EFIM20036: Panel Data
Spring 2024
Relevant Readings: Wooldridge; “Introduction to Econometrics, A modern Approach” Main content: Chapter 13-14.
Exercise 1 (Sample Final - Panel Data). Consider the panel dataset that collects informa- tion about medical expenditures (medexp), household income (income) . The data generating process follows:
medexpit = incomeit β1 + αi + λt + uit
where i = 1, ··· , 149 are household units and t = 1979, ··· , 1990. The variables uit , αi and λt are unboservable. We have a balanced panel with 149 units and 12 years, starting from year 1979 and ending with year 1990, inclusive. The uit are independent of the αi and λt . Moreover uit is serially uncorrelated: E[uit uis] = 0 for any s ≠ t. In addition, you may assume that E[uit |Xi] = 0 for all i and t.
a) First suppose that λt = 0 for all t so that there are no time effects. Give sufficient con- ditions such that Pooled OLS, Random Effects GLS, Fixed Effects OLS and First Difference OLS are all consistent estimators of β1 .
b) Suppose you doubt that the condition in a) is true. Among the four estimators we discussed: Pooled OLS, Random Effects GLS, Fixed Effects OLS and First Difference OLS, which ones are still consistent, regardless of whether the assumption you made in a) is true?
Now consider the case where λt is allowed to be nonzero. We specify a dummy variable model that includes both time and unit dummies in the following way:
medexpit = γ0 + incomeitγ1
+ δ1 D1it + δ2 D2it + ··· + δ148 D148it
+ ζ1979 L1979it + ζ1980 L1980it + ··· + ζ1989 L1989it + ϵit
where Djit and Ljit are dummy variables. Djit equal to 1 if i = j and 0 otherwise. Ljit is equal to 1 if t = j and 0 otherwise. For example, the D1 is equal to 1 only for unit i = 1, and 0 for all other units. Similarly L1985 is equal to 1 only for the year 1985 and 0 for all other years.
c) Observe that the dummies D149 and L1990 have not being included. Why?
d) Provide an expression for the following parameters of the dummy variable model, in terms of the underlying, unobserved parameters α 1 ,α2 , ··· ,α 149 ,λ1879 ,λ1980 , ··· ,λ1990 . What are:
γ0 =?
δ1 =?
δ148 =?
ζ1979 =?
ζ1989 =?
e) For each of the parameters in the dummy variable model listed above, give an inter- pretation (in words) that is consistent with your answer in d) .