Using the data on food in the Demand Functions data set, the following regressions were run, each with the logarithm of food as the dependent variable: (1) an OLS regression on a time trend T defined to be 1 in 1959, 2 in 1960, etc.; (2) an AR(1) regression using the same specification; and (3) an OLS regression on T and the logarithm of food lagged one time period, with the results shown in the table (standard errors in parentheses):
Discuss why each regression specification appears to be unsatisfactory. Explain why it was not possible to perform a common factor test.