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An Asymptotically F-Distributed Chow Test in the Presence of Heteroscedasticity and Autocorrelation
For predictive modeling and policy analysis using time series data, it is important to check whether a structural relationship is stable over time. The CHOW1960 test is designed to test whether a break takes place at a given period in an otherwise stable relationship. The test is widely used in empirical applications and has been included in standard econometric textbooks. This paper considers the Chow test in the presence of heteroscedasticity and autocorrelation. There is ample evidence that the Chow test can have very large size distortions if heteroscedasticity and autocorrelation are not accounted for (e.g., KRAMER1989 and GILES1992). Even if we account for them using heteroscedasticity and autocorrelation robust (HAR) variance estimators (e.g., NW1987 and Andrews1991), the test can still over-reject the null hypothesis by a large margin if chi-square critical values are used\footnote{When the Chow test is performed on a single coefficient, normal critical values are typically used on the t statistic. For now, we focus only on the Wald-type Chow test for more than one coefficients so that chi-square critical values are used.}. This is a general problem for any HAR inference, as the chi-square approximation ignores the often substantial finite sample randomness of the HAR variance estimator. To address this problem, the recent literature has developed a new type of asymptotics known as fixed-smoothing asymptotics (see, e.g., KV2002a, KV2002b, KV2005 for early seminal contributions). It is now well known that the fixed-smoothing asymptotic approximation is more accurate than the chi-square approximation. This has been confirmed by ample simulation evidence and supported by higher-order asymptotic theory\ in J2004 and \citet*{SPJ2008} .
In this study, we employ the series HAR variance estimator to implement the Chow test in a time series regression where the regressors and regression errors are autocorrelated. This type of HAR variance estimator is the series counterpart of the kernel HAR variance estimator. The advantage of using the series HAR variance estimator is that we can design the basis functions so that the fixed-smoothing asymptotic distribution is the standard F distribution. This is in contrast to commonly used kernel HAR variance estimators where the fixed-smoothing asymptotic distributions are nonstandard and critical values have to be simulated\footnote{In the series case, fixed-smoothing asymptotics holds the number of basis functions fixed as the sample size increases. In the kernel case, fixed-smoothing asymptotics holds the truncation lag parameter fixed at a certain proportion of the sample size.}.
To establish the asymptotic\ F theory for the Chow test under fixed-smoothing asymptotics, we have to transform the usual orthonormal bases such as sine and cosine bases using the Gram--Schmidt orthonormalization. This is because, unlike the HAR inference in a regression with stationary regressors and regression errors, using the usual bases as in S2013 does not lead to a standard fixed-smoothing asymptotic distribution, since the regressors in the regression for the structural break test are identically zero before or after the break point and are thus not stationary. The Gram--Schmidt orthonormalization ensures that the transformed bases are orthonormal with respect to a special inner product that is built into the problem under consideration. The asymptotic F test is very convenient to use, as the F critical values are readily available from standard statistical tables and programming environments.
Monte Carlo simulation experiments show that the F test based on the transformed Fourier bases is as accurate as the nonstandard test based on the usual Fourier bases. The F test and nonstandard test have the same size-adjusted power as the corresponding chi-square tests but much more accurate size. Given its convenience, competitive power, and higher size accuracy, we recommend the F test for practical use.
Our F test theory generalizes the classical Chow test in a linear normal regression where the F distribution is the exact finite sample distribution. The main departures are that we do not make the normality assumption and that we allow for heteroscedasticity and autocorrelation of unknown forms. Without restrictive assumptions such as normality and strict exogeneity, it is in general not possible to obtain the exact finite sample distribution. Instead, we employ the fixed-smoothing asymptotics to show that the Wald statistic is asymptotically F distributed.
This study contributes to the asymptotic F test theory in the HAR literature. The asymptotic F theory has been developed in a number of papers including S2011, SK2012, S2013, HS2017, LLSW2018, LS2018, WS2019, MSW2019. However, none of these studies considers the case where the regressors take the special form of nonstationarity as we consider here. Cho_Vogelsang2017 consider fixed-b asymptotics for testing structural breaks, but they consider only kernel HAR variance estimators. As a result, the fixed-smoothing asymptotic distributions they obtained are highly nonstandard.
The rest of this paper is organized as follows. Section (ref) presents the basic setting and introduces the test statistics. Section (ref) establishes the fixed-smoothing asymptotics of the F and t statistics. Section (ref) develops asymptotically valid F and t tests. Section (ref) extends the basic regression model to include other covariates whose coefficients are known to be stable over time. Section (ref) reports the simulation evidence. The last section concludes. Proofs are given in the appendix.
Given the time series observations $\left\{ X_{t}\in\mathbb{R}^{m},Y_{t} \in\mathbb{R}\right\} _{t=1}^{T},$ we consider the model \[ Y_{t}=X_{t}\cdot1\left\{ t\leq\left[ \lambda T\right] \right\} \cdot \beta_{1}+X_{t}\cdot1\left\{ t\geq\left[ \lambda T\right] +1\right\} \cdot\beta_{2}+u_{t}, \] for $t=1,2,\ldots,T$ where the unobserved $u_{t}$ satisfies $E\left( X_{t}u_{t}\right) =0.$ In the above, $\lambda$ is a known parameter in $(0,1)$ so that $\left[ \lambda T\right] $ is the period where the structural break may take place. The effects of $X_{t}$ on $Y_{t}$ before and after the break are $\beta_{1}\in\mathbb{R}^{m}$ and $\beta_{2}\in \mathbb{R}^{m},$ respectively. We allow $X_{t}u_{t}$ to exhibit autocorrelation of unknown forms. In particular, we allow $u_{t}$ to be heteroskedastic so that $E(u_{t}^{2}|X_{t})$ is a nontrivial function of $X_{t}.$
We are interested in testing the null of $H_{0}$: $\mathcal{R}\beta _{1}=\mathcal{R}\beta_{2}$ against the alternative $H_{1}:$ $\mathcal{R} \beta_{1}\neq\mathcal{R}\beta_{2}$ for some $p\times m$ matrix $\mathcal{R}.$ When $\mathcal{R}$ is the $m\times m$ identity matrix, we aim at testing whether $\beta_{1}$ is equal to $\beta_{2}.$ For the moment, we consider the case that all coefficients are subject to a possible break. In Section (ref), we consider the case that some of the coefficients are known to be time invariant.
Let \[ X_{1t}=X_{t}\cdot1\left\{ t\leq\left[ \lambda T\right] \right\} \text{, }X_{2t}=X_{t}\cdot1\left\{ t\geq\left[ \lambda T\right] +1\right\} . \] Note that both $X_{1t}$ and $X_{2t}$ are nonstationary. The form of the nonstationarity makes the problem at hand unique. Let $\beta=\left( \beta _{1}^{\prime},\beta_{2}^{\prime}\right) ^{\prime}$ and $\tilde{X}_{t} =(X_{1t},X_{2t}).$ Then \[ Y_{t}=\tilde{X}_{t}\beta+u_{t}, \] and the hypotheses of interest become $H_{0}:R\beta=0$ and $H_{1}:R\beta\neq0$ for $R=[\mathcal{R},-\mathcal{R}]\in\mathbb{R}^{p\times2m}.$
Denote $\tilde{X}=(\tilde{X}_{1}^{\prime},\ldots,\tilde{X}_{T}^{\prime })^{\prime}$, $Y=\left( Y_{1},\ldots,Y_{T}\right) ^{\prime},$ and $u=\left( u_{1},\ldots,u_{T}\right) ^{\prime}.$ We estimate $\beta$ by OLS: \[ \hat{\beta}=(\tilde{X}^{\prime}\tilde{X})^{-1}\tilde{X}^{\prime}Y. \] The OLS estimator $\hat{\beta}$ satisfies \[ \sqrt{T}(\hat{\beta}-\beta)=\hat{Q}^{-1}\frac{1}{\sqrt{T}}\sum_{t=1}^{T} \tilde{X}_{t}^{\prime}u_{t}, \] where \[ \hat{Q}=\frac{\tilde{X}^{\prime}\tilde{X}}{T}=\left(
\right) \] and $O$ is a matrix of zeros. To make inferences on $\beta$ such as testing whether $R\beta$ is zero, we need to estimate the variance of $T^{-1/2} \sum_{t=1}^{T}\tilde{X}_{t}^{\prime}u_{t}.$ To this end, we first construct the residual $\hat{u}_{t}=Y_{t}-\tilde{X}_{t}\hat{\beta},$ which serves as an estimate for $u_{t}.$ Given a set of basis functions $\left\{ \phi_{j}\left( \cdot\right) \right\} _{j=1}^{K},$ we then construct the series estimator of the variance as \[ \hat{\Omega}=\frac{1}{K}\sum_{j=1}^{K}\left[ \frac{1}{\sqrt{T}}\sum_{t=1} ^{T}\phi_{j}\left( \frac{t}{T}\right) \tilde{X}_{t}^{\prime}\hat{u} _{t}\right] ^{\otimes2}, \] where, for a column vector $a,$ $a^{\otimes2}$ is the outer product of $a,$ that is, $a^{\otimes2}=aa^{\prime}.$ The asymptotic variance of $R\sqrt {T}(\hat{\beta}-\beta)$ is then estimated by $R\hat{Q}^{-1}\hat{\Omega}\hat {Q}^{-1}R^{\prime}.$
The Wald statistic for testing $H_{0}:R\beta=0$ against $H_{1}:R\beta\neq0$ is \[ F_{T}=T\cdot(R\hat{\beta})^{\prime}\left[ R\hat{Q}^{-1}\hat{\Omega}\hat {Q}^{-1}R^{\prime}\right] ^{-1}(R\hat{\beta}). \] When $p=1$ and we test $H_{0}:R\beta=0$ against a one-sided alternative, say, $H_{1}:R\beta>0$, we can construct the t statistic: \[ t_{T}=\frac{\sqrt{T}\cdot R\hat{\beta}}{\left[ R\hat{Q}^{-1}\hat{\Omega} \hat{Q}^{-1}R^{\prime}\right] ^{1/2}}. \] The forms of the F and t statistics are standard.
To establish the asymptotic distributions of $F_{T}$ and $t_{T},$ we maintain the following three assumptions:
Note that $\frac{1}{\lambda}\int_{0}^{\lambda}dW_{m}\left( \lambda\right) $ and $\frac{1}{1-\lambda}\int_{\lambda}^{1}dW_{m}\left( \lambda\right) $ are the average changes of the Brownian motion over the intervals $\left[ 0,\lambda\right] $ and $\left[ \lambda,1\right] ,$ respectively. Lemma (ref) shows that $\sqrt{T}(\hat{\beta}_{1}-\beta_{1})$ and $\sqrt {T}(\hat{\beta}_{2}-\beta_{2})$ are (matrix) proportional to the average changes.\ Given the independence of these changes over any non-overlapping intervals, $\sqrt{T}(\hat{\beta}_{1}-\beta_{1})$ and $\sqrt{T}(\hat{\beta} _{2}-\beta_{2})$ are asymptotically independent.
Note that $\bar{\phi}_{j,1}$ can be regarded as an average of $\phi_{j}\left( \cdot\right) $ over the interval $\left[ 0,\lambda\right] .$ Similarly, $\bar{\phi}_{j,2}$ can be regarded as an average of $\phi_{j}\left( \cdot\right) $ over the interval $\left[ \lambda,1\right] .$ So $\phi _{j}\left( r\right) -\bar{\phi}_{j,1}$ and $\phi_{j}\left( r\right) -\bar{\phi}_{j,2}$ are the demeaned versions of $\phi_{j}\left( r\right) $ over the intervals $\left[ 0,\lambda\right] $ and $\left[ \lambda,1\right] ,$ respectively.
Using Lemma (ref), we can prove our main theorem below.
Like the finite sample distributions, the limiting distributions of $F_{T}$ and $t_{T}$ depend on $\lambda$ and the number and form of the basis functions. This is an attractive feature of the fixed-smoothing approximations, as they capture the effects of all these factors. More importantly, the fixed-smoothing approximations capture the randomness of the HAR variance estimator, which clearly affects the finite sample distributions of $F_{T}$ and $t_{T}.$ This is why the fixed-smoothing asymptotic approximations are more accurate than the chi-square or normal approximations.
The limiting distributions $F_{\infty}$ and $t_{\infty}$ in Theorem (ref) are pivotal but nonstandard. We can approximate the nonstandard distributions using a chi-square or t distribution. We can also design a new set of basis functions so that $F_{\infty}$ and $t_{\infty}$ become the standard F and t distributions after some multiplicative adjustment.
Define \[ \tilde{\phi}_{0}\left( r;\lambda\right) =\frac{1}{\lambda}1\left\{ r\leq\lambda\right\} -\frac{1}{1-\lambda}1\left\{ r>\lambda\right\} . \] Then
and so
As a result, \[ \lambda\left( 1-\lambda\right) F_{T}\rightarrow^{d}\eta_{0}^{\prime}\left( \frac{1}{K}\sum_{j=1}^{K}\eta_{j}\eta_{j}^{\prime}\right) ^{-1}\eta_{0}\text{ and }\sqrt{\lambda\left( 1-\lambda\right) }t_{T}\rightarrow^{d}\eta _{0}^{\prime}\left( \frac{1}{K}\sum_{j=1}^{K}\eta_{j}\eta_{j}^{\prime }\right) ^{-1/2}, \] where \[ \eta_{j}=\int_{0}^{1}\tilde{\phi}_{j}\left( r;\lambda\right) dW_{p}\left( r\right) \text{ for }j=1,\ldots,K. \]
When $K$ is relatively large, it is reasonable to approximate $K^{-1} \sum_{j=1}^{K}\eta_{j}\eta_{j}^{\prime}$ by its mean: \[ E\left[ \frac{1}{K}\sum_{j=1}^{K}\eta_{j}\eta_{j}^{\prime}\right] =I_{p}\cdot\frac{1}{K}\sum_{j=1}^{K}\int_{0}^{1}\left[ \tilde{\phi} _{j}\left( r;\lambda\right) \right] ^{2}dr. \] With such an approximation, we have
where `$\thicksim^{a}$' signifies distributional approximations. As a result, we can employ the following approximations:
where $\tilde{\phi}_{j,T}\left( r;\lambda\right) $ is the finite sample version of $\tilde{\phi}_{j}\left( r;\lambda\right) $ given by
It is important to point out that the chi-square and normal approximations are not based on the original Wald and t statistics but rather on their modified versions $F_{T}^{\ast}$ and $t_{T}^{\ast}$. To a great extent, the chi-square and normal approximations we propose here improve upon the conventional chi-square and normal approximations that are applied directly to the original Wald and t statistics.
Note that the chi-square distribution and standard normal distribution in ((ref)) and ((ref)) are not the asymptotic distributions of $F_{T}^{\ast}$ and $t_{T}^{\ast}$ for a fixed $K.$ The fixed-$K$ asymptotic distributions are given by
These follow directly from Theorem (ref). The chi-square distribution and standard normal distribution are only approximations to the above nonstandard fixed-K asymptotic distributions.
To obtain convenient fixed-K asymptotic approximations, we note that for each $j=0,1,\ldots,K,$ $\eta_{j}$ is normal. For each $j\neq0,$ we have
So $\eta_{0}$ is independent of $\eta_{j}$, $j=1,\ldots,K.$ In addition, \[ cov\left( \eta_{j_{1}},\eta_{j_{2}}\right) =\int_{0}^{1}\tilde{\phi}_{j_{1} }\left( r;\lambda\right) \tilde{\phi}_{j_{2}}\left( r;\lambda\right) dr. \] Therefore, if $\{\tilde{\phi}_{j}\left( r;\lambda\right) \}$ are orthonormal, then $\eta_{j}$ for $j=0,1,\ldots,K$ are independent standard normals. In this case, $\lambda\left( 1-\lambda\right) F_{\infty}$ is a quadratic form in a standard normal vector with an independent weighting matrix. After some adjustment, we can show that $\lambda\left( 1-\lambda \right) F_{\infty}$ is equal to a standard F distribution and that $F_{T}$ converges to the F distribution. Similarly, $\sqrt{\lambda\left( 1-\lambda\right) }\cdot t_{T}$ converges to Student's t distribution.
This is a very convenient result, as the fixed-smoothing asymptotic approximations are standard distributions and there is no need to simulate critical values.
When $\{\tilde{\phi}_{j}\left( r;\lambda\right) \}$ are orthonormal, we have $K^{-1}\sum_{j=1}^{K}\int_{0}^{1}\left[ \tilde{\phi}_{j}\left( r;\lambda\right) \right] ^{2}dr=1.$ In view of this, we can see that the definitions of $\tilde{F}_{T}^{\ast}$ and $\tilde{t}_{T}^{\ast}$ are similar to those of $F_{T}^{\ast}$ and $t_{T}^{\ast}$ given in ((ref)) and ((ref)). The only difference is that there is an additional degrees-of-freedom-adjustment factor in $\tilde{F}_{T}^{\ast}$ when $p>1.$
To design the basis functions such that $\{\tilde{\phi}_{j}\left( r;\lambda\right) \}$ are orthonormal, we need the following lemma.
Let
be the transformed Brownian motion. Then we have \[ \int_{0}^{1}\tilde{\phi}_{j_{1}}\left( r;\lambda\right) dW_{p}\left( r\right) =\int_{0}^{1}\phi_{j}\left( r\right) dW_{p}(r;\lambda), \] and \[ E\left[ dW_{p}\left( r;\lambda\right) dW_{p}^{\prime}\left( s;\lambda \right) \right] =I_{p}\cdot C(r,s;\lambda)drds. \] Therefore, $C(r,s;\lambda)$ can be regarded as the covariance kernel function for the transformed Brownian motion.
To design the basis functions $\left\{ \phi_{j}\left( r\right) \right\} $ such that $\{\tilde{\phi}_{j}\left( r;\lambda\right) \}$ are orthonormal on $L^{2}[0,1],$ we require that $\left\{ \phi_{j}\left( r\right) \right\} $ be orthonormal with respect to the covariance kernel function $C(r,s;\lambda ),$ that is,
This can be achieved by applying the Gram--Schmidt orthonormalization to any set of basis functions on $L^{2}[0,1]$. The chart below illustrates the procedure:
In the above, $\left\{ \phi_{j}\right\} $ is the initial set of basis functions, and $\{\phi_{j}^{\ast}\}$ is the Gram-Schmidt orthonormalized set. \textquotedblleft$\phi_{j}\rightarrow\tilde{\phi}_{j}$\textquotedblright\ and \textquotedblleft$\phi_{j}^{\ast}\rightarrow\tilde{\phi}_{j}^{\ast} $\textquotedblright\ reflect the effect of the estimation error in estimating $\beta:$ had we known $\beta,$ we would have used the true $u_{t}$ instead of $\hat{u}_{t}$ in constructing the variance estimator, and the key elements of the weighting matrix in ((ref)) in Theorem (ref) would have been $\int_{0}^{1}\phi_{j}\left( r\right) dW_{p}\left( r\right) $ instead of $\int_{0}^{1}\tilde{\phi}_{j}\left( r;\lambda\right) dW_{p}\left( r\right) .$ The Gram-Schmidt orthonormalization ensures that $\{\phi_{j}^{\ast}\}$ are orthonormal with respect to the covariance kernel $C(r,s;\lambda):\int_{0}^{1}\int_{0}^{1}\phi_{j_{1}}^{\ast}\left( r\right) \phi_{j_{2}}^{\ast}\left( s\right) C(r,s;\lambda)drds=1\left\{ j_{1} =j_{2}\right\} .$ In view of \[ \int_{0}^{1}\int_{0}^{1}\phi_{j_{1}}^{\ast}\left( r\right) \phi_{j_{2} }^{\ast}\left( s\right) C(r,s;\lambda)drds=\int_{0}^{1}\tilde{\phi}_{j_{1} }^{\ast}\left( r\right) \tilde{\phi}_{j_{2}}^{\ast}\left( r\right) dr, \] we have: $\{\tilde{\phi}_{j}^{\ast}\}$ are orthonormal on $L^{2}\left[ 0,1\right] .$
If we use $\{\phi_{j}^{\ast}\}$ in constructing the variance estimator, then \[ \lambda\left( 1-\lambda\right) F_{T}\rightarrow^{d}\eta_{0}^{\prime}\left( \frac{1}{K}\sum_{j=1}^{K}\eta_{j}\eta_{j}^{\prime}\right) ^{-1}\eta_{0} \] for $\eta_{j}=\int_{0}^{1}\tilde{\phi}_{j}^{\ast}\left( r;\lambda\right) dW_{p}\left( r\right) \thicksim iidN(0,I_{p})$ because $\{\tilde{\phi} _{j}^{\ast}\}$ are orthonormal on $L^{2}\left[ 0,1\right] .$ Moreover, for $j=1,\ldots,K,$ $\eta_{j}$ is independent of $\eta_{0}.$ Therefore, the asymptotic F theory in Proposition (ref) holds. Similarly, the asymptotic t theory holds.
Instead of searching for the basis functions that satisfy ((ref)), we search for their discrete versions: the basis vectors. For each basis function $\phi_{k}\left( r\right) ,$ the corresponding basis vector is defined as \[ \boldsymbol{\phi}_{k}=\left( \phi_{k}\left( \frac{1}{T}\right) ,\phi _{k}\left( \frac{2}{T}\right) ,\ldots,\phi_{k}\left( \frac{T}{T}\right) \right) ^{\prime}. \] Let $\mathbf{C}_{T}:=\mathbf{C}_{T}\left( \lambda\right) $ be the $T\times T$ matrix whose $\left( i,j\right) $-th element is equal to
By definition, $\mathbf{C}_{T}$ is symmetric and positive-definite. It is the discrete version of $C(r,s;\lambda).$ For any two vectors $r_{1},r_{2} \in\mathbb{R}^{T}$, we define the inner product
Then the discrete analogue of ((ref)) is
Given any basis vectors $\boldsymbol{\phi}_{1},\ldots,\boldsymbol{\phi}_{K},$ we now apply the Gram--Schmidt orthonormalization via the Cholesky decomposition. Let $\boldsymbol{\phi}=(\boldsymbol{\phi}_{1},\mathbf{\ldots },\boldsymbol{\phi}_{K})$ be the $T\times K$ matrix of basis vectors. Let $U_{T}\in\mathbb{R}^{K\times K}$\ be the upper triangular factor in the Cholesky decomposition of $\boldsymbol{\phi}^{\prime}\mathbf{C}_{T} \boldsymbol{\phi}/T^{2}$ such that $\boldsymbol{\phi}^{\prime}\mathbf{C} _{T}\boldsymbol{\phi}/T^{2}=U_{T}^{\prime}U_{T}.$ Define \[ \boldsymbol{\phi}^{\ast}=\boldsymbol{\phi}U_{T}^{-1}:=(\boldsymbol{\phi} _{1}^{\ast},\mathbf{\ldots},\boldsymbol{\phi}_{K}^{\ast}). \] We then have \[ (\boldsymbol{\phi}^{\ast})^{\prime}\mathbf{C}_{T}\boldsymbol{\phi}^{\ast }/T^{2}=\left( U_{T}^{\prime}\right) ^{-1}\boldsymbol{\phi}^{\prime }\mathbf{C}_{T}\boldsymbol{\phi}U_{T}^{-1}/T^{2}=\left( U_{T}^{\prime }\right) ^{-1}U_{T}^{\prime}U_{T}U_{T}^{-1}=I_{K}. \] That is, the columns of the matrix $\boldsymbol{\phi}^{\ast}$ satisfy the conditions in ((ref)).
Note that the $\left( k_{1},k_{2}\right) $-th element of $\boldsymbol{\phi }^{\prime}\mathbf{C}_{T}\boldsymbol{\phi}/T^{2}$ satisfies
This implies that $U_{T}$ converges to the upper triangular factor of the Cholesky decomposition of var$(\eta_{1},\ldots,\eta_{K}).$ As a result, every transformed basis vector is approximately equal to a linear combination of the original basis vectors. The implied basis functions are thus equal to linear combinations of the original basis functions. Therefore, if Assumption (ref) holds for the original basis functions, it also holds for the transformed basis functions. It then follows that Proposition (ref) holds when $\left\{ \boldsymbol{\phi}_{1}^{\ast },\mathbf{\ldots},\boldsymbol{\phi}_{K}^{\ast}\right\} $ are used as the basis vectors in constructing the asymptotic variance estimator. More specifically, if we estimate $\Omega$ by \[ \hat{\Omega}=\frac{1}{K}\sum_{j=1}^{K}\left[ \frac{1}{\sqrt{T}}\sum_{t=1} ^{T}\phi_{j,t}^{\ast}\tilde{X}_{t}^{\prime}\hat{u}_{t}\right] ^{\otimes2}, \] where $\phi_{j,t}^{\ast}$ is the $t$-th element of the vector $\boldsymbol{\phi}_{j}^{\ast},$ then the asymptotic F and t results in Proposition (ref) hold.
Suppose there is another covariate vector $Z_{t}\in\mathbb{R}^{\ell}$ whose effect on $Y_{t}$ does not change over time so that we have the model: \[ Y_{t}=X_{t}\cdot1\left\{ t\leq\left[ \lambda T\right] \right\} \cdot \beta_{1}+X_{t}\cdot1\left\{ t>\left[ \lambda T\right] +1\right\} \cdot\beta_{2}+Z_{t}\gamma+u_{t}. \] Let $Z=(Z_{1}^{\prime},\ldots,Z_{T}^{\prime})^{\prime}$ and $M_{Z} =I_{T}-Z(Z^{\prime}Z)^{-1}Z^{\prime}.$ Then \[ M_{Z}Y=M_{Z}\tilde{X}\beta+M_{Z}u. \] The OLS estimator of $\beta=\left( \beta_{1}^{\prime},\beta_{2}^{\prime }\right) ^{\prime}$ is now \[ \hat{\beta}=(\tilde{X}^{\prime}M_{Z}\tilde{X})^{-1}\tilde{X}^{\prime}M_{Z}Y. \] Let $\hat{u}=\left( \hat{u}_{1},\ldots,\hat{u}_{T}\right) ^{\prime} =M_{Z}Y-M_{Z}\tilde{X}\hat{\beta}=M_{Z}u-M_{Z}\tilde{X}(\hat{\beta}-\beta)$ and $\tilde{X}_{z}=(\tilde{X}_{z,1}^{\prime},\ldots,\tilde{X}_{z,T}^{\prime })^{\prime}=M_{Z}\tilde{X}.$ Define \[ \hat{Q}_{\tilde{X}\cdot Z}=\frac{\tilde{X}^{\prime}M_{Z}\tilde{X}}{T}\text{ and }\hat{\Omega}=\frac{1}{K}\sum_{j=1}^{K}\left[ \frac{1}{\sqrt{T}} \sum_{t=1}^{T}\phi_{j}\left( \frac{t}{T}\right) \tilde{X}_{z,t}^{\prime} \hat{u}_{t}\right] ^{\otimes2}. \] The Wald statistic for testing $H_{0}:R\beta=0$ against $H_{1}:R\beta\neq0$ takes the same form as before: \[ F_{T}=T\cdot(R\hat{\beta})^{\prime}\left[ R\hat{Q}^{-1}\hat{\Omega}\hat {Q}^{-1}R^{\prime}\right] ^{-1}(R\hat{\beta}). \] When $p=1,$ we construct the t statistic: \[ t_{T}=\frac{\sqrt{T}\cdot R\hat{\beta}}{\left[ R\hat{Q}^{-1}\hat{\Omega} \hat{Q}^{-1}R^{\prime}\right] ^{1/2}}. \]
To establish the asymptotic distributions of $F_{T}$ and $t_{T},$ we maintain the two assumptions below, which are analogous to Assumptions (ref) and (ref).
We partition $Q$ and $\Lambda$ according to \[ Q=\left(
\right) and \Lambda=\left(
\right) , \] where $Q_{XX}\in\mathbb{R}^{m\times m},Q_{ZZ}\in\mathbb{R}^{\ell\times\ell},$ $\Lambda_{X}\in\mathbb{R}^{m\times(\ell+m)},$ and $\Lambda_{Z}\in \mathbb{R}^{\ell\times(\ell+m)}.$
Theorem (ref) shows that the limiting distributions of the Wald statistic and t statistic are the same as in the case without the extra covariate $Z_{t}.$ The asymptotic F and t limit theory can be developed in exactly the same way as in Section (ref). We present the result formally as a corollary.
In this section, we investigate the finite sample properties of the proposed F test. We consider the linear regression model with $m=2$ and $X_{t} =(1,q_{t}).$ The regressor $q_{t}$ follows an AR(1) process, and the error $u_{t}$ follows an independent AR(1) or ARMA(1,1) process. That is,
where\ both $\epsilon_{q,t}$ and $\epsilon_{u,t}$ are iid $N(0,1)$ and $\left\{ \epsilon_{q,t},t=1,\ldots,T\right\} $ are independent of $\left\{ \epsilon_{u,t}:t=1,2,\ldots,T\right\} .$ Note that the AR parameter $\rho$ is the same for $q_{t}$ and $u_{t}.$
We consider the sample sizes $T=100,200$, and $500.$ We let $\lambda=0.4.$ Without the loss of generality, we set $\beta_{1}=(0,0)^{\prime}$ and $\beta_{2}=(0,0)^{\prime}$ under the null. We consider testing $H_{0} :\beta_{1}=\beta_{2}$ against $H_{1}:\beta_{1}\neq\beta_{2}$ so that $p=2$.
We consider two pairs of different tests, both of which are based on the series variance estimators. The first pair uses the (usual) Fourier bases
Each test in this pair is based on the same test statistic $F_{T}^{\ast}$ defined in ((ref)) but uses different reference distributions. The first test uses the chi-square approximation ($\chi_{2}^{2}$) while the second test uses the nonstandard fixed-smoothing approximation given in ((ref)). We refer to the two tests as \textquotedblleft $\chi^{2}:$ Fourier Bases\textquotedblright\ and \textquotedblleft$F_{\infty }^{\ast}:$ Fourier Bases,\textquotedblright\ respectively. The nonstandard critical values are simulated. We approximate the standard Brownian motion in the nonstandard distribution using scaled partial sums of 1000 iid $N(0,1)$ random variables. To compute the nonstandard critical values, we use 10,000 simulation replications.
The second pair of tests uses the transformed Fourier bases via the Gram--Schmidt orthogonalization given in Section (ref). Each of the two tests in this pair is based on the same test statistic $\tilde{F} _{T}^{\ast}$ defined in Proposition (ref). The first test uses the standard F approximation, and the second test uses the rescaled chi-square distribution $Kp\left[ K-p+1\right] ^{-1}\chi_{2}^{2}.$ Equivalently, the second test in this pair employs the test statistic $\tilde{F}_{T} =\lambda\left( 1-\lambda\right) \cdot F_{T}$ and the standard chi-square approximation ($\chi_{2}^{2}$). We refer to the two tests as \textquotedblleft $\chi^{2}:$ Transformed Bases\textquotedblright\ and \textquotedblleft$F:$ Transformed Bases,\textquotedblright\ respectively. The chi-square test in the second pair is used to illustrate the effectiveness of the F approximation in reducing the size distortion.
The nominal level of all tests is $5\%.$ The number of simulation replications is 10,000. Figures (ref) and (ref) report the null rejection probability for each test for the sample sizes $T=100$ and $T=500$ when $q_{t}$ and $u_{t}$ follow independent AR(1) processes with the same AR parameter $\rho.$ Several patterns emerge from these two figures:
Figure (ref) reports the null rejection probabilities when the sample size $T$ is $200$ and when the error process may have an MA component and the AR parameter may be negative. As in Figures (ref) and (ref), the same patterns emerge.
Next, we consider the size properties of the tests with a data-driven $K.$ Note that \[ R\sqrt{T}(\hat{\beta}-\beta)=\frac{1}{\sqrt{T}}\sum_{t=1}^{T}R\left( \frac{\tilde{X}^{\prime}\tilde{X}}{T}\right) ^{-1}\tilde{X}_{t}^{\prime} u_{t}=\frac{1}{\sqrt{T}}\sum_{t=1}^{T}v_{t}+o_{p}\left( 1\right) \] where $v_{t}=RQ^{-1}\tilde{X}_{t}^{\prime}u_{t}.$ Then \[ R\hat{Q}^{-1}\hat{\Omega}Q^{-1}R^{\prime}=\frac{1}{K}\sum_{j=1}^{K}\left[ \frac{1}{\sqrt{T}}\sum_{t=1}^{T}\phi_{j}\left( \frac{t}{T}\right) \hat {v}_{t}\right] ^{\otimes2}, \] where $\hat{v}_{t}=R\hat{Q}^{-1}\tilde{X}_{t}^{\prime}u_{t}.$ So $R\hat {Q}^{-1}\hat{\Omega}Q^{-1}R^{\prime}$ can be viewed as the series variance estimator of the long run variance of $\left\{ v_{t}\right\} .$ We can follow Phillips2005 and choose $K$ to minimize the mean square error (MSE) of $RQ^{-1}\hat{\Omega }Q^{-1}R^{\prime}.$ We fit a VAR(1) model to $\hat{v}_{t}$ and use the fitted model to compute the data-driven MSE-optimal $K.$
Table (ref) reports the null rejection probabilities and the average values of $K$ used with data-driven choice of $K$ for different sample sizes. The qualitative observations from Figures (ref)-- (ref) continue to hold with the data-driven $K.$ In particular, the nonstandard test and standard F test are more accurate than the corresponding chi-square tests, especially when the latter have large positive size distortion. The null rejection probabilities of the nonstandard test and the standard F test are close to each other. Similarly, the null rejection probabilities of the two chi-square tests are close to each other. As expected, the average value of $K$ decreases with the persistence of the underlying processes. The higher the persistence, the smaller the average $K$ value, and the more effective the nonstandard test and standard F test in reducing the size distortion.
To simulate the power of the tests, we let $\beta_{1}=\left( 0,0\right) $ and $\beta_{2}=\left( \delta,\delta\right) .$ Figure (ref) presents the size-adjusted power curves as functions of $\delta$ when the sample size is 200 and when both $q_{t}$ and $u_{t}$ follow AR(1) processes. The figure is representative of other cases. For the two tests in each pair, the size-adjusted powers are the same, as they are based on the same test statistic. Thus, we need only report two power curves: one for the usual Fourier bases and the other for the transformed Fourier bases. The basic message from Figure (ref) is that the size-adjusted powers associated with the two sets of bases are very close to each other. This, coupled with its size accuracy and convenience to use, suggests that we use the F test in empirical applications.
This study proposes asymptotic F and t tests for structural breaks that are robust to heteroscedasticity and autocorrelation. The tests are based on a special series HAR variance estimator where the basis functions are crafted via the Gram--Schmidt orthonormalization. Monte Carlo simulations show that the F test is much more accurate than the corresponding chi-square test.
This study assumes that there is a single known break point. The asymptotic F and t theory can be extended to the case with multiple but known break points. The theory can also be extended to allow for a linear trend or other deterministic trends, but we need to redesign the basis functions. In principle, the tests based on series HAR variance estimation can be extended to accommodate the case with an unknown break point along the line of Cho_Vogelsang2017. All the basic ingredients have been established in the study. We only need to take the supremum (or other functionals) of the Wald or t statistic over $\lambda$ as the test statistic. However, the convenient F approximation is lost, as the supremum of the standard distributions is not standard any more. Therefore, it is not clear whether there is still an advantage of using series HAR variance estimators rather than kernel HAR variance estimators.