Expectations and monetary neutrality: an empirical reexamination.

I. Introduction

Modern monetary business cycle models feature rational expectations and natural rate assumptions which imply that unanticipated changes in nominal aggregates affect real output, but anticipated changes do not. Initial tests conducted by Barro [3] and by Barro and Rush [4] supported the neutrality of anticipated money growth. In a particularly influential study, Mishkin [21; 22] rejected the neutrality of anticipated policy. Mishkin's paper [21] has become one of the most frequently cited studies in empirical macroeconomics. The influence of the study probably derives in large part from the unequivocal nature of the results. In particular, Mishkin showed that i) the hypothesis that anticipated policy has no effect on output is strongly rejected; ii) anticipated nominal changes have a greater estimated effect on output than do unanticipated nominal changes; iii) lags as long as 17 quarters have significant coefficients, implying that Barro's eight-lag model was underspecified; iv) in some specifications, the estimated sign of the effect of nominal innovations is opposite to the prediction of monetary business cycle theory. Frydman and Rappoport [9] used Mishkin's empirical model to provide further evidence against the neutrality hypothesis by showing that the data provide no evidence for distinguishing between the output effects of anticipated policy and those of unanticipated policy.

Although several subsequent studies have supported the neutrality hypothesis [7; 12], Mishkin's careful study and a small number of additional studies suggest that the neutrality hypothesis is not robust. The empirical rejection of the neutrality hypothesis contradicts a wide range of modern monetary models, including those based on imperfect information [14; 18; 19], and those based on imperfect competition [8; 20].(1) As a consequence, the focus of macroeconomists has in recent years shifted to non-monetary models in which money responds passively to changes in aggregate output [16; 17].

This paper explores the robustness of Mishkin's results using an updated data set. The model yields results that are more consistent with the neutrality hypothesis than those of Mishkin. In particular, the results contradict each of Mishkin's four salient findings described above. The study also presents evidence which suggests rejection of the Frydman-Rappoport hypothesis regarding the irrelevance of the distinction between the output effects of anticipated and unanticipated policy.

Section II describes the empirical model. In particular, I describe the approach to specifying forecasting equations for M1 and inflation. The forecasting equations allow decomposition of nominal changes into anticipated and unanticipated components. The extended data set yields specifications that differ from those obtained by Mishkin. Section III presents and interprets the results and section IV concludes.

II. Econometric Issues

The Model

I test the neutrality hypothesis in the context of the model employed by Mishkin and Frydman and Rappoport. The model consists of two equations:

[Mathematical Expression Omitted], (1a)

and

[X.sub.t] = [Z.sub.t][Gamma] + [[Mu].sub.t]. (1b)

Equation (1a) is the output equation, where [y.sub.t] is the log of real GNP and [Mathematical Expression Omitted] is the natural rate. Equation (1b) is the forecasting equation for the nominal aggregate. The variable [X.sub.t] represents the rate of growth of the nominal aggregate, and [Mathematical Expression Omitted] is the expectation of [X.sub.t] at time t - 1. The term [Z.sub.t] is a matrix of variables used to forecast [X.sub.t], and [Gamma] is a vector of parameters. The variables in [Z.sub.t] are known in period t - 1, and [v.sub.t] is assumed uncorrelated with any information available in period t - 1. These assumptions imply that the fitted values from equation (1b) represent optimal forecasts of [X.sub.t], hence

[Mathematical Expression Omitted]. (2)

Following Mishkin, I represent [Mathematical Expression Omitted] by a linear time trend, and allow [u.sub.t] to follow a second-order autoregressive process.(2) Equation (1a) can be written more specifically as

[y.sub.t] = [Alpha]t + [summation over i] [[Beta].sub.i]([x.sub.t-i] - [Z.sub.t-i][Gamma]) + [summation over i] [[Delta].sub.i][Z.sub.t-i][Gamma] + [[Rho].sub.1][u.sub.t-1] + [[Rho].sub.2][u.sub.t-2] + [[Epsilon].sub.t], (1a[prime])

where t is time and [[Epsilon].sub.t] is normally distributed and uncorrelated. The neutrality hypothesis states that expected changes in the nominal aggregate leave real output unaffected, which implies [[Delta].sub.i] = 0 for all i.

I explore Mishkin's claim that a specification which includes long lags rejects the neutrality hypothesis. Hence I extend the lag distributions of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] to twenty quarters. Mishkin found that lags of nominal shocks as long as seventeen quarters significantly affect output. Barro's model, which includes only eight quarterly lags, risks leaving out significant lags which can cause inconsistent estimates and incorrect test statistics.(3)

I estimate two different specifications of the model: one with M1 growth and the other with inflation as the nominal aggregate, [X.sub.t]. (Mishkin used M1 in one study [21], and inflation in another study [22]). Like Mishkin, I restrict the coefficients [[Beta].sub.i] and [[Delta].sub.i] to follow a fourth-degree polynomial distributed lag (PDL) with the endpoint constrained to zero. The fourth-degree PDL represents a relatively flexible lag distribution which the data rarely reject in practice. The neutrality hypothesis predicts zero values for the four PDL coefficients on anticipated policy.

Consistent estimation of the output equation requires exogeneity of M1 and inflation with respect to output. Negative correlation between unanticipated inflation and the equation (1) error term follows directly from the Equation of Exchange: cetera paribus, an increase in the growth rate of real income implies a reduction in the rate of inflation.(4) We can question the exogeneity of M1 as well, but most economists would probably consider the problem less severe for M1 than for inflation. In order to closely adhere to the spirit of Mishkin's test, I proceed as if M1 and inflation can be considered exogenous. A Monte Carlo study by Hoffman, Low, and Schlagenhauf [15] suggests that the issue might not be a cause for concern. Hoffman, Low, and Schlagenhauf examined Mishkin-Barro tests and found that non-zero correlation between errors in the forecasting and output equations does not increase the incidence of rejections of the correct null hypothesis.

Forecasting Equations

A remaining issue concerns the choice of variables in [Z.sub.t] used to forecast [X.sub.t]. Sargent [25] shows that an observational equivalence arises between the effects of anticipated and unanticipated policy when [Z.sub.t] contains only lagged values of the nominal variable. Identification of the model requires specification of (1b) as something other than a simple autoregression. Mishkin [21; 22] employs an atheoretical statistical procedure to choose the variables in [Z.sub.t]. The procedure retains in the forecasting equation only variables which prove statistically significant. A number of researchers, including Cover [5] and Frydman and Rappoport [9], have adopted Mishkin's specifications of the forecasting equations.

I use Mishkin's atheoretical approach to specification, but not his particular specifications. To determine the specification I regress M1 and inflation on their own four lagged values as well as on four lagged values of a set of macro variables, including: the growth rate of M2, the growth rates of real and nominal GNP, the unemployment rate, the 90-day T-bill rate, the real full-employment surplus, the real federal deficit, and net real exports.(5) Mishkin retains in the forecasting equation only variables whose four lagged values were jointly significant at the five percent level in the final specification. I include only variables whose lagged values jointly meet Amemiya's [2] criterion for minimizing the mean squared prediction error. In this case, Mishkin's and Amemiya's criteria produce the same specification. Using U.S. data from 1953.I to 1991.III, the procedure yields a specification of M1 growth as a function of its own lagged values and lagged values of the T-bill rate and the unemployment rate. The estimated inflation equation includes lagged inflation and lags of the T-bill rate. The procedure generates the following estimates:

[Mathematical Expression Omitted]; (3a)

N = 151, [R.sup.2] = .653, S.E. = .0060,

[Mathematical Expression Omitted]; (3b)

N = 151, [R.sup.2] =.680, S.E. =.0038,

where M represents the growth rate of M1, R is the 90-day T-bill rate, [Pi] stands for the growth rate of the GDP deflator, and UN represents the unemployment rate.(6) Parentheses contain standard errors.

Note that the specifications meet Sargent's criterion for identification since the regressors include variables other than lags of the dependent variable. Furthermore, the specifications differ from those obtained by Mishkin for the period 1954I to 1976IV. Mishkin's M1 equation included the full-employment surplus and excluded the unemployment rate; his specification for inflation included the growth rate of M2. Mishkin's specification procedure fails to replicate his particular specifications over the extended data set.

The credibility of equations (3a,b) as rational forecasts hinges on the stability of the parameters over time. Shifting parameters suggest changes in policy rules or general changes in the process generating the nominal aggregate. As a consequence, a number of researchers have demonstrated concern for the stability of the forecasting equations, including Mishkin [21; 22], Frydman and Rappoport [9], and Dutkowsky and Atesoglu [6].(7) I perform a test of general parameter stability over the sample as a whole. The test involves forming a t-statistic for the mean of the re-cursive residuals, which has expectation zero under the null hypothesis of parameter stability [13].(8) The procedure yields a t-statistic of 0.42 for the money equation and 1.69 for the inflation equation. Thus the tests fail to reject parameter stability at the .05 level. The result suggests that the forecasting equations (3a,b) accurately and parsimoniously represent the processes generating money growth and inflation.

[TABULAR DATA FOR TABLE III OMITTED]

III. Results

Maximum likelihood estimation of the two-equation model (1a,b) yields the Wald test statistics presented in Table I.(9) The table displays results of tests of the individual significance as well as the joint significance of anticipated and unanticipated policy.

The M1 specification supports the neutrality hypothesis. Unanticipated money is significant at the 5 percent level while anticipated money is not significant at conventional levels. The insignificance of anticipated inflation also concurs with the neutrality hypothesis. These results contradict Mishkin's rejection of the neutrality of anticipated policy. Note that unanticipated inflation is insignificant as well, which does not agree with the theoretical prediction that output fluctuations arise from nominal misperceptions. Hence monetary business cycle models receive relatively less support from the inflation specification.

Tables II and III present the individual lag coefficients and standard errors implied by the estimated PDL coefficients. The tables illustrate two main results: unanticipated policy performs better than anticipated policy, and the inflation specification fits the data poorly in comparison to the M1 specification. Contemporaneous unexpected money as well as the first two lags of unexpected money are significant at the 1 percent level. None of the anticipated M1 coefficients are significant at the 1 percent level, although the coefficients on lags 3 through 10 prove significant at the 5 percent level. The third and fourth lags of unanticipated M1 also reach significance at the 5 percent level. None of the individual lag coefficients on anticipated or unanticipated inflation demonstrate significance at the 1 percent level, and only contemporaneous unanticipated inflation is significant at the 5 percent level.

The unanticipated inflation coefficients are positive, which is consistent with monetary business cycle theory. This result contradicts Mishkin's [22] finding of negative output effects of unanticipated inflation. The estimated coefficients on anticipated inflation are generally negative. It follows that the coefficients on unanticipated inflation exceed those on anticipated inflation, which agrees with the theory. The question arises as to whether this difference is significant, i.e., whether the Frydman-Rappoport hypothesis of the irrelevance of the anticipated/unanticipated distinction applies to the inflation specification. I use the Wald criterion to test the Frydman-Rappoport hypothesis that the anticipated inflation PDL coefficients equal the unanticipated inflation PDL coefficients. The resulting [[Chi].sup.2] (4) equals 8.04 which indicates rejection of the null hypothesis of equal coefficients at the 10 percent level. Hence even the inflation specification provides some evidence for rejecting the Frydman-Rappoport hypothesis.

Table II. Maximum Likelihood Estimates of Output Equation Lag
Coefficients

             Anticipated M1 Growth      Unanticipated M1 Growth
Lag          Coeff.       Std. Err.     Coeff.         Std. Err.

0            .007           (.173)      .387(**)        (.153)
1            .191           (.146)      .599(**)        (.205)
2            .333           (.179)      .694(**)        (.261)
3            .439(*)        (.211)      .697(*)         (.298)
4            .513(*)        (.229)      .633(*)         (.323)
5            .558(*)        (.237)      .522            (.345)
6            .578(*)        (.239)      .383            (.367)
7            .577(*)        (.239)      .232            (.390)
8            .559(*)        (.240)      .083            (.411)
9            .527(*)        (.242)     -.053            (.426)
10           .484(*)        (.243)     -.168            (.432)
11           .433           (.242)     -.254            (.428)
12           .376           (.240)     -.309            (.414)
13           .317           (.235)     -.332            (.392)
14           .258           (.228)     -.322            (.367)
15           .200           (.221)     -.285            (.343)
16           .147           (.212)     -.226            (.322)
17           .099           (.200)     -.154            (.302)
18           .060           (.180)     -.081            (.274)
19           .029           (.146)     -.020            (.226)
20           .009           (.089)     -.014            (.140)

Notes: 151 quarterly observations, 1954I to 1991III.

* and ** indicate significance at the five and one percent levels.

We can perhaps attribute the poor performance of the inflation components to the endogeneity of inflation with respect to output. Such a bias might lead the estimated coefficient to differ insignificantly from zero when, in fact, a significant positive relationship exists. The oil price shocks of the 1970s make the data used here especially vulnerable to the endogeneity problem. An oil-price shock couples unanticipated inflation with declining output, which contradicts the prediction of monetary business cycle theory. Oil shocks might represent exceptional events specific to the 1970s which obscure the general relationship between output and monetary factors. A solution might involve the inclusion of the real price of oil or an index of energy prices in the output specification [12]. Also, we could measure inflation with the core CPI rather than the GDP deflator. The core CPI excludes food and energy prices and might more accurately reflect price level changes due exclusively to shifts in monetary factors.

To further explore the relationship between output and unanticipated nominal aggregate demand, I estimate the equation

[Mathematical Expression Omitted], (4)

for both the M1 and inflation specifications.(10) A Wald test of the hypothesis that the PDL coefficients on unanticipated M1 growth equal zero using the maximum likelihood estimates yields a [[Chi].sup.2] statistic of 12.79 with four degrees of freedom. The test rejects the neutrality of unanticipated M1 at about the 2 percent significance level. The inflation specification produces a [[Chi].sup.2] (4) of 9.22, implying rejection of the neutrality of unanticipated inflation at about the 6 percent level. Table IV displays the individual lag coefficients and standard errors implied by the estimated PDL coefficients. Contemporaneous and four lagged values of nominal shocks each prove significant at better than the 5 percent level for both the inflation and M1 specifications. The positive estimated effect of nominal shocks on output agrees with monetary business cycle theory. Some of the estimated lag coefficients are negative at relatively long lags (9 quarters or more), but none are statistically significant. The results again counter Mishkin's estimation of significant negative coefficients.

Table III. Maximum Likelihood Estimates of Output Equation Lag
Coefficients

            Anticipated Inflation      Unanticipated Inflation
Lag        Coeff.       Std. Err.      Coeff.        Std. Err.

0           .736         (.672)         .341(*)        (.172)
1          -.125         (.544)         .409           (.336)
2          -.724         (.577)         .557           (.549)
3         -1.10          (.642)         .755           (.775)
4         -1.30          (.683)         .976          (1.01)
5         -1.35          (.691)        1.20           (1.26)
6         -1.29          (.669)        1.40           (1.50)
7         -1.14          (.623)        1.57           (1.72)
8          -.937         (.562)        1.70           (1.90)
9          -.700         (.493)        1.77           (2.03)
10         -.451         (.427)        1.78           (2.10)
11         -.207         (.376)        1.72           (2.12)
12         -.014         (.354)        1.61           (2.07)
13          .203         (.371)        1.45           (1.95)
14          .349         (.417)        1.24           (1.78)
15          .446         (.474)         .992          (1.56)
16          .490         (.522)         .732          (1.30)
17          .480         (.545)         .476          (1.02)
18          .420         (.523)         .246           (.730)
19          .314         (.439)         .069           (.456)
20          .170         (.273)        -.024           (.211)

Notes: 151 quarterly observations, 1954I to 1991III.

* and ** indicate significance at the five and one percent levels.

The results also contradict Mishkin's finding of significant lag coefficients at relatively long lags. Mishkin estimated significant lag coefficients on M1 at lags of as many as 15 quarters and on inflation at lags of as many as 17 quarters. None of the estimated output equations discussed above contain significant lag coefficients beyond the 10th lag and only one specification (anticipated M1) features significant lag coefficients beyond the fourth lag. This result suggests misplaced emphasis on models with long lags. Instead, a relatively short and freely estimated lag distribution might be appropriate.

Table IV. Maximum Likelihood Estimates of Equation (4)
Lag Coefficients

         Unanticipated M1 Growth     Unanticipated Inflation

Lag      Coeff.         Std.Err.     Coeff.        Std. Err.

0        .350(**)        (.123)       1.23(**)        (.165)
1        .552(**)        (.189)       1.45(**)        (.287)
2        .651(**)        (.251)       1.51(**)        (.415)
3        .671(*)         (.291)       1.43(**)        (.519)
4        .630(*)         (.317)       1.25(*)         (.603)
5        .546            (.336)       1.01            (.673)
6        .436            (.354)        .74            (.734)
7        .312            (.371)        .45            (.785)
8        .187            (.387)        .17            (.826)
9        .070            (.399)       -.07            (.854)
10      -.032            (.403)       -.28            (.867)
11      -.112            (.399)       -.44            (.863)
12      -.167            (.387)       -.55            (.841)
13      -.197            (.369)       -.60            (.801)
14      -.201            (.348)       -.59            (.743)
15      -.182            (.327)       -.54            (.670)
16      -.145            (.308)       -.44            (.585)
17      -.097            (.288)       -.32            (.489)
18      -.046            (.260)       -.20            (.385)
19      -.005            (.213)       -.08            (.268)
20       .016            (.131)       -.01            (.147)

Notes: 151 quarterly observations, 1954I to 1991III.

* and ** indicate significance at the five and one percent levels.

IV. Concluding Remarks

Mishkin's strong rejection of the neutrality hypothesis does not hold over the updated data set. The results support the neutrality hypothesis when M1 growth defines the nominal aggregate. Monetary models receive less support from the inflation specification, but Mishkin's rejection of neutrality again does not occur. The result suggests that the neutrality of anticipated policy remains an open question.

The different results must arise from the extension of the data by nearly fifteen years, since the technique employed here does not substantially differ from that employed by Mishkin. The larger data set potentially increases the sharpness of the estimates and the power of the tests. Furthermore, the estimated lag distributions presented here have more plausible and consistent shapes than those estimated by Mishkin. The extended data set also fails to replicate Mishkin's particular specifications for the monetary forecasting equations. Each of the forecasting equations used in this study passes tests of parameter stability, whereas Mishkin's inflation equation did not.

Future research must strive to establish robust results with respect to alternative specifications. The present paper finds support for the neutrality hypothesis using M1 as the aggregate demand variable. But financial innovation during the period in question suggests that M1 might have had a different effect on output in the 1950s than in the 1980s. As an alternative, M2 can serve as the measure of aggregate demand.

Economists have generally come to view the neutrality hypothesis as incompatible with the empirical evidence. Research has shifted to models of real fluctuations with money playing only a passive role. This paper by no means presents conclusive evidence on the neutrality hypothesis, but it does indicate that monetary models remain a valid field of research.

1. Misperceptions models predict that discretionary monetary policy has no effect on output, whereas models of imperfect competition allow a possible role for monetary policy. Several authors observe, however, that the assumption of rational expectations, combined with a natural rate hypothesis, leads both types of model to predict the same time series relationship between nominal innovations and output; that is, only unanticipated nominal innovations affect output [1; 10; 12].

2. Higher-order terms proved insignificant.

3. A Monte Carlo study by Hoffman, Low, and Schlagenhauf [15] finds that ignoring relevant lags significantly biases the Mishkin and Barro tests toward over-rejection of the neutrality hypothesis, while including superfluous lags causes only slight bias.

4. Anticipated inflation is predetermined since it depends only on lagged values.

5. All series are taken from the Citibase data bank, with the exception of the full employment surplus which is from Gordon [11].

6. The presence of lagged dependent variables in equations (3) implies that consistency of the estimated parameters requires serially uncorrelated disturbances. I perform a Breusch-Godfrey test for serial correlation of up to eighth-order. The test produces [[Chi].sup.2](8) = 11.02 for the M1 equation and [[Chi].sup.2](8) = 11.04 for the inflation equation. The ten percent critical value is 13.36, so the tests fail to reject the null hypothesis of no serial correlation at conventional significance levels.

7. On the other hand, the Monte Carlo study of Hoffman, Low, and Schlagenhauf [15] finds that failure to model a structural break in the forecasting equation does not necessarily bias the results, especially in samples of 120 or more observations.

8. The nth recursive residual is the scaled forecast error for the nth observation based on a regression involving the first n - 1 observations.

9. Barro [3] and Barro and Rush [4] estimate the model using a two-step procedure: they obtain OLS estimates of the forecasting equation and use the residual and fitted values as regressors in an OLS estimation of the output equation. As Pagan [23] shows, the two-step estimates are not efficient, but they are consistent. I obtain maximum likelihood estimates from the consistent estimates through one iteration of Newton's method (method of scoring). Consistency of the initial estimates implies that further iteration produces no gain in efficiency [24]. Consistent estimation of the output equation requires the contemporaneous error term, [[Epsilon].sub.t], to be uneorrelated with contemporaneous and lagged nominal surprises, [v.sub.t-i], since the latter appear as regressors in the output equation. For the maximum likelihood computations, I further assume that the disturbance vectors for the two equations are normally distributed and uncorrelated. This assumption implies that output disturbances are uncorrelated with future nominal surprises, which concurs with the rationality postulate that future nominal surprises cannot be forecast from currently available data.

10. Equation (8) corresponds to Mishkin's [21] model no. 4.1 and [22] model 6.1.

References

1. Akerlof, George A. and Janet L. Yellen, "A Near-Rational Model of the Business Cycle, with Wage and Price Inertia." Quarterly Journal of Economics, supplement 1985, 823-38.

2. Amemiya, Takeshi, "Selection of Regressors." International Economic Review, June 1980, 331-54.

3. Barro, Robert, "Unanticipated Money, Output, and the Price Level in the United States." Journal of Political Economy, August 1978, 549-80.

4. ---- and Mark Rush. "Unanticipated Money and Economic Activity," in Rational Expectations and Economic Policy, edited by Stanley Fischer. Chicago: University of Chicago Press, 1980, pp. 23-54.

5. Cover, James Peery, "Asymmetric Effects of Positive and Negative Money-Supply Shocks." Quarterly Journal of Economics, November 1992, 1261-83.

6. Dutkowsky, Donald H. and H. Sonmez Atesoglu, "Unanticipated Money Growth and Unemployment: Post-Sample Forecasts." Southern Economic Journal, October 1986, 413-21.

7. Fackler, James S. and Randall E. Parker, "Anticipated Money, Unanticipated Money, and Output: 1873-1930." Economic Inquiry, October 1990, 774-87.

8. Fischer, Stanley, "Long-Term Contracts, Rational Expectations and the Optimal Money Supply Rule." Journal of Political Economy, February 1977, 191-205.

9. Frydman, Roman, and Peter Rappoport, "Is the Distinction between Anticipated and Unanticipated Money Growth Relevant in Explaining Aggregate Output?" American Economic Review, September 1987, 693-703.

10. Gordon, Robert J. "Comment," on Robert J. Barro and Mark Rush, "Unanticipated Money and Economic Activity," in Rational Expectations and Economic Policy, edited by Stanley Fischer. Chicago: University of Chicago Press, 1980, pp. 55-63.

11. ----. Macroeconomics, 6th ed. New York: Harper Collins, 1992.

12. Gray, Jo Anna, and David E. Spencer, "Price Prediction Errors and Real Activity: A Reassessment." Economic Inquiry, October 1990, 658-81.

13. Harvey, Andrew C. The Econometric Analysis of Time Series, 2nd ed. Cambridge, Mass.: MIT Press, 1990.

14. Haubrich, Joseph G. and Robert G. King, "Sticky Prices, Money, and Business Fluctuations." Journal of Money, Credit, and Banking, May 1991, 243-59.

15. Hoffman, Dennis L., Stuart A. Low, and Don E. Schlagenhauf, "Tests of Rationality, Neutrality, and Market Efficiency: A Monte Carlo Analysis of Alternative Test Statistics." Journal of Monetary Economics, November 1984, 339-63.

16. King, Robert G. and Charles I. Plosser, "Money, Credit, and Prices in a Real Business Cycle." American Economic Review, June 1984, 363-80.

17. Kydland, Finn E. and Edward Prescott, "Time to Build and Aggregate Fluctuations." Econometrica, November 1982, 1345-69.

18. Lucas, Robert E., "Expectations and the Neutrality of Money." Journal of Economic Theory, April 1972, 103-24.

19. ----, "An Equilibrium Model of the Business Cycle." Journal of Political Economy, December 1975, 1113-44.

20. Mankiw, N. Gregory, "Small Menu Costs and Large Business Cycles: A Macroeconomic Model of Monopoly." Quarterly Journal of Economics, May 1985, 529-37.

21. Mishkin, Frederic, "Does Anticipated Monetary Policy Matter? An Econometric Investigation." Journal of Political Economy, February 1982, 22-50.

22. ----, "Does Anticipated Aggregate Demand Policy Matter? Further Econometric Results." American Economic Review, September 1982, 788-802.

23. Pagan, Adrian, "Econometric Issues in the Analysis of Regressions with Generated Regressors." International Economic Review, February 1984, 221-47.

24. Rao, C.R. Linear Statistical Inference and Its Applications. New York: Wiley, 1965.

25. Sargent, Thomas J., "The Observational Equivalence of Natural and Unnatural Rate Theories of Macroeconomics." Journal of Political Economy, June 1976, 631-40.