EFIM20036: Panel Data I
Spring 2024
Relevant Readings: Wooldridge; “Introduction to Econometrics, A modern Approach”
Main content: Chapter 13-14.
In a panel data model we have the following structure:
yit = xit β + vit
vit = αi + uit
uit = ρui,t−1 + ϵit
We consider |ρ| < 1. The idiosyncratic shocks ϵit are i.i.d for all i and all t, with E[ϵit] = 0, E[ϵi(2)t] = σϵ(2) . The individual-specific, time invariant αi are also i.i.d with E[αi] = 0, E[αi(2)] = σα(2), Hence, the errors vit are independent across units but potentially dependent for the same unit over time. Moreover E[αi |Xi] ≠ 0, E[ϵit |Xi] = 0. Both αi and ϵit errors are are also conditionally homoskedastic. Moreover, αi ⊥ uit. Given the information above:
a) Construct the nT × nT variance-covariance matrix E[vvT |X].
b) Is the RE GLS estimator consistent?
c) Is the FE OLS estimator consistent?
d) Is the inference from the FE OLS correct, if so, give a sufficient condition for it to be true?
e) Suppose you derive the FD estimator for β according to the following equation:
yit − yi,t−1 = (xit − xi,t−1)β + (uit − ui,t−1)
Here the residual is ∆uit := uit −ui,t−1 . Construct the nT ×nT variance-covariance matrix E[∆u∆uT |X] using the assumptions above. Hint: To obtain Cov(uit −ui,t−1, ui,t−s −ui,s−t−1) use bi-linearity of Cov(· , ·) and what you know about the autocovariance function of an AR(1) process.
f) If you are unsure of the true structure of the errors uit can you suggest away to obtain valid standard errors for your β estimator?