Testing slope homogeneity in panel data models with a multifactor error structure

Abstract

Based on the common correlated effects (CCE) method and the Lagrange multiplier (LM) principle, this paper proposes a slope homogeneity test in a panel data model with a multifactor error structure that allows unobserved factors to be correlated with explanatory variables. The CCE method is first used to transform the regression equation to control for the unobserved common factors. Then, we adopt the idea of an LM-type test to conduct a homogeneity test. Our asymptotic analysis indicates that the test statistic is asymptotically normally distributed under the null hypothesis of homogeneity, regardless of the errors’ normality or homoskedasticity, as both N and T go to infinity, with $T^{2/3}N^{-1}\rightarrow 0$ and $T^{2}N^{-1}\rightarrow \infty$. It is also proved that the test is asymptotically powerful under a sequence of Pitman local alternatives. Monte Carlo simulations indicate that the test has good finite sample properties for all combinations of N and T, with the exception of a large N / T. The simulation results also suggest that the proposed test is robust to the errors’ non-normality and conditional heteroskedasticity.

Appendices

Appendix 1: Auxiliary lemmas

Lemma 1

Suppose that $\left\| {\varvec{\beta } _i } \right\| <K$ for each i, and Assumptions A1–A4 hold. Then

• (i) ${\bar{{\varvec{H}}}}'\bar{{\varvec{H}}}/T=\bar{{{\varvec{P}}'}}{\varvec{G}}'G\bar{{P}}/T+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$

• (ii) ${\bar{{\varvec{H}}}}'\varvec{F}/T=\bar{{{\varvec{P}}'}}({\varvec{G}}'\varvec{F}/T)+\mathrm{O}_p ((NT)^{-1/2})$;

• (iii) ${\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}/T={\varvec{X}}'_i \bar{{\varvec{M}}}_q F/T+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$

• (iv) $\varvec{Q}_{\textit{iT}} ={X}'_i \bar{{\varvec{M}}}_q X_i /T+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$,

• (v) $\varvec{\xi } _{\textit{iT}} =T^{-1/2}{\varvec{X}}'_i \bar{{\varvec{M}}}_{\varvec{q}} \varvec{v}_{\varvec{i}} +\mathrm{O}_p (T^{1/2}N^{-1})$

• (vi) $\bar{{\varvec{\varepsilon }}}^{*\prime }v_i /T =\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$.

Proof

These results are directly from Eqs. (36)–(38), (43), (44) and (A.13) in Pesaran (2006). $\square$

Lemma 2

Suppose that Assumptions A1–A5 hold, then

• (i) ${\varvec{\lambda }}'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i /T=\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$

• (ii) ${\varvec{\lambda }}'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{v}_i /T=\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$;

• (iii) ${\varvec{v}}'_i \bar{{\varvec{M}}}\varvec{v}_i /T={\varvec{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i /T+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$;

• (iv) $T^{-1/2}{\varvec{X}}'_{\varvec{i}} \bar{{\varvec{M}}}_{\varvec{q}} \varvec{v}_{\varvec{i}} =\mathrm{O}_p (1)$ and $(NT)^{-1/2} \sum \nolimits _{i=1}^{N} {{\varvec{X}}'_{\varvec{i}} \bar{{\varvec{M}}}_{\varvec{q}} \varvec{v}_{\varvec{i}}} =\mathrm{O}_p (1)$.

Proof

• (i) First noting that $\bar{{\varvec{H}}}=G\bar{{\varvec{P}}}+\bar{{\varvec{\varepsilon } }}^{*}$ and $\bar{{\varvec{M}}}_{\varvec{q}} =\varvec{I}_T -G\bar{{\varvec{P}}}({\bar{{\varvec{P}}}}'{\varvec{G}}'\varvec{G}\bar{{\varvec{P}}})^{-}{\bar{{\varvec{P}}}}'{\varvec{G}}'$, we have ${\varvec{\lambda }}'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i /T={\varvec{\lambda }}'_i {\varvec{F}}'\varvec{F}\varvec{\lambda }_i /T-{\varvec{\lambda }}'_i {\varvec{F}}'\bar{{\varvec{H}}}({\bar{{\varvec{H}}}}'\bar{{H}})^{-}{\bar{{H}}}'F\lambda _i /T= {\lambda }'_i {F}'\bar{{\varvec{M}}}_q \varvec{F}\varvec{\lambda } _i /T+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT))^{-1/2},$ where the last equality uses Lemma 1 (i) and (ii). When Assumption A5 is satisfied, using the results on generalized inverse, $\bar{{\varvec{M}}}_q =\varvec{M}_g$. Since $\varvec{F}\subset \varvec{G},$ then $\bar{{\varvec{M}}}_q \varvec{F}=\varvec{M}_g \varvec{F}=0$. Therefore $\varvec{{\lambda }}'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i /T=\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$.

• (ii) Since ${\varvec{\bar{{H}}}}'\varvec{v}_i /T={\bar{{\varvec{P}'}}{G}'v}_i /T+{\bar{\varvec{\varepsilon }}^{*\prime }v}_i /T$, then ${\varvec{\bar{{H}}}'v_i} /T={\bar{{\varvec{P}'}}{G}'v}_i /T+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})$ follows from Lemma 1 (vi). Using the same line of reasoning as in Lemma 2 (i), the result is proved to be true.

• (iii) The result can be shown in the same way as (i) and (ii).

• (iv) Since $v_i$ is independent of $\bar{{\varvec{M}}}_q$ and $\varvec{X}_i$, then $E\left( {T^{-1/2}\varvec{{X}}'_i \bar{{\varvec{M}}}_q v_i } \right) =0$ and $E\left( {T^{-1/2}\varvec{{X}}'_i \bar{{\varvec{M}}}_q v_i } \right) ^{2}=T^{-1}\sum \nolimits _{t=1}^T {a_{\textit{it}} E(v_{\textit{it}}^2 } )=\mathrm{O}_p (1)$ where $a_{\textit{it}}$’s are coefficients. Therefore, $T^{-1/2}\varvec{{X}}'_i \bar{{\varvec{M}}}_q v_i =\mathrm{O}_p (1)$. Then it is easily to prove that $(NT)^{-1/2}\sum \nolimits _{i=1}^N {\varvec{{X}}'_i \bar{{\varvec{M}}}_q v_i } =\mathrm{O}_p (1)$.

$\square$

Lemma 3

Suppose that Assumptions A1–A5 hold. Then under $\varvec{H}_0$, we have

• (i) $\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}} _i^2 ={v}'_i \bar{{\varvec{M}}}_q v_i /(T-k-2)+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2})=\sigma _i^2 +\mathrm{O}_p (T^{-1/2})+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((TN)^{-1/2})$;

• (ii) $N^{-1/2}\sum \nolimits _{i=1}^N {\varvec{{\xi }}'_{\textit{iT}} \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}} /\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^2 =N^{-1/2}\sum \nolimits _{i=1}^N {\tilde{z}_{\textit{iT}} } +\mathrm{O}_p (N^{1/2}T^{-1})+\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (N^{-1/2^{}})+\mathrm{O}_p (T^{1/2}N^{-1})+\mathrm{O}_p (T^{-1/2})$,

where $\tilde{z}_{\textit{iT}} =(T-r-1)\varvec{{v}}'_i \varvec{D}_i \varvec{v}_i /\varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i$ with $\varvec{D}_i =\bar{{\varvec{M}}}_q \varvec{X}_i (\varvec{{X}}'_i \bar{{\varvec{M}}}_q \varvec{X}_i )^{-1}\varvec{{X}}'_i \bar{{\varvec{M}}}_q$.

Proof

• (i) According to the definition of $\overset{\scriptscriptstyle \smile }{{\varvec{U}}} _i$ in Sect. 2.3, it is easily seen that

  $$\begin{aligned} \overset{\scriptscriptstyle \smile }{{\varvec{U}}} {\prime }_i \overset{\scriptscriptstyle \smile }{{\varvec{U}}} _i= & {} \left( {\bar{{\varvec{M}}}X_i (\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )+\bar{{\varvec{M}}}(F\lambda _i +v_i )} \right) ^{\prime }\\&\quad \times \,\left( {\bar{{\varvec{M}}}X_i (\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )+\bar{{\varvec{M}}}(F\lambda _i +v_i )} \right) \\= & {} \left( {T(\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} {)}'Q_{\textit{iT}} (\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )} \right. +2(\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} {)}'T^{1/2}\xi _{\textit{iT}} \\&\quad +\,2(\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} {)}' {X}'_i \bar{{\varvec{M}}}F\lambda +{v}'_i \bar{{\varvec{M}}}v_i +{\lambda }'_i F^{\prime }\bar{{\varvec{M}}}F\lambda _i +\left. {2{\lambda }'_i F^{\prime }\bar{{\varvec{M}}}v_i } \right) . \end{aligned}$$

  Under $\varvec{H}_0$,

  $$\begin{aligned} \beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} =(NT)^{-1/2}Q_{NT}^{-1} \xi _{NT} +(NT)^{-1}Q_{NT}^{-1} \sum \nolimits _{i=1}^N {{X}'_i \bar{{\varvec{M}}}F\lambda _i } . \end{aligned}$$

  By Lemmas 1 (v) and 2 (iv),

  $$\begin{aligned} \varvec{\xi } _{NT} =\mathrm{O}_p (1)+\mathrm{O}_p (T^{1/2}N^{-1/2}). \end{aligned}$$

  Combining Assumptions A4–A5 and Lemma 1 (iii), we have

  $$\begin{aligned} \beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} =\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2}). \end{aligned}$$

  Using Lemmas 1 and 2, we establish that

  $$\begin{aligned} \overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}} _i^2 =\overset{\scriptscriptstyle \smile }{{\varvec{U}}} {\prime }_i \overset{\scriptscriptstyle \smile }{{\varvec{U}}} _i /(T-k-2)={v}'_i \bar{{\varvec{M}}}_q v_i /(T-k-2)+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((NT)^{-1/2}). \end{aligned}$$

  In addition, we can further develop the relationship between $\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}} _i^2$ and $\varvec{\sigma }_i^2$ . Since $\varvec{v}_i$ is independent of $\bar{{\varvec{M}}}_q$, combining the law of iterated expectation and Theorem 1 in Bao and Ullah (2010), we have

  $$\begin{aligned} E(\varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i )= & {} \sigma _i^2 Tr(\bar{{\varvec{M}}}_q )=\sigma _i^2 (T-r-1),\\ E(\varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i )^{2}= & {} E(\varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i )E(\varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i )+2Tr(\bar{{\varvec{M}}}_q \bar{{\varvec{M}}}_q )\\= & {} \sigma _i^4 (T-r-1)^{2}+2(T-r-1). \end{aligned}$$

  Hence, for fixed k,

  $$\begin{aligned} E[\varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i /(T-k-2)-\sigma _i^2 ]^{2} =\mathrm{O}_p (T^{-1}), \end{aligned}$$

  which implies

  $$\begin{aligned} \varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i /(T-k-2)=\sigma _i^2 +\mathrm{O}_p (T^{-1/2}). \end{aligned}$$

  Therefore

  $$\begin{aligned} \overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}} _i^2 =\sigma _i^2 +\mathrm{O}_p (T^{-1/2})+\mathrm{O}_p (N^{-1})+\mathrm{O}_p ((TN)^{-1/2}). \end{aligned}$$

• (ii) Combining Lemmas 1 and 3, we get

  $$\begin{aligned} N^{-1/2}\sum \nolimits _{i=1}^N{\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{{\xi }}'_{\textit{iT}} \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}}= & {} N^{-1/2}(T-k-2)/(T-r-1)\\&\quad \times \,\sum \nolimits _{i=1}^N (T-r-1)\varvec{{\xi }}'_{\textit{iT}} \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}} /(\varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i )\\&\quad +\,\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{-1/2})\\= & {} N^{-1/2}\sum \nolimits _{i=1}^N {(T-r-1)\varvec{{\xi }}'_{\textit{iT}} \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}} /(\varvec{{v}}'_i \bar{{\varvec{M}}}_q \varvec{v}_i )}\\&\quad +\,\mathrm{O}_p (N^{1/2}T^{-1})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{-1/2})\\= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\tilde{z}_{\textit{iT}} } +\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (T^{1/2}N^{-1})\\&\quad +\,\mathrm{O}_p (N^{1/2}T^{-1})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{-1/2}). \end{aligned}$$

  The first equality follows from Lemma 3 (i), and the last equality follows from Lemma 1 (iv)–(v).$\square$

Lemma 4

Suppose that Assumptions A1–A5 hold, then under $\varvec{H}_0$

$$\begin{aligned} N^{-1/2}S_{lm}^f= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\tilde{z}_{\textit{iT}} } +\mathrm{O}_p (N^{1/2}T^{-1})+\mathrm{O}_p (TN^{-3/2})\\&+\,\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{1/2}N^{-1})+\mathrm{O}_p (T^{-1/2}). \end{aligned}$$

Proof

According to the definition of $S_{lm}^f$ in Sect. 2.3, we have

$$\begin{aligned} N^{-1/2}S_{lm}^f= & {} T^{-1}N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big ( {X_i (\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}}} _{CCEP} )+\varvec{F}\varvec{\lambda } _i +\varvec{v}_i \big )^{\prime }\nonumber \\&\times \,\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} \varvec{{X}}'_i \bar{{\varvec{M}}}\big ( {\varvec{X}_i (\beta _i-\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}}} _{CCEP} )+\varvec{F}\varvec{\lambda } _i +\varvec{v}_i \big )\nonumber \\= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} [\varvec{{\xi }}'_{\textit{iT}} \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}} +T(\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )'\varvec{Q}_{\textit{iT}} (\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP})\nonumber \\&+\,T^{-1}\varvec{\lambda } _i ^{\prime }\varvec{{F}}'\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} \varvec{{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i\nonumber \\&+\,2(\varvec{\beta } _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )'\varvec{{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i\nonumber \\&+\,2T^{1/2}(\varvec{\beta } _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )'\varvec{\xi } _{\textit{iT}} +2T^{-1/2}{\varvec{\lambda } }'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}} ]. \end{aligned}$$

(A.1)

For the last five terms in the above equality, first note $\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}} _i^2 =\mathrm{O}_p (1)$, $\varvec{Q}_{\textit{iT}} =\mathrm{O}_p (1)$, and $\varvec{\xi } _{NT} =\mathrm{O}_p (1)+\mathrm{O}_p (T^{1/2}N^{-1/2})$. Using Lemmas 1 and 3, it follows that

$$\begin{aligned}&N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} T(\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP})'\varvec{Q}_{\textit{iT}} (\varvec{\beta } _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )\\&\qquad \qquad \qquad =\,\mathrm{O}_p (TN^{-3/2})\nonumber +\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{1/2}N^{-1}), \end{aligned}$$

(A.2)

$$\begin{aligned}&N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} T^{-1}{\varvec{\lambda } }'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} \varvec{{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i\nonumber \\&\qquad \qquad \qquad =\,\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{1/2}N^{-1}),\nonumber \\&N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}}_i^{-2} (\varvec{\beta } _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )'\varvec{{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i\nonumber \\&\qquad \qquad \qquad =\,\mathrm{{O}}_p (TN^{-3/2})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{1/2}N^{-1}), \end{aligned}$$

(A.3)

$$\begin{aligned}&N^{-1/2}\sum \nolimits _{i=1}^N {T^{1/2}\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} (\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}}_{CCEP} )'\varvec{\xi } _{\textit{iT}}\nonumber \\&\qquad \qquad \qquad =\,\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p(N^{-1/2})+\mathrm{O}_p (T^{1/2}N^{-1}), \end{aligned}$$

(A.4)

$$\begin{aligned}&N^{-1/2}\sum \nolimits _{i=1}^N {T^{-1/2}\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\lambda }}'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}}\\&\qquad \qquad \qquad =\,\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{1/2}N^{-1}). \end{aligned}$$

Using Lemma 3 (ii) and the above five equations, we have

$$\begin{aligned} N^{-1/2}S_{lm}^f= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\tilde{z}_{\textit{iT}} } +\mathrm{O}_p (N^{1/2}T^{-1})+\mathrm{O}_p (TN^{-3/2})\\&\quad +\,\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{1/2}N^{-1})+\mathrm{O}_p (T^{-1/2}). \end{aligned}$$

Lemma 5

Let $\varvec{\Phi }$ be a $T\times T$ symmetric matrix and $\varvec{\Gamma }$ a positive definite $T\times T$ matrix, and suppose that $\varvec{\upsilon }\sim {\textit{IID}}(0,\varvec{I}_T )$, $\varvec{\upsilon }=(\varvec{\upsilon }_1 ,\varvec{\upsilon }_2 ,\ldots ,\varvec{\upsilon }_T )'$. Denote the pth cumulant of $\varvec{{\upsilon }'\varvec{\Gamma } \varvec{\upsilon }}$ by $\kappa _p$, and the $\gamma =1+m$ order, $\delta =r+m$ degree generalized cumulant of $(\varvec{{\upsilon }'\Phi \upsilon })^{r}$ and $\varvec{{\upsilon }'\Gamma \upsilon }$ by $\kappa _{rm}$ and assume that the following conditions hold:

Condition A: for $p=1,2,\ldots ,\kappa _p =\mathrm{O}(T)$; Condition B: $r=1,2,\ldots ,\kappa _{r0} =E(\varvec{{\upsilon }'\Phi \upsilon })^{r}=\mathrm{O}(T^{r})$;

Condition C: $r,m=1,2,\ldots ,\kappa _{rm} =\mathrm{O}(T^{l})$ with $l\le r$.

Then the Laplace approximate expansion for the rth moment of $\varvec{{\upsilon }'\Phi \upsilon }/\varvec{{\upsilon }'\Gamma \upsilon }$ is given by

$$\begin{aligned} E(\varvec{{\upsilon }'\Phi \upsilon }/\varvec{{\upsilon }'\Gamma \upsilon })^{r}=E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}]/[E(\varvec{{\upsilon }'\Gamma \upsilon })]^{r}+\phi _{1T} +\phi _{2T} +\mathrm{O}_p (T^{-3}), \end{aligned}$$

where $\phi _{1T} =\frac{r(r+1)}{2}\left[ {\frac{E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}]\kappa _2 }{[E(\varvec{{\upsilon }'\Gamma \upsilon })]^{r+2}}} \right] -r\frac{\kappa _{r1} }{[E(\varvec{{\upsilon }'\Gamma \upsilon })]^{r+1}}$,

$$\begin{aligned} \phi _{2T}= & {} \frac{r(r+1)}{2}\frac{\kappa _{r2} }{[E({\upsilon }'\Gamma \upsilon )]^{r+2}}-\frac{r(r+1)(r+2)}{2}\left[ {\frac{3E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}]\kappa _3 +\kappa _{r1} \kappa _{r2} }{[E(\varvec{{\upsilon }'\Gamma \upsilon })]^{r+3}}} \right] \\&\quad +\,\frac{r(r+1)(r+2)(r+3)}{8}\left[ {\frac{E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}]\kappa _2^2 }{[E(\varvec{{\upsilon }'\Gamma \upsilon })]^{r+4}}} \right] , \end{aligned}$$

And $\kappa _{r1} =E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}\varvec{{\upsilon }'\Gamma \upsilon }]-E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}]E(\varvec{{\upsilon }'\Gamma \upsilon })$,

$$\begin{aligned} \kappa _{r2}= & {} E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}({\varvec{\upsilon }'\Gamma \upsilon })^{2}]-2E(\varvec{{\upsilon }'\Gamma \upsilon })E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}(\varvec{{\upsilon }'\Gamma \upsilon })]\\&\quad -\,E[(\varvec{{\upsilon }'\Gamma \upsilon })^{2}]E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}]+2[E(\varvec{{\upsilon }'\Gamma \upsilon })]^{2}E[(\varvec{{\upsilon }'\Phi \upsilon })^{r}]. \end{aligned}$$

Proof

See Lieberman (1994). $\square$

Remark

Pesaran and Yamagata (2008) extends the results that allow $\varvec{\Gamma }$ being a semi-positive definite matrix.

Appendix 2: Proof of Theorems 1–2 and Corollary 1

Proof of Theorem 1

We first consider $E(\tilde{z}_{\textit{iT}} )$ and $Var(\tilde{z}_{\textit{iT}}$ ). By definition, $\bar{{\varvec{M}}}_q$ and $\varvec{D}_i$ are all idempotent matrices and independent of $v_i$. Using results of Bao and Ullah (2010), we have $E[v_i ^{\prime }\varvec{D}_i \varvec{v}_i /\sigma _i^2 ]=Tr(\varvec{D}_i )=k$, $E[v_i ^{\prime }\bar{{\varvec{M}}}_q \varvec{v}_i /\sigma _i^2 ]=Tr(\bar{{\varvec{M}}}_q )=T-r-1$. Conditions A, B and C of Lemma 5 are satisfied. After some tedious algebra, we have $\phi _{i1T} =\mathrm{O}_p (T^{-1})$, and $\phi _{i2T} =\mathrm{O}_p (T^{-2})$. Hence, $E(\tilde{z}_{\textit{iT}} )=k+\mathrm{O}_p (T^{-1})$. Similarly, we can obtain $Var(\tilde{z}_{\textit{iT}} )=2k+\mathrm{O}_p (T^{-1})$. Then by Lemma 4,

$$\begin{aligned} LM_s^c= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\left( {\tilde{z}_{\textit{iT}} -E(\tilde{z}_{\textit{iT}} )+E(\tilde{z}_{\textit{iT}} )-k} \right) /\sqrt{2k}} +\mathrm{O}_p (N^{1/2}T^{-1})\\&\quad +\,\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{-1/2})+\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (T^{1/2}N^{-1})\\= & {} N^{-1/2}\sum \nolimits _{i=1}^N {(\tilde{z}_{\textit{iT}} -E(\tilde{z}_{\textit{iT}} ))/\sqrt{2k}} +\mathrm{O}_p (N^{1/2}T^{-1})+\mathrm{O}_p (N^{-1/2})\\&\quad +\,\mathrm{O}_p (T^{-1/2})+\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (T^{1/2}N^{-1}). \end{aligned}$$

Applying central limit theorem (CLT) to the first term of the last equality above, we finally establish that

$LM_s^c \Rightarrow N(0,1),$ as $(N,T)\rightarrow \infty$ such that $T^{2/3}N^{-1}\rightarrow 0$ and $T^{2}N^{-1}\rightarrow \infty$. $\square$

Proof of Theorem 2

For the modified PY test, under $\varvec{H}_0$, we have

$$\begin{aligned} \overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{i,{\textit{CCE}}} -\tilde{\beta }_{\textit{WCCE}}= & {} T^{-1/2}\varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}} +T^{-1}\varvec{Q}_{\textit{iT}}^{-1} \varvec{{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i-\left( \sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{Q}_{\textit{iT}}\right) ^{-1}\\&\quad \times \,\left( {T^{-1/2}\sum \nolimits _{i=1}^N \overset{\scriptscriptstyle \frown }{{\varvec{\sigma }}} _i^{-2} \varvec{\xi } _{\textit{iT}} } +T^{-1}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i \right) . \end{aligned}$$

Use the definition of $\tilde{S}_{cce}$, it is easily seen that

$$\begin{aligned} N^{-1/2}\tilde{S}_{cce}= & {} \sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \left( {\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}}} _{i,CCE} -\tilde{\beta }_{WCCE} \right) ^{\prime }\varvec{{X}}'_i \bar{{\varvec{M}}}\varvec{X}_i \left( {\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}}} _{i,CCE} -\tilde{\beta }_{WCCE}\right) \\= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{{\xi }}'_{\textit{iT}} \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}} -N^{-1/2}T^{-1}\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\lambda } }'_i \varvec{{F}}'\bar{{\varvec{M}}}\varvec{X}_i } \right) \\&\quad \times \,\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{Q}_{\textit{iT}} } \right) ^{-1}\left( \sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i \right) \\&\quad +\,N^{-1/2}\sum \nolimits _{i=1}^N {T^{-1}\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\lambda } }'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} \varvec{{X}}'_i \bar{{\varvec{M}}}\varvec{F}\lambda _i -N^{-1/2}\\&\quad \times \,\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\xi } }'_{\textit{iT}} } \right) \left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{Q}_{\textit{iT}} } \right) ^{-1}\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{\xi } _{\textit{iT}} } \right) \\&\quad +\,2N^{-1/2}\sum \nolimits _{i=1}^N {T^{-1/2}{\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\lambda } }'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} \varvec{\xi } _{\textit{iT}} } +2(TN)^{-1/2}\\&\quad \times \,\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\xi } }'_{\textit{iT}} } \right) \left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{Q}_{\textit{iT}} } \right) ^{-1}\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i } \right) \end{aligned}$$

Combining Lemmas 1–3, it follows that

$$\begin{aligned}&N^{-1/2}T^{-1}\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\lambda } }'_i {\varvec{F}}'\bar{{\varvec{M}}}\varvec{X}_i } \right) \left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{Q}_{\textit{iT}} } \right) ^{-1} \left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i } \right) \\&\qquad \qquad \qquad =\,\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (T^{1/2}N^{-1})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{-1/2}),\\&N^{-1/2}\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\xi } }'_{\textit{iT}} } \right) \left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{Q}_{\textit{iT}} } \right) ^{-1}\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{\xi } _{\textit{iT}} } \right) \\&\qquad \qquad \qquad =\,\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (T^{1/2}N^{-1})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{-1/2}),\\&(TN)^{-1/2}\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{\xi } }'_{\textit{iT}} } \right) \left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{Q}_{\textit{iT}} } \right) ^{-1}\left( {\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i } \right) \\&\qquad \qquad \qquad =\,\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{-1/2}). \end{aligned}$$

Using Eqs. (A.1)–(A.4) and the above three equations, we obtain

$$\begin{aligned} \tilde{\Delta }_{adj}^{cce} -LM{}_s^c= & {} (N^{-1/2}\tilde{S}_{cce} -N^{-1/2}S_{lm}^f )/\sqrt{2k}=\mathrm{O}_p (TN^{-3/2})\\&\quad +\,\mathrm{O}_p (T^{1/2}N^{-1})+\mathrm{O}_p (N^{-1/2})+\mathrm{O}_p (T^{-1/2}). \end{aligned}$$

$\square$

Proof of Corollary 1

Under $H_{1,NT}$,

$$\begin{aligned} \overset{\scriptscriptstyle \smile }{{\varvec{U}}} _i =\bar{{\varvec{M}}}\varvec{Y}_i -\bar{{\varvec{M}}}\varvec{X}_i \overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} =\bar{{\varvec{M}}}\varvec{X}_i (\beta _i -\overset{\scriptscriptstyle \frown }{{\varvec{\beta }}} _{CCEP} )+\bar{{\varvec{M}}}(\varvec{F}\varvec{\lambda } _i +v_i )=\varvec{\gamma } _{iNT} +\varvec{\chi } _{iNT} , \end{aligned}$$

where $\gamma _{iNT} =\bar{{\varvec{M}}}\varvec{v}_i +\bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i -(NT)^{-1}\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{NT}^{-1} \sum \nolimits _{i=1}^N {({\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{v}_i +{\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i )}$, and $\varvec{\chi } _{iNT} =N^{-1/4}T^{-1/2}\bar{{\varvec{M}}}\varvec{X}_i \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]$ with $\tilde{\varvec{\delta } }_x =N^{-1}\sum \nolimits _{i=1}^N {\varvec{Q}_{\textit{iT}} \varvec{\delta } _i } =O_p (1)$. Then

$$\begin{aligned} N^{-1/2}S_{lm}^f= & {} T^{-1}N^{-1/2}\sum \nolimits _{i=1}^N {(\varvec{\gamma } _{iNT} +\varvec{\chi } _{iNT} )^{\prime }\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} {\varvec{X}}'_i \bar{{\varvec{M}}}(\varvec{\gamma } _{iNT} +\varvec{\chi } _{iNT} )\\= & {} T^{-1}N^{-1/2}\sum \nolimits _{i=1}^N {\varvec{\gamma } _{iNT} ^{\prime }\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{\gamma } _{iNT}\\&\quad +\,T^{-1}N^{-1/2}\sum \nolimits _{i=1}^N {\varvec{\chi } _{iNT} ^{\prime }\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{\chi } _{iNT} \quad \\&\quad +\,2T^{-1}N^{-1/2}\sum \nolimits _{i=1}^N {\varvec{\chi } _{iNT} ^{\prime }\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{\gamma } _{iNT}. \end{aligned}$$

For the first term in the above equality, we use the same line of reasoning in the proof of Lemma 4 and establish that

$$\begin{aligned} T^{-1}N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{\gamma } _{iNT} ^{\prime }\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{\gamma } _{iNT} ^{\prime }= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\tilde{\varvec{z}}_{\textit{iT}}}+\mathrm{O}_p (N^{1/2}T^{-1})\\&\quad +\,\mathrm{O}_p (TN^{-3/2})+\mathrm{O}_p (N^{-1/2})\\&\quad +\,\mathrm{O}_p(T^{1/2}N^{-1})+\mathrm{O}_p (T^{-1/2}). \end{aligned}$$

For the second term,

$$\begin{aligned} T^{-1}N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{\chi } _{iNT} ^{\prime }\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{\chi } _{iNT}= & {} N^{-1}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]'\\&\quad \times \,\varvec{Q}_{\textit{iT}} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]\\\ge & {} 0. \end{aligned}$$

And for the last term, using Lemmas 1 (v) and 2 (iv), we get

$$\begin{aligned}&T^{-1}N^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \varvec{\chi } _{iNT} ^{\prime }\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{\textit{iT}}^{-1} {\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{\gamma } _{iNT}\\&\qquad \qquad \qquad =\,N^{-3/4}T^{-1/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]'{\varvec{X}}'_i \bar{{\varvec{M}}}[\bar{{\varvec{M}}}\varvec{v}_i +\bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i\\&\qquad \qquad \qquad \quad -\,(NT)^{-1}\bar{{\varvec{M}}}\varvec{X}_i \varvec{Q}_{NT}^{-1} \sum \nolimits _{i=1}^N {({\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{v}_i +{\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i )} ]\\&\qquad \qquad \qquad =N^{-3/4}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]'\varvec{\xi } _{\textit{iT}} -N^{-7/4}\\&\qquad \qquad \qquad \quad \times \,\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]'\bigg [\varvec{Q}_{\textit{iT}} \varvec{Q}_{NT}^{-1} \sum \nolimits _{i=1}^N {\varvec{\xi } _{\textit{iT}} } \bigg ]\\&\qquad \qquad \qquad \quad +\,N^{-3/4}T^{3/2}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta }}_x \big ]'{\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i -N^{-7/4}T^{3/2}\\&\qquad \qquad \qquad \quad \times \,\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta }}_x \big ]'\bigg [\varvec{Q}_{\textit{iT}} \varvec{Q}_{NT}^{-1} \sum \nolimits _{i=1}^N {{\varvec{X}}'_i \bar{{\varvec{M}}}\varvec{F}\varvec{\lambda } _i } \bigg ]\\&\qquad \qquad \qquad =\,O_p (N^{-1/4})+O_p (N^{-3/4}T^{1/2}). \end{aligned}$$

Therefore

$$\begin{aligned} N^{-1/2}S_{lm}^f= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\tilde{z}_{\textit{iT}} } +N^{-1}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]'\\&\times \,\varvec{Q}_{\textit{iT}} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]+\mathrm{O}_p (N^{1/2}T^{-1})+\mathrm{O}_p (N^{-3/2}T)+\mathrm{O}_p (T^{-1/2})\\&+\,O_p (N^{-1/4})+O_p (N^{-3/4}T^{1/2}), \end{aligned}$$

and

$$\begin{aligned} LM_s^c= & {} N^{-1/2}\sum \nolimits _{i=1}^N {\left( {\tilde{z}_{\textit{iT}} -E(\tilde{z}_{\textit{iT}} )+E(\tilde{z}_{\textit{iT}} )-k} \right) \big /\varvec{\sqrt{2k}}} +N^{-1}\\&\times \,\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]'\varvec{Q}_{\textit{iT}} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ] /\varvec{\sqrt{2k}}\\&+\,\mathrm{O}_p (N^{1/2}T^{-1})+\mathrm{O}_p (N^{-3/2}T)+\mathrm{O}_p (T^{-1/2})+O_p (N^{-1/4})\\&+\,O_p (N^{-3/4}T^{1/2}). \end{aligned}$$

Finally, apply the CLT to the first term of the last equality above and conclude that $LM_s^c \Rightarrow N(\psi _{NT} /\varvec{\sqrt{2k}},1),$ as $(N,T)\rightarrow \infty$ such that $T^{2/3}N^{-1}\rightarrow 0$ and $T^{2}N^{-1}\rightarrow \infty$, where $\psi _{NT} =\mathop {\lim }\limits _{N\rightarrow \infty } N^{-1}\sum \nolimits _{i=1}^N {\overset{\scriptscriptstyle \smile }{{\varvec{\sigma }}}} _i^{-2} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ]'\varvec{Q}_{\textit{iT}} \big [\varvec{\delta } _i -\varvec{Q}_{NT}^{-1} \tilde{\varvec{\delta } }_x \big ] \ge 0$. $\square$