Consider the three-way error component model described in problem 3.15. The panel data can be unbalanced and the matrices of dummy variables are \(\Delta=\left[\Delta_{1}, \Delta_{2}, \Delta_{3}\right]\) with

\[u=\Delta_{1} \mu+\Delta_{2} \lambda+\Delta_{3} \eta+u\]

where \(\mu, \lambda\), and \(u\) are random variables defined (9.31) and the added random error \(\eta\) has mean zero and variance \(\sigma_{\eta}^{2}\). All random errors are independent among themselves and with each other. Show that \(P_{[\Delta]}=P_{[A]}+\) \(P_{[B]}+P_{[C]}\) where \(A=\Delta_{1}, B=Q_{[A]} \Delta_{2}\), and \(C=Q_{[B]} Q_{[A]} \Delta_{3}\). This is Corollary 1 of Davis (2002). 

\[\begin{equation*}
u=\Delta_{1} \mu+\Delta_{2} \lambda+u \tag{9.31}
\end{equation*}\]

Data From Problem 3.15:

Ghosh (1976) considered the following error component model:

\[u_{i t q}=\mu_{i}+\lambda_{t}+\eta_{q}+u_{i t q}\]

where \(i=1, \ldots, N ; T=1, \ldots, T\); and \(q=1, \ldots, M\). Ghosh (1976) argued that in international or interregional studies, there might be two rather than one cross-sectional component; for example, \(i\) might denote countries and \(q\) might be regions within that country. These four independent error components are assumed to be random with \(\mu_{i} \sim \operatorname{IID}\left(0, \sigma_{\mu}^{2}\right), \lambda_{t} \sim \operatorname{IID}\left(0, \sigma_{\lambda}^{2}\right), \eta_{q} \sim \operatorname{IID}\left(0, \sigma_{\eta}^{2}\right)\) and \(u_{i t q} \sim \operatorname{IID}\left(0, \sigma_{u}^{2}\right)\). Order the observations such that the faster index is \(q\), while the slowest index is \(t\), so that

\[\begin{aligned}
u^{\prime}= & \left(u_{111}, \ldots, u_{11 M}, u_{121}, \ldots, u_{12 M}, \ldots, u_{1 N 1}, \ldots,\right. \\
& \left.u_{1 N M}, \ldots, u_{T 11}, \ldots, u_{T 1 M}, \ldots, u_{T N 1}, \ldots, u_{T N M}\right)
\end{aligned}\]