Proving consistency of a two-way random effects model estimator

Question

In the standard two-way random effects model, it is assumed that

$$ X_{ij} = A_i + U_{ij}, (j = 1,\ldots ,m\,;~i = 1, \ldots ,s)$$

with the A’s and U’s all independently, normally distributed with

$$\begin{aligned} E(A_i) &= \xi ~, &\mathrm{Var}(A_i) &= \sigma_A^2 \\ E(U_{ij}) &= 0~, & \mathrm{Var}(U_{ij}) &= \sigma^2\end{aligned}$$

Prove that

$$\bar{X} = \frac{\sum \sum X_{ij}}{sm}$$

is:

1) Not consistent for $\xi$ if $m → \infty$ and $s$ remains fixed,

2) Consistent for $\xi$ if $s → \infty$ and m remains fixed.

Where I'm stucked:

i) From the sufficient condition for $Y_n \xrightarrow{p} c$ (Convergence in probability), i.e. $E(Y_n - c)^2 \to 0$ :

$$\bar{X} \xrightarrow{p} \xi \quad \text{if} \quad E(\bar{X} - \xi)^2 \to 0, n \to \infty$$ Therefore $$\mathrm{Var}(\bar{X}) = \mathrm{Var} \left( \frac{\sum \sum X_{ij}}{sm} \right) = \frac{1}{s^2m^2} \mathrm{Var}(\sum \sum X_{ij}) = \frac{1}{s^2m^2} \mathrm{Var} \left( \sum \sum A_i + U_{ij} \right)$$

Let $h_{ij} = \mathrm{Var}(\sum \sum A_i + U_{ij})$, then I have to prove that

$$h_{ij} = o(s^2 m^2)$$

Answer 1

Solution

Consistency of a estimator can be assessed throughout it's convergence in probability, i.e. if $\bar{X} \xrightarrow{p} \xi$, then $\bar{X}$ is a consistent estimator of $\xi$.

Using convergence in quadratic mean as a sufficient condition to ensure consistency, we will have:

\begin{equation*} E[\bar{X} - \xi]^2 \to 0, \quad\text{as}\quad n \to \infty \end{equation*}

\begin{equation*} \begin{split} E[\bar{X} - \xi]^2 & = Var[\bar{X}] \\ & = Var\left[ \frac{1}{s m} \sum \sum X_{ij} \right] \\ & = \frac{1}{s^2 m^2} Var\left[ \sum \sum X_{ij} \right] \\ & = \frac{1}{s^2 m^2} Var\left[ \sum \sum (A_i + U_{ij}) \right] \\ & = \frac{1}{s^2 m^2} \left\lbrace Var\left[\sum \sum A_i\right] + Var\left[ \sum \sum U_{ij} \right] \right\rbrace \\ & = \frac{1}{s^2 m^2} \left\lbrace Var\left[m \sum A_i \right] + Var\left[ \sum \sum U_{ij} \right] \right\rbrace \\ & = \frac{1}{s^2 m^2} \left\lbrace m^2 Var\left[\sum A_i \right] + Var\left[\sum \sum U_{ij} \right] \right\rbrace \\ & = \frac{1}{s^2 m^2} \left\lbrace m^2 s Var\left[A_i \right] + s m Var\left[ U_{ij} \right] \right\rbrace \\ & = \frac{m^2 s}{s^{2} m^2} Var[A_i] + \frac{s m }{s^{2} m^{2}} Var[U_{ij}] \\ & = \frac{\sigma_A^2}{s} + \frac{\sigma^2}{s m} \end{split} \end{equation*}

Now checking for the cases 1. and 2.

1. Not consistent for $\xi$ if $m \to \infty$ and $s$ remains fixed.

$lim_{m \to \infty}\left\lbrace \frac{\sigma_A^2}{s} + \frac{\sigma^2}{s m} \right\rbrace = \frac{\sigma_A^2}{s}$

2. Consistent for $\xi$ if $s \to \infty$ and $m$ remains fixed.

$\lim_{s \to \infty}\left\lbrace \frac{\sigma_A^2}{s} + \frac{\sigma^2}{s m} \right\rbrace = 0$

$\therefore \bar{X}$ is consistent as $s \to \infty$ and $m$ remains fixed.