Question

1. Let Ct be consumption and Xt be a predictor of consumption. Suppose you have quarterly data on C and X. Let D1t , D2t , D3t , and D4t be dummy variables such that D1t takes the value 1 in quarter 1 and 0 otherwise, D2t takes the value 1 in quarter 2 and 0 otherwise, etc. Which of the following, if any, suffer from perfect multicollinearity and why? a) Ct = α + βXt + γ1XtD1t + γ2XtD2t + γ3XtD3t + γ4XtD4t + ut b) Ct = α + βXt + γ1XtD1t + γ2XtD2t + γ3XtD3t + γ4Xt(1 − D1t − D2t − D3t) + ut c) Ct = α + δ1D1t + δ2D2t + γ1XtD1t + γ2XtD2t + γ3XtD3t + γ4XtD4t + ut d) Ct = α + δ1D1t + δ2D2t + βXt + γ1XtD1t + γ2XtD2t + γ3XtD3t + ut (There’s nothing special about labeling the parameters with α, β, γ1, γ2, δ1, δ2, etc. We could have labeled them β0, β1, β2, . . . without changing their interpretation.)

a) In models (a)–(d) of question 1, what are the slope coefficients of Xt in each of the 4 quarters? b) Suppose you estimate model (c) and wrote down the estimated slope coefficients for Xt in each of the 4 quarters. You then estimate model (d) and write down the estimated slope coefficients for Xt in each of the 4 quarters. Do your estimates change? Why or why not?

Answer 1