' : “fez. n3.“ ‘5‘ "° ~’ 3 1 This is to certify that the thesis entitled THE RATIONAL EXPECTATIONS HYPOTHESIS: A FRAMEWORK FOR SOLUTIONS WITH ECONOMETRIC IMPLICATIONS presented by Dennis Lee Hoffman has been accepted towards fulfillment of the requirements for Ph.D. degree in Economics Major professor BMW 4/77? 0-7639 © Copyright by Dennis Lee Hoffman 1978 THE RATIONAL EXPECTATIONS HYPOTHESIS: A FRAMEWORK FOR SOLUTIONS NITH ECONOMETRIC IMPLICATIONS BY Dennis Lee Hoffman A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1978 (El 0:11 *y‘ ABSTRACT THE RATIONAL EXPECTATIONS HYPOTHESIS: A FRAMEWORK FOR SOLUTIONS WITH ECONOMETRIC IMPLICATIONS By Dennis Lee Hoffman This study investigates the impact of the rational expecta- tions hypothesis (REH) on economic models by constructing a frame- work for analyzing rational expectations solutions, outlining an approach to estimation and tests of hypothesis regarding struc- tures which incorporate the REH, and surveying the recent literature to evaluate the use or misuse of the theory of rational expecta- tions in previous studies. The need for an analysis of the REH is derived from two .distinct factors. First, the expanding role of expectations in economic models warrants the development of an explanation for how these generally unobservable perceptions of future events are formed. Second, the REH, which is one explanation, is not based upon a well developed theoretical foundation. Specifically, neither the guidelines for incorporating the REH into general models nor the econometric implications of applying the theory to economic models have been explicitly stated in previous studies. In an effort to eliminate these deficiencies in the theoretical development of the REH, the present study adopts the Dennis Lee Hoffman following format. First, a framework for the implementation of rational expectations in general models is constructed. Particular emphasis is centered upon the conditions under which one may Ob- tain an observable expression for the expectation terms. This expression is designated as a rational expectations solution (RES). Second, the study reveals the econometric implications of replacing the expectations in the original structure with the RES suggested by the theory of rational expectations. Finally, a literature review compares the methodology employed in some recent treatments of the REH with that adopted in the present study. The pursuit of this format yields a number of significant contributions. In the first place, the framework provides guide- lines for the application of the REH to general models and accentuates the major complications a researcher is likely to encounter. Specifically, the analysis reveals that the RES de— pends upon the specification of stability Conditions and the nature of the processes assumed to generate the exogenous variables con- tained in the original structure under consideration. A second contribution of this study is the examination of the econometric significance of replacing the expectations terms with a relevant RES to generate a structure which is void Of unobservable variables. In this econometric analysis the restric- tions implied by the particular functional form of this re- formulated system are revealed and a method for testing their validity is outlined. Consequently, this study provides a pro- cedure for testing the validity of the REH as an explanation for individual's perceptions of future events. Dennis Lee Hoffman Finally, the literature review reveals that many recent studies have avoided the significant aspects of the REH which are emphasized in the present study. This neglect generally stems from either a misinterpretation Of the REH or the use of special cases which enable researchers to avoid many of the complexities inherent in the application of the theory Of rational expectations to general economic models. to my mother Bea Hoffman and my wife Cindy ii ACKNOWLEDGEMENTS l I would like to thank my thesis committee; Robert H. I Rasche, Peter Schmidt, James Johannes, and Mark Ladenson for their 4 helpful comments in regard to earlier drafts of this study. I am especially grateful to Peter Schmidt for his consultation in re- gard to the econometric section. Special thanks goes to my chairman, Robert H. Rasche, who provided constant guidance and encouragement at all stages of my research and for all the evenings and weekends he spent reading the manuscript. Also, I wish to acknowledge the graduate students at Michigan State University for providing an atmosphere in which it was easy to exchange ideas as we expanded our knowledge Of economics. Noralee Burkhardt and Linda Morrone deserve the credit for the careful typing of both the preliminary and final drafts. Finally, I wish to thank my wife Cindy for her patience during the hours I spent doing research as well as her expertise in editing a number of earlier drafts. Chapter I II III IV TABLE OF CONTENTS INTRODUCTION ................... l.l The Problem ................. l.2 Need for the Study ............. l.3 Format ................... l.4 Limitations ................. MODELS NITH CURRENT PERIOD ENDOGENOUS EXPECTATIONS ................... 2.l An Outline of Model I Structures ...... 2.2 A Rational Expectation Solution for Model I . 2.3 Observable Reduced Form for Model I ..... 2.4 Summary ................... MODELS WITH MULTI-PERIOD FUTURE EXPECTATIONS . . . iv 3.l An Outline of Model II Structures ...... 3.2 A Rational Expectation Solution for Model II 3.3 Observable Reduced Form for Model II . 3.4 An Example of a Model II Structure ..... 3.5 Summary ................... MODELS NITH MULTI-PERIOD FUTURE EXPECTATIONS AND LAGGED ENDOGENOUS VARIABLES ......... 4.l An Outline of Model III Structures ..... 4.2 A Rational Expectations Solution for Model III .................. 4.3 An Observable Reduced Form for Model III 4.4 An Example of a Model III Structure ..... 4.5 An Outline of the Multivariate Extension of Model III Analysis ............ 4.6 Summary ................... THE ECONOMETRIC IMPLICATIONS OF MODELS WITH RATIONAL EXPECTATIONS .............. S.l The Approach ................ 5.2 Identification ............... 5.3 Estimation ................. 5.4 Testing the REH ............... 5.5 Summary ................... Page th—I @0303 l3 13 IS 15 22 23 26 27 27 28 47 52 52 54 54 56 67 7T 8T Chapter Page VI AN ANALYSIS OF THE RECENT LITERATURE ON THE THEORY OF RATIONAL EXPECTATIONS .......... 83 6.l Examples which Deviate from the Approach Employed in the Current Study ........ 84 6.2 Examples of Models with Lead Expectations . . 9l 6.3 The Natural Rate Hypothesis ......... 95 6.4 Muth's Approach ............... 99 6.5 Shiller's Approach ............. 100 6.6 Summary ................... l03 VII SUMMARY AND CONCLUDING COMMENTS ......... 105 LIST OF REFERENCES ................ 108 APPENDIX ..................... lll A-l A Description of the Examples Considered ............... 111 A-2 REH Restrictions in Simple Theoretical Models ........... 113 A-3 Comparing the REH Restrictions with the Rule of Chapter V ......... 127 / CHAPTER I INTRODUCTION l.l The Problem This study investigates the impact of the rational expecta- tions hypothesis (REH) on economic models by constructing a frame- work for analyzing rational expectations solutions in general models, outlining an approach to the estimation and testing structures which incorporate the REH and surveying the recent literature to evaluate the use or misuse of the theory of rational expectations in pre- vious studies. A Study of rational expectations requires the considera- tion of models containing explanatory variables which appear as expectations of future endogenous variables. Since data is, for the most part, unavailable, one needs to determine how agents formulate these perceptions of future events. The theory of rational ex- pectations, originally advanced by Muth (l96l), is one explanation. This theory suggests that the rational agent equates expectations with the conditional forecasts of the "relevant economic theory“.1 The extensive application of this theory to models in current literature is the primary motivation of this analysis. 1See Muth (1961), p. 3l6. 1.2 Need for the Study The need for an analysis of the role of the REH is derived from two distinct factors. The first concerns the increasing emphasis placed upon expectations in recent studies coupled with an ad hoc explanation for how expectations are formed. Second, few studies have recognized the theoretical and econometric implications of the REH in general models. Since the analysis of Fisher (T930), the significance of expectations of price changes has been understood. Within the past decade new emphasis on price expectations has accompanied the natural rate hypothesis (NRH) which renders long run stabilization policy impotent [Friedman (l968), Lucas (l972)]. Sargent and Wallace (l975, l976) demonstrate that the addition of the REH to the NRH models preempts the role of short term stabilization policy, en- abling the results of the long run natural rate proposition to hold in the short run as well. This controversial result is most responsible for the movement away from the ad hoc notion of extra- polative or adaptive expectations first employed by Cagan (l956).2 2Extrapolative expectations pertains to the idea that expectations are adjusted according to the amount realized values deviate from previously formulated expectations: For example let t-lyt repre- sent the expectation Of the value of y in period t formulated on the basis of information in period t-l; then * * * {t-lyt ‘ t-Zyt-l} = (I ' °){yt-1 ' t-2yt-l} 0 < a < 1 describes an extrapolative scheme. This leads to an expression for expectations in terms of past values of y alone; * w i-l t-lyt ‘ (I " “’ Zi=l °‘ _yt-i‘ This framework has often been criticized for its lack of theoretical foundation as well as its suggestion that only knowledge of past values of the variable in question enter into the formation of ex- pectations. However, price expectations are not the only concern of rational expectations theorists. Theories of consumption and income, for example, the permanent income hypothesis; and interest rates, for example, the expectations theory of the term structure, place considerable emphasis upon expectations. An examination of the proper employment and validity of the REH is essential if rational expectations is to serve as a viable alternative to the extra- polative schemes employed in the previous analysis of these concepts. The second problem area which this study addresses is the deficiency of theoretical developments applying the REH to general models, specifically models with both multi-period future (lead) expectations and lagged endogenous variables. An outline of the solution procedure applicable to general models and an analysis of the conditions under which rational expectations solutions may be obtained would clearly aid both previous and forthcoming studies. A method of estimating and testing models which incorporate these general solutions is essential in augmenting the theoretical de- velopment of rational expectations. Finally, an investigation of the use of rational expectations in recent studies is warranted to alleviate the discord generated by the Muth article and assist in developing a uniform interpretation of the REH.3 l.3 Format To meet these problem areas, this study applies the REH to models of increasing generality in Chapters II, III, and IV. In 3The simple examples used by Muth in illustrating the REH have been the source of some confusion. This is discussed in chapter VI. each chapter the model under consideration is specified and the steps leading to the reformulated structure suggested by the REH are out- lined in detail. The individual treatment accentuates the different procedures required to obtain rational expectations solutions and . outlines the various restrictions under which stable solutions are obtained in different modeis.4 Chapter V offers an approach to estimation and hypothesis testing which is consistent with the REH according to the general framework outlined in the introductory chapters. Chapter VI supplies a discussion of a number of applications of the REH in current literature, highlighting both the improper interpretations of Muth's theory and the various assumptions which serve to circumvent the complexities of coping with the REH in more general specifications. l.4 Limitations Although this study provides an extensive treatment of rational expectations in economic models, it is not without limita- tions. It provides no explanation of how agents obtain the informa- tion required to form perceptions of future events which are rational in the sense of Muth. This difficult problem requires a general theory of rational expectations which has yet to be developed and is outside the scope of this study. Furthermore, the analysis offers no mechanism for how rational expectations adjust when the relevant theory Changes or is 4Stability is referred to, not in the probabilistic sense, but in the difference equation context. expected to change. Some advances in this direction have been made in recent studies by Shiller (1978) and Taylor (l975). However, these endeavors overlook the significant problems encountered when the relevant theory is static. These problems are revealed in the present analysis. Finally, although the test outlined in chapter V may prove useful in future applied work, the present study-offers no motive for employing rational expectations in economic models, but examines the implications of this choice for most of the models a researcher is likely to encounter. CHAPTER II MODELS WITH CURRENT PERIOD ENDOGENOUS EXPECTATIONS This chapter examines a specific class of models which contain expectations of endogenous variables. The expectations are expressed, following the REH, as functions of Observable vari- ables. Then,the reformulated version of the original model, incorporating the REH, is derived. 2.l An Outline of Model I Structures Perhaps the most common usage of expectations in economic models involves the assumption that present period (t) levels of endogenous variables are explained by perceptions of those variables formed by agents on the basis of information available one period earlier (t-l). Examples of this class are prevalent in the current literature and the general implications of the REH for these simple models are discussed in some of these studies. Nevertheless, these structures will be examined in the present analysis to provide a complete framework for analyzing the impact of the REH upon economic models. A model which contains the characteristics of these simple structures will be designated, Model I, and may be represented: _ e _ Ayt - B(L)xt + W(L)yt + et-lyt + at t - l,2,...,T (2.l.l) where; clilE/I\CC i) A is an m x m matrix of structural co- efficients, ii) yt is an m x l vector of endogenous variables, iii) B(L) is an' m x n matrix with elements consisting of finite polynomials in the lag operator L. Hence, typical elements are: q Ii 3.. = Z 3.. L9; Lgx = x _ 9:0 139 t t g i = l,2,...,m; j = l,2,...,n, is the order of the lag of the th a. qij jth exogenous variable in the i equation, h b. B.. is the gt coefficient in the 1J9 polynomial lag of order q, J jth exogenous variable in the i for the th equation, iv) xt is an n x 1 vector of exogenous variables,1 v) V(L) is an m x m matrix with elements con- sisting of finite polynomials in the lag operator L. Similarly:7 1The term exogenous pertains to those variables which are deter- mined outside the system (2.l.l) while endogenous variables are de- termined by the simultaneous interaCtion of the system. Including expectations of exogenous variables wOuld be a trivial extension of (2.l.l). The ensuing analysis would be unaltered and the problem created by their unobservability would be eliminated by an assumption comparable to (2.2.6) in the following section. i,j = l,2,...,m, a. r.. and v. are analogous to 91 1.1 iih J’ and Bijg above, b. wijo = O for all i,j, vi) 6 is an m x m matrix of structural co- efficients, vii) t_]y§ is an m x l vector of the values the endogenous variables are expected to take on in period t formulated on the basis of all in- formation available as of period t-l, viii) at is an m x 1 vector of structural dis- turbances which follow a stationary, multivariate, ARMA process: E(L)et = 5(L)Ut. where both 5(L) and 6(L) are m x m matrices whose elements are finite order polynomials in the lag operator L. ”ti is independently and identically distributed N{O, oEIT} for each i = l,...,m. -Following Zellner and Palm (l974) this may be expressed as an infinite order moving average process: = e“(L)a(L>u (O(L)Ut , 8t t provided the roots Of |5(L)1 = 0 lie out- side the unit circle. The final assumption (viii) accommodates any order of auto- correlation in the vector Of disturbances at and is amenable to the moving average structures examined by Muth in his seminal article. Incorporating this general expression for disturbances, (2.l.l) may be written: Ayt e B(L)xt + v(L)yt + et_]yi + C(L)ut, t = l,2,...,T. (2.1.2) The system described by (2.l.2) is the most general repre- sentation of a simultaneous equations system that includes expecta- tions of current period endogenous variables formed from informa- tion available last period. 2.2 A Rational Expectation Solution for Model I A rational expectatiOns solution (RES) is an expression for' rational expectations in terms of observable variables Obtained by the application of the REH to a specific model. To obtain the RES for a general Model I structure, consider the reduced form for (2.1.2): ' I I - .. - e - yt = A 1B(L)xt + A v(L)yt + A 1Ot_]yt + A w(L)u (2.2.1) t' This system represents the "relevant economic theory" for the m endogenous variables in the system. The theory of rational expectations suggests: IO 9 -_y = E{le_}E EY: t 1 t t_1 t t 1 t_1 t where It-l pertains to the set of information from which expecta- tions are formulated. Hence, rational expectations are the condi- tional forecasts of the relevant theory. Making conditional forecasts from a Model I structure assumed to contain rational expectations yields: E yt = E A“B(L)xt + E A"v(L)yt + E A’19 E yt + E A"O(L)ut.(2.2.2) t-l t-l t-l t-l t-l t-l This expression may be simplified by noting; 1) EElzt 1 = Zt-i for any variable 2 = l 2, t-i’ ii) Lag Operators affect realization dates, not expectation formation dates, therefore, k E L z = E z , k = 1,2,... t-l t t- 1 t k for any variable Zt’ iii) V110 = O for all i,j; to obtain: -1 -1 . -1 -1 [I - A e] E yt= E A B(L)xt + A v(L)yt + E A C(L)ut (2.2.3) t- 1 t-l . t-l Assuming A“e is of full rank yields: ' _ -1 -1 -1 -1 -1 E y - {I - A e} I E A B(L)x + A Y(L)y + E A w(L)U }. (2.2.4) t-l t t-l t t t-l t Therefore, the application of REH to Model I yields an ex- pression for the rational expectations of endogenous variables II which contains expectations of both current exogenous variables and the disburbance term, plus all the lagged variables in the original structure. Particular functions of the original structural para- meters form the coefficients of this expression. However, (2.2.4) becomes a RES only when the expectation terms on the right side of (2.2.4) are expressed in terms of Observ- able variables.2 Assumption viii, section 2.l, insures that: while all other values of E A'1w(L)ut are observable reduced form disturbances realized in prEDious periods. Additional assumptions are required to deal with the expectations of exogenous variables. Assume all exogenous variables follow known deterministic rules. This allows current exogenous variables to be predicted with certainty on the basis Of last period's information. This assump- tion allows (2.2.4) to be written as:3 E I e}']{A']B(L)xt + A"v(L)yt + A'In'(L)ut_1}, (2.2.5) t 1 yt={I-A which is a (RES) to a general Model I structure subject to the stated assumptions. A more comprehensive assumption is that the exogenous vari- ables follow some identifiable, stable, stochastic process! 2This is the definition of a rational expectation solution employed in the present context. 3Writing w(L) = woLo + NIL] +... where O1 is the matrix of co- efficients on the lag Of order i on u in (2.1.2) then t w'(L) a w1L° + OZL] +... 12 i.e. xt = r(L)xt_1 + nt , (2.2.6) where r(L) is a diagonal matrix whose elements are finite poly- nomials in the lag operator L and nt ~ N{O,zn}. Re-expressing the expectations term in (2.2.4), and follow- ing (2.2.6) yields: 4 -1 _ E A B(L)xt - t-l = A'1B E x + A"B'(L)x O t t-l t-l _ '1 '1 I - A BOF(L)Xt_1 + A B (L)xt~1 _ '1 l - A {BOF(L) + B (L)}xt_1 . . (2 2.7) The substitution of (2.2.7) into (2-2-4) VIéIdSI E y = {I - A"e}“IA‘1(B r(L) + B'(L))x + A"v(L)y t‘] t . O t‘]‘ t + A'Ie'(L)ut_1}. (2-2-8) I which is a RESfOr a Model I structure conditional upon (2.2.6). Hence, a valid RES for systems which have the charac- teristics of Model I, as defined above, depends upon two distinct factors. First, it demands the exact specification Of the relevant theory, including the nature of the reduced form disturbances. Second, the nature of the process from which agents form'expecta- tions of the exogenous variables is required. 4 Writing B(L) = BOL0 +...+ Bqu, q = Max(q1j) over all i,j where B1 is the matrix of coefficients on the lag of order i gon xt in (2.l.2), then: 1 = O 1 q'1 B (L) 81L + BZL +...+ BqL . T3 2.3 Observable Reduced Form for Model I Having derived the RES, the Observable reduced form is ob- tained by substituting (2.2.8) for' t_1y: in (2.2.1) to obtain:5 _ -1 -1 yt - A B(L)xt + A v(L)yt + A']e{I - A']e}-]{A-](BOF(L) + B'(L))}Xt-] + A-1e{I - A‘IOI‘IIA‘WHMt + A‘1w'(L)ut_1} + A"u(L)ut . (2.3.1) This expression represents the reformulated, reduced form structure obtained by replacing the expectations with the RES derived from' the application of the REH to the original reduced form system. Consequently, adding the REH to (2.2.1) yields an alternative structure which is void of the unobservable variables. 2.4 Summary The analysis of Model I structures reveals several important factors. First, when the only expectations terms encountered in a model are perceptions of present period endogenous variables formed 5Similarly an observable reduced form form (2.1.2) when the exogenous variables follow a known deterministic rule is obtainable, - making (2.3.1), I l l --1 - yt - A B(L)xt + A + A-]w(L)yt + A' -1 - - - v(1.)yt + A eII - A ‘91 {A B(L)x l t + A-1w(L)Ut . I4 one period earlier (Model I), the RES and consequent observable reduced form are computationally easy to obtain regardless of the 6 The investigation Of models magnitudes of structural parameters. whiCh contain expectations of endogenous variables for future periods, pursued in ensuing chapters, reveals that this is not always the case. In addition, section 2.2 demonstrates that the REH alone is insufficient to Obtain expressions for expectations of endogenous variables in terms of observable variables. Hence, the RES is con- ditional upon assumptions about the nature Of the processes gen- erating the exogenous variables and disturbances. Finally, the application of the REH to a Model I structure yields a reformulation which is a function of a particular set of variables whose coefficients are, in turn, specific functions of the orfiginal structural parameters. Thus, the observable reduced form suggested by the REH is distinguishable from any other reformulation of (2.2.l) obtained under an alternative assumption about how expectations are formed. 6Recall that the only restriction imposed upon structural para- meters to obtaina RES was that A‘1e be of full rank. CHAPTER III MODELS WITH MULTI-PERIOD FUTURE EXPECTATIONS This chapter investigates a class of models which contain expectations of endogenous variables for a finite number of future periods. The RES and resulting Observable reduced form are derived according to the format outlined in the previous chapter. 3.l An Outline of Model II Structures In the event that economic agents base current decisions upon expectations (formed in period t-l) of endogenous variables in future time periods t+l, t+2,..., the Model I framework may be broadened to include these multi-period future expectations. A few simple examples of these structures appear in the current literature.1 However, a general treatment of these structures is not explicitly stated in any of the previous studies of rational expectations. Models which possess these Characteristics will be designated Model II structures and may be represented as:2 Ayt = B(L)xt + e(F)t_1y: + m(L)ut, t = l,2,...,T (3.1.1) 1Some of these are discussed in Chapter VI. gThe effect of introducing lagged endogenous variables into (3.1.1) 15 examined in Chapter III. TS T6 where; i) A, yt, B(L), xt, t_1y:, w(L), ut are defined as in Chapter II, ii) 6(F) is an m x m matrix with elements con- sisting of finite polynomials in the lead operator F; hence, typical elements are, 6.. = X1J 9.. Fk° Fk ye = e 13 k=O ijk ’ t-l t t-lyt+k’ i,j = l,2,...,m, a) s.. is the order of the lead of the 13 expectations of the jth endogenous variable in the ith equation. th coefficient in the b) eijk is the k lead polynomial equation for the expectation of the jth th endogenous vari- able in the i equation, c) Since no information is available for periods later than (t-l), lead operators apply to realization dates and not ex- pectation formation dates. The system described by (3.l.l) is more general than (2.l.l) in the sense that it allows future expectation terms to enter as explanatory variables. However, it cannot accommodate lagged endogenous variables which could appear in Model I. 17 3.2 A_Rational Expectation Solution for Model I;_ As in Model I, the construction Of the reduced form is the first step toward replacing the expectations terms with expressions which are free of unobservables. Hence, Consider: 1B(L)x + A“e(E) + A“a(t)u (3.2.1) _. ' e yt ‘ A t-lyt t t' This system represents the "relevant economic theory" for the m endogenous variables in the system. Following the REH, the conditional forecasts from (3.2.1) are equated with the expectations in (3.2.1) to obtain: 1 E yt = E A‘ B(L)xt + E A"e(f) E yt + E A‘1a(L)ut. (3.2.2) t-l t-l t-l t-l t-l Combining the expectations terms yields: {I - A"e(F)} E yt = E A"B(L)xt + E A“a(L)ut. (3.2.3) t-l t-1 t-1 At first glance, this expression resembles the Model I analogue (2.2.3) which is a system of m equations in m unknown, current period, rational expectations. However, closer investiga- tion reveals that the Model II expression, (3.2.3) above, contains In equations and up to m(s+l), s = max sij’ unknown rational expectations. This structure, which represents current period rational expectations ( E t-l tions in later periods (51 yt+], tEl yt+2,...) contains m, finite order, difference equations in leads as opposed to lags. Yt) as functions of rational expecta- 18 The typical equation of this system may be Obtained by applying Nold's Chain Rule of forecasting to (3.2.3)3, yielding; -1 -1 {I - A“e(F)1 E y t-l = A E B(L) t-1 + A E CO(L) t-l j = 0,1,... t+j xt+j Ut+j’ (3.2.4) The relevant solution procedure for this system is analogous to the calculation of the "final form" of a simultaneous equations model.4’ In a "final form" all lagged endogenous variables are eliminated by substituting recursively. The lead expectation ——-_._. .# terms in (3.2.3) may be eliminated analogously, by recursive substitutions, utilizing the structures defined in (3.2.4) to obtain:5 E l )3 {I - A-16(F)}'1{A']B(L)xt + A‘ t 1 yt = E w(L)ut} . (3.2.5 t 1 The condition which guarantees the stability of this solu- tion is that the roots (with respect to F) of the determinantal 3wo1d (l938), Chapter 3. 4See Theil and Boot (1952), p. 135-152. 5The substitution procedure implied by (3.2.5) insures that all terms in periods t, t+l,..., later periods, appear as expecta- tions terms while those in t-l, t-2,..., and earlier represent actual realized values. This interpretation is unambiguously Inaintained when lag and lead operators are manipulated prior to expectation operations. Hence, the expectation term is positioned outside of the inverted lead coefficient. Fk E 2t = E szt for all variables 2 t-1 t 1 t in this analysis. 19 16(F)1 = 0, lie outside the unit Circle.6 Nhen equation, [I - A' this condition is satisfied, the system (3.2.5) expresses current period rational expectations as functions of all the lagged exogenous variables and disturbances in the original structure, plus expectations of exogenous variables and disturbances for all future periods. Following the definition employed in Chapter II, (3.2.5) becomes a RES when the expectations terms on the right hand side of (3.2.5) are replaced by observable variables. As in the Model I analysis, assumption viii, section 2.1, insures that all future disturbances have zero expectatiOn. Consequently: 1e(F)}“A‘1u(L)u = E {I - A t t-l DII - A'le(f)i"A“a(L)ut where, {I‘A-]9(F)}-]A-]w(L)Ut for periods w(L)Ut = t-l,t-2,...,earlier, OII-A“e(F)1'1A" O for periods t,t+l,t+2,...,later. This notation accentuates the fact that (3.2.5) contains no un- observable expectations of disturbances.7 6This condition is derived from considering an analogous example in Zeller and Palm (1974), p. l9. —_ ,‘Hht ~v..f.-v_- _. ’.—. .A—-‘.-.._.—.-— The lag and lead operators affect the time period for the dis- ttrrbance term. Therefore if {I-A'19(F)}.]A-1w(L)ut is depicted as: 20 The expectations of exogenous variables are eliminated by utilizing the assumptions Of Chapter II. As discussed in the Model I analysis, when exogenous variables follow known deterministic rules, they may be predicted with certainty for all future periods. The RES subject to this assumption is obtained by equating actual and expected future exogenous variables: E yt = {I - A’19(F)}‘1{A“B(L)xt} + t-1 1 -1 DII - A' (F)} w(L)U (3.2.6) t . An alternative assumption is that the exogenous variables follow the process described by (2.2.6). Taking conditional fore- casts on (2.2.6) and leading 2 periods obtains expressions for the R-period forecasts of the n exogenous variables: / 11 >0 tEIXit ‘ it( = Yilxi-t-l + Yi2xit-2 +'°'+ Yipxit-p tE1Xit+l Xit(1) ‘ Iiixit(0) I Yizxit-I +°°'+ Yipxit-p+1 tflxit+£ ' xit(£) = YiTXit(““) I Yi2xit(£-2) +"°+ Yipxit(“‘p) i = 1,2,...,n 1;-“ C1ut+1 ; C1 m x m matrix for all 1. Then; m -1 D Z Ciut+i 1:-” i=§m Ciut+i 21 where; i) 2 is arbitrary. ii) P is the order of the autoregressive process which generates all exogenous variables, iii) i1t(-j) = x1t_j Following Box and Jenkins (1976), the forecasts for lead times ; j = 1,2,... 2 :10 may be expressed in terms of the observable lagged exogenous variables to obtain:8 E x. e i. (t) = E Y(“i x (3 2 7) t-l 1,t+£ T,t 1:] 1,3 l,t-J (2) - 2 (R-h) YisJ' ‘ Yi.j+2+ .2. Yi.h+1Yi.I h-O (0) - *id ‘ Yip” Substituting (3.2.7) into (3.2.5) obtains tié‘RES when exogenous variables adhere to (2.2.6): ’ E yt = {I - A"e(F)}‘1{A"B(L)xt(t)} + t 1 - 1 -1 + D{I - A- (F)} {w(L)ut} (3.2.8) where; i) xt(2) represents the n-dimensional vector of R-period forecasts of exogenous variables when A 3_O, and realized, lagged, exogenous vari- ables when” I < O, 8These expressions are obtained by Box and Jenkins (l976), pp. 141—142. The notation is altered in (3.2.7) to accommodate 2 period forecasts. 0 22 ii) Fkxt(2) =.£t(t + k), iii) kat(2) = Xt(l - k) , In retrospect, the RES obtained for models which exhibit the characteristics of Model II, depend upon thesame factors required in the Model I solution; namely the exact specification of both the structural model and process generating the exogenous variables. However, the present analysis reveals that models with lead endogenous expectations warrant the use of extremely intricate sub- stitution procedures to Obtain the weights on lagged variables which appear in the RES, as well as consideration for the conditions which guarantee the stability of the solution. 3.3 Observable Reduced Form for Model II Following the format of the previous chapter, the observable reduced form is obtained by the substitution of the RES (3.2.8) into (3.2.1) to obtain:9 Yt = A']B(L)xt + A“e(F)II - A"e(F)1"IA“B(L)§t(t)} + A“e(F)O(I - A“e(F)}‘1{A'1a(L)ut} + A“u(L)ut . (3.3.1) This expression denotes the reformulated reduced form obtained by the application Of the REH to a general Model II structure. This reformulation suggests that the original endogenous variables may be expressed as a function of all the predetermined variables 1"_PPP”- gThe observable reduced form for Model II when all exogenous vari- ables follow known deterministic rules is obtained by substituting xt+£ for xt(2). 23 system (3.1.1) plus the lagged terms inherent in the processes which generate the exogenous variables. This result is not unlike that obtained for Model I, in (2.3.l), in regard to the designa- tion of the menu of variables which appear in the observable re- duced form. However, the coefficients of (3.3.1) are considerably more complex functions of structural parameters than those in the observable reduced form (2.3.1) obtained for Model I. A simple example exhibiting the characteristics of a Model II structure illustrates the steps leading to a Model II RES. Con- sider the single equation model: - . e z yt - B xt + eFt-Iyt + at t 1,2,...,T where from (3.1.1); // 1 yt is a scalar, 11 B(L) = 8' is a l x n vector, ) ) 111) xt is a n x 1 vector, ) ) ) 1v 6(F) = OF is a scalar, e . v t-lyt 15 a scalar, v1 w(L)Ut = 6t ~ N{O,X€} scalar. The structure (3.4.1) describes the "relevant economic theory" for the variable yt. Therefore, following the procedure outlined in 3.2, the application of the REH yields an expression analogous to (3.2.5): 24 -l E y E {1 - 6F} 8' x t-l t+l t-l t+l ” i i E Z O F 8' x t-l i=0 FT] E Z 913' x .. (3.4.2) t-l i=0 t+1*‘ Assume all exogenous variables are generated by first order, autoregressive processes. Hence, F(L), in (2.2.6), is a diagonal, n-dimensional, matrix Of zero order polynomials, r; where; F.. = y. l = 1,2,...,n. Therefore:10 E t_1 xt+1+i = I xt_, i = O.1,2,..., (3.4.3) The RES for this example may be obtained by the substitution of (3.4.3) into (3.4.2); yielding: ! - l . 1+2 E yt+1 - i 0 e B T Xt‘I’ (3.4.4) t 1 11MB :an the notation of (3.2.7), =« A+l t5] X18t+£ xi,t(l) i,j xjgt'I 2 . - + . . Since; Ygf} = Y1,j+£+ % OYi,h+lYifjh) = 7%,} for all 1 1f the x's are generated by first order processes. Hence, 71,1 = Yi,h+1 = O for j = 2,3,... and h = 1,2,3,... 25 provided the stability condition, |e| < l, is satisfied}1 Since (3.4.4) is a geometric progression, it may be simplified to obtain:12 E _ . 2 -1 t 1 yt+1 - B r {I - er} Xt-l‘ (3.4.5) Finally, the observable reduced form for this simple example is obtained by the substitution of (3.4.5) into (3.4.1): 2 _ . . -1 yt 3 xt + as r {I - er} Xt-l + at . (3.4.6) This example reveals two aspects of the application of the REH to Model II structures. First, the RES requires consideration of all future expectations of the exogenous variables, even when only one period lead expectations appear in the original model. Also, both the coefficients and stability conditions for the RES may, in some cases, be simple functions of the original structural parameters; regardless of the complicated expressions obtained in the analysis of a general Model II RES. 11Recall the condition defined in 3.2 requires that the roots of II - A'1e(F)| = 0 lie outside the unit circle. In this example the stability condition becomes, - |l-eF|=O ;|F|>l OY‘ [Hz—[RT ; |F1>1 Therefore the single root lies outside the unit circle when |e| < 1. 12lim r1 = D null matrix since |y11| < l for all i = l,...,n i-roo because (2.2.6) is assumed to be a stable, stochastic process. 26 3.5 Summary The implications of the REH for Model II structures are re- vealed by a comparison of the results Obtained in Chapters II and III. First, the existence of a RES in models with lead expecta- tions depends upon specific stability conditions which are, in turn, satisfied when structural parameters lie within particular intervals. This result differs from the Model I analysis which leads to a RES for all possible values of the original structural parameters. Second, as in the Model I analysis, the particular RES ob- tained depends upon the assumption made about the process which generates the exogenous variables. However, the RES for Model II demands more extensive forecastingirfoture exogenous variables since the solution contains expectations of all future exogenous variables. Finally, the arguments of the observable reduced form for models with lead expectations correspond with those obtained in the study of Model I. Also, the REH suggests a functional form for the weights on these variables that allows one to distinguish the observable reduced form, implied by the REH, from that obtained by employing an alternative expectations generating scheme. However, these weights are generally more complicated functions of the original structural parameters than those obtained in the observable reduced form for Model I. This complexity stems from the substitution procedure required to obtain a Model II RES. CHAPTER IV MODELS WITH MULTI-PERIOD FUTURE EXPECTATIONS AND LAGGED ENDOGENOUS VARIABLES This chapter examines a Class of structures which allow lagged endogenous variables to accompany multi-period future endogenous expectations as explanatory variables, thereby avoiding the simplification employed in the Model II analysis. 4.1 Afl_0utline gj_Model III Structures In an effort to analyze the impact of the REH on the most general models that a researcher is likely to encounter, lagged endogenous variables are added to the models with multi-period lead expectations to obtain a class of structures denoted as Model III. The general expression for this class is: =NUx+wuwt+MHbfli+MU%, (Ahn t t where all terms have been defined in previous analyses. This class is the most general representation of models which contain expectations formed from information available in period t-1.1 Unlike Model I and Model II, which may be expressed 1Shiller (1978), p. 29, deals with a more general version of (4.1.1) by including period t-2,t-3,... expectations of endogenous vari- ables. The import of his analysis is discussed in Chapter VI. 27 28 as special cases of Model III, the details for obtaining a RES for models with both lagged endogenous variables and future endogenous expectations have yet to be explicitly stated in the literature. The present analysis intends to fill this void. 4.2 A_Rational Expectations Solution for Model III Following the format employed in previous chapters, the first step toward a RES is to obtain the reduced form: 1 1 B(L)x + A‘ + A‘1u(L)u t (4.2.1) RUA+AJNH = - e yt A t-lyt t' According to the REH, the expectations in (4.2.1) are equated with the conditional forecasts from the "relevant theory" [in this case (4.2.1)]. These forecasts are generated by taking expectations on (4.2.1) to obtain: yt = E A“3(L)x + E A" t i(L)yt + E A‘]e(F) E y t-T E e - t-l t-l t t 1 t-l + E A’1 t-1 w(L)U (4.2.2) t . Combining the expectations terms with the lagged endogenous vari- ables and noting; E E yt+S = E yt+s for all t-1 t-1 t-1 1 yt+s: 1 1 1 E {I - A- t-l 6(F) - A- B(L)x + E A- V(L)}yt = E A t t-l t-1 w(L)Ut. (4.2.3) As with Model II, rational expectations solutions for the expected endogenous \ariables (up to m(s+l) in number) may be 29 obtained by leading (4.2.3) j-periods and substituting to obtain ex- preSSTOns for tEl yt, tEl yt+],...,tE yt+S in terms of Observable variables. The typical lead equation for (4.2.3) may be obtained by using the chain rule:2 1 1 E {I - A- t-1 O(F) - A’ V(L) = E B(L) t-1 E w(L) }yt+j-l _, ”t+j-1' x . + t+j-l t J = 1,2,... (4.2.4) Inspection of (4.2.4) reveals that the Model III rational expecta- tion for period t+j depends upon rational expectations of endo- genous variables in both later (t+j+l, t+j+2,...) and earlier (t+j-l, t+j-2,...) periods. Furthermore, rational expectations for periods t+j: j 5_r depend upon lagged endogenous variables.3 Therefore, there is simultaneous feedback through time among the expre551ons for tEl yt,...,tE1 yt+j in (4.2.4), i.e. yt+j-1 depends upon E yt+j E t 1 - t 1 depends upon E y and E - t-l y . t 1 t+3 t+j-1' This result is not obtained in the Model II analog (3.2.4) where the relationship among rational expectations expressions is shown to be strictly recursive through time, i.e. 2No1d, Chapter 3. 3This result follows from section 2.2, assumption i, expectations of variables in period t-l and before equal actual values of those variables. 30 E yt+j~l depends upon E y 12-1 t-I t+j but E. . ‘ ' . , t-l yt+J is independent of tEl yt+j-l As a result, the substitution procedure required to obtain a RES for Model III may not be described, by the inversion Of lead operator processes, the mechanism employed in the Model II analysis. Even though the simultaneous substitutions involved in ob- taining a Model III RES from (4.2.4) may not be characterized by convenient notation, the solution may be expressed in general terms as: ' } E yt = Rl{yt-r"°"yt-lixt-q""’Xt-litE xt,..., E xt+n-1;KO""’Ks-l t-l -1 t-1 (4.2.5) E y = R {y ,...,y ;x _ ,...,x _ ; E x ,..., E x _ ;K ,...,K _ } t-l t+s s+l t-r t-l t q t l t-l t t-l t+n l O s l where; i) R1, j = l,...,s+l, are linear functions which describe the RES for tE1 yt+j_1. solving the system of lead equations in (4.2.4) for These may be obtained by E yt,..., E t-l t-l ii) Kk; k = 0,1,...,s-l, are the values of distant future yt+s. endogenous expectations which invariably appear in R1,...,RS‘], iii) the effect of K0’°"’Ks-l in the solution for ' ' = ..., + , ' ' ishes tEl yt+j~l in (4.2.5), J 1,2, 5 l dimin 4 as n increases; hence, the solution is stable. 4An analogous assumption is imposed in the Model II analysis. The condition which insures that the effect Of distant future values Of endogenous expectations attenuate as the time horizon lengthens, 31 Two separate issues will be addressed in the analysis Of this general RES for Model III structures. The first concerns the determination of the functional form for R1, j = l,...,s+l, in (4.2.5) by analyzing the nature of the substitution procedure re- quired to obtain a RES for Model III. The second is the Specifica- tion of conditions which guarantee that the effect of distant future endogenous variables K0,...,Ks_1 in (4.2.5) declines as n in- creases, thereby insuring the stability of the solutions. Both of these objectives may be achieved by considering a simplified version of (4.2.1). Assume (4.2.1) is a single equation model; A = l, B(L) = B(L), B(L) and xt are n x l vectors, all other vari- ables are scalars, and at - N{O,z€}.5 Therefore (4.2.1) becomes: .. I e yt - B(L) xt + V(L)yt + 6(F)t-1yt + at . (4.2.6) The functional form of the solutions described in (4.2.5), when the relevant theory follows (4.2.6), may be obtained by the following procedure. 1) Assume initially that E yt+n’ E yt+n+l""’ - t-l t l tEl yt+n+S-l are known. Denote these as . 6 K0,...,KS-1 respectively. is that the roots of the characteristic equation 11 - A’]e(F)| = 0 lie outside the unit circle. S\II(L) is a polynomial of order r in the lag operator L. 6(F) is a polynomial of order s in the lead operator F. 6Models with S lead endogenous expectations require S of these assumptions. 32 ii) Construct a system of n-equations in n-"unknowns," E yt,..., E yt+n-l’ by making j-period forecasts t-1 t-1 outlined in (4.2.4) where, j = l,...,n. iii) Solve this system Of equations by inverting the nth- order coefficient matrix for this system. When the relevant theory is (4.2.6), the particular system of n-equations Obtained by leading (4.2.3) may be described as: J r = -v for i-j = g; g = 1,2,...,r = 1-60 for l-J = O = -Oh for i-j = h; h = -l,-2,...,-s =0 for i-jin+3,n+3)°"° (n+sin+sli k = S-2; E e ("-1) = (-l)s'2 (n) h=s-1 han-(h-(s-l)),j p(n+l,j)(n+2,n+1)---- (n+S-l,n+s-2) (n+s,n+s), l _ n- _ S-1 (n) 5’1’ 6Sunni _ ('1) p(n+1,J)in+2,n+1)"°'~(“+5’“+5‘1)' 7? II The subscript on 9(n) denotes the S pairs of rows and columns th deleted from {(r,S)A(n+S)} to form the particular cofactor (n order determinant) that equals the weighted sum of cofactors ((n-l)St order determinants from {(r,s)A(n)} ) which constitute the numerators of the coefficients on Kk’ k = O,...,S-l ih (4.2.10). The Sign (-l)k is necessary since the sign included in the cofactor, , (n) p(n+l,j)(n+2,n+l)-°-(n+k+l,n+k)(n+k+2,n+k+2)°'°(n+s,n+s)s k = 0,1,...,S-1, corresponds to (-l)n+3 only when k is even.10 Turning to the denominators of the Coefficients in (4.2.10), the determinant of I ~(r s)A(n)} may also be expressed as a cofactor A m . (5.2.7) Therefore, the condition for identification requires that the number of period t exogenous variables, for which at least one lagged value (lag i P) does not appear in the original system, exceeds the number of equations. Clearly, the condition could be modified to account for the possibility that not all endogenous variables appear in expectation form. In this case (5.2.7) would be: nl-n2>m1 9Only the rank of this submatrix is of concern since the last n - n1 columns of n? are null vectors. Hence, independent rows * and columns in the last n - 111 rows and columns of P (L) will * not augment pIn?P (L)}. 63 where, m1 is the number of endogenous expectations in each equa- tion. This result generalizes Wallis' identification condition for models with lagged variables.10 However, when lagged variables are considered, two conclusions reached by Wallis must be modified. First, his identification condition no longer applies to general models. Second, the nature of the autoregressive process does have a role in the identification analysis, contrary to Wallis' contention. The following analysis demonstrates these results. In the simple models employed by Wallis, the condition for identification of RF coefficients from those in the ORF is that the number of exogenous variables exceed the number of expectations terms. This condition is a special case of (5.2.7). When no lagged terms exist, the elements of n3 are the only terms which are not immediately identified from knowledge of the ORF coefficients. * Also, n1 - n, r (L) = r(L) and r12 = 0. Therefore; * pfn?F (L)} étmin{m, n}. Hence, Wallis' identification condition, n > m, (5.2.8) is confirmed. However, expression (5.2.7) reveals that this state- ment is invalid when some of the explanatory variables in the original RF are lagged exogenous or lagged endogenous variables. Clearly, the ORF coefficients on lagged values provide no Iowa111s, p. 25. 64 information for identifying the elements of N3 11 ' An additional result obtained from the Wallis study must be modified in light of the present analysis. Wallis maintains that substituting the autoregressive processes for exogenous variables into the ORF (as opposed to merely calculating a single value for the optimum forecast) is of "no assistance" in identifying the elements of n3.I2 This result is confirmed for Wallis' simple model by considering (5.2.8). The rank of r*(L) will be n, regardless of the orders of the individual auto~ regressive processes for the n exogenous variables. However, (5.2.7) reveals that 112 corresponds to the number of current period exogenous variables which are generated by a particular set of lagged exogenous variables that also appear in the original RF equation. The previous analysis demonstrates that when auto- regressive processes for these exogenous varialbes are sub- stituted into the ORF, the form of the resulting coefficients is not conducive to identifying elements of n3. Hence, the value of 112 depends upon the particular lag structure of the auto- regressive processes. As a result, when the model under investi- gation contains lagged variables, the condition for identification does depend upon the nature of the process generating the exogenous variables. However, when m > n], the additional 1IExamp1e III in the Appendix offers an illustration of this fact. IzWallis, p. 25. 65 information which may be gained by extending the order of the pro- cess assumed to generate the exogenous variables will never-be great enough to identify the elements of n3. When the question of identifying RF parameters from those in the ORF is extended to models with lead expectations, the analysis is complicated by two factors. One stems from the increased number of expectations terms contained in models with lead expectations. As a result, the number of elements in «3 increased from a maximum of m2 to a maximum of mZIS + l}. Secondly, the ORF coefficients are much more complicated functions of the RF parameters. The ensuing analysis contains an outline of the general approach for determining the conditions for identification for Model II and Model III structures. Initially, consider an unrestricted version of any Model II or Model III ORF as a function, linear in both parameters and vari- ables, which contains the set of variables appearing in the ORF of the model under investigation. Hence, let: yt =:p(L)zt + Vt (5.2.9) where; . ' 1 I) 21: {xti‘yt-l} ’ ii) tp(L) is an m x n + m matrix whose elements are 13 polynomials in the lag operator L. The length 13The elements of :p(L) are defined analogous to those of B(L) in section 2.1, assumption ii. 66 of the lags will depend upon the lags in «1(L), n2(L), and F(L), iii) forecasts from i = F(L)x have been entered to t t-l eliminate all "future" values. The general ORF structure obtained by an analysis of either Model II or Model III, illustrated in (3.3.1) and (4.3.4) respectively may be designated: yt = 1r(L)2t + vt (5.2.10) where; i) zt is defined in (5.2.9), ii) n(L) is an m x n + m matrix whose elements correspond to those in either (3.3.1) or (4.3.4) when expressions for the lead forecasts of 2t(2) have been substituted for the expected future values of exogenous variables. Since (5.2.9) and (5.2.10) contain identical sets of explanatory variables, the conditions for identification may be examined by equating the coefficients from (5.2.9) with the functions of original RF parameters which constitute the coefficients of (5.2.10). The resulting system is a functional relation from the elements of «(L) to the elements of 1 1<]-a<1, Since a < 0 (restriction of Cagan's model), Clearly, this stability condition is equivalent to the terminal con- dition obtained by Sargent and Wallace. Sargent and Wallace confront another problem emphasized in the Model II framework: the specification of the process which gen- erates the exogenous variables. They demonstrate that the adaptive scheme employed by Cagan will be "rational" only under specific restrictions on both the disturbances and the stochastic process governing the growth of the money stock.8 Hence, Sargent and Wallace obtain a reformulated structure which corresponds to the form of the RES suggested in the Model 11 framework. .Furthermore, they state the conditions under which stable 8Sargent and Wallace (1973), p. 336. 93 solutions may be obtained and accentuate the role of specifying the nature of the sequence of exogenous expectations generated by applying the REH to models with lead expectations. The recent studies by Wickens and Turnovsky each contain theoretical sections which outline the procedure for dealing with rational expectations in a simultaneous equation model that contains one period lead endogenous expectations but no lagged endogenous variables. Turnovsky employs a solution procedure which is identical, except for the omission of lead operator notation, to that outlined in the Model II framework. An analogous stability condition and corresponding ORF are obtained. An innovative aspect of Turnovsky's analysis is the computa- tion of the effects of "incremental" and sustained" changes in gexogenous (policy) variables on the endogenous variables in the system.9 This is accomplished through a comparative static analysis of Turnovsky's system of equations. In principle, Wickens obtains identical results, but the methodology employed varies somewhat. Wickens' solution procedure may be outlined as follows; i) eliminate expectations from the model by utilizing: _ e Yt+1 ‘ Yt+1,t I nt+1,t where; 9Turnovsky, p. 855. 94 Yt+1 is a vector of endogenous variables, b. YE+l,t is the forecast for period t+l for these variables based upon knowledge of the struc- ture in period t, c. nt+l,t is the forecast error, observed in period t+l, for a forecast made in period t, ii) solve this original structure as a system of dif- ference equations, by "successive substitution or ' utilizing a lead operator," to obtain current Y as t a function of future exogenous variables, disturbances and forecast errors under a specified stability con-I dition,IO iii) make conditional forecasts on this reformulated struc- tUre and simplify to obtain the desired RES. Therefore, Wickens' approach involves purging the lead endogenous variables from the original structure prior to making conditional forecasts. Nevertheless, the solution obtained by Wickens is Similar to that suggested by the Model II framework.H The presence of expected forecast errors ("2+l,t) in the Wickens solution could be accounted for in the Model II RES by allowing the existence of forecast errors in the conditional expectations on the original reduced form. This is a trivial extension of the Model II solution procedure. Hence, IOWickens, p. 8. HWickens, p. 8. 95 the Model II, Turnovsky and Wickens approaches yield identical results when the conditional forecasts are set equal to the rational .expectations. Therefore, the Sargent and Wallace, Turnovsky, and Wickens studies compare closely with the methodology and emphasis of the Model II framework. However, they do contain several limitations. The omission of lead operator notation is not conducive to extending the results of these studies to multi-lead models (i.e. more gen- eral Model II Structures). Also, the solution procedures offer no guidelines for obtaining a Model III RES. Finally, none of the studies suggest estimation procedures comparable to Section 5.3, which account for the Specific functional form of the RES. 6.3 The Natural Rate Hypothesis Perhaps the most notable application of the REH is Sargent and Wallace's (1975) simple "textbook" macro model which demon- strates that the level of output is independent of the choice of the deterministic money supply rule. This controversial result warrants a comparison of Sargent and Wallace's treatment of the REH with the framework outlined in the current analysis. Initially, Sargent and Wallace specify the "relevant theory" .for prices from their structural model to obtain: pt 3 Jo E pt T J1 tE pt+1 I JzI'It I Xt t-l —l where; price level _a v U fl "I money stock 96 iii) E pt 5 expectation of prices based upon information t-l as of period t-l; assumed to be rational. iv) Xt 2 reduced form disturbances. Clearly, this single equation reduced form designated equation (15) by Sargent and Wallace, is a scalar Model II. Therefore the rational expectation for this structure may be expressed as a func- tion of current and future expected Mt and Xt. However, output in this model is presumed to be affected by the difference between expected and actual prices (the natural rate hypothesis). This is reflected in the structural equation for real output designated by Sargent and Wallace as equation (1): * yt = a1Kt-1 + azIpt ‘ tpt-l) + u1t where; i) yt a real output, ii) Kt-l a capital stock, iii) ”1t a structural disturbance, iv) tPI-l a agent's expectations of the price level for period t formulated as of period t-l. Following the theory of rational expectations, conditional forecasts obtained from the relevant theory (15) are equated with the expecta— tions in (1). However, the natural rate hypothesis allows an ' alternative to the standard solution procedure. By taking expecta- tions on (15) and subtracting the resulting forecast from (15), one obtains Sargent and Wallace's equation (16): pt - E pt = szMt - E Mt} + Xt - E x . t-l t-l t t-1 97 Since the intent of the analysis is to obtain an observable expres- sion for the expectations terms in (l), casual inspection reveals that an observable expression for the entire difference, * pt ’ tpt-l’ would be equally valuable for this model. But this is supplied by (16). Assuming the rule that generates the money supply is deter- ministic Mt - tEl Mt = O, and (16) is substituted into (1) to obtain: yt = a1 t-l + a2 xt ' t5] xt + ”lt' Since K is exogenous in this model, Sargent and Wallace con- t-l clude that output is independent of the money supply rule.I2 Therefore, when actualland expected values appear in dif- ference form, as in the structure which displays the natural rate hypothesis, the issues emphasized in the current study are no longer significant. Clearly, the question of stability is no longer crucial, and, when exogenous variables are initially assumed to be deter- ministic, Sargent and Wallace demonstrate that the solution is in- dependent of the exogenous variable generating rule. This result agrees with that obtained by following the guide- lines of the current study when actual and expected values appear 12Sargent and Wallace (1975), p. 247. 98 as a difference. Moreover, this simplification may be employed even when the relevant theory for pt, (15), appears as a Model III struc- ture. Hence, the examination of the Sargent and Wallace article reveals that a researcher may avoid the complex problems of obtain- ing an RES by imposing certain restrictions on the coefficients of the model in question. A brief comment on Sargent and Wallace's policy prescriptions is noteworthy in light of the current examination of the natural rate hypothesis and the role of rational expectations. Many critics have attacked the use of the REH in Sargent and Wallace's study since policy rules regain their potency if expectations are assumed to be generated by an autoregressive scheme. However, the natural rate hypothesis is equally culpable since Sargent and Wallace's result hinges upon the occurance of pt - tPE-l in the original structural equation. For example, assume output is affected by the individual levels of prices and expected prices (the long run 13 Phillips Curve is not vertical). Therefore, (1) becomes: * yt ‘ a1Kt-1 * azpt ‘ a3 tPt-1+ “1t: Applying a Model II solution procedure from a reformulated version of (15) obtains: y = fIK _ , p , x , E x ,..., M , E M ,..., U 1 , t t 1 t t t_] t t t_, t 1t where the deterministic money supply rule now obviously affects the I3No theoretical basis is provided for this alternative structure. It is constructed only for expositional purposes. level of output. 99 I4 Clearly, the REH alone is not responsible for the controversial policy conclusions derived in the Sargent and Wallace investigation. 6.4 Muth's Approach The difficulties encountered in obtaining a RES by follow- ing the guidelines of the framework outlined above are mitigated by . utilizing the solution procedure employed in Muth's original study. This is revealed by comparing Muth's approach with that employed in the current study. Muth suggests the following RES procedure: 1') ii) iii) iv) v)~ express endogenous variables, expectations, and disturbances as output of a white noise process, substitute these expressions into the "relevant theory" (reduced form), assume knowledge of the weights on the process gen- erating the disturbances, since the "relevant theory" must hold for all shocks, obtain the weights on the white noise process for endogenous variables from those on the process generating disturbances, solve for the "rational expectation? in terms of past disturbances. This procedure contains a number of limitations. Initially, endogenous variables may be expressed as functions of white noise 14 The relevant theory for prices would no longer be (15) due to the change in (1). However, only the coefficients would be different. 100 only when the model is void of exogenous variables. Second, the solutidn obtained is totally dependent upon knowledge of the weights in the process assumed to generate the disturbances. And finally, the solution is expressed in terms of "past realizations as opposed to current state variables."15 In the unlikely event that Muth's solution procedure is applicable (when all elements of B(L) equal zero), assumption (viii) section 2.1 insures that the framework of Chapters II, III and IV is able to accommodate the assumptions required for Muth's RES. Furthermore, the solution procedures yield identical results under similar assumptions. Therefore, Muth provides the definition for rational ex- pectations in his seminal paper; but his suggested solution pro- cedure, useful only for restricted structural models, offers little insight for coping with the issues raised in the current study. As a result, researchers who consider models which are restricted to accommodate Muth's solution procedure do not confront the problems of stability; specifying exogenous variable generating processes; and the particular function form of the solution, which are inherent in obtaining a RES for more general structures. 6.5 Shiller's Approach Shiller provides an extensive treatment of the REH in his recent study. A solution procedure for general models is outlined. IsThe last point is attributed to Lucas (1970). P- 55- 101 It is suggested that the most general models (for example, Model 111) may be handled by solving them as partial difference equation I6 Shiller emphasizes the difficulty in with variable coefficients. obtaining solutions in this case. A comparison of Shiller's approach with the framework outlined in the present study reveals that the overall scope of the two analyses is very similar but there is a distinct divergence in emphasis. Shiller questions the uniqueness of a "rational expectations 17 equilibrium." Since agents gain information each period, addi- tional knowledge on the structure of the relevant theory and the 18 process generating the exogenous variables is obtained. This prompts Shiller to examine models of the form.I9 yt = th + V(L)yt + 91(F) E yt +...+ 6K(F) E yt + 8t (26) t l t-K Therefore, current endogenous variables are not only influenced by expectations of future values of endogenous variables based upon information available last period - (t-l), but aIso expectations of future endogenous variables formulated up to K periods in the past. Since there are innumerable ways to model the manner in which I55hi11er (1978), p. 30. 17Shiller (1978), p. 4; this term is analogous to "rational expecta- tion solution." 18Shiller incorporates the idea of Taylor (1975), noting that the RES is influenced by the influx of information. IgThis is Shiller's equation (26) expressed in the notation outlined in Chapters II, III and IV. 102 agents alter their perceptions about the future in response to new information, Shiller argues that an infinite number of solutions, or rational expectations equilibria, exist for models with lead endogenous expectations. Therefore, he concludes that "the existence of so many solutions to the rational expectation model implies a funda- mental indeterminacy for these models."20 In comparison with the Shiller approach, the framework out- lined in the preceding chapters does not take issue with the notion that the economy will reach a unique rational expectations equili- brium. Instead, emphasis is centered upon the examination of the specific stability conditions required for a solution's existence, the specification of an exogenous variable generating scheme to in- sure that the RES is void of unobservable variables, and the econo- metric implications of the particular functional form of the re- formulated structure suggested by the REH. These issues originate from simply extending Muth's definition to more general models. Therefore, while Shiller emphasizes the difficulty of obtaining unique rational expectations solutions when agents' perceptions change in light of new information, he fails to note the problems stressed in the current study which are significant even when the information set is fixed.ZI 20Sh111er (1978), p. 33. 2IShiller's point is analogous to the questions raised by Lucas (1976) for economic models that did not contain expectations. Lucas emphasizes the difficulty in monitoring the effects of policy actions due to the constant changes in the structure of the system in light of the new information precipitated by the policy action itself. 103 6.6 Summary The applications of the REH which appear in recent litera- ture have taken on a number of different forms. However, to varying degrees, all studies neglect the issues which were emphasized in the development of the framework constructed in Chapters II, III, IV and V. These issues involved the specification of stability conditions which guarantee the existence of the solutions derived in the Model II and Model III frameworks, the process assumed to gen- erate the exogenous variables, and taking advantage of the specific functional form of the reformulated structure to fully realize the econometric implications of the REH. In spite of this common theme, past researchers have been led to de-emphasize these issues by following diverse approaches to the theory of rational expectations. Some have overlooked these problems by failing to specify the steps leading to a RES (Haley, McCallum). Others have confined the scope of their analyses to simple models; thereby, not realizing the theoretical and econo- metric implications of applying the theory to more general models (Sargent and Wallace (1973), Lucas, Turnovsky, Wickens). Still others confront models with such severe parameter restrictions that all the difficulties of obtaining general solutions vanish (Sargent and Wallace (1975), Muth). Finally, some articles deal with such Since rational expectations are conditional forecasts of the relevant theory, this constant evolution of the structure of economic models poses complications for the theory of rational ex- pectations. Shiller's analysis focuses upon some of these prob- lems. 104 a broad scope that other difficulties of coping with the REH take precedence over the issues raised in the present framework (Shiller, Taylor). Therefore, the general approach to the REH provided in the present framework provides insight which may not be gained by analyzing individual studies. This framework, which accommodates many types of specifications, provides the foundation for determining the validity of the theory of rational expectations as an explanation for the perceptions of economic agents. CHAPTER VII SUMMARY AND CONCLUDING COMMENTS The goal of this study is to examine the implications of the theory of rational expectations as an explanation of indi- viduals' perceptions of future events. The investigation con-a centrates upon construction of a framework for applying the REH to various economic models, examination of the econometric implications of incorporating the REH into a structural model, and briefly reviews the manner in which rational expectations have been employed in recent studies. The framéwork provides an analysis of three categories of models (Model I, Model II, and Model III) which contain rational expectations formed on the basis of all the information available I'prior to the current period. Model I refers to those structures which contain expectations of current period values of endogenous variables. Models which allow expectations of a finite number of future periods to appear as explanatory variables, but omit lagged endogenous variables, are conSidered in the analysis of Model II. Model III structures possess both multi-period future expectations and lagged endogenous variables. The framework follows a uniform format in its investigation of all of these structures. First, the expectations are equated with conditional forecasts from the 105 l06 model in question to obtain observable expressions for the ex- pectations terms. These expressions are referred to as rational expectations solutions (RESs). Second, these RESs are substituted into the original model to obtain a structure which no longer con- tains unobservable variables. Several significant implications of the REH are revealed in this framework. First, restrictions on the parameters of the structural model are necessary to insure the stability of the solutions. Second, a RES will invariably be dependent upon the nature of the process assumed to generate the exogenous variables in the model under investigation. Finally, the observable struc— ture obtained in the analysis is a function of the predetermined variables in the model and the variables inherent in the exogenous variable generating process. Moreover, the coefficients of this expression are known functions of the parameters in the original model under investigation. ' The econometric implications of the REH are,realized by pursuing three issues which arise due to the particular functional form of the observable structure suggested by the REH. The study examines the conditions which insure that the original reduced form Coefficients can be identified from knowledge of those in the observable reduced form. An estimation procedure which accounts for the functional form of the observable structure is develOped. Finally, the analysis reveals the restrictions implied by this reformulated observable structure and outlines a procedure for testing the theOry of rational expectations. 107 The survey of current literature confirms the import of the present analysis since the significant issues discussed through- out this study and briefly summarized above are notably absent from many previous applications of the REH. The literature review accentuates this omission and offers explanations for the de- emphasis of some of these important aspects of the theory of rational expectations. In the final analysis, this study provides a number of significant contributions to the continuing study of the REH as an explanation for the expectations of economic agents. The framework of Chapters II, III, and IV accentuates the problems of specifying rational expectations solutions which must be con- sidered in any application of rational expectations. Moreover, the complexity of these solutions illustrates how difficult it may be to formulate a general theory which explains the process whereby agents obtain enough information to form "rational" ex— pectations. Also, the framework establishes guidelines for the uniform interpretation of the REH in all economic models. This may eliminate potential disagreements about what is meant by assum- ing expectations are "rational" in the sense of Muth. Finally, this study provides the details for a test of the theory of rational expectations, thereby establishing a method for deter- mining the validity of the hypothesis as an explanation of indi- vidual's perceptions of future events. LIST OF REFERENCES LIST OF REFERENCES Bard, Y. 1974. Nonlinear Parameter Estimation, pp. 61-71. New York: Academic Press. Box, G.E.P., and Jenkins, G.M. 1976. Time series analysis: forecasting and control, pp. 126-70. San Francisco: Holden-Day. Cagan, P. 1956. The monetary dynamics of hyperinflation. In Studies in the quantity theory of money. Edited by Milton Friedman. Chicago: University of Chicago Press. Fisher, I. 1930. The theory of interest rates. New York: Macmillan. Friedman, M. 1968. The role of monetary policy. American Economic Review 58:1-17.. Hadley, G. 1964. Nonlinear and dynamic programming, pp. 47-9. Reading, Massachusetts: Addison-Wesley Publishing Company, Inc. Haley, N.J. 1976. An empirical investigation into price adjust- ments in the cherry industry. Journal of Canadian Agricultural Economics 24:57-63. Hammarling, S.J. 1970. Latent roots and latent vectors, pp. 153-4. Toronto: The University of Toronto Press. Kmenta, J. 1971. Elements of econometrics, pp. 430-51. New York: Macmillan. Lucas, R.E. 1970. Econometric testing of the natural rate hypo- thesis. In The econometrics of price determination con- ference. pp. 50-9. Edited by Otto Eckstein. Washington: Board of Governors of the Federal Reserve System. . 1972. Expectations and the neutrality of money. Journal of Economic Theogy 4:103-04. . 1976. Econometric policy evaluation. In the Phillips curve and labor markets. Edited by Karl Brunner. Supple- ment to the Journal of Monetary Economics 2:19-46. 108 ~ 109 McCallum, B.T. 1972. Inventory holdings, rational expectations, and the Law of Supply and Demand. Journal of Political Economy 80:386-93. . 1974. Competitive price adjustments: an empirical study. American Economic Review 64:56-65. . 1975. Rational expectations and the natural rate hypo- thesis: some evidence for the United Kingdom. Ihg_ Manchester School 43:56-67. . 1976. Rational expectations and the natural rate hypo- thesis: some consistent estimates. Econometrica 44:43-52. . 1977. The role of speculation in the Canadian exchange market: some estimates using rational expectations. Review of Economics and Statistics 59:145-51. Muth, J.F. 1961. Rational expectations and the theory of price movements. Econometrica 29:315-35. Nelson, C.R. 1975. Rational expectations and the predictive efficiency of economic models. Journal of Business 87: 331-43. Ramsey, J.B. "Graduate economic theory." Department of Economics, Michigan State University. Processed-1976. Sargan, J.D., and Sylwestrowicz, J.D. 1976. "A comparison of alternative methods of numerical optimization in estimating simultaneous equation econometric models." London School of Economics. Sargent, T.J., and Wallace, N. 1973. Rational expectations and the dynamics of hyperinflation. International Economic Review 14:328-50. . 1975. Rational expectations, the optimal monetary in- strument and the Optimal money supply rule. Journal of Political Economy 83:241-55. . 1976. Rational expectations and the theory of economic policy. Journal of Monetary Economics 2:169-84. Shiller, R.J. 1978. Rational expectations and the dynamic structure of macroeconomic models. Journal of MonetarygEconomics 4:1-44. Silvey, 5.0. 1970. Statistical inference, pp. 68-86. Baltimore: Penguin Books. 110 Taylor, J. 1975. Monetary policy during the transition to rational expectations. Journal of Political Economy 83:1009-21. Thei1, H., and Boot, J.C.G. 1962. The final form of econometric equations systems. Review of the International Statistical Institute 30:136-52. TUrnovsky, S.J. 1977. Structural expectations and the effective- ness of government policy in a short run macroeconomic model. American Economic Review 67:851-66. Wallis, K.F. "Econometric implications of the rational expecta- tions hypothesis." Paper presented at the Econometric Society European Meeting, Vienna, September 1977. Wickens, M.R. l976. "Rational expectations and the efficient estimation of econometric models." Essex University. Working paper no. 35. Wold, H.0. 1954. A study in the analysis of stationary time series, Chapter 3.‘_Uppsa1a: Almquist and Wicksell Zellner, A., and Palm, F. 1973. Time series analysis and simultaneous equation econometric models. Journal of Econometrics 2:17-53. APPENDIX FOR CHAPTER V The actual form of the restrictions generated by the REH may be examined by considering a number of simple examples. The distinction between type I and type II restrictions is noted and the number of restrictions obtained is compared with the restriction rule developed in Chapter V. A-1 A Description of the Examples Considered Following the notation of Chapter V: 1') ii) iii) iv) vi) m is the total number of equations and the number of expectations terms contained in each, n is the number of exogenous variables in each equation, n1 is the number of period-t exogenous variables in each equation, n2 is the number of these "1 period -t exogenous variables for which every lagged value, lag §.P, appears in the original RF, P is the order of the auto-regressive process gen- erating all of the exogenous variables, c; is the number of lagged values of the jth period t exogenous variable, j = l,...,n], lag §.P, which appear in the original RF; i.e. n] * zj=1 cj = k 1, 111 112 vii) k1 is the total number of lagged variables the original RF which can be obtained by lagging members of n], viii) k2 is the total number of lagged exogenous vari- ables that do not have corresponding period t values in the original RF. The examples which will be considered may be described: Ia: m = 1, n = n1 = 1, P = 1, ki = 0 for all i, Ib: m = l, n = n1 = 2, P = l, ki = 0 for all i, Ic: m = 1, n = n1 = l, P = 2, ki = 0 for all, i, IIa: m = 2, n = n1 = 2, P = 1, k1 = 0 for all i, IIb: m = 2, n = n] = 3, P = l, ki = 0 for all i, IIc: m = 2, n = n1 = 2, P = 2, k1 = 0 for all i, IId: m = 2, n = r1 = 1, P = 2, k = 0 for all i, III: m = l, n = 4, n1 = 3, n2 = 1, k1 = 3, k2 = l, k = 1, P Two additional comments clarify the following analysis. First, the disturbances in all the models considered are assumed to be individually and identically distributed normal -- the standard disturbance assumption. Also, the notation employed in the single equation examples corresponds to the symbols used to denote structural parameters in Chapters II, III, IV, since the reduced form and structural model are indistinguishable in the single equation case. 113 A-2 REH Restrictions in Simple Theoretical Models EXAMPLE I Ia: Assume the ”relevant theory" may be expressed as: e yt th + eyt + 5t: (Ia) where; m = 1, n l, assume P = 1, Applying the REH to (Ia), obtain the resulting RES: -1 -1 E y = (1 - e) B E x = (1 - e) Byx . t-l t t-l t t“ - Hence, the ORF is: yt = th + 6(1 - 9)-]BYXt-1 + at. (Ia-ORF) The unrestricted version of this structure is: yt = mlxt + ¢2xt_] + at. (Ia-ORF-HA) Clearly there are no restrictions implied by the application of the REH to Ia since the number of RF parameters in (Ia-ORF) equals the number of coefficient estimates obtainable from (Ia-ORF-HA). However, the RF parameters are-identified from knowledge of (Ia-ORF-HA) since: u: u cl’1 6 = ($1Y + ¢z)-1¢2 where y can be obtained from (AR-l). 114 Ib: Assume the relevant theory of Ia is altered by adding an exogenous variable: y = B x + B x + eye + 8 (lb) t 1 1t 2 2t t t where; m = 1, n = 2, assume P = 1, xlt = lelt-l I ”it ; th ‘ Y2x2t-l T ”2t ° Applying the REH to Ia, obtain the resulting RES: : _ -1 tEIYt ‘1 9’ (Bllelt-l + 82Y2x2t-1)° Hence, the ORF is: _ -1 yt ‘ lelt I B2x2t I 9(] ’ e) (Blylxlt-l + 82Y2x2t-1) + 8t' (Ib-ORF) The unrestricted version of this structure is: Vt ‘ ”lxlt I szzt T P3Xit-i + P4X2t-i + at' (Ib‘ORF'HA) The numberof coefficient estimates from (Ib-ORF-HA) exceeds the number of RF parameters. The restriction may be revealed by equating the coefficients in the two structures: 41 = 8]. m2 = 82. $3 = 6(1 - 9)-1B]Y1, 44 = 9(1 - 9)-]BZY2- v v v Clearly; -—3- = —1--—1-. P4 32 Y2 115 There may be a number of ways to express this single restriction. The restriction is a type I, or overidentifying, restriction since it stems from the fact that this four equation system contains two independent solutions for e in terms of the ”i and vi: -1 I (PiYi I P3) P3 —J ‘0 CD I and 9 = (@2Y2 I $4)-1W4° Ic: Assume the relevant theory is (Ia) but the process generating the exogenous variables is changed: _ e yt ‘ th I 6yt I et 1; but now P = 2, where; m = 1, n I°e° Xt I Illxlt-l I Yizxit-z I ”t /’ (AR'Z) Applying the REH to Ic, obtain the resulting RES: _ '1 Hence, the ORF is: - -1 _ The unrestricted version of this structure is: yt I q’i"it I ¢2x1t-1 I q’3"it--2 I Et' (IC'ORF‘HA) Following the procedure outlined above, equate the coefficients in (Ic-ORF) and (Ic-ORF-HA) to reveal the restriction: 116 W]: BI, W2 ‘ 8(1 ‘ GI-IBIYII’ '1 W3 T 9(1 ' 9) B1Y12' 32.. 311. Clearly; . I3 Y12 This restriction may be classified as a type II since it is obtained from the fact that the ORF coefficients on the lagged values of the single exogenous variable differ only by a factor of ., i = 1,2. Yli EXAMPLE II IIa: Assume the relevant theory is: _ e - e ylt ‘ "iixit I "izxzt I "13y1t I "14y2t I Vlt’ - e e y2t ‘ "lelt I Izzxzt I "23y1t I "24’2t I Vzt I N U 3 N N U .0 N ...—a H where; m - I°e' Xlt I lelt-l I ”lt’ th I szzt-i I “2t' Applying the REH to obtain a RES: tflylt (‘ ‘ "24) "14 I "11 "12 -1 = 0 Elth 1‘23 (I ' "13) "21 "22 . ‘1 - where, D - (I - n24)(1 - n13) - n23n14. (IIa) lelt-l szzt-i 117 1 tflylt I D {(1 ‘ "24)"11 I "14"21}Y1X1t-1 -1 I D {(1 ‘ I24)"12 I "i4I22IY2X2t-i’ = -1 - E th D I"23"11 I (I "13)"21IY1x1t-1 t 1 -1 I D {I23Iiz I (I ' "13)"22}Y2x2t-1' The resulting ORF is: yit I "iixit I "izxzt + D-I{n13((1-n + ) + 24)"11 Ii4I21 "14("23Iii I (I‘I13)I21)}Yixit-i -1 I D {"13((I‘I24)“12 I "i4I22) I "14("23"12 I (I' "13)"22)IY2x2t-i I Vlt’I th I "lelt I Tr22X2t -l I D {"23((I'"24)"11 I "14I21) I 1T24("23Iii I (I‘I13)"21Ile1t-l 1 I D I"23((“I24)I12 I "14"22) I Tr24("23Iiz I (I‘Ii3)I22}Y2x2t-i + V2t. The unrestricted version of this structure is: ylt I Piixit I opizxzt I Piaxit-i I cPiaxzt-i I Vlt’ th I Izixzt I I22x2t I stxit-i I 324x2t-i I Vlt' Equating the coefficients in the restricted and unrestricted ver- sions obtains: 118 Xit‘ I11 ‘ "11 x2t‘ I12 I "12 . _ -1 - xlt-l' 313 ’ D ("13((I'IZ4ID11 I "14"21) I Tr14("23"ii I (I'Ii3)"2i)}Yi . -‘I . x2t-i' 914 ‘ D {"13III'I24)"12 I I14I22) I 1T14("23"12 I (I'Iis)“2z)}Y2’ xit‘ 921 I "21 x2t‘ Ip22 I “22 ° = -1 - xlt-l' q’23 D ("23((I "24)"11 I "14”21) I 1T24("23Iii I (I'D13)"21)}Y1 , - -1 x2t-1' I’24 ' D {"23((I‘"24)"12 I "l4"22) I Tr24("23Iiz I (I'I13II22)}Y2’ Opii I "11’ 912 I "12 GPi3 I {c11D11 I chIZlIYl D14 I {C11"12 I c12I22IY2 C11 I ("13(I ‘ "24) I "14"23ID.1 c12 I "140-1 q’21 I "21’ 0P22 I T‘22 up23 I (c21"11 I C22"21)Yi q’24 I (CaiIiz I c22"22)Y2 c21 I "23 c22 I ("23"14 I "24‘1 ‘ “23))D-1° Hence, the eight unrestricted ORF coefficient estimates can be ex- pressed in terms of eight RF parameters. Therefore, no restrictions eixst. However, the RF parameters can be identified from these eight equations; n13, n14, n23, «24 can be obtained from the solutions for c1], C12, c2], c22. 119 IIb: Assume Case a) is altered by the addition of one exogenous variable; m = 2, n = n1 = 3, P = 1 the relevant theory is: = ‘ e e ylt "10 x0t I "llxlt I Iizxzt I "13y1t I "14y2t I Vlt (IIb) = e e y2t "zoXOt I "zixit I 1T22x21.- I "zayit I "24Y2t I V2t° Following the procedure from IIa, obtain a RES and resulting ORF. Then equate the coefficients with an unrestricted ORF. Hence, for the coefficient on; XDtI qpio I Trio "11 "12 1 Xit‘ q’11 x2tI IP12 x0t-1I GPi3 I D- {I13II‘"24)"10 I "14Izo) I "14("23IIO I (I'"13)"2o)IYo xlt-1I $14 ' 0-]{D13II'"24)"11 I "14"21) I "14("23Dll I ‘1'"13II21)I71 , -1 x2t-1' Dis ‘ D {I13II'I24)“12 I "14I22) I "14("23Iiz I (I‘Ii3)I22)}Y2’ XOt‘ P20 ‘ 1T20 xltI cp21 I "21 xzt‘ 1’22 I 1T22 x0t-i‘ IP23 I D ]{"23((I’"24)"10 I "i4“2o) I Tr24(I23Iio I (I‘Ii3)I20)IYo . g ‘1 - xlt-l' q’24 D {"23IIIID24II11 I I14I21) I "24("23"11 I (I "13)"21)IY1 . g ‘I - x2t-1‘ IP25 D {"23((I'"24)"12 I “i4"22) I 1T24(I23Iiz I (I "13)"22)II2’ 07‘; "10 ’ opii I "ii ’ op12 I "12 (CiiIio I c12"20)Io I10 cPi3 120 914 I (CiiIii I ci2I2i)Yi q’15 I (c11D12 I ci2"22)72 _ -l cii ‘ D ("13(I ‘ I24) I "i4I23) - -1 C12 ‘ D ("14) 920 I "20’ 921 I "21’ fiz I Tr22 923 I (c21"10 I c22I20)Yo 924 I (c21Dll I C22I2i)Yi 925 I (c21"12 I c22"22)Y2 where; -1 C2i D ("23): _ -1 c22 ‘ D ("23"14 I "24(IIIi3))' Eliminating C11 and c12 from the first system reveals: 913 GPii Yi ‘ 914 fio Y0 = II’13 opiz Y2 ‘ opis q’io Yo Pii op20 Yi '.fio q’21 Yi 912 320 Y2 ' 310 322 Y2 Similarly, eliminating c2] and c22 from the second system yields: 323 Iii Yi ' I24 I10 Y0 = OP23 qpi2 Y2 ‘ 325 Rio Yo . Pii i’20 Yi ’ CI’io I21 Yi 912 I’20 Y2 ‘ Irio W22 Y2 These two restrictions are type I, or overidentifying, since there are three independent equations for C11 and c12 and three independent equations for C2] and c22' As a result there are six independent expressions for four of the RF parameters n13, n14, n23, n24 -- resulting in two restrictions. 121 11c: Assume the "relevant theory” is IIa but the processes generating the exogenous variables are altered: Xit I "llxlt-l I Iizxit-z I Dlt’ th I I2ix2t-i I "22x2t-2 I D2t' Again, following the previous format, obtain a RES and ORF. Then equate the coefficients with an unrestricted ORF to obtain: For; xit‘ Iii Iii xzt‘ Iiz I12 -1 D {Ii3(("I24)IiiIIi4I21 "14("23"11+(I Ii I2i)}Yii . -‘I ‘ x1t-2' I’i32 ' D {Ii3III'I24)IiiIIi4I2i -l Xit-i‘ Ii3i )II ) th-i‘ Ipi4i I D {I13III‘I24)I12IIi4I22) ) 4.1'. II Ii4(I23I 22I("Ii3 I22)}I2i -1 31f) Ii4(I 23 IiiI(I'Ii3) I21)}I12 )W xzt-z‘ I142 I D {"13IIIIDZ4)"12+"14"22 IIi4II23IizIII'I13)I22)IY22’ x2tI II22 I I22 ’ . ,‘I - x1t-1‘ Ip231 D {I 23((I‘I24II 11+"14"21)+"24("23"11I(I I13)I21)}Iii xit-2‘ I232 I D :II23III'I24II 11+"14"21)+"24("23"1l+(1'"13)"21)IY12 x2t-2‘ I’24i I D {I23I(I'I24)I12IIi4I22)II24II23Ii2III‘Ii3)I22)II21 . -‘I - x2t-2' I242 ' D ”23‘(1"I24)Ii2IIi4I22)II24(I23I12I(1 I13)I22)}Y22’ 07‘; Iii I Iii q’i2 I Iiz @131 : {CIIWII + C12W21}Y11 C11, C12 dEfIHEd as in bOth q’132 I IciiIii I ci2I2iIIi2 III and 11b 122 I141 I IciiIiz I ci2I22IY2i I142 I {C11D12 I C12I22IY22’ Izi I I21 I22 I I22 I23i I Ic21"11 I C22IziIYii $232 = {c21w11 + C22w211v12 c2], c22 defined as in both $24] = {C21W12 + c22n221v21 11a and 11b. I242 I {CziIiz I C22I22IY22 ° I Y 4 Y Clearly; _1§1.= .11.; .111.= .21.. I132 712 I142 I22 ’ cp232 I12 I242 Y22 As before there are numerous ways to express the four independent restrictions implied by this system. The restrictions originate from the similarity (up tofla factor of Yij) of the coefficients for the lagged values of each of the exogenous variables in each equation. Analogous to Ic, these restrictions are not over- identifying. Although there are eight equations in the four un- knows, (c11, c12’ c2], c22), it is obvious that only four of these are independent. Consequently, the values for n13, n14, n23, n24 may be obtained from exactly four expressions which link the cij with unrestricted ORF coefficients. IId: Assume the relevant theory now contains only one exogenous variable -- but the process generating it is second order: 123 - e e It ‘ Iiixt I Iiayit I "14y2t I Vlt’ y = n x + n ye + w ye + v t 21 t 23 1t 24 2t 2t' Once again, following the REH and comparing the resulting ORF with an unrestricted version obtains: For; t‘ ' I11 I "ii = -1 - D {"13((‘ I24)IiiIIi4I21)IIi4II23IiiIII‘I13)I21)IIii , _ -l xt-2' I122 ‘ D {"13III'I24)"11+"14"21)ID14("23"11+(I'D13)"21)IYlZ’ xt-i‘ Iizi Xt I21 I I21 , _ -1 xt-l' I221 ' D {I23III'I24)IiiIIi4I21)II24II23IiiIII'Ii3II2i)IIii ' = -1 - - xt-2' I222 D {"23“1 "24)"11ID14DZl)+"24("23"11+(I I13)I21)}Iiz° ¢ Y W C]ear]y; ijzl=_ll.=._ggl . 122 Yi2 I222 These two restrictions, like those in IIc, are type II. This is especially clear in this case since the values of «13, n14, n23, "24 cannot be identified from knowledge of the ORF coefficients in the above system. Despite the existence of six equations and six RF parameters, there are clearly only four independent equations in the above system. Nevertheless, the REH does generate testable re- strictions in this case. Example III Assume the "relevant theory" contains lagged variables: yt I Dlxlt I szzt I B3X3t I B4X2t-i I 85x3t-1 I D6X3t-2 e I D7X4t-l I Iyt-i I Dyt I 8t‘ m=1,n1=4,n1=3,k1=3,k2=13k=1. Assume all exogenous variables are generated by second order pro- (2855852 I I 2’ xit I Yiixit-i I Iizxit-z I Dit’ x2t I Yzixzt-i I Y22x2t-2 I Dit’ x3t I Y3iX3t-i I Y32x3t-2 I Dit’ x4: I Y4ix4t-i I Y42x4t-2 I ”it“ n2 = 1 Apply the REH to obtain the resulting RES for this model: E _ -l t 1y: ‘ (I‘D) {BlYllxlt-l I BiYizxit-z I 82Y21x2t-i I B2I22X2t-2 I DsIsixat-i I 83*32X3t-2 I B4X2t-i I B5x3t-i I szat-z I 87x4t-l I Iyt-iI ° Combining terms and substituting the RES into the relevant theory yields the ORF: yt I Dlxlt I szzt I D3X3t )‘I + [v + 9(1 - e vlyt_] I GEIIDJ-IIDlyllxlt-1 I BiIizxit-z I 82Y22x2t-2} + [34 + 9(1 - e)'I(82v21 + 84)Jx3t_1 + [85 + 9(1 - 9)-I(83v3] + 85)]x3t_1 I [35 I 9(‘ ' 9)-1(83I32 I 86)]x3t-2 + £87 + 9(1 - e)‘Is7Jx4t_1 + It' In the notation of section 5.2, the coefficients on; 125 x x x corres ond to 0' 1t’ 2t’ 3t D ”1’ -1 . yt_1 - correspond to (I - n3) n2(L), _ * XIt-I’ XIt-Z’ X2,t-2 correspond to n3(I - n3) In?F (L); since III] + Y12L 0 0 0 * 0 Y L 0 0 0 0 0 0 3 0 0 0 0 and, I _ ** x2t_], x3t_], X3t-2’ x4t_1 correspond to n3(I - n3) Ingr (L) + (I - n3)“n:(L). Therefore, following the notation used in the general example from Cahpter V, page 60, the (ORF-HA) for this example is: It I Ip0ixit I Ioéxzt I I02x3t I Ith-i I I21xit-i I I22x2t-2 I Izaxat-z I IP3iX2t-i I I32x3t-1»+;I33x3t-2 I I34x4t-i I It where; the unrestricted ORF coefficients; $01: i = 1,2,3 apply to period t exogenous variables; o1 apply to lagged endogenous variables; ¢2i’ i = 1,2,3 apply to lagged values whose coefficients appear in r*(L); I3i’ i = 1,2,3,4 apply to lagged values which appear in the original RF. To examine the conditions for identification and reveal the REH restrictions, equate the restricted and unrestricted ORF coefficients: 126 I01 I 8i I02 I 82 I03 I 83 m1 = [v + e(l - e)'Iv3 921 = 9(1 - 9)-181Y11 $22 = 6(1 - e)'Ie1v12 923 = 9(1 - 6)'182722 v3] = [84 + 9(1 - e) 1(82v21 + 84)] $32 = [85 + 6(1 - 6) 1(8373] + 85)] 433 = [86 + 9(1 - a)' (83v32 + 86)] $34 = [87 + 6(1 - e)-IB7J The condition for the identification of the nine RF para- meters from the relevant ORF coefficients is satisfied since; In the example above, 6 corresponds to n3. Clearly, a solution for e, in terms of p and v , can be obtained from any of the expressions, oz], 922: 923- This solution may then be substituted into the expressions for m], 931» i = l,...,4 to identify the remaining RF parameters. The restrictions implied by this structure are: '8 N .4 —-l 1 22 1 22 39.1. lu..I21 I02 Y22 23 .e .4 The restrictions are obtained by analyzing the expressions for 127 pg], 922: $23 which are the coefficients contained in the ex- pression analogous to (5.2.5) for this example. The first restriction is type I, or overidentifying, since the system contains ten independent equations and nine RF para- meters. The second restriction is obtained by noting the similarities of the coefficients on lagged values of xlt’ hence, it is a type II. A-3 Comparing the REH Restrictions with the Rule of Chapter V The rule for counting restrictions is: Type I = m - Max{n1 - n2 - m, 0}, n I * Type II = m X Max{P - c. - l, 0}, .=1 3 J n I * and R = m{ X {P - c.} - Min(n1 - n2, m)}. i=1 3 * Both n2 and cj equal zero for examples I and II due to the absence of lagged variables. Example Ia: m = l, n = n1 = l, P = 1, type I = 1 Max{D, 0} = 0. type II = l Max{O, 0} = 0. R = l{l - Min(l, 1)} = 0. Rule agrees with restrictions obtained. 128 type I = l{Max(1, 0)} = 1. type II = l{Max{0, O} + Max{D, 0}} = 0. R = 1{1 + 1 - Min(2, 1)} = 1. Rule agrees with restrictions obtained. type I = l{Max(D, 0)} = 0. type II = l{Max(1, 0)} = l. R = 1{2 - Min(l, 1)} = 1. Rule agrees with restrictions obtained. IIa: m = 2, n = n1 = 2, P = 1, type I = 2{Max(0, 0)} = 0. type II = 2{Max(0, 0) + Max(0, 0)} = 0. R = 2{1 + 1 - Min(2, 2)} = 0. Rule agrees with restrictions obtained. IIb: m = 2, n = n1 = 3, P = 1, type I = 2 - Max{1, 0} = 2. type II = 2(Max(0, 0) + Max(0, 0) + Max(0, 0)) = 0. R = 2C1 + l + l - Min(3, 2)] = 2. '. Ru1e agrees with restrictions obtained. IIc: m R = 2, n = n1 = 2, P = 2, type I = 2 - Max{O, 0} = 0. type II = 2{Max(1, 0) + Max(l, 0)} 4. 2(2 + 2 - Min(2, 2)} = 4. Rule agrees with restrictions obtained. 2. = 2, n = n1 = 1, P = 2, type I = 2 - MaxI-l, 0} = 0. type II = 2 ~ {Max(l, 0)} R = 2 - {2 - Min{1, 2}} = 2. Rule agrees with restrictions obtained. III: m 3 O wimfl'dfl-N O 0 type I type II R 1; P,:'2’ 4 : x],x2,x3,x4. 3 Z X-I.X2,X3. 1 : x3. 0. l. 2. 1 - Max{3 - l - l, 0} = l - Max{2 - 0 - 1, 0} + 1 - Max{2 - 1 - 1, 0} + l . Max{2 - 2 - 1, 0} = 11(2-0) + (2-1) + (2-2) Ru1e agrees with restrictions 1. - Min(2,l)} = 2. obtained. "I7'11IIIIIIIIIIIIII