_. . «-—..._.._..,..’ THE TERM STRUCTURE OF TNTEREST RATES. THE EXPECTATIONS HYPOTHESIS, AND THE FORMULATION ‘ 0F EXPECTED INTEREST RATES Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY FRANK J. BONELLO 1968 LIBRARY Michigan State. nivetsity This is to certify that the thesis entitled The Term Structure Of Interest Rates, The Expectations Hypothesis, And The Formulation Of Expected Interest Rates presented by Frank Bonello has been accepted towards fulfillment of the requirements for flak/sweat” Major profess / \ DatJZ/(féé/ /é/7éf Kfl ‘ ABSTRACT THE TERM STRUCTURE OF INTEREST RATES, THE EXPECTATIONS HYPOTHESIS AND THE FORMULATION OF EXPECTED INTEREST RATES by Frank J. Bonello In several empirical investigations into the EXpec- tations Hypothesis of the term structure of interest rates, alternative theories concerning the formulation of expected future interest rates or, more briefly, expecta- tions mechanisms are employed. This study is an investi- gation of these expectations mechanisms. Conceptually three categories of mechanisms are established: the regressive, the extrapolative, and the cyclical. Opera- tionally two additional categories are necessary: the error learning mechanism and the combined regressive- extrapolative mechanism. It is established that these five types of mechanisms constitute unique and, therefore, competing hypotheses concerning the formulation of expected future interest rates. It is demonstrated that the impli- cations of the mechanisms are contingent upon the defini- tions of the exogenous variables employed in each of the mechanisms and upon the types of changes in these variables. The mechanisms are tested against the one-, three-, five-, seven-, and nine-year forward rates derived from yield curves for fixed maturity U. S. Government securities for the 1953-1967 period. Both regression analysis and a |lllll|ll Frank J. Bonello t’" newly devised set of specification error tests are utilized } in the empirical analysis. A criterion is established for ‘ the selection of the "correct" mechanism for each set of forward rates: a mechanism is the "correct" mechanism for a particular forward rate if it is the only mechanism free of specification error and consistent with the data. The empirical results reveal that none of the mechanisms tested represents the "correct" mechanism for the one-, three-, five-, and seven-year forward rates. In each of these instances there are at least two mechanisms which are free of specification error and consistent with the data. It appears that this latter result is due to an independent variable which is common to several of the mechanisms. In the case of the nine-year forward rate all the mechanisms except one are misspecified. The correctly specified mechanism, an extrapolative mechanism in which the recent trend in rates is defined as the difference between the current nine-year rate and the previous period's nine-year rate, is also consistent with the data. Consequently this mechanism is selected as the "correct" mechanism for the nine-year forward rate. THE TERM STRUCTURE OF INTEREST RATES, THE EXPECTATIONS HYPOTHESIS, AND THE FORMULATION OF EXPECTED INTEREST RATES By Frank J. Bonello A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1968 saw»? Copyright FRANK J. BONELLO 1968 ii ACKNOWLEDGMENTS I wish to acknowledge the assistance of my guidance committee which included Professor Jan Kmenta, Professor Paul Smith, and Professor William Russell. Professor Russell, who acted as chairman of the committee, was a constant source of encouragement and without his theoret- ical contributions this study would never have even been begun. I also wish to thank Professor James Ramsey who devised the specification error tests and provided me with a COpy of the computer program that constitutes the source of all the empirical results presented here. Professor Ramsey also answered several statistical questions for me. I must also acknowledge the programing done for me by Gerald Musgrave; his patience was unlimited. Jeffery Roth was also kind enough to answer several programing questions for me. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . . . . . . . . iii LIST OF TABLES . . . . . . . . . . . . . . v Chapter I. INTRODUCTION . . . . . . . . . . . . 1 II. THE TERM STRUCTURE OF INTEREST RATES, THE EXPECTATIONS HYPOTHESIS, AND DECISION- - MAKING UNDER UNCERTAINTY . . . . . . . 5 III. EXPECTATIONS MECHANISMS . . . . . . . . 21 IV. LONG-RUN AND SHORT-RUN IMPLICATIONS OF THE MECHANISMS . . . . . . . . . "9 V. EMPIRICAL ANALYSIS . . . . . . . . . . 71 VI. CONCLUSIONS AND IMPLICATIONS FOR FURTHER RESEARCH . . . . . . . . . . 10“ APPENDIX A Review of Empirical Investigations into the Expectations Hypothesis . . . . . . . . 109 BIBLIOGRAPHY OF WORKS CITED . . . . . . . . . 132 iv Table l. 10. ll. l2. 13. LIST OF TABLES Conditions for Types of Expected Changes in the n-year Rate . . . . . . . Sources of the Various Types of Changes in Rates . . . . . . . . . . . General Equations, Alternative Definitions, and Specific Regression Equations for the Tested Mechanisms . . . . . . Regression Results for the One-year Forward Rate . . . . . . . . Results of the Specification Error Tests for the One-year Forward Rate . . . . Regression Results for the Three-year Forward Rate . . . . . . . . . Results of the Specification Error Tests for the Three-year Forward Rate . . . . Regression Results for the Five-year Forward Rate . . . . . . . . . Results of the Specification Error Tests for the Five-year Forward Rate . . . . . Regression Results for the Seven-year Forward Rate . . . . . . . . . Results of the Specification Error Tests for the Seven—year Forward Rate . . . . Regression Results for the Nine—year Forward Rate . . . . . . . . . Results of the Specification Error Tests for the Nine-year Forward Rate . . . . . Page 67 69 83 89 9O 93 9A 96 97 99 100 102 103 CHAPTER I INTRODUCTION This study is an empirical investigation of interest rate expectations mechanisms. These mechanisms are devices employed or said to be employed by investors in forecasting the future values of interest rates. Expectations play a part in determining the current values of almost all economic variables.1 Generally, however, economic analysis ignores or relegates to a role of secondary importance the effect of expectations on economic behavior and examines the influence of other aspects of economic phenomena. One of several exceptions is the traditional theory of the term structure of interest rates where expectationally motivated behavior is considered as the determinant of the term structure. This traditional theory is, of course, the Expectations Hypothesis (EH). In recent years the term structure has been the subject of extensive investigation by economists and a number of attempts have been made to ascertain the validity of the EH. These empirical studies take either of two approaches. Under one approach evidence of successful lKenneth Boulding, Economic Analysis, Rev. Ed. (New York: Harper, l9A8), p. 832. forecasting is seen as the criterion for acceptance of the EH while under the second such evidence is not required.2 In the latter instance, however, there must be an indepen- dent specification of expected future interest rates. It is within this context that interest rate expectations mechanisms arise for they represent the devices that allow independent specification of expected future interest rates. To a large degree empirical verification of the initially develOped mechanism was accepted as verification of the EH,3 but in the past few years other mechanisms have been developed. A more current view is that any test of the EH which includes an expectations mechanism is, more properly, a test of that particular expectations mechanism.“ 2For a discussion of the problem of whether or not ,/ successful forecasting is a valid criterion for accepting “1 or rejecting the EH see: David Meiselman, The Term Structure of Interest Rates (Englewood Cliffs: Prentice Hall, 1962), pp. 6—17; J. B. Michaelson, "The Term Struc- ture of Interest Rates: Comment," Quarterly Journal of Economics, LXXVII (February, 1963), pp. 166-1745’Reu5en A. Kessel, The Cyclical Behavior of the Term Structure of Interest Rates (New Yofk: NationaITBureau OI:Economic Research, 1063), pp. 12-22. 3For an example of such an interpretation see Thomas E. Holland,"A Note on the Traditional Theory of the Term Structure of Interest-Rates on Three- and Six-Month Treasury Bills," International Economic Review, VI (September, 1965), pp. 330-336. “G. O. Bierwag and M. A. Grove, "A Model of the Term Structure of Interest Rates," Review of Economics and Statistics, XLIX (February, 1967), pp. 50-63. Although there has been a steady increase in the number of interest rate expectations mechanisms, as of this date there has been no attempt to compare the various mechanisms either theoretically or empirically. The major difficulty with theoretical comparison is a lack of stand- ards as to what constitutes a conceptually sound expec- tations mechanism. Several problems impede a comparison ...- 7 ._.- x, 11“ of the mechanisms on the basis of already completed tests. 5 These obstacles include differences in data as well as 6 This investigation will attempt differences in format. to overcome these difficulties by restructuring the mechanisms in such a way that they are all explaining the same set of dependent variables and by using a single set of data in the statistical analysis. The first step towards this end, Chapter II, is a discussion of the term structure, the EH, and decision making under uncertainty. This discussion is necessary 5The importance of differences in data is best illus- trated by extensive criticism of data employed by various investigators. The most detailed criticism of the Durand data, which have been used in several studies, is given by J. A. G. Grant, "Meiselman on the Structure of Interest Rates: A British Test," Economica, XXXI (February, 1964), pp. 51-71. A criticism of the data used by Grant is given by Douglas Fisher, "The Term Structure of Interest Rates: A Comment," Economica, XXXI (November, 1964), pp. AlZ-AlQ. 6The use of different formats had lead to confusion concerning the nature of the various investigations. Only very recently has it been recognized that the major differ— ences between various studies resides in the eXpectations mechanism employed; Bierwag and Grove, Op. cit., p. 60. because these topics represent the foundation common to the construction of each mechanism and constitute the basis for a correction of differences in format. In Chapter III categories of mechanisms are defined and descriptions of various mechanisms within each category are given. Chapter IV examines the long-run and short-run implications of the mechanisms under two alternative sets of assumptions. In Chapter V the tests used to compare the mechanisms are described and the results of these tests are presented. Chapter VI contains a summary and several implications for further research. CHAPTER II THE TERM STRUCTURE, THE EXPECTATIONS HYPOTHESIS, AND DECISION MAKING UNDER UNCERTAINTY The Term Structure The term structure of interest rates is the relation— ship of yields to maturity, or, identically, yields to redemption prevailing on securities of different maturity as of a moment in time.7 The yield to m aturity_ of a 9““— security is the discount rate which equates the payments w-h\ \ ..__.' dH/ made to the holder of the security, i.e., periodic interest payments and cash value at maturity, to the current market price of the security. In the discount formula: C C C F (l 1) Pi = 1+Rl) + (1+Ri)2 + ... + (1+: )1 + (1+: )1 i J/ A where Pi = the current market price of the i-year security, C1 = the annual interest payments made in years 1, 2, . . ., i, Fi = cash value at maturity of the i-year security, and Ri = the discount rate on the i-year security; 7 The restriction is usually added that the securities in the different maturity classes are generally alike; that is, there are no differences in default risk between the maturitylclasses or in call options. Differences in coupon rates of interests and redemption values are, however, allowed. the yield to maturity is equal to R1.8 The term structure \w/ is the relationship between the R 's where i represents the i maturity of various securities. The term structure is graphically represented by the yield curve. In constructing this curve, maturities are \_/ represented along the horizontal axis and yields along the vertical axis. Historically the yield curve has assumed three basic shapes: horizontal, ascending, and descending.9 In the latter two cases, the curve has assumed various rates of ascendency and descendency. Instances can be found of a single curve which combines two or all three of the basic patterns (e.g., humped or "U" shaped yield curves). Any theory of the term structure must be able to explain any possible shape of the yield curve, i.e, why a given rela- tionship of yields as of a moment in time. There are three major classes of term structure theories: the Unbiased Expectations Hypothesis (EH), Hicksian Liquidity PremIEmgMgdel, and the Segmented:Market lO Hypothesis. There are, to be sure, alternative formulations 8For a discussion of the computations involved in determining yields to maturity see Lester V. Chandler, The Economics of Money and Bankin , Ath ed. (New York: Harper and Row, 196A). pp. A2-A8 or Meiselman, op. cit., pp. 1-3. 9For a three- dimensional representation of past yield curves see Burton Malkiel, The Term Structure of Interest Rates (Princeton, N. J. Princeton University Press, 195551313 8-.9 10The three major classes of term structure theories have been given different labels by different writers; for example see Meiselman, op. cit., pp. 9—17; Kessel, op. cit., within a given class and even an approach which combines the EH and the Segmented-Market Hypothesis.11 The basic differ- ence between these competing theories is the raEionale which each provides for the demand and supply of securities.12 The EH postulates that the term structure is deter- mined by investops attempting to maximize their returns on the basis of expectations about the future level of interest -. ‘.__ rates; speculative activity dominates the securities market.13 As commonly understood this kind of investor behavior implies that the term structure is <2[in equilibgium when expected rates/of return from holdingvdefault free secur- ities are equal for every chosen interval of time. An e9 \ell expected holding period rate of return (EHPRR) is the ratio ,fl_k~jflmwflflg,fl of interest received during the holding period plus (minus) \ any discounted capital gain (loss) to the current market price of the security. According to the EH the equality of 3‘ pp. l-A; and Malkiel, op. cit., pp. 17-28. In this study the titles of the major c asses are selected on the opinion that these titles are the most self—descriptive. llF. Modigliani and R. Sutch, "Innovations in Interest Rate Policy," American Economic Review, Supplement, LVI (May, 1966), pp. 178-197. 12Perhaps not surprisingly the different theories can be used to explain the same term structure; see "Changing Structure of Interest Rates," Federal Reserve Bank of St. Louis Review, A9 (June, 1967), pp. 2-5. 13Joseph w. Conard, An Introduction to the Theory of Interest (Berkely: University of California Press, I959) especially Part III and F. A. Lutz "The Structure of Inter- est Rates," Quarterly Journal of Economics, LV (November, 1940), pp. 36- 63. The latter is reprinted in Readings in the Theory of Income Distribution, eds. w. Fellner and B. Haley. Philadelphia (Blakiston Co.), l9A9, pp. “99-529. EHPRREE is a necessary and sufficient condition for equil- ibrium. In the absence of equal EHPRR's, investors have a motive for switching--exchanging a security with a given maturity for a security with a different maturity--and, therefore, will engage in switching and thereby induce changes in yields. The Hicksian Liquidity-Premium Model rests on the premise that there is a "constitutional weakness" in the 1“ This "constitutional weakness" is an securities market. imbalance in the supply and demand for securities. It is argued that the imbalance arises because borrowers have a desire to borrow long in order to insure a steady avail— ability of funds while lenders prefer to restrict their activities to short-term securities in order to avoid risk. To induce lenders to enter the long-term market, borrowers would have to offer better terms in the long- term market, to offer an extra incentive to lenders to undertake a long-term obligation which is less liquid and, therefore, more risky. The amount of this extra incentive is the liquidity premium. According to this theory the normal equilibrium relationship is for long-term rates to exceed short-term rates; "normal backwarigiation?"exists.~ "flux ‘F‘fl The prOponents of the Segmented-Market Hypothesis recognize the existence of speculative behavior in the market for securities but maintain that non-speculative fl-H —_4w——‘ 1“J. R. Hicks, Value and Capital, 2nd ed. (London: Oxford University Press, 1946), pp. lAl-IUS. investmenpwdpmipapes the market and consequently determines the yield curve.15 Non-speculative investment is the selec- tion of a portfolio structure suited to the financial requirements of the investor. In support of this position the following arguments are often made. The securities portfolio of large institutional investors is usually either 'predominately,long-term or predominately short-term. This preference for either long-term or shert-term securities is determined by the natppemof the financial claims which the institution itself must meet. For example, because the claims against commercial banks are short-term, commercial banks will invest the majority of their funds in short- term securities. In the same way, because financial claims_L against lifg/insurance companies, pension funds, etc., are A long-term, the securities portfolio of these institutions‘ will be largely long-term. Secondly the same type of argument is applied to the issuance of securities. If the borrower issues a long-term security, it reflects a long- term need for funds; short-term securities are issued only“ when the need for funds is shOrt-term or temporary in In nature. On the basis of these two points it is argued that for both borrowers and lenders securities of different - maturities are not perfect substitutes and, therefore, 15J. M. Culbertson, "The Term Structure of Interest Rates," Quarterly Journal of Economics, LXXI (November, 1957), pp. ABS-517. ' lO securities of different maturities constitute non-competing -groups. The yield curve, under this hypothesis, is deter- mined by the pressures of supply and demand within each of the segmented markets. \‘ “ -—‘_ —\ The Expectations Hypothesis The usual interpretation of the EH is that long- term market rates of interest are a function of the current short rate and a series of expected short rates.l6 Specif- ically, the long-term rate plus unityzksthe geometric mean of the current short rate plus unity and a series of eXpected short rates plus unity with the averaging process extending over the maturity of the particular long-term security. Thus, (l + Rl,t) = (1 + trl,t) 2- (1.2) (1 + R2,t) - (1 + trl,t)(l + t+lrl,t) n— .0. (l + Rn,t) ‘ (l + trl,t)(l + t+lrl,t) (1 + t+n-lrl,t) 16 I Most of the discussion presented in this section Ctraws quite heavily from the work of William R. Russell which is; presented in a series of three Michigan State University E3(30nometrics Workshop papers. The three papers are: "The TWIeoretical Foundations of the Unbiased Expectations I‘iypothesisfl WorkshOp Paper No. 6505; "The Security Selection Approach to the Term Structure of Interest Rates," “kbrkshop Paper No. 6608; and "The Elastic Expectations' Nkbdels of the Term Structure of Interest Rates," WorkshOp Paper No 6505. ' 11 where Ri t = the actual i-year rate prevailing at time t ’ as calculated according to equation (1.1) where i > 0 and t+jri,t = the expected rate for an i-year seucrity startépgngpapipimpij based on expectations preva \Mpwwhere j >0 and i > 0. In this way the EHPRR for the first one—year period is the same for all securities and equal to the current one-year rate of interest. In order to insure the equality of the first one-year EHPRR for every security, the market rates of interest currently expected to prevail at the end of the year must be such that: (l + R1,t) = (l + trl,t) 2 - (l + R2,t) ‘ (l + tE1,t)(l + t+1Ei,t) I! V, (1.3) ‘ 3 _ 2 (l + R3,t) ‘ (l + t?1,t)(l + t+1r2,t) : :’ )/ . n _ . 2 n-l (l + Rn,t) ‘ (l T trl,t)(l + t+lrn-l,t) If equations (1.2) are to be consistent with equations (1.3), it is required that: (1.u) l + t+lrl,t = l + t+lrl,t 2 _ (l + t+lrl,t) ' (l + t+lr1,t)(l + t+2 l,t) o n-l - o . ... l + r (1 + t+1rn-l,t) ' (l + t+1rl,t)(l T t+2r1,t) ( t+11-1 ' 12 The difference between equations (1.2) and (1.3) is a difference in the specification of independent variables. In equations (1.2) the independent variables are the current u.“ . . ”4—--...-- short rate or, more generally, any current rate and the series of expected future one-year rates. In equations (1.3) \ \w-- w - the independent variables are a current rate and the team, A,_-— structure expected to prevail at the end of the year. Either \ __--- _ ” ’- .. -— .- ._. ‘5‘ specification is ESHéleeehe QIEhwthe sh: There is an addi- tional specification of the independent variables compatible with the EH. In this third case the independent variables are a current rate and a time series of expected consol rates. This time series of expected consol rates implies a set of expected short rates and is, therefore, equivalent to equations (1.2). These alternative sets of independent variables in conjunction with the EH all yield the same basic implication of equality of EHPRR's and are, therefore, empirically indistinguishable. Simply, the above discussion indicates that investors may attempt to forecast different sets of variables, i.e., to forecast a series of future short rates, the future term structure, or a series of future m.— fl consol rates, but the basic implicationfipf the EH is _ , WA; ' F:!.m‘~rw-.--—-I-—-”‘ I‘ve..- " hay-v unchanged. The EH itself says nothing about which set of variables is employed although the specification of a current rate and a series of expected short rates as the independent variables [equations (1.2)] is often taken, 13 as synonymous with the EH. Furthermore, the EH says nothing about how expectations are formed. In each of the three sets of independent variables one of the independent variables is a current rate. Without the knowledge of a current rate the absolute structure of rates cannot be determined. At best the given expected rates allow only the determination of the relative rate structure. This specification of a current rate as an independent variable is necessary because the partial equilibrium model of the term structure cannot determine the level of what the literature calls "the interest rate." The equilibrium structure of rates determined by the partial equilibrium model of the EH, however, must be in equilibrium with "the interest rate" or, more generally, the variables in the broader economic system. Thus, to insure equilibrium a current market rate is assigned an equilibrium value determined outside the model of the term structure. The current equilibrium term structure will change if either or _..~_.—.._. ...._. —-- -- -. both of the independent variables, the independent current rate or the set of expected rates, change. But the designation of the current term structure as an equilibrium term structure does not necessarily imply that the term structure will not change over time even if the independent variables are constant. Take equations (1.3) for example. 14 This set of rates implies that without a change in the independent variables the term structure at time t+l will be (1 + Rl,t+l) (l + t+lr1,t) 2 2 (1+R2,t+1) (l + t+lr2,t) (1.5) 3 (l + R3,t+1)3 = (l + t+lr3,t) n: n (l + Rn,t+l) (l + t+lrn,t) The term structures at times t and t+l will be the same if and only if (1 + t+1r1,t) z (1 + trl,t) 2 _ (l + t+lr2,t) (1+ tr1,t)(l + t+lrl,t) (1.6) 3 2 (l + t+1r3,t) = (l + 131,12”l + t+lr2,t) n _ . .., n-l (l + t+lrn,t) ‘ (l + trl,t) (l + t+lrn-l,t) If equations (1.6) do not hold then the two term structures will not be the same even though the independent variables are unchanged and both structures yield equal EHPRR's. Consequently a distinction may be drawn between permanent and temporary equilibrium term structures. The former satisfies equations (1.6) so that the term structure will change over time if and only if one or both of the 15 independent variables changes. A temporary equilibrium term structure does not satisfy equations (1.6) so that change can occur over time even though the independent variables are unchanged. In this case the movement is along an equilibrium path and if either or both of the independent variables change a new equilibrium path is defined. As was stated previously, there are three major classes of term structure theories and the rationale which ——_ m is. each provides for the demand and supply of securities m represents the basic difference between them. In most dise;eeions and in the remainder of this study these dif— ferences are narrowed to differences in the demand for securities. At this point it is necessary to demonstrate the various assumptions which can be made concerning the demand for securities so that the EH obtains. There are four alternate assumptipns which insure the implications of the EH. These alternate assumptions are: expectations are confidentlygheld, investors utility functions are lipear, a sufficient number of speculators exist, and the distribution of investor time horizons is matched by the maturity distribution of securities. To indicate the nature of these assumptions, it is beneficial to reduce the security selection problem to its bare essentials: the investor selects the security or securities which he feels will yield him the greatest utility. With this basic behavioral postulate, it is not surprising that 16 the first two assumptions restrict the utility function of all investors while the third states that there are enough investors with a particular type of utility function to cause the market to be a reflection of their behavior. With the assumption that expectations are confidently held, all characteristics of the securities except their EHPRR's become irrelevant. In effect the term structure is no longer a problem of decision making under uncer— tainty. If utility is a function of EHPRR's then the investor maximizes utility by choosing the security with the maximum EHPRR. If investor utility is a function of EHPRR's and the utility function is linear, again all characteristics of the securities except their respective EHPRR's become irrelevant and investors maximize their utility by choosing the security with the maximum EHPRR. With the third assumption speculators are defined as investors who are completely indifferent to all charac- teristics of securities exceptEHPRR's. If there are a suffiEient number of these investors, then any differential in EHPRR's arising in the market will be bid away and interest rates will move to those levels which yield equal EHPRR's for all securities. The argument for the fourth assumption follows from the fact that the security which has a term to maturity equal to the investor's time horizon is the security which 17 yields him a certain (i.e., definite) return. If differ- entials in EHPRR's occur in equilibrium, it must be true that at the market prices yielding equal EHPRR's there must be an excess demand for those securities with the lower equilibrium EHPRR. If it is assumed that investors are riskhaverters and thus prefer the certain return to the uncertain return, then the equilibrium term structure in which the EHPRR's are equal for all securities occurs when the terms to maturity of the issued securities matches the time horizon of investors. In summary then, there are three alternate sets of independent variables and four alternate assumptions con- cerning investment behavior consistent with the basic implication of the EH. Combining any of the sets of independent variables with any of the alternate assumptions is sufficient to insure equal EHPRR's. Decision Making Under Uncertainpy According to the EH the term structure is determined by investors attempting to maximize their returns on the basis of expectations about the future level of interest rates. Because this future level is unknown, security selection under the EH represents a problem of decision making under uncertainty. 18 The broad topic of uncertainty represents an area under which there is an ever-expanding body of literature. The contribution of economics to this literature may be considered as consisting of two parts. First there is the general problem of choice under uncertainty at the theoret- ical level or, more precisely, the attempts to establish the apprOpriate decision criterion within an uncertainty situation. The second part may be labeled uncertainty quantification and refers to the methods by which the values of the variables which enter the decision criterion may be calculated. At this time there appears to be something approaching a consensus of Opinion concerning the apprOpriate decision criterion; an economic unit attempts to maximize an eXpected utility function whenever it finds itself in an uncertainty situation. There also appears to be something of a consensus concerning uncertainty quantification; a probability distribution can be used to determine the values of the variables that enter the decision criterion. Shackle has been a major critic of this approach to 17 In brief his concep- decision making under uncertainty. tualization of a decision made under uncertainty represents the result of two distinct stimulation functions. The 17G. L. S. Shackle, Expectations in Economics (Cambridge: Cambridge University Press, 1939), Chapter VII. l9 focusing stimulation function is the process by which a person is mentally stimulated to focus his attention on a pair of values; the focus gain and the focus loss.18 The action stimulation function is concerned with the choice between projects from a consideration of the focus gain and the focus loss. In substance then the focusing stim- ulation function is the means by which expectations are formed and replaces the notion of probability distributions while the action stimulation function replaces the expected utility function as the decision criterion. Shackle's main argument in support of his scheme is that proba- bility analysis is unrealistic for in most uncertainty situations the individual has neither the means nor the ability to construct probability distributions. In any case there are two requisites for a decision in any uncertainty situation: a basis of choosing between alternatives and the quantification of uncertainty. In the many discussions of the EH the expected utility function, either explicitly or implicitly, is most frequently employed as the decision criterion used by the investor. However, these discussions do not employ probability distributions for the quantification of uncertainty, 18B. R. Williams, "The Impact of Uncertainty on Economic Theory with Special Reference to the Theory of Profits," in Uncertainty and Business Decision: A Symposium, C. Carter, G. Meredith, and G. L. S. Shackle eds. (Liverpool University Press, 1954), pp. 58-71. 20 although occasionally there is a suggestion to that effect. Consequently expected interest rates are not considered to be the expected values of probability distributions but are derived through other methods. These other methods repre- sent the expectations mechanisms which are the subject matter of this study. The question immediately arises as to why, in this particular area, probability distributions have not been employed. Inasmuch as there have been no explicit argu- ments against probability analysis any reasons offered here would be pure conjecture. None the less possible reasons may be offered: a general feeling that the secur- ities market does not reflect those characteristics con- ducive to probability analysis; the lack of evidence that such analysis is undertaken by investors, and the belief* that expectational formation occurs along the lines incor- porated by the various expectations mechanisms. In any event the expectations mechanisms are the devices said to be employed by investors in forecasting expected interest rates. And as long as some method of determining expected interest rates is specified in combi- nation with any one of the four alternate assumptions concerning investor behavior, then the basic implication of the EH, equal EHPRR's, is assured. CHAPTER III EXPECTATIONS MECHANISMS Before defining the various interest rate expectations mechanisms and describing various mechanisms within each category, it is desirable to illustrate the need for and the use of such mechanisms by investors. This is best done by portraying the security selection problem which the investor faces. Assume that the investor is at a point t in time and has a one-year holding period. Further assume that the investor must choose from securities with maturities equal to or greater than one year. According to the EH the investor wishes to select that security or group of secur- ities which will yield him the maximum EHPRR. To make a selection the investor must compute the EHPRR's on the various securities and this can be done only after the investor has forecasted a term structure for t+l, the end of his holding period. The investor can do this either directly or indirectly. In the latter case the investor forecasts the time series of expected short rates or the time series of expected consol rates and transforms either series into the term structure eXpected to prevail at time t+l. Interest rate expectations mechanisms are simply 21 22 the different methods the investor might employ in arriving at any series of expected rates. In this study the assump- tion will be that the investor proceeds directly and con- sequently is concerned only with the term structure expected to prevail at the end of his holding period. Within this context then the various interest rate expectations mech— anisms are simply the different methods which the investor might use in forecasting the term structure expected to prevail at the end of his holding period. This approach is assumed throughout the remainder of this study. Categories of Expectations Mechanisms Conceptually there are any number of ways in which investors may formulate expectations about future interest rates. In the literature a number of methods have been mentioned and several of these have been employed in empir- ical studies. In order to facilitate the current discussion it is helpful to establish categories of interest rate mechanisms. These categoreis are not be to-considered definitive but rather a device for logical grouping. The listing of the mechanisms within the categories is not to be interpreted as an exhaustive inventory of all possible mechanisms. It does include, however, all the mechanisms that have been used in empirical studies of the EH. 23 With this in mind the present discussion is structured in terms of three categories of mechanisms: regressive, extrapolative, and cyclical.19 A regressive mechanism is typically defined as the view which holds that the market eXpects interest rates to move towards a normal level which is based on past exper- ience. The belief that recent changes in rates will lead to furtherchanges in rates is usually identified as the basic premise for the extrapolative mechanisms. Proponents of cyclical mechanisms hold that investors take as their expectational guidepost some sort of conceptualization of a normal cyclical movement of rates. x—w-g——- ”n.— 'q—n— - - In order to draw as fine a distinction as possible between the three categories of mechanisms, expectational formation can be divided into three separate but not unrelated elements: a general market outlook, current assessment of the market, and specific expectational formation. The opinion that the market is continually moving towards a normal position is the general market outlook 19The terms "regressive" and "extrapolative" are used by Modigliani and Sutch, op. cit., p. 181 and Frank DeLeeuw, "A Model of Financial Behavior," in the Brookings Quarterlnyconometric Model of the United States, J. S. Duesenberry, et. al., eds. (Chicago: Rand McNally. 1965), pp. A65-530. A cyclical expectations mechanism is suggested by J. M. Culbertson, "The Interest Rate Structure: Toward Completion of the Classical System," in The Theory of Interest Rates, F. H. Hahn and R. P. R. Breckling (London: MacMiIlan, 1965), p. 176. 2A characteristic of the regressive theme. The investor is said to visualize a normal yield curve or, what is the same thing, a normal rate for each maturity. If the current yield curve does not coincide with the normal yield curve, changes in the yield curve are anticipated and seen as movements towards the normal yield curve. Obviously the current assessment of the market involves a comparison of the existing yield curve with the normal yield curve or the current rate for a particular maturity with the normal rate for that maturity. For a particular maturity if the current rate is above (below) the normal rate, the general expectation will be that the rate will fall (rise). If the current rate is equal to the normal rate then the general expectation is that the rate will not change. In deter- mining the specific expected rate the investor also uses the normal rate. If the current rate does not equal the normal rate the investor will visualize the future rate as either the normal rate itself or as a rate which is between the current rate and the normal rate. In the case of an extrapplative mechanism the general market outlook is that future changes in rates will repre- sent extensions of recent changes. The investor is said to \‘s .9 . - ._._ - ..v~" --.....-—-_4-—o.~—-—.—-~o—v-’ ' visualize a current trend in the market or, again what is the same thing, a current trend in the rates for each maturity. Unless there have been no recent changes in the rates for the various maturities, changes in the rates for 25 the various maturities are anticipated and seen as movements extending the trends for the various maturities. Clearly in order to assess the current state of the market the investor must ascertain the direction of recent changes in rates for the various maturities. If recent changes in a rate for a particular maturity have been positive (negative) the general expectation of the investor is that the change in this rate over the period t to t+1 will also be positive (negative). If the rate has not changed then the investor will anticipate no change in the rate over the period t to t+1. In estimating the specific rate for t+1 the investor takes the position that the rate will be equal to the current rate plus some multiple (it may be less than or greater than 1 but always greater than 0) of the recent change. Under the cyclical framework the general outlook is one of continuous cyclical movement in the term structure, a continuous cyclical movement of rates for each maturity. The investor is said to foresee a repgtipiqnmgfmpast cyples. The investor employing a cyclical mechanism faces a more complex process in evaluating the current state of the market. If recent changes in rates have been positive the investor may conclude that the market is in the upswing of a cycle. But if the current rate is equal to the peak of the nopmal cycle, the investor may conclude that the market at t+l will represent the beginning of the downswing 26 in the cycle. The opposite would occur if recent changes in rates were negative but the current rate were equal to the lowest rate in the normal cycle. The estimation of a specific expected rate is also complex. Having determined the phase of the current cycle the investor refers to a corresponding phase of the normal cycle. Using this point on the normal cycle as a reference, the investor calculates the change in the rate over a year as determined by the normal cycle. The specific rate for t+1 is taken as the sum of the current rate plus some multiple of the yearly change along the normal cycle (where the current rate and the reference point on the normal cycle are considered as in agreement). Clearly then the mechanisms can be distinguished on these three features: general market outlook, basis of assessing the current state of the market, and specific expectational formation. For a regressive mechanism the general market outlook is movement towards a normal level; for an extrapolative mechanism, a continuation of recent changes; and for the cyclical mechanisms, a continuation of past cyclical behavior. With a regressive mechanism the assessment of the current state of the market is accomplished by comparison of a current rate with the normal rate and this difference is the basis for the deter- mination of the specific expected rate. An extrapolative 27 mechanism assessment involves recent changes as does Specific expectational formation. With a cyclical mechanism both recent changes and levels are employed in assessment and the normal cycle is used in calculating specific expected rates. These features are the differences in the mechanisms. There is, however, a basic similarity between them: in each tQS_EE§t is the basis of expectational formation; that is, each mechanism involves a backward time horizon. For a regressive mechanism an examination of past rates leads to a conceptualization of a normal rate. In the case of an extrapolative mechanism, the investor examines recent changes in rates in order to ascertain a trend in rates. With a cyclical mechanism the investor constructs a normal interest rate cycle by reviewing the past movement of rates. This discussion has attempted to define the general nature of the three categories of interest rate expecta- tions mechanisms. There are, of course, different formu- lations of the behavior assumed by each category. The remainder of this chapter is devoted to a description of several of these alternate formulations. 28 Regressive Mechanisms The view that investors expect interest rates to regress towards a normal level can be attributed to J. M. Keynes,2O Joan Robinson,21 and B. Malkiel,22 and recognition of the possible influence of such expectational behavior has been made by Van Horne.23 In each of these instances the regressive position suggests that future changes in the term structure will represent a movement towards a normal level. Thus the expected rate on an n-year security prevailing at time t is seen as equal to the current n-year rate plus all or some part of the difference between the current n-year rate and the normal rate on n-year securities. Algebraically: (2'1) t+lrn,t= Rn,t + Fntfinfi - Rn,tJ where R = the normal rate (level) for securities with a n,t n-year term to maturity and Fn = a measure of the extent to which the n-year rate adjusts to the difference between the current n-year rate and the normal n-year rate where 0 < Fn < l. 20 J. M. Keynes, The General Theory of Employment, Interest, and Money (New York: Harcourt Brace, 1936), pp. gal-20“. 21Joan Robinson, "The Rate of Interest," Econometrica, XIX (April, 1951). p. 102. 22 Malkiel, OB. Cite, pp. 82’880 23James Van Horne, "Interest Rate Risk and the Term Structure of Interest Rates," Journal of Political Economy, LXXIII (August, 1965), pp. 334-351. 29 If R > R then r > R if the normal rate is n,t n,t t+l n,t n,t; greater than the current rate then investors expect that the n-year rate will increase. If Rn,t < Rn,t then t+lrn,t < Rn,t5 if the normal rate is less than the current rate then investors expect the n-year rate will decrease. Equation (2.1) represents a general specification of the regressive mechanism. In this equation there are two independent variables: Rh,t, and Rn,t but the latter is given by the term structure prevailing at time t. Conse- quently only R need be determined independently. There n,t are as many versions of equation (2.1) as there are inter- pretations of Rn,t' There are a number of ways in which Rh t may be 3 specified which are consistent with the regressive view. Assuming the investor has an m period backward time horizon and therefore an m set of R where i ranges from 1 n,t-1's to m, the investor may take Rh to be any measure of the t 3 central tendency of this set. The investor may interpret Rn,t as the mean, mode, or median of the set. In addition, if the near distant past is more important than the far distant past, Rh’t may be the weighted average of this set. However, m itself may or may not be constant. If m is held constant (the investor considers only the past m rates from any current date), as a period passes the most recent rate is added to the set while the least recent rate is elim- inated. If m is allowed to increase, as time passes then 30 an additional rate is added to the set without the elimin- ation of a past rate. Including both views compounds the number of interpretations of Rh,tand the number of possible tests of the regressive mechanism. The use of various measures of central tendency does not exhaust the number of ways in which Rh,tmay be calcu- lated. Malkiel's normal range model,‘2Ll can be construed as an alternative means of determing Rh,t. Malkiel develOped the normal range model in order to explore the effects of subjective expectations of a normal price (rate) range on the structure of rates when the -market believes that normal fluctuations will be contained within the range. The first step in the suggested process is the determination of what the prices of the various securities would be if either the lower or the upper limit of the price range should prevail and the resulting price adjustments. The second step is the computation of the mathematical expectation of gain. Assuming that it is equally likely that either the upper or lower limit will prevail, the mathematical eXpectation of gain is equal to the difference between the possible gain [( .5)(price gain)] and the possible loss [(.5)(price loss)]. The third step is to determine the market prices equalizing the 2uMalkiel, op. cit. 31 mathematical expectation of gain. Under this process the equilibrium term structure will be the structure which equalizes these mathematical gains. Retaining the assumption that is is equally likely that either the upper or lower limit of the price range will prevail, Malkiel's model can be summarized as follows: (2.2) PE - PC = [(.5)(PH - PC)] - [(.5)(PC - PL)1 where PE = expected price, PC = current price, PE - PC = mathematical expectation of gain, PH = upper limit of the price range, and PL = lower limit of the price range. This model can be converted into one which generates expected interest rates. Equation (2.3) represents the Malkiel normal range model when applied to interest rates: (2.3) t+lrn t = R t = [<.5>1 — [<.51 n, S 3 n where R: = the lower limit of the interest rate range and H Rn = the upper limit of the range. Equation (2.3) reduces to 32 Equation (2.4) is a special case of equation (2.1); and two equations are equivalent if it is assumed that H + R Rh t = —2—§——2- and the adjustment to the normal level is 3 foreseen as being completed by the end of the period (FN = v.25 The specification of a normal interest rate range requires a backward time horizon. If the investor has a m period backward time horizon then Hi and RE may be consid- ered as the absolute extremes experienced during the past m periods. If the m period backward time horizon contains time units such as years, R: and R: may be any measure of the central tendency of the yearly highs and lows. Clearly there are many alternative definitions of the normal rate for a particular maturity. But regardless of the definition of the normal rate, the process by which the term structure expected to prevail at the end of the investor's holding period is determined remains unchanged. That is, by applying the behavior indicated by equation (2.1) to each maturity, mutatis mutandis, the term structure expected to prevail at the end of the period is: t+lrl,t = R1,t + F1(R1,t ' R1,t) (2‘5) t+lr2,t = R2,t + F2(R2,t ‘ R2,t) t+lrn,t = Rn,t + Fn(Rn,t ’ Rn,t) 25 In an empirical version of his model Malkiel actually employs a weighted average. Malkiel, op. cit., pp. 83-85. 33 Extrapolative Mechanisms Although extrapolative mechanisms have many forms they all contain the basic premise that future changes in rates will represent a continuation of recent changes in rates. Taking, perhaps, the simplest form, the extrapolative mechanism can be expressed as: (2'6) t+1rn,t = Rn,t + Gn(Rn,t ' Rn,t:13 where Rn t-l = the previous period's n-year rate and .9 Gn = a measure of the extent to which expected n—year rate adjusts to recent changes in the n—year rate where Gn > 0 The View that recent changes in economic variables will induce further changes in the same direction is common in economic analysis and it is not surprising to find this same idea expressed in interest rate expectations mechanisms. Because such mechanisms can be constructed rather easily, it is understandable that versions of this mechanism are employed in studies where there is a need for the specifi- cation of the future values of interest rates but where attention is directed at some problem other than the term structure. An example of this is a portfolio analysis undertaken by Farrar.26 In his study two different 26D. E. Farrar, The Investment Decision Under Uncertainty (Englewood Cliffs: Prentice-Hall, 1962), Chap. 1. 3A extrapolative mechanisms are used. In one the expected change in a rate is taken as equal to the previous period's change. (2‘7) t+1rn,t ’ Rn,t = Rn,t ‘ Rn,t-1 Equation (2.7) is equivalent to equation (2.6) if in the latter GN = 1. In the other the expected percentage change is assumed equal to the previous period's percentage change: (2.8) t+lrset ' Rnlt = Rn,t Rn,t—i Rn.t Rn,t—1 or R - R t n t-l (209) r = R + G n) . 3 R . L" .J Equations (2.7) and (2.9) can be labeled the simple airth- metic extrapolative mechanism and the simple percentage extrapolative mechanism respectively. In both of these versions of the extrapolative mechanism there are two independent variables Rn,t and R Both of these variables are known at time t and n,t-1' therefore there is no need for a device which generates these variables. However the equations (2.7) and (2.9) do not exhaust the list of possible extrapolative formu- lations. 35 There is also the extrapolative view that the recent trend in rates is represented by the difference between the current rate and some measure of the central tendency of recent past rates. In this instance, the expected change in a n-year rate is equal to the difference between the current n—year rate and the selected measure of central tendency: (210) r -R =G R-Rk ‘ t+1 n,t n,t n n n,t-l or (211) r =R +G R-Rk ‘ t+1 n,t n,t n n n,t-l where Rfi t—l = a measure of the central tendency of the past 3 k rates. Equation (2.11) may be termed the complex arith- metic extrapolative mechanism. k In equation (2.11) the independent variable Rh t-l ’ must be calculated. The calculation of this variable requires the specification of a backward time horizon but the horizon involved here is typically assumed to be shorter than that considered in the discussion of regressive mechanisms; say k periods. Thus —:,t-l may be interpreted as the mean, mode, median, or weighted average of the k set of past rates, of R where i ranges from 1 to k. t-i's 36 The three equations, (2.7), (2.9), and (2.11) represent alternative versions of the extrapolative mechanism. For the sake of simplicity let equation (2.11) represent the general equation for the extrapolative category of mechanisms. With this in mind the term struc- ture expected to prevail at the end of the investor's holding period can be generalized as: - wk t+lrl,t ‘ R1,t + G1(R1,t ‘ R1,t-l) (2 12) r = R + G (R — Rk ) ' t+1 2,t 2,t 2 2,t 2,t-1 r = R + G (R - fik ) t+1 n,t n,t n n,t n,t-l For the sake of convenience two other mechanisms are to be included within the extrapolative category. Actually convenience is not the only justification which may be used for both of the following mechanisms use to some degree, recent changes in rates as an explanatory variable. How- ever, these two mechanisms are not aptly described by the general equation [equation (2.11)] postulated for the extrapolative category. Because of this in latter dis- cussions each of the two formulations which follow will be treated as separate classes of mechanisms. Meiselman has developed what is known as the error- learning mechanism.27 He begins with the outlook of the EH kl fij 27Meiselman, 0p. cit. 37 characterized by equations (1.2). In this fashion the term structure at time t is: (1 + R1,t) =(l + tr1,t) (l + R2,t)2 = (l + trl,t)(l + t+lrl,t) (2.13) (l + R3,t)u = (1 + trl,t)(l + t+1r’1,t)(l + t+2rl,t) (l + Rn,t)n = (1 + t;1,t)(l + t+1rl,t)(l + t+2E1,t)"° (l + t+n-lrl,t) and the term structure at time t+1 is: (2.1A) (1 + R1,t+1) = (l + t+lrl,t+l) (l + R2,t+1)2 _ (l + t+lrl,t+l)(l + t+2rl,t+l) (1 + R3,t+l)3 = (l + t+11"1,t+1)(l + t+2rl,t+l)(l + t+3rl,t+l) (l + Rn,t.+1)n = (l + t+1r'1+1:)(l + t+2r'1,t+1.)(l + t+3r1,t+I).°. (1 + ). t+nrl,t+l 38 Meiselman distinguishes between anticipated and unanticipated changes in expected short rates. The anticipated change in a short rate expected a given number of years from the current date is the change that is to occur because of the passage of time. That is: (2‘15) iAt+l = t+irl,t ' t+1-lrl,t where iAt+1 = The anticipated change in the short rate eXpected to prevail 1 years from the current date t as the current date changes from t to t+1. The unanticipated change in the expected short rate is the change that results from a change in expectations about future short rates. In this way, (2'16) 1Ut+l = t+irl,t+l ‘ t+ir1,t where iUt+1 = the unanticipated change in the short rate expected to prevail 1 years from the current date. Meiselman's error learning mechanism attempts to explain the unanticipated change in expected short rates. Before stating his hypothesis a further elaboration of the unanticipated change may be helpful. For example take the and Now both of these rates refer rates t+2r1,t r+2r1,t+1’ to the same expected short rate, same in the sense that the date at which both rates are expected to become the actual short rate is the same, on the date t+2. The difference between the two rates is the expectational base period. 39 The rate r is based on expectations prevailing at t+2 1,t time t while is formulated on the basis of expec- t+2rl,t+l tations prevailing at time t+1. Meiselman argues that the difference between these rates represents an unanticipated revision in expectations and such revisions are related to errors in forecasting the current short rate. Therefore, (2.17) U = D i t+1 R ' i( 1,t+1 t+lr1,t) where Rl,t+l"trl,t = the error made in forecasting the current short rate. This error learning model is an attempt to explain unanticipated changes in expected future short rates but it can be converted into a mechanism which determines the term structure expected to prevail at the end of the period. If expectations are unrevised--there is no error in forecasting 28 t+2rn,t+l = t+2rn,t° If the current short rate-—then 28From equations (1.A), it is clear that: )"- (1 + (1 + ')(1 + t+2rn,t t+2rl,t t+2rl,t) (l + t+nrl,t) and (1 + (1 + n - t+2rn,t+l) ‘ t+2rl,t+1)(l + t+3r1,t+l) (l + t+nrl,t+l) A0 there is an error in forecasting the current short rate then all the one-year forward rates included within t+2tn t are 3 revised and the new n-year rate will be systematically related to the forecasting error: (2°18) t+2rn,t+l = t+2rn,t + Dn(Rl,t+l ' t+lrl,t) where Dn represents the extent to which the n-year rate adjusts to the error in forecasting the current short rate and is greater than zero. This equation represents the general equation for the error learning mechanism. According to the error learning mechanism the term structure expected to prevail at the end of the investor's holding period is: But according to the error learning mechanism, if there is no error in forecasting the current short rate then: t+2rl,t = t+2rl,t+l t+3rl,t = t+3r1,t+l t+nrl,t = t+nrl,t+l and therefore: (1+ >“=<1+ >n t+lrn,t t+lrn,t+l OI” t+lrn,t = t+lrn,t+1‘ Al + D1(R t+1rl,t = t+lrl,t—l 1,t ‘ trl,t-l) (2.19) t+1r2,t = t+1r2,t-l 1,t tr1,t-1) + D (R n t+lrn,t = t+lrn,t-l 1,t ’ trl,t-l) Whenever there are two Opposing view a synthesis is usually attempted. This is the case with the expectations mechanisms. There are two separate studies which combine both the regressive and extrapolative views.29 Because the more recent work includes a good deal of the original reconciliation, the present discussion will be restricted to this latter construction. Modigliani and Sutch began with the regressive element and state that the normal level (Rh,t) can be approximated by some average of rates for the past m periods and a constant which represents a very long run normal level: (2.20) R = v u R _ + (l - v)c 0 < v < l where ui's = weights adding up to 1. In this way the eXpected change in Rn t is: 3 - R (2.21) t+1rn,t n,t = YlERn,t ’ Rn,t] They are: DGLGGUW O . 0113., and Modigliani and ’ # Sutch, OE. Cit. A2 01" m R Z (2.22) t+lrn,t - n,t = Y1 Vk=1ui + (l - V)C -R R n,t-i n,t Their extrapolative View is that the recent trend in rates maytxeapproximated by the difference between the current rate and some weighted average of recent past rates: k (2.23) t+lrn — Rn’t = Y2 0. Of course, this represents only one method by which the investor may utilize a cyclical conceptualization of interest rate movement in his forecasts. Another approach may be the use of average cyclical changes and certain rules to determine the current phase of the interest rate cycle. For example, assume that the investor constructs a normal interest rate cycle for a particular maturity on the basis of past experience and calculates the positive average change along the upward phase and the negative average change along the downward phase. Whether he employes the positive average change or the negative average change depends upon his assessment Of the current phase of the cycle for the particular maturity. Further assume that the investor employs the following rules: (i) if the current rate is above (below) the peak (trough) rate of the normal cycle always forecast a negative (positive) average A6 change; (ii) if the current rate is between the peak rate and the trough rate then the average change must agree in sign with the general pattern of change unless the two most recent changes have been in a direction opposite to the general pattern. In the latter case the average change must agree in sign with that of the two most recent changes. In this fashion then: (2.30) t+lrn,t = Rn,t + En(ACC) where ACC = average cyclical change and E = the extent to which the n-year rate adjusts to the cyclical change, En > 0. For purposes of this discussion it will be assumed that equation (2.29) represents the general equation for the cyclical mechanism. Thus the term structure expected to prevail at the end of the investor's holding period is: _ c c t+lrl,t ' R1,t + Hl(t+lrl,t ‘ Rl,t) r = R + H ( r° — Rc ) (2.31) t+1 2,t 2,t 2 t+1 2,t 2,t R + H ( rc Rc ) t+lrn,t z n,t n t+1 n,t - n,t A7 Summary This chapter has attempted to define three categories Of interest rate expectations mechanisms. Because two of the models included within the extrapolative category cannot be aptly described by the general equation for that category. five general equations are needed to describe the mechanisms discussed. These equations are: (1) t+1rn,t = Rn,t T Fn(§n,t ' Rn,t the regressive mechanism, _ —k (2) t+lrn,t I Rn,t + Gn(Rn,t - Rn,t-l) the extrapolative mechanism, (3) t+1rn,t T t+lrn,t-l T Dn(R1,t ’ trl,t-l) the error learning mechanism, n,t n n,t - Rn,t) + Gn Rn,t the“ t+lrn,t > n,t A I = R but n,t t+1rn,t - t+lrn,t . This follows directly from the behavioral premise of the mechanism (that 0 < Fn E 1). Therefore: > t+2rn,t+1 ‘ t+lrn,t ‘ The successive expected n-year rates continue to represent movement in the given direction until t+lrn,t ’ Rn,t' t+jrn,t+j-l ' t+jrn,t ’ This conclusion only holds when the normal level is defined as a constant. If the normal level varies over time, as might be the case if the normal level were defined asxa moving average, then no such conclusion can be drawn. 0'“... I I - h... 6"“ "HAW—F. - _. Before demonstrating this, the manner in which the normal rate experiences anticipated changes needs to be illus— trated. The difference in a moving average between two successive points in time arises because a component is added to the set of variables from which the moving average is computed while another component is eliminated. For example, the difference between Rn,t and Rn,t+l in equations (3.2) arises because the rate R is added to the set n,t+l 53 which determines Rn,t+l while a rate, say Rn,t-m is eliminated. The difference between Rn,t and t+lrn,t arises in much the same way only the rate is the added t+lrn,t rate. Because this rate is known during time t, the normal level for the period t+1 is known during time t. Now it must be demonstrated that by defining the normal level as a moving average, the conclusion that the series of expected n-year rates is either continually increasing, continually decreasing,or constant does not hold. This is most easily done by example. Suppose that According conditions at time t are such that Rh > Rn t 2 to the regressive mechanism then t+lrn,t > R .t' n,t' In this case, as in the previous case, the direction of change in the n-year rate is determined by the sign of the difference R - R n n t In the previous case the restriction that 3 st. > * = - t+lrn,t _ t+lrn,t Rn,t insured that the sign of was the same as the sign of Rh t - R 3 t+lrn,t ' t+lrn,t n,t° < In the current case the restriction that t+lrn,t — t+lrn,t remains, but because the normal level is a moving average VIA R then R . For example, if t+lrn,t < n,t-m t+lrn,t n,t t+lrn,t < Rn,t‘ Simply, in this instance there is no 5A guarantee that the sign of t+lrn,t - t+lrn,t is the same as the sign of Rfi - R All this serves to demonstrate ,t n,t° one thing: the implications of the mechanism can be quite different depending on the definition of the normal level. The implications of the regressive mechanism when unanticipated changes in the independent variables are allowed are derived under equations (3.2). In brief the consequence of an unanticipated change cannot be ascer- tained until the nature and magnitude of the change are known. By comparing equations (3.2) and 3.3) one can demon- strate that the mechanism itself can account for unantici- pated revisions in expected rates. This can be illustrated by examining the n—year rate expected to prevail during the period t+2. As stated in the previous chapter the unantici— pated revision in this rate can be defined as: (3°u) nUt+l = t+2rn,t+l ' t+2,rn,t The first term on the right hand side of equation (3.A) is determined under the same conditions which are assumed in deriving equations (3.2) while the second term on the right hand side of equation (3.A) is derived under the same conditions as those assumed in the derivation of equations (3.3). Clearly the unanticipated revision in the expected n—year rate is caused by the unanticipated changes in the 55 independent variables; either an exogeneous influence has entered the market or there has been an unanticipated revision in expectations. Unanticipated revisions in expectations in this mechanism are limited to a change in the normal level other than that induced by the substitution of a current rate for an expected rate while the difference between the current rate and the corresponding expected rate, i.e., Rn,t and trn,t-l’ represents the force of 30 exogeneous influence. The Extrapolative Mechanism The general equation for the extrapolative mechanism is: _ -k (3'5) t+lrn,t I Rn,t + Gn(Rn,t I Rn,t-l) Using this mechanism a series of expected n-year rates can be generated: _ -k t+lrn,t ‘ Rn,t T Gn(Rn,t ' Rn,t-l) r = R + c (R - Rk (3.6) t+2 n,t+l n,t+l n n,t+l n,t) r . = R + G (R - Rk t+1 n,t+i-l n,t+i-l n n,t+i-l n,t+l—2) 30 An exogeneous influence has two effects but the second follows from the first. The first effect is difference between Rn,t+l and t+lrn,t while the second effect is the change in the normal rate caused by this difference. This change in the normal rate is removed in order to keep the influence of unanticipated revision in eXpectations separate from that of exogeneous forces. 56 Equations (3.6) represent the series of eXpected n-year rates derived when allowing unanticipated changes in the independent variables. When the assumption of no unantici—' pated changes is imposed then equations (3.6) can be rewritten as: _ —k t+lrn,t ‘ Rn,t T Gn(n,t ‘ Rn,t-l) r =r+G(r-r—k) (3.7) t+2 n,t+l t+1 n,t n t+1 n,t t+1 n,t r. = O r + G ( r . - Fk ) t+1 n,t+i-1 t+1-l n,t n t+i-l n,t t+1-l n,t The implications of equation (3.7) depend on the definition of the recent change in the n-rate. If the recent change in the n-year rate is defined as the difference between the current n-year rate and the previous period's rate [as in equation (2.6)] then the series of n-year rates defined by equations (3.7) must be continually increasing, contin- ually decreasing, or constant. The type of change depends on initial difference between the current n-year rate and the previous period's n-year rate. For example: —k if Rn,t > Rn,t—l then r > R = RR and t+1 n,t n,t t+1 n,t’ therefore, t+rrn,t+1 > t+1rn,t and so on. In this case there is no limit to the one-way movement in the n—year rate. 57 However, this conclusion does not hold if the recent change in the n-year rate is defined as the difference between a moving average of recent n-year rates and the previous period's n-year rate [as in equations (2.13)]. In the same way in which the normal rate in the regressive mechanism can change without violating the assumptions of no unanticipated changes in the independent variables, the moving average of recent past rates can change without violating the same assumption. The argument that a change in definition changes the conclusion is also similar to the one employed in the discussion of the regressive mechanism but should be demonstrated nonetheless. Assume that initial conditions are such that R > Rk n,t n,t-1° . Before deter— Consequently t+lrn,t > Rn,t mining the rate must be computed. This t+2rn,t+l’ t+lrn,t computation involves the substitution of r for some t+1 n,t —k other rate, say Rn,t-k‘ If t+lrn,t < Rn,t—k then t+lrn,t may be less than Rk n t 13 that is, the recent change in rates , - for period t is positive while the recent trend in rates for period t+1 may be negative. Therefore if the recent trend in rates is defined as the difference between a moving average of recent rates and the previous period's n-year 58 rate, it can no longer be concluded that the series of expected n-year rates, derived under the assumption of no unanticipated changes in independent variables, must be continually increasing, decreasing, or constant. As was stated the implications of the extrapolative mechanism when unanticipated changes in the independent variables are allowed are derived in equations (3.6). The consequences of an unanticipated change within the framework of this mechanism cannot be established until the nature and magnitude of the change are known, as was true in attempting to draw the long run implications under the same conditions for the regressive mechanisms. By comparing equations (3.6) and (3.7) it can be demonstrated that this mechanism also can account for unanticipated revisions in expectations. Again the n-year expected to prevail during t+2 is used for the illustration: (3'8) nUt+l = t+2rn,t+l ‘ t+2rn,t The first term on the right hand side of equation (3.8) assumes the same conditions as those prevailing in deriving equations (3.6) while the second term in equations (3.8) arises under the same conditions as those prevailing in the derivation of equations (3.7). Again the unanticipated revision in the expected n-year rate is caused by unantici— pated changes in the independent variables; either an 59 exogeneous influence has entered the market or there has been an unanticipated revision in expectations. Also as before the unanticipated revisions in expectations are limited to the change in the moving average of recent n-year rates other than that induced by the substitution of a current rate for an expected rate. The Error-Learning Mechanism The error-learning mechanism does not determine expectations pgp se but rather revisions in expectations. As formulated by Meiselman the mechanism attempts to explain unanticipated changes in expected short rates or unantici- pated shifts in the time series of expected short rates. Even using the reformulation presented in the previous chapter there is no way of examining the long run implica- tions of this mechanism unless the prevailing expected rates are known, a constraint which is unique to this mechanism. Clearly then the error learning mechanism represents a unique device and therefore cannot be used within the same frame of reference as the other mechanisms. The adjustment process it implies is no longer necessary when the other mechanisms are employed for, as has been shown for the regressive and extrapolative mechanisms and as will be shown for the combined regressive-extrapolative mechanism and the cyclical mechanism, these mechanisms themselves account for unanticipated revision in expected interest rates through unanticipated changes in the respec— tive independent variables for each mechanism. 60 The Combined Regressive-Extrapolative Mechanism The general equation for the combined regressive and extrapolative mechanism is: _ —k (3.9) t+1rn,t T Rn,t n n,t T Rn,t) + Gn(Rn,t T Rn,t-l) Using this mechanism a series of expected n-year rates can be generated: (3.10) r = R + F (R - R ) + G (R - 'R'k t+1 n,t n,t n n,t n,t-l n n,t n,t-l) r = R + F (T' - R ) + G (R — Rk t+2 n,t+l n,t+l n Rn,t+l n,t+l n n,t+l n,t) O O O O 9 . O o O O O t+irn,t+i-l T Rn,t+i—1 T Fn(Rn,t+i-l T Rn,t+i-l) T Gn(Rn,t+i-l T —k Rn,t+i-2) Equations (3.10) correspond to the series of expected n-year rates derived when unanticipated changes in the independent variables are allowed. If unanticipated changes in the independent variables are not allowed then equations (3.10) can be rewritten as follows: 61 (3.11) T R T Rn,t—l t+lrn,t n,t T Fn(Rn,t T Rn,t-l) T Gn(Rn,t t+2rn,t+l T t+lrn,t T Fn(t+lrn,t T t+lrn,t) T Gn(t+1rn,t T trn,t) t+irn,t+1-l T t+1-lrn,t+1-2 T Fn(t+irn,t T t+1-lrn,t> T Gn (t+1-lrn,tTt+i-lrn,t)' In general equations (3.11) indicate that the series of expected n-year rates is determined by the net pressures of the regressive and extrapolative elements. To be sure alternate definitions of normal level and recent changes in rates can affect the series but the purpose of the present discussion is to explore the workings of the mechanism in very general terms. Take the case where the initial starting point is such that both the regressive and extrapolative elements are exerting positive pressure. Consequently t+1rn t > Rn t' 3 3 Unless then in the following period t+lrn,t < t+;rn,t again both elements are exerting positive pressure and therefore If then t+2rn,t+l > t+1rn,t' t+lrn,t T t+lrn,t t+2rn,t+1 may be equal to, greater than, or less than r ; it all depends on the relative strength of the two, t+1 n,t ‘ now countervailing forces. If the extrapolative element dominates then the difference between the n-year rate and its normal level may be increasing and thus the strength of 62 the negative regressive pressure increases. At some point in time the regressive pressure may become greater than the extrapolative pressure and the movement in the n-year rate reverses its direction. So even the use of simple defini- tions for the independent variables does not restrict the movement of the expected n-year rates to one and only one direction. Using alternate definitions in this case has no dramatic effect on this conclusion. The implications of the combined regressive- extrapolative mechanism when unanticipated changes in the independent variables are allowed are indicated by equations (3.10). In this case before implications of the unantici- pated change can be drawn, the nature and the magnitude of the change must be known. As was the case with the regressive and extrapolative mechanism separately, this mechanism also accounts for unanticipated revisions in expected rates. Again this type of revision occurs because of unanticipated changes in the independent variables and involves the comparison of an expected n-year rate derived under the same conditions as those prevailing when equations (3.10) are derived and an expected n-year rate derived under conditions similar to those employed in deriving equations (3.11) 63 The Cyclical Mechanism The general equation for the cyclical mechanism is _ C (3.12) t+lrn,t - Rn,t T Hn(t+lrn,t T Rn.t) Using this mechanism a series of expected n-year rates can be generated: (3.13) _ c c t+1 rn,t T Rn,t T Hn(t+1rn,t T Rn,t) _ c c t+2rn,t+l T Rn,t+l T Hn(t+2rn,t+l T Rn,t+l) . I c 3 c t+irn,t+i-l Rn,t+i—i T Hn(t+irn,t+l-l T Rn,t+i-l) Equations (3.13) correspond to the series of expected n-year rates that are obtained when unanticipated changes in the independent variables are allowed. When no such changes are allowed then equations (3.13) may be rewritten as follows: (3.1A) _ c _ c t+lrn,t — Rn,t + Hn(t+lrn,t Rn,t) _ c c t+2rn,t+l T t+lrn,t T Hn(t+2rn,t T t+lrn,t) r . = r + H ( rc - rc ) t+1 n,t+i—l t+1-l n,t+i-2 n t+i n,t t+1-l n,t 6A Equations (3.1A) indicate that the series of expected n-year rates will be a reflection of the normal cycle regard- less of the definition of the normal cycle. In this case the definition of the normal cycle does affect the nature of the expected series, but the basic conclusion that this series of expected rates will be cyclical is unaltered. With the assumption of no unanticipated changed in independent variables, the long run implications of this mechanism follow from equations (3.13). As with the previous mechanisms the effects of an unanticipated change are not known until the nature and magnitude Of the unanticipated changes in these variables are known. Care should be taken in this case to point out the possible ramifications of an unanticipated change in the independent variables. Take, for example, the rate This rate can be determined t+2rn,t+l' only after a reference point on the normal cycle.(R: t) is 2 established. As equations (3.1A) indicate, without an unanticipated change in the independent variables this reference point would simply be an extension of the refer- ence point employed in the determination Of the rate r . When unanticipated changes in the independent t+1 n,t variables are allowed, however, the reference point is no longer restricted, it can be any point on the normal cycle depending on the nature and magnitude of the unanticipated change. 65 This mechanism, as with the other mechanisms, accounts for the unanticipated revisions in expected rates. It again simply involves a comparison of an eXpected rate derived under the assumption of no unanticipated changes in the independent variables with an expected rate derived when such changes are allowed. These then are the long-run implications of the interest rate expectations mechanisms. With the regressive and extrapolative mechanisms the definitions of the inde- pendent variables alter drastically the implications of these mechanisms when no unanticipated changes in the independent variables are allowed. Four mechanisms-~the regressive, the extrapolative, the combined regressive extrapolative and the cyclical--can themselves explain how unanticipated revisions in expected rates can occur; in each case they arise because of unanticipated changes in the independent variables. When such mechanisms are employed, it is not necessary to use an auxillary device such as the error-learning mechanism to explain such unanticipated revisions in expected rates. Short-run Implications Three alternative approaches may be taken in deriving the short—run implications of the interest rate expectations mechanisms. One approach is to examine unanticipated revisions in expected rates (t+lrn,t - t+lrn,t-l)‘ Another 66 is to inquire concerning the anticipated revision in expected rates (t+2rn,t - t+lrn,t)‘ The third alternative is to analyze the expected, or equivalently, the anticipated change - R in the current rate ( The unanticipated t+lrn,t n,t)‘ revision in expected rates (caused by unanticipated changes in the independent variables) and the anticipated revision in expected rates (explained by anticipated changes in the independent variables) have been discussed in the previous section of this chapter. Thus the discussion of the short run implications of the various mechanisms will be limited to an examination of the expected change in the current rate. In this discussion attention is directed at the conditions which each of the mechanisms must satisfy in order to indicate a particular type of expected change in current rates. 'There are three possible values for the difference between the n-year rate expected to prevail at the end of the investor's holding period and the current n-year rate. This difference maytmezero ( = Rn 1), positive t+lrn,i , > R ), or negative ( < R ). The problem (t+lrn,i n,i t+lrn,i n,i then is to state the conditions under which each mechanism will indicate each type of change. These conditions are given in Table l. 67’ a a a 3.. a a a. q a q o c o cna+n o :x u . ena+o o c; n an+o o c no cna+o c o : no an+o v :+ m HNOHHozo p.: u.c : t A at: xv ovfl cl UV a p32 . cm o.: u.cr .1 {MA r CCQ 3v r 3 a a. a ova ml t m as 3 c: p L: c, 1 c: , (V D A; :3! av 9AA; cl xv m uSL u.: p.: . u.: ”.2 c u.c B.r: c. . . . . .cm . c .n :I c :u. an r 37 . c c c c U m A kw r A ”(Tow .mv r. r i RV P u l L visuvu W UCm u mAu x .m vac..lpac C .I I A xm mv o + a a a a a. a L . c c c s . s c 52.0 H Hm H J. x .m U :«HMP .L. ”A...“ L Q.M+ MU om Mac: paws... vim was. ”was. . JnC UQC C Mac VAC HATU I N..- \ l A .LA we. at; t W" a m a A. .< .I al u A; El my 0+ an L A a: )n a a. 0.. a )n )n o c u c u : t a u a c c , c c . ; . one u m v em .a t :wu : nzm .n a .H p x.nt x rem I an» r .H c>wpmaocmcuxmlo>fimmocuix pmcHoEoo ' tv.—| .- I II. p.: o.c: p.c: p.:. u.c: p.c , p.:. p.:. : .u.c: u.: +u Cm v n Cu n J :3 A m A .El mv 0+ fl" LH e_| x.l ..| NA] m>fipmH0dwcpxm .2 .2 .2 .C o m v s n o a u o r o.cr A L.no o.c e.c c n.: p.: H+p. _ I A an my m+ mu c O>Hmmocwmm n.cn v n.cnfi+n o.c; o.c.a+o n.:: A o.csa+n ( . c u i u O>Hummmz oath O>Huwm0d Emficmcoms .Oumm cmmwuz ecu :H mowcmco UOOOdem no hoaxe pow RCOHpHpcooII.H mqm¢9 68 Note that in Table l the error-learning mechanism is omitted. The reason for this, as indicated in the previous section of this chapter, is that the error-learning mechanism explains only unanticipated revision in expected rates. Because of this, the conditions under which this mechanism implies a particular type of expected change in a current rate has largely nothing to do with the mechanism itself. For example, if the current error in forecasting the current short rate is known, the difference between current n-year rate and the n-year rate expected to prevail at the end of the period cannot be determined. This again points out the distinctive nature of this mechanism and why it is to be kept separate from the other, more general, expectations mechanisms. Summary This chapter has explored the long-run and short-run implications of several interest rate expectations. In the course of this discussion three types of changes in rates were mentioned: unanticipated revisions in expected rates, ant1019§£§§m§eyisions in expected rates, and eXpected changes_ipmpur§ent rates; and sources of such changes were indicated. Table 2 represents a brief recapitulation Of of this discussion. (39 whdpvdcflvm H LL30 map >9 ac>flu r4 ’1) 1) p ,. n.c: o.: H+o A on x is v c Ohms mocOLOLOc can Eocu zmzm UOHLOQ wcwpaoc m.LOBmm>:fi ecu mH coax: macho Hmecoc on» co mums oz» 0cm cacao awesoc on» so cums coco Icmwmc on» consumn zOHHmSOocH weaupomduo on no: pmzn moocmsadcfi 03p on» son» choc ea monmmlcoc ma soap: wepmc CH aces» ucmomc m LO\pcm when Hmscoc oz» coozpmn muHHmzdocH o.a. x r than ps03w peepcss on» mafihmmomaoc CH Locum I a H o Hen modem IHcm> accusedoccfi ecu :H newcmco popdeOHucmc: means IHcm> pampcmqoccfi ecu CH mowcezo cmmeAOMpcwc: izsuojcpm FLO» usmgsno LLB >n CO>HO that Local: on» c“ mowcmzo poquHOHuc< mums cmozlc Ono Cw moecmzo pepquOHpc< A a cl C AH o Hno I o Hmv a+ Huo.cse+o u o.cna+o wcficpmoq coccm Ac.cm I o.cna+ovcz + e.cm u c.cna+o o _ o . . Hmofiaoso .: .c c an em I u my 0+ Ae.cm I o.:mvca u c.cna+o O>HomHOQmprmIo>meonwom pmcHnEoo uac unC A am i av moans ocomIcoc ma IHnm> sneezedmccfl ecu ohms EmmaI: on» .c .c .c .c seas: wmumc CH econ» pcoomc < CH woacmgo pmpmcHOHBCHCD CH mmncwco pmmeHOHuc< A» am I u xv u u u ca+u p.c: p.c: . A n x mv O>Huwaoomhuxm mums natal: moans package can new comp hmmmuc Iwnm> occUCOdmccH ecu mum; pmozlc Oz» .2 ac : .: Hashes who comzoOp zhfiamswmcH CH moacmco doomsflofiucmc: cfi mmocmco powwoflofioc< A» m I p mv m n u ca+u O>Hmmmcmmm . a I a a a a p cm I p cca+p H p cca+p.I p cca+u o aha+u I u ccm+p nepmm wepmm pcuoqum CH neumm.pOooooxm CH . pathcso CH mmwcmcu pmpoodxm mcoama>mm pmudeOHpcwcs mcoflmfl>om poquHOHpc< Emficmnooz. mowcmco no momma .moumm.:H newcwzo no momma escapm> en» no moouaomII.m canoe 70 The following points have been established: (1) the several mechanisms discussed represent competing hypotheses concerning the formulation of expected interest rates, (11) the error-learning mechanism stands apart from the other mechanisms for it attempts to eXplain unanticipated changes in expected interest rates%l(iii) the long-run implications of the mechanisms are contingent upon the definitions of the independent variables employed in the various mechanisms, and (iv) the short-run implications of the mechanisms are determined by the current values of the independent variables except for the error learning mechanism which also requires the knowledge of expectations prevailing during the previous period. 31As was shown the other mechanisms account for such changes by themselves. CHAPTER V EMPIRICAL ANALYSIS The purpose Of this study is to determine which of the interest rate expectations mechanisms provides the "best explanation" of expected interest rates. The previous chapter has attempted to indicate, in general terms, the long- and short-run implications of each class of mechanism. The present chapter aims at the empirical evaluation of the mechanisms. However, before the empirical analysis can be accomplished, a number of methodological issues must be settled. The Methodology It was previously stated that the development of interest rate expectations mechanisms arose because a particular approach to the testing of the EH requires an independent specification of expected interest rates. This implies, rather indirectly, that the term structure itself provides clues concerning the nature of interest rate expectations. If this is the case, then the sets of expected interest rates (one set for each maturity) gener- ated by each of the various mechanisms can be compared to the sets of expected rates implied by the term structure. 71 72 Before examining this approach to evaluating the mechanisms, the derivation of expected interest rates from the term structure should be demonstrated. . Assuming that the first one—year EHPRR on every security is equal (viz., the EH obtains), then the term structure prevailing at time t can be described as: (l + Rl,t) = (l + trl,t) (l T R2,t) T (1 T trl,t) (l T t+lrl,t) (A.1) (1+R)3=(1+ r )(1+ r )2 3,t t 1,t t+1 2,t ° n _ . . n-l (1 T Rn,t) T (1 T tr1,t) (l T t+lrn-l,t)' From equations (A.l) it follows that: (l + R2 t)2 = ’ .- t+lrl,t (1 + R1 1:) l ’ 2 I t+1r2 t T 3’t T 1 ’ (1 + R1,t) n— (1 + Rn t)n r = 4— - l t+1 n-l,t (l + Rl t) ’ 73 The rates derived in this fashion in conjunction with the assumption that the EH obtains can be interpreted as rates on securities of various maturities expected to prevail one-year from the date t. For example, t+lrn-l,t is the rate expected to prevail during period t+1 for securities with an n-l term to maturity according to expectations existing during period t. It should be pointed out that the interpretation of the rates derived in the manner illustrated by equations (A.2) depends upon the assumption made concerning investor behavior. For example, if it is assumed that the Hicksian Liquidity Premium Model rather than the EH obtains then, these same derived rates are no longer expected rates but exceed expected rates by an amount equal to a liquidity premium. For the sake of clarity in eXposition, the rates derived in the fashion indicated by equations (A.2) will be called forward rates while the expression expected rate will be reserved for those rates generated by the interest rate mechanisms. Because the assumption here is that the EH prevails, the term "forward rate" is equivalent to the term "expected rate" and not an expected rate plus a liquidity premium. As was mentioned above, an obvious approach in evaluating the mechanisms is to compare the sets of expected rates with the sets of forward rates. However, 7A before any of the mechanisms can generate a single forward rate, the adjustment variable in the mechanisms must be specified. Because of this a more straight forward approach (in the sense of bypassing the specification of each of the adjustment variables) for evaluating the mechanisms would be to regress each set of forward rates against each of the mechanisms. For example, with the regressive mechanism one can evaluate the explanatory ability of this mechanism through the regression equation: (R - R + e ) T b n,t n,t) n = a + b (R (A.3) t+lrn,t n n,l n,t n,2 where an, bn,l’ and bn,2 are defined as the usual regression coefficients and en is the stochastic disturbance term. Regression equations for alternative definitions of the normal level and for the other mechanisms and their alter- native forms can be constructed in similar fashion. This, of course, implies that the evaluation of the mechanisms is to be accomplished solely through a compar- ison of the correlation and regression statistics of the various mechanisms. This, however, is not the case. Rather tests have been defined under general conditions for the specification errors of omitted variables, incorrect functional form (of the regressors only), errors in variables, simultaneous equation problems, and 75 heteroskedasticity.32 Consequently both sets of statis- tical tools, correlation and regression analysis and the Specification error tests, are employed in the evaluation of the mechanisms. An empirical evaluation of several conceptually acceptable but competing hypotheses can be considered as a two-step process. The first step is the determination of whether or not a particular hypothesis is consistent with the data. When a particular hypothesis is evaluated through correlation and regression analysis, this deter- mination involves a comparison of the size and sign of the estimated regression coefficients with the size and sign of the coefficients as implied by the theoretical frame- work of the hypothesis. It also involves an assessment Of the significance of these coefficients as well as the extent and significance of the overall estimated relation- ship. A particular hypothesis is accepted, judged as consistent with the data,if it satisfies these conditions. The second step in the evaluation process is the determination of whether or not one of two or more hypotheses which are consistent with the data is superior or, as conventionally interpreted, provides a better fit. Normally this involves a comparison of the coefficients 32James B. Ramsey, "Tests for Specification Errors in Classical Least Squares Regression Analysis," Michigan Stzte University Econometrics Workshop Paper NO- 66019 19 7. 76 of determination yielded by each of the competing hypotheses. Obviously there is no need to proceed to this second step if only one of the competing hypotheses is consistent with the data. The usual interpretation of regression and correlation statistics, however, presumes the absence of specification error. That is, the least squares regression estimates of the parameters of a particular regression model cannot be interpreted in the usual manner if specification error prevails in that model. But the existence of the specifi- cation error tests aside from testing for the presence of specification error may allow greater discrimination between two competing hypotheses. For example, assume that two competing hypotheses yield approximately equiv- alent conclusions concerning their acceptability on the basis of the regression statistics, but one of the regres- sion models contains specification error. Under these conditions it would appear that the hypothesis free of specification error can be accepted as the "correct" hypothesis. With the inclusion of the specification error tests, the possibility arises that all the competing hypotheses contain specification error. Before discussing the nature of the evaluation process under these circumstances, a brief discussion of the effects of the various types of specifi- cation errors and the specification error tests themselves is necessary. 77 With the specification error Of omitted variables, bias is determined by the extent to which the included explanatory variable(s) is correlated with the omitted variable(s).33 If the explanatory variable is not corre— lated with the omitted variable then unbiased estimates are obtained. In the case of incorrect functional form, the estimated regression will represent a bad approximation of the true relationship.3u The seriousness of this problem is reduced when the values of the explanatory variable(s) in the population are of a similar order of magnitude as the values of the variable(s) in the sample. With errors in variables there are several cases. If both variables contain measurement error or if the error is restricted to the explanatory variable then the estimates obtained through least squares regression analy- sis are inconsistent. If, however, the error is restricted to the dependent variable then the estimates are consistent. In the case of simultaneous equation problems, the bias arises because of the dependence between the explana- tory variable(s) and the disturbance caused by the exis- tence of other relationships involving the explanatory 33E. Malinvaud, Statistical Methods of Econometrics (Chicago: Rand McNally and Company,Tl966), pp.T263-266. 3LTIbid” pp. 266-271. 78 variable(s). The nature of the bias is determined by the nature of the relationships involving the explanatory variable. If heteroskedasticity prevails then the least squares estimates are unbiased but not efficient and the results of significance tests conducted with conventionally calculated standard errors are invalid. With the specification errors of omitted variables, \ incorrect functional form, errors in variables, or simul- taneous equation problems the distribution of the least squares residual vector is asymptotically normal with non- 35 For these same specification errors the null mean. asymptotic distribution of the squared residuals is non- central chi-square with one degree of freedom. With heteroskedasticity the mean of the least squares residual vector is null but the distribution of the squared residuals is scaled central chi-square with one degree of freedom. Each of these specification errors represents a violation of the full ideal conditions that are required to be satisfied by a classical unbiased linear regression model. In total four specification error tests have been developed: RESET, RASET, KOMSET, AND BAMSET. RESET, RASET, and KOMSET are designed to detect the specification 35Ramsey, Op. cit. 79 errors of omitted variables, incorrect functional form, errors in variables, and simultaneous equation problems but do not discriminate between the four types of specifi- cation error. BAMSET is designed to detect the specifi— cation error of heteroskedasticity and, thus, there is some ability to discriminate between the heteroskedasticity on the one hand and omitted variables, incorrect functional form, errors in variables, and simultaneous equation problems on the other. All four tests are asymptotic and each assumes that the observations on the variables are statistically independent and that the distribution of the least squares residuals in the misspecified model is normal.35 The testing procedure is as follows: (i) a partic— ular mechanism is estimated regression analysis and (11) all four specification error tests are applied at a chosen level of statistical significance. A particular mechanism is rejected, interpreted as containing specification error, if any one or more of the tests statistics is within the 36 statistically critical region. For RESET the test 35Ramsey, op. cit. 36Although the tests are only asymptotically effi- cient, the power of the tests rise rather rapidly with increases in sample size; the power of the joint test would be close to one for sample size fifty. See James B. Ramsey, Tests for Specification Errors in Least Squares Regression Analysis (unpublished Ph. D. dissertation, University of Wisconsin, Madison, 1967), Appendix C. 80 statistic is an F ratio where the null hypothesis of no spec- ification error is accepted when the calculated F ratio is less than the F value for a particular significance level. The test statistic for RASET is a t ratio with the null hypothesis being distributed as Student's "t". BAMSET results in the test statistic M and under the null hypoth- esis M is asymptotically distributed as chi-square. KOMSET has no specific test statistic but involves the evaluating of the significance of several inequalities; under the null hypothesis none of the inequalities is significant. Note that the null hypothesis of no specification error is accepted only if all four tests indicate the acceptance of the null hypothesis. Given this joint use of the tests and assuming the four tests are statistically independent,37 then the probability of Type I error (rejection of a correct null hypothesis) is .OA when the significance level for each of the tests is set at 1% and .18 if the significance level is 5% for each test. Now returning to the case where two regression models yield equivalent conclusions on the basis of their correlation and regression statistics and both models contain Specification error. It would appear that a tentative selection could be made on the basis of the 37Ibid. Note that this study also contains tests that give some indication that the four tests are indepen- dent. 81 number of tests indicating specification error and/or the strength of the specification error. The procedure in this study, however, is to accept a model as the "correct" model if it is the only model which is free of specifica- tion error and consistent with the data. In all other cases, because no completely unambiguous selection of the "correct" model can be made, no attempt will be made to select the "correct" model. It should be pointed out that the testing design is such that the question is not which mechanism represents a "correct" explanation of forward rates but rather which mechanism represents a "correct" explanation of a partic- ular set of forward rates (i.e., which mechanism correctly explains the one—year forward rate, which mechanism correctly explains the two-year rate, and so on). In this way the possibility that different mechanisms are employed in forecasting rates for the different maturities is allowed. The remaining methodological issue concerns the data to be employed in this study. Here the primary data, the data from which the dependent and independent variables are constructed, are derived from the yield curve for fixed-maturity U. S. Treasury securities published monthly in the Treasupy Bulletin. These yield curves are smooth curves fitted by eye to closing bid quotations. The primary data are formulated by reading off rates for each 82 maturity from these smooth curves. The closing bid quota- tions themselves cannot be used because they do not provide a continuing series for each maturity and without such a series the empirical tests described above could not be undertaken. Although the primary data are available monthly only quarterly observations are employed here beginning with the third quarter of 1953 and extending through the second quarter of 1967 for a total of 55 observations. Unfortunately for the first 13 quarters only the first five forward rates can be calculated. For the remaining quarters the first nine forward rates can be calculated. The empirical tests use only five of the nine available forward rates; the one-, three-, five-, seven-, and nine-year forward rates. Specification of the Mechanisms Previous discussion has established that there are five competing categories of interest rate expectations mechanisms. Moreover, within a given category there are alternative versions of the same mechanism and these alter- native versions arise thorugh the use of different defini- tions of the independent variables included within the mechanisms. In order to undertake an empirical analysis of the mechanisms it is necessary to define these alter- native forms. Table j3 is a summary of all the formula- tions tested in this study. Note that Table 3 contains 83 TABLE 3.--General Equations, Alternative Definitions, and Specific Regression Equations for the Tested Mechanisms. Class of I Mechanism Identification Alternative Definitions Specific Regression Equations Regressive R.1 15 150 n'tTT r - a + b (R ) + b (R l) t e R.1 - -——TTr——— - Rn,t t+1 n,t n n,1 n,t n,2 ' n 16 T Rn t i a ' - R.2 R.2 . TTTIITTTT - Rn t t+1rn,t ' an T bn,l(Rn,t) T bn,2(fl'2) T en ‘ l E t a 1 tiv I. I - u x r no a e P 1 l - Rn,t Rn,t-1 t+lrn,t an + bn.1(Rn,t) + bn.2(E.l) + en 3 T Tn t-l ,' I I.2 I 2 - ’n,t - TTTTHT—T— t+1rn,t T 8n T bn,1(Rn,t) T bn,2(E'2) T en A x Rn t 1 . I=1 ' T L.3 1.3 - TH,C TTTTTTTTT t+1rn,t T 8n T bn,1(Hn,t) T bn,2(E'3) T en 15 Combined "9F.1 CHE.) a I‘ 'I - F e + T. + rezressive- 1.001 n,t-1 ”UL t*1rnot - an bnul(Hnot) bn02(CRh 1) en extrapolative T lb (‘RI'~ " (‘73.? I X n. - R . i-lfll n.t_1 n.t ,,1rn.t - an + bn.1(Rn.t) . bn.2(CRE.2) + en car 3 CRH.3.1 A 1"“5 IE .ASTTTR r - a o b (R I o b (an: 3 1) 1_.u5 1_1 n,t-1 tel n,t n b,1 n,t n,2 ' ' if 11 + bn.3(CRE.3.2) + Tn (W 3 '> = 1"“ z C'iTl'I . .. .I. ——_FIT . ) 'n,L-i 1-.vi i-l Error- Lea n1 L . i . = I - I r ”g L 1 'L 1 T1,t trl,t-1 t+1rn,t 8n T bn,1(t-1”n,t-1) + bn.2(EL.1) t en c ciIcai 0.1 c. - ° - y . 1 rn,t t+1rn,t 8n T bn,l(Rn,t) T bn,2(c’1) T en Inertia I. I . ' 1 I I 1 t+1rn,t-1 t+1rn,t Tn + bn,l(1'1) I 102 o - I I 2 Rn.t t+1’n,t an + bn.1(I.2) 84 six categories of mechanisms. The new category is labeled the inertia category and contains two alternative formula- tions. In the first (Identification 1.1) the current n-year forward rate is regressed against the previous period‘s n-year forward rate while in the second (Identification 1.2) the current n-year forward rate is regressed against the current n-year rate. There are two versions of the regressive mechanism and the difference between the two resides in the definition of the normal level (fih ). With the first regressive ’ t mechanism (R.l) the current n-year rate is included within the set of rates from which the normal level is computed while in the second (R.2) the current n-year rate is not included within the set. In both instances, however, a 16 quarter backward time horizon is used. There are two . reasons for using a 16 quarter backward time horizon. On the one hand several other investigators have assumed that the same time span or a shorter time span is appropriate for this type of mechanism. On the other hand any sub— stantially longer time span would seriously hamper the testing of the mechanisms by reducing sample size. Three alternative definitions of the recent trend in rates are used for the extrapolative mechanism. Actually k the differences reside in the definition of fih This t. ’ variable is defined as the previous period's n-year rate (E.l), a four quarter moving average which includes the 85 current n-year rate (E.2), and a four quarter moving average which excludes the current n-year rate (E.3). There are three alternative versions of the combined regressive-extrapolative mechanism. In the first two such mechanisms (CRE.1 and CRE.2) the use of a set of weights that approximates a fourth degree polynomial is held to account for both the regressive and extrapolative pressures.38 The difference between the two is that one (CRE.l) includes the current n-year rate in the set of past rates while in the other (CRE.2) the current n-year rate is excluded. In the third combined regressive-extapolative mechanism (CRE.3), the different weights, .45 and .65, transform the same set of past rates into the equivalent of an average of past rates (with .65) and into the most recently experienced rate (with .HS). 38The weights as given by Modigliani and Sutch, op. cit., p. 192, are: 01 = .0229 09 = .osuu 02 = .0293 010 = .0603 03 = .0373 011 = .0537 o“ = .0458 012 = .Ouu9 05 = .0536 013 = .03“? °6 = .0599 014 = .0239 07 = .06U1 015 = .0136 08 = .0656 016 = .0051 86 There is only one version of the error learning mechanism and one version of the cyclical mechanism. The former corresponds exactly to the previous discussions of this mechanism. In the case of the cyclical mechanism, the first cycle observed for each set of actual rates was assumed to be the normal cycle for that particular rate. Using the normal cycle the average cyclical change during the upward phase of the cycle and the average cyclical change during the downward phase of the cycle were calcu- lated.39 Whether the average cyclical change during the upswing or the downswing was used in the mechanism was determined according to the discussion of Chapter II. If each of the alternative forms of the mechanisms is considered as a separate mechanism then a total of twelve mechanisms are evaluated. Included within the twelve are three mechanisms that correspond to previously utilized mechanisms (CRE.2, CRE.3, and EL.1) while another (R.2) differs from a previously utilized mechanism only in terms of the assumed backward time horizon.“0 39The average cyclical changes for the various forward rates are:, one-year forward rate, .25 and -.87; three-year forward rate, .20 and -.61; five-year forward rate, .17 and -.73; seven-year forward rate, .30 and -.62; and the nine- year forward rate, .26 and -.55. uoThe mechanisms and the studies to which they corre- spond are: CRE.2, Modigliani and Sutch; CRE.3, DeLeeuw, EL.1, Meiselman; and R.2, Malkiel. 87 Empirical Results This section contains the results of the regression analysis and the specification error tests for the various mechanisms.”1 The procedure here is to examine the mech- anisms relative to a particular forward rate. Before discussing these results, it should be pointed out that for one version of the combined regressive- extrapolative mechanism (CRE.3) the regressor matrix was found to be singular. This was the case regardless of which forward rate as used as the dependent variable. In addi- tion for the nine—year forward rate data limitations precluded the testing of the error-learning mechanism (EL.1) and the first inertia mechanism (I.l). Thus, eleven mechanisms, not twelve as implied by Table 3, were tested against the one—, three-, five-, and seven-year forward rates and nine mechanisms were tested against the nine-year forward rate. It should also be pointed out that the significance levels employed for the four specification error tests are: RESET, 1%; RASET, 1%; KOMSET, 5%;“2 and BAMSET, 1%. ‘r‘ ulAll the estimates for the regression analysis and the specification error tests were obtained through Pro- gramme Datgen. A description of this program is contained in: James B. Ramsey, "Programme Datgen: A Computer Pro- gramme to Calculate the Regression Specification Error Tests: RESET, RASET, BAMSET, and KOMSET," Michigan State University Econometrics Workshop Paper No. 670fl, 1967. NZ program. This significance level is set by the computer 88 Again assuming the four tests are independent then with these significance levels the probability of Type I error under the Joint use of the four tests is .08. The One-gear Forward Rate The results of the regression analysis for each of the eleven mechanisms when the one-year forward rate was used as the dependent variable are presented in Table 4 while the results of the specification error tests for these regressions are shown in Table 5.”3 The results of the specification error tests are consistent between the mechanisms. For each mechanism RESET and RASET indicate the acceptance of theruul hypoth— esis of no specification error while BAMSET and KOMSET, in those instances where mustests are well-defined, also indicate the acceptance of the null hypothesis. Thus none of the mechanisms can be eliminated on the basis of speci— fication error. The regression analysis indicates that each of the mechanisms has a high coefficient of determination (R2), the lowest being obtained with the first inertia hypothesis (1.1). The signs of the coefficients are as anticipated with the exception of those for the second independent u3The notation I.D. indicates that the particular Specification error test is ill-defined. The conditions under which BAMSET and KOMSET are ill-defined can be found in Ramsey, "Program Datgen . . ." pp. 23-24. Table U.-Regression Results for the One-year Forward Rate. Number of Obs. Estimated Equation Identifi— Class of cation Mechanism 00 006,(7(Rel) (.ow) Regressive 3 3) (.002) (.07 (J'\ |;7\ (‘6 m0\ t\— ('1: 3 8.2 (h Li\ 1 I: la. a Extrapolative (DR MR 89 G'\ Ll”\ (\J (3 ‘1 NC CRE.1 Combined ON (V) L Regressive- Extrapolative (‘J ON ON (0 m L13 U 3) (.133 (.00 a) ; 0 I EL.l Error- 5 Learning .91 Cyclical .72 C) i—i Inertia .93 50 90 panama .Q.H .Q.H Bmmoo< Emmoo< Emmoo< Emmoo< Bdmoo< Emmoo< .Q.H Bdmoo< Bdmoo< so pdooo< Emwzox Edmoo< Bamoo< Emmoo< Edmoo< pommom so pdmoo< mm mm mm mm sooootm mo mmmpwoa .Q.H .Q.H oa.om .Q.H mw.3m mm.mm .Q.H .Q.H .Q.H mm.mm .Q.H ones 2 9mmz.*’ .007 + ,383(3 t) (.002) (.QAZ) J’ .307 + .80B(r, t) ( .011 r.0:3) -,» .' ”b + .6743 NL ) (.091) (.0:3) ,, .006 + .SVR(R. L) (.001) (.(23) J’ .005 + .131(E t) (.002) (.002) J’ .012"; + (.003) (. .010 + . (.001) (. 003 + . (.003) ( .007 + . (.001) (. 935(_ r- O“ ‘3’ ) 930(I.l) .081) 8Ul(l.8) 02:) t+1 5,“ E.l) .186(CHE.2 (.077) .3.hti(liI.._L) (.003) 001(c.1> u.) 40 39 50 .97 .97 .97 .98 .97 94 cccmcm .Q.H Emmoo< .Q.H .Q.H .Q.H .Q.H .Q.H .Q.H .Q.H Bdeo< .Q.H to pdmoo< emmzom BmMQQ< pommmm Lo pdmoo< mm Eopmmsm do moopwma .Q.H .Q.H .Q.H .Q.H .Q.H .Q.H .Q.H .Q.H .Q.H mm.om .Q.H node Z Emmz¢m Emmoog Emmoo< Edmoo< Bdmuu< mmoo< Damoo< Emmooq Bdeo< Bmeo< Bayoo< Bdmoo< powwom so poooo< 03 Q3 3m m3 3m mm m3 m3 m3 3m mm Eocccsm no mootwoa mmmH.H mmmm.fi @33H.H moon. mmmm.a mamm. omm3. mmmw. ammo. m3mm.H womm. oauwm p Bmm DO .533 v '\ 3" d L ' D, kovlv‘l) K. .L:/ , (ov‘pt. Combined 095.1 Y: t = .004 + .flfiliR t) + .177(CRE.1) A0 .99 Regressive- ” (.001) (.030) “’ (.und) Extrapolative v~ 3 . .1 — '7 a { '1 .j’. ‘ . y . » 3 , ‘3 C flit . 2 7" ‘, = o "-1 I? i7." + o 9 I 8’ ‘\:‘x f- L ) + o l L 1‘ (t 3:1 :1 0 I2 ) 59 I :O ) ’J ('33:) ("2.3) ,1, (.0111 Error- Learning E‘ZL.1 7:: . = .01". + ,;{)9(:‘_ 1f .. + ,1..;-i§‘i-;L,1) “3.; .LL: Cyclical 3.1 Y, t = .030 + .513(R, t) + .0-1(C.1) ‘0 .'» "' (.011) (.033) 3’ (.Jgfi) Inertia 1.1 YE f = .034 + .le(Z.l) 57 .33 b .031) (.031) /\ 1.2 Y = .005 + .890(I.2) 50 .40 (.001) (.018) 97 .Q.H {awooa Bdmooa Renaud .m.3 .Q.3 .a.3 .Q.3 .Q.H .Q.H .Q.H powwom co pdmoo< emmzox aamooa aamooa coc3cm so cocoo< 33 m3 Eoooocd do mmopmoa .3.H mm.3© .Q.3 3m.m3 .Q.3 .Q.3 .Q.H .Q.3 .a.H .a.H .Q.H once 2 Emm53m Emmoo< Hammo< Bdmoo< Baumoq Bdmuoq admouq Bdmou< acmooq Hauoo< emmooa Edmooq powwow to oomoo< Q3 03 3m m3 3m mm m3 m3 m3 3m mm eoccacm - do moonmoo mmwo.3 Oman. mmmo.m memm.3 oom3.3 3333.3 m333. mwmm. m3mm.3 oomm.3 3mmo.3 O3umm u emm|m>3m one how momma Loshm :03pmo393oodm one we mp3zmomll.m m3nme 98 The Seven-Year Forward Rate The results of the regression analysis for the seven— year forward rate are contained in Table 10 and the speci- fication error test results for these regressions are shown in Table 11. The latter indicates that three of the mechanisms suffer fmxnspecification error and all three mechanisms are extrapolative mechanisms (E.l, E.2, and E.3). The remaining eight mechanisms yield high coeffi- cients of determination except for the first inertia mechanism. The sign of the second independent variable for the cyclical mechanism is opposite to that implied by the discussion of the mechanism while the signs of the coefficients for all the other variables are as antici- pated. The standard error for these coefficients indi— cate that the second independent variable for mech- anisms R.1, R.2, and C.l are insignificant. This however leaves five mechanisms which are free of specification error and consistent with the data. Although one of the five, mechanism 1.1, as an R2 sub- stantially below that of the other mechanisms, there is no basis for selecting one of the other four mechanisms as the "correct" mechanism for the seven-year forward rate. Table 10.—-Regression Results for the Seven-year Forward Rate. Class‘of Mechanism Identifi- cation Estimated Equation Number of Obs. Regressive Extrapolative Combined Regressive- Extrapolative Error- Learning Cyclical Inertia m w CRE.1 C) 33 {'1 I") EL.1 .002 + 1.06(R .002) (.09L) . t .003) (.080)“1 7’t'1 .036(E.3) (.021) .2:9 030.1) (.110) .018(c.1) (.019) ) + .Al§(EL.1) (.050) 25 2A 39 37 30 L» \D 40 .99 .99 .99 .99 .99 .65 10C) powwom .Q.H .a.3 .Q.3 .Q.3 .a.3 .Q.H .Q.3 .Q.H Damoo< .Q.3 .Q.H no pdmood Bumsox .Q.H .Q.H .Q.3 .Q.3 .u.3 .Q.H .Q.3 .Q.H .Q.3 .Q.H .Q.H once 5 Emmzqm Emmoo< Emmoo< Hmmoo< Bamoo< Dmemm 91mmo< Edmoo< eduoo< Bamooq edmoo< Bmwoo< powwom so udmoo< om mm mm 3m 03 cm 3m mm 3m m3 om Eoeooam do mwopmmo oomm.3 330m.m mwwo.3 33Qo.m m3mm.3 omng.3 3cm3. mm3m.3 mmsm.3 mmmm.3 mmwo.3 03pmm o emmom map you momma noshm :03owOHM3oodm on» mo mp33mwmll.33 039mb 101 The Nine-Year Forward Rate Nine mechanisms were tested against the nine-year forward rate. The results of these regressions are presented in Table 12 and the results of the specification error tests for these regressions are shown in Table 13. In this instance the specification error tests indicate that eight of the nine mechanisms are misspeci- fied. The one correctly specified mechanism is the first extrapolative mechanism E.l. The signs of the regression coefficients for this mechanism are as anticipated and the standard error for these coefficients indicate that the coefficients are significant. Thus, for the nine— year forward rate this mechanism can be considered as the "correct" mechanism. Summary The empirical evidence presented in the previous section indicates that for four of the five sets of depen- dent variables, there could be no selection of the "correct" mechanism. For the remaining set all but one of the mechanisms tested was misspecified. Because the first extrapolative mechanism (E.l) was free of specification error and consistent with the data it was accepted as the "correct" mechanism for the nine-year forward rate. 102? Ao3o.v A3oo.v a oo. o: Am.3oomo. + soo. u o ow m.3 mfipomcH Ao3o.v o.o Ao3o.v A3oo.v p.o mo. om A3.ovomo. I A mooso. + moo. u y 3.o 3m0H303o A533.V p.o Asoo.v Amoo.o p.o mo. 3m Am.mxoossm. + A mooo.3 + 3oo. u m m.mmo m>3pm3oomppxm Am33.v o.o Aooo.v Ammo.o p.o um>flmmmommm om. mm A3.mmooomm. + A- mvmo.3 + moo. n y 3.mmo omchsoo Ao3o.v p.o Avo.v A3oo.v p.o oo. om Am.uvomo. + A mvmom. + ooo. n » m.m Asmo.v p.o Ao3o.v A3oo.v o.o om. om Am.mvmmo. + A mommo. + moo. n 3 m.m Ammo.v o.o Ao3o.v A3oo.v s.o om. om A3.mvooo. + A momMo. + goo. u 3 3.o m>3pm30oocpxm Aooo.v o.o Aooo.v Amoo.v o.o om. om Am.mvom3. + A mv3so. + moo. u A m.m Amoo.v p.o Aomo.v Amoo.v p.o a mm. mm A3.mosm3. + A mvsmo. + moo. u o 3.x m>3mmmcamm m .WMC COHUNSUM UQUGE m COHDMO EWHCMEOQE m smassz 3p m -HofipcmoH oo mmMHo .mpmm ULMZLom mezlw23z on» pom mp33mmm 203mmmpwmmll.m3 m3pme 103 .o.3 .o.3 .o.3 .o.3 .o.3 .o.3 .o.3 eomooo .o.3 pomomm mo pomood emmzom .o.3 .Q.3 .Q.3 .o.3 .Q.3 .Q.H .Q.3 .Q.H .Q.3 once 2 emms¢m Homomm Emmoo¢ Emmoo< Emmoo< Emmoo¢ Emmoo¢ BmMoo¢ Emmoo¢ Emmoo< poohmm no uqooo< mm mm m3 om 3m mm mm m3 om Eooooam mo moonwmo ooo3.m mmmm. mmzm. comm. 3m33. mmm3.3 mmo:.m mmmz. m33m.3 03pmm p . Bmm¢m Emmoo< Bomomm Homomm Homomm Bomomm Homomm Emmoo< Homomm Homomm pomnmm no pamoo< :m.: mm.: 53.: o3.: mm.: om.: mm.: s3.: o3.: Eoommcm mo mommwom Nmmo.m :omm.m mmmm.w ommn.o mom:.O3 mo::.mm 3mmm.3 o:mm.s ozm3.o 03pmm m Bmmmm m.3 3.0 m.mmo 3.mmo m.m m.m 3.m m.m 3.x pmme nobpm Em3cwsooz :03pm03m3omom .mwmm pnmzpom hmmwlmcflz on» 30% mumme gophm coaumOHQHoQO map mo mp33mmmll.m3 mqm¢9 CHAPTER VI CONCLUSIONS AND IMPLICATIONS FOR FURTHER RESEARCH This study has investigated five categories of hypotheses concerning the formulation of expected interest rates as found in the literature on the EH and the term structure of interest rates. In order to focus attention directly on these interest rate expectations mechanisms, it was necessary to begin with an examination of the EH and the term structure of interest rates. It was demon- strated that there are alternative specifications of the independent variables that, according to the EH, determine the term structure. One of these alternative specifica- tions was selected and employed in all subsequent dis- cussions: the independent variables consist of a current rate and the term structure expected to prevail at the end of the period. Next the categories of interest rate expectations mechanisms were discussed. It was shown that the categories-— regressive, extrapolative, combined regressive-extrapolative, error-learning, and cyclical--can be distinguished in terms of three features: general market outlook, basis of assessing the current state of the market, and specific 10A 105 expectational formation. It was also demonstrated that each of the categories uniquely determines the term structure expected to prevail at the end of the period; that is, each category is conceptually unique and, there— fore, the five categories represent competing hypotheses. It was further demonstrated that the long-run impli- cations of the mechanisms are contingent upon the defi- s nitions of the independent variables employed in the mechanisms and upon the effects of unanticipated changes in the independent variables. Further it was demonstrated ‘~LJ l'. '.O-=ST.-—J*Mn. yup...“ r. 1!... that the short-run implications of the mechanisms are determined by the current values of the variables in each of the respective mechanisms and that at least one inde- pendent variable in each of the mechanisms was constructed from past interest rates. The question then was asked: given that these mechanisms are competing hypotheses, which of the mechanisms represents the "correct" mechanism for each of the sets of forward or expected rates considered? In order to answer this question correlation and regression analysis and specification error tests were employed in an empirical analysis of the mechanisms including at least one version of each category and two inertia mechanisms. The criterion for a "correct" mechanism was established: a mechanism is the "correct" mechanism for a particular forward rate if it is the only mechanism free of specification error and 106 consistent with the data. The empirical results indicated that none of the mechanisms tested could be selected as the correct mechanism for the one-, three-, five-, and seven- year forward rates. In each of these instances there were at least two mechanisms which were free of specification error and consistent with the data. In the case of the nine- year forward rate all the mechanisms except one were found to suffer from specification error. The one mechanism, an extrapolative mechanism in which the recent trend in rates was defined as the difference between the current nine—year rate and the previous period's nine-year rate, was also con- sistent with the data. Consequently this mechanism was selected as the "correct" mechanism for the nine-year forward rate. The empirical results also indicate that four cate- gories of mechanisms--the regressive, the extrapolative, the combined regressive-extrapolative, and the cyclical-- all yield high coefficients of determination because each of these mechanisms contains a common independent variable. Indeed when only this variable was used as the sole inde- pendent variable there was no significant loss in explana- tory ability. It would appear, therefore, that although these categories of mechanisms are conceptually unique they are not unique operationally. This eXplains why one investi- gator,“5 when employing a regressive mechanism, found no differences in his results regardless of how he defined the normal level. It also explains how two studies relatively uSMalkiel, op. cit. 107 similar in format but one employing a regressive mechanism and the other a combined regressive-extrapolative mechanism obtained similar results.146 As a word of caution it should be pointed out that there is a basic difference between the conceptual frame- work of the analysis and the statistical approach employed. m According to the EH forward or expected rates are deter- 4 mined initially and then current rates are determined. However, if a mechanism that included a current rate for m-_i:"4 ' each maturity were used to forecast each and every for- ward rate then all the current rates would have to be known initially. That is, all the current rates cannot be deter- mined by the EH which serves as the theoretical foundation of this study because all the current rates would have to be known prior to the determination of the forward rates. This problem, of course, does not represent an invalidation of the approach used in the empirical analysis for the problem concerns the general applicability of the mechanisms in forecasting the entire term structure while the empirical analysis was concerned with the reliability of the mechan- isms in explaining selected forward rates. There are several implications for further research. It is possible to establish additional criteria for selec- tion of the "correct" mechanism. For example one might ”sThe two studies are Malkiel, op. cit. and Modigliani and Sutch, op. cit. 108 also compare the signs of predicted changes in rates with the signs of actual changes. There is also the possibility that a change in the testing format may allow the selection of the "correct" mechanism. In this instance one might examine the ability of the mechanisms to explain anticipated changes in expected rates. At a more general level there is the problem of the widely conflicting evidence concerning the validity of the EH. Those studies that reject the EH, typically, are structured in such a way that they require evidence of v I m‘mm-“ ‘14.. -H’. in? successful forecasting for acceptance of the EH while those studies that accept the EH do not employ this decision criterion. It would appear that one's view on the validity of the.EH depends on whether or not he accepts evidence of successful forecasting as the appropriate decision criterion. However, it may be argued that the evidence concerning successful forecasting is incomplete. Previous studies have been concerned with their realized rates of return on selected securities over short periods of time or the success of the market in predicting future short rates. Perhaps by examining the ability of the market to predict the entire term structure over various lengths of time a reconciliation of the conflicting evidence on the EH can be accomplished. APPENDIX A Review of Previous Empirical Investigations of the Expectations Hypothesis 109 w: A Review of Previous Empirical Investigations of the Expectations Hypothesis This appendix is divided into two parts. Part A con— tains a description of those investigations which employ evidence of successful forecasting as the criterion for acceptance or rejection of the EH. Part B, logically enough, includes a description of investigations which do not employ such a criterion. In part A a strict chrono— ." fir l1. .1 r." .-'. 1 logical presentation is made beginning with the least '.'.~ A recent investigation while part B is subdivided into w T-r-‘fu‘nfi. sections, sections determined by a particular approach to the term structure problem. A. Tests Requiring Evidence of Successful Forecasting for Acceptance a? the ’ Expectations Hypothesis Macauly argued that if the EH is valid then the pronounced and well known seasonal variation in call money rates that occurred before the establishment of the Federal Reserve System should have been anticipated by variations in the time money rategLl7 that is an upturn in rates on loans from one to six months, loans similar to call money loans, should have preceeded the seasonal upturn in call money rates. Comparing the two series Macauly found that the time money rate did move before the call money rate and, therefore, concluded that there was evidence of defi- nite and relatively successful forecasting. “7Frederick R. Macauly, The Movements of Interest Rates, Bond Yields, and Stock Prices in the United States Since 1856 (New York: NationalPBureau of Economic Research, 1938). llO lll Hickman in his endeavor to find evidence of successful forecasting made three different tests.)48 In one test Hick- man compared the yield curves implied by the term structure during the l935—19H2 period and subsequently observed actual rates. The second test involved a comparison of the signs of predicted and actual changes in one-year spot rates. The third test was a comparison of the actual changes in a long rate with the change in the same rate as implied by the term structure. In all three cases, Hickman could find no evidence of successful forecasting and, thus, rejected the EH. Walker, like Macauly, isolated an instance where the expectations of the market could be presumed to be accurate.“9 In Walker's case it was the behavior of the securities market before and after the enactment of the interest rate stabil- ization policy of World War II. The prestabilization term structure implied that future short-term rates would be higher than the then current short-term rate while the stabilization policy dictated that future short-term rates would be the same as the then current short-term rate. If the financial community became convinced that the monetary “8W. Braddock Hickman, "The Term Structure of Interest Rates: An Exploratory Analysis" (New York: National Bureau of Economic Research, 19u2). ugCharles E. Walker, "Federal Reserve Policy and the Structure of Interest Rates on Government Securities," Quarterly Journal of Economics, LXVIII, February, 1954, pp. 19’n20 —Y 112 authorities would actively support the stabilization policy then, according to the EH, there should have been a switch out of shorts and into longs. The fact that such a switch did occur led Walker to conclude that the EH does prevail. According to Culbertson,50 if the EH is valid then holding period rates of return on short-term and long-term securities should be the same for identical periods of time. Culbertson calculated realized rates of return, including capital gains and losses, to investors who held either long-term Treasury bonds or Treasury bills for one- week and three month holding periods during 1953. When he found marked differences in these rates of return, Culbertson concluded that it would be foolish to argue that speculators would Operate in the goverment securities market and predict as badly as his results suggested. Therefore, he rejected the EH. Kessel examined the relationship between forward two-, four-, six-, eight—, nine-, and thirteen-week Treasury 51 He bill rates and subsequently observed actual rates. analyzed each rate for the frequency of high and low pre- dictions and calculated the average size of the prediction error. Since there were a larger number of high predictions 50John M. Culbertson, "The Term Structure of Interest Rates," Quarterly Journal of Economics, LXXI, November, 51Kessel, op. cit. .4541- V, 4’ ' ‘ -u -. . 113 for each rate and all average errors were positive, Kessel concluded that forward short rates were biased and high estimates of future short-term rates and, therefore, the EH was, at best, an incomplete explanation of the term structure. Michaelson attempted to test for the following alternative theories concerning securities markets:52 (a) uniform expectations—-no risk aversion, (b) uniform expectations with risk aversion, (c) diverse expectations with risk aversion, and (d) segmented markets. He developed three hypotheses concerning realized yields: (1) realized yields closely approximate anticipated yields, (2) when (1) does not hold realized yields pri- marily reflect changes in expectations, and (3) realized yields primarily reflect shifts in non-expectational factors. For acceptance of (a), the uniform expectations-— no risk aversion theory then, (1) must hold. Hypothesis (2) is compatible with any of the first three theories while hypothesis (3) implies the segmented market theory. Michaelson calculated the yields for weekly periods between January 5, 1951, to December 28, 1962, for selected Treasury bills, each series from one to thirteen weeks, and Treasury bonds with 2%, 5, 7%, and 10 year maturities. He then examined the means of the time series of realized v7 52Jacob B. Michaelson, "The Term Structure of , Interest Rates and Holding Period Yields on Government Securities," Journal of Finance, XX, September, 1965, pp. AAA-463. P 1f?" 11a yields and found that they tended to increase with increases in term to maturity. This was interpreted as supporting the uniform eXpectations with risk aversion theory. The standard deviations of these series also tended to increase with increases in term to maturity. None of the theories provides any explanation for this. Finally Michaelson correlated the various time series with each other. His results suggest that unanticipated price changes are, primarily, the result of positively correlated changes in expected short rates. These later results are in accord with the three expectationally oriented theories. B. Tests Not Requiring Evidence of Successful Forecasting for Acceptance ofvthg EH 1. Application of the Error Learning Mechanism Meiselman begins with the general hypothesis that changes in interest rates can be related to factors which systematically cause revisions in expectations.53 Two specific hypotheses follow from this general hypothesis: unanticipated changes in eXpected short rates are related to the error in forecasting the current short rate: (A°l) t+lrl,t ‘ t+lrl,t-1 = a + b (Rl,t ’ trl,t-l) 53Meiselman, 0p. cit. -"-"‘l' tong-'10. J- ‘ I; .. - . ":w ._ . , . .‘VJ #1:." 3‘. in} 115 and, because a long-term rate is an average of current forward short rates, unanticipated changes in the long term rate are also based on errors made in forecasting the current short rate: ° t+lrn,t ' t+1rn,t-l = a + b (R (A'Z) l,t ' trl,t-l) where n > 1. Since data limitations prevented the calculation of unantic- ipated changes in the long rate, actual changes in the long- rate were used in the tested version of equation (A.2); that is: (A.3) AR = a + b (R n,t 1,t ‘ trl,t-l) In addition to the testing of equations (A.1) and (A.3), Meiselman compared the sign of change as predicted by the error learning mechanism with the sign of the actual sign. Using the Durand data Meiselman was able to calculate eight forward one-year rates over the 1901-1954 period. In tests of equation (A.l) he found that unanticipated change in each of the forward rates is highly correlated with the forecast error although the correlation coefficients decreased as maturity increased. For all eight versions of equation (A.l) the constant term was not significantly different from zero. In the tests of equation (A.3) Meiselman used the thirty-year rate. Again the correlation unlit” 3-x ' Y” . .0 n b" 116 between changes in this rate and the forecast error was high. In comparing the signs of predicted changes and actual changes, Meiselman found a high degree of synchron- ization. On the basis of this evidence, Meiselman concluded that expectationally motivated behavior dominates the market. Grant examined the Meiselman hypothesis in detail and pointed out certain deficiencies in data used by Meiselman.514 Grant then applied the Meiselman model to British data. The data used by Grant were supplied to him by a British brokerage firm and covered the years 192N-l962 by quarters. Four one—year forward rates were derived and used in testing. equation (A.l). In these tests Grant found that the error learning hypothesis did not achieve a high degree of explanatory power. A third degree parabola as well was fitted to the data but with only a slight improvement in results. Grant also examined the extent to which the actual signs of unanticipated changes in the forward rate were synchronized with the signs of the changes as predicted by the error-learning mechanism. Again the evidence was not convincing. Thus, Grant concluded that the usefulness of this hypothesis in explaining observable behavior was low. SuGrant, op. cit. 117 Wood agreed with Meiselman that expected future short- term are related to changes in prevailing short-term rates.55 However, he expanded the Meiselman model. Briefly, the expansion is as follows: (A.u) = fn (R t+nrl,t ' t+nrl,t-l 1, t ’ trl,t-l)’ but if the EH prevails then any n-period market rate can be looked at as a geometric mean of the implied short-term rates spanning n-periods. Thus: (A.5) Rn,t = {(1 + t R1 ,t)[1 + a2 + b2(Rl,t ‘ trl,t-l) + t+1r1,t—1]"° A. _ n- - 000 [l + an + b n(R1, t ' trl,t-l)+ t+n-lrl,t-l] ’n ' 1’2’3 or (A.6) Rn,t = [(1 + t R1 ,t)(1 + a2 ,t + b2,tRl,t) _1_ n-l (1 + an,t + bn,tRl,t)] where b2, b3, . . . bn are measures of the responsiveness to R1,t of t+lrl,t . . . t+n-lrl,t and a2,t . . an,t vr V fi 55John H. Wood, "Expectations, Erorrs, and the Term Structure of Interest Rates, " Journal of Political Economy, LXXI, April, 1963, pp. 160-171. 118 encompass all the factors besides Rl t which influence 3 expectations of future short rates. Through further assump— tion Wood evolves the regression equation (A17) ARn,t = an + BnAR1,t + e A A where d and B are leastsquare estimators and e1 is the random error term. Wood tested equation (A.7) against both the Durand data and on monthly averages of daily rates on U. S. Government securities for the period l9u7-l962. With the former data 16 tests of equation (A.7) were made where n ranged from 2 to Ho. The correlations were high although again declining as n increased. With the U. S. Government security data three tests were made using A-, 16-, and 50- year maturities. In this instance the fits were not as good. Wood concluded, therefore, that future long rates can be expected to change less than future short rates because of changes in current short rates but nonetheless future long-rates will still change by a significant amount. Van Horne applied the error learning model56 to Treasury yield curve data that appear monthly in the Treasury Bulletin. His tests covered the 1954-1963 period and include 11 forward one-year rates. Applying the error learning model [equation (A.l)], he found that a significant 56Van Horne, op. cit. 119 portion of the movements in forward rates for Treasury securities can be explained by the errors in forecasting the current one-year rate. However, Van Horne also found that the constant term in the regression equation for each forward rate was positive and significantly different from zero. This implied that an additional explanatory variable was needed. Because a constant term significanlty differ- ent from zero in equation (A.l) implies the existence of a risk premium, Van Horne constructed a variable which might explain the extent of risk premium, The added variable was the deviation of the actual forward rate from its accus- tomed level where the accustomed level was defined as an average of beginning levels or, what is the same thing, of past forward rates. The results of the multiple regression analysis were viewed asconsistent with the supposition that interest rate risk varies inversely with the level of interest rates.57 Bierwag and Grove devise and test variations of the 58 These variations, Meiselman error learning mechanism. however, are derived from their own model of the term structure. In order to present meaningfully the results 57This Van Horne study has several shortcomings which are discussed by Richard Roll, "Interest-Rate Risk and the Term Structure of Interest Rates: Comment," Journal of Political Economy, LXXIV, December, 1966, pp. 629—632. 58 Bierwag and Grove, op. cit. 120 of their test, it is necessary to briefly reconstruct their own theoretical structure. Bierwag and Grove begin with an attack upon the implicit assumption concerning investor behavior within the Meiselman model: the assumption of single valued expecta- tions means that investors would never diversify among securities. Thus forward rates cannot be interpreted as unbiased expected rates. The crux of their argument is that identical single-valued expectations means that the yield structure cannot be determined since no investor would undertake to purchase an infinite supply of bonds forth- coming if the common predicted rate were less than the ruling forward rate or to issue bonds to satisfy the infinite demand if the common predicted rate exceed the ruling forward rate. According to Bierwag and Grove therefore, the forward rate in equilibrium is a weighted average of investors' predicted rates. The importance of the i th investor's opinion in the determination of the equilibrium forward rate varies directly with the size of his investment fund, his audacity (willingness to tolerate variance of return), and the confidence he has in his prediction. Algebraically (A.8) D R i( ll M'U = 96 t+lrn,t 1 1 1,t+1 )Vi 121 = the individual investor's prediction of the next period's one-year rate and * where E o .M .c '2' - ." '? I L i's weights adding up to one but declining as 130 where n should be appreciably smaller than m and the weights 81 would decline rather rapidly. Combining equations (A.28) and (A.27) they Obtain (A.29) ARg = -aR + b R i 2,t-i + dc "MS 1 where a = (cl - a2), 1 b a vU - a U with S defined as zero for i > n and 1 l 2 i i . d = d1 (1 - v) f Substituting the last equation into equation (A.2A) then, i m (A.30) RL’t - Rs’t — -BaRL,t + ingbistt‘i + B dc + Ft. The summation term in this last equation represents the difference between two lag structures which is pre- sumably similar to a fourth degree polynomial. But M-S do not test equation (A.30). Instead of using a distributed lag of the long rate, they use a distributed lag on the short rate (to avoid possible bias without, presumably, altering the basic behavioral relationships), and use the spread as the dependent variable. m (A.31) R - RS = d + B R + 2 B R 131 They applied this last question to quarterly U. S. 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