Heteroskedasticity testing through a comparison of Wald statistics

Abstract

This paper shows that a test for heteroskedasticity within the context of classical linear regression can be based on the difference between Wald statistics in heteroskedasticity-robust and nonrobust forms. The test is asymptotically distributed under the null hypothesis of homoskedasticity as chi-squared with one degree of freedom. The power of the test is sensitive to the choice of parametric restriction used by the Wald statistics, so the supremum of a range of individual test statistics is proposed. Two versions of a supremum-based test are considered: the first version does not have a known asymptotic null distribution, so the bootstrap is employed to approximate its empirical distribution. The second version has a known asymptotic distribution and, in some cases, is asymptotically pivotal under the null. A simulation study illustrates the use and finite-sample performance of both versions of the test. In this study, the bootstrap is found to provide better size control than asymptotic critical values, namely with heavy-tailed, asymmetric distributions of the covariates. In addition, the use of well-known modifications of the heteroskedasticity consistent covariance matrix estimator of OLS coefficients is also found to benefit the tests’ overall behaviour.

Appendix

Proof of Lemma 1

The scaled difference between Wald statistics can be successively written as

$$\begin{array}{rll} n^{-1/2}s^{2}\left( W_{R}-W_{NR}\right)&=&n^{-1/2}r\left( b\right) ^{\prime }\left[ s^{2}\left( T^{\prime }D_{e}T\right) ^{-1}-\left( T^{\prime }T\right) ^{-1}\right] r\left( b\right)\\ &=& n^{-1/2}r\left( b\right) ^{\prime }\left( T^{\prime }T\right) ^{-1}T^{\prime }\left( s^{2}I_{n}-D_{e}\right) T\left( T^{\prime }D_{e}T\right) ^{-1}r\left( b\right)\\ &=&c_{1}\left( b\right) ^{\prime }\left[ n^{-1/2}X^{\prime }\left( s^{2}I_{n}-D_{e}\right) X\right] c_{2}\left( b\right)\\ &=&c_{1}\left( b\right) ^{\prime }\left[ n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) x_{i}x_{i}^{\prime }\right] c_{2}\left( b\right)\\ &=&c_{1}\left( b\right) ^{\prime }\left[ n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left( x_{i}x_{i}^{\prime }-M_{n}\right) \right] c_{2}\left( b\right) \text{.}\end{array}$$

White (1980, Theorem 2) shows that, under homoskedasticity, the elements of

$$n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left( x_{i}x_{i}^{\prime }-M_{n}\right)$$

are asymptotically normally distributed with null means. Also, under White’s Assumptions, $n^{-1}X^{\prime }X=\Sigma _{X}+O_{p}\left ( n^{-1/2}\right )$ and $n^{-1}X^{\prime }D_{e}X=\Xi _{X}+O_{p}\left ( n^{-1/2}\right )$, so

$$\begin{array}{rll} nT^{\prime }T &=&\Sigma _{T}+O_{p}\left( n^{-1/2}\right) =R\left( \beta \right) \Sigma _{X}^{-1}R\left( \beta \right) ^{\prime }+O_{p}\left( n^{-1/2}\right) \text{,} \\ nT^{\prime }D_{e}T &=&\Xi _{T}+O_{p}\left( n^{-1/2}\right) =R\left( \beta \right) \Sigma _{X}^{-1}\Xi _{X}\Sigma _{X}^{-1}R\left( \beta \right) ^{\prime }+O_{p}\left( n^{-1/2}\right) \text{.} \end{array}$$

From the definitions of $\gamma _{j}$ and $c_{j}\left ( b\right )$, $j=1,2$ [see Eqs. 2 and 5], it then follows that $c_{j}\left ( b\right ) =\gamma _{j}+O_{p}\left ( n^{-1/2}\right )$.

Considering now each case in Lemma 1:

• (i) $r\left ( \beta \right ) \neq 0$ at the true value of the regression parameters: false auxiliary restriction. In this case, $\gamma _{j}\neq 0$, $j=1,2$, so one can write

  $$\begin{array}{lll} && n^{-1/2}s^{2}\left( W_{R}-W_{NR}\right) \\&&\qquad=\gamma _{1}^{\prime }\left[ n^{-1/2}\sum\limits_{i=1}^{n}{}\left( s^{2}-e_{i}^{2}\right) \left( x_{i}x_{i}^{\prime }-M_{n}\right) \right] \gamma _{2}+o_{p}\left( 1\right) =\delta _{n}+o_{p}\left( 1\right) \text{,} \end{array}$$

  where

  $$\delta _{n}\equiv \gamma _{1}^{\prime }\left[{} n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left( x_{i}x_{i}^{\prime }-M_{n}\right){}\right] \gamma _{2}=n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left( \alpha _{i}-\bar{\alpha}\right) \text{.}$$

  This is a linear combination of asymptotically normal elements, so, under $H_{0},\upsilon ^{-1/2}\delta _{n}\overset {D}{\longrightarrow }\mathcal {N}\left ( 0,1\right )$, with⁶

  $$\upsilon \equiv n^{-1}\sum\limits_{i=1}^{n}E\left \{ \left[ \left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \left( \alpha _{i}-\bar{\alpha}\right)\right] ^{2}\right \} \text{.}$$

  (14)

  Replacing $\gamma _{j}$ with $c_{j}\left ( b\right )$, $j=1,2$, and recalling the definition of $a_{i}$ [see (4)], one has $\upsilon ^{-1/2}n^{-1/2}s^{2}\left( W_{R}-W_{NR}\right) \overset{D}{\longrightarrow }\mathcal{N}\left( 0,1\right)$ as well. Then, the result in Part (i) of Lemma 1 follows.

• (ii) $r\left ( \beta \right ) =0$ at the true value of the regression parameters: true auxiliary restriction. In this case, $\gamma _{j}=0$ so $c_{j}\left ( b\right ) =O_{p}\left ( n^{-1/2}\right )$, $j=1,2$. Thus,

  $$\begin{array}{lll} && n^{1/2}s^{2}\left( W_{R}-W_{NR}\right)\\ &&\quad = n^{1/2}c_{1}\left( b\right) ^{\prime }n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left( x_{i}x_{i}^{\prime }-M_{n}\right) n^{1/2}c_{2}\left( b\right)\\ &&\quad = n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) n^{1/2}c_{1}\left( b\right) ^{\prime }\left( x_{i}x_{i}^{\prime }-M_{n}\right) n^{1/2}c_{2}\left( b\right) =O_{p}\left( 1\right) \text{.}\end{array}$$

  (15)

  A standard second-order Taylor expansion of the l-th component of $r\left ( b\right )$ around $\beta$ yields

  $$\begin{array}{rll} r_{l}\left( b\right)&=&r_{l}\left( \beta \right) +R_{l}\left( \beta \right) \left( b-\beta \right) +\frac{1}{2}\left( b-\beta \right) ^{\prime }D_{l}\left( \beta ^{\ast }\right) \left( b-\beta \right)\\ &=& R_{l}\left( \beta \right) \left( b-\beta \right) +O_{p}\left( n^{-1}\right) \text{, }l=1,...,j\text{,} \end{array} $$

  where $R_{l}\left ( \beta \right ) \equiv \partial r_{l}\left ( \beta \right ) /\partial \beta ^{\prime }$, $D_{l}\left ( \beta ^{\ast }\right ) \equiv \partial ^{2}r_{l}\left ( \beta ^{\ast }\right ) /\partial \beta \partial \beta ^{\prime }$ is evaluated at some convex combination of b and $\beta$ , and the last equality results from $r\left ( \beta \right ) =0$ and the fact that $b-\beta =O_{p}\left ( n^{-1/2}\right )$. Consequently [recall (5)],

  $$\begin{array}{rll} n^{1/2}c_{1}\left( b\right) &=&\Sigma _{X}^{-1}R\left( \beta \right) ^{\prime }\Sigma _{T}^{-1}R\left( \beta \right) \Sigma _{X}^{-1}n^{-1/2}X^{\prime }\varepsilon +O_{p}\left( n^{-1/2}\right) \text{,} \\ n^{1/2}c_{2}\left( b\right) &=&\Sigma _{X}^{-1}R\left( \beta \right) ^{\prime }\Xi _{T}^{-1}R\left( \beta \right) \Sigma _{X}^{-1}n^{-1/2}X^{\prime }\varepsilon +O_{p}\left( n^{-1/2}\right) \text{.} \end{array}$$

  Plugging these results in (15) and given that

  $$\begin{array}{rll} s^{2}-e_{i}^{2}&=&\sigma ^{2}+O_{p}\left( n^{-1}\right) -\left[ \varepsilon _{i}^{2}-2\varepsilon _{i}x_{i}^{\prime }\left( b-\beta \right) +\left( b-\beta \right) ^{\prime }x_{i}x_{i}^{\prime }\left( b-\beta \right) \right]\\ &=&\sigma ^{2}-\varepsilon _{i}^{2}+O_{p}\left( n^{-1/2}\right) \end{array}$$

  uniformly in i, one can write

  $$ n^{1/2}s^{2}\left( W_{R}-W_{NR}\right) =\delta _{n}^{\ast }+O_{p}\left( n^{-1/2}\right) \text{,}$$

  (16)

  with

  $$\delta _{n}^{\ast }\equiv n^{-1/2}\sum\limits_{i=1}^{n}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \left( \alpha _{i}^{\ast }-\bar{\alpha} ^{\ast }\right) \text{.}$$

  –see (6).

  Noting that $\alpha _{i}^{\ast }$ is a quadratic form in the n elements of $\varepsilon$, write the last summation as $\delta _{n}^{\ast }=n^{-1/2}\sum _{i=1}^{n}\delta _{ni}^{\ast }$, where

  $$ \delta _{ni}^{\ast }=n^{-1}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \sum\limits_{l=1}^{n}\sum\limits_{m=1}^{n}d_{ilm}\varepsilon _{l}\varepsilon _{m}\text{,}$$

  (17)

  and $d_{ilm}$ denotes the element $\left ( l,m\right )$ of the matrix $D_{i}$ , defined in (7). Under $H_{0}$ the mean of $\delta _{ni}^{\ast }$ is given by

  $$\begin{array}{rll} E\left( \delta _{ni}^{\ast }\right) &=&n^{-1}E\left[ \left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \sum\limits_{l=1}^{n}\sum\limits_{m=1}^{n}d_{ilm}\varepsilon _{l}\varepsilon _{m}\right]\\ &=&n^{-1}E\left( \left( \sigma ^{2}-\varepsilon _{i}^{2}\right) d_{iii}\varepsilon _{i}^{2}\right) +n^{-1}E\left[ \left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \underset{l\neq i\vee m\neq i}{ \sum\limits_{l=1}^{n}\sum\limits_{m=1}^{n}}d_{ilm}\varepsilon _{l}\varepsilon _{m}\right] \text{,} \end{array}$$

  which is $O\left ( n^{-1}\right )$, uniformly in i, under some bounded moment assumptions, homoskedasticity and independence. Under such assumptions, one can also show that $E\left ( \delta _{ni}^{\ast \text { } 2}\right ) =O\left ( 1\right )$ and $E\left ( \delta _{ni}^{\ast }\text { } \delta _{nl}^{\ast }\right ) =O\left ( n^{-1}\right )$, uniformly in i and l, $i\neq l$. Consequently,

  $$\lim\limits_{n\rightarrow \infty }V\left( \delta _{n}^{\ast }\right) =\lim\limits_{n\rightarrow \infty }n^{-1}\left[ \sum\limits_{i=1}^{n}V\left( \delta _{ni}^{\ast }\right) +\sum\limits_{i=1}^{n}\sum\limits_{l\neq i}^{n}COV\left( \delta _{ni}^{\ast },\delta _{nl}^{\ast }\right) \right]$$

  exists nonnull and finite. Thus, $\upsilon ^{{\ast }-1/2}\delta _{n}^{\ast }\overset {D}{\longrightarrow }\mathcal {N}\left ( 0,1\right )$ with

  $$ \upsilon ^{\ast }\equiv V\left( \delta_{n}^{\ast }\right) =n^{-1}\sum\limits_{i=1}^{n}E\left \{ \left[ \left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \left( \alpha _{i}^{\ast }-\bar{\alpha}^{\ast }\right) \right] ^{2}\right \} \text{.}$$

  (18)

  As $n^{1/2}s^{2}\left ( W_{R}-W_{NR}\right ) =\delta _{n}^{\ast }+O_{p}\left ( n^{-1/2}\right ), \upsilon ^{\ast -1/2} n^{1/2}s^{2}\left ( W_{R}-W_{NR}\right ) \overset {D}{\longrightarrow }\mathcal {N}\left ( 0,1\right )$ as well. This yields the result in part (ii).

□

Proof of Remark 1

Under homokurticity of $\varepsilon _{i}$ one can write [see (14)]

$$\upsilon =\left( \mu _{4}-\sigma ^{4}\right) n^{-1}\sum\limits_{i=1}^{n}E\left[ \left( \alpha _{i}-\bar{\alpha}\right) ^{2}\right] \text{,}$$

estimable by

$$n^{-1}\sum\limits_{i=1}^{n}\left( e_{i}^{2}-s^{2}\right) ^{2}n^{-1}\sum\limits_{i=1}^{n}\left( a_{i}-\bar{a}\right) ^{2}\text{.}$$

This factorization allows an asymptotically equivalent version of the test statistic (8) to be written as

$$\begin{array}{lll}&& \frac{\left[ n^{-1/2}\sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left( a_{i}-\bar{a}\right) \right] ^{2}}{n^{-1}\sum_{i=1}^{n}\left( e_{i}^{2}-s^{2}\right) ^{2}n^{-1}\sum_{i=1}^{n}\left( a_{i}-\bar{a}\right) ^{2}} \\ &&\quad= n\left[ \frac{\sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left( a_{i}-\bar{a }\right) }{\sum_{i=1}^{n}\left( a_{i}-\bar{a}\right) ^{2}}\right]^{2}\frac{ \sum_{i=1}^{n}\left( a_{i}-\bar{a}\right) ^{2}}{\sum_{i=1}^{n}\left( e_{i}^{2}-s^{2}\right) ^{2}}\text{,} \end{array}$$

readily recognized as $nR^{2}$ from the OLS regression of $e_{i}^{2}$ on a constant and $a_{i}$ [this reasoning closely follows the Proof of Corollary 1 of White (1980)].

In case $r\left ( \beta \right ) =0$ (true auxiliary restriction), the deduction of a similar “direct test” would require factorization of $E\left \{ \left [ \left ( \sigma ^{2}-\varepsilon _{i}^{2}\right ) \left ( \alpha _{i}^{\ast }-\bar {\alpha } ^{\ast }\right ) \right ] ^{2}\right \}$ into $\left ( \mu _{4}-\sigma ^{4}\right ) E\left [ \left ( \alpha _{i}^{\ast }-\bar {\alpha }^{\ast }\right ) ^{2}\right ]$, say [see (18)]. However, recalling the Assumptions of error independence and $E\left ( \varepsilon _{i}|x_{i}\right ) =0$, careful inspection of the expression of $\alpha _{i}^{\ast }-\bar {\alpha }^{\ast }$ in (17) allows one to write

$$\begin{array}{lll} && E\left \{ \left[ \left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \left(\alpha _{i}^{\ast }-\bar{\alpha}^{\ast }\right) \right] ^{2}\right \}\\&&\quad=n^{-1}E\left[ \left( \sigma ^{2}-\varepsilon _{i}^{2}\right) ^{2}\left( p_{i}\varepsilon _{i}^{4}+\varepsilon _{i}^{2}\sum_{l=1,l\neq i}^{n}q_{il}\varepsilon_{l}^{2}\right.\right.+\left.\left.\varepsilon _{i}\sum_{l=1,l\neq i}^{n}r_{il}\varepsilon _{l}^{3}\right) \right] \text{,} \end{array}$$

where the coefficients $p_{i}$, $q_{il}$ and $r_{il}$ do not involve the error terms (only the regressors). Consequently, validity of the suggested factorization requires an assumption of the sort $E\left ( \varepsilon _{i}^{p+q}|x_{i}\right ) =E\left ( \varepsilon _{i}^{p}|x_{i}\right ) E\left ( \varepsilon _{i}^{q}|x_{i}\right )$, $\forall i$, $q=1,2$, $p=2,4$ {the term $p_{i}\varepsilon _{i}^{4}$ is neglected because only $\lim _{n\rightarrow \infty }V\left ( \delta _{n}^{\ast }\right )$ is of interest – not $V\left ( \delta _{n}^{\ast }\right )$ per se – and $\lim _{n\rightarrow \infty }n^{-1}E\left [ \left ( \sigma ^{2}-\varepsilon _{i}^{2}\right ) ^{2}p_{i}\varepsilon _{i}^{4}\right ] =0$}. Adopting such an assumption, hardly ever justifiable in practice, would greatly restrict the range of admissible error distributions. Thus, application of Remark 1 in this case requires rather more stringent assumptions than those needed for false auxiliary restrictions. □

Proof of Corollary 1

Note, first, the following general feature of the test when $r\left ( \cdot \right )$ is a scalar function. In this case T is a column $n-$vector and $a_{i}=r\left ( b\right ) ^{2}T_{\left ( i\right ) }^{2}/\left ( T^{\prime }TT^{\prime }D_{e}T\right )$, where $T_{\left ( i\right ) }\equiv x_{i}^{\prime }\left ( X^{\prime }X\right ) ^{-1}R\left ( b\right ) ^{\prime }$ denotes the i-th element of T. Let $\overline {T^{2}}\equiv n^{-1}\sum _{i=1}^{n}T_{\left ( i\right ) }^{2}$; cancelling out constant terms, the statistic in (8) becomes

$$\frac{\left[\sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) a_{i}\right] ^{2}}{ \sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) ^{2}\left( a_{i}-\bar{a}\right) ^{2}}=\frac{\left[ \sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) T_{\left( i\right) }^{2}\right] ^{2}}{\sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) ^{2}\left( T_{\left( i\right) }^{2}-\overline{T^{2}}\right) ^{2}}\text{,} $$

which does not involve $r\left ( b\right )$ [it depends on $\beta$ only through $R\left ( b\right )$].

Considering now each statement in the Corollary:

• (i) If $r\left ( \beta \right ) =R\beta -r$, a scalar, then $\partial r\left ( \beta \right ) /\partial \beta ^{\prime }=R$, a vector of constants not involving $\beta$. Thus, from the above, the test statistic (8) no longer depends on $\beta$ and, consequently, is asymptotically pivotal (it is not pivotal, because its finite sample distribution depends upon the error distribution). The second part of the statement is obvious, given the asymptotic equivalence of Eqs. 8 and 9 with false auxiliary restrictions.

• (ii) The result is a direct consequence of the fact that $u_{i}^{2}$ in (10) is proportional to $\left [ x_{i}^{\prime }\left ( X^{\prime }X\right ) ^{-1}R^{\prime }\right ] ^{2}$ when $r\left ( \cdot \right )$ is a scalar affine function. To see this, start by writing the reparameterized model in matrix form as $y^{\ast }=X^{\ast }\beta ^{\ast }+\varepsilon$, where

  $$\begin{array}{rll}X&\equiv& \left[ \underset{n\times \left( k-1\right) }{X_{1}}\vdots \text{ }\underset{n\times 1}{x_{2}}\right] \text{, \ \ }x_{2}^{\ast }\equiv \left(1/R_{2}\right) x_{2}\text{, \ \ }y^{\ast }\equiv y-x_{2}^{\ast }\text{, \ }\beta ^{\ast }\equiv\left( \beta _{1}^{\prime },\theta \right) ^{\prime }\text{,} \\ X^{\ast }&\equiv&\left[ X_{1}^{\ast }\text{ }\vdots \text{ }x_{2}^{\ast }\right] =XA\text{, \ \ \ }\underset{k\times k}{A}=\left[\begin{array}{cc}I_{k-1} & 0 \\-\left( 1/R_{2}\right) R_{1} & 1/R_{2}\end{array}\right] \text{,}\end{array}$$

  with $\theta$ and R defined in the main text and $I_{k-1}$ the $\left ( k-1\right )$-identity matrix. Under the reparameterized model the auxiliary restriction becomes $\theta =R^{\ast }\beta ^{\ast }=0$, $R^{\ast }\equiv RA$.

As $r\left ( \cdot \right )$ is a scalar affine function [and $r\left ( \beta \right ) \neq 0$], the test can be computed as in (9), taking $nR^{2}$ from the OLS regression of $e_{i}^{2}$ on an intercept and the regressor $\left [ x_{i}^{\prime }\left ( X^{\prime }X\right ) ^{-1}R^{\prime }\right ] ^{2}$. In matrix form, the n-vector with generic element $x_{i}^{\prime }\left ( X^{\prime }X\right ) ^{-1}R^{\prime }$ can be written $X\left ( X^{\prime }X\right ) ^{-1}R^{\prime }$. As A is invertible,

$$X\left( X^{\prime }X\right) ^{-1}R^{\prime }=XA\left( A^{\prime }X^{\prime }XA\right) ^{-1}A^{\prime }R^{\prime }=X^{\ast }\left( X^{\ast \prime }X^{\ast }\right) ^{-1}R^{\ast \prime }\text{.}$$

The residuals from the original and reparameterized model ($e^{\ast }$) are equal, because

$$e^{\ast }=M^{\ast }y^{\ast }=My^{\ast }=My-\left( 1/R_{2}\right) Mx_{2}=My=e \text{,}$$

where $M^{\ast }=I-X^{\ast }\left ( X^{\ast \prime }X^{\ast }\right ) ^{-1}X^{\ast \prime }=I-X\left ( X^{\prime }X\right ) ^{-1}X^{\prime }=M$ and $Mx_{2}=0$, since M projects onto the space orthogonal to the space spanned by the columns of X. Thus, the direct test can also be computed as $nR^{2}$ from the OLS regression of $e_{i}^{\ast 2}$ on an intercept and the regressor $\left [ x_{i}^{\ast \prime }\left ( X^{\ast \prime }X^{\ast }\right ) ^{-1}R^{\ast \prime }\right ] ^{2}$. From the definition of $R^{\ast }$ and the usual formulae for the inverse of partitioned matrices, the n -vector with generic element $x_{i}^{\ast \prime }\left ( X^{\ast \prime }X^{\ast }\right ) ^{-1}R^{\ast \prime }$ is given by

$$X^{\ast }\left( X^{\ast \prime }X^{\ast }\right) ^{-1}R^{\ast \prime }=\left( x_{2}^{\ast \prime }M_{1}^{\ast }x_{2}^{\ast }\right) ^{-1}M_{1}^{\ast }x_{2}^{\ast }\text{,}$$

where $M_{1}^{\ast }\equiv I_{k-1}-X_{1}^{\ast }\left ( X_{1}^{\ast \prime }X_{1}^{\ast }\right ) ^{-1}X_{1}^{\ast \prime }$ and $X_{1}^{\ast }\equiv X_{1}-x_{2}^{\ast }R_{1}$. This is proportional to the vector of OLS residuals from the regression of $x_{2}^{\ast }$ on $X_{1}^{\ast }$, proportional, in turn, to $M_{1}^{\ast }x_{2}$, the n-vector of OLS residuals from the regression of $x_{2}$ on $X_{1}^{\ast }$. Thus, $\left [ x_{i}^{\ast \prime }\left ( X^{\ast \prime }X^{\ast }\right ) ^{-1}R^{\ast \prime }\right ] ^{2}$ and $u_{i}^{2}$ are proportional, so the regression of $e_{i}^{2}$ on an intercept and $\left [ x_{i}^{\prime }\left ( X^{\prime }X\right ) ^{-1}R^{\prime }\right ] ^{2}$ and regression (10) yield the same $nR^{2}$ statistic. □

Proof of Lemma 2

(i) $r\left ( \beta \right ) \neq 0$: false auxiliary restriction. Consider again the relation $s^{2}-e_{i}^{2}=\sigma ^{2}-\varepsilon _{i}^{2}+O_{p}\left ( n^{-1/2}\right )$ and consequently

$$ n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left( x_{i}x_{i}^{\prime }-M_{n}\right)=n^{-1/2}\sum\limits_{i=1}^{n}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \left( x_{i}x_{i}^{\prime }-M_{n}\right)+O_{p}\left( n^{-1/2}\right) \text{.}$$

Under $H_{1}$, the leading term in this expression can be written as

$$n^{-1/2}\sum\limits_{i=1}^{n}\left[ -\sigma ^{2}n^{-1/2}h\left( x_{i}\right) -u_{i} \right] \left( x_{i}x_{i}^{\prime }-M_{n}\right) \text{,} \qquad u_{i}\equiv \varepsilon _{i}^{2}-E\left( \varepsilon _{i}^{2}|x_{i}\right) \text{.} $$

Under some bounded moment assumptions and independence, the asymptotic mean of this random matrix can be written

$$\mu \equiv -\sigma ^{2}p\lim_{n\rightarrow \infty }n^{-1}\sum\limits_{i=1}^{n}h\left( x_{i}\right) \left( x_{i}x_{i}^{\prime }-M_{n}\right) \text{.}$$

Thus, under $H_{1}$, the elements of $n^{-1/2}\sum _{i=1}^{n}\left ( s^{2}-e_{i}^{2}\right ) \left ( x_{i}x_{i}^{\prime }-M_{n}\right )$ are asymptotically normal with means the corresponding elements of $\mu$. It immediately follows that $\upsilon ^{-1/2}\delta _{n}\overset {D}{ \longrightarrow }\mathcal {N}\left ( \gamma _{1}^{\prime }\mu \gamma _{2},1\right)$, with $\upsilon$ defined in (14). Then, $\upsilon ^{-1/2}n^{-1/2}s^{2}\left ( W_{R}-W_{NR}\right ) \overset {D}{ \longrightarrow }\mathcal {N}\left ( \gamma _{1}^{\prime }\mu \gamma _{2},1\right )$ as well. Standard results from Statistics ensure that, under the sequence $H_{1}$,

$$\frac{1}{\upsilon }\left[ n^{-1/2}s^{2}\left( W_{R}-W_{NR}\right) \right] ^{2}\overset{D}{\longrightarrow }\chi _{1}^{2}\left( \lambda \right) \text{,}$$

with noncentrality parameter $\lambda \equiv \left ( \gamma _{1}^{\prime }\mu \gamma _{2}\right ) ^{2}/\upsilon$. (ii) $r\left ( \beta \right ) =0$: true auxiliary restriction.

Recall expression (16) with $\delta _{n}^{\ast }$ written as

$$\delta _{n}^{\ast }=n^{-1/2}\sum\limits_{i=1}^{n}n^{-1}\sum\limits_{l=1}^{n}\sum\limits_{m=1}^{n}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) d_{ilm}\varepsilon _{l}\varepsilon _{m} \text{.} $$

Under bounded moment assumptions and independence, its asymptotic mean is

$$\begin{array}{rll} \mu ^{\ast }\equiv p\lim\limits_{n\rightarrow \infty }\delta _{n}^{\ast }&=&p\lim\limits_{n\rightarrow \infty }n^{-3/2}\sum\limits_{i=1}^{n}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \sum\limits_{l=1}^{n}d_{ill}\varepsilon _{l}^{2}\\ &=&p\lim\limits_{n\rightarrow \infty }n^{-3/2}\sum\limits_{i=1}^{n}d_{iii}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \varepsilon _{i}^{2}\\ &&+p\lim\limits_{n\rightarrow\infty }n^{-3/2}\sum\limits_{i=1}^{n}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \sum\limits_{l=1\text{, }l\neq i}^{n}d_{ill}\varepsilon _{l}^{2}\\ &=&p\lim\limits_{n\rightarrow \infty }n^{-3/2}\sum\limits_{i=1}^{n}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \sum\limits_{l=1\text{, }l\neq i}^{n}d_{ill}\varepsilon _{l}^{2}\text{,} \end{array}$$

with the first equality resulting from the fact that, under independent errors, for $l\neq m$,

$$E\left[ \left( \sigma ^{2}-\varepsilon _{i}^{2}\right) d_{ilm}\varepsilon _{l}\varepsilon _{m}\right] =0\text{,}$$

and the last equality due to

$$p\lim\limits_{n\rightarrow \infty }n^{-3/2}\sum\limits_{i=1}^{n}d_{iii}\left( \sigma ^{2}-\varepsilon _{i}^{2}\right) \varepsilon _{i}^{2}=0\text{,}$$

because of the factor $n^{-3/2}$. As before, replace $\varepsilon _{i}^{2}$ by $\sigma ^{2}\left [ 1+n^{-1/2}h\left ( x_{i}\right ) \right ] +u_{i}$, $u_{i}\equiv \varepsilon _{i}^{2}-E\left ( \varepsilon _{i}^{2}|x_{i}\right )$ . Then,

$$\begin{array}{rll} \mu ^{\ast }&=&p\lim\limits_{n\rightarrow \infty }n^{-3/2}\sum\limits_{i=1}^{n}\sum\limits_{l=1 \text{, }l\neq i}^{n}d_{ill}\left[ -\sigma ^{2}n^{-1/2}h\left( x_{i}\right) +u_{i}\right] \\ &&\times~\left \{ \sigma ^{2}\left[ 1+n^{-1/2}h\left( x_{l}\right) \right]+u_{l}\right \}\\ &=&-\sigma ^{4}p\lim\limits_{n\rightarrow \infty }n^{-2}\sum\limits_{i=1}^{n}\sum\limits_{l=1\text{, }l\neq i}^{n}d_{ill}h\left( x_{i}\right)\\ &&-~\sigma ^{4}p\lim\limits_{n\rightarrow \infty }n^{-5/2}\sum\limits_{i=1}^{n}\sum\limits_{l=1\text{, }l\neq i}^{n}d_{ill}h\left( x_{i}\right) h\left( x_{j}\right)\\ &&-~\sigma ^{2}p\lim\limits_{n\rightarrow \infty }n^{-2}\sum\limits_{i=1}^{n}\sum\limits_{l=1\text{, } l\neq i}^{n}d_{ill}h\left( x_{i}\right) u_{l}+\sigma ^{2}p\lim\limits_{n\rightarrow \infty }n^{-3/2}\sum\limits_{i=1}^{n}\sum\limits_{l=1\text{, }l\neq i}^{n}d_{ill}u_{i}\\ &&+~\sigma ^{2}p\lim\limits_{n\rightarrow \infty }n^{-2}\sum\limits_{i=1}^{n}\sum\limits_{l=1\text{, } l\neq i}^{n}d_{ill}u_{i}h\left( x_{l}\right) +p\lim\limits_{n\rightarrow \infty }n^{-3/2}\sum\limits_{i=1}^{n}\sum\limits_{l=1\text{, }l\neq i}^{n}d_{ill}u_{i}u_{l}\\ &=&-\sigma ^{4}p\lim\limits_{n\rightarrow \infty }n^{-2}\sum\limits_{i=1}^{n}\sum\limits_{l=1\text{, }l\neq i}^{n}d_{ill}h\left( x_{i}\right) \end{array}$$

(because all $p\lim$’s are null, except the first). This last expression can be written as $-\sigma ^{4}p\lim _{n\rightarrow \infty }n^{-2}\sum _{i=1}^{n}\left [ tr\left ( D_{i}\right ) -\left ( D_{i}\right ) _{i,i} \right ] h\left ( x_{i}\right )$, with $\left ( D_{i}\right ) _{i,i}$ the diagonal i-th element of the matrix $D_{i}$ defined in (7). Finally, given that $\sum _{i=1}^{n}\left ( D_{i}\right ) _{i,i}h\left ( x_{i}\right ) =O_{p}\left ( n\right )$,

$$\mu ^{\ast }=-\sigma ^{4}p\lim\limits_{n\rightarrow \infty }n^{-2}\sum\limits_{i=1}^{n}tr\left( D_{i}\right) h\left( x_{i}\right) \text{.} $$

Thus, $\upsilon ^{\ast \text { }-1/2}\delta _{n}^{\ast }\overset {D}{ \longrightarrow }\mathcal {N}\left ( \mu ^{\ast },1\right )$, with $\upsilon ^{\ast }$ defined in (18). Obviously, $\upsilon ^{\ast \text {}-1/2}n^{1/2}s^{2}\left ( W_{R}-W_{NR}\right ) \overset {D}{\longrightarrow } \mathcal {N}\left ( \mu ^{\ast },1\right )$ as well. Again, standard Statistics results ensure that under the sequence $H_{1}$,

$$\frac{1}{\upsilon ^{\ast }}\left[ n^{1/2}s^{2}\left( W_{R}-W_{NR}\right) \right] ^{2}\overset{D}{\longrightarrow }\chi _{1}^{2}\left( \lambda ^{\ast }\right) \text{,} $$

with noncentrality parameter $\lambda ^{\ast }\equiv \mu ^{\ast 2}/\upsilon ^{\ast }$. □

Proof of Lemma 3

Consider the g-th auxiliary restriction; write the corresponding element of the vector $n^{-1/2}s^{2}wd$ as

$$\begin{array}{l} n^{-1/2}s^{2}\left[ W_{R}^{\left( g\right) }-W_{NR}^{\left( g\right) }\right] =\delta _{n}^{\left( g\right) }+o_{p}\left( 1\right) \text{,} \\ \delta _{n}^{\left( g\right) }\equiv n^{-1/2}\sum\limits_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) \left[ \alpha _{i}^{\left( g\right) }-\overline{ \alpha ^{\left( g\right) }}\right] \text{,} \end{array}$$

where, as mentioned, $\alpha _{i}^{\left ( g\right ) }$ is defined analogously as $\alpha _{i}$ or $\alpha _{i}^{\ast }$, if, respectively, $r_{g}\left ( \beta \right ) \neq 0$ or $r_{g}\left ( \beta \right ) =0$, $g=1,...,m$, at the true value of the regression parameters [$\overline {\alpha ^{\left ( g\right ) }}$ is defined as the sample average of $\alpha _{i}^{\left ( g\right ) }$, $i=1,...,n$]. Under present Assumptions, with homoskedastic errors, the multivariate Liapounov central limit theorem can be applied to the random m -vector $n^{-1/2}\left ( \delta _{n}^{\left ( 1\right ) },\ldots ,\delta _{n}^{\left ( m\right ) }\right ) ^{\prime }$, that is,

$$\Psi \times n^{-1/2}\left( \delta _{n}^{\left( 1\right) },\ldots ,\delta _{n}^{\left( m\right) }\right) ^{\prime }\overset{D}{\longrightarrow } \mathcal{N}\left( 0_{m},I_{m}\right) \text{,}$$

with $\Psi$ a symmetric pd matrix such, that $\Psi ^{2}=\Upsilon ^{-1}$ and $\Upsilon$ the average covariance matrix defined in (11). The Assumptions of the present paper (namely uniform boundedness of moments of products of $\varepsilon _{i}$ and components of $x_{i}$, up to order eight) and White’s Assumption 6 ensure that $\Upsilon$ is a pd matrix with uniformly bounded elements for sufficiently large n. This ensures existence of the matrix $\Psi$. The asymptotic equivalence between each $n^{-1/2}s^{2}\left [ W_{R}^{\left ( g\right ) }-W_{NR}^{\left ( g\right ) }\right ]$ and $\delta _{n}^{\left ( g\right ) }$ then yields the statement in the Lemma. □

Proof of Corollary 2

Consider the components of the $m\times 1$ vector $\Psi n^{-1/2}s^{2}wd$,

$$\left( \Psi _{1}^{\prime }n^{-1/2}s^{2}wd,\ldots ,\Psi _{m}^{\prime }n^{-1/2}s^{2}wd\right) ^{\prime }\text{.} $$

According to Lemma 3, these components are asymptotically uncorrelated standard normal rv’s, so they are asymptotically independent. Thus, the corresponding squared variables, $\left ( \Psi _{g}^{\prime }n^{-1/2}s^{2}wd\right ) ^{2}$, are asymptotically independent chi-squared with one df. The desired result immediately follows from the well-known fact that, for independent rv’s $t_{g}$, $g=1,...,m$,

$$\Pr \left( \sup \left \{ t_{1},...,t_{m}\right \} \leq t\right) =\Pr \left( t_{1}\leq t,...,t_{m}\leq t\right) =\prod_{g=1}^{m}\Pr \left( t_{g}\leq t\right) \text{.}$$

□

Proof of Corollary 3

Consider, first, the following definitions, for scalar auxiliary restrictions ($j=1$):

If $r_{g}\left ( \beta \right ) \neq 0$,

$$\begin{array}{rll} a_{i}^{\left( g\right) }&\equiv& r_{g}\left( b\right) ^{2}T_{g\left( i\right) }^{2}/\left( T_{g}^{\prime }T_{g}T_{g}^{\prime }D_{e}T_{g}\right) \text{,} \\ T_{g}&\equiv& X\left( X^{\prime }X\right) ^{-1}R_{g}\left( b\right) ^{\prime } \text{,} \end{array}$$

with $T_{g\left ( i\right ) }$ the i-th element of $T_{g}$; if $r_{g}\left ( \beta \right ) =0$,

$$\begin{array}{rll} a_{i}^{\left( g\right) }&\equiv& nr_{g}\left( b\right) ^{2}T_{g\left( i\right) }^{2}/\left( T_{g}^{\prime }T_{g}T_{g}^{\prime }D_{e}T_{g}\right) \text{,} \\ T_{g}&\equiv& n^{1/2}X\left( X^{\prime }X\right) ^{-1}R_{g}\left( b\right) ^{\prime }\text{.} \end{array}$$

Thus, if $r_{g}\left ( \beta \right ) \neq 0$,

$$s^{2}\left[ W_{R}^{\left( g\right) }-W_{NR}^{\left( g\right) }\right] =r_{g}\left( b\right) ^{2}/\left( T_{g}^{\prime }T_{g}T_{g}^{\prime }D_{e}T_{g}\right) \sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) ^{2}T_{g\left( i\right) }^{2} $$

and, if $r_{g}\left ( \beta \right ) =0$,

$$s^{2}\left[ W_{R}^{\left( g\right) }-W_{NR}^{\left( g\right) }\right] =nr_{g}\left( b\right) ^{2}/\left( T_{g}^{\prime }T_{g}T_{g}^{\prime }D_{e}T_{g}\right) \sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) ^{2}T_{g\left( i\right) }^{2}\text{.}$$

Define the $m\times m$ diagonal matrix $DR\left ( b\right )$ with g-th diagonal entry $r_{g}\left ( b\right ) ^{2}/\left ( T_{g}^{\prime }T_{g}T_{g}^{\prime }D_{e}T_{g}\right )$ or n times this [if $r_{g}\left ( \beta \right ) \neq 0$ or $r_{g}\left ( \beta \right ) =0$, respectively]. Then, the matrix V can be written as

$$ V=DR\left( b\right) \times M\times DR\left( b\right) \text{,}$$

(19)

where M is $m\times m$ symmetric and pd for large enough n, with generic element

$$M_{gh}\equiv \sum_{i=1}^{n}\left( s^{2}-e_{i}^{2}\right) ^{2}\left[ T_{g\left( i\right) }^{2}-\overline{T_{g}^{2}}\right] \left[ T_{h\left( i\right) }^{2}-\overline{T_{h}^{2}}\right] \text{, \ \ }g,h=1,...,m\text{.} $$

Given the definition of $T_{g}$, $g=1,...,m$, it is clear that M depends on $\beta$ only through the derivatives $R_{g}\left ( b\right ) \equiv \partial r_{g}\left ( b\right ) /\partial b^{\prime }$ – not through the functions $r_{g}\left ( b\right )$ themselves.

From (19),

$$V^{-1}=DR\left( b\right) ^{-1}\times M^{-1}\times DR\left( b\right) ^{-1} \text{,}$$

where $DR\left ( b\right ) ^{-1}$ is diagonal with g-th entry $\left ( T_{g}^{\prime }T_{g}T_{g}^{\prime }D_{e}T_{g}\right ) /r_{g}\left ( b\right ) ^{2}$ or this quantity divided by n [if $r_{g}\left ( \beta \right ) \neq 0$ or $r_{g}\left ( \beta \right ) =0$, respectively]. Thus, considering PM, the symmetric square root matrix of $M^{-1}$,

$$V^{-1}=DR\left( b\right) ^{-1}\times PM^{2}\times DR\left( b\right) ^{-1}= \left[ DR\left( b\right) ^{-1}PM\right] ^{2}=P^{2}\text{,} $$

from which $P=DR\left ( b\right ) ^{-1}PM$, where PM depends on $\beta$ only through the derivatives $R_{g}\left ( b\right )$, $g=1,...,m$.

Then, as PM is symmetric, $P\times wd=PM\times \left [ DR\left ( b\right ) ^{-1}wd\right ]$, which is an m-vector depending on the auxiliary restrictions only through $R_{g}\left ( b\right )$, $g=1,...,m$, because the functions $r_{g}\left ( b\right )$ are canceled out in $DR\left ( b\right ) ^{-1}wd$. Thus, when all the functions $r_{g}\left ( \cdot \right )$, $g=1,...,m$, are scalars, the vector $Pn^{-1/2}s^{2}wd$ does not depend directly on the value of the $r_{g}\left ( \beta \right )$.

Now, if $r_{g}\left ( \beta \right )$ is scalar affine, that is $r_{g}\left ( \beta \right ) =R_{g}\beta -q$, then $R_{g}\left ( b\right ) =R_{g}$, $g=1,...,m$ vectors of constants not involving b. Thus, M (and PM) do not involve b, the only link of $P\times wd$ to $\beta$. Therefore $P\times wd$ is a vector of asymptotically pivotal statistics. From Lemma 3, these statistics are asymptotically independent.

For independent rv’s $t_{g}$, $g=1,...,m$, $\Pr \left ( \sup \left \{ t_{1},...,t_{m}\right \} \leq v\right ) =\prod_{g=1}^{m}\Pr \left ( t_{g}\leq t\right )$. If every $t_{g}$ is asymptotically pivotal for all DGP’s in $H_{0}$, then $\Pr \left ( t_{g}\leq t\right )$, $g=1,...,m –$ and so $\Pr \left ( \sup \left \{ t_{1},...,t_{m}\right \} \leq t\right ) –$ is invariant under all DGP’s in $H_{0}$. Thus, $\sup \left \{ P_{1}^{\prime }wd,\ldots ,P_{m}^{\prime }wd\right \}$ is asymptotically pivotal because the $P_{g}^{\prime }wd$, $g=1,...,m$, are asymptotically pivotal independent statistics. Obviously, this statement applies to $\sup \left \{ \left ( P_{1}^{\prime }wd\right ) ^{2},\ldots ,\left ( P_{m}^{\prime }wd\right ) ^{2}\right \}$ as well. □