Rank-based test for slope homogeneity in high-dimensional panel data models

Abstract

A large number of existing high-dimensional panel data analyses are established based on normal or nearly normal distribution assumptions, which may be not robust to severe departures of normality. Since the observed data may not follow the normal distribution in some specific applications, it is necessary to design robust tests to departures of normality. On this ground, we propose a rank-based score test for testing slope homogeneity in high-dimensional panel data regressions, where robust tests to departures of normality are still rare. Both theoretical and numerical results demonstrate the advantage of the proposed test in robustness to departures of normality.

Appendix

First, we consider the asymptotic property of ${\hat{\varvec{\beta }}}_{RWFE}$. According to Hettmansperger and McKean (1998) or Rashid et al. (2012), we have

$$\begin{aligned} {\hat{\varvec{\beta }}}_{RWFE}=&\,\varvec{\beta }_0+\left( \sum _{i=1}^N \tau _i^{-2}{\varvec{\Sigma }}_i\right) ^{-1}\sum _{i=1}^N\tau _i^{-2}{\varvec{\Sigma }}_i\delta _i\nonumber \\&+\left( \sum _{i=1}^N \tau _i^{-2}{\varvec{\Sigma }}_i\right) ^{-1}\sum _{i=1}^N \mathbf{M}\mathbf{X}_i \psi (F_i({\varvec{u}}_i))/\tau _i+o_p(T^{-1/2}N^{-1/2}). \end{aligned}$$

(9)

Define $\xi _{it}=\sqrt{12}\left( \frac{R(u_{it})}{T+1}-\frac{1}{2}\right)$ and recall that $e_{it}=\sqrt{12}\left( \frac{R({\hat{\varepsilon }}_{it})}{T+1}-\frac{1}{2}\right)$. We can decompose ${\hat{Q}}_{RS}$ in (7) as follows:

$$\begin{aligned} {\hat{Q}}_{RS}=&\,\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum } {{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} e_{it}e_{is}}\\ ={}&\,\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \xi _{is}\xi _{it}+\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} (e_{it}-\xi _{it})(e_{is}-\xi _{is}), \\&+2\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} (e_{it}-\xi _{it})\xi _{is}\\ \equiv {}&\,A_{1}+A_{2}+A_{3}. \end{aligned}$$

First, we consider the first part $A_{1}$. Let $\eta _{it}= \sqrt{12}(F_i( u_{it})-\frac{1}{2})$. Below, we will prove that

$$\begin{aligned} A_{1}=\frac{T(T-1)}{(T+1)^2}\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \eta _{is}\eta _{it}+\frac{kTN}{(T+1)^2}+o_p(N^{1/2}). \end{aligned}$$

(10)

Obviously,

$$\begin{aligned}&\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \xi _{is}\xi _{it}\\&\quad =\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \frac{12}{(T+1)^2}\sum _{k=1}^T\sum _{l=1}^T \left( I( u_{ik}\le u_{is})-\frac{1}{2}\right) \left( I( u_{il}\le u_{it})-\frac{1}{2}\right) \\&\quad =\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \frac{12}{(T+1)^2}\sum _{k=1}^T \left( I( u_{ik}\le u_{is})-\frac{1}{2}\right) \left( I( u_{ik}\le u_{it})-\frac{1}{2}\right) \\&\qquad +\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \frac{12}{(T+1)^2}\underset{k\not =l}{\sum \sum } \left( I( u_{ik}\le u_{is})-\frac{1}{2}\right) \left( I( u_{il}\le u_{it})-\frac{1}{2}\right) \\&\quad \equiv D_{1}+D_{2}, \end{aligned}$$

where

$$\begin{aligned} {\mathbb {E}}(D_{1})&=\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\mathbb {E}}({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} )\frac{12}{(T+1)^2}\sum _{k=1}^T {\mathbb {E}}\{(\frac{1}{2}-F_i( u_{ik}))^2\}=\frac{pTN}{(T+1)^2}, \\ \mathrm {var}(D_{1})&=O(T^{-1}{\mathbb {E}}(D_{1}^2))=O(N^2/T^3)=o(N), \end{aligned}$$

due to $N=o(T^3)$, and

$$\begin{aligned} D_{2}&=\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \frac{12}{(T+1)^2}\underset{k\not =l}{\sum \sum } \left( I( u_{ik}\le u_{is})-\frac{1}{2}\right) \left( I( u_{il}\le u_{it})-\frac{1}{2}\right) \\&= \frac{T(T-1)}{(T+1)^2}\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it}\eta _{is}\eta _{it}\\&\quad +\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \frac{12}{(T+1)^2}\underset{k\not =l}{\sum \sum } \left( I( u_{ik}\le u_{is})-F_i( u_{is})\right) \left( I( u_{il}\le u_{it})-\frac{1}{2}\right) \\&\quad +\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \frac{12}{(T+1)^2}\underset{k\not =l}{\sum \sum } \left( I( u_{ik}\le u_{is})-F_i( u_{is})\right) \eta _{it}\\&\equiv \frac{T(T-1)}{(T+1)^2}\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it}\eta _{is}\eta _{it}+D_{21}+D_{22}. \end{aligned}$$

Then, we have that ${\mathbb {E}}(D^2_{21})=O(T^{-1}N)$ and ${\mathbb {E}}(D^2_{22})=O(T^{-1}N)$, based on which we complete the proof of (10).

Next, we consider the second part $A_{2}$. We have that

$$\begin{aligned}&{\mathbb {E}}(A_{2}|\mathbf{X}_1,\ldots ,\mathbf{X}_N)\\&\quad =\frac{12}{(T+1)^2}\sum _{i=1}^N {\mathbb {E}}_{\mathbf{X}}\bigg \{\underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \sum _{l=1}^T\sum _{k=1}^T(I( u_{il}\le u_{is}+(\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))\\&\qquad -I( u_{il}\le u_{is}))\times I( u_{ik}\le u_{it}+(\mathbf{X}_{it}-\mathbf{X}_{it})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))-I( u_{ik}\le u_{it}))\bigg \}\\&\quad =\frac{12}{(T+1)^2}\sum _{i=1}^N{\mathbb {E}}_{\mathbf{X}}\bigg \{\underset{1\le s<t\le T}{\sum \sum } {{\mathbb {E}}\bigg [}{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \sum _{l=1}^T \sum _{k=1}^T(I( u_{il}\le u_{is}+(\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))\\&\qquad -I( u_{il}\le u_{is}))\times I( u_{ik}\le u_{it}+(\mathbf{X}_{it}-\mathbf{X}_{it})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))-I( u_{ik}\le u_{it}))\big |{u_{is}, u_{it}\bigg ]}\bigg \}\\&\quad =\frac{12}{(T+1)^2}\sum _{i=1}^N{\mathbb {E}}_{\mathbf{X}}\bigg \{\underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \sum _{l=1}^T\sum _{k=1}^T(F_i( u_{is}+(\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))-F_i( u_{is}))\\&\qquad \times (F_i( u_{it}+(\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))-F( u_{it}))\bigg \}\\&\quad =\frac{12}{(T+1)^2}\sum _{i=1}^N{\mathbb {E}}_{\mathbf{X}}\bigg \{\underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \sum _{l=1}^T\sum _{k=1}^T\bigg (\int f_i^2(x)dx ((\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))\\&\qquad +\int f_i^{'}(\xi _{isl})f_i(x)dx((\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))^2\bigg )\\&\qquad \times \left( \int f_i^2(x)dx ((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))\right. \\&\qquad \left. +\int f_i^{'}(\xi _{itk})f_i(x)dx((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))^2\right) \bigg \}\\&\quad =\frac{12}{(T+1)^2}\sum _{i=1}^N{\mathbb {E}}_{\mathbf{X}}\bigg \{\underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \sum _{l=1}^T\sum _{k=1}^T\left( \int f_i^2(x)dx\right) ^2\\&\qquad \times (\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE})(\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE})\bigg \}\\&\qquad +\frac{24}{(T+1)^2}\sum _{i=1}^N{\mathbb {E}}_{\mathbf{X}}\bigg \{\underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \sum _{l=1}^T\sum _{k=1}^T\int f_i^2(x)dx\int f_i^{'}(\xi _{itk})f_i(x)dx \\&\qquad \times (\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE})((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))^2\bigg \}\\&\qquad +\frac{12}{(T+1)^2}\sum _{i=1}^N{\mathbb {E}}_{\mathbf{X}}\bigg \{\underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \sum _{l=1}^T\sum _{k=1}^T\left( ((\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))^2\right. \\&\qquad \left. \times ((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE}))^2\right) \\&\qquad \times \left( \int f_i^{'}(\xi _{isl})f_i(x)dx\int f_i^{'}(\xi _{itk})f_i(x)dx\right) \bigg \}\\&\quad \equiv A_{21}+A_{22}+A_{23}, \end{aligned}$$

where ${\mathbb {E}}_{\mathbf{X}}$ denotes the conditional expectation given $\mathbf{X}_1,\ldots ,\mathbf{X}_N$, i.e. ${\mathbb {E}}(\cdot |\mathbf{X}_1,\ldots ,\mathbf{X}_N)$, and $\xi _{itk}$ is a variable between $u_{it}$ and $u_{it}+(\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RFE})$. By (9), we have

$$\begin{aligned} {\mathbb {E}}(A_{21})=&{T}\left\{ \sum _{i=1}^N \tau _i^{-2}\delta _i^\top {\varvec{\Sigma }}_i \delta _i-\sum _{i=1}^N\tau _i^{-2}\delta _i^\top {\varvec{\Sigma }}_i\left( \sum _{i=1}^N \tau _i^{-2}{\varvec{\Sigma }}_i\right) ^{-1}\sum _{i=1}^N\tau _i^{-2}{\varvec{\Sigma }}_i\delta _i\right\} \\&+o(N^{1/2}). \end{aligned}$$

Define $\gamma _i=\left( \delta _i-\left( \sum _{i=1}^N \tau _i^{-2}{\varvec{\Sigma }}_i\right) ^{-1}\sum _{i=1}^N\tau _i^{-2}{\varvec{\Sigma }}_i\delta _i\right) /\tau _i$. We have that

$$\begin{aligned} {\mathbb {E}}(A_{22})&\le c_1\sum _{i=1}^N {\mathbb {E}}\left( T^2{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it}(\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE}))^2\right) \\&\le c_2T\sum _{i=1}^N\sqrt{{\mathbb {E}}\left( (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})^\top {\varvec{\Sigma }}_i(\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})\right) {\mathbb {E}}\left( (\mathbf{X}_{it}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE}))^4\right) }\\&\le c_3\sum _{i=1}^NT(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^{3/2}=o(N^{1/2}), \end{aligned}$$

and

$$\begin{aligned} {\mathbb {E}}(A_{23})&\le c_4 T^2\sum _{i=1}^N {\mathbb {E}}\left( {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it}((\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE}))^2\right. \\&\quad \left. \times ((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE}))^2)\right) \\&\le c_5\sum _{i=1}^NT^2\sqrt{{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2\right) {\mathbb {E}}\left( (\mathbf{X}_{is}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE}))^4(\mathbf{X}_{it}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE}))^4\right) }\\&\le c_6T\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^2=o(N^{1/2}), \end{aligned}$$

where $c_i$, $i=1, \ldots , 6$, are all positive constants that are independent of the samples. Thus, ${\mathbb {E}}(A_{2})=\psi _{RS}+o(N^{1/2})$ under the alternative.

Then, we consider the variance of $A_{2}$. We have that

$$\begin{aligned}&\mathrm {var}(A_{2})\\&\quad =\mathrm {var}\Biggl (\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum } {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} (e_{it}-\xi _{it})(e_{is}-\xi _{is})\Biggr )\\&\quad =O(T^{2})\sum _{i=1}^N{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} )^2 (e_{it}-\xi _{it})^2(e_{is}-\xi _{is})^2\right) \\&\qquad +O(T^{3})\sum _{i=1}^N {\mathbb {E}}\left( {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{ik} (e_{is}-\xi _{is})^2(e_{it}-\xi _{it})(e_{ik}-\xi _{ik})\right) \\&\qquad +O(T^{-1}){\mathbb {E}}^2(A_{2})\\&\quad \equiv B_{1}+B_{2}+o(N). \end{aligned}$$

Similar to the arguments in the calculation of ${\mathbb {E}}(A_{2})$, we have that

$$\begin{aligned} B_{1}=&\,O(T^{2})\sum _{i=1}^N\Big \{{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2((\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right. \\&\left. \times ((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right) \\&+{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2((\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^4\right) \\&+{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2((\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^4((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^4\right) \Big \}\\ \equiv {}&\, O(T^{2})(B_{11}+B_{12}+B_{13}). \end{aligned}$$

Here,

$$\begin{aligned} B_{11}=&\,\sum _{i=1}^N{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2((\mathbf{X}_{is}-\mathbf{X}_{il})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2((\mathbf{X}_{it}-\mathbf{X}_{ik})^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right) \\ =&\,\sum _{i=1}^N{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2({\mathbf{X}}_{is}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2(\mathbf{X}_{it}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right) \\&+2\sum _{i=1}^N{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2(\mathbf{X}_l^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2(\mathbf{X}_{it}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right) \\&+\sum _{i=1}^N{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2(\mathbf{X}_{il}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2(\mathbf{X}_{ik}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right) , \end{aligned}$$

where

$$\begin{aligned}&\sum _{i=1}^N{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2({\mathbf{X}}_{is}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2(\mathbf{X}_{it}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right) \\&\quad =\,\sum _{i=1}^N{\mathbb {E}}(({\mathbf{X}}_{is}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2 E((\mathbf{X}_{it}^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2|{\mathbf{X}}_{is}))\\&\quad =\,O(T^{-2}\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^2),\\&{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2(\mathbf{X}_l^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2(\mathbf{X}_j^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right) \\&\quad =O(T^{-2}\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^2),\\&{\mathbb {E}}\left( ({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it})^2(\mathbf{X}_l^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2(\mathbf{X}_k^\top (\varvec{\beta }_i-{\hat{\varvec{\beta }}}_{RWFE})))^2\right) \\&\quad =O(T^{-2}\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^2). \end{aligned}$$

Hence, $T^2B_{11}=O\left( \sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^2\right) =o(N)$ under the local alternative (8). Taking the same procedure as $B_{11}$, we can show that

$$\begin{aligned}&B_{12}=O\left( T^{-2}\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^3\right) ,~~B_{13}=O\left( T^{-2}\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^4\right) ,\\&B_{2}=O\left( T\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^2\right) . \end{aligned}$$

Thus, under the local alternative (8), $\mathrm {var}(A_{2})=o(N)$, which indicates that $A_{2}=\psi _{RS}+o_p(N^{1/2})$.

Similarly, we have that ${\mathbb {E}}(A_{3})=0$ and

$$\begin{aligned} \mathrm {var}(A_{3})=O(T\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i))+O(T\sum _{i=1}^N(\gamma _i^\top {\varvec{\Sigma }}_i\gamma _i)^2)=o(N), \end{aligned}$$

under the local alternative (8). Thus, we have $A_{3}=o_p(N^{1/2})$.

Finally, we will prove that

$$\begin{aligned} \frac{1}{\sqrt{2pN}}\sum _{i=1}^N \underset{1\le s<t\le T}{\sum \sum }{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it} \eta _{is}\eta _{it} \mathop {\rightarrow }\limits ^{d}N(0, 1). \end{aligned}$$

Define ${\tilde{W}}_{Tt}=\sum _{s=2}^t Z_{Ts}$, where $Z_{Tt}=\sum _{s=1}^{t-1}\sum _{i=1}^N\eta _{is}\eta _{it}{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it}$. Let ${\mathcal {F}}_t=\sigma \{\varvec{\varepsilon }_{1.}, \ldots , \varvec{\varepsilon }_{t.}\}$ be the $\sigma$-field generated by $\{\varvec{\varepsilon }_{s.}, s\le t\}$ and $\varvec{\varepsilon }_{s.}\equiv (u_{1s},\ldots ,u_{Ns})^\top$ for each $s\le t\le T$.

It is easy to show that ${\mathbb {E}}(Z_{Tt}|{\mathcal {F}}_{t-1})=0$ and it follows that $\{{\tilde{W}}_{Ts}, {\mathcal {F}}_s; 2\le s\le T\}$ is a zero mean martingale. Let $v_{Tt}={\mathbb {E}}(Z_{Tt}^2|{\mathcal {F}}_{t-1})$, $2\le t\le T$, and $V_T=\sum _{t=2}^T v_{Tt}$. The central limit theorem (Hall and Heyde 1980) will hold if we can show

$$\begin{aligned} \frac{V_T}{\mathrm {var}({\tilde{W}}_{TT})}\mathop {\rightarrow }\limits ^{p}1 \end{aligned}$$

(11)

and

$$\begin{aligned} \sum _{i=2}^T{\mathbb {E}}(Z_{Tt}^4)=o(N^2). \end{aligned}$$

(12)

In fact, it can be shown that

$$\begin{aligned} v_{Tt}=\sum _{i=1}^N\left( \sum _{s=1}^{t-1}\eta _{is}^2{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{is}+2\sum _{1\le s<k < t}\eta _{is}\eta _{ik} {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{ik}\right) , \end{aligned}$$

which results in

$$\begin{aligned} \frac{V_n}{\mathrm {var}({\tilde{W}}_{nn})}={}&\frac{2}{T(T-1)pN}\sum _{i=1}^N\left\{ \sum _{s=1}^{T-1}s\eta _{is}^2{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{is}+2\sum _{1\le s<k\le T}\eta _{is}\eta _{ik}{\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{ik}\right\} \\ \equiv&C_{1}+C_{2}. \end{aligned}$$

Simple algebras lead to ${\mathbb {E}}(C_{1})=1$ and

$$\begin{aligned} \mathrm {var}(C_{1})=\frac{4}{T^2(T-1)^2p^2N^2}\sum _{i=1}^N {\mathbb {E}}\left( \sum _{s=1}^{T-1}s^2 (\eta _{is}^4({\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{is})^2-p)\right) =O(T^{-1}N^{-1}). \end{aligned}$$

Hence, $\mathrm {var}(C_{1})\rightarrow 0$ and then $C_{1}\mathop {\rightarrow }\limits ^{p}1$. Similarly, ${\mathbb {E}}(C_{2})=0$ and

$$\begin{aligned} \mathrm {var}(C_{2})&=\frac{8}{T^2(T-1)^2p^2N^2}\sum _{i=1}^N\left( \sum _{s=3}^T \frac{s(s-1)}{2}+\sum _{s=3}^{T-1}\frac{s(T-s)(s-1)}{2}\right) \\&=O(N^{-1}) \rightarrow 0, \end{aligned}$$

which implies that $C_{2} \mathop {\rightarrow }\limits ^{p}0$. Thus, (11) holds. On the other hand,

$$\begin{aligned} \sum _{t=2}^T {\mathbb {E}}(Z_{Tt}^4)=\sum _{t=2}^T {\mathbb {E}}\left( \left( \sum _{i=1}^N\sum _{s=1}^{t-1}\eta _{is}\eta _{it} {\tilde{\mathbf{X}}}_{is}^\top {\tilde{\mathbf{X}}}_{it}\right) ^4\right) , \end{aligned}$$

which can be decomposed as $3Q+P$ with

$$\begin{aligned} Q&=O(1)\sum _{i=1}^N\sum _{t=2}^T \sum _{k\not =l}^{t-1}E\left( {\tilde{\mathbf{X}}}_{it}^\top {\tilde{\mathbf{X}}}_{ik}{\tilde{\mathbf{X}}}_{ik}^\top {\tilde{\mathbf{X}}}_{it}{\tilde{\mathbf{X}}}_{it}^\top {\tilde{\mathbf{X}}}_{il}{\tilde{\mathbf{X}}}_{il}^\top {\tilde{\mathbf{X}}}_{it}\right) =O(T^{-1}N), \\ P&=O(1)\sum _{i=1}^N\sum _{t=2}^T \sum _{s=1}^{t-1}E\left( ({\tilde{\mathbf{X}}}_{it}^\top {\tilde{\mathbf{X}}}_{is})^4\right) =O(T^{-2}N). \end{aligned}$$

Thus, we have $Q=o(N^2)$ and $P=o(N^2)$. Hence (12) holds. These complete the proof of Theorem 1.