Estimation of dynamic mixed double factors model in high-dimensional panel data

Abstract

This paper endeavors to develop some dimension reduction techniques in panel data analysis when the numbers of individuals and indicators are very large. We use principal component analysis method to represent a large number of indicators via minority common factors in the factor models. We propose the dynamic mixed double factor model (DMDFM for short) to reflect cross section and time series correlation with the interactive factor structure. DMDFM not only reduces the dimension of indicators but also deals with the time series and cross section mixed effect. Different from other models, mixed factor models have two styles of common factors. The regressors factors reflect common trend and the dimension reducing, while the error components factors reflect difference and weak correlation of individuals. The results of Monte Carlo simulation show that generalized method of moments estimators have good properties of unbiasedness and consistency. Simulation results also show that the DMDFM can improve the prediction power of the models effectively.

Appendix: Proof of theoretical results

A. Proof of Theorem 4.1.

Denote $b(z,\beta )=Z_{i}\Delta \epsilon _{i}$, where $\beta =(\beta _{L}^{'},\beta _{F}^{'})_{'}$. From Eq. (4.8), we have $E[b(z,\beta )]=0$. We calculate partial derivative for each parameter to be estimated, $\partial b(z,\beta )/\partial \beta$, then let

$$\begin{aligned} Db(\beta _{L},\beta _{F})=\left( \partial b(b(z,\beta )/\partial \beta _{L}^{'},\partial b(z,\beta )/\partial \beta _{F}^{'}\right) ^{'} \end{aligned}$$

because the uniform consistency of random disturbance term, using Taylor series expansion around $\beta _{L}$ and $\beta _{F}$:

$$\begin{aligned} b(z,{\hat{\beta }})= & {} b(z,\beta )+Db\left( \beta _{L}^{*},\beta _{F}^{*}\right) (b(z, {\hat{\beta }})\nonumber \\&-\,b(z,\beta ))+o(b(z,\beta )) \end{aligned}$$

(6.1)

where ${\hat{\beta }}=({\hat{\beta }}_{L}^{'},{\hat{\beta }}_{F}^{'})^{'}$, $\beta _{L}^{*}$, $\beta _{F}^{*}$ are between $\beta _{L}$, ${\hat{\beta }}_{L}$, and $\beta _{F}$, ${\hat{\beta }}_{F}$, respectively, multiplied by weighting matrix A simultaneously:

$$\begin{aligned} Ab(z,{\hat{\beta }})= & {} A b(z,\beta )+ADb\left( \beta _{L}^{*},\beta _{F}^{*}\right) (b(z,{\hat{\beta }})\nonumber \\&-\,b(z,\beta ))+o(b(z,\beta )) \end{aligned}$$

(6.2)

Given the following three items:

• (i) From assumptions as before, given optimal weighting matrix $A_{O}$, we can obtain unique optimal estimator of $\beta$. $\beta$ is continuous vector defined on Euclid space $R^{n}$, and space $\Theta$ constituted by $\beta$ is a subset of $R^{n}$, and is closed and bounded.

• (ii) For $b(z,\beta )=Z_{i}\Delta \epsilon _{i}$, $\forall \epsilon >0$, from (6.1)

  $$\begin{aligned} E(b(z,{\hat{\beta }}))=b(z,\beta ) \end{aligned}$$

  so,

  $$\begin{aligned} \big |b(z,{\hat{\beta }})-b(z,\beta )\big |\xrightarrow {p}0 \end{aligned}$$

  (6.3)

  for given matrix A, denote

  $$\begin{aligned} \hat{S}_{N}(\beta )=b\big (z,{\hat{\beta }}\big )^{'}\hat{A}bv(z,{\hat{\beta }}\big ) \end{aligned}$$

  and

  $$\begin{aligned} S_{0}(\beta )=b(z,\beta )^{'}Ab(z,\beta ) \end{aligned}$$

  from (A.3), $S_{0}(\beta )$ is continuous.

• (iii) Next, prove $S_{0}(\beta )$ convergence with probability 1.

$$\begin{aligned}&\left| \hat{S}_{N}(\beta )-S_{0}(\beta )\right| =\left| b(z,{\hat{\beta }})^{'}\hat{A}b (z,{\hat{\beta }})-b(z,\beta )^{'}Ab(z,\beta )\right| \\&\quad =\,\bigg |\big (b(z,{\hat{\beta }})-b(z,\beta )\big )^{'}\hat{A}\big (b(z,{\hat{\beta }})-b(z,\beta )\big )\\&\qquad +\,b(z,\beta )^{'}\hat{A}(b(z,{\hat{\beta }})-b(z,\beta ))\\&\qquad +\,b(z,\beta )^{'}\hat{A}b(z,\beta )-b(z,\beta )-b(z,\beta )^{'}Ab(z,\beta )\bigg |\\&\quad =\,\bigg |\big (b(z,{\hat{\beta }})-b(z,\beta )\big )^{'}\hat{A}\big (b(z,{\hat{\beta }})-b(z,\beta )\big )\\&\qquad +\,b(z,\beta )^{'}\hat{A}(b(z,{\hat{\beta }})-b(z,\beta ))^{'}\hat{A} (b(z,\beta )\\&\qquad +\,b(z,\beta )^{'}(\hat{A}-A)b(z,\beta )\bigg |\\&\quad =\,\bigg |\big (b(z,{\hat{\beta }})-b(z,\beta )\big )^{'}\hat{A}\big (b(z,{\hat{\beta }})-b(z,\beta )\big )\\&\qquad +\,b(z,\beta )^{'}(\hat{A}+\hat{A}^{'})(b(z,{\hat{\beta }})\\&\qquad -\,b(z,\beta ))+b(z,\beta )^{'}(\hat{A}-A)b(z,\beta )\bigg | \end{aligned}$$

Using triangle inequalities

$$\begin{aligned}&\le \bigg |(b(z,{\hat{\beta }})-b(z,\beta ))^{'}\hat{A}(b(z,{\hat{\beta }})\\&\quad -b(z,\beta ))\bigg |+\bigg |b(z,\beta )^{'}(\hat{A}+\hat{A}^{'})(b(z,{\hat{\beta }})\\&\quad -b(z,\beta ))\bigg | +\bigg |b(z,\beta )^{'}(\hat{A}-A)b(z,\beta )\bigg | \end{aligned}$$

Using Cauchy–Schwartz inequalities

$$\begin{aligned}&\le \big \Vert b(z,{\hat{\beta }})-b(z,\beta )\big \Vert ^{2}\big \Vert \hat{A}\big \Vert +2\big \Vert b(z,\beta )\big \Vert \big \Vert b(z,{\hat{\beta }})\\&\quad -\,b(z,\beta )\big \Vert \big \Vert \hat{A}\big \Vert +\big \Vert b(z,\beta )\big \Vert ^{2}\big \Vert \hat{A}-A\big \Vert \end{aligned}$$

because

$$\begin{aligned}&b(z,{\hat{\beta }})-b(z,\beta )\xrightarrow {p}0\\&\quad \hat{A}-A\xrightarrow {p}0 \end{aligned}$$

we have

$$\begin{aligned} \left| \hat{S}_{N}(\beta )-S_{0}(\beta )\right| \xrightarrow {p}0 \end{aligned}$$

By Newey and Mcfadden (1994), following uniform convergence theorem, the conclusion is obtained. $\square$

B. Proof of Theorem 4.2.

• (1) Because

  $$\begin{aligned}&\partial R_{1}(\beta _{L},\beta _{F})/\partial \beta =\partial \left( b(z,\beta )^{'}Ab(z,\beta )\right) /\partial \beta \\&\quad =\partial \left( b(z,\beta )^{'}/\partial \beta Ab(z,\beta )\right) +\partial \left( b(z,\beta )^{'}/\partial \beta Ab(z,\beta )\right) \\&\quad =2\partial \left( b(z,\beta )^{'}/\partial \beta Ab(z,\beta )\right) \end{aligned}$$

  where $\beta =(\beta _{L},\beta _{F})^{'}$ for notation simplicity. Following this notation, in order to estimate GMM, we solve first-order condition, so we obtain that

  $$\begin{aligned} R_{1}({\hat{\beta }})^{'}Ab(z,{\hat{\beta }})=0 \end{aligned}$$

  (6.4)

  from (6.1), for optimal matrix $A_{O}$, we have

  $$\begin{aligned} R_{1}(\beta )^{'}A_{O}b(z,{\hat{\beta }})= & {} R_{1}(\beta )^{'}A_{O}\sqrt{N}b (z,{\hat{\beta }})\nonumber \\&+\,o(b(z,\beta )) \end{aligned}$$

  (6.5)

  using Taylor series expansion around $\beta$

  $$\begin{aligned}&R_{1}(\beta )^{'}A_{O}b(z,{\hat{\beta }})=R_{1}(\beta )^{'}A_{O} \left( \sqrt{N}b(z,\beta ) \right. \\&\quad \left. +\,R_{1}(\beta )\sqrt{N}\left( {\hat{\beta }}-\beta \right) \right) +o(b(z,\beta )) \end{aligned}$$

  from (6.4),we have

  $$\begin{aligned}&R_{1}(\beta )^{'}A_{O}R_{1}(\beta )\sqrt{N}\left( {\hat{\beta }}-\beta \right) \\&\quad =-R_{1}(\beta )^{'}A_{O}\sqrt{N}b(z,\beta )+o(b(z,\beta )) \end{aligned}$$

  so

  $$\begin{aligned}&\sqrt{N}({\hat{\beta }}-\beta )=-\left( R_{1}(\beta )^{'}A_{O}R_{1}(\beta )\right) ^{-1}\\&\quad R_{1}(\beta )^{'}A_{O}\sqrt{N}b(z,\beta )+o(b(z,\beta )) \end{aligned}$$

  by Eq. (4.15) as previous, we have

  $$\begin{aligned} \sqrt{N}b(z,\beta )\xrightarrow {d}N(0,D_{1}) \end{aligned}$$

  and

  $$\begin{aligned} \left( R_{1}(\beta )^{'}A_{O}R_{1}(\beta )\right) ^{-1}R_{1}(\beta )^{'}A_{O} \end{aligned}$$

  is a determined matrix, so

  $$\begin{aligned} \sqrt{N}\left( {\hat{\beta }}-\beta \right) \xrightarrow {d}N(0,\Sigma _{1}) \end{aligned}$$

  i.e.,

  $$\begin{aligned} \sqrt{N}\left( \left( {\hat{\beta }}_{L},{\hat{\beta }}_{F}\right) -(\beta _{L},\beta _{F})\right) \xrightarrow {d}N(0,\Sigma _{1}) \end{aligned}$$

• (2) Similar to the proof of (1), omitted.

$\square$