Article

Expectations of Inflation and Interest Rate Determination

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1984
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The relationship between nominal interest rates and expectations of inflation has received widespread attention over the years.1 Although the theoretical issue is not a new one, the relatively high rates of inflation and nominal interest rates experienced in recent years have brought the question into sharp focus. The concern of policymakers has been directed not only at the large fluctuations in nominal and real interest rates but also at real interest rates that are very high relative to historical levels and have risen rapidly to those levels.

The concern over the behavior of real interest rates stems from the prominent role they play in macroeconomic analysis and policy formulation. First and foremost, it is widely believed that high real interest rates discourage productive investment and thereby lead to economic slowdown, which hinders growth.2 Second, the movements in real interest rates constitute the main link in the economic transmission mechanism whereby shifts in monetary policy induce changes in economic activity—assuming that the monetary authority is, indeed, able to effect changes in real interest rates. Third, it is often argued that policies resulting in higher real interest rates tend to increase the external value of the domestic currency owing both to the induced capital inflows and to the strengthening of the current account position resulting from the dampening effect of higher real rates of interest on economic activity.

Although the importance of real interest rates in economic analysis is widely recognized, not much research has been undertaken on their determination, and the theory developed by Irving Fisher over half a century ago remains the foundation of most current investigations of the subject. Nevertheless, Fisher’s theory is a controversial one, and particularly when it is interpreted as suggesting a constant real rate. The controversy is due, in part, to the fact that real interest rates are not observed directly, but must be inferred from the behavior of the rates of nominal interest and inflation. Because the concept of a real interest rate involves expectations of future rates of inflation, any analysis of real rates requires assumptions regarding the formation of expectations. Consequently, hypotheses tested, and conclusions arrived at, regarding real interest rates depend upon the validity of the assumed mechanisms of expectations formation.

This paper adopts a statistical approach to the determination of expectations. One advantage of this approach is that it minimizes the amount of economic theorizing one must do in order to obtain a meaningful series on expected rates of inflation that can then be used to generate time series on real rates of interest. An estimated system of equations is then used to derive a representation of the unobserved ex ante real rates of interest as a linear function of the “information” set presumed to be known by economic agents. The methodology developed here is a general one and could be used with alternative specifications of the information vector. In this sense, the present paper is preliminary, developing the methodological framework and providing a simple application.

This paper is organized as follows. Section I contains a selective review of the literature, with discussion of some theoretical issues pertaining to the behavior of real rates of interest. Section II deals with the appropriate econometric methodology to be used in testing the theory. Section III presents the empirical results, which are relevant not only for the information they convey regarding real rates of interest but also regarding the dynamic structure of the economies examined. Section IV summarizes the conclusions of the study.

I. Theoretical Issues

role of real interest rates

According to neo-Keynesian theory, a change in the real rate of interest alters the demand for investment by changing the present value of the future stream of returns to capital goods. A rise in the real rate of interest lowers the value of the existing stock of capital, because the future stream of returns is discounted more heavily. As the value of existing capital falls, relative to its replacement cost, the demand for investment is reduced, leading to a fall in asset demand and employment.

The link between the real rate of interest and investment demand is also a crucial aspect of the mechanism through which monetary policy affects the real side of the economy. Within the Keynesian framework, changes in the rate of monetary expansion primarily induce changes in nominal interest rates, through the portfolio allocation behavior of asset holders. To the extent that changes in nominal interest rates are translated into changes in real interest rates, and to the extent that investment demand responds to these changes, monetary policy is able to affect economic activity.

An understanding of the behavior and the determinants of real rates of interest is also crucial for an understanding of foreign exchange markets, particularly in terms of the impact of alternative policies upon exchange rates. Higher real interest rates cause an appreciation of the domestic currency, both through increased capital inflows and through the current account impact of the economic slowdown. It is thus frequently argued that excessive reliance on monetary measures to combat inflation induces large fluctuations in exchange rates owing to the response of real interest rates to restrictive monetary policy.

The foregoing analysis is essentially Keynesian, in that it is based, at least implicitly, on the assumption that the elasticity of real interest rate with respect to money is non-zero, which, in turn, implies that nominal interest rates and expectations of inflation respond differently to changes in policy. Although one can give several justifications for this differential response, the most common is that expectations adjust slowly to a changing economic environment. It is also assumed that nominal wages are fixed in the short run. With the sluggish response of expectations, changes in monetary policies will indeed alter the real interest rates, thereby allowing the monetary authority to attenuate cyclical fluctuations in economic activity through the impact of variations in the real interest rate upon aggregate demand. An implication of the sluggish response of price expectations and other rigidities is that policymakers face a trade-off between unemployment and inflation.

The two crucial elements of the mechanism through which the policy actions of the monetary authorities affect employment, non-zero elasticities of real interest rates and real wages with respect to the money supply, have been criticized by the monetarists. Earlier critics emphasized that any trade-off between inflation and unemployment is only temporary, resulting from a slower response of wages than prices to changes in nominal demand. According to this view, as elaborated by Milton Friedman3 in his presidential address to the American Economic Association,4 monetary expansion results in an initial fall in real wages, which will indeed stimulate employment, expanding it beyond the natural rate. However, as expectations begin to adjust to higher rates of inflation, the increase in nominal wages will accelerate. Owing to the presence of excess demand for labor in the economy, the rise in nominal wages will exceed the rate of inflation, pushing real wages up. Real wages will continue to rise until equilibrium is restored at the natural rate of employment with a higher rate of price and wage inflation.

In a similar fashion, monetarists argue that the impact of changes in monetary policy upon the nominal interest rate is related to the rate at which expectations adjust to the new economic policy. In view of the expansionary monetary policy, the public comes to expect a higher rate of inflation, and nominal interest rates will increase rather than decrease because, as first proposed by Fisher, lenders anticipating demand will increase the rates and borrowers will be willing to pay a higher rate of interest.5 It is clear, however, that Fisher, in formulating the relationship between the money rate of interest and the value of money, was thinking mainly in theoretical terms. He believed that in practice, the changes in the value of money would not be fully reflected in the nominal interest rates, mainly owing to a lack of foresight. Furthermore, he suggested that “in so far as there exists any adjustment of the money rate of interest to the changes in the purchasing power of money, it is for the most part (1) lagged and (2) indirect.”6

measuring the fisher effect

The renewed prominence of Fisher’s proposition following Friedman’s presidential address stimulated a substantial amount of research on the relationship of nominal interest rates to expectations of inflation. Many of the earlier studies were based on the theoretical expression whereby the nominal interest rate was expressed as the sum of the real interest rate, the expected rate of inflation, and a residual term distributed independently of the real rate and expectations of inflation. A fundamental difficulty with empirical implementation of this theoretical expression is the lack of independent data on either the real rate of interest or expectations of inflation. The approach adopted by Fisher and others was to maintain specific hypotheses about real rates and formation of expectations by the public. Generally, it was assumed that the real rate of interest, being a function of deeper economic variables, like the rate of time preference and various marginal rates of substitution and transformation, moved slowly and therefore could be approximated as constant. Expectations of inflation, though, were modeled within an error-learning framework in which the public was assumed to form its expectations about future rates of inflation as a distributed lag of past rates of inflation. Much of the earlier econometric work has proceeded along these lines, differing mainly about the exact specification of the distributed lag that was presumed to generate the expectations of inflation and about the presence of other economic variables that might play a role in the determination of interest rates.

In these earlier studies, the objective was to estimate the magnitude of the “Fisher effect,” which was defined as the amount of change in nominal interest rates resulting from a change in the expected rate of inflation. Given this objective, it was natural to proceed by regressing nominal interest rates on a measure of the expected rate of inflation. If the estimated coefficient were significantly different from unity, it was argued, then the monetary authority could cause systematic variations in real interest rates and, therefore, stabilize fluctuations in output. A number of studies conducted along these lines estimated that the magnitude of the Fisher effect was less than unity, thus rejecting Fisher’s hypothesis.7 These findings were interpreted as suggesting that an expansionary monetary policy leads to a fall in real interest rates because the nominal interest rates fail to adjust sufficiently to the higher rate of anticipated inflation. As real interest rates fall, aggregate demand is stimulated by the ensuing increase in investment demand.

Recent studies suggest, however, that the method used to estimate the magnitude of the Fisher effect is not entirely appropriate and that the results are likely to be biased. Often, the mechanism of expectation formation was proxied by a distributed lag of the actual rate of inflation, with the sum of distributed-lag coefficients constrained to be 1. Under this restriction, one can easily show that in a regression of nominal interest rates on current and lagged rates of inflation, the sum of estimated coefficients is, indeed, an estimate of the magnitude of the Fisher effect. This particular restriction, however, is arbitrary and does not follow from any firm economic reasoning. It is like a neutrality result whereby a permanent increase in the actual rate of inflation leads to an equivalent increase in the expected rate of inflation after a sufficiently long adjustment period. There is little dispute about the requirement that permanent increases in the actual rate of inflation ought to lead to permanent increases in the expected rate of inflation; this is a consistency requirement. What is disputed is that consistency implies the sum-to-unity restriction. As it turns out, this is not the case, and the sum of the distributed-lag coefficients in the formation of expectations could have any value without violating the consistency requirement.

Apart from the questions regarding identification, a more important issue is the doubt that the magnitude of the Fisher effect can shed any light on the ability of the monetary authority to implement systematic changes in real interest rates. This point may be illustrated by the following interpretation of the test procedure.

Consider an economy in which borrowers and lenders can draw up contracts denominated in terms of either money or a composite commodity called the consumption basket. Let the interest on loans denominated in terms of the consumption basket be denoted by ρt, and the interest on loans denominated in terms of monetary units by rt. Furthermore, suppose that economic agents anticipate that the price of the consumption basket in terms of money will increase at the rate πt, over the period, at the end of which the loan contracts mature. If these markets are dominated by risk-neutral agents and are efficient, then the anticipated real yields will differ from each other by only a random term. Let ρtb denote the ex ante real rate of interest on a contract denominated in nominal terms. Risk is expressed by ρtbρt=ut,, where ut denotes a random-term orthogonal8 to the information available to economic agents at the time the contracts are drawn up. The efficiency of the market can be tested by regressing ρtbρt on the anticipated rate of inflation. The estimated equation will be of the form

and the null hypothesis to be tested is α = 0. If πt, is added to both sides of the equation and the terms are rearranged, one obtains

which reduces to a regression equation under the assumption that the yield on contracts denominated in units of the consumption basket is constant. Under the present interpretation, an estimate of a that is significantly different from zero will indicate that the joint hypothesis of risk neutrality and market efficiency could be rejected. Such a result implies that, at equilibrium, the real interest rate differential between the two assets varies systematically with the anticipated rate of inflation, but the result does not convey much information regarding the effectiveness of monetary policy to induce changes in real rates of interest. In fact, unless one entertains some additional assumptions, it is not possible to infer, from the estimates of a, the direction of change in either ρt, or ρtb.

Under an alternative, and somewhat more orthodox, interpretation, a regression of nominal interest rates on a measure of the expected rate of inflation derives its content from the definition of the anticipated real yield on an instrument as being equal to the difference between the fixed nominal yield and the anticipated rate of inflation.

In equation (3), rt and ρt denote the nominal and ex ante real yields on the same instrument, and πt denotes the expected rate of inflation. Note that equation (3), being the definition of ex ante real yield, is exact when πt is the true (but unobserved) expected rate of inflation. The projection of ex ante real yield on a constant and on the expected rate of inflation is given by

where ut is orthogonal to the constant and πt. Substituting equation (4) into equation (3), one obtains the following representation of nominal interest rate:

Because ut is orthogonal to πt, equation (5) is the projection of nominal interest rates on a constant and on the expected rate of inflation, and consequently c and α can be estimated consistently by ordinary least squares through equation (5). If α is significantly different from zero, then Fisher’s hypothesis is rejected by the data for the instrument in question.

Often these conclusions are extended to imply that variations in the expected rate of inflation cause variations in real rates of interest, and that monetary policy is a potent instrument of stabilization because alternative policy measures result in variations in real interest rates and thus affect real variables. Neither of these extensions are justified, however. The fact that the estimate of a in equation (5) is significantly different from zero implies only that anticipated inflation and real interest covary, without providing any evidence as to the direction of causality. Similarly, it is not possible to pass judgment on the efficacy of monetary policy from the estimates of α. For example, if both real interest rates and expectations of inflation respond to a set of real exogenous variables, whereas only expectations respond to nominal variables, real interest rates will be correlated with the expected rate of inflation, resulting in an estimate of a which is significantly different from zero, even though monetary policy is ineffective in controlling real interest rates.9

The foregoing considerations indicate that alternative formulations aimed at measuring the magnitude of the Fisher effect are best considered as variations on a single theme: measuring the correlation between the real rate of interest and the expected rate of inflation, both of which are unobserved by the econometrician. Moreover, Fisher’s proposition that an x percent increase in the expected rate of inflation results in an x percent increase in the nominal interest rate is vacuous unless some hypothesis is maintained regarding the behavior of real interest rates.10 In order to convert Fisher’s proposition into a hypothesis with testable implications, it is sufficient to assume that real interest rates are orthogonal to the expected rate of inflation. This is an interesting hypothesis with some implications for economic behavior, although, it must be emphasized, it conveys no information regarding the potency of monetary policy. A test of this hypothesis is formulated in Section III and implemented for France, the Federal Republic of Germany, Japan, the United Kingdom, and the United States.

role of expectations

A further set of issues regarding the expectation proxies emerged from analysis of the rational-expectations hypothesis, which shifted the focus of investigations from estimation of the magnitude of the Fisher effect to the determination of criteria that will help decide what patterns of distributed-lag coefficients are plausible. A major econometric lesson arising from this analysis is that any identifying restrictions should be derived from the same principles that are assumed to guide economic behavior. If such behavior derives from a process of optimization, then one ought to assume that agents optimize not only with respect to their decision variables but also with respect to their techniques of forecasting. Such an assumption implies that forecast errors should have no systematic behavior relative to the information set which formed the basis of the expectations in the first place. In the more specific case of linear decision rules, the foregoing implication translates into the orthogonality principle, which states that forecast errors must be orthogonal to those variables, a linear combination of which gives the forecast itself.

This orthogonality principle was a key ingredient in Fama (1975), which generated substantial controversy. His formulation is based on expressing the ex post rate of inflation as the sum of the expected rate of inflation and a forecast error. Hence, the expected rate of inflation is equal to the actual rate of inflation minus the forecast error. In this formulation, equation (3) can be written as

where xt+1 denotes the actual rate of inflation between times t and t + 1, and et+1 denotes the forecast error. The coefficient ψ, whose theoretical value is 1, is added to the equation in order to have an estimable equation. Under the maintained hypothesis that the real rate of interest ρt is constant, equation (6) reduces to

Equation (7), in turn, can be solved for xt+1, yielding the form that was actually estimated

The orthogonality principle, jointly with the maintained hypothesis of constant real rate of interest, ensures that equation (8) can be consistently estimated by ordinary least squares. Fama’s results, based on monthly data for the United States and covering 1953–71, support the hypothesis that ex ante real rate of interest is constant. Furthermore, his estimate of ψ is not significantly different from unity and therefore fails to reject Fisher’s hypothesis.

Fama’s paper was criticized on a number of points, particularly for an apparent lack of robustness in its results with respect to the sample period.11 Subsequent work, using identical data and methodology, indicates that as the period of estimation is extended, the estimates of ψ decline, accompanied by increases in R2 and the estimated first-order serial correlation of residuals.12 This observed pattern of estimates is symptomatic of a bias that varies slowly but systematically over time.

Such a bias could result if the ex ante real interest rate is not exactly equal to a constant, but has a stochastic component ut. It can be easily shown that in the presence of a stochastic component ut, the least-squares estimate of β in equation (8) will be biased by an amount equal to βσu2/σr2. These criticisms notwithstanding, Fama’s paper was influential because of its novel application of the orthogonality principle.

In addition to the orthogonality principle, the rational-expectations hypothesis has a number of other concrete implications that can be used to derive testable propositions from maintained hypotheses in general, and from the Fisher hypothesis in particular. Under the rational-expectations hypothesis, subjective expectations of the public are equal to the conditional expected values of the same variables based on the information available to the public at the time expectations are formed. Consequently, under rational expectations, the expected rate of inflation πt is equal to E(xt + 1 | Ωt), where xt+1 denotes the rate of inflation between times t and t + 1; Ωt, denotes the information set available to, and utilized by, the public in forming its expectations; and E denotes the conditional-expectations operator. In general, it must be recognized that Ωt, includes historical values, up to and including period t, of readily available time series, such as interest rates, rates of inflation, and money supply. It must also be recognized that Ωt, probably includes other variables, some of which may not be quantifiable. However, for the purpose of hypothesis testing, it is not necessary to know the contents of Ωt; it is sufficient to have knowledge of a subset of Ωt, say Ht.

With these preliminaries taken care of, equation (3) can be rewritten, using the new notation, as

If one takes conditional expectations on both sides of equation (9), conditional on Ht, one obtains

It is a well-known result in statistics that E[(E(xt+1t) Ht] = E(xt +i | Ht) whenever Ht is a subset of Ωt.13 Under the assumption of a constant real rate of interest, one obtains

The message conveyed by equation (11) is clear: projection of the nominal interest rate r, on a set of variables should be equal, up to an additive constant, to the projection of the rate of inflation between times t and t + 1—xt+1—on the same set. This is clearly a testable proposition.14 All that is necessary is to regress rt, and xt+1 on a given set of variables and test whether the two regressions are identical. Of course, the power of such a test depends upon the extent to which information contained in Ωt but excluded from Ht helps to predict the rate of inflation. Let It be the complement of Ht in Ωt. Then, under the null hypothesis, residuals from the regression of nominal interest rates on Ht are equal to E(xt+1|It), and those from the regression of the inflation rate on Ht are equal to E(xt+1|It) + et+1 where et+1 denotes the forecast error in inflation when these forecasts are based on Ωt. If the variance of E(xt+1|It) is high (indicating that some important variables are excluded from Ht), the power of the test is reduced accordingly.

This line of reasoning was initially developed and empirically implemented by Shiller (1972). Although he did not conduct a formal statistical test, Shiller’s results suggest that Fisher’s hypothesis is a reasonable working hypothesis and seems to work fairly well.15

nature of the problem

Although Fama and Shiller both used certain principles derived from the rational-expectations hypothesis, they failed to take into account the general-equilibrium nature of Fisher’s hypothesis. As Sargent (1973a), and subsequently Begg (1977), have forcefully argued, the proper framework within which to study Fisher’s hypothesis belongs to macroeconomic analysis, and the econometric methodology must take into account the general equilibrium nature of the problem. A principal reason for this follows from the maintained hypothesis that expectations are formed rationally. Under rational expectations, it is presumed that the agents understand the economic environment in which they operate and the implications of alternative policies upon this environment. If, for example, alternative monetary policies induce alternative time paths for the price level, then the agents will try to predict the likely courses of action by the policymakers and will incorporate such predictions into their expectations. Consequently, the mechanism through which the forecasts are generated is intertwined with the linkages between the various sectors of the economy.

Once the macroeconomic nature of the relationship between the interest rate and expectations of inflation is realized, the appropriate econometric methodology to investigate this relationship becomes more complex. Generally, tests based on single-equation methods are inappropriate. This follows from the fact that macroeconomic models are characterized by simultaneities, and, unless one holds strong a priori beliefs to the contrary, it must be assumed that all contemporaneous variables are correlated with all structural disturbances. Under these circumstances, single-equation methods, where some of the regressors are dated the same as the dependent variable, will yield biased estimates.

There are, of course, ways to alleviate the problem and obtain consistent parameter estimates. One can use any one of the family of instrumental variable estimators. The basic idea here is to substitute linear combinations of some exogenous variables for the endogenous variables that appear as regressors. If the exogenous variables are uncorrelated with the residuals, then so will any linear combination of them. Thus, replacement of the endogenous variables with linear combinations of exogenous variables results in consistent parameter estimates. For the application of these methods, the choice of instruments is crucial, particularly when the exogenous variables in the system are stochastic.

An alternative, and somewhat more attractive, strategy to overcome the problems inherent in single-equation methods would be to investigate Fisher’s hypothesis within a general-equilibrium framework. The strategy in conducting such a test would be to construct a complete macroeconomic model that incorporates Fisher’s hypothesis as a restriction and to test the model against data. To the extent that the model is not rejected, the hypothesis will be verified.

This approach was used by Sargent (1973 c) to test Fisher’s hypothesis jointly with the natural-rate hypothesis and the rational-expectations hypothesis. Within the context of a macroeconomic model, Sargent conducts four alternative tests,16 two of which call for rejection of the composite hypothesis. Although there is some degree of ambiguity in interpreting his results, the evidence against the composite hypothesis seems to be stronger, since the tests which call for rejection are more powerful.17

Although the strategy of testing the Fisher hypothesis within the context of a macroeconomic model is appealing, it has a number of drawbacks. First of all, when a complete macroeconomic model is specified, it is comprised of a number of maintained hypotheses, and what is being tested is the resultant composite hypothesis. If the data does not call for its rejection, then there are no problems. However, if any one of the individual hypotheses is at variance with the data, the composite hypothesis will be rejected without providing the researcher with a clue to what aspect, or aspects, are causing the rejection.

Second, for the purposes of hypothesis testing, it is necessary to identify structural parameters. Identification of the structure, however, is not easily accomplished. The inappropriate manner in which econometricians approach structural identification was elucidated recently by Sims (1980). He argues that in order to be able to achieve identification of the structure, it is common practice to make shrewd aggregations and exclusion restrictions. Such restrictions are ad hoc and have no firm foundations in economic theory. As a result of these identification restrictions, macroeconomic models tend to be parsimoniously parameterized. Although such restrictions and parsimonious parameterizations may be reasonable for partial-equilibrium analysis, the general-equilibrium properties of models thus identified may be extremely distorted. Since identification of large-scale macro models lacks credibility, and therefore should not be taken seriously, such models are inappropriate for testing alternative economic theories.

An equally damaging assessment emerges regarding the classification of variables as exogenous and endogenous. Upon closer examination, it is seen that variables that have traditionally been considered exogenous turn out not to be so. For example, policy variables are determined not in isolation from the economic environment but rather in response to it. Furthermore, the rational-expectations theorists have forcefully argued that policy formulation should take the form of setting rules for systematically changing the policy variables as economic conditions change. Doing this amounts to explicitly endogenizing the policy variables and further weakens the credibility of identification.

Nevertheless, macroeconomic models are useful tools for policy analysis and forecasting, as Sims carefully stresses, in spite of the fact that reduced forms of such models will be distorted by false identification restrictions. The reason is that the restricted estimators can actually produce forecasts or projections with a smaller error than unrestricted estimators, provided that the restrictions are not grossly inappropriate.

The foregoing considerations have important implications for research methodology. The macroeconomic nature of Fisher’s hypothesis renders partial-equilibrium analysis inappropriate. Simultaneity is an essential feature of macroeconomic analysis, and it necessitates a general-equilibrium treatment of macroeconomic questions. By extension, the econometric methodology used in empirical work on macroeconomic questions must also reflect their general-equilibrium nature. The methodology of traditional macroeconometric analysis, however, is also inappropriate for hypothesis testing.18

In this paper, an alternative methodology suggested by Sims has been adopted. This methodology, discussed in some detail in the next section, involves estimating systems of equations as unrestricted reduced-form equations, treating all variables as endogenous. The main advantage of this approach is that it minimizes the amount of a priori economic theorizing necessary to perform the estimation. Treating all variables as endogenous relieves one of the burden of classifying them as exogenous or endogenous, whereas treating all equations as unrestricted reduced-form equations eliminates the need to impose identifying restrictions. Following this preliminary stage of estimation, specific hypotheses can be formulated and tested in terms of the restrictions they imply over the parameter space.19

II. Methodological Issues20

vector autoregressions

The statistical work in this paper is based on vector auto-regressions. Every component of a vector of economic variables is regressed (projected) on lagged values of every variable in the system. In principle, the regressions are unconstrained. The resulting model of the economy is that of a linear stochastic difference equation in the vector of economic variables. The model is dynamic because of the dependence of the current vector of variables on their lagged values. Dynamics of the model arise solely from the autoregressive components and not from the serial-correlation properties of the residuals.

The main advantage of vector-autoregressive specification is that it is very general, capable of exhibiting a wide variety of dynamic behavior. Any vector-stochastic process that is covariance stationary can be represented arbitrarily well by a vector autoregression.21 Moreover, many of the traditional macroeconomic models may be viewed as vector autoregressions, subject to sets of restrictions which are implied either by exclusionary identifying restrictions on the structural equations or by the separation of variables into exogenous and endogenous groups. These restrictions effectively limit the admissible parameter space and result in parsimonious parameterizations. Conversely, when vector autoregressions are unrestricted, they tend to be profligately parameterized. This is, in fact, the main weakness of vector auto-regressions. The number of free parameters increases with the square of the number of variables in the system; and even for moderate-sized systems, degrees of freedom are exhausted rapidly. Nevertheless, it is still feasible to estimate small vector auto-regressive systems, particularly if the serial correlation of the variables declines rapidly.

Formally, it is proposed to estimate the following system of vector autoregressions:

where yt denotes the vector of variables in which one is interested; Ats are square matrices, conformable in size with the vector yt; and ut denotes the vector of disturbances whose properties are to be specified shortly. The order of autoregression, m(T), is, in principle, a function of the number of observations, which increases with the sample size in such a way that the ratio of the number of estimated parameters to the sample size approaches zero. The estimation of vector autoregressions constitutes the first stage and essentially amounts to a preliminary data transformation, whereby the information contained in the time series could be extracted and interpreted more readily. Afterward, at the second stage, specific hypotheses could be formulated and tested. At the third stage, outcomes of hypothesis tests could be used to restrict the parameter space, and the system of equations could be re-estimated under these restrictions.

Systems of equations like (12) should best be viewed as approximations to infinite projections given by

The sequence ut will have the following properties:22

where’ denotes transpositions. In statistical terminology, yt is a linear regression on its past (yt-1,yt-2) plus an innovation ut at time t. Thus, equation (13) describes an autoregressive scheme. It is well known that innovations from a finite mth order auto-regression, utm converge to ut, in the mean-square sense, as m increases.23 This convergence forms the basis of the approximation. Now, the following matrix-valued function is defined:

The determinant of the matrix A(z) will be a scalar-valued function of z expressed in the form of a power series b(z)=Σi=1bizi. If the zeros of the function b(z) are all outside of the unit circle, then there exists a matrix-valued function D(z)=Σi=1Dizi converging on the unit disk such that D(z)A(z) = I.D(z) is called the inverse of A(z) under convolution and satisfies the following system of equations:

If both sides of equation (13) are convoluted by D(z), the result is

which can equivalently be expressed as

However, by virtue of equation (15), all the terms involving yM, j > 0 will cancel out, and equation (17) will reduce to

This is the moving-average representation of yt. The element on the jth row and kth column of the matrix Di measures the impact on variable j of an innovation in variable k after i periods.

The moving-average representation shows the source of the dynamic variables yt. The vector of variables, yt, is represented as a linear combination of serially uncorrelated random variables that lack any dynamics of their own. Consequently, any dynamic behavior of the vector yt comes from the impulse-response function D(Z). This function, through delays and scaling, distributes the impact of innovations over time, giving rise to the observed amplitude variations and phase shifts in the original variables.

The moving-average coefficient matrices, Djs, in equation (18) trace out the system’s response to innovations of unit magnitude, whereas for comparison purposes one needs to find the system’s response to “typical” innovations. Because each element of the vector ut can have a different variance, innovations of unit magnitude are not equally probable events and therefore they are not typical. In addition, the innovations are contemporaneously correlated. For these reasons, it is necessary to transform the innovations so that they are orthogonal across equations and have unit variances. There is no unique way of doing this. One way to proceed, which has a natural interpretation, is to obtain normalization by representing ut as a linear combination of standard variates with a triangular coefficient matrix.

where E(etet)=I and L is lower triangular. Using equation (19), the covariance matrix of innovations can be represented as

Thus, the coefficient matrix L corresponds to the factor in the Choleski decomposition of the covariance matrix of innovations. The moving-average representation of yt in terms of the normalized innovations becomes

expectations mechanism

In order to investigate the relationship between nominal interest rates and expectations of inflation, it is necessary to generate reasonable proxies for the public’s subjective expectations. It is proposed to do this by using estimated vector autoregressions. As was remarked earlier in Section I, in the subsection entitled “role of expectations,” reasonable proxies for expected inflation series could be generated on the basis of a subset of the actual information set used by the public. It is assumed that observations on the vector yt are included in the information as Ωt, and these observations are taken as the subset of Ht to proxy the expectations.24

Let rt denote the nominal interest rate on bonds with N periods to maturity. The real rate of return on this bond is defined as the difference between the nominal interest rate and the expected rate of inflation over its lifetime. Thus, the real interest rate can be expressed as

The expression Etxt+1 denotes the expected rate of inflation between periods t+j – 1 and t + j, based on information available at time t. If it is assumed that expectations are formed on the basis of Ht =(yt,yt-1, yt-2,….) then the real interest rate series, ρt, could be generated using the estimated vector autoregressions. Forecasts of one period ahead could be formed using equation (12). Let the first elements of the vector yt be the rate of inflation. Then, a forecast of inflation one period ahead is given by

where f1 denotes a row vector of the same size as yt, and has 1 as its first element and 0 otherwise.25 Forecasts of increasing horizons could be formed by using the chain rule of forecasting in conjunction with equation (12).

In this study’s statistical work, equation (22) is approximated as follows:

with 0<λ<1. This approximation is done primarily for computational reasons. By replacing the term N1Σj=1NEtxt+j with [(1λ)/(λ)]Σj=1λjEtxt+j it is possible to obtain simple closed-form expressions that facilitate computations. Moreover, it is well known that forecasts lose their precision as they are extended into the future. Therefore, equation (24) has the appealing implication that in determining the current real interest rate, expectations in which there is more confidence have a greater weight than expectations in which there is less confidence.

The choice of λ is dictated by the need to have the mean horizon covered by equation (24) be the same as the mean horizon covered by equation (22). The mean horizon of equation (22) is equal to (N + 1)/2, whereas that of equation (24) is equal to 1/(1 - λ). In order for the two mean horizons to be equal, λ should be set equal to (N - 1)/(N + 1).

With these preliminaries taken care of, closed-form expressions are now derived to represent ex ante real interest rates in terms of the observed variables. Consider the vector autoregression of yt.

with the moving-average representation given by

Optimal forecasts of vector y are given by:

where [ ]+ means that negative powers of z should be ignored.26 The term in brackets in equation (27) can be expressed as

If both sides of equation (27) are multiplied by λk and summed over k, one gets

One can simplify the right-hand side of equation (29) to obtain

Using the relationship between the autoregressive and moving-average representations, equation (30) can be simplified to

where B(z)=Σj=0m1Bjzj and the coefficient matrices Bj satisfy the following recursion equation:

The real rate of interest is then given by

Without loss of generality, it can be assumed that the nominal interest rate is the second variable in the vector yt, so that equation (33) reduces to

or, equivalently,

Equation (35) gives the representation of ex ante real interest rates as a linear combination of vectors yt through yt-m+1. Note, however, that equation (35) is not a regression equation; it is an exact relationship. This reflects the fact that the relationship between the nominal interest rate, the real interest rate, and expectations of inflation is exact. Whenever any two of these variables are given, the third one is determined by equation (3).

III. Some Empirical Results

In this section, the methodology described in the previous section is used to analyze the behavior, and derive the properties, of real interest rates in five industrial countries—the United States, France, the Federal Republic of Germany, Japan, and the United Kingdom. (The data used in the empirical analysis is discussed in Appendix I.) Ex ante real interest rates are computed as the difference between nominal interest rates and optimal forecasts of inflation. For each country, optimal forecasts of inflation are generated on the basis of estimated systems of autoregressions using short-term nominal interest rates and annual rates of change in the monetary base, narrow money, the index of industrial production, and the consumer price index. The estimated systems of autoregressions are not presented here, for reasons discussed in Appendix I, except for summary statistics, which are reported in Table 1.

Table 1.Five Industrial Countries: Summary Statistics of Vector Autoregression
Dependent VariableF-Statistics1
CountryInflation

rate
Interest

rate
Industrial

production
Base

money
System F

marginal significance
United States
Standard error1.350.330.691.112.6048.3
Serial correlation0.01–0.030.010.01–0.011.0
France
Standard error2.280.260.672.305.4449.3
Serial correlation0.030.040.030.01–0.011.0
Germany, Fed. Rep. of
Standard error1.310.270.532.224.9045.8
Serial correlation–0.050.080.000.04–0.031.0
Japan
Standard error2.331.020.411.611.9777.4
Serial correlation0.000.010.000.000.011.0
United Kingdom
Standard error1.730.760.592.261.6184.7
Serial correlation0.000.010.01–0.020.041.0

The empirical questions addressed in this paper can be categorized into two groups:

(1) There are questions pertaining to movements in real interest rates over time, such as what stylized facts can be established and how they relate to significant economic developments. In this connection, the first subsection examines historical movements in real interest rates with a view toward improving understanding of their macroeconomic significance.

(2) There are questions regarding the statistical relationship between real interest rates and expectations of inflation—more specifically, the extent to which real interest rates are correlated with rates of inflation and whether such correlations can be used to effect systematic changes in real interest rates through policy changes. To this end, in the second subsection, the co-movements between real interest rates and expectations of inflation are studied; and it is suggested that although these two variables are correlated, such correlations can be used neither to formulate monetary policy nor to influence the time path of real interest rates.

movements in real interest rates

In macroeconomic analysis, it is common practice to postulate real interest rates as being constant, particularly in medium- and long-term analyses. Results obtained in this paper strongly refute that premise. The estimated real-interest-rate series, which are plotted in Charts 15, show substantial short- and long-run fluctuations in all five countries under investigation, with standard deviations ranging from 2.1 for France to 4.6 for the United Kingdom. Moreover, real-interest-rate series are very highly autocorrelated in all of the countries (see Chart 6). Autocorrelation coefficients decay slowly, indicating that unexpected changes in real interest rates persist over time. Such persistent behavior is inconsistent with the characterization of real interest rates as random movements around a constant.

Chart 1.United States: Real Interest Rates, 1968–82

Chart 2.France: Real Interest Rates, 1968–82

Chart 3.Federal Republic of Germany: Real Interest Rates, 1968–82

Chart 4.Japan: Real Interest Rates, 1968–82

Chart 5.United Kingdom: Real Interest Rates, 1968–82

Chart 6.Five Industrial Countries: Serial Correlation Coefficients of Real Interest Rates

A number of other salient features of real interest rates are also worth mentioning. Starting approximately in the middle of 1973, there was a sharp fall in real interest rates in all five countries, resulting in negative real rates. Although the extent to which interest rates became negative in real terms and the duration of negative real interest rates differed among these countries, their experience during 1973–75 was clearly out of line with their experience during 1968–72. For example, in the United States, real interest rates averaged 0.7 percent during 1968–72, but fell to -1.8 percent during 1973–75. Although real interest rates recovered partially by the end of 1975, they remained negative until the second half of 1980. France and the Federal Republic of Germany had broadly similar experiences, except that for Germany real interest rates became positive relatively rapidly. In Japan and the United Kingdom, the fall in the real interest rates was much sharper, with rates as low as -10 percent not uncommon during much of 1973–75.

A second episode of sharp changes in real interest rates occurred in the middle of 1980, when real interest rates rose rapidly. The speed with which real interest rates rose was particularly pronounced in France and the United States. By 1981, real interest rates were positive in all five countries. Moreover, the increase in real rates of interest took place against a background of high rates of inflation; therefore, nominal interest rates climbed to unprecedented levels. Accompanying these high interest rates was also an apparent increase in the volatility of interest rates as was manifested by large month-to-month (even week-to-week) fluctuations in interest rates.

It should be recalled that both episodes of large movements in real interest rates occurred at times of high inflation. Moreover, both episodes also coincided with substantial increases in oil prices and large reductions in industrial production. The two episodes are, however, distinguished by the speed at which the rate of inflation declined following large movements in real interest rates. After the 1973–75 episode, underlying inflationary pressures persisted, notwithstanding a gradual deceleration in prices; whereas after the 1979–81 episode, the inflation rate declined rapidly. It is reasonable to conjecture that examination of real interest rate movements might foster an improved understanding of the differences between the two episodes.

In both 1973–74 and 1979–80, large increases in oil prices took place against a background of weakening real economic activity and strengthening inflationary pressures in industrial countries. The magnitude of oil price increases and the scale of relative-price adjustment that was called for led to a sudden fall in aggregate output. This was an unexpected aggregate supply shock. In the first episode, the supply shock was followed, after a short lag, by a sharp fall in real interest rates, which induced an increase in current expenditures and a reduction in saving. The resulting increase in aggregate demand, at a time when aggregate supply was falling, led to the intensification and entrenchment of the inflationary process. Only when the momentum of the initial aggregate supply shock declined sufficiently, did the expansionary effect of negative real interest rates start to contribute to economic recovery.

In contrast, following the second round of oil price increases, industrial countries adopted non-accommodative economic policies to bring inflation under control. Nominal interest rates rose, and real interest rates turned positive. The increase in real interest rates discouraged current expenditures and stimulated savings, leading to a fall in aggregate demand and aggregate supply, thereby eliminating inflationary pressures. Thus, the increase in real interest rates after the second round of oil price increases was largely responsible for the speed at which inflation slowed.27

The stylized facts suggest that variations in real interest rates lead primarily to changes in aggregate demand by inducing shifts in time paths of expenditures. Moreover, aggregate supply shocks result in relatively persistent movements in real output. Consequently, following negative aggregate supply shocks, policies that temporarily reduce real interest rates are likely to be inflationary without having a significant positive impact on real output.

correlation between real interest rate and expected rate of inflation

This study now returns to a topic that was discussed earlier in Section l’s second subsection—namely, the test of whether real interest rates are correlated with expected rates of inflation. If market participants’ subjective expectations of inflation were observed, the computation of correlation coefficients would have been a straightforward exercise. However, given the fact that “true” expectations are not observed, a more roundabout procedure is necessary. Lack of observed time series on expectations implies that optimal forecasts of inflation that are based on estimated vector autoregressions would represent true expectations, subject to an error of measurement. This error of measurement, in turn, will render the correlation coefficient computed on the basis of optimal forecasts of inflation, a biased estimate of the true correlation coefficient. Nevertheless, it is still possible to formulate a test in the form of upper and lower bounds under the assumption that errors of measurement are independent of true expectations. Technical details of this test are presented in Appendix II. Because it is computationally more convenient to do so, the test is formulated in the form of bounds on the regression coefficient of the real interest rate on the expected rate of inflation, rather than on the coefficient of correlation between these two variables.28

Table 2 presents upper and lower bounds on the regression coefficient of the real interest rate on expected rates of inflation, together with the estimated standard errors of these bounds. These results show very clearly that real interest rates are correlated with expected rates of inflation in all five countries, and the hypothesis that real interest rates are independent of expected rates of inflation is strongly rejected by the data.29

Table 2.Five Industrial Countries: Upper and Lower Bounds on Regression Coefficients of Real Interest Rate on Expected Rates of Inflation1
CountryLower BoundUpper Bound
United States–0.327–0.254
(0.024)(0.021)
France–0.255–0.196
(0.020)(0.016)
Germany, Fed. Rep. of0.1580.269
(0.036)(0.033)
Japan–0.721–0.601
(0.045)(0.034)
United Kingdom–0.763–0.629
(0.039)(0.032)

However, caution must be exercised before reaching conclusions about the structure of economies or deriving policy prescriptions. It is well known that a property of Keynesian economic models is a contemporaneous correlation between the real rate of interest and expectations of inflation. Nevertheless, this property cannot be used to distinguish between alternative economic models. For example, it is possible to show that a class of rational-expectations models in which monetary policy is not a potent instrument of stabilization also permits a contemporaneous correlation between the real rate of interest and the expected rate of inflation.

More importantly, at least from a policy point of view, is whether nominal interest rates adjust fully to changes in expected rates of inflation, and, if they do, what the duration of the adjustment process is.30 In order to shed some light on these important questions, nominal interest rates were regressed on lagged values of real interest rates and current and lagged values of expected rates of inflation with the objective of using the sum of distributed-lag coefficients as an indicator of the speed of adjustment. Table 3 presents some summary statistics of these regressions, the details of which are omitted here. First of all, it is obvious that the fit of regressions is very satisfactory, with high correlation coefficients, the absence of serial correlation in residuals, and the expected sign and magnitude of the sum of distributed-lag coefficients. In all five countries, the sum of distributed-lag coefficients of both expected rates of inflation and real interest rates is very close to unity. Given the fact that the highest-order lag included in the regressions is six, these results very strongly support the hypothesis that nominal interest rates adjust almost fully to changes in both expected rates of inflation and real rates of interest, and that the adjustment process is completed in six months.

Table 3.Five Industrial Countries: Summary Statistics of Regressions of Nominal Interest Rates on Real Interest Rates and Expected Rates of Inflation1
Sum of Coefficients
πρR2D-W
United States1.000.950.951.97
France0.990.940.962.00
Germany, Fed. Rep. of0.990.930.981.98
Japan0.960.940.981.99
United Kingdom0.970.960.972.02

The hypothesis that the sum of the distributed-lag coefficients on expected rates of inflation is equal to unity cannot be rejected at any conventional level of significance for the United States, France, and the Federal Republic of Germany. For Japan and the United Kingdom, the sum is significantly different from one at the 5 percent level of significance, but this difference is unimportant from a policy perspective in view of the closeness of these sums to one.31 The sums of the distributed-lag coefficients on lagged real rates of interest, however, are all significantly different from one at the 5 percent level, though again the differences are rather small.

IV. Concluding Remarks

One major conclusion that emerges from this paper’s findings is that real interest rates are not constant over time. In all five countries examined, real interest rates exhibit substantial variation and serial correlation. The presence of both variability and serial correlation is inadmissible under the hypothesis of constant real interest rates. The fact that real interest rates are not constant over time is further verified by the correlation between real interest rates and expectations of inflation. If ex ante real interest rates had been constant, the correlations would have been zero.

A second conclusion is that policies that change expected rates of inflation systematically leave real interest rates changed, with nominal interest rates adjusting fully to the new time path of expected rates of inflation. By extension, it is highly unlikely that expansionary monetary policies will succeed in lowering nominal interest rates, provided that the implications of such policies are well understood by economic agents and therefore result in an upward adjustment of their expectations of inflation.

Finally, consideration should be given to investigating the determination of real interest rates within a richer specification, to account for the impact of other important variables, such as wages, the exchange rate, and the fiscal deficit. In estimating these larger systems of vector autoregressions, parameter space must be restricted, perhaps using Bayesian methods, in order to preserve the degrees of freedom. Such an approach seems to be a promising area for future research.

APPENDICES
I. Some Notes on Data and Estimated Autoregressions

Data

In order to implement the methodology that is outlined in Section II, vector autoregressions were estimated for five countries: the United States, France, the Federal Republic of Germany, Japan, and the United Kingdom. The variables included in the vector autoregressions are the annual rates of change of the money stock, the monetary base, the index of industrial production, and the consumer price index, as well as a short-term interest rate. The interest-rate variable for the United States and the United Kingdom is the three-month treasury bill rate; for France and the Federal Republic of Germany, it is the three-month interbank lending rate; and for Japan, it is the two-month lending rate. The source of most data is the Fund’s monthly publication, International Financial Statistics, while interest rate data were supplied by the Board of Governors of the (U.S.) Federal Reserve System. The period of estimation is 1968 through mid-1982.

The data are not seasonally adjusted, despite the fact that some of the variables display some seasonal variation. It is assumed here that seasonal variation is part of the information set used by economic agents. Rational behavior of such agents dictates that they utilize this information in forming their forecasts, and such behavior should manifest itself in the observed behavior of the variables. As demonstrated by Saracoglu and Sargent (1978), seasonal behavior of the information variables results in a complicated pattern for the response variables, whereby the former variables are mapped into the latter with their phase shifted and amplitude scaled. It is true that the resulting response variables also display some seasonal variation. However, this variation is substantially different from that of the information variables.

Estimated vector autoregressions

In estimating any autoregressive process, a decision has to be made about the order of autoregression. There are no straightforward ways of accomplishing this. Because the statistical foundations of vector autoregressions are based on asymptotic distribution theory, a crucial consideration is the degrees of freedom. On the one hand, when the order of autoregression is such that the number of estimated parameters is of the same order of magnitude as the remaining degrees of freedom, the applicability of asymptotic theory becomes doubtful.32 On the other hand, the order of autoregression should be high enough to capture the dynamic behavior of variables. This study’s experiments with different orders suggested that six lags accounted sufficiently for the serial correlations, while still leaving enough degrees of freedom to render attributions to asymptotic theory meaningful.33

There is a problem in reporting the results of estimated vector autoregressions. For each country, the model to be estimated has 170 free parameters including 150 autoregressive coefficients, 5 constants, and 15 parameters for the residual covariance matrix. It seems unreasonable to report all the estimated parameters for all the countries. The autoregressive coefficients, themselves, are relatively uninformative. They tend to oscillate, and many of them are not significantly different from zero, which makes them difficult to interpret. Moreover, the estimated coefficients have little relevance to the Fisher hypothesis. In view of these considerations, standard errors of estimate and the first-order serial-correlation coefficient of residuals for each equation are reported in Table 1, together with F-statistics that test the goodness of fit for each country. These statistics indicate that the estimated vector autoregressions fit the data closely in every country. First-order serial-correlation statistics of residuals show that they are spectrally white, as required by the theory. Moreover, the estimated residual-covariance matrices are approximately diagonal, indicating that innovations in different variables are contemporaneously uncorrelated. In all countries, the residuals of inflation and interest rates have substantially smaller variances than the residuals of other variables, suggesting that large, unexpected fluctuations in the rate of inflation and the rate of interest are much less likely than such fluctuations in the money stock, industrial production, or base money.

The estimated systems of vector autoregressions were also tested for stability. Stability is a necessary condition for the transformation of the autoregressive representation into the moving-average representation. To test for stability, the polynominal determinants of the autoregressive system were computed and their roots calculated. These roots were uniformly outside the unit circle, indicating that the estimated systems were stable. However, the presence of complex roots that are close to each other in magnitude and have small imaginary parts suggests the possibility of mildly nonstationary behavior in all of the countries. No attempt was made, however, either to investigate this behavior or to account for it in the estimation.34 Following these preliminary tests, the estimated vector autoregressions were used to compute vectors h of equation (35) in order to generate time series on ex ante real interest rates for all five countries.

II. Derivation of Bounds of Regression Coefficient of Real Interest Rate on Expected Inflation Rate

In this appendix, tests are formulated in the form of upper and lower bounds around the regression coefficient of real interest rates on expected rates of inflation.

Let xt* be the public’s true (and unobserved) expectations of inflation based on the complete information set Ωt. Let πt be a proxy for xt* based on the information set Ht, which is a subset of Ωt. Then xt* can be written as

where nt is orthogonal to πt, and therefore to it, as well. Furthermore, we have the relationship

In equations (36) and (37), only rt and πt are observed. From equation (37), the covariance between rt and xt* is given by

whereas from equation (36), one obtains

If the nominal interest rate is included in Ht, then σrn = 0, with the result that

From equation (39’) and the equality of the actual rate of inflation, xt+1, and the sum of expected rate of inflation, xt*, and the forecast error ut+1, the following inequalities are obtained:

Equations (40) and (41) jointly imply the following inequalities:

Moreover, if σρx*>0, then

Otherwise,

Note that the regression coefficient of the real interest rate on the expected rate of inflation is given by σρx*σx*2. By virtue of equations (42), (42’), (43), and (43’), this coefficient is bounded both from above and from below by observable magnitudes.

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Mr. Saracoglu, economist in the Developing Country Studies Division of the Research Department, received his doctorate from the University of Minnesota and also studied at the Middle East Technical University in Ankara. He was formerly Assistant Professor of Economics at Boston College.

Some examples are Begg (1977), Fama (1975, 1976), Makin and Tanzi (1984), Mishkin (1981), and Sargent (1973 a, 1973 b, 1973 c, 1976).

In formulations of this view, investment demand depends upon the ratio of the value of existing stocks of capital to their replacement cost. This ratio, known as Tobin’s q, is equal to the ratio of the net marginal product of capital to the real rate of interest.

Friedman’s presidential address was delivered to the Eightieth Annual Meeting of the American Economic Association in Washington on December 29, 1967.

Fisher’s ideas on the relationship between the purchasing power of money and the nominal interest rates were stated clearly in his book The Theory of Interest, especially in Chapters II and XIX. In writing about the effects of changes in the purchasing power of money (p. 37), he stated that “the influence of such changes in the purchasing power of money on the money rate of interest will be different according to whether or not that change is foreseen. If it is not clearly foreseen, a change in the purchasing power of money will not, at first, greatly affect the rate of interest expressed in terms of money.” However, to the extent that changes in the purchasing power of money are foreseen, it is possible, at least theoretically, to make allowances for the expected change in the unit of value. “To offset a foreseen appreciation, therefore, it is necessary only that the rate of interest be correspondingly lower, and to offset a foreseen depreciation, that it be correspondingly higher.”

Fisher, however, concedes (p. 38) that because of ignorance and indifference, appreciations and depreciations are never fully foreknown and therefore they are only partially provided for in the rate of interest itself.

It is interesting that Keynes’s criticism of Fisher’s theory of the distinction between the money rate of interest and the real rate of interest is based on a false assertion. Keynes asserts that “It is difficult to make sense of this theory as stated, because it is not clear whether the change in the value of money is or is not assumed to be foreseen.” (Keynes (1936), p. 142.) Fisher, however, was a very careful writer who made it clear whether the changes in the value of money were, or were not, assumed to be foreseen.

Fischer (1930), p. 494.

That is, completely uncorrelated with the information variables determining expectations.

In fact, in a class of rational-expectations models, one can easily demonstrate that the ex ante real interest rate will be correlated with the expected rate of inflation, even though the monetary policy cannot induce systematic movements in output or the real interest rate. Conversely, one can construct models where monetary policy is a potent instrument of stabilization, but the estimates of a in equation (5) will be zero.

See Dwyer (1981) for an elaboration of this point.

Examples are Carlson (1977), Joines (1977), Nelson and Schwert (1977), and Begg (1977). Also see Fama (1977) for his response to some of the criticisms.

This result is known as the law of iterated projections or the law of nested conditional expectations. Its proof is simple and can be found in any statistics textbook.

The test will be valid as long as the contemporaneous interest is excluded from Ht. Otherwise, the left-hand side of equation (11) will reduce to the singular projection of rt on itself.

See Shiller (1972), especially page 113. The distributed-lag coefficients he reports in Tables 5–22 through 5–24 tend to support Fisher’s hypothesis.

These tests, however, are indirect tests of Fisher’s hypothesis because the statistics are formulated in terms of the natural-rate hypothesis.

Nevertheless, Sargent states that “I imagine that the evidence would not be sufficiently compelling to persuade someone to abandon a strongly held prior belief in the natural rate hypothesis.” (Sargent (1973 c), p. 462) Also see comments and discussion following Sargent’s paper.

This basically reflects Sims’s point of view. For an excellent paper that defends some of the traditional methodology, see Malinvaud (1981).

For examples of this, see Sargent (1979 a), Saracoglu (1980), and Dwyer (1981).

For alternative expositions of methodology, see Litterman (1980) and Sims (1980).

Let yt, denote a vector-stochastic process with E(yt) = 0. The autocovariance of yt is given by K(s,t)=E(ytys). If K(s, t) = K(s - t) for all s and t, then yt is a covariance-stationary stochastic process.

For expositional purposes, it is assumed that the vector yt has an expected value of 0.

See Sargent (1979 b), pp. 256–62.

If information is valued by market participants, it is reasonable to assume that they utilize any relevant information that is readily and costlessly available.

Vector f1 “picks” the rate of inflation from the vector yt.

For a detailed exposition of representation theory and optimal forecasting, see Whittle (1963).

There were, of course, a number of other related factors. The accommodative policies of 1973–75 were probably unexpectedly lax and therefore contributed to the subsequent economic upturn. Similarly, the policies adopted during the latter episode were probably unexpectedly strict, particularly in view of the policies of the earlier episode, and probably contributed to the length and severity of the economic slowdown.

The regression coefficient, as is well known, is proportional to the correlation coefficient.

From the estimated bounds, it is possible to form a range for the magnitude of the Fisher effect. In the United States, this range is 0.673 to 0.746. These values are in complete accord with those reported by Makin and Tanzi (1984), Table 2.

The time it takes for nominal interest rates to adjust to changes in expected rates of inflation is important, because if it takes a long time for the adjustment to be completed, then alternative policies will have semipermanent effects, even though nominal interest rates will fully adjust eventually.

Note that Japan and the United Kingdom are also the two countries where the contemporaneous impact of expected rates of inflation on real interest rates is largest, as evidenced in Table 2.

See Sims (1980) for a detailed discussion of these issues.

No attempt was made, however, to find the “optimal” order of autoregressions for each variable of every country. This would have required a systematic and costly search but would not have affected the results substantially.

It is possible to estimate vector autoregressions with time-varying coefficients. See, for example, Sims (1982).

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