Question

2. Let Ct be consumption and X; be a predictor of consumption. Suppose you have quarterly data on C and X. Let Dit, D2t, D3t, and D4t be dummy variables such that Dit 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? \text { b) } C_{t}=\alpha+\beta X_{t}+\gamma_{1} X_{t} D_{1 t}+\gamma_{2} X_{t} D_{2 t}+\gamma_{3} X_{t} D_{3 t}+\gamma_{4} X_{t}\left(1-D_{1 t}-D_{2 t}-D_{3 t}\right)+u_{t} \text { d) } C_{t}=\alpha+\delta_{1} D_{1 t}+\delta_{2} D_{2 t}+\beta X_{t}+\gamma_{1} X_{t} D_{1 t}+\gamma_{2} X_{t} D_{2 t}+\gamma_{3} X_{t} D_{3 t}+u_{t} \text { c) } C_{t}=\alpha+\delta_{1} D_{1 t}+\delta_{2} D_{2 t}+\gamma_{1} X_{t} D_{1 t}+\gamma_{2} X_{t} D_{2 t}+\gamma_{3} X_{t} D_{3 t}+\gamma_{4} X_{t} D_{4 t}+u_{t} \text { c) } C_{t}=\alpha+\delta_{1} D_{1 t}+\delta_{2} D_{2 t}+\gamma_{1} X_{t} D_{1 t}+\gamma_{2} X_{t} D_{2 t}+\gamma_{3} X_{t} D_{3 t}+\gamma_{4} X_{t} D_{4 t}+u_{t}

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