Asymptotic distribution of quasi-maximum likelihood estimation of dynamic panels using long difference transformation when both N and T are large

Abstract

This note shows that the asymptotic properties of the quasi-maximum likelihood estimation for dynamic panel models can be easily derived by following the approach of Grassetti (Stat Methods Appl 20:221–240, 2011) to take the long difference to remove the time-invariant individual specific effects.

Appendix

Since

$$\begin{aligned}&\sqrt{NT}\left( \hat{\gamma }_{MLE}-\gamma \right) \nonumber \\&\quad =\left( \frac{1}{NT} \sum _{i=1}^{N}\tilde{{\mathbf {y}}}_{i,-1}^{\prime }\Omega ^{-1}\tilde{{\mathbf {y}}}_{i,-1}\right) ^{-1}\left( \frac{1}{\sqrt{NT}}\sum _{i=1}^{N}\tilde{{\mathbf {y}}}_{i,-1}^{\prime }\Omega ^{-1}\left( -\xi _{i}{\mathbf {1}}_{T}+{\mathbf {u}} _{i}\right) \right) , \qquad \qquad \end{aligned}$$

(6.1)

and from (2.8),

$$\begin{aligned} \tilde{y}_{it}=-\xi _{i}\frac{1-\gamma ^{t}}{1-\gamma } +\sum _{s=1}^{t}\gamma ^{t-s}u_{is}. \end{aligned}$$

The denominator of (6.1) is of the form

$$\begin{aligned} \frac{1}{NT}\sum _{i=1}^{N}\tilde{{\mathbf {y}}}_{i,-1}^{\prime }\Omega ^{-1} \tilde{{\mathbf {y}}}_{i,-1}= & {} \frac{1}{\sigma _{u}^{2}}\frac{1}{NT} \sum _{i=1}^{N}\tilde{{\mathbf {y}}}_{i,-1}^{\prime }\left( \left[ I_{T}-\frac{ \sigma _{\xi }^{2}}{\sigma _{u}^{2}+T\sigma _{\xi }^{2}}{\mathbf {1}}_{T} {\mathbf {1}}_{T}^{\prime }\right] \right) \tilde{{\mathbf {y}}}_{i,-1} \nonumber \\= & {} \frac{1}{\sigma _{u}^{2}}\frac{1}{NT}\sum _{i=1}^{N}\sum _{t=1}^{T-1}\tilde{ y}_{it-1}^{2}\nonumber \\&-\frac{\sigma _{\xi }^{2}}{\sigma _{u}^{2}\left( \sigma _{u}^{2}+T\sigma _{\xi }^{2}\right) }\frac{1}{NT}\sum _{i=1}^{N}\left( \sum _{t=1}^{T-1}\tilde{y}_{it-1}\right) ^{2}, \end{aligned}$$

(6.2)

By continuous substitution, the first term on the right hand side of (6.2)

$$\begin{aligned} \frac{1}{\sigma _{u}^{2}}\frac{1}{NT}\sum _{i=1}^{N}\sum _{t=1}^{T-1}\tilde{y}_{it-1}^{2}= & {} \frac{1}{\sigma _{u}^{2}}\frac{1}{NT}\sum _{i=1}^{N} \sum _{t=1}^{T-1}\frac{\left( 1-\gamma ^{t}\right) ^{2}}{\left( 1-\gamma \right) ^{2}}\xi _{i}^{2}\nonumber \\&+\;\frac{1}{\sigma _{u}^{2}}\frac{1}{NT} \sum _{i=1}^{N}\sum _{t=1}^{T-1}\left( \sum _{s=1}^{t}\gamma ^{t-s}u_{is}\right) ^{2} \nonumber \\&-\;2\frac{1}{\sigma _{u}^{2}}\frac{1}{NT}\sum _{i=1}^{N}\sum _{t=1}^{T-1} \sum _{s=1}^{t}\frac{1-\gamma ^{t}}{1-\gamma }\xi _{i}\gamma ^{t-s}u_{is}\nonumber \\= & {} \frac{1}{\left( 1-\gamma \right) ^{2}\sigma _{u}^{2}}\frac{1}{N} \sum _{i=1}^{N}\xi _{i}^{2}+\frac{1}{\sigma _{u}^{2}}\frac{1}{NT} \sum _{i=1}^{N}\sum _{t=1}^{T-1}\left( \sum _{s=1}^{t}\gamma ^{t-s}u_{is}\right) ^{2}\nonumber \\&+\;o_{p}\left( 1\right) \nonumber \\&\overset{p}{\rightarrow }\frac{\sigma _{\xi }^{2}}{\left( 1-\gamma \right) ^{2}\sigma _{u}^{2}}+\frac{1}{1-\gamma ^{2}}, \end{aligned}$$

(6.3)

For the second term of the right hand side of (6.2), we note that for all i

$$\begin{aligned} \frac{1}{T}\sum _{t=1}^{T-1}\tilde{y}_{it-1}=-\xi _{i}\frac{1}{T} \sum _{t=1}^{T-1}\frac{1-\gamma ^{t}}{1-\gamma }+\frac{1}{T} \sum _{t=1}^{T-1}\sum _{s=1}^{t}\gamma ^{t-s}u_{is}, \end{aligned}$$

with

$$\begin{aligned} \frac{1}{N}\sum _{i=1}^{N}\left( \frac{1}{T}\sum _{t=1}^{T-1}\tilde{y}_{it-1}\right) ^{2}= & {} \frac{1}{N}\sum _{i=1}^{N}\xi _{i}^{2}\left( \frac{1}{T }\sum _{t=1}^{T-1}\frac{1-\gamma ^{t}}{1-\gamma }\right) ^{2}\\&+\,\frac{1}{N} \sum _{i=1}^{N}\left( \frac{1}{T}\sum _{t=1}^{T-1}\sum _{s=1}^{t}\gamma ^{t-s}u_{is}\right) ^{2}\\&-\,\frac{1}{1-\gamma }\frac{1}{N}\sum _{i=1}^{N}2\xi _{i}\frac{1}{T} \sum _{t=1}^{T-1}\sum _{s=1}^{t}\gamma ^{t-s}u_{is}+o_{p}\left( 1\right) \\&\overset{p}{\rightarrow }\frac{\sigma _{\xi }^{2}}{\left( 1-\gamma \right) ^{2}}, \end{aligned}$$

then

$$\begin{aligned} \frac{\sigma _{\xi }^{2}}{\sigma _{u}^{2}\left( \sigma _{u}^{2}+T\sigma _{\xi }^{2}\right) }\frac{1}{NT}\sum _{i=1}^{N}\left( \sum _{t=1}^{T-1}\tilde{y }_{it-1}\right) ^{2}= & {} \frac{T\sigma _{\xi }^{2}}{\sigma _{u}^{2}\left( \sigma _{u}^{2}+T\sigma _{\xi }^{2}\right) }\frac{1}{N}\sum _{i=1}^{N}\left( \frac{1}{T}\sum _{t=1}^{T-1}\tilde{y}_{it-1}\right) ^{2} \nonumber \\&\overset{p}{\rightarrow }\frac{\sigma _{\xi }^{2}}{\sigma _{u}^{2}\left( 1-\gamma \right) ^{2}}, \end{aligned}$$

(6.4)

as $\left( N,T\right) \rightarrow \infty$. Combining (6.3) and (6.4) yields

$$\begin{aligned} \frac{1}{NT}\sum _{i=1}^{N}\tilde{{\mathbf {y}}}_{i,-1}^{\prime }\Omega ^{-1} \tilde{{\mathbf {y}}}_{i,-1}\overset{p}{\rightarrow }\frac{1}{1-\gamma ^{2}}, \end{aligned}$$

(6.5)

as $\left( N,T\right) \rightarrow \infty$.

For the limit of the numerator of (6.1), we note that

$$\begin{aligned} E\left[ \tilde{{\mathbf {y}}}_{i,-1}^{\prime }\Omega ^{-1}\xi _{i}{\mathbf {1}}_{T} \right] =\frac{\sigma _{u}^{-2}}{1+T\sigma _{\xi }^{2}\sigma _{u}^{-2}}E \left[ \tilde{{\mathbf {y}}}_{i,-1}^{\prime }{\mathbf {1}}_{T}\xi _{i}\right] =- \frac{\sigma _{\xi }^{2}\sigma _{u}^{-2}}{1+T\sigma _{\xi }^{2}\sigma _{u}^{-2}}\sum _{t=1}^{T-1}\frac{\left( 1-\gamma ^{t}\right) }{1-\gamma } \end{aligned}$$

(6.6)

and

$$\begin{aligned} E\left( \tilde{{\mathbf {y}}}_{i,-1}^{\prime }\Omega ^{-1}{\mathbf {u}}_{i}\right)= & {} \sigma _{u}^{-2}E\left( \tilde{{\mathbf {y}}}_{i,-1}^{\prime }{\mathbf {u}} _{i}\right) -\frac{\sigma _{\xi }^{2}\sigma _{u}^{-4}}{1+T\sigma _{\xi }^{2}\sigma _{u}^{-2}}E\left( \tilde{{\mathbf {y}}}_{i,-1}^{\prime }{\mathbf {1}}_{T}{\mathbf {1}}_{T}^{\prime }{\mathbf {u}}_{i}\right) \nonumber \\= & {} -\frac{\sigma _{\xi }^{2}\sigma _{u}^{-2}}{1+T\sigma _{\xi }^{2}\sigma _{u}^{-2}}\sum _{t=1}^{T-1}\frac{1-\gamma ^{t}}{1-\gamma }, \end{aligned}$$

(6.7)

Combining (6.6) and (6.7) yields

$$\begin{aligned} E\left( \tilde{{\mathbf {y}}}_{i,-1}^{\prime }\Omega ^{-1}\left( -\xi _{i} {\mathbf {1}}_{T}+{\mathbf {u}}_{i}\right) \right) =0, \end{aligned}$$

This suggests that the MLE of $\gamma$ is asymptotically unbiased either N or T or both tend to infinity.

Furthermore, under assumptions 1–2, the variance of the MLE $\hat{\gamma }_{MLE}$ is given by

$$\begin{aligned} V_{MLE}^{-1}=-\frac{1}{NT}E\left( \frac{\partial ^{2}\log L}{\partial \gamma ^{2}}\right) =\frac{1}{NT}\sum _{i=1}^{N}E\left( \tilde{{\mathbf {y}}}_{i,-1}^{\prime }\Omega ^{-1}\tilde{{\mathbf {y}}}_{i,-1}\right) =\frac{1}{ 1-\gamma ^{2}}. \end{aligned}$$