Skewness-based test diagnosis of technical inefficiency in spatial autoregressive stochastic frontier models

Abstract

In the Spatial Autoregressive (SAR) Stochastic Frontier (SF) model, the inefficiency term, which distinguishes it from the SAR model, can capture the effects of technical inefficiency. To determine whether inefficiency significantly exists in the cross-sectional SARSF model, this paper proposes a skewness-based test. This test does not rely on the normality assumption for the disturbances and allows inefficiency to follow an unknown one-sided distribution. We establish the asymptotic theory of the test statistic under spatial near-epoch dependent properties. Furthermore, we extend this test to the panel SARSF data model, accounting for both individual and time fixed-effects. Additionally, Monte Carlo simulations demonstrate the robustness of our test against non-normal disturbances and its satisfactory performance with different one-sided distributions for inefficiency. Finally, we provide an empirical application using data from 137 dairy farms in Northern Spain to illustrate the presence of technical inefficiency in production according to our test.

Appendices

Appendix A. Useful definition and lemma

This appendix lists some useful definitions and lemmas.

Definition 1

(L_p-norm and L_p-NED). For any random variable t with a finite p^th absolute moment, where p≥1, denote its L_p-norm by $\parallel t{\parallel }_{p}={\left[{{{\rm{E}}}}| t{| }^{p}\right]}^{1/p}$. Let $g=\left\{{g}_{ni},i\in {D}_{n},n\ge 1\right\}$ and $v=\left\{{v}_{ni},i\in {D}_{n},n\ge 1\right\}$ be two random fields, where D_n satisfies Assumption 3. Assume that $\mathop{\sup }\nolimits_{i,n}{\left\Vert {g}_{ni}\right\Vert }_{p} < \infty$. The random field g is said to be L_p-NED on v if

$${\left\Vert {g}_{ni}-{{{\rm{E}}}}\left({g}_{ni}| {{{{\mathcal{F}}}}}_{ni}(s)\right)\right\Vert }_{p}\le {d}_{ni}\psi (s)$$

for some arrays of finite positive constants $\left\{{d}_{ni},i\in {D}_{n},n\ge 1\right\}$ and for some sequence ψ(s) ≥ 0 such that $\mathop{\lim }\nolimits_{s\to \infty }\psi (s)=0$, where ${{{{\mathcal{F}}}}}_{ni}(s)$ is the σ-field generated by the random variables v_nj ’s with units j ’s located within the ball B_i(s) that is centered at i with radius s, and ψ(s) is the NED coefficient. If we further have $\mathop{\sup }\nolimits_{n}\mathop{\sup }\nolimits_{i\in {D}_{n}}{d}_{ni}\, < \,\infty$, then g is said to be uniformly L_p-NED on v.

Lemma 1

(Theorem 1 in Jenish and Prucha 2012). Let $\left\{{D}_{n}\right\}$ be a sequence of arbitrary finite subsets of D such that $\left\vert {D}_{n}\right\vert \to \infty$ as n → ∞, where $D\subset {{\mathbb{R}}}^{d},d\ge 1$ is as in Assumption 3, and let T_n be a sequence of subsets of D such that D_n ⊆ T_n. Suppose further that $Z=\left\{{Z}_{i,n},i\in {D}_{n},n\ge 1\right\}$ is L₁-NED on $\varepsilon =\left\{{\varepsilon }_{i,n},i\in {T}_{n},n\ge 1\right\}$ with the scaling factors d_i,n. Suppose that Z and ε satisfy: (a) there exist nonrandom positive constants $\left\{{c}_{i,n},i\in {D}_{n},n\ge 1\right\}$ such that Z_i,n/c_i,n is uniformly L_p-bounded for some p > 1, i.e., $\mathop{\sup }\nolimits_{n}\mathop{\sup }\nolimits_{i\in {D}_{n}}E{\left\vert {Z}_{i,n}/{c}_{i,n}\right\vert }^{p}\, < \,\infty$, and (b) the α-mixing coefficients of the input field ε satisfy $\bar{\alpha }(u,r,v)\le \varphi (u,v)\hat{\alpha }(r)$ for some function φ(u, v) which is nondecreasing in each argument, and some $\widehat{\alpha }(r)$ such that $\mathop{\sum }\nolimits_{r = 1}^{\infty }{r}^{d-1}\widehat{\alpha }(r) < \infty$. Then

$$\frac{1}{{M}_{n}\left\vert {D}_{n}\right\vert }\mathop{\sum}\limits_{i\in {D}_{n}}\left({Z}_{i,n}-{{{\rm{E}}}}{Z}_{i,n}\right)\mathop{\to }\limits^{{L}_{1}}0,$$

where ${M}_{n}=\mathop{\max }\limits_{i\in {D}_{n}}\max \left({c}_{i,n},{d}_{i,n}\right)$.

Lemma 2

(Proposition 1 in Xu and Lee 2015). Suppose the parameter space Θ of θ is compact. Consider the SAR model Y_n = λW_nY_n + X_nβ + V_n.

• (1) Under Assumption 2, if $\mathop{\sup }\nolimits_{1\leqslant\, j\,\leqslant k,i,n}{{{\rm{E}}}}{\left\vert {x}_{ij,n}\right\vert }^{p}\, < \,\infty$ and $\mathop{\sup }\nolimits_{i,n}{{{\rm{E}}}}{\left\vert {\epsilon }_{ni}\right\vert }^{p}\, < \,\infty$ for some p ⩾ 1, then ${\left\{{y}_{ni}\right\}}_{i = 1}^{n}$ and ${\left\{{{{{\bf{w}}}}}_{i,n}{Y}_{n}\right\}}_{i = 1}^{n}$ are uniformly L_p bounded.

• (2) Under Assumptions 3-4(a) and 5, ${\left\{{y}_{ni}\right\}}_{i = 1}^{n}$ is geometrically L₂-NED on ${\left\{{{{{\bf{x}}}}}_{i,n},{v}_{ni}\right\}}_{i = 1}^{n}:{\left\Vert {y}_{ni}-{{{\rm{E}}}}\left[{y}_{ni}| {{{{\mathcal{F}}}}}_{ni}(s)\right]\right\Vert }_{2}\leqslant C({\zeta }^{1/{\bar{d}}_{0}})^{s}$ for some C > 0 that does not depend on i and n. The same conclusion also holds for ${\left\{{{{{\bf{w}}}}}_{i,,n}{Y}_{n}\right\}}_{i = 1}^{n}$.

• (3) Under Assumptions 3-4(b) and 5, ${\left\{{y}_{ni}\right\}}_{i = 1}^{n}$ is L₂-NED on ${\left\{{{{{\bf{x}}}}}_{i,n},{v}_{ni}\right\}}_{i = 1}^{n}:{\left\Vert {y}_{ni}-{{{\rm{E}}}}\left[{y}_{ni}| {{{{\mathcal{F}}}}}_{ni}(s)\right]\right\Vert }_{2}\leqslant C/{s}^{\alpha -d}$ for some C > 0 that does not depend on i and n. The same conclusion also holds for ${\left\{{{{{\bf{w}}}}}_{i,n}{Y}_{n}\right\}}_{i = 1}^{n}$.

Lemma 3

(Generalization of Corollary 4.3(b) in Gallant and White 1988). If $\left\{{Y}_{i,n}\right\}$ and $\left\{{Z}_{i,n}\right\}$ are both uniformly L_2r bounded for some r > 2, and uniformly and geometrically L₂-NED, then $\left\{{Y}_{i,n}{Z}_{i,n}\right\}$ is uniformly and geometrically L₂-NED.

Appendix B. Technical details in Section 2 and 3

This appendix provides some expressions in Section 2 and main proofs for the results in Section 3.

• Π^† in Assumption 2 by the best GMME, or by the QMLE under H₀

  For the best GMME ${\tilde{\theta }}_{n,bgmme}^{{\dagger} }$,

  $${\Pi }^{{\dagger} -1}=\mathop{\lim }\limits_{n\to \infty }\frac{1}{n}{{{\rm{E}}}}\left[\begin{array}{rr}{{{\rm{tr}}}}({G}_{n1}^{ds}{G}_{n})+\frac{1}{{\sigma }^{{\dagger} 2}}{({G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} })}^{{\prime} }({G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} })&\frac{1}{{\sigma }^{{\dagger} 2}}{({G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} })}^{{\prime} }{X}_{n}^{{\dagger} }\\ \frac{1}{{\sigma }^{{\dagger} 2}}{X}_{n}^{{\dagger} {\prime} }({G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} })&\frac{1}{{\sigma }^{{\dagger} 2}}{X}_{n}^{{\dagger} {\prime} }{X}_{n}^{{\dagger} }\end{array}\right],$$

  (B.1)

  where ${G}_{n1}^{ds}=[{G}_{n}-{{{\rm{Diag}}}}({G}_{n})]+{[{G}_{n}-{{{\rm{Diag}}}}({G}_{n})]}^{{\prime} }$, and Diag(G_n) is the diagonal matrix generated by the diagonal elements of G_n. Under H₀, for the QMLE ${\tilde{\theta }}_{n,mle}^{{\dagger} }$,

  $${\Pi }^{{\dagger} -1}=\mathop{\lim }\limits_{n\to \infty }\frac{1}{n}{{{\rm{E}}}}\left[\begin{array}{rr}{{{\rm{tr}}}}({G}_{n2}^{ds}{G}_{n})+\frac{1}{{\sigma }^{{\dagger} 2}}{({G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} })}^{{\prime} }({G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} })&\frac{1}{{\sigma }^{{\dagger} 2}}{({G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} })}^{{\prime} }{X}_{n}^{{\dagger} }\\ \frac{1}{{\sigma }^{{\dagger} 2}}{X}_{n}^{{\dagger} {\prime} }({G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} })&\frac{1}{{\sigma }^{{\dagger} 2}}{X}_{n}^{{\dagger} {\prime} }{X}_{n}^{{\dagger} }\end{array}\right],$$

  (B.2)

  where ${G}_{n2}^{ds}=[{G}_{n}-({{{\rm{tr}}}}({G}_{n})/n){I}_{n}]+[{G}_{n}-({{{\rm{tr}}}}({G}_{n})/n){I}_{n}]^{{\prime} }$. No matter in (B.1) or (B.2), the following fact holds.

  Fact: the second column of Π^†−1 is $(1/\sigma^{\dagger 2}){\mathbf{h}}_2^{\dagger \prime} = (1/\sigma^{\dagger 2})\lim_{n\to\infty} n^{-1} {\mathrm{E}} [{\mathbf{1}}_n^\prime G_n X_n^{\dagger} \beta^{\dagger}, {\mathbf{1}}_n^\prime X_n^{\dagger}]^\prime$.

  The above fact is useful in Corollary 1, and more details can be found in the proof of the corollary.

• Elements of ${\Sigma }_{n}^{{\dagger} }$ and Σ_n in Proposition 2

  ${\Sigma }_{n}^{{\dagger} }$ is a symmetric 3 × 3 matrix, whose (i, j)^th element is denoted as ${\delta }_{ij,n}^{{\dagger} }$. Recall that ${\sigma }^{{\dagger} 2}={{{\rm{E}}}}[{\epsilon }_{ni}^{{\dagger} 2}],{\mu }_{s}^{{\dagger} }={{{\rm{E}}}}[{\epsilon }_{ni}^{{\dagger} s}]$ for s = 3, 4, 5, 6, and ${{{{\bf{h}}}}}_{1}^{{\dagger} }$ and ${{{{\bf{h}}}}}_{2}^{{\dagger} }$ are defined in (3.2). For the diagonal elements of Σ_n,

  $$\begin{array}{ll}{\delta }_{11,n}^{{\dagger}}={\sigma }^{{\dagger} 2}+{{{{\bf{h}}}}}_{2}^{{\dagger} }{\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{2}^{{\dagger} {\prime} }-2{{{{\bf{h}}}}}_{2}^{{\dagger} }{C}_{\epsilon ,1n},\\{\delta }_{22,n}^{{\dagger} }={\mu }_{4}^{{\dagger} }-{\sigma }^{{\dagger} 4}+4{\sigma }^{{\dagger} 4}{{{{\bf{h}}}}}_{1}^{{\dagger} }{\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} {\prime} }-4{\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{1}^{{\dagger} }{C}_{\epsilon ,2n},\\ {\delta }_{33,n}^{{\dagger} }={\mu }_{6}^{{\dagger} }-{\mu }_{3}^{{\dagger} 2}-6({\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{2}^{{\dagger} }+{\mu }_{3}^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} }){{{{\bf{h}}}}}_{2}^{{\dagger} }{C}_{\epsilon ,3n}\\\qquad+9({\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{2}^{{\dagger} }+{\mu }_{3}^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} }){\Pi }^{{\dagger} }{({\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{2}^{{\dagger} }+{\mu }_{3}^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} })}^{{\prime} }.\end{array}$$

  (B.3)

  For the off-diagonal elements,

  $$\begin{array}{l}{\delta }_{21,n}^{{\dagger} }={\delta }_{12,n}^{{\dagger} }={\mu }_{3}^{{\dagger} }-{{{{\bf{h}}}}}_{2}^{{\dagger} }{C}_{\epsilon ,2n}-2{\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{1}^{{\dagger} }{C}_{\epsilon ,1n}+2{\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{1}^{{\dagger} }{\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{2}^{{\dagger} {\prime} },\\ {\delta }_{31,n}^{{\dagger} }={\delta }_{13,n}^{{\dagger} }={\mu }_{4}^{{\dagger} }+3({\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{2}^{{\dagger} }+{\mu }_{3}^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} }){\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{2}^{{\dagger} {\prime} }-{{{{\bf{h}}}}}_{2}^{{\dagger} }{C}_{\epsilon ,3n}-3({\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{2}^{{\dagger} }+{\mu }_{3}^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} }){C}_{\epsilon ,1n},\\ {\delta }_{32,n}^{{\dagger} }={\delta }_{23,n}^{{\dagger} }={\mu }_{5}^{{\dagger} }-{\mu }_{3}^{{\dagger} }{\sigma }^{{\dagger} 2}-2{\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{1}^{{\dagger} }{C}_{\epsilon ,3n}-3({\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{2}^{{\dagger} }+{\mu }_{3}^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} }){C}_{\epsilon ,2n}+6{\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{1}^{{\dagger} }{\Pi }^{{\dagger} }({\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{2}^{{\dagger} }+{\mu }_{3}^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} })^{{\prime} }.\end{array}$$

  (B.4)

  Here, ${C}_{\epsilon ,jn}={{{\rm{E}}}}[{n}^{-1/2}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} j}\cdot {n}^{1/2}({\tilde{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} })]$ as the (k + 1)-dimensional expectation vector for any j ∈ {1, 2, 3}. Under ${H}_{0},{\mu }_{3}^{{\dagger} }={\mu }_{5}^{{\dagger} }=0$, then ${\Sigma }_{n}^{{\dagger} }$ is reducible to Σ_n.

Proof of Proposition 1.

The probability limit of ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} }$ is shown in p. 5–6. We now only analyze that of ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 3}$, and ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 2}-{\sigma }^{{\dagger} 2}={o}_{p}(1)$ can be proved by a similar argument. By the expansion, it follows that

$$\begin{array}{rl}&\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 3}-\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 3}\\=&\frac{3}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\tilde{\theta }}_{n}^{{\dagger} })+\frac{3}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} }[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\tilde{\theta }}_{n}^{{\dagger} })]^{2}\\&+\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\tilde{\theta }}_{n}^{{\dagger} })]^{3}.\end{array}$$

(B.5)

Under Assumptions 1–5, {z_i,n} is uniformly L_r-norm bounded for some r = 4 + ι > 4 and is uniformly L₂-NED on ${\{{{{{\bf{x}}}}}_{i,n}^{{\dagger} },{\epsilon }_{ni}^{{\dagger} }\}}_{i = 1}^{n}$, as the argument below (3.1). Assumption 1 tells us that E∣v_ni∣^6+ι < ∞ for some ι > 0, so {ϵ_ni} must be uniformly L_r-norm bounded for some r > 4 under H₀. Hence, according to Lemma 3 (in Appendix A), $\{{\epsilon }_{ni}^{{\dagger} }{{{{\bf{z}}}}}_{i,n}\}$ is uniformly L₂-NED on ${\{{{{{\bf{x}}}}}_{i,n}^{{\dagger} },{\epsilon }_{ni}^{{\dagger} }\}}_{i = 1}^{n}$ under H₀. Similarly, under H₀, all of ${\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{z}}}}}_{i,n},{\epsilon }_{ni}^{{\dagger} }{{{{\bf{z}}}}}_{i,n}{{{{\bf{z}}}}}_{i,n}^{{\prime} }$ and $\{{z}_{ij,n}{{{{\bf{z}}}}}_{i,n}{{{{\bf{z}}}}}_{i,n}^{{\prime} }\}$ are also uniformly L₂-NED on ${\{{{{{\bf{x}}}}}_{i,n}^{{\dagger} },{\epsilon }_{ni}^{{\dagger} }\}}_{i = 1}^{n}$. Besides, let h_n,ij be either 1 or any element of ${{{{\bf{z}}}}}_{i,n}^{{\prime} }$ for j = 1, 2, 3. According to the generalized Hölder’s inequality, under Assumptions 1–5,

$$\begin{array}{l}\quad\mathop{\sup }\limits_{i,n}{{{\rm{E}}}}| {h}_{n,i1}{h}_{n,i2}{h}_{n,i3}| \\\le \mathop{\sup }\limits_{i,n}\left\{{({{{\rm{E}}}}| {h}_{n,i1}{| }^{3})}^{1/3}{({{{\rm{E}}}}| {h}_{n,i2}{| }^{3})}^{1/3}{({{{\rm{E}}}}| {h}_{n,i3}{| }^{3})}^{1/3}\right\}\, < \,\infty\end{array}$$

and

$$\begin{array}{l}\quad\mathop{\sup }\limits_{i,n}{{{\rm{E}}}}| {v}_{ni}^{k}{h}_{n,i1}{h}_{n,i2}| \\\le \mathop{\sup }\limits_{i,n}\left\{{({{{\rm{E}}}}| {v}_{ni}{| }^{3k})}^{1/3}{({{{\rm{E}}}}| {h}_{n,i1}{| }^{3})}^{1/3}{({{{\rm{E}}}}| {h}_{n,i2}{| }^{3})}^{1/3}\right\}\, < \,\infty\end{array}$$

hold for k = 0, 1, 2. By making use of the LLN in Jenish and Prucha (2012) again, ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{z}}}}}_{i,n}^{{\prime} }={{{\rm{E}}}}[{n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{z}}}}}_{i,n}^{{\prime} }]+{o}_{p}(1)={O}_{p}(1)$. Similarly, ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} }{{{{\bf{z}}}}}_{i,n}{{{{\bf{z}}}}}_{i,n}^{{\prime} }={n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{{{{\bf{z}}}}}_{i,n}{{{{\bf{z}}}}}_{i,n}^{{\prime} }{z}_{n,ij}={O}_{p}(1)$. Working with ${\theta }^{{\dagger} }-{\tilde{\theta }}_{n}^{{\dagger} }={o}_{p}(1)$, it yields that ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\tilde{\theta }}_{n}^{{\dagger} })={o}_{p}(1),{n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} }{[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\tilde{\theta }}_{n}^{{\dagger} })]}^{2}={o}_{p}(1)$, and ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\tilde{\theta }}_{n}^{{\dagger} })]}^{3}={o}_{p}(1)$. Therefore, $\frac{1}{n}\mathop{\sum }\nolimits_{i = 1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 3}-\frac{1}{n}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 3}=\frac{1}{n}\mathop{\sum }\nolimits_{i = 1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 3}-{\mu }_{3}^{{\dagger} }={o}_{p}(1)$.

Under H₀, by the Lindeberg-Lévy CLT, ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 3}={O}_{p}({n}^{-1/2})$ since ${{{\rm{E}}}}[{\epsilon }_{ni}^{{\dagger} 3}]={{{\rm{E}}}}[{v}_{ni}^{3}]=0$. The proof is accomplished. □

Proof of Proposition 2.

To check the correctness of (3.3), we take the term regarding ${n}^{-1}{\sum \nolimits_{i = 1}^{n}}{\tilde{\epsilon}}_{ni}^{{\dagger}3}$ as the example. Applying the mean-value theorem to ${n}^{-1}{\sum \nolimits_{i = 1}^{n}}{\tilde{\epsilon }}_{ni}^{{\dagger} 3}={n}^{-1}{\sum \nolimits_{i = 1}^{n}}({y}_{ni}-{\bf{z}}_{i,n}^{\prime}{\tilde{\theta}}_{n}^{{\dagger} })^{3}\triangleq \,{n}^{-1}{\sum \nolimits_{i = 1}^{n}}{\epsilon }_{ni}^{{\dagger} 3}({\tilde{\theta }}_{n}^{{\dagger}})$ at θ^†, it yields that

$$\begin{array}{ll}\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 3}=\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 3}-\frac{3}{{n}^{3/2}}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 2}({\bar{\theta }}_{n}^{{\dagger} })[{{{{\bf{w}}}}}_{i,n}{Y}_{n},{{{{\bf{x}}}}}_{i,n}^{{\dagger} {\prime} }]\cdot \sqrt{n}({\tilde{\theta }}_{n}-{\theta }^{{\dagger} })\\ =\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 3}-\frac{3}{{n}^{1/2}}\left[\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 2}({\bar{\theta }}_{n}^{{\dagger} }){{{{\bf{w}}}}}_{i,n}{Y}_{n},\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 2}({\bar{\theta }}_{n}^{{\dagger} }){{{{\bf{x}}}}}_{i,n}^{{\dagger} {\prime} }\right]\cdot \sqrt{n}({\tilde{\theta }}_{n}-{\theta }^{{\dagger} }),\end{array}$$

where ${\bar{\theta }}_{n}^{{\dagger} }$ satisfies $\parallel {\bar{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} }\parallel \le \parallel {\tilde{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} }\parallel$ and therefore, ${\bar{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} }={o}_{p}(1)$. By expansion we can obtain

$$\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 2}({\bar{\theta }}_{n}^{{\dagger} }){{{{\bf{w}}}}}_{i,n}{Y}_{n}=\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{w}}}}}_{i,n}{Y}_{n}+\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\bar{\theta }}_{n}^{{\dagger} })]^{2}{{{{\bf{w}}}}}_{i,n}{Y}_{n}+\frac{2}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} }{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\bar{\theta }}_{n}^{{\dagger} }){{{{\bf{w}}}}}_{i,n}{Y}_{n}.$$

As mentioned above, any element of $\{{{{{\bf{z}}}}}_{i,n}:{{{{\bf{z}}}}}_{i,n}^{{\prime} }=[{{{{\bf{w}}}}}_{i,n}{Y}_{n},{{{{\bf{x}}}}}_{i,n}^{{\dagger} {\prime} }]\}$ are uniformly L_2r-norm bounded for some r > 2, and uniformly L₂-NED on $\{{{{{\bf{x}}}}}_{i,n}^{{\dagger} {\prime} },{\epsilon }_{ni}^{{\dagger} }\}$. Based on this, we can further conclude that $\{{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{w}}}}}_{i,n}{Y}_{n}\}$ is uniformly L₄-norm bounded, and uniformly L₂-NED on $\{{{{{\bf{x}}}}}_{i,n}^{{\dagger} {\prime} },{\epsilon }_{ni}^{{\dagger} }\}$, thereby ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{w}}}}}_{i,n}{Y}_{n}-{\sigma }^{{\dagger} 2}{{{\rm{E}}}}[{n}^{-1}{{{{\bf{1}}}}}_{n}^{{\prime} }{G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} }]-{\mu }_{3}^{{\dagger} }{n}^{-1}{{{\rm{tr}}}}({G}_{n})={o}_{p}(1)$ by the LLN in Jenish and Prucha (2012) and the fact that ${{{\rm{E}}}}[{n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{w}}}}}_{i,n}{Y}_{n}]={{{\rm{E}}}}[{n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{w}}}}}_{i,n}{S}^{-1}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} }]+{{{\rm{E}}}}[{n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{w}}}}}_{i,n}{S}^{-1}{\epsilon }_{n}^{{\dagger} }]={\sigma }^{{\dagger} 2}{{{\rm{E}}}}[{n}^{-1}{{{{\bf{1}}}}}_{n}^{{\prime} }{G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} }]+{\mu }_{3}^{{\dagger} }{n}^{-1}{{{\rm{tr}}}}({G}_{n})$. Ditto for ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{z}_{ij,n}^{{\prime} }{{{{\bf{w}}}}}_{i,n}{Y}_{n}$ and ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} }{z}_{ij,n}{{{{\bf{w}}}}}_{i,n}{Y}_{n}$. Taking with ${\theta }^{{\dagger} }-{\bar{\theta }}_{n}^{{\dagger} }={o}_{p}(1)$ together, it follows that ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\left[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\bar{\theta }}_{n}^{{\dagger} })\right]}^{2}{{{{\bf{w}}}}}_{i,n}{Y}_{n}$ and ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} }\left[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{\dagger}{-}{\bar{\theta }}_{n}^{{\dagger} })\right]{{{{\bf{w}}}}}_{i,n}{Y}_{n}$ are o_p(1). Therefore, ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}({\bar{\theta }}_{n}^{{\dagger} }){{{{\bf{w}}}}}_{i,n}{Y}_{n}{-}{\sigma }^{{\dagger} 2}{{{\rm{E}}}}[{n}^{-1}{{{{\bf{1}}}}}_{n}^{{\prime} }{G}_{n}{X}_{n}^{{\dagger} }{\beta }^{{\dagger} }]={o}_{p}(1)$. By a similar argument for

$$\begin{array}{*{20}{l}}\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}\epsilon_{ni}^{\dagger 2}(\bar{\theta}_n^{\dagger}){\mathbf{x}}_{i,n}^{\dagger \prime} = \frac{1}{n}\mathop{\sum }\limits_{i = 1}^{n}\epsilon_{ni}^{\dagger 2}{\mathbf{x}}_{i,n}^{\dagger \prime} + \frac{1}{n}\mathop{\sum }\limits_{i = 1}^{n}[{\mathbf{z}}_{i,n}^{\prime}(\theta^{\dagger} - \bar{\theta}_n^{\dagger})]^{2}\\ \times\, {\mathbf{x}}_{i,n}^{\dagger \prime} + \frac{2}{n}\mathop{\sum }\limits_{i = 1}^{n}\epsilon_{ni}^{\dagger}{\mathbf{z}}_{i,n}^{\prime}(\theta^{\dagger} - \bar{\theta}_n^{\dagger}){\mathbf{x}}_{i,n}^{\dagger \prime},\end{array}$$

the last two terms on the right-hand side satisfy ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\bar{\theta }}_{n}^{{\dagger} })]^{2}{{{{\bf{x}}}}}_{i,n}^{{\dagger} {\prime} }={o}_{p}(1)$ and $(2/n)\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} }[{{{{\bf{z}}}}}_{i,n}^{{\prime} }({\theta }^{{\dagger} }-{\bar{\theta }}_{n}^{{\dagger} })]{{{{\bf{x}}}}}_{i,n}^{{\dagger} {\prime} }={o}_{p}(1)$. Working with ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}{{{{\bf{x}}}}}_{i,n}^{{\dagger} {\prime} }-{{{\rm{E}}}}[{n}^{-1}{\sigma }^{{\dagger} 2}{{{{\bf{1}}}}}_{n}^{{\prime} }{X}_{n}^{{\dagger} }]={o}_{p}(1)$ by the Chebyshev LLN, it yields that ${n}^{-1}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} 2}({\bar{\theta }}_{n}^{{\dagger} }){{{{\bf{x}}}}}_{i,n}^{{\dagger} }={{{\rm{E}}}}[{n}^{-1}{\sigma }^{{\dagger} 2}{{{{\bf{1}}}}}_{n}^{{\prime} }{X}_{n}^{{\dagger} }]+{o}_{p}(1)$. Therefore, we obtain

$$\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 3}=\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 3}-\frac{3}{\sqrt{n}}({\sigma }^{{\dagger} 2}{{{{\bf{h}}}}}_{2}^{{\dagger} }+{\mu }_{3}^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} })\sqrt{n}({\tilde{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} })+{o}_{p}({n}^{-1/2}).$$

Similarly, the other two equalities in (3.3) hold.

To derive the asymptotic distribution of (3.4), it is equivalent to deriving that of

$$\begin{array}{ll}{F}_{n}\triangleq \frac{{c}_{3}}{n}\mathop{\sum }\limits_{i=1}^{n}({\epsilon }_{ni}^{{\dagger} 3}-{\mu }_{3}^{{\dagger} })+\frac{{c}_{2}}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} 2}+\frac{{c}_{1}}{n}\mathop{\sum }\limits_{i=1}^{n}{\epsilon }_{ni}^{{\dagger} }\\\qquad\,\,\,\,+\,\frac{1}{\sqrt{n}}{{{\bf{f}}}}({\bf{c}})\sqrt{n}({\tilde{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} })\end{array}$$

(B.6)

with f(c) being defined in (3.5), which depends on the limiting distribution of $\sqrt{n}({\tilde{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} })$. By Assumption 2, we know ${n}^{1/2}({\tilde{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} })={g}_{n}({\theta }^{{\dagger} })+{o}_{p}(1)$, where g_n(θ^†) is quadratic forms in ${\epsilon }_{ni}^{{\dagger} }$. Define ${A}_{n}^{d}={A}_{n}-[{n}^{-1}{{{\rm{tr}}}}({A}_{n})]\cdot {I}_{n}$. Therefore, without lose of generality, we can assume ${g}_{n}({\theta }^{{\dagger} })={n}^{-1/2}D{[{\epsilon }_{n}^{{\dagger} {\prime} }{P}_{1}^{d}{\epsilon }_{n}^{{\dagger} },\cdots ,{\epsilon }_{n}^{{\dagger} {\prime} }{P}_{s}^{d}{\epsilon }_{n}^{{\dagger} },{\epsilon }_{n}^{{\dagger} {\prime} }Q]}^{{\prime} }\in {{\mathbb{R}}}^{k+1}$, where $D=[{d}_{1},\cdots \,,{d}_{s},{{{{\bf{d}}}}}_{q}]\in {{\mathbb{R}}}^{(k+1)\times (s+q)},P\in {{\mathbb{R}}}^{n\times n}$, and $Q\in {{\mathbb{R}}}^{n\times q}$ are constant IV matrix with q and s being arbitrary constants. Accordingly, (B.6) becomes

$$\begin{array}{lll}{F}_{n}=\frac{{c}_{3}}{n}\mathop{\sum }\limits_{i=1}^{n}({\epsilon }_{ni}^{{\dagger} 3}-{\mu }_{3}^{{\dagger} })+\frac{{\epsilon }_{n}^{{\dagger} {\prime} }}{n}[{c}_{2}{I}_{n}+{{{\bf{f}}}}({\bf{c}})\mathop{\sum }\limits_{j=1}^{s}{d}_{j}{P}_{j}^{d}]{\epsilon }_{n}^{{\dagger} }\\\qquad\;\;+\,\frac{1}{n}\left[{c}_{1}{\bf{1}}_{n}^{{\prime} }+{{{\bf{f}}}}({\bf{c}}){{{{\bf{d}}}}}_{q}{Q}^{{\prime} }\right]{\epsilon }_{n}^{{\dagger} }\\\quad\;\; \triangleq \frac{{c}_{3}}{n}\mathop{\sum }\limits_{i=1}^{n}({\epsilon }_{ni}^{{\dagger} 3}-{\mu }_{3}^{{\dagger} })+\frac{1}{n}{\epsilon }_{n}^{{\dagger} {\prime} }{A}_{n}{\epsilon }_{n}^{{\dagger} }+\frac{1}{n}{{{{\bf{b}}}}}_{n}^{{\prime} }{\epsilon }_{n}^{{\dagger} }.\end{array}$$

The cubic CLT¹¹ tells us that

$$\sqrt{n}{F}_{n} \sim AN\left(0,\mathop{\lim }\limits_{n\to \infty }{{{{\bf{c}}}}}^{{\prime} }{\Sigma }_{n}^{{\dagger} }{{{\bf{c}}}}\right),$$

(B.7)

where ${\Sigma }_{n}^{{\dagger} }$ is defined as in Appendix B. Based on (B.7), taking (3.4), (B.6) and (B.7) together, (3.6) follows from the Slutsky theorem. Under ${H}_{0},{\mu }_{3}^{{\dagger} }={\mu }_{5}^{{\dagger} }=0$ and ${\Sigma }_{n}^{{\dagger} }$ reduces to Σ_n. The proof is accomplished. □

Proof of Proposition 3.

Employing the delta method for (3.6), we can obtain

$$\begin{array}{ll}&{n}^{1/2}[{S}_{n}({\tilde{\epsilon }}_{n}^{{\dagger} })-{S}^{{\dagger} }]\\=&{n}^{1/2}\left[\psi (\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} },\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 2},\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\tilde{\epsilon }}_{ni}^{{\dagger} 3})-\psi(0,{\sigma }^{{\dagger} 2},{\mu }_{3}^{{\dagger} })\right]\\ \sim & AN\left(0,\frac{\partial \psi (0,{\sigma }^{{\dagger} 2},{\mu }_{3}^{{\dagger} })}{\partial [a,b,c]}\mathop{\lim }\limits_{n\to \infty }{\Sigma }_{n}^{{\dagger} }\frac{\partial \psi (0,{\sigma }^{{\dagger} 2},{\mu }_{3}^{{\dagger} })}{\partial {[a,b,c]}^{{\prime} }}\right),\end{array}$$

where $(\partial \psi (0,{\sigma }^{{\dagger} 2},{\mu }_{3}^{{\dagger} })/\partial [a,b,c])=[-3/{\sigma }^{{\dagger} },-3{\mu }_{3}^{{\dagger} }/(2{\sigma }^{{\dagger} 5}),1/{\sigma }^{{\dagger} 3}]={{{{\bf{r}}}}}_{1}^{{\prime} }.$

Further under H₀, we have ${S}^{{\dagger} }={\mu }_{3}^{{\dagger} }=0$, and $\partial \psi (0,{\sigma }^{{\dagger} 2},0)/\partial [a,b,c]=[-3/{\sigma }^{{\dagger} },0,1/{\sigma }^{{\dagger} 3}]={{{{\bf{r}}}}}_{2}^{{\prime} }$. So ${n}^{1/2}{S}_{n}({\tilde{\epsilon }}_{n}^{{\dagger} }) \sim AN\left(0,\mathop{\lim }\nolimits_{n\to \infty }{{{{\bf{r}}}}}_{2}^{{\prime} }{\Sigma }_{n}{{{{\bf{r}}}}}_{2}\right)$. The proof is accomplished. □

Proof of Corollary 1.

It follows from Proposition 3 that under Assumptions 1–5, ${n}^{1/2}[{S}_{n}({\tilde{\epsilon}}_{n}^{\dagger})-{S}^{\dagger}]\cdot ({\bf{r}}_{1}^{\prime} {\Sigma}_{n}^{\dagger}{\bf{r}}_{1})^{-1/2} \,{\sim}\, {AN}\left(0,1\right)$. Then, since ${\tilde{\sigma }}^{{\dagger} }-{\sigma }^{{\dagger} }={o}_{p}(1)$ and ${\tilde{\Sigma}}_{\ast}-{\Sigma }_{n}^{{\dagger}}={o}_{p}(1),{\sqrt{n}}[{S}_{n}({\tilde{\epsilon }}_{n}^{\dagger})-{S}^{\dagger}]\cdot ({{\tilde {\bf{r}}}}_{1}^{\prime}{\tilde{\Sigma}}_{\ast }{{\tilde {\bf{r}}}}_{1})^{-1/2} \,{\sim}\, {AN}(0,1)$ holds by the Slutsky theorem. Under ${H}_{0},{\tilde{{{{\bf{r}}}}}}_{2}={\tilde{{{{\bf{r}}}}}}_{1}$. (3.9) follows from two facts that S^† = 0, and

$${n}^{1/2}{S}_{n}({\tilde{\epsilon }}_{n}^{{\dagger} })-\frac{n{\sum \nolimits_{i = 1}^{n}}{\tilde{\epsilon }}_{ni}^{{\dagger} 3}}{({\sum \nolimits_{i = 1}^{n}}{\tilde{\epsilon }}_{ni}^{{\dagger} 2})^{3/2}}={o}_{p}(1).$$

We now prove (3.10). As ${\tilde{\epsilon }}_{n}^{{\dagger} }$ is generated from either MLE (Lee 2004, Theorem 3.2) or the best GMM estimation (Lee 2007, Proposition 3) under ${H}_{0},{{{{\bf{h}}}}}_{2}^{{\dagger} }$ is the second column of the matrix σ^†2Π^†−1 (i.e., the fact below (B.2)), since ${X}_{n}^{{\dagger} }$ must have an intercept (as said in the footnote 3). Then, $(1/{\sigma }^{{\dagger} 2}){\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{2}^{{\dagger} {\prime} }={{{{\bf{e}}}}}_{2,k+1}$, where e_2,k+1 is the second column of the identity matrix I_k+1. By direct calculation, we obtain

$${{{{\bf{h}}}}}_{2}^{{\dagger} }{\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{2}^{{\dagger} {\prime} }={\sigma }^{{\dagger} 2}\,\,{{\mbox{and}}}\,\,{{{{\bf{h}}}}}_{1}^{{\dagger} }{\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{2}^{{\dagger} {\prime} }={{{{\bf{h}}}}}_{2}^{{\dagger} }{\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{1}^{{\dagger} {\prime} }=0,$$

since the second element of ${{{{\bf{h}}}}}_{2}^{{\dagger} }$ is 1 and that of ${{{{\bf{h}}}}}_{1}^{{\dagger} }$ is 0. Recall that ${C}_{\epsilon ,jn}={{{\rm{E}}}}[{n}^{-1/2}\mathop{\sum }\nolimits_{i = 1}^{n}{\epsilon }_{ni}^{{\dagger} j}\cdot {n}^{1/2}({\tilde{\theta }}_{n}^{{\dagger} }-{\theta }^{{\dagger} })]$ for j = 1, 2, 3. When v_ni’s are normal, it is from straight calculation that

$${C}_{\epsilon ,1n}={\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{2}^{{\dagger} {\prime} }={\sigma }_{v}^{2}{{{{\bf{e}}}}}_{2,k+1}\,\,{{\mbox{and}}}\,\,{C}_{\epsilon ,3n}=3{\sigma }_{v}^{2}{\Pi }^{{\dagger} }{{{{\bf{h}}}}}_{2}^{{\dagger} {\prime} }=3{\sigma }_{v}^{4}{{{{\bf{e}}}}}_{2,k+1}.$$

For the elements in (B.3) and (B.4), ${\delta }_{11,n}^{{\dagger} }={\delta }_{13,n}^{{\dagger} }=0$, ${\delta }_{33,n}^{{\dagger} }={\mu }_{6}^{{\dagger} }-9{\sigma }_{v}^{6}=6{\sigma }_{v}^{6}$, whereby we can derive

$${{{{\bf{r}}}}}_{2}^{{\prime} }{\Sigma }_{n}{{{{\bf{r}}}}}_{2}={{{{\bf{r}}}}}_{2}^{{\prime} }{\Pi }_{n}^{{\dagger} }{{{{\bf{r}}}}}_{2}={\delta }_{11,n}^{{\dagger} }{r}_{1}^{2}+{\delta }_{33,n}^{{\dagger} }{r}_{3}^{2}+2{\delta }_{13,n}^{{\dagger} }{r}_{1}{r}_{3}=6{\sigma }_{v}^{6}(1/{\sigma }_{v}^{3})^{2}=6.$$

where ${{{{\bf{r}}}}}_{2}\triangleq [{r}_{1},0,{r}_{3}]^{{\prime} }$. The proof is accomplished. □

Appendix C. Technical details in Section 4

We first show how to transform the panel SARSF model (4.1’) for eliminating individual and time fixed-effects.

Let $[{F}_{T,T-1},\frac{1}{\sqrt{T}}{{{{\bf{1}}}}}_{T}]$ be the orthogonal matrix consisting of the eigenvectors of J_T, where ${F}_{T,T-1}\in {{\mathbb{R}}}^{T\times (T-1)}$ corresponds to the eigenvalues of one and $\frac{1}{\sqrt{T}}{{{{\bf{1}}}}}_{T}\in {{\mathbb{R}}}^{T}$ corresponds to the eigenvalue of zero. So does the orthogonal matrix $[{F}_{n,n-1},\frac{1}{\sqrt{n}}{{{{\bf{1}}}}}_{n}]$ for J_n. For any n × T matrix [Z_n1, ⋯ , Z_nT], define the n × (T − 1) transformed matrix $[{Z}_{n1}^{* },\cdots \,,{Z}_{n,T-1}^{* }]=[{Z}_{n1},\cdots \,,{Z}_{nT}]{F}_{T,T-1}$. Similarly, we can obtain ${X}_{nt}^{* },{\alpha }_{to}^{* }$ and ${\epsilon }_{nt}^{{\dagger} * }$. Then, (4.1’) implies ${Y}_{nt}^{* }=\lambda {W}_{n}{Y}_{nt}^{* }+{X}_{nt}^{* }\beta +{\alpha }_{t}^{* }{{{{\bf{1}}}}}_{n}+{\epsilon }_{nt}^{{\dagger} * }$ for t = 1, 2, ⋯ , T − 1, where c_n is eliminated since ${{{{\bf{1}}}}}_{T}^{{\prime} }{F}_{T,T-1}={{{{\bf{0}}}}}_{T-1}^{{\prime} }$. After multiplying ${F}_{n,n-1}^{{\prime} }$ to the left on both sides of the transformed model, model (4.1’) can further be simplified to

$${Y}_{nt}^{* * }=\lambda {W}_{n}^{* }{Y}_{nt}^{* * }+{X}_{nt}^{* * }\beta +{\epsilon }_{nt}^{{\dagger} * * },t=1,2,\cdots \,,T-1,$$

(C.1)

where ${Y}_{nt}^{* * }={F}_{n,n-1}^{{\prime} }{Y}_{nt}^{* }\in {{\mathbb{R}}}^{n-1},{W}_{n}^{* }={F}_{n,n-1}^{{\prime} }{W}_{n}{F}_{n,n-1}\in {{\mathbb{R}}}^{(n-1)\times (n-1)},{X}_{nt}^{* * }={F}_{n,n-1}^{{\prime} }{X}_{nt}^{* }\in {{\mathbb{R}}}^{(n-1)\times k}$, and ${\epsilon }_{nt}^{{\dagger} * * }={F}_{n,n-1}^{{\prime} }{\epsilon }_{nt}^{{\dagger} * }\in {{\mathbb{R}}}^{n-1}$.

Remark 4

In the transformed panel SARSF model (C.1), the transformed time-effect term ${\alpha }_{t}^{* }{{{{\bf{1}}}}}_{n}$ is eliminated, because ${F}_{n,n-1}^{{\prime} }{{{{\bf{1}}}}}_{n}={{{{\bf{0}}}}}_{n-1}$. Thus, there does not exist an intercept term in model (C.1).

Recall ${{{{\boldsymbol{\epsilon }}}}}_{nT}^{{\dagger} }=[{\epsilon }_{n1}^{{\dagger} {\prime} },{\epsilon }_{n2}^{{\dagger} {\prime} },\cdots \,,{\epsilon }_{n,T}^{{\dagger} {\prime} }]$ and define ${{{{\boldsymbol{\epsilon }}}}}_{n-1,T-1}^{{\dagger} * * }=[{\epsilon }_{n1}^{{\dagger} * * {\prime} },{\epsilon }_{n2}^{{\dagger} * * {\prime} },\cdots \,,{\epsilon }_{n,T-1}^{{\dagger} * * {\prime} }]$. Both vectors satisfy the following condition:¹²

$$\begin{array}{*{20}{l}}{{{{\boldsymbol{\epsilon }}}}}_{n-1,T-1}^{{\dagger} * * }=({I}_{T-1}\otimes {F}_{n,n-1}^{{\prime} })[{\epsilon }_{n1}^{{\dagger} * {\prime} },{\epsilon }_{n2}^{{\dagger} * {\prime} },\cdots \,,{\epsilon }_{n,T-1}^{{\dagger} * {\prime} }]\\\qquad\quad\;\;\;=({F}_{T,T-1}^{{\prime} }\otimes {F}_{n,n-1}^{{\prime} }){{{{\boldsymbol{\epsilon }}}}}_{nT}^{{\dagger} }.\end{array}$$

(C.2)

It follows that

$$\begin{array}{*{20}{l}}&{{{\rm{Var}}}}[{{{{\boldsymbol{\epsilon }}}}}_{n-1,T-1}^{{\dagger} * * }]\\=&({F}_{T,T-1}^{{\prime} }\otimes {F}_{n,n-1}^{{\prime} }){{{\rm{Var}}}}[{{{{\boldsymbol{\epsilon }}}}}_{nT}^{{\dagger} }]{({F}_{T,T-1}^{{\prime} }\otimes {F}_{n,n-1}^{{\prime} })}^{{\prime} }\\ =&{\sigma }^{{\dagger} 2}({F}_{T,T-1}^{{\prime} }\otimes {F}_{n,n-1}^{{\prime} }){({F}_{T,T-1}^{{\prime} }\otimes {F}_{n,n-1}^{{\prime} })}^{{\prime} }\\=&{\sigma }^{{\dagger} 2}({I}_{T-1}\otimes {I}_{n-1}).\end{array}$$

(C.3)

This indicates even if the original panel SARSF model (4.1’) is transformed into (C.1), their centered composite errors do not only have the same variance σ^†2, but also are uncorrelated mutually. Next, for estimating the transformed model in (C.1), we consider QMLE as example in what follows. Given that QMLE treats ${{{{\boldsymbol{\epsilon }}}}}_{n-1,T-1}^{{\dagger} * * }$ as normal, the concentrated log-likelihood function of (C.1) with respect to λ is

$$\begin{array}{*{20}{l}}\ln L_{n,T}(\lambda) = -(1/2)(n-1)(T-1)\cdot\ln(2\pi + 1) \\\qquad\qquad\quad\;\;+\, (T-1)\cdot\ln \left|(1 - \lambda)^{-1} (I_n - \lambda W_n)\right|\\\qquad\qquad\quad\;\; -\,(1/2)(n-1)(T-1)\cdot\sigma_{nT}^{\dagger 2}(\lambda),\end{array}$$

(C.4)

whose derivation will be introduced below, where

$$\begin{array}{*{20}{l}}\quad\beta(\lambda) = \left[\mathop{\sum}\limits_{t=1}^{T} \tilde{X}_{nt}^\prime J_n \tilde{X}_{nt}\right]^{-1}\mathop{\sum}\limits_{t=1}^{T}\tilde{X}_{nt}^\prime J_n (I_n - \lambda W_n) \tilde{Y}_{nt}, \\\,\sigma^{\dagger 2}(\lambda) = \frac{1}{(n-1)(T-1)} \mathop{\sum}\limits_{t=1}^{T} \left[(I_n - \lambda W_n)\tilde{Y}_{nt} - \tilde{X}_{nt}\beta(\lambda)\right]^\prime J_n\\\qquad\qquad\; \times \left[(I_n - \lambda W_n)\tilde{Y}_{nt} - \tilde{X}_{nt}\beta(\lambda)\right]\end{array}$$

(C.5)

with ${\tilde{X}}_{nt}={X}_{nt}-{T}^{-1}\mathop{\sum }\nolimits_{t = 1}^{T}{X}_{nt}$ and ${\tilde{Y}}_{nt}={Y}_{nt}-{T}^{-1}\mathop{\sum }\nolimits_{t = 1}^{T}{Y}_{nt}$. The QMLE ${\tilde{\lambda }}_{nT}$ can be obtained by maximizing $\ln {L}_{n,T}(\lambda )$, as well as the plug-in estimator for β and σ^†2 are defined by ${\tilde{\beta }}_{nT}=\beta ({\tilde{\lambda }}_{nT})$ and ${\tilde{\sigma }}_{nT}^{{\dagger} 2}={\sigma }^{{\dagger} 2}({\tilde{\lambda }}_{nT})$, respectively.

We now discuss how to derive (C.4). First the log-likelihood function of the transformed panel SARSF model in (C.1) is

$$\begin{array}{ll}\ln L_{n,T}(\theta) = -(1/2)(n-1)(T-1)\cdot\ln(2\pi + \sigma^{\dagger 2})\\\qquad\qquad\quad\;\;+\, (T-1)\cdot\ln |I_{n-1} - \lambda W_n^*| \\\qquad\qquad\quad\;\,\,- \left[1/(2\sigma^{\dagger 2})\right]\cdot\sum\nolimits_{t=1}^{T-1}\epsilon_{nt}^{\dagger ** \prime}(\theta)\epsilon_{nt}^{\dagger **}(\theta), \end{array}$$

(C.6)

where $\theta ={[\lambda ,{\beta }^{{\prime} }]}^{{\prime} }$ and ${\epsilon }_{nt}^{{\dagger} * * }(\theta )=({I}_{n-1}-\lambda {W}_{n}^{* }){Y}_{nt}^{* * }-{X}_{nt}^{* * }\beta$. On the one hand, notice that

$$\begin{array}{*{20}{l}}\left| I_{n-1} - \lambda W_n^{\ast} \right| = \left| I_{n-1} - \lambda F_{n,n-1}^{\prime} W_n F_{n,n-1} \right| \\\qquad\qquad\quad\;\;\;\;= (1-\lambda)^{-1}\left| I_n - \lambda W_n \right|\end{array}$$

(C.7)

holds for the row-normalized W_n, which is assumed in Section 4. On the other hand, by (C.2), we obtain

$$\begin{array}{ll}&\mathop{\sum }\limits_{t=1}^{T-1}{\epsilon }_{nt}^{{\dagger} ** {\prime} }(\theta ){\epsilon }_{nt}^{{\dagger} ** }(\theta )\,\\=&\,{{{{\boldsymbol{\epsilon }}}}}_{nT}^{{\dagger} {\prime} }(\theta )({F}_{T,T-1}\otimes {F}_{n,n-1})({F}_{T,T-1}^{{\prime} }\otimes {F}_{n,n-1}^{{\prime} }){{{{\boldsymbol{\epsilon }}}}}_{nT}^{{\dagger} }(\theta )\\ =&\,{{{{\boldsymbol{\epsilon }}}}}_{nT}^{{\dagger} {\prime} }(\theta )({J}_{T}\otimes {J}_{n}){{{{\boldsymbol{\epsilon }}}}}_{nT}^{{\dagger} }(\theta )=\mathop{\sum }\limits_{t=1}^{T}{\tilde{\epsilon }}_{nt}^{{\dagger} {\prime} }(\theta ){J}_{n}{\tilde{\epsilon }}_{nt}^{{\dagger} }(\theta ),\end{array}$$

(C.8)

where ${\tilde{\epsilon }}_{nt}^{{\dagger} }(\theta )=({I}_{n}-\lambda {W}_{n}){\tilde{Y}}_{nt}-{\tilde{X}}_{nt}\beta$. Substituting (C.7) and (C.8) into (C.6) yields that

$$\begin{array}{ll}\ln L_{n,T}(\theta) = -(1/2)(n-1)(T-1)\cdot\ln(2\pi + \sigma^{\dagger 2}) \\\qquad\qquad\quad\;\;+\, (T-1) \cdot \ln \left|(1 - \lambda)^{-1}(I_{n} - \lambda W_{n})\right| \\\qquad\qquad\quad\;\;-\, [1/(2\sigma^{\dagger 2})]\cdot\sum\nolimits_{t=1}^{T}\tilde{\epsilon}_{nt}^{\dagger}(\theta) J_{n} \tilde{\epsilon}_{nt}^{\dagger}(\theta).\end{array}$$

(C.9)

Taking derivatives for $\ln {L}_{n,T}(\theta )$ with respect to σ^†2 and β, and setting such derivatives to zero, we have

$$\begin{array}{ll}&\frac{(n-1)(T-1)}{{\sigma }^{{\dagger} 2}}-\frac{1}{{\sigma }^{{\dagger} 4}}\mathop{\sum }\limits_{t=1}^{T}{\tilde{\epsilon }}_{nt}^{{\dagger} }(\theta ){J}_{n}{\tilde{\epsilon }}_{nt}^{{\dagger} }(\theta )=0\,\,{{\mbox{and}}}\,\\&\,\mathop{\sum }\limits_{t=1}^{T}{\tilde{X}}_{nt}^{{\prime} }{J}_{n}{\tilde{X}}_{nt}\beta -\mathop{\sum }\limits_{t=1}^{T}{\tilde{X}}_{nt}^{{\prime} }{J}_{n}({I}_{n}-\lambda {W}_{n}){\tilde{Y}}_{nt}=0,\end{array}$$

both which indicate (C.5) holds. Finally, the concentrated log-likelihood function in (C.4) follows from taking (C.5) instead of the counterpart in (C.9).