Adjusted P and F tests and the Keynesian-classical debate.

AuthorHine, Steven C.
  1. Introduction

    The debate over the manner in which changes in money growth rates affect real economic activity, and over the role that systematic monetary policy plays in this relationship, has been a protracted one, yet it remains unresolved. This paper shows that a model proposed by Fischer [13] in which multi-period-ahead expectational errors in money growth are crucial for the determination of real economic aggregates is empirically superior to two leading alternatives when the three are estimated on a sample of U.S. quarterly data spanning the 1970s. In addition, we show that the test used to demonstrate these results, a version of Davidson and MacKinnon's [9] P test which we adjust experimentally to remove small sample biases in its size, is, in this application, more powerful than the more commonly-used F-ratio form of the likelihood ratio test.(1) We use these results on relative power to explain and reconcile some conflicting decisions to which the two alternative tests lead.

    The alternative specifications against which the Fischer model is tested are an equilibrium business cycle model in which only expectational errors over a short time horizon matter for real variables,(2) and a model in which actual money growth affects the real economy regardless of whether it is anticipated or not.(3) The direct comparison of these three models is carried out using true ex ante forecasts of money growth rather than an ex post decomposition of money growth into an anticipated and an unanticipated part by way of the frequently-used auxiliary equation approach.

    Despite the many studies evaluating the New Classical and the AUDI models, neither the view that only unanticipated money growth matters nor the view that only actual money growth matters has appeared persuasive. It is well-known that these models and the Fischer model have radically divergent implications regarding the efficacy of policy conducted according to systematic rules,(4) rendering the choice of specification crucial from a policy analysis perspective. Our evidence of the superiority of Fischer's model thus supports the view that anticipatable monetary policy can have real effects, but that these effects differ from those following unanticipated policy changes.

  2. Model Specifications and Alternative Tests

    The models evaluated in this study are single-equation reduced form expressions which explain deviations in unemployment from its natural rate by means of a distributed lag on alternative measures of money growth. The forms of the three models of unemployment, which we identify as the Barro, Fischer, and AUDI models respectively, are given by the following equations:

    U[N.sub.t] = U[N.sub.nt] + [summation of] [[Beta].sub.1i](t - i - 1[m.sub.t - i] - [E.sub.t - i - 1][m.sub.t - i]) + [u.sub.1t] where i = 0 to [k.sub.1],(1)

    U[N.sub.t] = U[N.sub.nt] + [summation of] [[Beta].sub.2i](t - i - 1[m.sub.t] - [E.sub.t - i - 1][m.sub.t]) + [u.sub.2t] where i = 0 to [k.sub.2],(2)

    U[N.sub.t] = U[N.sub.nt] + [summation of] [[Beta].sub.3i] t - i - 1[m.sub.t - i] [u.sub.3t] where i = 0 to [k.sub.3],(3)

    where U[N.sub.t] is a transformation of the rate of unemployment, U[N.sub.nt] is the natural rate of the transformed unemployment rate, t - i - 1 [m.sub.t-j] is the annualized rate of change in the money supply between periods t - i - 1 and t - j, and [E.sub.t - i - 1][m.sub.t - j] is the anticipated value of t - i - 1[m.sub.t - j] based on information available in period t - i - 1. In each case, U[N.sub.nt] is represented by including a constant and the fiscal variable G/y, the ratio of federal government purchases to output,(5) in the estimated equations. The residuals [u.sub.jt] are specified to be second-order autoregressive transformations of independent and identically distributed normal random variables, a specification that adequately captures the serial correlation present in the quarterly data used.

    There are many studies which have tested versions of the Barro model in equation (1) and the AUDI model in equation (3), beginning with the papers by Barro [1; 2], but the evidence has been quite mixed. The conclusions reached by Barro include the findings that ". . . the hypothesis that only the unanticipated part of money growth is relevant to unemployment is accepted . . ." while ". . . the reverse hypothesis that the [unanticipated money growth] values are irrelevant to unemployment, given the [anticipated money growth] values, can easily be rejected [1,109]." Mishkin, however, claims that ". . . anticipated monetary policy does not appear to be less important than unanticipated monetary policy. In fact, the opposite seems to be the case [23, 118],"(6) although he does not directly test for the significance of unanticipated money growth given its anticipated component. Carns and Lombra [6] find that an ex ante measure of unanticipated money growth serves to overturn Barro's finding of insignificance of the anticipated part while supporting the conclusion that unanticipated money matters.

    Frydman and Rappaport consider the relative importance of unanticipated and anticipated money growth and claim that "raw money growth affects real output in the short run, irrespective of whether it is rationally anticipated or not [14, 702]." However, though they demonstrate that a model characterized by their AUDI hypothesis is not rejected by an equilibrium business cycle model, their testing procedure does not allow a test of whether the equilibrium business cycle model is rejected by the AUDI hypothesis.

    McAleer, Pesaran, and Beta [20] find that their "New Classical" model does not reject a "Keynesian" specification, and as their approach allows for a switching of the roles that the models play in the hypothesis test, they can also claim some support for an apparent rejection of the New Classical model by the Keynesian model. They point out, though, that this conclusion is suspect because of biases in their test statistic. In fact, in a related study, McAleer and McKenzie [21] present evidence that neither of two versions of the New Classical model can be rejected by Keynesian models on the basis of at least one test.(7) The recent exchange between Pesaran [27; 28] and Rush and Waldo [30] also serves to illustrate the tenuousness of the conclusions when these models are tested against each other.

    The empirical methods used in the above articles provide illustrations of the types of tests of non-nested non-linear models mentioned in the introduction. One approach involves embedding any two, or all three, of the above competing models in an artificial composite model, and then testing each of the original models against it. Such a procedure therefore yields a set of linear restrictions that the null model imposes on some of the parameters in the artificial composite model. For example, a test of the Barro model as the null hypothesis against the Fischer alternative(8) would require estimating the artificial model

    U[N.sub.t] = U[N.sub.nt] + [summation of] [[Beta].sub.1i](t - i - 1[m.sub.t - i] - [E.sub.t - i - 1][m.sub.t - i]) where i = 0 to [k.sub.1] + [summation of] [[Beta].sub.2j](t - j - 1[m.sub.t] - [E.sub.t - j - 1][m.sub.t]) + [u.sub.t] where j = 1 to [k.sub.2] (4)

    and then testing the restrictions [[Beta].sub.2j] = 0 for j = 1 through [k.sub.2]. The composite equations necessary for testing other competing specifications, and the implied restrictions, would be constructed similarly.

    Various tests of such restrictions are available, with those based on the likelihood ratio principle being frequently used in this literature. These tests involve a comparison of the values of the sum of squared residuals from the restricted and the unrestricted or composite models, based either on their difference or on a ratio of the two. One form of the likelihood ratio test is based on the F-ratio(9) F = [(RSSR - USSR)/q]/[USSR/(n - k)] where RSSR and USSR are the restricted and unrestricted sum of squared residuals respectively, q is the number of restrictions imposed by the null model on the composite equation, and n - k is the degrees of freedom in the composite equation. No exact small sample test statistic is available in the non-linear case, and the second-order autoregressive error process makes our models non-linear.(10) Since the chi-square form tends to reject more frequently than does the F form, the F-ratio was used as the likelihood ratio test. Tests were conducted in which each of the three models served as the null against alternative composite models which embedded the null and each of the two competing models individually. In addition to these six tests, each model served as a null against an alternative embedding all three models simultaneously. Thus, we carried out a total of nine F tests.

    The other procedure which we considered employs the P test appropriate for testing non-linear non-nested hypotheses, a test based on the suggestions of Cox [7; 8] and proposed by Davidson and MacKinnon [9], in which each specification is evaluated by its respective ability to explain the variation in the dependent variable left unexplained by the other specification. Examples of applications of the P test to situations in which the non-linearities in estimation arise from autoregressive specifications of the residuals are found in Bernanke, Bohn, and Reiss [5] and McAleer, Peseran, and Bera [20]. The test involves first using an appropriate non-linear least-squares estimation procedure to obtain estimates of the null and alternative models. Then ordinary least squares estimates of the coefficient [Alpha] and the vector b in the auxiliary equation

    [Mathematical Expression Omitted],

    which can be more conveniently written as

    [Mathematical Expression Omitted],

    are obtained. In these equations, [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the predicted values of unemployment under the null and alternative models respectively, and...

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