A matrix exponential spatial specification approach to panel data models

Abstract

This paper extends the matrix exponential spatial specification to panel data models. The matrix exponential spatial panel specification produces estimates and inferences comparable to those from conventional spatial panel models, but has computational advantages. We present maximum likelihood approach to the estimation of this spatial model specification and compare the results with the fixed effects spatial autoregressive panel model.

Appendix

1.1 The FESAR estimation

Usually, $\mathbf{X}$ has a vector of $1$’s for estimating the intercept. The fixed effect $\mathbf{c}$ is restricted so that $\mathbf{j}_N ' \mathbf{c}=0$. Thus, the interpretation of the fixed effect $c_i$ is the difference between the $i$th observation mean and the overall mean.

In practice, using the within transformation, the presence of intercept does not make any difference because all effects constants through time are dropped. Therefore, the model (1) can be rewritten as

$$\begin{aligned} \mathbf{Q}\mathbf{y}&= \mathbf{Q}\mathbf{X}\varvec{\beta } +\mathbf{Q}\varvec{\epsilon } \end{aligned}$$

(12)

$$\begin{aligned} \mathbf{Q}&= \mathbf{I}_{ NT }-\left( \frac{1}{T} \mathbf{j}_T \mathbf{j}_T '\otimes \mathbf{I}_N \right) \end{aligned}$$

(13)

The likelihood function of the FESAR model considering the within transformation is

$$\begin{aligned} \mathcal {L}(\varvec{\beta },\sigma ^2,\rho )=(2\pi \sigma ^2)^{-\frac{ NT }{2}}\hbox {exp}\left\{ -\frac{1}{2\sigma ^2} (\mathbf{SQy}-\mathbf{QX}\varvec{\beta })'(\mathbf{SQy}-\mathbf{QX}\varvec{\beta }) \right\} |\mathbf{S}| \end{aligned}$$

Fixing $\varvec{\beta }$ and $\sigma ^2$, we can find the likelihood function only for $\rho$. Thus, let $\tilde{\varvec{\beta }}=(\mathbf{X'QX})^{-1}\mathbf{X'Qy}$ and $\tilde{\sigma }^2=\frac{1}{ NT }\mathbf{y'S'PSy}$, where $\mathbf{P}$ is the orthogonal projections matrix given by $\mathbf{P} = \mathbf{I}_{ NT } - \mathbf{QX}(\mathbf{X'QX})^{-1}\mathbf{X'Q}$. Therefore, the likelihood for $\rho$ is given by:

$$\begin{aligned} \mathcal {L}(\rho ,\sigma ^2,\varvec{\beta })&= (2\pi \sigma ^2)^{-\frac{ NT }{2}}\hbox {exp}\left\{ -\frac{1}{2\sigma ^2} (\mathbf{y'S'PSy})'(\mathbf{y'S'PSy}) \right\} |\mathbf{S}|\nonumber \\ \mathcal {L}(\rho )&= (2\pi )^{-\frac{ NT }{2}}\left( \frac{1}{T} \mathbf{y'S'PSy} \right) ^{-\frac{ NT }{2} }\hbox {exp}\left\{ -\frac{1}{2} NT \right\} |\mathbf{S}| \end{aligned}$$

Note that $\mathbf{I}_{ NT }-\rho (\mathbf{I}_T\otimes \mathbf{W})=\mathbf{I}_T \otimes (\mathbf{I}_N-\rho \mathbf{W})$, and therefore, $|\mathbf{S}|=| \mathbf{I}_{T}\otimes (\mathbf{I}_N-\rho \mathbf{W}) | = | \mathbf{I}_N - \rho \mathbf{W}|^T$. So, we can write (14) as

$$\begin{aligned} \mathcal {L}(\rho )&= k(\mathbf{y'S'PSy})^{-\frac{ NT }{2}}|\mathbf{I}_N - \rho \mathbf{W} |^{T}\end{aligned}$$

(14)

$$\begin{aligned} l (\rho )&= \log \; k - \frac{ NT }{2}\log \; (\mathbf{y'S'PSy} )+ T \;\log \; | \mathbf{I}_N - \rho \mathbf{W}| \end{aligned}$$

(15)

Using eigenvalues, (15) can be written as

$$\begin{aligned} l (\rho )=\log \; k -\frac{ NT }{2}\log \; (\mathbf{y'S'PSy} ) +T \sum _{i=1}^N \log \; (1- \rho \omega _i) \end{aligned}$$

(16)

where $\omega _i$ is the $i$th eigenvalue of the matrix $\mathbf{W}$.

Note that

$$\begin{aligned} \mathbf{y'S'PSy}=\mathbf{y}'\mathbf{P}\mathbf{y}\!-\!\rho \mathbf{y}'\mathbf{P}(\mathbf{I}_T\otimes \mathbf{W})\mathbf{y} \!-\! \rho \mathbf{y}'(\mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}\mathbf{y} + \rho ^2\mathbf{y}'(\mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}(\mathbf{I}_T \otimes \mathbf{W})\mathbf{y} \end{aligned}$$

because

$$\begin{aligned} \mathbf{y}'\mathbf{S}'\mathbf{P}\mathbf{S}\mathbf{y}&= \mathbf{y}'(\mathbf{I}_{ NT } \otimes (\mathbf{I}_N-\rho \mathbf{W}))'\mathbf{P}(\mathbf{I}_{ NT } \otimes (\mathbf{I}_N-\rho \mathbf{W}))\mathbf{y} \\&= \mathbf{y}'(\mathbf{I}_{ NT } \otimes (\mathbf{I}_N-\rho \mathbf{W}'))\mathbf{P}(\mathbf{I}_{ NT } \otimes (\mathbf{I}_N-\rho \mathbf{W}))\mathbf{y} \\&= \mathbf{y}'(\mathbf{I}_{ NT }-\rho \mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}(\mathbf{I}_{ NT }-\rho \mathbf{I}_T \otimes \mathbf{W})\mathbf{y} \\&= (\mathbf{y}' - \rho \mathbf{y}' \mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}(\mathbf{y}-\rho \mathbf{I}_T \otimes \mathbf{W}\mathbf{y}) \\&= \mathbf{y}'\mathbf{P}(\mathbf{y}-\rho (\mathbf{I}_T \otimes \mathbf{W})\mathbf{y})-\rho \mathbf{y}'(\mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}(\mathbf{y}-\rho (\mathbf{I}_T \otimes \mathbf{W})\mathbf{y}) \\&= \mathbf{y}'\mathbf{P}\mathbf{y} - \rho \mathbf{y}'\mathbf{P}(\mathbf{I}_T\otimes \mathbf{W})\mathbf{y} \!-\! \rho \mathbf{y}'(\mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}\mathbf{y}\!+\!\rho ^2 \mathbf{y}'(\mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}(\mathbf{I}_T \otimes \mathbf{W})\mathbf{y} \end{aligned}$$

Consequently, the likelihood for $\rho$ is

$$\begin{aligned} l (\rho )&= \log \; k + T \sum _{i=1}^N \log \; (1-\rho \omega _i) \\&- \frac{ NT }{2} \log \; [\mathbf{y}'\mathbf{P}\mathbf{y} - \rho \mathbf{y}'\mathbf{P}(\mathbf{I}_T\otimes \mathbf{W})\mathbf{y} - \rho \mathbf{y}'(\mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}\mathbf{y}\\&+\rho ^2 \mathbf{y}'(\mathbf{I}_T \otimes \mathbf{W}')\mathbf{P}(\mathbf{I}_T \otimes \mathbf{W})\mathbf{y} ] \end{aligned}$$

So $\rho$ can be estimated by an iterative Newton-Raphson process as

$$\begin{aligned} \rho _{j+1}=\rho _j - \frac{ l (\rho ) }{ l '(\rho ) } \end{aligned}$$

(17)

where

$$\begin{aligned} l '(\rho )&= \frac{\partial l (\rho ) }{ \partial \rho }=-\mathbf{y}'\mathbf{P}(\mathbf{I}_T\otimes \mathbf{W})\mathbf{y} - \mathbf{y}'(\mathbf{I}_T\otimes \mathbf{W}') \mathbf{P} \mathbf{y} \\&+2\rho \mathbf{y}'(\mathbf{I}_T\otimes \mathbf{W}')\mathbf{P}(\mathbf{I}_T \otimes \mathbf{W})\mathbf{y} - T\sum _{i=1}^N \frac{\omega _i}{1-\rho \omega _i} \end{aligned}$$

If it is desirable to estimate the model with two fixed effects, then the matrix $\mathbf{Q}$ is replaced by

$$\begin{aligned} \mathbf{Q}=\mathbf{I}_N\otimes \mathbf{I}_T - \frac{1}{T}\mathbf{j}_T \mathbf{j}_T' \otimes \mathbf{I}_N - \frac{1}{N} \mathbf{I}_N\otimes \mathbf{j}_N \mathbf{j}_N' + \frac{1}{ NT }\mathbf{j}_{ NT } \mathbf{j}_{ NT }' \end{aligned}$$

and the estimation procedure is the same as of the one-way fixed effect.

1.2 Properties of the MESPS maximum likelihood estimators

Let the likelihood of the model $\mathbf{Sy}=\mathbf{X}\varvec{\beta }+\varvec{\epsilon }$ be

$$\begin{aligned} L (\mathbf{y},\varvec{\theta })&= (2\pi )^{-\frac{ NT }{2}}(\gamma )^{-\frac{ NT }{2}}|\mathbf{S}|\hbox {exp}\left\{ -\frac{1}{2\gamma } (\mathbf{Sy}-\mathbf{X}\varvec{\beta })'(\mathbf{Sy}-\mathbf{X}\varvec{\beta })\right\} \\ l (\mathbf{y},\varvec{\theta })&= -\frac{ NT }{2}\log (2\pi ) -\frac{ NT }{2}\log \; \gamma -\frac{1}{2\gamma } (\mathbf{Sy}-\mathbf{X}\varvec{\beta })'(\mathbf{Sy}-\mathbf{X}\varvec{\beta }) \end{aligned}$$

once $|\mathbf{S}|=1$ and $\varvec{\theta }=(\varvec{\beta }, \gamma ,\alpha )$.

Let, for instance, $H(\varvec{\theta })=- l (\mathbf{y},\varvec{\theta })$ and $G(\varvec{\theta })=\frac{\partial }{\partial \varvec{\theta }} H(\varvec{\theta })$. In a first-order Taylor series context, it is reasonable to write

$$\begin{aligned} G(\varvec{\theta })=G(\varvec{\theta }_0 )+\frac{\partial }{\partial \varvec{\theta }}G(\varvec{\theta }_0)(\varvec{\theta }-\varvec{\theta }_0)+ o \end{aligned}$$

Evaluating $\varvec{\theta }$ at $\hat{\varvec{\theta }}_{ML}$ and if $o$ is a neglectable error, then $G(\hat{\varvec{\theta }}_{ML})=G(\varvec{\theta }_0 )+\frac{\partial }{\partial \varvec{\theta }}G(\varvec{\theta }_0)(\hat{\varvec{\theta }}_{ML} -\varvec{\theta }_0)$.

Notice that $G(\hat{\varvec{\theta }}_{ML})=0$ because that is the optimality first condition.

Rewriting $G(\hat{\varvec{\theta }}_{ML})=G(\varvec{\theta }_0 )+\frac{\partial }{\partial \varvec{\theta }}G(\varvec{\theta }_0)(\hat{\varvec{\theta }}_{ML} -\varvec{\theta }_0)$, we have

$$\begin{aligned} -G(\varvec{\theta }_0)&= \frac{\partial }{\partial \varvec{\theta }} (\hat{\varvec{\theta }}_{ML}-\varvec{\theta }_0 ) \\ \hat{\varvec{\theta }}_{ML}-\varvec{\theta }_0&= \left\{ - \frac{\partial ^2}{\partial \varvec{\theta } \partial \varvec{\theta }'} l (\mathbf{y},\varvec{\theta })\right\} ^{-1}G(\varvec{\theta }_0) \end{aligned}$$

And $I (\varvec{\theta })=\left( - \frac{\partial ^2}{\partial \varvec{\theta } \partial \varvec{\theta }'} l (\mathbf{y},\varvec{\theta })\right) ^{-1}$ is the Fisher information matrix. Rearranging,

$$\begin{aligned} \hat{\varvec{\theta }}_{MV} - \varvec{\theta }_0=[n I (\varvec{\theta })]^{-1}\frac{\partial }{\partial \varvec{\theta }}[- l (\mathbf{y},\varvec{\theta })] \\ \end{aligned}$$

Notice that $l (\mathbf{y},\varvec{\theta })=\sum _{i=1}^{ NT } \log \; f(y_i)$ and $E \left[ \frac{\partial }{\partial \varvec{\theta }}\log \;f(y_i) \right] =0$ because

$$\begin{aligned} E \left[ \frac{\partial }{\partial \varvec{\theta }} l (\varvec{\theta }) \right]&= \frac{\partial }{\partial \varvec{\theta }} \sum _{i=1}^{ NT }\int _{-\infty }^{\infty } l (y_i,\varvec{\theta })f(y_i, \varvec{\theta })\mathrm{d}y_i=\sum _{i=1}^{ NT }\int _{-\infty }^{\infty } \frac{\partial }{\partial \varvec{\theta }} l (y_i,\varvec{\theta })f(y_i,\varvec{\theta })\mathrm{d}y_i\\&= \sum _{i=1}^{ NT }\int _{-\infty }^{\infty } \frac{f'(y_i,\varvec{\theta })}{f(y_i,\varvec{\theta })} f(y_i,\varvec{\theta })\mathrm{d}y_i= \sum _{i=1}^{ NT }\int _{-\infty }^{\infty } f'(y_i,\varvec{\theta }) \mathrm{d}y_i \\&= \sum _{i=1}^{ NT } \frac{\partial }{\partial \varvec{\theta }}\int _{-\infty }^{\infty } f(y_i,\varvec{\theta })\mathrm{d}y_i=\sum _{i=1}^{ NT } \frac{\partial }{\partial \varvec{\theta }} 1=0 \end{aligned}$$

and

$$\begin{aligned} \hbox {Var}[- l (\mathbf{y},\varvec{\theta })]=n I (\varvec{\theta }) \end{aligned}$$

The Central Limit Theorem may be used in the sense that

$$\begin{aligned} \sum _{i=1}^{ NT } \frac{\partial }{ \partial \varvec{\theta } } \log \; f(y_i) \rightarrow N(0,n I (\varvec{\theta })) \end{aligned}$$

$$\begin{aligned} E(\hat{\varvec{\theta }}_{ML}-\varvec{\theta })&\rightarrow [n I (\varvec{\theta })]^{-1}E \left[ - \frac{\partial }{\partial \varvec{\theta }} l (\mathbf{y}) \right] =\mathbf{0} \\ \hbox {Var}(\hat{\varvec{\theta }}_{ML}-\varvec{\theta })&\rightarrow [n I (\varvec{\theta })]^{-1}\hbox {Var} \left[ - \frac{\partial }{\partial \varvec{\theta }} l (\mathbf{y}) \right] [n I (\varvec{\theta })]^{-1} \\&\rightarrow [n I (\varvec{\theta })]^{-1}n I (\varvec{\theta }) [n I (\varvec{\theta })]^{-1}\\&\rightarrow [n I (\varvec{\theta })]^{-1}\\&\sqrt{n}(\hat{\varvec{\theta }}-\varvec{\theta })\rightarrow N(\mathbf{0}, I (\varvec{\theta })^{-1} ) \end{aligned}$$