Variance component estimation under misspecification. This is based on problem 91.3 .3 in Econometric Theory by Baltagi and Li (1991). This problem investigates the consequences of under- or overspecifying the error component model on the variance components estimates. Since the one-way and two-way error component models are popular in economics, we focus on the following two cases:

(1) Underspecification: In this case, the true model is two-way; see (3.1):

\[u_{i t}=\mu_{i}+\lambda_{t}+u_{i t} \quad i=1, \ldots, N \quad t=1, \ldots, T\]

while the estimated model is one-way; see (2.2):

\[u_{i t}=\mu_{i}+u_{i t}\]

\(\mu_{i} \sim \operatorname{IID}\left(0, \sigma_{\mu}^{2}ight), \lambda_{t} \sim \operatorname{IID}\left(0, \sigma_{\lambda}^{2}ight)\), and \(u_{i t} \sim \operatorname{IID}\left(0, \sigma_{u}^{2}ight)\) independent of each other and among themselves.

(a) Knowing the true disturbances \(\left(u_{i t}ight)\), show that the BQUE of \(\sigma_{u}^{2}\) for the misspecified one-way model is biased upwards, while the BQUE of \(\sigma_{\mu}^{2}\) remains unbiased.

(b) Show that if the \(u_{i t}\) are replaced by the one-way least squares dummy variables (LSDV) residuals, the variance component estimate of \(\sigma_{u}^{2}\) given in part

(a) is inconsistent, while that of \(\sigma_{\mu}^{2}\) is consistent.

(2) Overspecification: In this case, the true model is one-way, given by (2.2), while the estimated model is two-way, given by (3.1).

(c) Knowing the true disturbances \(\left(u_{i t}ight)\), show that the BQUE of \(\sigma_{\mu}^{2}, \sigma_{\lambda}^{2}\) and \(\sigma_{u}^{2}\) for the misspecified two-way model remain unbiased.

(d) Show that if the \(u_{i t}\) are replaced by the two-way (LSDV) residuals, the variance components estimates given in part

(c) remain consistent. Hint: see solution 91.3

.3 in Econometric Theory by Baltagi and Li (1992b).

Transcribed Image Text:

Uit=fi + + Vit i = 1,..., N t = 1,..., T (3.1)