The bias toward zero in identifying relationships: reply to Kennedy. (response to Peter Kennedy in this issue, p.382)n.

Fremling, Gertrud M. ; Lott, John R., Jr.

We will argue that rational expectations has considered only the "misestimation" type of error, which can "cancel out" in the aggregate, but that with errors in identifying relationships, there is no similar cancelling out effect (Fremling and Lott [1996, 276])

Kennedy [1999] misses much of our point. When he extends our two-variable case to a three-variable case, he views agents as exclusively making the traditional "specification errors." As is well known, specification errors concern what variables to include when estimating a regression. He fails to address our main contribution - that people sometimes do not identify that a relationship even exists. When describing our argument Kennedy goes so far as to always replace our references to "[model] identification error" with the term "specification error." For instance, the way he refers to our footnote 5 (p. 279) gives the misleading impression that we ourselves are using the term "specification errors." Our footnote clearly uses the term "identification errors,"(1) and it is only because he switches these terms that he is able to conclude that "Fremling and Lott have violated one of their own assumptions ...."

In our 1996 paper we showed the following. Individual actors sometimes are quite clueless as to an economic problem, failing to understand that two (or more) variables are related. We extensively described this "identification problem" on p. 278, and we stated: "The crucial argument in this paper is that at least some people fail to identify a true relationship, and therefore never take the next step, which is to estimate the strength of it. Failing to estimate the strength of a relationship is essentially equivalent to estimating it to be zero." Thus, many do not reach the regression-step at all. When aggregating across individuals making various mistakes, economic behavior at the group level resembles a situation where every individual estimates a regression but underestimates the regression coefficient. Even though all our actors are "rational" - and thus only make non-systematic mistakes in what variable(s) are omitted - the aggregate results could equally well have been generated by a group of non-rational actors systematically underestimating the regression coefficients. In other words, when observing aggregate behavior that contains aggregate systematic errors, one must be cautious not to erroneously infer irrationality on the part of the individuals.

Now to Kennedy's numerical example with two independent variables rather than one. How should our analysis be applied? We maintain that there exists a total of eight possible cases: one of correctly recognizing all three variables as being connected, three of missing only one, three of missing two, one of missing all. Kennedy only recognizes four of the cases. This makes an enormous difference. If Kennedy had also included the implicit zero coefficients in the remaining four cases, he would have found a substantial "bias toward zero": for the representative individual, [Alpha] would have been only 4.05 (not 8.09) compared to its true value of 6. Kennedy never mentions his estimate of [Beta], but as a careful reader can deduce, this estimate is only 2.16, a serious underestimate. And including the four other cases with zero's would have yielded a mere average estimate of 1.08, way below its true value of [Beta] = 4. In our view, Kennedy's Monte Carlo experiment merely confirms our results.(2)

Kennedy makes it appear that his numerical example corresponds to our macroeconomic modeling in section V. It does not, for he overlooks our explicit assumption that "the general price level cannot be perfectly and immediately observed by the workers ..." (p. 283). This a very classical assumption in macroeconomics and forms the basis for why changes in the nominal wage are sometimes confused with changes in the real wage. Kennedy's disregard for the dependent variable makes him overlook the other four cases in his numerical example. Certainly, it can be argued that the variables [Delta]M and [Delta]F are more frequently omitted, and a complex model with the different probabilities of omission could be set up. In the macroeconomic situation at hand, we would guess (and this is just a mere guess) that no more than 1% of workers in any given month would observe [Delta]M, 1% would observe [Delta]F, and at most 50% would observe [Delta]P. Depending on the exact combinations of who knows what, the exact number for the average estimates of [Alpha] and [Beta] could vary slightly, but it is crystal clear that with numbers like this the ignorant would totally dominate: at least 98% would never simultaneously have data on [Delta]P changes and either [Delta]M or [Delta]F. The cases of "implicitly zero estimates" would be overwhelming. Even if a handful of workers spuriously picked up some of the influence of [Delta]F when estimating the impact of [Delta]M, their estimates of [Alpha] would be drowned in the aggregate by zero's from those who are basically clueless.

As for Kennedy's second issue, prediction error, we have absolutely no disagreement. We demonstrated underestimation of coefficients, which certainly does not imply underestimation of prediction. Our political business cycle case exemplified this: we explicitly stated that the expected price temporarily exceeds the actual price (p. 286). Whether underestimation is somehow a more likely consequence than overestimation probably depends on the particular problem and whether "underestimation" refers to levels or to changes in levels. In our political business cycle example, under- and overprediction seem equally common when expressed in levels. However, the absolute value of changes tends to be underestimated. (Alternatively formulated, inertia is overestimated.)

To conclude, Kennedy portrays our analysis as merely rehashing the long-recognized econometrics problem of excluding explanatory variables. Our distinct contribution however is formulating a two-step process, where there can be severe errors in setting up a model, including errors in identifying the dependent variable. We argue that in many economic cases a high fraction of the public acts as if they were setting the coefficient to zero. Our results continue to hold when there are more than two variables involved.

We thank Scott Masten for his helpful comments.

1. We also made the distinction between the existence of a model and specification errors clear in several other different places mi our paper, and we referenced existing research that has dealt with specification error type issues. For example, on pages 277-78 we cite articles that discuss the traditional misspecification type problems and then point out that "our model shall ... focus exclusively on individuals making mistakes in identifying relationships."

2. Kennedy further stacks the cards in his favor by having the two explanatory variables being highly correlated. The overestimation of [Alpha] that he finds is due to picking up the influence of the other, more important variable whenever it is excluded. He does not mention that, according to his own figures, [Beta] was underestimated by slightly more than [Alpha] was overestimated.

REFERENCES

Fremling, Gertrud M., and John R. Lott, Jr. "The Bias Towards Zero in Aggregate Perceptions: An Explanation based on Rationally Calculating Individuals." Economic Inquiry, April 1996, 276-95.

Kennedy, Peter. "Specification Error, Prediction Bias, and Rational Expectations." Economic Inquiry, April 1999, 382-84.