Professor Campbell performed a signal service by pointing out that analyses of economic performance under Republican and Democratic presidents must control for the ups and downs of the business cycle that are beyond a president's control. He correctly noted in particular that Republican and Democratic presidents of the postwar era took office under very different economic conditions (Campbell 2011, 15-20). We strongly concurred with these contentions (Comiskey and Marsh 2012, 41-45).
Unfortunately, Campbell rejects all means of controlling for those ups and downs that do not involve the use of a dependent variable lagged one or two quarters--a flawed practice for two reasons. The first flaw is that ifa variable and its lagged value are closely related, one runs the risk of controlling for the dependent variable itself. The second difficulty is that controlling for the dependent variable lagged one quarter controls by extension for the factors that determined that dependent variable up to just three months ago. For these reasons, "lagged dependent variables can suppress the explanatory power of other independent variables" (Achen 2001, 1).
Campbell seeks to avoid the first of these dangers by claiming that economic variables and their lagged values are related to one another but not really related. On the one hand, Campbell writes--correctly--that "economic conditions are not neatly packaged into quarters. Since the economy is continuously in motion, the condition of the economy at time t should be expected to have an effect on the economy at time t+l" (Campbell 2012, 813). Yet Campbell also maintains that economic conditions in adjoining quarters are really not all that related "since the lagged measure is arrived at independently and is of economic activity over a nonoverlapping and significantly long period of time" (Campbell 2012, note 3; see also Campbell 2011, 11).
Campbell cannot have it both ways. As we noted previously, the fact is that economic conditions in quarters t and t + l are not independent of each other (Comiskey and Marsh 2012, 45). If they were, then by the terms of his own argument Campbell would have no need to control for conditions in the preceding quarter. Campbell seeks to escape this dilemma by claiming that "the dependent economic variable and the lagged economy ... are analytically independent of each other, but they are empirically related to one another" (emphasis in the original; Campbell 2012, note 6). This distinction strikes us as fanciful. They are analytically related for the reasons Campbell explains so well, and his statistical results demonstrate that they are also empirically related. (1)
The second way a lagged dependent variable "can suppress the explanatory power of other independent variables" (Achen 2001, 1) is that the lagged variable controls for the factors that made the dependent variable what it was in period t-1--the same factors that are impacting it in period t. In our case, one of those factors is the party of the president.
It can be shown algebraically that the lagged dependent variable in Campbell's Model 2 (2012) simply disguises the impact of the president's party on economic performance. In the following, Y denotes economic performance, D denotes the party of the president, and the parameters are those Campbell reports in Equation 2 of his Table 1. Going back n quarters from time t, Campbell's model is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
Substituting Equation (4) into the equation before it, and substituting the resulting equation into the one before that, etc., and continuing back to Equation (1), we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where c is some constant. …