the explanatory power of alternative theories of inflation and unemployment, 1895-1979

The Explanatory Power of Alternative Theories of Inflation and Unemployment, 1895-1979Author(s): John D. ReaSource: The Review of Economics and Statistics, Vol. 65, No. 2 (May, 1983), pp. 183-195Published by: The MIT PressStable URL: http://www.jstor.org/stable/1924484 .

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The Review of Economics and Statistics VOL. LXV MAY 1983 NUMBER 2

THE EXPLANATORY POWER OF ALTERNATIVE THEORIES OF INFLATION AND UNEMPLOYMENT,

1895- 1979

John D. Rea*

I. Introduction

A S a result of the past decade of inflation, there seems to be a steadily growing number of

economists who are convinced that the long-run Phillips curve is vertical. Nevertheless, their number has not yet reached unanimity and sharp differences do remain over the theoretical founda- tions of the observed relationship between inflation and unemployment. For example, some economists (Eckstein and Girola, 1978; Tobin, 1972) view the long-run Phillips curve as possibly having a negative slope. Still others (Meltzer, 1977; Stein, 1978) doubt that a Phillips-type relation exists at all and argue instead that monetary models best describe the past inflation. Finally, even those who accept the validity of the vertical Phillips curve disagree on whether policy can be used to lower unemployment temporarily with a higher inflation rate (Santomero and Seater, 1978, pp. 527-532).

The large number of empirical studies that have been conducted to resolve these different views have obviously not been successful. One reason is that most of these studies have focused narrowly on the estimation of a single theory and have not broadly compared the explanatory power of alternative theories. Consequently, the purpose of this paper is to compare the explanatory power of alternative theories of inflation and unemployment over the 1895-1979 time period.'

For purposes of this comparison, three models of inflation and unemployment have been selected. One is a Keynesian model containing a negatively sloped Phillips curve, a second is Stein's (1978) monetarist model containing no Phillips-curve relation, and a third is the natural rate model containing a vertical Phillips curve. Two versions of the third model are examined. One uses adaptive expectations, which implies policy can temporarily change the unemployment rate by varying the inflation rate, and the other uses rational expectations, which precludes such a policy. Together these models are broadly representative of different views on the subject and furthermore, each has received empirical support. A further discussion of these models is contained in section II of the paper.

To carry out the comparison, each model is estimated for the 1895-1956 period and then used to predict inflation and unemployment for the 1957-1979 period. The estimation techniques and the criteria used to evaluate their predictions are outlined in section III and the estimated models and their predictions are presented in sections IV and V, respectively. The results of this analysis indicate that no single model can by itself explain inflation and unemployment since 1895. Instead, for the 1895-1956 period, the model with a negatively sloped Phillips curve best describes inflation and unemployment, while Stein's monetarist model is superior for the 1957-1979 period. Possible explanations for these findings are offered in section VI. Received for publication March 12, 1980. Revision accepted

for publication July 29, 1982. * Oklahoma State University. Support for this research was provided by a grant from

the Dean's Excellence Fund, College of Business Administra- tion, Oklahoma State University. I wish to thank James L. Butkiewicz and the referees for their comments.

I In the past several years, interest in analyzing the data over longer intervals of time has re-emerged. See Eckstein and (irola (1978), Gordon (1980), Meltzer (1977), and Sachs (1980).

These studies have generally been concerned with whether a Phillips curve model is adequate to explain the data and whether the slope of the Phillips curve has decreased in the postwar period.

[ 183 1



184 THE REVIEW OF ECONOMICS AND STATISTICS

II. Theoretical Models

The three models of inflation and unemployment discussed in this section are presented in the form in which they will be estimated. In all cases, the formulations are either taken or adapted from previously estimated models. The first model considered is called the trade-off model, which refers to the hypothesis that the long-run Phillips curve is negatively sloped. An expression adapted from Eckstein and Girola (1978), which can be used to test this hypothesis, is

p = aO + a/UN + a2p(- 1) + a3pm

-a4PRD - a5WW2 + e. (1)

The variables which are all measured annually have the following definitions:

p is the rate of inflation in the implicit deflator for Gross National Product, in per cent;

UN is the civilian unemployment rate, in per cent;

pm is the rate of inflation in the wholesale price index of raw materials, in per cent;

PRD is the rate of change in labor productivity, measured as the ratio of real GNP to man hours worked of persons engaged in domestic production;

WW2 is a dummy variable for price controls during World War II and is 1 from 1942 to 1945;2

e is the disturbance.3

The parameters, ao through a5, are assumed to be positive; furthermore, a2 is assumed to be less than unity in order to produce a negatively sloped, long-run Phillips curve.4

In the natural rate model, the long-run Phillips curve is hypothesized to be vertical. A specification of the Phillips curve similar to that used by

Sargent (1976) for testing this hypothesis is

UN = bo-b( p-pe) + b2UN(-1) + e, (2)

where pe is the expected rate of inflation and b2 is assumed to lie between zero and one. For the Phillips curve to be vertical, p must equal pe in the long run; in this situation, the equilibrium or natural unemployment rate equals bo/(1 - b2). The condition for a vertical Phillips curve is satisfied by both rational and adaptive expectations. If expectations are formed rationally, then

pe = E(plI), (3)

which indicates that pe is the expected value of the inflation rate given all available information (Muth, 1961). If expectations are formed adaptively, then

pe = (1 -k) p (-1 ) + kpe (-1 ). (4)

Although both forms of expectations imply that the long-run Phillips curve is vertical, it is only in the case of adaptive expectations that policy can alter the unemployment rate in the short run through a higher inflation rate. Since the two specifications for expectations have different short-run implications, both will be used in the subsequent tests of the natural rate model.

In contrast to the trade-off model and the natural rate model, Stein's (1978) monetarist model is built upon the premise that no functional, Phil- lips-type relationship exists between inflation and unemployment. Instead, the prime determinant of changes in the inflation rate is the rate of change in the real money supply. Thus, Stein specifies the inflation equation as

p = (1 - c)p(- 1) + cm(- 1) + e, (5)

where m is defined as the growth rate of the broad money supply consisting of currency and bank demand and time deposits other than large certificates of deposit.5 To test whether this formulation is superior to a Phillips-curve relationship, Stein offers the general specification,

p = co- cUN(-1) + c2m(-1)

+ (1 - C3)p(- 1) + e,

which specializes to (5) when co and cl are zero and when c2 equals c3. To complete the model, Stein adds an equation for unemployment having

2 Other dummy variables for capturing the effects of World War I, the National Recovery Administration, and the Korean War were found to be insignificant. These dummy variables and the WW2 dummy variable were all insignificant in the other models discussed below and hence were excluded from their specifications.

3An appendix describing the data in full is available from the author. All rates of change are calculated as the first differences of the natural logarithms of the variable.

4In subsequent equations, all parameters are likewise assumed to be positive.

' Stein uses currency and demand deposits.



THEORIES OF INFLATION AND UNEMPLOYMENT 185

the form,

UN = do + (1 - d,)UN(- 1)

-d2 [m(-1) -p(-1)] + u, (6)

where u represents the disturbance. This equation, coupled with equation (5), implies that for a constant growth rate of the money supply, the unemployment rate and the inflation rate are inde- pendent, in which case the natural rate of unemployment is given by the expression do/d,. In the short run, however, any pattern between inflation and unemployment is possible.

Inclusion of the unemployment equation in the monetarist model points out that an aggregate demand equation is needed to complete the first two models. A simple formulation of this equation may be obtained by expressing the conven- tionally defined aggregate demand function, Y=

H(M/P, Z), in terms of the unemployment rate rather than real output Y. This is readily accom- plished by replacing Y with the production function, Y = F(UN, t), where the time trend captures the effect of technological change and growth in the capital stock and labor force. In the aggregate demand equation, M/P is the real money supply and Z is a vector of policy and exogenous variables. Combining these two equations and solving for UN yields the following form of the aggregate demand equation used in estimating the trade-off and the natural rate models:6

UN = ho-h,log(M/P) - h2log(G)

-h3log(X) + h4t + u. (7)

The new variables are defined as follows:

M is the broad money supply; P is the implicit deflator for Gross National

Product; G is real purchases of goods and services by

Federal, state, and local governments; X is real exports.

III. Empirical Procedures

The immediate objective of this study is to determine which of the three models best explains inflation and unemployment over the 1895-1979 period. However, in light of Lucas' (1976) analysis of model instability, the possibility that no single model can outperform the others for the entire period must be considered. Hence, the models' explanatory power during subperiods needs to be examined as well.

One logical point to break the sample is the mid-1950s. Two reasons can be cited. First, structural changes in either inflationary expectations (Klein, 1976) or the Phillips curve (Sachs, 1980) may have occurred at this time. Second, much of the present controversy over the theory of inflation reflects different interpretations of the postwar behavior of prices. It thus seems relevant to examine the ability of these three models estimated over an earlier time period to predict inflation since the mid-1950s.7

Consequently, the comparison of the models is carried out in two steps. In the first, each model is estimated using annual data for the 1895-1956 subperiod for the purpose of determining whether their theoretical properties are consistent with the data. Because the models are simultaneous, each one is also simulated over this subperiod and a comparison of their accuracy is made. In the second step, the estimated models are used to predict inflation and unemployment for the 1957-1979 subperiod with the objectives being first to determine the predictive power of each individual model and second to compare their predictive accuracy.

A. Methods of Estimation

With the exception of the trade-off model, the method used to estimate the models is full-information maximum likelihood, with corrections made, where necessary, for serial correlation. The presence of serial correlation, the form of which is approximated by an autoregressive process (Amemiya, 1973), is determined from preliminary 6 This form of the aggregate demand equation was selected

after some empirical experimentation. Inclusion of inflationary expectations yielded either an insignificant coefficient or one with an incorrect sign. Geometric distributed lags for the explanatory variables in (7) were also insignificant. Finally, a tax variable should have been included, but no meaningful measure of tax rates was available for the entire period.

7Likelihood ratio tests conducted on the estimated models also indicate a structural change occurred in each during the mid- 1950s.




instrumental variable regressions.8 If serial correlation is present, the structural equations are transformed to eliminate it, and the subsequent full-information maximum-likelihood estimation jointly estimates the structural and autoregressive parameters.

The method used to estimate each equation in the trade-off model is nonlinear, instrumental variables (Amemiya, 1974; Berndt et al., 1974). This method is used because the Phillips-curve equation is nonlinear and because the variables, pm and PRD, are assumed to be endogenous. The degree of serial correlation is estimated, as before, from preliminary instrumental variable regressions. If serial correlation is present, both the parameters of the structural equations and the autoregressive disturbance are jointly estimated.9

B. Model Comparisons

The properties of the models are compared both for the 1895-1956 subperiod used in estimating the models and for the 1957-1979 subperiod. In the estimation subperiod, the comparison is based upon the signs and significance of coefficients and goodness-of-fit criteria. In addition, each model is simulated over this subperiod, and the quality of their predictions is evaluated. Similarly, in the 1957-1979 subperiod, the comparison is based upon the simulations of these models."0

The evaluation of the predictions from these simulations is made on both an absolute and relative basis. In evaluating predictions on an absolute basis, the objective is to determine whether the predicted values contain any systematic errors that could be exploited to improve predictive accuracy. The presence of systematic errors is indicated either by the existence of serially correlated prediction errors or by the failure of the predictions to satisfy

the Mincer-Zarnowitz (1969, pp. 4-41) conditions of unbiasedness and efficiency." The implication of models that do contain systematic errors is that their specification is incomplete.

In evaluating models on a relative basis, the objective is to examine their predictive accuracy relative to one another. The simplest means of doing so is to compare root-mean-squared errors. That model with the smallest root-mean-squared error would represent the superior predictor. This form of comparison is further refined in the concept of a composite predictor (Nelson, 1972; Cooper and Nelson, 1975; Granger and Newbold, 1977, pp. 268-278), which is a linear combination of two or more predictors. A composite predictor is obtained by regressing the variable being predicted on the predictors. If the predictors are unbiased, then the intercept is zero and the slope coefficients sum to one. The slope coefficients of the predictors measure their relative importance; the larger the coefficient, the greater the contribu- tion of the predictor to improving predictive accuracy. If one predictor contains all the information possessed by the other predictors, its coefficient is one and the others are each zero. The dominance of one predictor clearly indicates model superior- ity; in its absence, no one model can be regarded as a complete theory.

IV. Model Estimates

A. Trade-Off Model

The estimated equations of the Phillips curve and aggregate demand function for the trade-off model, shown in table 1, are generally consistent with the theoretical formulations in equations (1) and (7). All parameter estimates have the correct signs and, with the exception of the intercept in the Phillips curve equation, are significant at the 5% level. Based upon adjusted coefficients of determination, the equations fit the data well. In addition, the estimated Phillips curve possesses a long-run trade-off between inflation and unemployment, as indicated by the coefficient of the lagged inflation rate being significantly less than one.

8 In these regressions, the instruments consist of current and lagged values of the exogenous variables for the particular model in which the equation being analyzed belongs. The order of the autoregressive process is then determined by inspecting the significance of the coefficients in the autoregressions in- volving the residuals from this regression. No allowance is made for serial correlation between disturbances in different equations of the same model. (See Fair, 1972; Fuller, 1976, pp. 435-446.)

9 In the nonlinear, instrumental estimation, the instruments consist of all predetermined variables in the model after the autoregressive transformation has been applied.

10All simulations are static and ex post. In those equations with serially correlated disturbances, lagged prediction errors are included.

" A predictor is said to be unbiased and efficient if in the least-squares regression of the actual values on the predicted, the intercept is zero and the slope coefficient is one.




TABLE 1. -ESTIMATES OF THE TRADE - OFF, NATURAL RATE, AND MONETARIST MODELS

1895-1956, ANNUAL

Trade-Off Model

p = -.09 + .30p(- 1) + 8.66/UN - .58PRD + .42pm (.71) (.06) (3.28) (.19) (.03)

-3.19WW2+e R 2=.84, SEE = 2.53 (1.64)

UN = 89.70 -.15 log(M/P) --.05 log G - .09 log X (18.52) (.06) (.02) (.02)

+.97t + u (.20)

u=.61u(-1) R 2 .79, SEE = 2.21 (.11)

Natural Rate Model-Adaptive Expectations

UN = .26- .30[p-p(-1)]+1.42UN(-1) (.34) (.04) (.08)

-.48UN(-2) + v (.08)

v = -.54v(- 1) -.32v(-2) R2 = .72, SEE = 2.54 (-.10) (.13)

UN = 103.35 - .22 log(M/P) - .02 log G - .08 log X (11.82) (.04) (.02) (.02)

+ 1.06t + u (.15)

u= .59u(-1) R 2 .75, SEE = 2.42,ISI= 19.90 (.12)

Natural Rate Model-Rational Expectations

UN = 3.40 - .25 log(P/Pe) + .4OUN(- 1) + e (2.08) (.06) (.1 1)

e= .77e(-1) R 2 .63, SEE = 2.93 (.07)

UN = 351.77 -.96 log(M/P ) +.01 log G -.07 log X (34.01) (.09) (.03) (.04)

+3.43t + u (.54)

u = .86u(- 1) R2 = -.28, SEE = 5.47, SI = 207.57 (.07)

Monetan'st Model

p =p(- 1) + .19[m(- 1) -p(- 1)] +e (.11)

e = -.39e(-2) - .25e(-4) R2 = .10, SEE = 6.02 (.11) (.11)

UN =1.72+ .72UN(-1)- .09[m(-1)-p(-1)]+u (1.18) (.19) (.07)

u=.30u(-1) R .72, SEE = 2.57,ISI= 180.89 (.22)

Note: With the exception of the standard error of the estimate, SEE, all standard errors are asymptotic. SEE is adjusted for degrees of freedom. A7 is the adjusted coefficient of determination and equals (I - SEE2)/s2, where s is the standard deviation of the dependent variable. ISI indicates the generalized variance of those models estimated by full-information maximum likelihood.

B. Natural Rate Model-Adaptive Expectations Version

The estimates of the natural rate model employ- ing adaptive expectations, presented in table 1, show the Phillips curve in a form different from that in equation (2). The equation in table 1 is obtained by applying a Koyck transformation to equation (2). Use of this transformation, which is necessary to eliminate the unobservable pe, results in the equation,

UN = bo(1 -k)-b[p-p(-1)] + (k + b2) UN(- 1)- b2kUN(-2) + v,

(8)

where v = e - ke(- 1). Since bo, b2, and k are not identified, separate estimates of them cannot be obtained and thus only the reduced-form estimates of equation (8) are reported.

The estimates themselves provide support for the natural rate hypothesis. All parameter estimates in both equations have the correct signs and only the intercept in the Phillips curve equation and the coefficient of government purchases in the aggregate demand equation are insignificant at the 5% level. Both equations fit the data well with the values of the adjusted coefficient of determination being in excess of 0.70. Moreover, the model is stable and the estimate of the natural rate of unemployment is 4.3%.

C. Natural Rate Model-Rational Expectations Version

To obtain the estimates of the rational expectations model shown in table 1, a procedure similar to that described by Hoffman and Schmidt (1978) and Wallis (1980) is employed. With the identities pe = log[Pe/P(- 1)] and p = log[P/P(- 1)] substituted into the Phillips curve equation (2), the procedure involves first solving equations (2) and (7) for the reduced form of log P. The expected value of this equation is then used to obtain an expression for log Pe that can be substituted into (2) to yield

UN = bo + bl(bo - ho)/h -b

x [log P - log Me

- (h2log Ge + h3log Xe + h4t)/h 1] +b2(1 + bl/h1)UN(- 1) + e. (9)




From equations (7) and (9), the parameters of the rational expectation version can be estimated. The variables log Me, log Ge, and log Xe in (9) are the expected values of log M, log G, and log X, respectively. Assuming that these exogenous variables each follow univariate autoregressive processes, their expected values required in (9) can be taken as the calculated values from their autoregressions.

The results of the estimation show that the rational expectations version of the natural rate model does not fit the data well, especially when compared to the adaptive expectations version. Although the coefficients, except that for log G, have the correct signs, the standard errors of the estimate and the generalized variance are considerably larger in the rational expectations version. This is true of both equations and thus it would seem that the adaptive expectations version is the better formulation of the natural rate model.

D. Monetarist Model

The monetarist model is not fully consistent with the data since the hypothesis can be rejected that the coefficient of lagged money growth m ( - 1) should equal one minus that of the lagged inflation rate p(- 1) in equation (5). The estimate of the former coefficient is 0.24, the estimate of the latter is 0.37, and the chi-square test statistic of the hypothesis is 18.4.12 In table 1, the results of applying the inappropriate restrictions are reported, and as can be seen, the inflation equation provides a very poor explanation of the data.

In spite of this shortcoming, the remaining restrictions on the coefficients in both the inflation and aggregate demand equation cannot be rejected, and in addition, all coefficients have the correct signs. Although the coefficient of the lagged rate of change in the real money supply is insignificant, the aggregate demand equation does fit the data almost as well as the corresponding equation in the trade-off model. Finally, the model is stable and implies that the natural rate of unemployment is 6.2%. '3

V. Evaluation of Predictive Accuracy

A. 1895-1956 Subperiod

Based upon the statistics obtained from the model simulations, none of the estimated models is completely free of systematic errors during the 1895-1956 subperiod. (See table 2.) The trade-off model, the rational expectations version of the natural rate model, and the monetarist model are inefficient predictors of the inflation rate. The rational expectations version also has serially correlated errors for both unemployment and inflation, and the adaptive expectations version has serially correlated errors for the unemployment rate.

On a relative basis, the trade-off model appears to be the best predictor of inflation and unemployment. This model produces the smallest root- mean-squared errors for both the rate of inflation and unemployment rate, although for the latter variable the difference over the adaptive expectations version is not great. Moreover, in the composite predictors formed using all four economic models, the trade-off model receives the greatest weight in predicting both inflation and unemployment. Indeed, for the composite predictor of unemployment, the hypothesis that the coefficient of the trade-off model is one and the other coefficients are zero cannot be rejected. This hypothesis is, however, rejected for the composite predictor of inflation.

The poorest predictors are the rational expectations version of the natural rate model and the monetarist model. Both have root-mean-squared errors that are larger than those of the other two models, with the difference being especially pro- nounced for predictions of inflation. Moreover, with one exception, these models are assigned small or even negative weights in the composite predictors.

The adaptive expectations version falls some- where between these two extremes. It predicts unemployment almost as well as the trade-off model, but is considerably less accurate in predicting inflation. However, it is far more accurate in predicting both variables than the rational expectations version and the monetarist model.

Based upon these results, if one model had to be chosen for the 1895-1956 subperiod, it would probably be the trade-off model. Its estimates are consistent with the underlying theory, it provides

12 The chi-square statistic is minus twice the likelihood ratio and is asymptotically distributed as a chi-square random variable with one degree of freedom.

13For comparison, autoregressive, moving average models have been estimated for the unemployment and inflation rates. Neither fit the data as well as the trade-off and natural rate models. In subsequent simulations, these time-series models likewise perform worse than the economic models.




TABLE 2.-STATISTICS FOR EVALUATING PREDICTIONS OF INFLATION ANI) UNEMPLOYMENT 1895- 1956

Model

Natural Rate

Adaptive Rational Statistic Trade-off Expectations Expectations Monetarist

Unemployment Rate

t-statistic for unbiasedness - 0.17 0.00 0.00 0.54

t-statistic for efficiency - 0.48 - 0.69 - 0.49 - 0.70

Autocorrelation coefficients of prediction error

Ist order 0.15 0.31a 0.02 0.06 2nd order -0.17 -0.07 -0.28a -0.19

Root-Mean-Squared Error 2.00 2.11 2.28 2.49

Coefficient in composite 0.61 0.26 0.34 - 0.21 predictorb (0.24) (0.27) (0.22) (0.22)

Inflation Rate

t-statistic for unbiasedness -0.65 0.00 0.00 - 1.42

t-statistic for efficiency -3.04a - 1.10 -4.37a - 3.24a


I st order 0.18 - 0.08 0.45a - 0.19 2nd order -0.01 -0.19 0.04 -0.13


Coefficient in composite 0.68 0.39 -0.13 0.07 predictorc (0.10) (0.14) (0.11) (0.08)

aSigniticant at the 57. level. R2 = 0.83; SEE = 1.99; DW = 1.64. This equation has been fitted subject to the restrictions that the slope coefficients sum to one and the intercept is zero.

The F-statistic for this constraint is 0.06 with 2 and 57 degrees of freedom. The F-statistic for the hypothesis that the intercept is zero, the coefficient of the trade-ofl' model is unity, and the remaining coefficients are zero is 0.75 with 5 and 57 degrees of freedom.

R2 = 0.80; SEE = 3.03; DW = 1.98. This equation has been fitted subject to the restrictions that the slope coefficients sum to one and the intercept is zero. The F-statistic for this constraint is 0.51 with 2 and 57 degrees of freedom. The F-statistic for the hypothesis that the intercept is zero, the coefficient of the trade-oft model is unity, and the remaining coefficients are zero is 3.47 with 5 and 57 degrees of freedom.

the best fit to the sample data, and it is the best predictor of inflation and unemployment. Never- theless, the model does have some deficiencies. For one, the model is an inefficient predictor of inflation. But more importantly, because the trade-off model does not subsume all the information about inflation contained in the other models, it cannot be regarded as a complete theory of inflation. Hence, the choice of the trade-off model is tenta- tive.

B. 1957-1979 Subperiod

Based upon the root-mean-squared errors from the simulations over the 1957-1979 subperiod, all the models predict better than in the 1895-1956 subperiod. (See table 3.) Nonetheless, none of the models are free of systematic error. The best of the

group is the monetarist model which only has biased predictions of inflation. In contrast, the others have multiple deficiencies. With one exception, they are biased and inefficient predictors of inflation and have serially correlated prediction errors. Moreover, in predicting unemployment the rational expectations and the trade-off models are inefficient, and the trade-off and adaptive expectations models have serially correlated errors.

On a relative basis, the monetarist model is by far the best predictor of both inflation and unemployment. It produces the smallest root-mean- squared errors, with the one for inflation being less than one-half as large as the next smallest. Fur- thermore, the predictions for the monetarist model receive the largest weights in the composite predictors. Indeed, for both composites, the hypothesis that the weight assigned to the monetarist model is




TABLE 3.-STATISTICS FOR EVALUATING PREDICTIONS OF INFLATION AND UNEMPLOYMENT

1957- 1979

Model

Natural Rate

Adaptive Rational Statistic Trade-off Expectations Expectations Monetarist

Unemployment Rate

t-statistic for unbiasedness 1.60 - 0.78 1.31 0.35

t-statistic for efficiency - 2.14a - 1.49 - 2.46 a - 1.29


I st order 0.59 036 - 0.27 - 0.34 2nd order 0.38 0.29 - 0.30 - 0.06


Coefficient in composite 0.12 0.10 - 0.36 1.15 predictor' (0.27) (0.39) (0.34) (0.44)

Inflation Rate

t-statistic for unbiasedness 1.00 2.31a 5.39a -2.16a

t-statistic for efficiency 6.58a - 7.52a - 3.06a 0.46


1st order 0.21 0.58a 0.33 -0.19 2nd order 0.39b 0.26 - 0.30 -0.15

Root-Mean-Squared Error 2.83 2.71 3.35 1.1(0

Coefficient in Composite 0.12 0.06 0.02 0.79 predictord (0.08) (0.11) (0.08) (0.08)

Signiticant at the 5% level. bSignificant at the 10% level. cR = 0.48; SEE = 0.93; DW = 2.46. This equation has been fitted subject to the restrictions that the slope coefficients sum to one and the intercept is zerco.

The F-statistic for this constraint is 0.53 with 2 and 18 degrees of freedom. The F-statistic for the hypothesis that the intercept is zero, the coefficient of the monetarist model is unity, and the remaining coefficients are zero is 0.58 with 5 and 16 degrees of freedom.

dR2 = 0.96; SEE = 0.99; DW = 2.34. This equation has been fitted subject to the restrictions that the slope coefficients sum to one and the intercept is zero. The F-statistic t'or this constraint is 1.02 with 2 and 18 degrees of freedom. The F-statistic for the hypothesis that the intercept is zero, the coefficient of the monetarist model is unity, and the remaining coefficients are zero is 2.08 with 5 and 18 degrees of freedom.

one and the other weights are zero cannot be rejected. Hence, the monetarist model subsumes all information in the other models.

As regards the other models, the adaptive expectations version of the natural rate model once again outperforms the rational expectations version in the 1957-1979 period. The former con- sistently produces smaller root-mean-squared errors and receives larger weights in the composite predictors. Finally, the trade-off model predicts less accurately than the adaptive expectations version, as indicated by the sizes of their root- mean-squared errors.

VI. Discussion of the Results

The results suggest the following conclusions about the models' explanatory power. During the 1895-1956 subperiod, the trade-off model has the

greatest explanatory power and the monetarist model has the least. Just the reverse is true of the 1957-1979 subperiod where the monetarist model has the greatest and the trade-off model, the least. In both subperiods, the adaptive expectations version of the natural rate model outperforms the rational expectations version and, in addition, falls between the trade-off and monetarist models.

Prior to considering some explanations for these findings, it is of interest to note that estimates of these models for the 1957-1979 subperiod confirm these conclusions.'4 (See tables A.1 to A.4 in the appendix.) As might be expected, all models un- dergo structural changes between the two subperi-

14 The 1895-1979 estimates are similar to the 1895-1956 estimates and consequently produce predictions of inflation and unemployment similar to those discussed in the previous section.




ods. But more importantly, the monetarist model is consistent in the 1957-1979 subperiod with all of its underlying hypotheses, whereas the estimates of the trade-off and natural rate models are not satisfactory. In particular, their problems center on the Phillips curve with the chief difficulty being that all variables, except the inflation terms, are insignificant. For the trade-off model, this includes the inverse of the unemployment rate, thereby implying independence between inflation and unemployment in both the short run and the long run. Finally, the adaptive expectations version fits the 1957-1979 subsample better than the rational expectations version. 15

As far as the trade-off model is concerned, there are two possible reasons for the variation in its explanatory power, both of which focus on structural shifts in the Phillips curve. One is that it is due to a change in inflationary expectations associ- ated with the change in 1934 from a gold to a fiduciary monetary standard. Although inflationary expectations do not directly appear in the Phillips curve, the lagged inflation rate can be interpreted as embodying a return-to-normal form of inflationary expectations.'6 Klein (1976) has argued that this form of expectations would be appropriate for a gold standard, but not a fiduciary standard where an extrapolative scheme would be more appropriate. Klein also argues that it was not until the mid-1950s at the earliest that individuals recognized the need to alter inflationary expectations. If Klein is correct, then the satisfactory results for the trade-off model during the 1895-1956 subperiod may be due to its use of the correct specification for inflationary expectations and its poor performance thereafter may reflect the change in expectations.

Sachs' analysis (1980) of the stability of the Phillips curve provides a second possible explanation for the poor performance of the trade-off model in the 1957-1979 subperiod. Sachs finds the slope of the Phillips curve to be flatter in the postwar period, a development which he attributes to greater economic stability and the spread of long-term labor contracts in the postwar period. As with changes in inflationary expectations, the implication of these developments is a structural change in the Phillips curve during the postwar period. Since the sample used to estimate the trade-off model largely excludes observations from the postwar period, such a structural shift would cause the model to predict poorly during the 1957-1979 subperiod.

Interestingly enough, both these explanations are partly corroborated by the 1957-1979 estimates of the trade-off model shown in table A. 1 in the appendix. As indicated earlier, the source of the model's instability is the Phillips curve. In addition, the coefficient of the lagged inflation rate does increase, which is consistent with Klein's contention regarding changes in inflationary expectations. Moreover, the estimate of the slope of the Phillips curve in the 1957-1979 subperiod is much flatter than the 1895-1956 estimate. In spite of the evidence, these explanations are not necessarily complete because the Phillips curve is structurally unstable between the 1895-1929 and 1930-1956 subperiods (see table A.1) and because the overall estimates for the 1957-1979 subperiod are not satisfactory.

Applying these two explanations to the monetarist model is not as straightforward a matter as in the case of the trade-off model, and consequently produces a much less satisfactory explanation of the model's variations in predictive power. As far as the role of inflationary expectations is concerned, Stein (1978) assumes in deriving the model that their influence is weak. However, the changed behavior of prices caused by the change in the monetary standard is consistent with the change in predictive performance of the inflation equation between the 1895-1956 and 1957-1979 subperiods. To understand why, first note, as Meltzer (1977) has pointed out, that the absence of a high degree of positive serial correlation in the inflation rate during the gold standard period implies that lagged money growth rates are better predictors of inflation than lagged inflation rates.

'5It might be thought that the poorer quality of the data prior to 1929 might be responsible for these findings. This does not, however, appear to be the case as the estimates for the 1930-1979 subperiod are very similar to those for the 1895-1979 period. (See tables A.1 to A.4 in the appendix.) Hence, it seems unlikely that the findings are affected by the inclusion of the pre-1930 data.

16To see this, suppose equation (1) had been formulated as p = a1 + ac1UN + pe + a3prn - a4PRD + e, and expectations were formed as pe - p- 1) n[ p* - p(- 1)]. By sub- stituting for pe, the first equation becomes p a'1 + np + a/1UN + (1 - n)p(- 1) + a3pm - a4PRD + e, which corre- sponds to the form of the estimated equation (1). In the expectation equation, p* is the expected long-run inflation rate and short-run expectations follow a regressive or return-to-normal pattern (Kane and Malkiel, 1976). The estimated value of n is 0.69.




In contrast, the high degree of positive serial correlation during the fiduciary standard implies just the opposite. As can be seen in table 1, the estimated inflation equation gives a small weight of 0.19 to the lagged money rate variable and a large weight of 0.81 to the lagged inflation rate. Hence, the behavior of inflation for most of the 1895-1956 subperiod has coincided with a pattern that the model is incapable of predicting while in the 1957-1979 subperiod, it has coincided with a pattern that the model can explain.

The flatter slope of the Phillips curve may also be related to the superior performance of the monetarist model in the 1957-1979 subperiod. In formulating the model, Stein (1978) begins with a Phillips-type relation in which the inflation rate depends positively upon excess labor demand and excess product demand. An increase in the unemployment rate reduces excess labor demand and raises excess product demand and consequently has an ambiguous effect on inflation. For his model, Stein hypothesizes that the net effect of these two influences is zero and hence the slope of the Phil- lips curve is zero. While Stein's reasoning is quite different from that used by Sachs to justify the reduction in the slope of the Phillips curve, it is noteworthy that the 1957-1979 estimates of the slope of the short-run Phillips curve from the trade-off model and some of Sachs' postwar estimates are quite close to zero. Hence, during the 1957-1979 subperiod, the weak relationship between inflation and unemployment may simply mean that other models such as Stein's, which de-emphasize the link between the two, are able to predict better.

The finding that the adaptive expectations version of the natural rate model outperforms the rational expectations version also deserves further comment. Although it may be tempting to do so, this result should not necessarily be taken as an indictment against rational expectations. As the results indicate, the model in which rational expectations are used may be the wrong model. But more importantly, the mechanical method used to calculate the expectation formula using the model's equations may not be appropriate when the pub- lic's recognition of policy changes takes time (Friedman, 1979). The change in the monetary standard aptly illustrates this point, which may, in addition, be one reason why the rational expectations model performs so poorly. In retrospect, it

might have been better to have used only the pre-1934 period in estimating the equations used to calculate the expectations of the exogenous variables. Similarly, when simulating the rational expectations model during the 1960s and the 1 970s, new equations for the exogenous variables might have been used. Thus, a more careful application of rational expectations may prove them to be useful methods of formulating expectations.

VII. Conclusions

The most important finding of this study is that no model by itself explains the past eighty years of inflation and unemployment. Strictly speaking, this finding applies only to the three models included in this paper and it may well be that a different model might provide a consistent explanation for this span of years. In this regard, Gordon (1980, 1981) has proposed and estimated one alternative for this same time period in which the rate of change in real output and expected inflation are the prime determinants of inflation. Although all the parameter estimates do not appear stable, this model should be tested further.

The research in this paper implies that there are at least two obstacles to finding a general theory of inflation and unemployment. The first, and per- haps least difficult, is changes in inflationary expectations. Not only must allowance be made in this regard for the effects of changes in the monetary standard and policy rules, but the time at which individuals realize the implications of these changes must be taken into account. The second obstacle is gradual changes like those in labor contracts and the economic environment that may actually make it impossible to find stable functional relationships. If so, then some method must be found that permits these developments to be incorporated into models as they occur rather than as special effects after they have happened. Without this framework, predictions of inflation and unemployment could not be made with any degree of confidence.

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APPENDIX

TABLE A. I - ESTIMATES OF THE TRADE - OFF MODEL

1895-1979 AND SELECTED SUBPERIODS

Variable/Statistics 1895-1979 1895-1956 1957-1979 1895-1929 1930-1956 1930-1979

Phillips Curve (dependent variable = p)

constant 0.40 - 0.09 1.90 -0.85 -0.31 0.19 p(- 1) 0.33a 0.30a 0.70a 0.32a 0.22a 0.30a 1/UN 6.79a 8.66a 1.41 9.17a 12.8la 10.46a PRD - 0.49a - 0.58a - 0.52 - 0.61a - 0.22 - 0.25 pm 0.42a 0.42a 0.09 0.40a 0.36 0.35a WW2c - 3.15 a - 3- lgb _ - 5 57a - 5.o4a NEPd - 4.56a - - 0.73 - - - 4.43a

R 2 0.81 0.84 0.81 0.90 0.81 0.76 SEE 2.47 2.53 1.04 2.23 2.51 2.24

Demand Curve (dependent variable UN)

constant 98.65a 89.70a 79.34 91.27a 60.35a 67.98a log( M/P) - 0.19a - 0.15a -0.24a -0.20a -0.02 -0.09 log G - 0.05a - 0.05a - 0.02 - 0.03 - 0.08a - 0.07a log X - 0.08a -0.09a -0.07a -0.07a -0.I la - 0.I la t 1.07a 0.97a l.4la 0.99a 0.49b 0.91a u(- 1) 0.69a 0.6 la 0.71a 0.47a 0.47a 0.8 a R 2 0.77 0.79 0.52 0.63 0.89 0.86 SEE 2.01 2.21 0.89 2.20 1.92 1.72

F-Statistics for Tests of Structure Stability

1895-1929/1930-56/1957-79 1895-1956/1957-79 1895-1929/1930-56 1930-56/1957-79 Phillips Curve 2.67a 2.59a 2.32a 3.27a Demand Curve 2.13 1.29 2.21 2.76a

aSignificant at the 5% level. bSignificant at the 10% level. c WW2 is a dummy variable used to capture the effects of price controls during World War II. It is I for 1942 to 1945. d NEP is a dummy variable used to capture the effects of price and wage controls during the Nixon Administration. It is 0.5 in 1971 and I for 1972 and 1973.

TABLE A.2.- ESTIMATES OF THE NATURAL RATE MODEL ADAPTIVE EXPECTATIONS VERSION


Variable/Statistics 1895- 1979 1895-1956 1957- 1979 1895- 1929 1930-1956c 1930-1979

Phillips Curve (dependent variable = UN)

constant 0.31 0.26 1.88b 0.46 - 0.03 - 0.03 p - P(- 1) -0.33a -0.30a -0.93a -0.23a -0.54a -0.53a UN( - 1) 1.45a l.42a 0.42 . 18a 1.58a 1.58a UN(- 2) - 0.50a - 0.48a - 0.29 -0.29a - 0.57b - 0.56a v(- 1) - 0.50a - 0.54a - 0.14 _0.59a - 0.33 - 0.34b v(- 2) - 0.32a - 0.32a - O.S lb 0.30b - 0.22 - 0.21 R 2 0.71 0.72 - 0.17 0.61 0.76 0.76 SEE 2.25 2.54 1.39 2.25 2.90 2.22

Demand Curve (dependent variable = UN)

constant 111.41a 103.35a 72.98a 120.13a 77.68a 77.95a log( M/P) - 0.25a - 0.22a - 0.20 - 0.30a -0.08 -0.09 log G - 0.02 -0.02 - 0.01 - 0.01 - 0.06 a _0.06 b log X - 0.08a - 0.08a - 0.06a - 0.08a 0. I la -0. la t 1.13a 1.06a l.12 a l.44a 0.71a 0.72a u(- 1) 0.69a 0.59a 0.64a 0.52a 0.48b 0.48a

R 2 0.74 0.75 0.47 0.58 0.87 0.57 SEE 2.16 2.42 0.93 2.36 2.11 2.96

x2-Statistics for Tests of Structural Stability

1895-1956/1957-79 1895-1929/1930-56/1957-79 1895-1929/1930-56 1930-56/1957-79 x2 84.14a 114.91a 30.78a 140.08a df 15 30 15 15

aSignificant at the 5% level. bSignificant at the 10% level. c Failed to converge.




TABLE A.3.- ESTIMATES OF THE NATURAL RATE MODEL RATIONAL EXPECTATIONS VERSION


Variable/Statistics 1895-1979 1895-1956 1957-1979 1895-1929 1930-1956c 1930-1979

Phillips Curve (dependent variable = UN)

constant 3.77a 3.40 4.05b 3.67 -5.10 2.79 log(P/Pe) -0.24a -0.25 a -0.43a -0.23a - 0. 17 a -0.22a UN(- 1) 0.37a 0.40a 0.28 0.06 0.67a 0.61a e(- 1) 0.79a 0.77a 0.59a 0.8 la 0.57 0.78a R2 0.63 0.63 -0.46 0.49 0.83 0.75 SEE 2.57 2.93 1.55 2.59 2.46 2.28

Demand Curve (dependent variable = UN)

constant 332.43a 351.77a 111.16a 167.86a 158.73 182.53a log(M/P) -0.87a -0.96a -0.38a -0.44a -0.73a -0.49a log G 0.00 0.01 0.03 -0.01 0.02 - 0.01 log X -0.08a -0.07b -O. oa -0.08a -0.05 -0.07a t 2.98a 3.43a 1.91a 2.05a 3.617 1.64a u(-1) 0.89a 0.86a 0.72a 0.77a 0.96a 0.84a R 2 -0.06 -0.28 0.42 0.44 0.04 0.62 SEE 4.33 5.47 0.98 2.72 5.79 2.78


1895-1956/ 1957-79 1895-1929/ 1930-56/ 1957-79 1895-1929/ 1930-56 1930-56/ 1957-79 X2 55.67a 79.Sla 23.84a 58.13a df 13 26 13 13

aSignificant at the 5% level. bSignificant at the 10% level. ' Failed to converge.

TABLE A.4.-ESTIMATES OF THE MONETARIST MODEL


Variable/Statistics 1895-1979 1895-1956 1957-1979 1895-1929 1930-1956 1930- 1979

Inflation Equation (dependent variable p)

p(- 1) 0.82a 0.81a 0.88a 0.77a 0.85a 0.86a rn(- 1) 0.18a 0.I9b 0.12a 0.23 0.15b 0.14a e(-2) -0.39a 0.3ga - 0.51 -0.29b -0.58a -0.57 a

e(-4) -0.25 a -0.25a 0.01 -0.33a -0.14 -0.14 R 2 0.16 0.10 0.81 -0.03 0.34 0.44 SEE 5.13 6.02 1.05 6.98 4.64 3.38

Unemployment Equation (dependent variable UN)

constant 1.83a 1.72 2.43b 1.69a 2.45 3.02

UN(- 1) 0.72a 0.72a 0.70a 0.78a 0.58 0.51b mn(- 1) -p(- 1) -0 .ob .09 - 0.26a -0.18a -0.01 -0.05

U(- 1) 0.28 0.30 -0.20 -0.15 0.8lb 0.80 a

R 2 0.72 0.72 0.63 0.59 0.79 0.80 SEE 2.22 2.57 0.78 2.31 2.69 2.04


1895-1956/ 1957-79 1895-1929/ 1930-56/ 1957-79 1895-1929/ 1930-56 1930-56/ 1957-79 x 2 88.84a 115.14a 26.30a 69.87a

df 10 20 10 10

a Significant at the 5% level. hSignificant at the 10% level.



the explanatory power of alternative theories of inflation and unemployment, 1895-1979

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