Quasi-maximum likelihood estimation of short panel data models with time-varying individual effects

Abstract

Since the commonly available time series on micro units are typically quite short, this paper considers a different estimation of linear panel data models where the unobserved individual effects are permitted to have time-varying effects on the response variable. We allow flexible possible correlations between included regressors and unobserved individual effects, and the model can accommodate both time varying and time invariant covariates. The quasi-maximum likelihood method is then proposed to obtain the estimates, which are easily executed by a simple iterative method. Two types of approaches to estimate the covariance matrix are introduced. The large sample properties are established when $n\rightarrow \infty$ and T is fixed. The estimates are efficient when both the individual effects and random errors follow normal distributions. Simulation studies show that our estimates perform well even when the correlations between the regressors and unobserved individual effects are misspecified. The proposed method is further illustrated by applications to a real data.

Appendix: Theoretical proofs

In this section, we give the proofs of Theorem 1 and 2.

Proof of Theorem 1

To distinguish between the arguments in the log quasi-likelihood function $\ell ({\varvec{\mu }}, {\varvec{\beta }}, \Omega )$ and the true values of the parameters, we use ${\varvec{\mu }}_0$, ${\varvec{\beta }}_0$ and $\Omega _0$ to denote the true values of ${\varvec{\mu }}$, ${\varvec{\beta }}$ and $\Omega$ only in the proof of this theorem.

It follows by (2.3) that

$$\begin{aligned} A({\varvec{\beta }}_0)\mathbf{z}_i= {\varvec{\mu }}_0 + \Lambda _0 {\varvec{\alpha }}_i + \mathbf{e}_i, \end{aligned}$$

and then

$$\begin{aligned}&\frac{1}{n}\sum \limits _{i=1}^n \mathbf{z}_i = [A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 + o_P(1),\\&\frac{1}{n}\sum \limits _{i=1}^n \mathbf{z}_i \mathbf{z}_i^{'} = [A({\varvec{\beta }}_0)]^{-1} \{ {\varvec{\mu }}_0 {\varvec{\mu }}_0^{'} + \Omega _0\}[A^{'}({\varvec{\beta }}_0)]^{-1} + o_P(1), \end{aligned}$$

by law of large numbers. Therefore,

$$\begin{aligned}&\frac{1}{n}\sum \limits _{i=1}^n [A({\varvec{\beta }})\mathbf{z}_i - {\varvec{\mu }}] [A({\varvec{\beta }})\mathbf{z}_i - {\varvec{\mu }}]^{'} \\&\quad = \frac{1}{n}\sum \limits _{i=1}^n A({\varvec{\beta }})\mathbf{z}_i \mathbf{z}_i^{'} A^{'}({\varvec{\beta }}) - \frac{1}{n} \sum \limits _{i=1}^n A({\varvec{\beta }})\mathbf{z}_i {\varvec{\mu }}^{'} - \frac{1}{n} \sum \limits _{i=1}^n {\varvec{\mu }}\mathbf{z}_i^{'} A^{'}({\varvec{\beta }}) + {\varvec{\mu }}{\varvec{\mu }}^{'} \\&\quad = A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} \Omega _0[A^{'}({\varvec{\beta }}_0)]^{-1}A^{'}({\varvec{\beta }}) \\&\qquad + \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \}^{'} + o_P(1). \end{aligned}$$

Moreover, we obtain by straightforward calculations that uniformly for any ${\varvec{\mu }}$, ${\varvec{\beta }}$ and $\Omega$ in the compact supports,

$$\begin{aligned} \frac{1}{n} \ell ({\varvec{\mu }}, {\varvec{\beta }}, \Omega )= & {} -\frac{T(p+1)}{2}\log 2\pi -\frac{1}{2}\log |\Omega | \nonumber \\&- \frac{1}{2}\text{ tr }\Big ([A({\varvec{\beta }}_0)]^{-1} \Omega _0[A^{'}({\varvec{\beta }}_0)]^{-1} A^{'}({\varvec{\beta }})\Omega ^{-1}A({\varvec{\beta }})\Big ) \nonumber \\&- \frac{1}{2} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \}^{'}\Omega ^{-1} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \} + o_P(1) \nonumber \\= & {} \frac{1}{n}E[\ell ({\varvec{\mu }}, {\varvec{\beta }}, \Omega )] + o_P(1) \end{aligned}$$

(A.1)

by Assumption 3 and Lemma 2.4 of Chapter 36 (Engle and McFadden 1994).

Let $Q({\varvec{\mu }}, {\varvec{\beta }}, \Omega )=\frac{1}{n}E[\ell ({\varvec{\mu }}, {\varvec{\beta }}, \Omega )]$. In the following, we will show the uniqueness of maximizer, that is $Q({\varvec{\mu }}, {\varvec{\beta }}, \Omega ) - Q({\varvec{\mu }}_0, {\varvec{\beta }}_0, \Omega _0)<0$ for any $({\varvec{\mu }}, {\varvec{\beta }}, \Omega )$ in the complement of an open neighborhood of $({\varvec{\mu }}_0, {\varvec{\beta }}_0, \Omega _0)$.

Define an auxiliary model: $Y_i^*= \mu _0 + \mathbf{e}_i$, with $\mathbf{e}_i \sim (\mathbf{0}, \Omega _0)$, and $Y_i^*=A({\varvec{\beta }}_0)\mathbf{z}_i$ with ${\varvec{\beta }}_0$ being known. Then its log quasi-likelihood function is

$$\begin{aligned} \ell _a({\varvec{\mu }}, \Omega )= -\frac{nT(p+1)}{2}\log 2\pi - \frac{n}{2}\log |\Omega | - \frac{1}{2}\sum \limits _{i=1}^n [ A({\varvec{\beta }}_0)\mathbf{z}_i - {\varvec{\mu }}]^{'}\Omega ^{-1} [ A({\varvec{\beta }}_0)\mathbf{z}_i - {\varvec{\mu }}] \end{aligned}$$

Set $Q_a(\Omega )= \max _{{\varvec{\mu }}} \frac{1}{n} E_a[\ell _a({\varvec{\mu }}, \Omega )]$, where $E_a$ is the expectation under the auxiliary model. Then it can be easily shown that

$$\begin{aligned} Q_a(\Omega )=-\frac{T(p+1)}{2}\log 2\pi - \frac{1}{2}\log |\Omega | - \frac{1}{2}\text{ tr }(\Omega ^{-1}\Omega _0). \end{aligned}$$

It follows by Jensen inequality, for all $\Omega$ in the compact support, $Q_a(\Omega )\le \frac{1}{n} E_a [\ell _a ({\varvec{\mu }}_0, \Omega _0)] =Q_a(\Omega _0)$. Hence,

$$\begin{aligned}&Q({\varvec{\mu }}, {\varvec{\beta }}, \Omega ) - Q({\varvec{\mu }}_0, {\varvec{\beta }}_0, \Omega _0) \\&\quad = Q_a(\Omega )- Q_a(\Omega _0) - \frac{1}{2} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \}^{'}\Omega ^{-1} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \} \\&\qquad + \frac{1}{2}\text{ tr }(\Omega ^{-1}\Omega _0) - \frac{1}{2}\text{ tr }\Big ([A({\varvec{\beta }}_0)]^{-1} \Omega _0[A^{'}({\varvec{\beta }}_0)]^{-1} A^{'}({\varvec{\beta }})\Omega ^{-1}A({\varvec{\beta }})\Big ) \\&\quad = Q_a(\Omega )- Q_a(\Omega _0) - \frac{1}{2} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \}^{'}\Omega ^{-1} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \} \\&\qquad - \frac{1}{2}\text{ tr }\Big ( A^{'}({\varvec{\beta }}) \Omega ^{-1} A({\varvec{\beta }}) \left\{ [ A({\varvec{\beta }}_0)]^{-1}\Omega _0 [A^{'}({\varvec{\beta }}_0)]^{-1} - [ A({\varvec{\beta }})]^{-1}\Omega _0 [A^{'}({\varvec{\beta }})]^{-1}\right\} \Big ) \end{aligned}$$

Since $\text{ cov }(\mathbf{z}_i)= [A({\varvec{\beta }}_0)]^{-1}\Omega _0 [A^{'}({\varvec{\beta }}_0)]^{-1}$ for the true model, we have

$$\begin{aligned} {[} A({\varvec{\beta }}_0)]^{-1}\Omega _0 [A^{'}({\varvec{\beta }}_0)]^{-1} - [ A({\varvec{\beta }})]^{-1}\Omega _0 [A^{'}({\varvec{\beta }})]^{-1} \end{aligned}$$

is semipositive definite. Therefore,

$$\begin{aligned} \text{ tr }\Big ( A^{'}({\varvec{\beta }}) \Omega ^{-1} A({\varvec{\beta }}) \left\{ [ A({\varvec{\beta }}_0)]^{-1}\Omega _0 [A^{'}({\varvec{\beta }}_0)]^{-1} - [ A({\varvec{\beta }})]^{-1}\Omega _0 [A^{'}({\varvec{\beta }})]^{-1}\right\} \Big ) \ge 0 \end{aligned}$$

by $\tau _{\min }(B)\text{ tr }(D) \le \text{ tr }(BD) \le \tau _{\max }(B)\text{ tr }(D)$ for nonnegative matrices B and D.

Moreover, it follows by straightforward calculations that

$$\begin{aligned} A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1}{\varvec{\mu }}_0 - {\varvec{\mu }}=\left( \begin{array}{cc} {\varvec{\lambda }}_0+[I_T\otimes ({\varvec{\beta }}_0-{\varvec{\beta }})^{'}]{\varvec{\delta }}_0 - {\varvec{\lambda }}\\ {\varvec{\delta }}_0- {\varvec{\delta }}\end{array}\right) \ne 0 \end{aligned}$$

when ${\varvec{\beta }}\ne {\varvec{\beta }}_0$ and ${\varvec{\mu }}\ne {\varvec{\mu }}_0$. Then we have

$$\begin{aligned} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \}^{'}\Omega ^{-1} \Big \{ A({\varvec{\beta }})[A({\varvec{\beta }}_0)]^{-1} {\varvec{\mu }}_0 - {\varvec{\mu }}\Big \}>0 \end{aligned}$$

provided that $\Omega$ is positive definite. Hence, $Q({\varvec{\mu }}, {\varvec{\beta }}, \Omega ) - Q({\varvec{\mu }}_0, {\varvec{\beta }}_0, \Omega _0)<0$ for any $({\varvec{\mu }}, {\varvec{\beta }}, \Omega )$ in the complement of an open neighborhood of $({\varvec{\mu }}_0, {\varvec{\beta }}_0, \Omega _0)$.

The consistency of ${\hat{{\varvec{\mu }}}}$, ${\hat{{\varvec{\beta }}}}$ and ${\hat{\Omega }}$, which maximize $\frac{1}{n} \ell ({\varvec{\mu }}, {\varvec{\beta }}, \Omega )$, then follows from the uniqueness of maximizer and (A.1) according to Theorem 3.4 of White (1994). $\square$

Proof of Theorem 2:

As $\Omega =\Lambda \Lambda ^{'} + \Sigma _e$, we have

$$\begin{aligned} \Omega ^{-1}= \Sigma _e^{-1} - \Sigma _e^{-1}\Lambda (I_q + \Lambda ^{'}\Sigma _{e}^{-1}\Lambda )^{-1}\Lambda ^{'} \Sigma _{e}^{-1} \equiv \Sigma _e^{-1} - \Sigma _e^{-1}\Lambda B \Lambda ^{'} \Sigma _{e}^{-1} \end{aligned}$$

(A.2)

and $B \Lambda ^{'} \Sigma _{e}^{-1} \Lambda = I_q - B.$ It follows that

$$\begin{aligned} \frac{1}{n} \sum \limits _{i=1}^n ({\mathcal {X}}_i^{*'}, \mathbf{0})\Omega ^{-1} \left( \begin{array}{c} {\mathcal {X}}_i^* \\ \mathbf{0} \end{array}\right)= & {} \frac{1}{n} \sum \limits _{i=1}^n {\mathcal {X}}_i^{*'} ( \Sigma _{\epsilon }^{-1} - \Sigma _{\epsilon }^{-1} D B D^{'} \Sigma _{\epsilon }^{-1}) {\mathcal {X}}_i^{*} \nonumber \\= & {} \frac{1}{n} \sum \limits _{i=1}^n \left( \begin{array}{ccc} {\mathcal {X}}_{i,1}^{*'}M{\mathcal {X}}_{i,1}^{*} &{} \cdots &{} {\mathcal {X}}_{i,1}^{*'}M{\mathcal {X}}_{i,p}^{*} \\ \vdots &{} \vdots &{} \vdots \\ {\mathcal {X}}_{i,p}^{*'}M{\mathcal {X}}_{i,1}^{*} &{} \cdots &{} {\mathcal {X}}_{i,p}^{*'}M{\mathcal {X}}_{i,p}^{*} \end{array}\right) , \end{aligned}$$

(A.3)

and for any $k=1, \ldots , p$,

$$\begin{aligned} {\mathcal {X}}_{i,k}^{*}=(I_T\otimes \mathbf{l}_k^{'}) [\Gamma {\varvec{\alpha }}_i + \mathbf{u}_i - (\Gamma {\bar{{\varvec{\alpha }}}} + {\bar{\mathbf{u}}})], \end{aligned}$$

(A.4)

where ${\bar{{\varvec{\alpha }}}}= \frac{1}{n}\sum \limits _{i=1}^n {\varvec{\alpha }}_i$, and ${\bar{\mathbf{u}}}= \frac{1}{n}\sum \limits _{i=1}^n \mathbf{u}_i$.

Therefore, for $j,k=1,\ldots ,p$,

$$\begin{aligned} \frac{1}{n}\sum \limits _{i=1}^n {\mathcal {X}}_{i,j}^{*} {\mathcal {X}}_{i,k}^{*'}= & {} (I_T\otimes \mathbf{l}_j^{'})\Big \{\frac{1}{n}\sum \limits _{i=1}^n(\Gamma {\varvec{\alpha }}_i + \mathbf{u}_i) (\Gamma {\varvec{\alpha }}_i + \mathbf{u}_i)^{'} \nonumber \\&\ \ - \frac{1}{n}\sum \limits _{i=1}^n(\Gamma {\varvec{\alpha }}_i + \mathbf{u}_i)(\Gamma {\bar{{\varvec{\alpha }}}} + {\bar{\mathbf{u}}})^{'} \nonumber \\&\ \ - \frac{1}{n}\sum \limits _{i=1}^n(\Gamma {\bar{{\varvec{\alpha }}}} + {\bar{\mathbf{u}}})(\Gamma {\varvec{\alpha }}_i + \mathbf{u}_i)^{'} + (\Gamma {\bar{{\varvec{\alpha }}}} + {\bar{\mathbf{u}}})(\Gamma {\bar{{\varvec{\alpha }}}} + {\bar{\mathbf{u}}})^{'} \Big \}(I_T\otimes \mathbf{l}_k) \nonumber \\= & {} (I_T\otimes \mathbf{l}_j^{'})(\Gamma \Gamma ^{'} + \Sigma _u)(I_T\otimes \mathbf{l}_k)+ o_P(1) \end{aligned}$$

by law of large numbers. Then, we have

$$\begin{aligned} \frac{1}{n} \sum \limits _{i=1}^n ({\mathcal {X}}_i^{*'}, \mathbf{0})\Omega ^{-1} \left( \begin{array}{c} {\mathcal {X}}_i^* \\ \mathbf{0} \end{array}\right) = \Psi + o_P(1). \end{aligned}$$

(A.5)

Furthermore,

$$\begin{aligned}&\frac{1}{n} \sum \limits _{i=1}^n \Vert ({\mathcal {X}}_i^{*'}, \mathbf{0})[{\hat{\Omega }}^{-1}- \Omega ^{-1}] \left( \begin{array}{c} {\mathcal {X}}_i^* \\ \mathbf{0} \end{array}\right) \Vert \nonumber \\&\quad \le \frac{1}{n} \sum \limits _{i=1}^n \Vert ({\mathcal {X}}_i^{*'}, \mathbf{0})\Vert \cdot \Vert \left( \begin{array}{c} {\mathcal {X}}_i^* \\ \mathbf{0} \end{array}\right) \Vert \cdot \Vert {\hat{\Omega }}^{-1}- \Omega ^{-1}\Vert \nonumber \\&\quad = \frac{1}{n} \sum \limits _{i=1}^n \text{ tr }({\mathcal {X}}_i^{*'}{\mathcal {X}}_i^*)\cdot \Vert {\hat{\Omega }}^{-1}- \Omega ^{-1}\Vert =o_P(1) \end{aligned}$$

(A.6)

using similar arguments as establishing (A.5), consistency of ${\hat{\Omega }}$ and boundedness of the parameters. Combing the results of (A.3), (A.5) and (A.6), we have

$$\begin{aligned} \frac{1}{n} \sum \limits _{i=1}^n ({\mathcal {X}}_i^{*'}, \mathbf{0}){\hat{\Omega }}^{-1} \left( \begin{array}{c} {\mathcal {X}}_i^* \\ \mathbf{0} \end{array}\right) = \Psi + o_P(1). \end{aligned}$$

(A.7)

Moreover, for any nonzero vector $\mathbf{b}=(b_1, \ldots , b_p)^{'} \in R^p$, we have by straightforward calculations and (A.4)

$$\begin{aligned}&\frac{1}{\sqrt{n}} \sum \limits _{i=1}^n \mathbf{b}^{'}({\mathcal {X}}_i^{*'}, \mathbf{0})\Omega ^{-1} [\Lambda {\varvec{\alpha }}_i + \mathbf{e}_i- (\Lambda {\bar{{\varvec{\alpha }}}} + {\bar{\mathbf{e}}})] \nonumber \\&\quad = \frac{1}{\sqrt{n}} \sum \limits _{i=1}^n \sum \limits _{k=1}^p b_k ({\varvec{\xi }}_i - {\bar{{\varvec{\xi }}}})^{'} F^{'} H_k^{'} R F({\varvec{\xi }}_i -{\bar{{\varvec{\xi }}}}) \nonumber \\&\quad = \frac{1}{\sqrt{n}} \sum \limits _{i=1}^n \sum \limits _{k=1}^p b_k {\varvec{\xi }}_i^{'} F^{'} H_k^{'} R F{\varvec{\xi }}_i - {\bar{{\varvec{\xi }}}}^{'} \frac{2}{\sqrt{n}} \sum \limits _{i=1}^n \sum \limits _{k=1}^p b_k F^{'} H_k^{'} R F{\varvec{\xi }}_i \nonumber \\&\qquad +\, {\bar{{\varvec{\xi }}}}^{'}\sqrt{n} \sum \limits _{k=1}^p b_k F^{'} H_k^{'} R F {\bar{{\varvec{\xi }}}}^{'} . \end{aligned}$$

(A.8)

Since $RFF^{'}=(I_T, \mathbf{0})$ and $RFF^{'}R^{'}=M$ by ( A.2), it follows that

$$\begin{aligned} E\left[ \sum \limits _{k=1}^p b_k{\varvec{\xi }}_i^{'} F^{'} H_k^{'} R F{\varvec{\xi }}_i \right] = \sum \limits _{k=1}^p b_k \cdot \text{ tr }\Big (H_k^{'} R F F^{'}\Big )=0, \end{aligned}$$

and

$$\begin{aligned}&\text{ var }\left( \sum \limits _{k=1}^p b_k {\varvec{\xi }}_i^{'} F^{'} H_k^{'} R F{\varvec{\xi }}_i \right) \\&\quad = \sum \limits _{k=1}^p \sum \limits _{j=1}^p b_k b_j \cdot \text{ cov }({\varvec{\xi }}_i^{'} F^{'} H_k^{'} R F{\varvec{\xi }}_i, {\varvec{\xi }}_i^{'} F^{'} H_j^{'} R F{\varvec{\xi }}_i) \\&\quad = \sum \limits _{k=1}^p \sum \limits _{j=1}^p b_k b_j \Big \{\sum \limits _{l=1}^{T(p+1)+q} (E\xi _l^4 - 3) ( F^{'} H_k^{'} R F)_{ll} (F^{'} H_j^{'} R F)_{ll} \Big ) \\&\qquad +\, \text{ tr }(F^{'} H_k^{'} R F F^{'} H_j^{'} R F) + \text{ tr }(F^{'} H_k^{'} R F F^{'} R^{'} H_j F) \Big \}\\&\quad = \sum \limits _{k=1}^p \sum \limits _{j=1}^p b_k b_j \Big \{ \sum \limits _{l=1}^{T(p+1)+q} (E\xi _{1l}^4 - 3) [ F^{'} H_k^{'} R F]_{ll} [F^{'} H_j^{'} R F]_{ll} \Big ) \\&\qquad +\, \text{ tr }\Big ((I_T\otimes \mathbf{l}_k)M (I_T\otimes \mathbf{l}_j^{'}) (\Gamma \Gamma ^{'} + \Sigma _u)\Big ) \Big \}. \end{aligned}$$

Then we obtain by central limit theorem for iid. random variables that

$$\begin{aligned} \frac{1}{\sqrt{n}} \sum \limits _{i=1}^n \sum \limits _{k=1}^p b_k {\varvec{\xi }}_i^{'} F^{'} H_k^{'} R F{\varvec{\xi }}_i {\mathop {\rightarrow }\limits ^{d}} N\Big (\mathbf{0}, \mathbf{b}^{'} (\Delta + \Psi ) \mathbf{b}\Big ). \end{aligned}$$

(A.9)

Similarly, it can be shown that $\frac{1}{\sqrt{n}} \sum \limits _{i=1}^n \sum \limits _{k=1}^p b_k F^{'} H_k^{'} R F{\varvec{\xi }}_i =O_p(1)$. Therefore, the second and third terms of (A.8) are both $o_p(1)$ using ${\bar{{\varvec{\xi }}}}=O_P(n^{-1/2})$. This, together with $\mathbf{z}_i^*= ({\mathcal {X}}_i^{*'}, \mathbf{0})^{'}{\varvec{\beta }}_0 + F ({\varvec{\xi }}_i - {\bar{{\varvec{\xi }}}})$, consistency of ${\hat{\Omega }}$ and (A.7), leads to the asymptotic distribution of Theorem 2. $\square$