Random Effects Decomposition

Motivation

I work through the error components econometric model outlined in Amemiya (1985). I use Hayashi (2000) as a reference text. I work through this example because I use this model in my working paper with Chris Mayer on bubble identification and I would like to work out the details as I didn’t spend much time on these sorts of models in my core econometrics courses.

In my paper with Chris, I develop a method of identifying relative mispricings between city specific markets in the US residential housing market using flows of speculative buyers between cities and assuming that city sizes are exogenous. Previously, analysts suspected that the housing bubble was due to credit supply factors. I use a random effects model to gauge the relative importance of $1)$ aggregate credit supply factors and $2)$ cross-city speculator flows in explaining mis-pricing in the housing market in our sample.

Econometric Framework

I characterize the random effects error components estimator outlined in Amemiya (1985, Ch. 6). Consider a balanced panel with $N$ panels and $T$ observations per panel. I study a regression specification of the following type:

(1) $\begin{align*} y_{n,t} \ &= \ \langle X_{n,t} \mid \beta \rangle \ + \ \mu_n \ + \ \lambda_t \ + \ \varepsilon_{n,t} \end{align*}$

I can vectorize this specification by stacking each of these $N \times T$ equations:

(2) $\begin{align*} \begin{split} \mathcal{U} \ &= \ \langle I_N \otimes 1_T \mid \mu \rangle \ + \ \langle 1_N \otimes I_T \mid \lambda \rangle \ + \ \mathcal{E} \\ Y \ &= \ \langle X \mid \beta \rangle \ + \ \mathcal{U} \end{split} \end{align*}$

Assumptions

I make the following assumptions about the shape of the errors:

Assumption: (Error Structure) I assume that:

1) Unbiased-ness: $\langle \mu_n \rangle = 0$ , $\langle \lambda_t \rangle = 0$ and $\langle \varepsilon_{n,t} \rangle = 0$

2) White-Noise: $\langle \mu_n \mid \lambda_t \rangle = 0$ , $\langle \lambda_t \mid \varepsilon_{n,t} \rangle = 0$ and $\langle \varepsilon_{n,t} \mid \mu_n \rangle = 0$

3) Homoskedasticity: $\vert \mu \rangle \langle \mu \vert = I_N \cdot \sigma^2_\mu$ , $\vert \lambda \rangle \langle \lambda \vert = I_T \cdot \sigma^2_\lambda$ and $\vert \varepsilon \rangle \langle \varepsilon \vert = I_{N \times T} \cdot \sigma^2_{\varepsilon}$

What are the key take-aways from these assumptions? First, assumption $1)$ means that there is a constant term in the explanatory $X$ variables. Assumption $2)$ is just the standard white noise assumption. Assumption $3)$ is the key restriction. This assumption says that the within and between effects are independent across time and panels respectively. The estimator I define below allows me to learn the values of $\sigma_\mu^2$ , $\sigma_\lambda^2$ and $\sigma_\varepsilon^2$ .

Estimation

How do I go about estimating these $3$ objects? First, I define some notation to make my life a bit easier and stave of carpel tunnel for a few more semesters:

(3) $\begin{align*} \begin{split} F \ &= \ \vert I_N \otimes 1_T \rangle \langle I_N \otimes 1_T \vert \\ G \ &= \ \vert 1_N \otimes I_T \rangle \langle 1_N \otimes I_T \vert \end{split} \end{align*}$

Also, let $H$ be an $(N \cdot T) \cdot (N \cdot T - N - T + 1)$ unit matrix. I name the error covariance matrix $\Omega$ , and then characterize it as a linear function of the $3$ variance terms of interest:

(4) $\begin{align*} \begin{split} \Omega \ &= \ \vert \mathcal{U} \rangle \langle \mathcal{U} \vert \\ &= \ \sigma_\mu^2 \cdot F \ + \ \sigma^2_\lambda \cdot G \ + \ \sigma_\varepsilon^2 \cdot I_{N \times T} \end{split} \end{align*}$

I can write out the inverse of the error covariance matix $\Omega$ as follows:

(5) $\begin{align*} \begin{split} \Omega^{-1} \ &= \ \frac{1}{\sigma_\varepsilon^2} \cdot \left( I_{N \times T} - \gamma_1 \cdot F + \gamma_2 \cdot G + \gamma_3 \cdot H \right) \\ \gamma_1 \ &= \ \frac{\sigma_\mu^2}{\sigma_\varepsilon^2 + T \cdot \sigma_\mu^2} \\ \gamma_2 \ &= \ \frac{\sigma_\lambda^2}{\sigma_\varepsilon^2 + N \cdot \sigma_\lambda^2} \\ \gamma_3 \ &= \ \gamma_1 \cdot \gamma_2 \cdot \left( \ \frac{2 \cdot \sigma_\varepsilon^2 + T \cdot \sigma_\mu^2 + N \cdot \sigma_\lambda^2}{\sigma_\varepsilon^2 + T \cdot \sigma_\mu^2 + N \cdot \sigma_\lambda^2} \ \right) \end{split} \end{align*}$

This formulation shows that the sample error covariance matrix will provide unbiased and consistent estimates if both $N \to \infty$ and $T \to \infty$ . In this not, I am not going to worry about what is the most consistent estimator for the parameters. Next, I want to decompose the error covariance matrix into within, between and indiosyncratic components. To do this I need $1$ last piece of notation:

(6) $\begin{align*} Q \ &= \ I \ - \ \frac{F}{T} \ - \ \frac{G}{N} \ + \ \frac{H}{N \cdot T} \end{align*}$

Think about this as an orthogonal decomposition of a unitary error covariance matrix into each of the $3$ components: within, between and idiosyncratic. Then, using this term, Amemiya (1971) shows that the following estimators for the parameter vector $\begin{bmatrix} \sigma_\mu^2 & \sigma_\lambda^2 & \sigma_\varepsilon^2 \end{bmatrix}$ :

(7) $\begin{align*} \begin{split} \hat{\mathcal{U}} \ &= \ Y \ - \ \langle X \mid \hat{\beta} \rangle \\ \hat{\sigma}_{\varepsilon}^2 \ &= \ \frac{\langle \hat{\mathcal{U}} \mid \langle Q \mid \hat{\mathcal{U}} \rangle \rangle}{(N-1) \cdot (T-1)} \\ \hat{\sigma}_{\mu}^2 \ &= \ \frac{\langle \hat{\mathcal{U}} \mid \langle \frac{T-1}{T} \cdot F - \frac{T-1}{N \cdot T} \cdot H - Q \mid \hat{\mathcal{U}} \rangle \rangle}{T \cdot (N-1) \cdot (T-1)} \\ \hat{\sigma}_{\lambda}^2 \ &= \ \frac{\langle \hat{\mathcal{U}} \mid \langle \frac{N-1}{N} \cdot G - \frac{N-1}{N \cdot T} \cdot H - Q \mid \hat{\mathcal{U}} \rangle \rangle}{T \cdot (N-1) \cdot (T-1)} \end{split} \end{align*}$