 — 99 2 1931 -15.6 - 9.068 - 4.671 2.907 - 9.4 - 13.1 1932 —26.3 —19.235 - 8.518 - 7.268 0.2 - 1? L 1933 - 2.1 -11.266 — 3.297 - 7.268 16.4 8.5 1934 15.4 6.320 5.221 17.444 - 8.4 - 7.3 1935 10.6 13.464 10.991 - 4.361 1.5 4 0 1936 28.0 12.640 9.892 22.290 — 6.9 — 4 1937 11.7 6 370 3 847 - . 4-3 7.8 10 ’ 1938 -10.3 - 0.550 - 1.374 27.620 — 37.4 - 31.5 1939 16.4 10.167 9.617 3.392 2.8 3.4 1940 17.8 15.662 14.563 9 691 - 7 f - 6 1941 36.1 ' 20.059 18.460 98.169 - 72.1 - 31.1 1942 —— 24.455 24.730 $82.417 —— —_ 1943 -— 51.658 46.438 210.78. -- —— 1944 —- 46.712 36.546 84.313 -- —- 1945 —— 54.406 37.920 -112.932 -— —— 1946 -_ f2,973 18,410 -474 t __ -_ 1947 -13.9 21 158 17.311 — .1 2»1 — 1. — - 1 1948 22.4 6.861 2.198 0 :1 14 1 1949 — 2.0 - 1.923 - 3 297 15.373 — 5 4 — 4‘ 1950 29.4 8.518 7 419 — “1.351 41 7 4- 1951 26.5 15.113 14 014 122.108 -11« 7 —111' 1952 9.8 23.356 16 487 61.776 — 83 . — ,. F 1953 15.3 17.586 9 068 28.104 _ 30 4 — 1 9 1954 - 8.6 16.212 4 946 43 610 1: 8 .7 1 1955 28.1 18.135 11 541 — 16.9(0 26 9 3 1956 7.3 8.793 4.390 8.207 — 10.7 - 4.. 1957 5.2 13.464 2 473 ".1IT - 33 5 - 7 1958 - 6.3 24.730 3.572 4 311 _ 3. 4 — 14 1959 27.0 24.455 12.915 - 6 784 9 3 L.1 1960 10.2 0.824 5 221 2 35 - 11 0 — 11.4 1961 7.8 30.775 5 770 31 012 - 9 0 — 39 H 1962 28.5 42.866 8 518 31 012 - 45 4 - 11.0 1963 19.0 51.384 11.816 15.506 — 47.9 - 8.3 1964 26.7 53.857 27.203 22.290 — 49.4 - 227 1965 31.9 66.222 17 311 2 907 — 37 2 11 . . A ”.351 I’m-h 9:21.}: 7 Varmr' 246 defined money supply, the narrowly defined money supply, and autonomous expenditures respectively. Column (5) shows the effects on real NNP given the values of all past exogenous variables and the current values of the exogenous variables other than MS and G. That is, if the current changes in MS and G had been equal to zero, the changes in real NNP would have been given by column (5). Similarly, column (6) indicates the changes in real NNP which would have occurred had there been no changes in either M1 or G. 1 Applying the goal of a $10 billion increase in yearly NNP (measured in constant 1958 dollars) to the changes given in column (5), we see that current changes in government expenditures were of the wrong sign 7 of the 34 years while the current changes in the money supply plus time deposits were of the wrong sign in 9 cases. If we look instead at column (6), changes in government ex- penditures were in the wrong direction 7 times and the alterations in the size of the money stock were in the wrong direction 13 times. Conclusions The relative effectiveness of the policy instru— ments is not clear cut when we examine the actual changes which occurred in these variables over the sample period. The cumulative absolute influence of government 247 expenditure on current income was less in a majority of cases when compared with the money supply defined as currency in circulation plus adjusted demand deposits plus time deposits; the absolute influence was greater, however, in the majority of cases when compared to the money supply defined as excluding time deposits. The "errors" found in monetary policy were somewhat smaller in number than those found in fiscal policy. In any case, our results do not seem to give strong support to the Friedman—Meiselman hypothesis that monetary policy has a decided advantage over fiscal policy as a tool for eco— nomic stabilization. Search for Crucial Coefficients that Determine Whether System is Stable and Whether Long Run Multipliers of M and G are Positive In an attempt to find a stable auxiliary equation of a fundamental dynamic equation that yields positive long run multipliers for the money supply and government expenditure, numerous variation of Klein Model III were considered. These variations involved combinations of the following: (a) truncating the model by allowing some endogenous variables to become exogenous, (b) drOpping variables from structural equations, and (c) dropping lagged variables from equations which express endogenous variables in terms of exogenous variables and endogenous variables not yet purged from the quasi-fundamental dynamic 248 equation. These eXperiments revealed some factors in- fluencing stability and the sign of the multipliers. In this section we first delineate the experiments that were run on Klein Model III and then summarize what they indi— cate as to the crucial coefficients in the model. Next we offer a more abstract analysis in terms of dynamic models in general. Variations of Klein Model III As shown earlier in this chapter, the first and second attempts at finding a suitable fundamental dynamic equation revealed, first, that retaining all statistically significant variables (and only those variables) leads to indeterminate long run multipliers for M and G. Second, allowing u3 to remain in equation (l2**) also produces t-l indeterminacy if both p and H are eliminated from the fundamental dynamic equation for (Y+T). Third, if only p is eliminated (H treated as exogenous) then (13a) repre- sents the left hand side of the fundamental dynamic equa- tion. The time path of (Y+T) is unstable. The next attempt paralleled the first two except that neither p nor H were purged from the fundamental dynamic equation for (Y+T). This resulted in an auxiliary equation with a largest root equal to 1.43. Truncating Klein Model III further by altering the status of X to an exogenous variable constituted the fourth trial. In 249 this case the largest root of the corresponding auxiliary equation was 1.23. So, both cases gave unstable time paths of (Y+T); the money supply long run multipliers were both negative, as well. Hence it became clear that merely truncating Klein Model III would not produce stability nor generate positive long run multipliers for M. Further investigation into Klein Model III involved dropping, once again, u from (l2**). Thus, p must be 3t—1 treated as exogenous if the long run multipliers are not to be indeterminate. We also convert H to an exogenous status. Hence, the relevant structure in this set of trials consists of (1'), (2'), (4') through (11'), (13'), (14'), and (15'). Trial 1 The first case in this new series of trials sets the coefficient of r in equation (7') equal to zero, t yielding a new expression for rt: (3I) rt = 0.06817 17151 + 0.00025 Y + 0.00070 13952 Yt- t 1 + 0.94768 42221 r + 0.00963 87966 t t—l S - 0.01591 32726 Nt_l. Equation (31) differs from the equation (3) developed earlier in that the coefficient of r in (31) is less t-l 250 than unity. Since it is this coefficient which is in— volved in the Koyck transformation to purge r from the expression for (Y+T), it was felt that a Koyck multiplier less than unity might contribute to stability. In fact, this adjustment did cause the new expression for (Y+T) to have a largest root equal to 1.38 after further elimi- nating I, K, i, and X from the expression for (Y+T). This compares favorably to the largest root, 1.43, ob- tained above when equation (3) was used rather than (31). Since setting the coefficient for rt-l in (7') reduces the degree of instability of the time path of (Y+T), we retain a variant of (3I) and seek other adjustments which will reduce the degree of instability even further. However, before we proceed we should note that elimi- nating rt-l from (2) using (31) has caused the sum of the coefficients of (Y+T) to change from positive to negative while keeping the sum of the coefficients of G positive. This arises from the fact that the long run multiplier (Just after r is eliminated) is 1(1 - 0.94768 42221) t—l while that for (Y+T) (Just after r is eliminated) is t-l (0.15245 - 0.01111 - 0.01111) (1 - 0.94768 42221) - 23.04778 (0.00025 + 0.00070 13952). The second term of the sum of the coefficients of (Y+T) enters through the substitution of (31) for r in (2) and then transferring t-l terms involving (Y+T) to the left hand side. Since the "Koyck constant" (= 0.94768 42221) is very close to one, 251 the negative second term more than offests the first term causing the sum of the coefficients of (Y+T) to become negative. The sum of the coefficients of G, however, remain positive, causing the long run multiplier for G to be negative. Since the new sum of coefficients of i is -l.82335 (l - 0.94768 42221) and since the money supply enters the expression for (Y+T) via 1 with a positive sum of coefficients (see equation (8)), the net effect is a negative sum of coefficients of M in the i expression for (Y+T). Naturally, this means the long run 3 multiplier for money is then positive. The signs of these multipliers do not change as we move to eliminate I, K, i, and X from the expression for (Y+T). This is due first of all to the "Koyck constants" involved in these elimination processes being positive but less than one. Secondly, eliminating I and K introduces no new terms involving either (Y+T) or M into the expres- sion for (Y+T) which might offset the first effect. (New terms for G are never brought into the expression for (Y+T) since G appears in only the structural equation (13').) Thirdly, when i is eliminated, the negative sum of coefficients in the expression for (Y+T) is rein- forced. Fourthly, when X is eliminated, the sum of the coefficients of (Y+T) is not affected. We explore the third and fourth effects further since they are not as apparent as the first two. First 252 we look at the elimination of i from the quasi fundamental dynamic equation. When i is eliminated the coefficients of (Y+T) in (8) are multiplied by each coefficient of i appearing in the eXpression for (Y+T) obtained Just prior to the elimination of 1. These new coefficients are then transferred to the left hand side of the new expression for (Y+T) and when added together form part of the total sum of coefficients of (Y+T). To determine the sign of this partial sum we (a) note the sign of the sum of coefficients of (Y+T) in (8), (b) note the sign of the sum of the coefficients of i appearing in the quasi fundamental dynamic equation Just prior to the elimina- tion of i, and (0) change the sign of the product of (a) and (b) to transfer this product to the left hand side of the new expression for (Y+T). Since the sum of coeffi- cients of (Y+T) in (8) is negative and the sum of coeffi— cients of i in the quasi fundamental dynamic equation is also negative, the negative sum of coefficients of the previous quasi fundamental dynamic equation (i.e. the equation which contained the variable i) is reinforced upon the elimination of i. The effect on the sign of the sum of coefficients of (Y+T) resulting from the elimination of X is more com- plex than that evolving from the elimination of 1. First, X does not enter the expression for (Y+T) until K is eliminated from that expression. However, because It is 253 defined as Kt - Kt-l’ the sum of coefficients of K in the expression for (Y+T) is equal to zero. Thus, even though X enters the new expression for (Y+T) when K is eliminated, the sum of the coefficients of X appearing in this new expression is equal to zero. Eliminating i from the ex- pression for (Y+T) does not bring in any other terms in— volving X. Therefore, performing a Koyck transformation on the expression.for (Y+T) does not alter the value of the sum of coefficients of X. Eliminating X from the quasi fundamental dynamic equation for (Y+T) yields a sum corresponding to (b) in the previous paragraph which is equal to zero. Thus, the product of (a) (which is now the sum of coefficients of (Y+T) in (9)) with (b) is also equal to zero and, therefore, X has no effect on the sign of the sum of the coefficients of (Y+T) appearing in the final fundamental dynamic equation. Trial 2 Next, it was decided to drop the variable Yt from equation (8'). This causes the expression for rt to lose the term 0.00025 Y in (31). That is, the expression for t r is now: t II) (3 r t = 0.06817 17151 + 0 00070 13952 Yt-l + 0.94768 42221 r + 0.00963 87966 t -1 - 0.01591 32726 N (rm ('1‘ -1' 254 This expression is the same as (3") in the previous sec- tion. Upon eliminating I, K, i, and X using the expres- sions for these variables outlined above, we obtained a largest root of 1.33 which makes the system only slightly less unstable. Also, the equilibrium multiplier for G remains negative. Trial 3 In the third attempt we retained equation (3") as the expression for r and altered equation (8) by delet- t ing the variable (Y+T)t_l. This alteration caused the lagged coefficients of (Y+T) in the fundamental dynamic equation to be smaller in absolute value. It was felt that reducing these coefficients would aid stability. In fact, this alteration did result in a stable system with the largest root equal to 0.96565. A problem arose, however, when we dropped (Y+T),“l from equation (8). The sum of coefficients of the (Y+T)'s in the fundamental dynamic equation became positive. Thus drOpping (Y+T)t-l was enough to alter the sign of the sum of coefficients. The result of this change in sign was that the government expenditures long run multiplier be- came positive (which is fine), but the money supply equilibrium multiplier became negative. “I?" .-T 255 Trial 4 We then attempted to reverse the sign of the equilib- rium money multiplier without simultaneously (a) changing the sign of the long run government expenditure multi- plier and (b) causing the system to become explosive. We truncated equation (8) this time by deleting MS as t-l well as (Y+T)t-l’ Thus, the expression for it became: LJ :1 (8 ) it = 5.42018 7806 - 0.31100 5231 Ms . ‘_._ t LT 4 g}, — 0.02724 59254 t + 36.72590 965 pt — 45.53528 182 pt_l + 0.09144 97031 (Y+T)t + 0.75424 0556 it_l. Using (8II) in the derivation of the fundamental dynamic equation gave the desired positive long run multiplier for MS and did not affect either the corresponding G multiplier or the stability of the time path of (Y+T). Thus, Trial 4 yielded the fundamental dynamic equation we decided to adopt and which we outlined in the previous section. Trial 5 Next, we tested the fundamental dynamic equation derived using (31) and (8"). The system once again was found to be stable and exhibited positive long run mul- tipliers for M and G. However, since the largest root 256 imithis case was 0.97225, we chose to stay with the equa— tion derived in Trial 4. Trial 6 Another attempt was made to include the price leVel as an endogenous variable in the model. This was done by taking the fundamental dynamic equation of Trial 4--given as equation (16) in the previous section--and then elimi- nating p using equation (13). It was to no avail for the largest root was complex with a modulus of 1.152. fl’flhfimgm‘w “hH‘IHS‘L Trial 7 Once again we tried to find a stable fundamental dynamic equation for (Y+T) which did not contain p. This attempt involved dropping the variable (Y+T)t-l from (13) and then proceeding as usual. However, the largest root associated with the resulting auxiliary equation was 1.4989. Other Trials Several other trials were run, but do not merit much attention. Three involved changing I to an exogenous status; all three gave an unstable time path for (Y+T). Two other trials are worthy of mention. Both were com- ;Nited fdumn an estimated structure for Klein Model III ‘which was different from that given in Chapter V. How- ever, tflus size of the coefficients in this version were 257 close enough to the estimated structural equations given in this paper so that the results of the two trials seem to be relevant. One trial attempted to find a fundamental dynamic equation while retaining all coefficients--even those in the structure that were not statistically signifi- cant. Once again the system was found to be explosive. The second trial on the different version of the estimated structure indicated that the variable (Yt + Yt-l + Yt—2) in equation (5') contributes to stability. Initial Findings of the Crucial Coefficients Which Determine Stability and Size of Policy Multipliers On the basis of the trials outlined above it may be tempting to assert that (a) the crucial variables for inherent stability of Klein Model III are the coefficients of (Y+T) in the expression for r, the coefficient of (Y+T)t_l in the equation for i, and the coefficients of rt-l ixi‘the expression for v; (b) the crucial variables that determine the sign of the long run multiplier for in the money are the coefficients of (Y+T)t-l and MS t-l expression for i; and (c) the crucial variables determin- ing the sign of the equilibrium multiplier for government expenditures are the coefficient of rt-l in the structural equation for v and the coefficient of (Y+T)t—l in the expression for 1. However, these variables altered stability and the signs of the multipliers only when we L "Iain-.04.“. onus-rum . I ‘1'.) ; l1 ' 258 inhifllcmher coefficients constant. There may in fact beoUwrcases which set an entirely different group of cmdfuumms equal to zero and which would also yield a thMflefundamental dynamic equation. That is, intui- thmlyitis not any one or two coefficients that are crmflaltmt rather it is the combinations of coefficients 4.. (pmmimflarly the sums) which matter most for the multi- plies mm.it is the relationships among the coefficients 0f(Y+T)in the fundamental dynamic equation that matter w; “Jam. f 1_ I A for stability. 4 Iheil and Boot have shown that in general stability of Unesystem depends on the coefficients of the current and lagged endogenous variables in the structure and that the long run multiplier associated with any given policy instrument depends on the structural coefficients of that policy instrument as well as the structural coefficients of all current and lagged endogenous variables. Appendix Onp. 227 we found that reinstating the variable (U3)t-l as it appears in equation (12“) in Chapter V and then eliminating H from the quasi fundamental dynamic equation for (Y+T) resulted in the sums of the coeffi- cients of each of M and G to equal zero. This arose be- cause the coefficients of Ht-l and Ht-2 in equation (15) sum exactly to one. We now show that these coefficients will always sum to one no matter what the estimated values of the structural coefficients are. To show this, we develop equation (15) analytically. Since the coeffi- cients of Ht-l and Ht—2 in (15) depend only on the coefficients of H, X, p, and u3 in the structural and derived equations for these variables, we simplify the derivation by abstracting from the constant terms and other variables which appear in these equations. We tnegin by expressing the structural equation for Ii t O‘ixt ‘ O‘th + “1Ht-1 + 0‘2Ht-1 01": (£1) (1 + dl)Ht = let + (a1 + 012)Ht_1 259 imim hunt-sum f r "i " .315! I P— .l.‘ (Elsi! I 260 Wnudngthe structural equation for Xt as (b) Xt = 81 pt. Therefore, (0) (l + 01) Ht = cl 81 pt + (cl + a2)Ht—l° It should be kept in mind that we are trying to solve for That is, p and X 1% in terms of lagged values of H only. have been already eliminated from the quasi-fundamental dynamic equation for (X+T). To eliminate p from (c) we write the structural equation for p as (d) 00 pt = Xt ’ Xt-1 + 00 pt-1 + C1 (“3)t-1' From equation (3'), (e) (u3)t-l = ’“1 Xt—l + (l + 0‘1) Ht—l ‘ (“1 + 0‘2) Ht-2’ Usiiug (e) to eliminate (u3)t_l from (d) yields 00 pt = 81 pt ' B1 pt—i + 01 (l + a1) Ht-l - 01 (a1 + a2) Ht- + 00 pt-l ‘ O1 0‘1 Bl pt—l 2 261 01": (f) p = C0 ' B1 ’ G1 “1 81p + 01 (l + a1)q t 00 - 81 t-1 00 - Bl t-l - 01 (a1 + 012)q - 00 - Bl t-2 Solving (c) for Ht: a B a + a l 1 1 2 ' = ———————— (C ) Ht 1 + alpt + l + cl Ht-l‘ Performing a Koyck transformation to eliminate pt from (c') we obtain: 0 l 1 1 1 l 1 1 l (a) H = [( ) + ( t 00 — 81 (l + cl) 00 - 81 + cl + a2] H [a1 81 01 (cl + c2) 1 + 01 t—l (l + al)(00 — 817 o - 8 — 0 a B a + a - 0 1 1 l 1* 1 2- + ( (00 - cl) )(1 + a1 )1 Ht-2 It is shown below that the sum of the coefficients of H and II is identically equal to unity-~provided that t-2 <31 + —l and 00 + 81. t-l “‘n 4' runs "V 262 We now see if 0 1 1 1 1 1 1 1 2 [( )+ ( )+( )l 0‘1 B1 C1 (”1 + “2) O0 ’ 81 ’ U1 “1 81 “1 ‘ [1 + a ( C _ B ) + ( U _ B )( l + ) 1 0 1 0 1 equals one. That is, does the following equality P'“ u (00 - 81 - oldlBl)(l+dl) + d18101(1+al) + (01+d2)(oO-Bl) (00 - Bl)(l + 01) F‘ . m.— -1 "1L - _ (GO—Bl)(l+dl) + alslol(al+a2) + (00 - Bl — olalsl)(al+a2) (00 - Bl)(l + a1) always hold? The above expression is equivalent to: (00 - Bl - oldlBl)(l+dl) + 018101 (1+al) + (dl+a2)(OO-Bl) (co-Bl)(l+dl) + clBlol (dl+d2) + (00 — Bl — oldlBl)(dl+d2) ? or: 01(00—81) + 02 (co—Bl) = (GO-81)(al+c2) or: (dl+d2)(oo—Bl) = (GO-Bl)(dl+d2) which is always true. Therefore, the sums of the coefficients of each of M and G in the fundamental dynamic equation are always equal to zero . FOOTNOTES--CHAPTER VII 1We did find another stable fundamental dynamic equation after we had completed the analysis of the dynamic multipliers and the causes of changes in NNP given later in this chapter and associated with equation (16). This fundamental dynamic equation resulted from retaining H as an endogenous variable but converting p to an exogenous status. Equation (14) was used to eliminate H from the expression for (Y+T) after removing the X's. This new equation had a largest root approximately equal to 0.9641 and long run money and government expenditure multipliers equal to 4.59 and 2.07 respectively. 2The reader should be cautioned that the figure 0.96565 represents an estimate, not the true value, of the largest root. If the standard error of this estimate is not rather small, the true value of the largest root may be more than one. See H. Theil and J. C. G. Boot, "The Final Form of Econometric Equation Systems," Read- ings in Economic Statistics and Econometrics, ed. by Arnold Zellner (Boston: Little, Brown and Company, 1968), pp. 624-627. 3Thomas R. Saving, "Monetary-Policy Targets and Indicators," The Journal of Political Economy, LXXV (No. 4, Part II; Supplement: August, 1967): 447. 4 Theil and Boot, pp. 611—630. 263 CHAPTER VIII CONCLUSION In this study we have extended the FM analysis to find whether FM's hypothesis (that the money supply is a more important determinant of total spending than is autonomous expenditure) is verified for revised data, alternative definitions of the policy variables, and more refined models of income determination. The focus of the study was on sample periods of as long a duration as possible in order to retain the flavor of FM's analysis. The simple one-equation models we tested initially did not contradict FM's findings that money is more highly correlated with income than are autonomous expen- ditures. However, this aspect of the study did reveal that FM's definition of the money supply consistently produced higher coefficients of determination than did a narrower definition of the money stock. It also showed that, in general, FM's definition of autonomous expendi- tures evoked lower coefficients of determination when regressed against income than did the alternative defini- tions offered by FM's critics. Since we questioned FM's criteria for deciding the definitions of the policy variables and pointed to likely 264 ‘mi' ':I: ‘3'... I-‘ I'- 265 biases resulting from their single—equation tests, we proceeded to carry out empirical tests with the help of more complete models of income determination. The first step, admittedly, was a small one for it was based on the highly simplified three equation Klein Model II. Testing this model for the period 1922-1941 and 1946—1965 and performing a dynamic analysis of the system disclosed evidence supporting the other side of the issue. The estimated long run government expenditure multiplier was greater than seven and its impact multiplier was nearly three. 0n the other hand, the equilibrium money supply multiplier was less than one half and the concurrent effect of a hypothetical increase in the money supply was precisely zero. An analysis of the dynamic properties of a revised version of Klein Model III yielded policy implications that were less apparent than those provided by the simpler models. While the estimates of the intermediate and long run multipliers for the money supply were larger than those estimated for government expenditures, the impact multiplier of the latter was greater. When we analyzed the causes of the actual changes in income during the sample period we found further evidence that it may be necessary to temper the FM claim that monetary policy is a more powerful stabilization device. Our findings suggested that the relative efficacy of the money stock Fflw.‘ melw o 3~‘. "'1" ‘v. I‘— '. 266 and government expenditures depends significantly on the particular definition of money adopted and on whether a comparison is based on cumulative or concurrent effects of the policy instruments. In short, we did not find a clear answer to the question of the relative effective— ness of monetary and fiscal policy as FM have claimed to have done. Possible Shortcomings of Klein Model III and Suggestions for Further Research The dynamic analysis of Klein Model III, which WTWTETSI‘3PFTW " .. 'u‘. at: ' occupied the greatest share of our attention in this study, gave results that are, of course, only as good as the estimated structural equations used to launch the analysis. A possible shortcoming of the structure is that it specifies the money stock in terms of nominal units while expressing government expenditures and income in real terms. Because of this, the comparisons made in Chapter VII may not have placed the policy instruments on equal footing. For instance, the relative absolute sizes of the cumulative effects of money on real income may have differed significantly from those presented in Table 11 had the money stock been deflated by a price index. The reason is that for a given increase in real income and a given increase in the nominal money stock, the effect of real money on income will be greater than that for 267 nominal money in years which experienced increases in the price level. The Opposite would be the case in those years during which the price level fell. Several possible specification errors exist in Klein Model III. For instance, permanent income may be more appropriate than the current level of income as an explanatory variable in the consumption function. Also, the money market seems rather sterile. A model of income determination incorporating a more elaborate theory of the supply of and demand for money may alter the find- ings substantially. The most damaging criticism of Klein Model III appears to be that it lacks a bond market. From Walras' law it is Justifiable to delete the bond market from a model of effective demand when we are concerned with static equilibrium analysis. However, in dynamic analysis the focus is on the time paths which the variables assume in approaching their equilibrium values. Thus, a com- plete dynamic analysis would consider the movements in the bond market and their influences on other components of economic activity. Of the models presently in existence, we found Klein Model III to be the most appropriate one to examine the FM hypothesis for a time period of as long a duration as possible. It was not entirely adequate, however. 268 Some of the most recent efforts to build models of income determination for the U. S. economy are exempli- fied by the Brookings and Federal Reserve-MIT quarterly econometric models. The Brookings model contains approxi- mately 150 non-linear equations.l Therefore it is ex- tremely difficult to examine its properties except by using simulation techniques. The Federal Reserve-MIT model, unlike the Brookings model, "has as its major pur— pose the quantification of monetary policy and its effect 2 on the economy." The formulators of the former model have tried to avoid the "puzzling results"3 found in other econometric models either in their financial sectors or in the responses to financial variables in other sectors . . . by concentrating most of . [their] . . . efforts on the treatment of financial markets and on the links between finaicial markets and markets for goods and services. DeLeeuw and Gramlich report that: the preliminary results suggest that both monetary and fiscal policy have powerful ef- fects on the economy though monetary policy operates with a longer lag. We also find that the response of money income to both monetary and fiscal policy changes is stronger than that implied by other large-scale econometric models. Hopefully, the further development of this model, and others like it, will contribute significantly to our understanding of the effects of policy instruments on the economy. FOOTNOTES-—CHAPTER VIII 1James S. Duesenberry et al., ed, The Brookings Quarterly Econometric Model of the United States (Chicago: Rand McNally & Company, 1965). 2Frank deLeeuw and Edward Gramlich, "The Federal Reserve—MIT Econometric Model," The Federal Reserve Bulle— tin, LIV (January, 1968), 11-40.II 3Ibid., p. 11. 14Ibid. 5Ibid., p. 12. 269 BIBLIOGRAPHY BIBLIOGRAPHY Books Christ, Carl F. Econometric Models and Methods. New York: John Wiley & Sons, Inc., 1966. Goldberger, Arthur S. Econometric Theory. New York: John Wiley & Sons, Inc., 1964. . Topics in Regression Analysis. New York: The MacMillan Co., 1968. Johnston, J. Econometric Methods. New York: McGraw- Hill Book Co., Inc., 1963. Klein, L. R. Economic Fluctuations in the United States, 1921—1941. Cowles Commission Monograph No. 11; New York: John Wiley & Sons, 1950. Klein, L. R., and Goldberger, A. S. An Econometric Model of the United States 1929-1952. Amsterdam: North Holland, 1955. Periodicals Ando, Albert, and Modigliani, Franco. "The Relative Stability of Monetary Velocity and the Investment Multiplier." American Economic Review, LV (September, 19657: . "Rejoinder." American Economic Review, LV (September, 1965), deLeeuw, Frank, and Gramlich, Edward. "The Federal Reserve—MIT Econometric Model." Federal Reserve Bulletin, LIV (January, 1968), DePrano, Michael, and Mayer, Thomas. "Tests of the Rela- tive Importance of Autonomous Expenditures and Money." American Economic Review, LV (September, 1965), "Rejoinder." American Economic Review, LV (September, 1965), 271 . In? -‘ -":' . “.1 u' 272 Fisher, Franklin. "Dynamic Structure and Estimation in Economywide Econometric Models." The Brookings Quarterly Econometric Model of the United States. Edited by J. S. Duesenberry, G. Fromm, L. R. Klein, and E. Kuh. Chicago: Rank McNally, 1965. Friedman, Milton, and Meiselman, David. "The Relative Stability of Monetary Velocity and the Investment Multiplier in the United States, 1897-1958." Stabilization Policies. Edited by E. C. Brown, et al. A Series of Research Studies Prepared for the Commission on Money and Credit; Englewood Cliffs: Prentice-Hall, 1963. . "Reply to Ando and Modigliani and to DePrano and Mayer." American Economic Review, LV (September, 1965), . "Reply to Donald Hester." Review of Economics and Statistics, XLVI (November, 1964),I Fromm, Gary, and Klein, Lawrence R. "The Complete Model: A First Approximation." The Brookings Quarterly Econometric Model of the United States. Edited by J.6S. Duesenberry, et al. Chicago: Rand McNally, l9 5. Hester, Donald D. "Keynes and the Quantity Theory: A Comment on the Friedman-Meiselman CMC Paper." Review of Economics and Statistics, XLVI (November, 19647, . "Rejoinder." Review of Economics and Statistics, XLVI (November, 1964I, Lewis, M, K. "Friedman and Meiselman and Autonomous Ex- penditures." American Economic Review, LVII (June, 1967). Morishima, Michio, and Saito, Mitsuo. "A Dynamic Analysis of the American Economy 1902-1952." International Economic Review, V (May, 1964), Saving, Thomas R. "Monetary-Policy Targets and Indicators." The Journal of Political Economy, LXXV (No. 4, Part II; Supplement: August, 1967), Theil, H., and Boot, J. C. G. "The Final Form of Econo- metric Equation Systems." Readings in Economic Statistics and Econometrics. Edited by Arnold Zellner. Boston: Little, Brown & Company, 1968. 273 Timberlake, R. H., Jr., and Forston, James. "Time Deposits in the Definition of Money." American Economic Re- view, LVII (March, 1967), Valavanis-Vail, Stefan. "An Econometric Model of Growth, U. S. A. 1869-1953." American Economic Review, XXXXV (May, 1955),