13.8 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.

1: OLS 2: AR(1) 3: OLS I 0.0181 0.0166 0.0024 (0.0005) (0.0021) (0.0016) LGFOOD(-1) 0.8551 (0.0886) Constant 5.7768 5.8163 0.8571 (0.0106) (0.0586) (0.5101) 0.8551 P (0.0886) R 0.9750 0.9931 0.9931 RSS 0.0327 0.0081 0.0081 PR d 0.2752 1.3328 1.3328 h 2.32