Specification Error, Prediction Bias and Rational Expectations
Kennedy, Peter, Economic Inquiry
In a recent issue of this journal Fremling and Lott  present a novel explanation of why aggregate forecasts can be biased, casting considerable doubt on the applicability of rational expectations. Their argument is based on some individuals making specification errors (which in a misuse of econometric terminology they call "identification errors") when formulating economic models, omitting relevant explanatory variables from estimation of economic relationships. This causes these individuals to produce biased coefficient estimates, so that when individual coefficient estimates are averaged to obtain the corresponding coefficients in the "aggregate" or "representative individual" forecasting relationship, the aggregate/representative coefficients are biased. This in turn creates prediction bias, offering an explanation for why empirical evidence tends not to support rational expectations. Unfortunately, Fremling and Lott's presentation of their argument is misleading. In particular, they lead readers to believe that the phenomenon they identify results in aggregate/representative coefficient estimates biased toward zero, and they lead readers to believe that this causes underprediction. The purpose of this note is to show that although these results are possible, Fremling and Lott's argument does not guarantee this - it is also possible for aggregate coefficient estimates to be biased away from zero and/or that overprediction could result.
II. BIAS TOWARD ZERO
Throughout their paper Fremling and Lott suggest that their argument shows that coefficient estimates of an aggregate relationship between variables or of a representative individual are biased toward zero. "Bias towards zero" appears in the title of their paper; the first sentence of their abstract talks of "downward biases in the aggregate that are equivalent to the public underestimating the strengths of the true relationships"; in summarizing their analytical result they state (p. 279) "there should be a bias toward zero for the 'representative' estimate"; on p. 280 they claim that "the importance of our argument lies in that the bias created by identification errors is always negative"; and in contrasting their theory to rational expectations they conclude on p. 283 that "our theory predicts that if mistakes occur, they will involve, as the evidence indicates, systematic underestimates of the relationship."
Fremling and Lott's explanation of this phenomenon rests on the reasonable assumption that some people make specification errors when estimating relationships, omitting relevant explanatory variables. Suppose, for example, P depends on M and F, as in the macroeconomic modeling example in section V of Fremling and Lott's paper, so that we predict [Delta]P by
[Delta][P.sup.*] = [[Alpha].sup.*][Delta]M + [[Beta].sup.*][Delta]F
where [[Alpha].sup.*] and [[Beta].sup.*] are ordinary least squares estimates of the corresponding parameters [Alpha] and [Beta]. Adopting the usual statistical assumptions, as noted by Fremling and Lott in their footnote 4, p. 278, if everyone estimates this relationship by regressing [Delta]P on [Delta]M and [Delta]F, the aggregate or representative relationship will have unbiased coefficient estimates. Fremling and Lott claim that if some people estimate this relationship omitting AM, in effect estimating [Alpha] by zero, they will cause the aggregate/representative relationship to have a coefficient estimate biased toward zero. This is their result shown in equation (3) on p. 279.
Unfortunately, in reaching this result Fremling and Lott have violated one of their own assumptions, namely that, as noted in their footnote 5, p. 279, the specification errors are random. They have forgotten to account for those individuals who estimate this relationship by omitting [Delta]F. These individuals could create upward-biased estimates of a that could cause the aggregate/representative estimate of [Alpha] to be biased away from zero! …