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Bootstrapping Structural Change Tests
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Linear models with endogenous regressors are commonly employed in time series econometric analysis.\footnote{For example, Brady:2008 examines consumption smoothing using by regressing consumption growth on consumer credit, the latter being endogenous because it depends on liquidity constraints. Zhang/Osborn/Kim:2008, Kleibergen/Mavroeidis:2009, Hall/Han/Boldea:2012 and Kim/Manopimoke/Nelson:2014 investigate the New Keynesian Phillips curve, where inflation is driven by expected inflation and marginal costs, both endogenous since they are correlated with inflation surprises. Bunzel/Enders:2010 and Qian/Su:2014 estimate the forward-looking Taylor rule, a model where the Federal fund rate is set based on expected inflation and output, both endogenous as they depend either on forecast errors or on current macroeconomic shocks. All these studies test for structural change in their estimated equations as part of their analysis.} In many cases, the parameters of these models are assumed constant throughout the sample. However, given the span of many economic time series data sets, this assumption may be questionable and a more appropriate specification may involve parameters that change value during the sample period. Such parameter changes could reflect legislative, institutional or technological changes, shifts in governmental and economic policy, political conflicts, or could be due to large macroeconomic shocks such as the oil shocks experienced over the past decades and the productivity slowdown. It is therefore important to test for parameter - or structural - change. Various tests for structural change have been proposed with one difference between them being in the type of structural change against which the tests are designed to have power. In this paper, we focus on the scenario in which the potential structural change consists of discrete changes in the parameter values at unknown points in the sample, known as break - (or change-) points. Within this framework, two types of hypotheses tests are of natural interest: tests of no parameter change against an alternative of change at a fixed number of break-points, and tests of whether the parameters change at $\ell$ break-points against an alternative that they change at $\ell+1$ points. These hypotheses tests are of interest in their own right, and also because they can form the basis of a sequential testing strategy for estimating the number of parameter break-points, see Bai/Perron:1998.
Hall/Han/Boldea:2012 (HHB hereafter) propose various statistics for testing these hypotheses in linear models with endogenous regressors based on Two Stage Least Squares (2SLS).\footnote{Perron/Yamamoto:2015 propose an alternative approach based on OLS.} Their tests are the natural extensions of the analogous tests for linear models with exogenous regressors estimated via OLS that are introduced in the seminal paper by Bai/Perron:1998. A critical issue in the implementation of these tests in a 2SLS setting is whether or not the reduced form for the endogenous regressors is stable. If it is then, under certain conditions, HHB�'s test statistics converge in distribution to the same distributions as their OLS counterparts and are pivotal, see HHB and Perron/Yamamoto:2014. However, if the reduced form itself is unstable and/or there is unconditional heteroskedasticity, then these limiting distributions no longer apply (HHB), and are, in fact, no longer pivotal (Perron/Yamamoto:2014, Perron/Yamamoto:2014). This is a severe drawback as in most cases of interest the reduced form is likely to be unstable. This problem has been circumvented in two ways. HHB suggest a testing strategy based on dividing the sample into sub-samples over which the RF is stable but this is inefficient compared to inferences based on the whole sample, and can be infeasible if the sub-samples are small. Perron/Yamamoto:2015 propose using a variant of HansenBE:2000's fixed regressor bootstrap to calculate the critical values of the test. Their simulation evidence suggests the use of this bootstrap improves the reliability of inferences but they do not establish the asymptotic validity of the method.\footnote{An alternative approach is to estimate the number and location of the breaks via an information criteria, see Hall/Osborn/Sakkas:2015. However, this approach has the drawback that inferences can be sensitive to the choice of penalty function.}
In this paper, we explore the use of bootstrap versions of 2SLS-based tests for parameter change in far greater detail than previous studies. We consider inferences based on two different types of bootstrap versions of the structural change tests, provide formal proofs of their asymptotic validity and report simulations results that demonstrate that the bootstrap tests provide reliable inferences in the finite sample sizes encountered in practice. More specifically, we consider the case where the right-hand side variables of the equation of interest contains endogenous regressors, contemporaneously exogenous variables, lagged values of both and lagged values of the dependent variable. This equation of interest is part of a system of equations that is completed by the reduced form for the endogenous regressors and equations for the contemporaneously exogenous variables. This system of equations is assumed to follow a Structural Vector Autoregressive (SVAR) model in which the parameters of the mean are subject to discrete shifts at a finite number of break-points in the sample. Both the number and location of the break-points are unknown to the researcher. These break-points define regimes over which the parameters are constant, and it is assumed that the implied reduced form VAR is stable within each such regime. The errors of the VAR are assumed to follow a vector martingale difference sequence that potentially exhibits both conditional and unconditional heteroskedasticity. Given this error structure, we explore methods for inference based on the wild bootstrap proposed by Liu:1988 because it has been found to replicate the conditional and unconditional heteroskedasticity of the errors in other contexts. In particular, we consider two versions of the wild bootstrap: the wild recursive bootstrap (which generates recursively the bootstrap observations) and the wild fixed-regressor bootstrap (which adds the wild bootstrap residuals to the estimated conditional mean, thus keeping all lagged regressors fixed). These bootstraps have been proposed by Goncalvez/Kilian:2004 to test the significance of parameters in autoregressions with (stationary) conditional heteroskedastic errors. Our primary focus is on bootstrap versions of $\sup$-$Wald$ - type statistics to test for structural changes in the parameters of the equation of interest (with endogenous variables) estimated by 2SLS, but our validity arguments also extend straightforwardly to analogous $\sup$-$F$-type statistics.
While our primary focus is on models where the reduced form for the endogenous regressors is unstable, our results also cover the case where this reduced form is stable. In the latter case, the test statistics have a pivotal limiting distribution under conditions covered by our framework, specialized to errors that are unconditionally homoskedastic. For these situations, the bootstrap methods we propose are expected to provide a superior approximation to finite sample behaviour compared to the limiting distribution because the bootstrap, by its nature, incorporates sample information. Thus bootstrap versions of the tests are attractive in this setting as well.
In the case where there are no endogenous regressors in the equation of interest, our framework reduces to a linear model estimated by Ordinary Least Squares (OLS). For this set-up, HansenBE:2000 proposes the wild fixed-design bootstrap to test for structural changes using a $\sup$-$F$ statistic. Very recently, \\ Georgievetal:2018 (GHLT, hereafter) consider HansenBE:2000's bootstrap for versions of $\sup$-$F$ type tests for parameter variation in predictive regressions with exogenous regressors. Both HansenBE:2000 and GHLT establish the asymptotic validity of this bootstrap within the settings they consider.\footnote{In fact, GHLT demonstrate that HansenBE:2000's proof of the asymptotic validity of the bootstrap needs an amendment when the predictive regressors are (near-) unit root processes.} There are some similarities and important differences between our framework (specialized to the no endogenous regressor case) and those in HansenBE:2000 and GHLT. We adopt similar assumptions about the error process to GHLT and like both HansenBE:2000 and GHLT consider fixed regressor bootstrap tests of a null of constant parameters versus an alternative of parameter change. Important differences include: GHLT allow for strongly persistent variables whereas our framework assumes the system is stable within (suitably defined) regimes; our analysis covers tests for additional breaks in the model, the use of the recursive bootstrap and also inferences based on $\sup$-$Wald$ tests. Thus our results for this case complement those of HansenBE:2000 and GHLT.\footnote{The wild fixed-regressor bootstrap is also included in the recent simulation study exploring the finite sample properties of inference methods about the location of the break-point in models estimated via OLS reported in Chang/Perron:2018.}
Although the frameworks are different, HansenBE:2000, GHLT and our own study all find their bootstrap versions of the structural change tests work well in finite samples. Interestingly, Chang/Perron:2018 find that bootstrap-based inferences about the location of breaks have similar advantages in finite samples.\footnote{Chang/Perron:2018 report results from a comprehensive simulation study that investigates the finite sample methods of various methods for constructing confidence intervals for the break fractions in linear regression models with exogenous regressors. They consider variants of the intervals based on i.i.d., wild and sieve bootstraps.} Collectively, our paper and these other recent studies suggest the use of the bootstrap can yield reliable inferences in linear models with multiple break-points in the sample sizes encountered in practice.
An outline of the paper is as follows. Section 2 lays out the model, test statistics and their bootstrap versions. Section 3 details the assumptions and contains theoretical results establishing the asymptotic validity of the bootstrap methods. Section 4 contains simulation results that provide evidence on the finite sample performance of the bootstrap tests. Section 5 concludes. Appendix A contains all the tables for Section 4, with additional simulations relegated to a Supplementary Appendix. Appendix B contains the proofs, with some background results relegated to the same Supplementary Appendix.
Notation: Matrices and vectors are denoted with bold symbols, and scalars are not. Define for a scalar $N$, the generalized vec operator $\mbox{\bf vect}_{s=1:N}(\bm A_s) = (\bm A_1', \ldots, \bm A_N')'$, stacking in order the matrices $\bm A_s, (s=1,\ldots, N)$, which have the same number of columns. Let $\mbox{\bf diag}_{s=1:N}(\bm{A}_s) = \mbox{\bf diag}(\bm A_1, \ldots, \bm A_{N})$ be the matrix that puts the blocks $\bm A_1, \ldots, \bm A_N$ on the diagonal. If it is clear over which set $\mbox{\bf vect}$ and $\mbox{\bf diag}$ operations are taken, then the subscript $s=1:N$ is dropped on these operators. If $N$ is the number of breaks in a quantity, $T_{1},\ldots, T_{N}$ are the ordered candidate change-points and $T$ the number of time series observations, where $\tau_{0}=0$, $\tau_{N+1}=1$, and where $\bm \tau_N=(\tau_{0},\mbox{\bf vect}_{s=1:N}(\tau_s)',\tau_{N+1})$ is a partition of the time interval $[1,T]$ divided by $T$, such that $[T \tau_{s}] = T_{s,\bm \tau_N}$ for $s=1,\ldots N+1$. Define the regimes where parameters are assumed constant as $ I_{s,\bm \tau_N} = [T_{s-1}+1, T_{s}]$ for $s=1, \ldots, N$. Below the breaks in the structural equation are denoted by $\bm \tau_N=\bm \lambda_m$, and those in the reduced form by $\bm \tau_N=\bm \pi_h$, where $m$ an $h$ are the number of breaks in each equation. A superscript zero on any quantity refers to the true quantity, which is a fixed number, vector or matrix. For any random vector or matrix $\bm Z$, denote by $||\bm Z||$ the Euclidean norm for vectors, or the Frobenius norm for matrices. Finally, $\boldsymbol{0}_a$ and $\boldsymbol{0}_{a\times a}$ denote, respectively, an $a\times 1$ vector and a $a\times a$ matrix of zeros, and $1_A$ denotes an indicator function that takes the value one if event $A$ occurs.
This section is divided into three sub-sections. Section 2.1 outlines the model. Section 2.2 outlines the hypotheses of interest and the test statistics. Section 2.3 presents the bootstrap versions of the test statistics.
Consider the case where the equation of interest takes the form
where $\bm w_t=\mbox{\bf vect}(\bm x_t, \bm z_{1,t})$, $\bm z_{1,t}$ includes the intercept, $\bm r_t$ and lagged values of $y_t$, $\bm x_t$, and $\bm r_t$, and $\bm \beta_{(i)}^0$ are parameters in regime $i$. The key difference between $\bm x_t$ and $\bm r_t$ is that $\bm x_t$ represents the set of explanatory variables which are correlated with $u_t$, and $\bm r_t$ represents the set of explanatory variables that are uncorrelated with $u_t$. We therefore refer to $\bm x_t$ as the {\it endogenous} regressors and $\bm r_t$ as the {\it contemporaneously exogenous} regressors.\footnote{This terminology is taken from Wooldridge:2006[p.349] and reflects that fact $\bm r_t$ may be correlated with $u_n$ for $t\neq n$.} Equation ((ref)) can be re-written as:
where $\bm \beta_t^0 = \bm \beta_{(i)}^0$ if $t \in I_{i,\bm \lambda_m^0}, i=1,\ldots,m$ and similar notation holds for $\bm \beta_{\bm x, t}$ and $\bm \beta_{\bm z, t}$. For simplicity, we refer to ((ref)) as the “structural equation” (SE).
The SE is assumed to be part of a system that is completed by the following equations for $\bm x_t$ and $\bm r_t$. The reduced form (RF) equation for the endogenous regressors $\bm x_t$ is a regression model with $h$ breaks ($h+1$ regimes), that is:
The vector $\bm z_t$ includes the constant, $\bm r_t$ and lagged values of $y_{t}$, $\bm x_t$ and $\bm r_t$. It is assumed that the variables in $\bm z_{1,t}$ are a strict subset of those in $\bm z_t$ and therefore we write $\bm z_t=\mbox{\bf vect}(\bm z_{1,t},\bm z_{2,t})$. Equation ((ref)) can also be rewritten as:
where $\bm \varDelta_t^0 = \bm \varDelta_{(i)}^0$ if $t\in I_{i,\bm \pi_h^0}$, $i=1,\ldots, h+1$. The contemporaneously exogenous variables $\bm r_t$ are assumed to be generated as follows,
where $\bm z_{3,t}$ includes the constant and lagged values of $\bm r_t$, $y_t$ and $\bm x_t$.
Equations ((ref)), ((ref)) and ((ref)) imply $\tilde{\bm z}_t=\mbox{\bf vect}(y_t, \bm x_t, \bm r_{t})$ evolves over time via a SVAR process whose parameters are subject to discrete shifts at unknown points in the sample. To present the reduced form VAR version of the model, define $n=\dim(\tilde{\bm z}_t)$ and let $\bm \tau_N$ denote the partition of the sample such that all three equations have constant parameters within the associated regimes.\footnote{For example, suppose $m=1$, $h=2$ and $d=1$ with $\bm \lambda_m^0=[0,0.5,1]^\prime$,$\bm \pi_h^0=[0,0.3,0.5,1]^\prime$ and $\bm \omega_d^0=[0,0.7,1]^\prime$, then $N=3$ and $\bm \tau_N=[0,0.3,0.5,0.7,1]^\prime$.} We can then write equations ((ref)), ((ref)) and ((ref)) as:
where $\bm e_t=\bm A_s^{-1}\bm \epsilon_t$,
$\bm \beta_{\bm r, s}^{0\prime}$ denotes the sub-vector of $\bm \beta_{s}^{0\prime}$ that contain the coefficients on $\bm r_t$ in ((ref)) ($\bm \beta_{\bm r, s}^{0\prime}$ and $\bm \beta_{s}^{0\prime}$ are the values of $\bm \beta_{\bm r, t}^{0\prime}$ and $\bm \beta_{t}^{0\prime}$ for $[\tau_{s-1}T]+1\,\leq t\,\leq [\tau_s T]$); $\bm \varDelta_{\bm r,s}^{0'}$ denotes the sub-matrix of $\bm \varDelta_{s}^{0'}$ that contains the coefficients on $\bm r_t$ in ((ref)) ($\bm \varDelta_{\bm r, s}^{0\prime}$ and $\bm \varDelta_{s}^{0\prime}$ are the values of $\bm \varDelta_{\bm r, t}^{0\prime}$ and $\bm \varDelta_{t}^{0\prime}$ for $[\tau_{s-1}T]+1\,\leq t\,\leq [\tau_s T]$), and $\bm \epsilon_t=\mbox{\bf vect}(u_t,\bm v_t,\bm \zeta_t)$. For ease of notation, we assume the order of the VAR is the same in each regime, but our results easily extend to the case where the order varies by regime.
As stated in the introduction, this paper focuses on the issue of testing for structural change in the SE. Within the model described above, there are two types of test that are of particular interest. The first tests the null hypothesis of no parameter change against the alternative of a fixed number of parameter changes in the sample that is, a test of $H_0:\, m=0$ versus $H_1:\,m=k$. The second tests the null of a fixed number of parameter changes against the alternative that there is one more, that is, it tests $H_0:\, m=\ell$ versus $H_1:\,m=\ell+1$. We consider appropriate test statistics for each of these scenarios in turn below.
As the tests are based on the Wald principle, calculation of the test statistics here requires 2SLS estimation of the SE under $H_1$. On the first stage, the RF is estimated via least squares methods. If the number and location of the breaks in the RF are known then this estimation is straightforward. However, in general, neither the number or location of the breaks is known and so they must be estimated. For our purposes here, it is important that both $h$ and $\bm \pi_h^0$ are consistently estimated and that $\hat \bm \pi_h$, the estimator of $\bm \pi_h^0$, converges sufficiently fast (see Lemma (ref) in the Appendix B). These properties can be achieved by estimating the RF either as a system or equation by equation, and using a sequential testing strategy to estimate $h$; see, respectively Qu/Perron:2007 and Bai/Perron:1998. Provided the significance levels of the tests shrink to zero slowly enough, $\hat h$ approaches $h$ with probability one as the sample size $T$ grows; {\it e.g.} see Bai/Perron:1998[Proposition 8]. The same consistency result holds if we estimate $h$ via the information criteria; {\it e.g.} see Hall/Osborn/Sakkas:2013. For this reason, in the rest of the theoretical analysis, we treat $h$ as known. However, we explore the potential sensitivity of the finite sample performance of the tests for structural change in the SE to estimation of $h$ in our simulation study. Let $\hat \bm \varDelta_{(j)}$ be the estimator of $\bm \varDelta_{(j)}^0$, $\hat \bm \varDelta_t=\sum_{j=1}^{h+1}\hat \bm \varDelta_{(j)}1_{t\in \hat I_j^*}$, where $\hat{I}_j^*=\left\{[\hat{\pi}_{j-1}T]+1, [\hat{\pi}_{j-1}T]+2, \ldots,[\hat{\pi}_{j}T]\right\}$, and $\hat{\bm x}_t=\hat \bm \varDelta_t\bm z_t^\prime$ that is, $\hat{\bm x}_t$ is the predicted value for $\bm x_t$ from the estimated RF.\\[0.1in]
{\it Case (i): $H_0:\, m=0$ versus $H_1:\,m=k$}\\ Under $H_1$, the second stage estimation involves estimation via OLS of the model,
for all possible $k$-partitions $\bm \lambda_k$. Let $\hat \bm \beta_{(i)}$ denote the OLS estimator of $\bm \beta_{(i)}$ in ((ref)), $\hat \bm \beta_{\bm \lambda_k}$ denote the OLS estimator of $\mbox{\bf vect}_{i=1:k+1}(\hat\bm \beta _{(i)})$ in ((ref)) (that is, $\hat \bm \beta_{\bm \lambda_k}$ is the OLS estimator of $\mbox{\bf vect}_{i=1:k+1}(\bm \beta_{(i)})$ based on partition $\bm \lambda_k$).\footnote{Strictly, $\hat \bm \beta_{(i)}$ depends on $\bm \lambda_k$ but we have suppressed this to avoid excessive notation.} To present the $\sup$-$Wald$ test, we define $\bm R_k=\tilde{\bm R}_k\otimes \bm I_{p}$ where $\tilde{\bm R}_k$ is the $k\times(k+1)$ matrix whose $(i,j)^{th}$ element, $\tilde{R}_k(i,j)$, is given by: $\tilde{R}_k(i,i)=1$, $\tilde{R}_k(i,i+1)=-1$, $\tilde{R}_k(i,j)=0$ for $i=1,2,\ldots, k$, and $j\neq i$, $j \neq i+1$. Also let $\bm \varLambda_{\epsilon,k}=\{\bm \lambda_k: |\lambda_{i+1}-\lambda_i|\ge\epsilon,\lambda_1\ge\epsilon,\lambda_k\le 1-\epsilon\}$. With this notation, the test statistic is:
where:
and $\bm \beta_{\bm x}^0$ is the true value of $\bm \beta_{\bm x,(i)}^0$ for $i=1,2,\ldots,m+1$ under $H_0$.
As mentioned in the introduction, our framework assumes the errors are a m.d.s. that potentially exhibits heteroskedasticity, and so the natural choice of $\hat{\bm M}_{(i)}$ is the Eicker-White estimator, see Eicker:1967 and White:1980. This can be constructed using the estimator of $\bm \beta_{x,(i)}$ in (ref) under either $H_0$ or $H_1$, where $\bm \beta_{x,(i)}$ are the elements of $\bm \beta_{(i)}$ containing the coefficients on $\hat \bm x_t$. For the purposes of the theory presented below, it does not matter which is used because the null hypothesis is assumed to be true. However, the power properties may be sensitive to this choice. In our simulation study reported below, we use the Eicker-White estimator based on $\hat \bm \beta_{\bm x,(i)}$, the estimator of $\bm \beta_{x,(i)}$ under $H_1$, that is, $$ \hat{\bm M}_{(i)}\;=\;\widehat{EW}\left[\,\hat\bm \varUpsilon_t^\prime \bm z_t(\hat u_t+\hat\bm v_t^\prime \hat \bm \beta_{\bm x,(i)});\,I_{i,\bm \lambda_k}\,\right], $$ where $\hat{u}_t=y_t-\bm w_t^\prime\hat\bm \beta_{(i)}$ for $t\in I_{i,\bm \lambda_k}$, $\hat\bm v_t=\bm x_t-\hat \bm \varDelta_t \bm z_t^\prime$, $\hat\bm \varUpsilon_t=[\hat \bm \varDelta_t,\bm \varPi]$ and, for any vector $\bm a_t$ and $I\subseteq\{1,2,\ldots,T\}$, $\widehat{EW}\left[\bm a_t;\,I\,\right]=T^{-1}\sum_{t\in I}\bm a_t\bm a_t^\prime$.\\[0.1in]
{\it Case (ii): $H_0:\, m=\ell$ versus $H_1:\,m=\ell+1$}\\ Following the same approach used by Bai/Perron:1998 for OLS based inferences, suitable tests statistics can be constructed as follows. The model with $\ell$ breaks is estimated via a global minimization of the sum of squared residuals associated with the second stage of the 2SLS estimation of the SE. For each of the $\ell+1$ regimes of this estimated model, the $\sup$-$Wald$ statistic for testing no breaks versus one break is calculated. Inference about $H_0:m=\ell$ versus $H_1:\,m=\ell+1$ is based on the largest of these $\ell+1$ $\sup\text{-}Wald$ statistics.
More formally, let the estimated SE break fractions for the $\ell$-break model be $\hat{\bm \lambda}_\ell$ and the associated break points be denoted $\{\hat{T}_i\}_{i=1}^\ell$ where $\hat{T}_i=[T\hat{\lambda}_i]$. Let $\hat{I}_i=I_{i,\hat{\bm \lambda}_\ell}$, the set of observations in the $i^{th}$ regime of the $\ell$-break model and partition this set as $\hat{I}_i=\hat{I}_i^{(1)}(\varpi_i)\cup\hat{I}_i^{(2)}(\varpi_i)$ where $\hat{I}_i^{(1)}(\varpi_i)=\{t:\,[\hat{\lambda}_{i-1}T]+1,[\hat{\lambda}_{i-1}T]+2,\ldots,[\varpi_iT]\}$ and $\hat{I}_i^{(2)}(\varpi_i)=\{t:\,[\varpi_iT]+1,[\varpi_iT]+2,\ldots,[\hat{\lambda}_iT]\}$. Consider estimation of the model
for all possible choices of $\varpi_i$ (where for notational brevity we suppress the dependence of $\bm \beta_{(j)}$ on $i$). Let $\hat{\bm \beta}(\varpi_i)=\mbox{\bf vect}\left(\hat{\bm \beta}_{(1)}(\varpi_i), \hat{\bm \beta}_{(2)}(\varpi_i)\right)$ be the OLS estimators of $\mbox{\bf vect}(\bm \beta_{(1)},\bm \beta_{(2)})$ from ((ref)). Also let $\mathcal{N}(\hat{\bm \lambda}_\ell)=[\hat{\lambda}_{i-1}+\epsilon,\hat{\lambda}_{i}-\epsilon]$. The $\sup\text{-}Wald$ statistic for testing $H_0:\,m=\ell$ versus $H_1:\,m=\ell+1$ is given by
where\footnote{The comment above about the calculation of $\hat{\bm M}_{(i)}$ apply equally to $\hat{\bm M}_{j}(\varpi_i)$.}
In this section, we introduce the bootstrap analogues of the test statistics presented in the previous section. As noted above, our framework assumes the error vector $\bm \epsilon_t$ to be a m.d.s that potentially exhibits conditional and unconditional heteroskedasticity, and so we use the wild bootstrap proposed by Liu:1988 because it has been found to replicate the conditional and unconditional heteroskedasticity of the errors in other contexts.\footnote{Liu:1988 developed the wild bootstrap has been developed in Liu:1988 following suggestions in Wu:1986 and Beran:1986 in the context of static linear regression models with (unconditionally) heteroskedastic errors.} We consider both the wild recursive (WR) bootstrap and the wild fixed (WF) bootstrap. These procedures differ in their treatment of the right-hand side variables in the bootstrap samples as described below.\\[0.1in]
{\it Generation of the bootstrap samples:}\\ Let $\tilde{\bm z}^b_t=\mbox{\bf vect}(y_t^{b}, \bm x_t^{b}, \bm r_{t})$ where $y_t^{b}$ and $\bm x_t^{b}$ denote the bootstrap values of $y_t$ and $\bm x_t$. Note that because $\bm r_t$ is contemporaneously exogenous its sample value is used in the bootstrap samples. In all cases below, the bootstrap residuals are obtained as $u_t^b=\hat u_t\nu_t$ and $\bm v_t^b=\hat\bm v_t\nu_t$, where $\hat u_t$ and $\hat \bm v_t$ are the (non-centered) residuals under the null hypothesis and $\nu_t$ is a random variable that is discussed further in Section 3.
For the WR bootstrap, $\{y_t^{b\prime}\}$ and $\{\bm x_t^{b\prime}\}$ are generated recursively as follows:
where the vector $\bm z_t^b$ contains a constant, $\bm r_t$ and lags of $y_t^b$, $\bm x_t^b$ and $\bm r_{t}$; $\hat \bm \beta_{\bm x,t}$ and $\hat \bm \beta_{\bm z,t}$ are the sample estimates of $\bm \beta_{\bm x,t}^0$ and $ \bm \beta_{\bm z,t}^0$ under $H_0$ of the test in question.
For the WF bootstrap, $\bm z_t$ is kept fixed and, following Goncalvez/Kilian:2004, the bootstrap samples are generated as follows:
where again $\hat \bm \beta_{\bm x,t}$ and $\hat \bm \beta_{\bm z,t}$ are the sample estimates of $\bm \beta_{\bm x,t}^0$ and $ \bm \beta_{\bm z,t}^0$ under $H_0$ of the test in question.
{\it Case (i): $H_0:\,m=0$ vs $H_1:\,m=k$}\\ First consider the WR bootstrap. 2SLS estimation is implemented in the bootstrap samples as follows. On the first stage, the following model is estimated via OLS $$ \bm x_t^b\; =\; \bm z_t^{b\prime} \bm \varDelta_j\, +\,\mbox{error}, \qquad t\in\hat{I}_j^*,\qquad j=1,2,\ldots,h+1, $$ to obtain $\hat{\bm \varDelta}_j^b=\left\{\sum_{t\in \hat I_j^*}\bm z_t^b\bm z_t^{b\prime}\right\}^{-1}\sum_{t\in \hat I_j^*} \bm z_t^b \bm x_t^{b\prime}$. Define $\hat \bm \varDelta_t^b = \sum_{j=1}^{\hat h+1}1_{t\in \hat I_j^*}\hat{\bm \varDelta}_j^b$, $\hat \bm x_t^{b\prime}= \bm z_t^{b\prime}\hat\bm \varDelta^b_t$, and $\hat \bm w_t^b=\mbox{\bf vect}(\hat \bm x_t^{b}, \bm z_{1,t}^{b})$. For a given $k$-partition $\bm \lambda_k$, the second stage of the 2SLS in the bootstrap samples involves OLS estimation of
and let $\hat \bm \beta_{\bm \lambda_k}^b$ be the resulting OLS estimator of $\mbox{\bf vect}_{i=1:k+1}(\bm \beta_{(i)})$. The WR bootstrap version of the $\sup$-$Wald_T$ statistic is:
where:
Now consider the WF bootstrap, for which $y_t^b$ and $\bm x_t^b$ are generated via (ref)-(ref). The first stage of the 2SLS involves LS estimation of $$ \bm x_t^b\; =\; \bm z_t^{\prime} \bm \varDelta_j\, +\,\mbox{error}, \qquad t\in\hat{I}_j^*,\qquad j=1,2,\ldots,h+1, $$ to obtain $\hat{\bm \varDelta}_j^b=\left\{\sum_{t\in \hat I_j^*}\bm z_t\bm z_t^{\prime}\right\}^{-1}\sum_{t\in \hat I_j^*} \bm z_t \bm x_t^{b\prime}$. Now re-define $\hat \bm \varDelta_t^b = \sum_{j=1}^{\hat h+1}1_{t\in \hat I_j^*}\hat{\bm \varDelta}_j^b$, $\hat \bm x_t^{b\prime}= \bm z_t^{\prime}\hat\bm \varDelta^b_t$, and $\hat \bm w_t^b=\mbox{\bf vect}( \hat \bm x_t^{b}, \bm z_{1,t})$. For a given $k$-partitions $\bm \lambda_k$, the second stage of the 2SLS in the bootstrap samples involves OLS estimation of (ref) and let $\hat \bm \beta_{\bm \lambda_k}^b$ be the resulting OLS estimator of $\mbox{\bf vect}_{i=1:k+1}(\bm \beta_{(i)})$. The WF bootstrap $\sup\text{-}Wald$ statistic is defined as in (ref) with $Wald_{T\bm \lambda_k}^b$ defined as in ((ref)) only with $\hat{\bm w}_t$ and $\hat{\bm \varDelta}_t^b$ redefined in the way described in this paragraph, and $\hat{\bm M}_{(i)}^b$ in (ref) replaced by $\hat{\bm M}_{(i)}^b=\widehat{EW}\left[ \hat\bm \varUpsilon_t^{b\prime} \bm z_t(u_t^b+\bm v_t^{b\prime}\hat\bm \beta_{\bm x,(i)}^b);\,I_{i,\bm \lambda_k}\,\right]$.\\[0.1in]
{\it Case (ii): $H_0:\, m=\ell$ versus $H_1:\,m=\ell+1$}\\ For each bootstrap the first stage of the 2SLS estimation and the construction of $\hat{\bm w}_t$ is the same as described under {\it Case (i)} above. Let $\hat{I}_{i}^{(j)}$ be defined as in the discussion of {\it Case (ii)} in Section (ref), and consider
for all possible choices of $\varpi_i$ (where, once again, we suppress the dependence of $\bm \beta_{(j)}$ on $i$). Let $\hat{\bm \beta}^b(\varpi_i)=\mbox{\bf vect}\left(\hat{\bm \beta}_{(1)}^{b}(\varpi_i), \hat{\bm \beta}_{(2)}^{b}(\varpi_i)\right)$ be the OLS estimators of $\mbox{\bf vect}(\bm \beta_{(1)},\bm \beta_{(2)})$ from ((ref)). The bootstrap version of $\sup\text{-}Wald_T(\ell+1\,|\,\ell)$ is given by
where
and $\hat{\bm M}_{j}^b(\varpi_i)\, =\,\widehat{EW}\left[\,\hat\bm \varUpsilon_t^{b\prime} \bm z_t^b(u_t^b+\bm v_t^{b\prime}\hat\bm \beta_{\bm x,(i)}^b); \hat{I}_{i}^{(j)}\, \right]$ for WR, and $\hat{\bm M}_{j}^b(\varpi_i)\, =\,\widehat{EW}\left[\,\hat\bm \varUpsilon_t^{b\prime} \bm z_t(u_t^b+\bm v_t^{b\prime}\hat\bm \beta_{\bm x,(i)}^b); \hat{I}_{i}^{(j)}\,\right]$ for WF.
In this section, we establish the asymptotic validity of the bootstrap versions of the test statistics described above. To this end we impose the following conditions.
Before presenting our main theoretical results, we discuss certain aspects of the assumptions.
The following theorems establish the asymptotic validity of the bootstrap versions of the $sup\text{-}Wald$ tests.
In this section, we investigate the finite sample performance of the bootstrap versions of the $\sup$-$Wald$ and $\sup$-$F$ statistics. We consider a number of designs that involve stability or instability in the SE and/or the RF. In all the designs the variable $x_t$ is endogenous and the SE is estimated by 2SLS. Recalling from above that $h$ and $m$ denote the true number of breaks in the RF and SE respectively, the four scenarios we consider are as follows.
For the four scenarios above we consider the following choices for $u_t$, $\bm v_t$ and $\bm r_t$:
In {\it Case A}, the errors $u_t$ and $v_t$ are homoskedastic and the contemporaneous exogenous regressors $\bm r_t$ are stable. In {\it Case B}, the errors are conditionally heteroskedastic. In {\it Case C} the errors have a contemporaneous upward shift in the unconditional variance, while in {\it Case D} there is also an upward shift in the variance of $\bm r_t$.
In our simulations we consider the behavior of the bootstrap tests both under their null and alternative hypotheses. For scenarios $(h,m)=(0,0)$ and $(h,m)=(1,0)$ we consider the behavior of the $\sup$-$Wald_T$. For scenarios $(h,m)=(0,1)$ and $(h,m)=(1,1)$ we consider the performance of the $\sup$-$Wald_T(2|1)$. In order to assess the power of our bootstrap tests we also consider the case when the null hypotheses are not true and there is an additional break in the SE parameters at $[T/2]$. More exactly we consider in all the four scenarios described above the following:
with $g$ a constant; $i=1$ and $\tilde T=T$ for scenarios $(h,m)=(0,0)$ and $(h,m)=(1,0)$, and the equation for $y_t$ for $t<[T/2]+1$ is the same as that given in the two scenarios $(h,m)=(0,0)$ and $(h,m)=(1,0)$; $i=2$ and $\tilde T=[3T/4]$ for scenarios $(h,m)=(1,0)$ and $(h,m)=(1,1)$, and the equation for $y_t$ for $t<[T/2]+1$ and $t>[3T/4]$ is the same as that given in the two scenarios $(h,m)=(1,0)$ and $(h,m)=(1,1)$. When $g=0$, the null hypothesis is satisfied. We illustrate the behavior of the tests under the alternative hypothesis for the following values of $g$: $g=-0.007, -0.009$ for scenario $(h,m)=(0,0)$; $g=-0.05,-0.07$ for scenario $(h,m)=(1,0)$; $g=0.3,0.4$ for scenario $(h,m)=(0,1)$; $g=-0.5,0.5$ for scenario $(h,m)=(1,1)$.
For scenarios $(h,m)=(1,0)$ and $(h,m)=(1,1)$ we have tested for the presence of max $2$ breaks in the RF for $x_t$ (in (ref)-(ref) and (ref)-(ref) respectively) prior to testing for breaks in the SE. More exactly we tested the null hypothesis $H_0:h=\ell$ against $H_1:h=\ell+1$, $\ell=0,1$ using the WR and WF bootstrap $\sup$-$Wald$ for OLS. If the bootstrap $p$-value (given by the fraction of bootstrap statistics more extreme than the $\sup$-$Wald$ based on the original sample) was larger than $5\%$, then we imposed the $\ell$ breaks (assumed under null $H_0:h=\ell$) in the RF and estimated their locations which were subsequently accounted for in the estimation of the SE.
We now describe other features of the calculations before discussing the results. For the WR and the WF bootstraps the auxiliary distribution (from Assumption (ref)) is the Rademacher distribution proposed by Liu:1988 which assigns 0.5 probability to the value $\nu_t=-1$ and 0.5 probability to $\nu_t=1$, $t=1,\ldots,T$. The same $\nu_t$ is used to obtain both the bootstrap residuals $u_t^b=\hat u_t\nu_t$ and $v_t^b=\hat v_t\nu_t$ in order to preserve the contemporaneous correlation between the error terms. We consider $T=120,240,480$ for the sample size and $B=399$ for the number bootstrap replications. All results are calculated using $N=1,000$ replications.
The reported rejection rates of the WR and WF bootstraps are calculated as: $N^{-1}\sum_{j=1}^N 1_{t_j\geq t^b_{1-\alpha_1,j}}$, where $\alpha_1=0.10,0.05,0.01$ are the nominal values of the tests; $t_j$ is the statistic (sup-$Wald$) computed from the original sample; $t^b_{1-\alpha_1,j}$ is $1-\alpha_1$ quantile of the bootstrap distribution calculated as $(1-\alpha_1)(B+1)$ bootstrap order statistic from the sample of bootstrap statistics in simulation $j=1,\ldots, N$.
For the WR bootstrap, the bootstrap samples were generated recursively with start-up values for $y^b_1$ and $x^b_1$ being given by the first observations from the sample ($x_1,y_1$); see Davidson/MacKinnon:1993.
In all settings, the bootstrap samples are generated by imposing the null hypothesis. The value of $\epsilon$, the trimming parameter in Assumption (ref), is set to $0.15$ which is a typical value used in the literature.
We now turn to our results. We present results for the $\sup$-$Wald$ test under both the null and alternative hypotheses in Tables (ref)-(ref) of the paper. In Tables (ref)-(ref) of the Supplementary Appendix we also present similar results for the $\sup$-$F$ test. The first two columns of these tables give the rejection rates of the tests under the null hypothesis, while columns 3-6 give the rejection rates of the tests under the alternative hypothesis.\footnote{ The rejection rates under the alternative are not level-adjusted, but since we have used the same sequence of random numbers for repetition $i$, $i=1,\ldots,N$, in the experiments under both null and the alternative hypotheses, one can always subtract (or add) the positive (or negative) size discrepancy (relative to the nominal size) from the rejection rate under the alternative in order to obtain the level-adjusted power of the test; see Davidson:1998.}.
From the first two columns of Tables (ref)-(ref), it can be seen that the WR bootstrap works better in general than the WF bootstrap. The latter has large size distortions for scenarios $(h,m)=(0,0)$, $(h,m)=(0,1)$ and $(h,m)=(1,0)$ whether the errors are conditionally homoskedastic, are conditionally heteroskedastic or have a break in the unconditional variance. For scenario $(h,m)=(1,1)$, the WF bootstrap is only slightly undersized or oversized. Regarding the behavior of the $\sup$-$Wald$ test under the alternative hypothesis, the main conclusion that emerges from columns 3-6 of Tables (ref)-(ref) is that the power is influenced in small sample ($T=120$) by the number of breaks in RF and SE, the distribution of the errors $u_t$ and $v_t$, the distribution of $r_t$, as well as the number of breaks in the variance of the errors and in the variance of $\bm r_t$. When there is a break in SE, we need a larger $g$ in (ref) to be able to see an increase in the power of the test, compared with scenarios with no break in SE ($g=0.3, 0.4$ for scenario $(h,m)=(0,1)$, and $g=-0.5,0.5$ for scenario $(h,m)=(1,1)$, while $g=-0.007,-0.009$ for scenario $(h,m)=(0,0)$ and $g=-0.05,-0.07$ for scenario $(h,m)=(1,0)$). This can be explained by the fact that the second break in the SE is tested over smaller samples than the first break in the SE. Moreover, the power is lower for the smallest sample ($T=120$) when the error terms have an upward shift in the variance ($Case$ $C$ in Tables (ref)-(ref)) and the contemporaneous exogenous regressors also have an upward shift in their variance ($Case$ $D$). However, for $T=240, 480$ the power increases sharply in all cases.
In Tables (ref) and (ref) we have sequentially tested for the presence of max $2$ breaks in the RF for $x_t$ (in (ref)-(ref) and (ref)-(ref) respectively) using the WR/WR $\sup$-$Wald$ for OLS, and the resulting number of RF breaks was imposed in each simulation prior to estimating the RF and SE and computing the test statistics for 2SLS. The fraction of times that $0,1,2$ breaks were detected in RF (out of 1,000 replications of the scenarios), is given in Tables (ref)-(ref) from Section (ref) of the Supplemental Appendix. To assess the impact of the pre-testing in RF (in the first two columns of Tables (ref) and (ref)), we have obtained the rejection frequencies of the bootstrap tests when the number of breaks in the RF is held at the true number; see (the first two columns of) Tables (ref) and (ref) from Section (ref) of the Supplemental Appendix. To complement our results, we have also considered a break in RF of smaller size than the one mentioned after (ref)-(ref) by taking $\bm \delta_{r,(1)}^0=(1,1,1,1)'$ (and the rest of the parameter values are as mentioned after (ref)-(ref)); see Tables (ref) and (ref) from Section (ref) of the Supplemental Appendix.
Looking at the results for the $\sup$-$Wald$ our results suggest that in the smaller samples ($T=120$, $240$) the recursive bootstrap is clearly to be preferred over the fixed regressor bootstrap. In the larger sample ($T=480$), the case for the WR over the WF is more marginal as the latter yields only slightly oversized tests. This relative ranking of the two methods is intuitive from the perspective of Davidson:2016's (Davidson:2016) first “golden rule" of bootstrap, which states: “The bootstrap DGP [...] must belong to the model [...] that represents the null hypothesis.” The fixed regressor bootstraps treat the lagged dependent variables in the RF and SE as fixed across bootstrap samples, and as such do not seem to replicate the true model that represents the null hypothesis. This would seem to point toward a recommendation to use the WR but it is important to note an important caveat to our results: our designs involve models for which both recursive and fixed bootstraps are valid. As discussed in Section (ref), the fixed regressor bootstrap is asymptotically valid under weaker conditions than the recursive bootstrap. Therefore, while the recursive bootstrap works best in the settings considered here, there may be other settings of interest in which only the fixed bootstrap is valid and so would obviously be preferred.
In this paper, we analyse the use of bootstrap methods to test for parameter change in linear models estimated via Two Stage Least Squares (2SLS). Two types of test are considered: one where the null hypothesis is of no change and the alternative hypothesis involves discrete change at $k$ unknown break-points in the sample; and a second test where the null hypothesis is that there is discrete parameter change at $l$ break-points in the sample against an alternative in which the parameters change at $l+1$ break-points. In both cases, we consider inferences based on a $\sup$-$Wald$-type statistic using either the wild recursive bootstrap or the wild fixed regressor bootstrap. We establish the asymptotic validity of these bootstrap tests under a set of general conditions that allow the errors to exhibit conditional and/or unconditional heteroskedasticity and the regressors to have breaks in their marginal distributions. While we focus on inferences based on $\sup$-$Wald$ statistics, our arguments are easily extended to establish the asymptotic validity of inferences based on bootstrap versions of the analogous tests based on $\sup$-$F$ statistics; see the Supplementary Appendix.
Our simulation results show that the wild recursive bootstrap is more reliable compared to the wild fixed regressor bootstrap, yielding $\sup$-$Wald$-type tests with empirical size equal or close to the nominal size. The gains from using the wild recursive bootstrap are quite clear in the smaller sample sizes, but are more marginal in the largest sample size ($T=480$) in our simulation study. This would seem to point toward a recommendation to use the wild recursive bootstrap but it is important to note that the wild fixed bootstrap is asymptotically valid under less restrictive conditions than the wild recursive bootstrap. Thus, while both bootstraps are valid in our simulation design, there may be other circumstances when the recursive bootstrap is invalid and the fixed bootstrap would be preferred. The powers of the bootstrap tests are affected in small sample by the characteristics of the error distribution, but in moderate sample sizes often encountered in macroeconomics, there is a very sharp increase in power.
Our analysis covers the cases where the first-stage estimation of 2SLS involves a model whose parameters are either constant or themselves subject to discrete parameter change. If the errors exhibit unconditional heteroscedasticity and/or the reduced form is unstable then the bootstrap methods are particularly attractive because the limiting distributions are non-pivotal. As a result, critical values have to be simulated on a case-by-case basis. In principle it may be possible to simulate these critical values directly from the limiting distributions presented in our Supplementary Appendix replacing unknown moments and parameters by their sample estimates but this would seem to require knowledge (or an estimate of) the function driving the unconditional heteroskedasticity. In contrast, the bootstrap approach is far more convenient because it involves simulations of the estimated data generation process using the residuals and so does not require knowledge of the form of heteroskedasticity. Furthermore, our results indicate that the bootstrap approach yields reliable inferences in the sample sizes often encountered in macroeconomics.