Consider the panel data regression in equation (15.1) for \(N\) cross-sectional units with \(T=3\) time-series observations. Assume that FE1-FE5 hold.

a. Apply the first-difference transformation to model (15.1). What is the resulting specification? Is there unobserved heterogeneity in this model? Explain.

b. Let \(\Delta e_{i t}=\left(e_{i t}-e_{i, t-1}\right)\). Find the variance of \(\Delta e_{i t}\) for \(t=2\) and \(t=3\).

c. Assuming that the idiosyncratic error \(e_{i t}\) is serially uncorrelated, show that the correlation between \(\Delta e_{i 3}\) and \(\Delta e_{i 2}\) is \(-1 / 2\).

d. What must the serial correlation for \(e_{i t}\) be in order for \(\Delta e_{i 3}\) and \(\Delta e_{i 2}\) to be uncorrelated?

Data From Equation 15.1:-

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Yit = B +2x2it + w; + (u; + e ;) = + B2 x 2 + Wi + vit (15.1)