Bootstraps for Dynamic Panel Threshold Models

Threshold regression models have been widely used by empirical researchers, which have been more fruitful because of their extensions to the panel data context. Estimation and inference methods for the threshold model in non-dynamic panels were developed by Hansen, 1999b and Wang, (2015). Dynamic panel threshold models were considered by Seo and Shin, (2016), which proposes the generalized method of moments (GMM) estimation by generalizing the Arellano and Bond, (1991) dynamic panel estimator. A latent group structure in the parameters of the panel threshold model was investigated by Miao et al., 2020b .

Applications of the panel threshold models cover numerous topics in economics. The effect of debt on economic growth is a well-known example that has been analyzed using the panel threshold models, e.g., Adam and Bevan, (2005), Cecchetti et al., (2011) and Chudik et al., (2017). Another example is the threshold effect of inflation on economic growth such as the works by Khan and Senhadji, (2001), Rousseau and Wachtel, (2002), Bick, (2010), and Kremer et al., (2013). The benefit of foreign direct investment to productivity growth that depends on the regime determined by absorptive capacity is studied by Girma, (2005) using firm-level panel data.

It is common practice to make inference in threshold regression models based on an assumption about whether the model is continuous or not. Continuous threshold models that have kinks at the tipping points have received active research attention, e.g., Hansen, (2017); Kim et al., (2019) and Yang et al., (2020). In the literature, kink threshold models are analyzed for estimators that impose the continuity restriction as in Chan and Tsay, (1998), Hansen, (2017), and Zhang et al., (2017). On the other hand, unrestricted estimators are commonly used for discontinuous threshold models as in Hansen, (2000). However, Hidalgo et al., (2019) showed that the unrestricted least squares estimator possesses a different asymptotic property in the absence of discontinuity. Specifically, while the unrestricted model is not misspecified under continuity, failing to impose the restriction results in incorrect inference without proper care.

In the empirical literature, there has been mixed use of kink/discontinuous threshold models without much consideration of a possible specification error. Among the empirical examples referred to previously, Khan and Senhadji, (2001) use a continuous threshold model and impose continuity on their estimation procedure. They claim that the continuous model is desirable to prevent small changes in inflation rate from yielding different impacts around the threshold level. On the other hand, Bick, (2010) claims that the discontinuous threshold model is more appropriate for the same research question since overlooking a regime-dependent intercept can result in omitted variable bias. However, both of them do not provide econometric evidence that supports their choice of models.

For the dynamic panel threshold model, asymptotic normality of the GMM estimator is derived by Seo and Shin, (2016) under the fixed $T$ scheme. However, the asymptotic normality is valid only for the discontinuous models since it requires a full rank condition on the Jacobian of the population moment, which is violated in continuous models. Although the continuity-restricted estimator described in Kim et al., (2019) is asymptotically normal, it may be problematic since empirical researchers often do not agree about whether their threshold models should have a kink or a jump at the threshold as in Khan and Senhadji, (2001) and Bick, (2010). Therefore, we are focusing on the unrestricted GMM estimator and bootstrap inference methods which do not require any pretest on continuity or prior knowledge about continuity of true models.

We first show that when the true model is continuous, the asymptotic normality of the unrestricted GMM estimator breaks down and the convergence rate of the threshold estimator becomes $n^{1/4}$-rate, which is slower than the standard $\sqrt{n}$-rate. Moreover, the standard nonparametric bootstrap is inconsistent in this case because the Jacobian from the bootstrap distribution does not degenerate fast enough due to the slow convergence rate of the threshold estimator.

We propose two different bootstrap methods to obtain confidence intervals for the parameters that are consistent regardless of whether the true model is continuous or not. One is for the threshold location, and the other is for the coefficients. The two bootstrap methods achieve the consistency irrespective of the continuity of the model by adaptively setting the recentering parameter at the bootstrap for GMM introduced by Hall and Horowitz, (1996). This means that our bootstrap moment function achieves zero not at the sample estimator but at the parameter values that we propose. In the bootstrap for the threshold location, we employ a grid bootstrap to fix the recentering parameter. The grid bootstrap was originally proposed by Hansen, 1999a for inference on an autoregressive parameter and applies the test inversion principle. In case of the bootstrap for the coefficients, the recentering parameter is set to adjust the unrestricted estimator by a data driven criterion on the model’s continuity. We also introduce a bootstrap test of model continuity.

Furthermore, we establish the uniform validity of the grid bootstrap for the unknown continuity (or discontinuity) of the threshold model. The importance of uniform validity is well recognized in the literature, notably in the works of Mikusheva, (2007), Andrews and Guggenberger, (2009), and Romano and Shaikh, (2012), among others, who have studied the uniformity of resampling procedures. In particular, Mikusheva, (2007) showed the uniform validity of the grid bootstrap for linear autoregressive models. Our work extends this advantage of the grid bootstrap to a different class of nonstandard inference problems involving continuity of the model.

A set of Monte Carlo simulations demonstrates that the grid bootstrap performs favorably for inference on the threshold location, not only when the model is continuous but also when it includes a jump for various jump sizes. However, inference on the coefficients turns out to be more challenging. Bootstrap confidence intervals for the coefficient, based on percentiles of bootstrap distributions, tend to exhibit severe undercoverage. Nevertheless, our residual bootstrap method improves upon the standard nonparametric bootstrap in both cases.

We apply our inference methods to the dynamic firm investment model, whose static version has been studied by Fazzari et al., (1988) or Hansen, 1999b . It takes financial constraints into account via the threshold effect to determine a firm’s investment decision.

In the literature, Dovonon and Renault, (2013) and Dovonon and Hall, (2018) also deal with the degeneracy of the Jacobian in the context of the common conditional heteroskedasticity testing problem. And a bootstrap based test for the common conditional heteroskedasticity feature was proposed by Dovonon and Goncalves, (2017). However, their works do not deal with a discontinuous criterion function and their null hypothesis of interest always induces the degeneracy of the first-order derivative. That is, they are only concerned with a hypothesis testing and do not consider the confidence intervals. So, they do not have to address the uncertainty associated with the potential degeneracy of the Jacobian.

Meanwhile, there is also a substantial body of literature on singularity-robust inference such as Andrews and Cheng, (2012, 2014) and Han and McCloskey, (2019), among many others. They are motivated by weak or non-identification problems, where models are not point identified. In contrast, we focus on the inference problem that does not involve identification failure even though the Jacobian of the moment restriction can become singular. Andrews and Guggenberger, (2019) study more general singular cases than non-identification, but their approach requires differentiability of sample moments for the subvector inference. Since our model exhibits discontinuity, the method of Andrews and Guggenberger, (2019) is not applicable.

This paper is organized as follows. Section 2 explains the dynamic panel threshold model. Section 3 presents the asymptotic distribution theories of the estimators and test statistics related to the threshold location and continuity. Section 4 proposes bootstrap methods. Section 5 reports Monte Carlo simulation results. Section 6 contains an empirical application. Section 7 concludes. The mathematical proofs and technical details are left to the Appendix.

We consider the dynamic panel threshold model,

where $1\leq i\leq n$, $1\leq t\leq T$, and $x_{it}\in\mathbb{R}^{p}$ is a regressor vector that includes $y_{i,t-1}$ and $q_{it}$. The threshold variable $q_{it}\in\mathbb{R}$ is allowed to be endogenous and is the last element of $x_{it}$.¹¹1Our analysis still holds if researchers have two sets of regressors $x_{1it}$ and $x_{2it}$ such that $y_{it}=x_{1it}^{\prime}\beta+(1,x_{2it}^{\prime})\delta 1\{q_{it}>\gamma\}+\eta_{i}+\epsilon_{it}$ where $q_{it}$ is an element of $x_{2it}$. However, this paper sticks to the current form to keep the exposition simple. Then, we partition $x_{it}$ and write $x_{it}=(\xi_{it}^{\prime},q_{it})^{\prime}\in\mathbb{R}^{p}$.

When $x_{it}$ consists of the lagged dependent variables, the model becomes the well-known self-exciting threshold autoregressive (TAR) model popularized by Chan and Tong, (1985). The static version where the lagged dependent variables are excluded from $x_{it}$ was considered by Hansen, 1999b , while the current dynamic model was studied by Seo and Shin, (2016).

The parameter $\gamma\in\Gamma$ denotes the threshold location, where $\Gamma$ is a compact set in $\mathbb{R}$, and $\alpha=(\beta^{\prime},\delta^{\prime})^{\prime}\in A\subset\mathbb{R}^{2p+1}$ denotes the collection of coefficients. Let $\theta=(\alpha^{\prime},\gamma)=(\beta^{\prime},\delta^{\prime},\gamma)^{\prime}\in\Theta=A\bigtimes\Gamma$ denote the vector of all the parameters. The fixed effect $\eta_{i}$ is constant across time for each individual in the panel data. It is not identified but is eliminated after first-differencing for the GMM estimation. The idiosyncratic error $\epsilon_{it}$ is independent across individuals.

For the estimation, we use the GMM after the first-difference transformation

where

Let $z_{it}$ denote a set of instrumental variables at time $t$ such that $E[z_{it}\Delta\epsilon_{it}]$ becomes a zero vector, which may include lagged dependent variables $y_{it-2},...,y_{i1}$ and certain lagged variables of covariates $x_{it}$ and/or $q_{it}$, depending on the assumptions regarding exogeneity of those variables.

Then, we can define a vector of moment functions for the GMM estimation,

where $k\geq dim(\theta)=2p+2$ and $t_{0}\geq 2$ is the earliest period that the regressor and instrument can be defined. For example, $k=(T-1)(T-2)/2$ when $z_{it}=(y_{it-2},...,y_{i1})^{\prime}$ and $t_{0}=3$. Denote the population moment by $g_{0}(\theta)=E[g_{i}(\theta)]$ and the sample moment by

We write $g_{i}$ instead of $g_{i}(\theta_{0})$ for simplicity of notations.

We consider the two-stage GMM estimation of the dynamic panel threshold model. In the first stage, we get an initial estimate by $\hat{\theta}_{(1)}=\arg\min_{\theta\in\Theta}\bar{g}_{n}(\theta)^{\prime}\bar{g}_{n}(\theta)$ to compute a weight matrix

and obtain the second stage estimator

where $\hat{Q}_{n}(\theta)=\bar{g}_{n}(\theta)^{\prime}W_{n}\bar{g}_{n}(\theta)$. Seo and Shin, (2016) proposed averaging of a class of GMM estimators that are constructed from randomized first stage estimators. We do not pursue the averaging since our primary goal is the bootstrap inference.

In practice, the grid search algorithm is employed to compute the estimates. Note that when $\gamma$ is given, $\hat{\alpha}(\gamma)=\arg\min_{\alpha\in A}\hat{Q}_{n}(\alpha,\gamma)$ can be easily computed because the problem becomes the estimation of the linear dynamic panel model. Then, $\hat{\gamma}$ minimizes the profiled criterion $\tilde{Q}_{n}(\gamma)=\hat{Q}_{n}(\hat{\alpha}(\gamma),\gamma)$ over the grid of $\Gamma$.

Let $\theta_{0}=(\alpha_{0}^{\prime},\gamma_{0})^{\prime}=(\beta_{0}^{\prime},\delta_{0}^{\prime},\gamma_{0})^{\prime}$ denote the true parameter value that lies in the interior of $\Theta$. For the point identification of $\theta_{0}$, $g_{0}(\theta)=0_{k}$ should hold if and only if $\theta=\theta_{0}$, where $0_{k}=(0,...,0)^{\prime}\in\mathbb{R}^{k}$. Let

and $M_{i}(\gamma)=\left[\begin{array}[]{c;{2pt/2pt}c}M_{1i}&M_{2i}(\gamma)\end{array}\right]$. Additionally, define $M_{0}(\gamma)=E[M_{i}(\gamma)]$, $M_{10}=E[M_{1i}]$, $M_{20}(\gamma)=E[M_{2i}(\gamma)]$, $\bar{M}_{n}(\gamma)=n^{-1}\sum_{i=1}^{n}M_{i}(\gamma)$, $\bar{M}_{1n}=n^{-1}\sum_{i=1}^{n}M_{1i}$, and $\bar{M}_{2n}(\gamma)=n^{-1}\sum_{i=1}^{n}M_{2i}(\gamma)$. We write $M_{0}$, $M_{20}$ and $\bar{M}_{n}$ instead of $M_{0}(\gamma_{0})$, $M_{20}(\gamma_{0})$ and $\bar{M}_{n}(\gamma_{0})$, respectively, for simplicity of notation. The identification condition is stated in Theorem 1 that follows.

Theorem 1 (i) is the identification condition for the coefficients once the true threshold location is identified. This means that instruments should be relevant to the first-differenced regressors appearing in $\eqref{eq:fd}$ when $\gamma=\gamma_{0}$.

Theorem 1 (ii) is for the identification of the threshold location, which excludes the possibility of $\delta_{0}=0_{p+1}$. In the standard GMM problem, it is usually assumed that the Jacobian of $g_{0}(\theta)$ at $\theta_{0}$ is of full column rank for both the point identification and the asymptotic normality of the GMM estimator. The condition (ii) does not require the full rank condition on the Jacobian, which is related to the presence of a jump in the threshold model, and thus it generalizes the identification conditions in Seo and Shin, (2016). When the model is continuous and has a kink at the threshold location, the last column of the Jacobian matrix, which is the first-order derivative with respect to $\gamma$ at the true parameter, becomes a zero vector. This degeneracy does not violate the condition (ii), but it fails the asymptotic normality of the standard GMM estimator, which relies on the linearization of $g_{0}(\theta)$ near $\theta_{0}$ as in Newey and McFadden, (1994).

To define the continuity, recall that $q_{it}$ is the last element of $x_{it}$ such that $x_{it}=(\xi_{it}^{\prime},q_{it})^{\prime}\in\mathbb{R}^{p}$. Accordingly, partition $\delta=(\delta_{1},\delta_{2}^{\prime},\delta_{3})^{\prime}$, where $\delta_{2}\in\mathbb{R}^{p-1}$ and $\delta_{1},\delta_{3}\in\mathbb{R}$, and $\delta_{0}=(\delta_{10},\delta_{20}^{\prime},\delta_{30})^{\prime}$. Hence, $\delta_{3}$ is the change in the coefficient of the threshold variable when the threshold variable surpasses the tipping point. Likewise, $\delta_{2}$ and $\delta_{1}$ are the changes in the coefficients for the other regressors, $\xi_{it}$, and the intercept, respectively. The continuity of the dynamic panel threshold model is formally given in Definition 1.

Note that this definition of continuity requires that $\delta_{3}\neq 0$; otherwise, $\delta=0_{p+1}$.

The rank of the first-order derivative matrix, say $D_{1}$, of $g_{0}(\theta)$ at $\theta=\theta_{0}$ is crucial to the standard asymptotic normality of the GMM estimator. Let $G$ denote the first-order derivative of $g_{0}(\theta)$ with respect to $\gamma$ at $\theta=\theta_{0}$. Then,

where the conditional expectation $E_{t}[\cdot|q]=E[\cdot|q_{it}=q]$ and the density function $f_{t}(\cdot)$ of $q_{it}$ are assumed to exist. The derivation of $G$ is provided in the proof of Lemma D.1. Note that the first-order derivative of $g_{0}(\theta)$ with respect to $\alpha$ at $\theta=\theta_{0}$ is $M_{0}$. The linear independence of $G$ from the other columns in $D_{1}$ is required for the standard linear approximation

Recall that the vector $G$ can be written as the product of the matrix $G_{0}$ and the vector $\delta_{0}$, (5), and the first and last columns of $G_{0}$ are linearly dependent since $q_{it}=\gamma_{0}$ for all $t$ due to the conditioning. Then, the standard rank condition on the first derivative matrix $D_{1}$ can follow from a more primitive rank condition on $\left[\begin{array}[]{c;{2pt/2pt}c}M_{0}&G_{0,-(p+1)}\end{array}\right]$, that is, the linear independence of all the columns in $M_{0}$ and all but the last column of $G_{0}$, for the discontinuous case. Even if the primitive condition is met, however, the continuity restriction makes $G=0_{k}$ since $E_{s}[z_{it}(1,x_{is}^{\prime})\delta_{0}|\gamma_{0}]=(\delta_{10}+\delta_{30}\gamma_{0})E_{s}[z_{it}|\gamma_{0}]=0$ for $s\leq t$, which leads to degeneracy of $D_{1}$.

When the rank condition fails due to the continuity, the expansion becomes

where

The detailed derivation is given in the proof of Lemma D.1. It is worth noting that $H$ is identical to the first column of $G_{0}$ up to a constant multiple. Then, the rank condition on $\left[\begin{array}[]{c;{2pt/2pt}c}M_{0}&H\end{array}\right]$ is implied by the rank condition on $\left[\begin{array}[]{c;{2pt/2pt}c}M_{0}&G_{0,-(p+1)}\end{array}\right]$. Thus, the rank condition on $\left[\begin{array}[]{c;{2pt/2pt}c}M_{0}&G_{0,-(p+1)}\end{array}\right]$ can be viewed as a sufficient condition for both Assumptions LK and LJ in the next section, apart from the continuity restriction on $\theta$. Next section formalizes this discussion and presents the asymptotic distribution of the GMM estimator $\hat{\theta}$ under the continuity.

This section considers the asymptotic analysis when $T$ is fixed, the data are independent and identically distributed across $i$, and $n\rightarrow\infty$. Specifically, the data for each individual $i$ is determined by the realization of $\{(z_{it},x_{it},\epsilon_{it})_{t=1}^{T},y_{i0},\eta_{i}\}$, where $y_{i0}$ denotes the initial value. We make the following assumptions.

Assumptions G and D are similar to Assumptions 1 and 2 in Seo and Shin, (2016) except for the differentiability conditions in D which allow the second-order derivative of the population moment to be defined. Since the regressors include lagged dependent variables, G requires the individual fixed effects and initial values to have finite fourth moments, too. The assumption also includes the conditions in Theorem 1. LK is a rank condition for a nondegenerate asymptotic distribution when the underlying model is continuous. This condition may be viewed as less restrictive than the standard rank assumption as discussed in the preceding section where $G$ and $H$ are defined. For easy reference, we restate the standard full rank assumption for the asymptotic normality of the GMM estimator for the discontinuous threshold regression below.

In a simple model, where $y_{it}=x_{it}^{\prime}\beta+(\delta_{1}+\delta_{3}q_{it})1\{q_{it}>\gamma\}+\eta_{i}+\epsilon_{it}$, LK is equivalent to LJ because $G=(\delta_{10}+\delta_{30}\gamma_{0})G_{01}$ while $H={\delta_{30}}G_{01}/2$, where $G_{01}$ is the first column of $G_{0}$ in (5).

Theorem 2 below establishes the asymptotic distribution of the GMM estimator when the dynamic panel threshold model is continuous.

We observe that the convergence rate of $\hat{\gamma}$ is $n^{1/4}$, which is slower than the standard $\sqrt{n}$-rate. Meanwhile, Seo and Shin, (2016) show the $\sqrt{n}$-convergence rate for $\hat{\gamma}$ when the model is discontinuous. Intuitively, it would be more difficult to detect the precise threshold location when there is a kink than when there is a jump at the tipping point. More technically, when the threshold model is discontinuous and the Jacobian is not singular, the limit of the GMM objective function admits a quadratic approximation with respect to $\gamma$ at the true value, while the limit admits a quartic approximation for the continuous model. Hence, the limit objective function becomes flatter in $\gamma$ at the true value resulting in the slower convergence rate. Hidalgo et al., (2019) also showed in the least squares context that when the model is continuous, the convergence rate of the threshold estimator slows down to $n^{1/3}$, while it is superconsistent $n$-rate when the model is discontinuous.

Moreover, we can observe that the asymptotic distribution of $\hat{\alpha}$ is also shifting to a non-normal distribution. Hence, standard inference methods based on the asymptotic normality become invalid for the continuous dynamic panel threshold model.

The asymptotic distribution of the GMM estimator is identical to the distribution reported in Theorem 1 (b) in Dovonon and Hall, (2018), which studies a smooth GMM problem with the degeneracy of the Jacobian. Theorem 2 shows that even though the criterion of our threshold model is discontinuous with respect to the parameter $\gamma$, the same asymptotic distribution as that of Dovonon and Hall, (2018) appears.

The censored normal distribution also appears in Andrews, (2002) which studies the estimation of a parameter on a boundary. Heuristically, because our analysis depends on the second-order derivative of $\gamma$ for the local polynomial expansion of $g_{0}(\theta)$ near $\theta_{0}$, only the asymptotic distribution of $(\hat{\gamma}-\gamma_{0})^{2}$ can be derived. Since $(\hat{\gamma}-\gamma_{0})^{2}$ should be nonnegative, the asymptotic censored normal distribution appears as in Andrews, (2002). Meanwhile, Dovonon and Goncalves, (2017) show that the standard nonparametric bootstrap becomes invalid when the Jacobian degenerates. To address this issue, we propose different bootstrap methods in Section 4 for the inference of the parameters.

The asymptotic distribution in Theorem 2 can be used for parameter inference when the true model is continuous, but the estimator is obtained without imposing the continuity restriction. As discussed in Seo and Shin, (2016), $M_{0}$ and $\Omega$ can be consistently estimated, while $H$ can be nonparametrically estimated similarly to $G$. It is then straightforward to simulate the limit distribution from Theorem 2 by generating random numbers for $U$ and $V$. However, there are several drawbacks to that approach, and hence we do not recommend it. First, empirical researchers might construct confidence intervals based on Theorem 2 when they cannot reject the continuity. However, Leeb and Pötscher, (2005) show that confidence intervals after model selection are subject to size-distortion. Second, even if the true model is known to be continuous, the continuity-restricted estimator explained in Kim et al., (2019) is more efficient and asymptotically normal. Therefore, using the continuity-restricted estimator for estimation and inference is preferable. Finally, the nonparametric estimation of $H$ requires a tuning parameter and has a slower convergence rate.

Seo and Shin, (2016) derived the asymptotic distribution of the GMM estimator and propose an inference method when the underlying model is discontinuous. When the true model is discontinuous and Assumptions G, D, and LJ hold,

$\Omega$ can be estimated by $\hat{\Omega}=\frac{1}{n}\sum_{i=1}^{n}[g_{i}(\hat{\theta})g_{i}(\hat{\theta})^{\prime}]-\bar{g}_{n}(\hat{\theta})\bar{g}_{n}(\hat{\theta})^{\prime}$. Note that $D_{1}=\left[\begin{array}[]{c;{2pt/2pt}c}M_{0}&G\end{array}\right]$, and $M_{0}$ can be estimated by $\bar{M}_{n}(\hat{\gamma})$, while the estimation of $G$ involves nonparametric estimation of the conditional means and densities. See section 4 of Seo and Shin, (2016) for more details. Note that $(\hat{D}_{1}\hat{\Omega}^{-1}\hat{D}_{1})^{-1}$ diverges when the model is continuous since the last column of $\hat{D}_{1}$ converges to a zero vector when it is consistent. This paper does not analyze the issue and leaves it for future research.

As usual, the superscript “*” denotes the bootstrap quantities or the convergence of bootstrap statistics under the bootstrap probability law conditional on the original sample. For example, $E^{*}$ denotes the expectation with respect to the bootstrap probability law conditional on the data. “$\xrightarrow{d^{*}}$, in $P$” denotes the distributional convergence of bootstrap statistics under the bootstrap probability law with probability approaching one. We write “$\nu_{n}^{*}=O_{p}^{*}(1)$, in $P$” if a sequence $\nu_{n}^{*}$ is stochastically bounded under the bootstrap probability law with probability approaching one. More details are written in Section B.1. Let $\widehat{F}^{*-1}_{n}(\varphi;S^{*})$ denote the empirical $\varphi$ quantile of a bootstrap statistic $S^{*}$.

This section introduces three different bootstrap schemes. The first bootstrap is for constructing bootstrap confidence interval(CI)s for the threshold, while the second bootstrap is for constructing bootstrap CIs for the coefficients. Both methods aim to provide valid inferences, regardless of whether the model is continuous or not. The third bootstrap is for testing continuity of the threshold model. The three bootstrap methods can be represented by means of Algorithm 1 with suitable choices of $\theta_{0}^{*}=(\beta_{0}^{*\prime},\delta_{0}^{*\prime},\gamma_{0}^{*})^{\prime}$.

In step 1, we resample the regressors, the instruments, and the residuals jointly to maintain the dependence among them, unlike in the usual residual bootstrap. See e.g., Giannerini et al., (2024) for the description of the standard residual bootstrap, which resamples the residuals only, and the wild bootstrap for the testing of linearity in the threshold regression. There could be other ways of resampling not mentioned here and we do not attempt to decide which is the best here.

The parameter $\theta_{0}^{*}$ is used in step 2 of Algorithm 1 to generate the dependent variables in the bootstrap samples. In step 4, recentering of the bootstrap sample moment is done by subtracting $\bar{g}_{n}(\hat{\theta})=(\frac{1}{n}\sum_{i=1}^{n}z_{it_{0}}^{\prime}\widehat{\Delta\epsilon}_{it_{0}},...,\frac{1}{n}\sum_{i=1}^{n}z_{iT}^{\prime}\widehat{\Delta\epsilon}_{iT})^{\prime}$. Note that the expectation of $\bar{g}_{n}^{*}(\theta)$ by the bootstrap probability law conditional on the data becomes zero when $\theta=\theta_{0}^{*}$ due to the recentering, which can be easily checked from the following equations: $g_{it}^{*}(\theta_{0}^{*})=z_{it}^{*}(\Delta y_{it}^{*}-\Delta x_{it}^{*}\beta_{0}^{*}-1_{it}(\gamma_{0}^{*})^{\prime}X_{it}^{*}\delta_{0}^{*})=z_{it}^{*}\widehat{\Delta\epsilon}_{it}^{*}$ and $E^{*}[g_{it}^{*}(\theta_{0}^{*})]=n^{-1}\sum_{i=1}^{n}z_{it}\widehat{\Delta\epsilon}_{it}$ for $t=t_{0},...,T$.

A different choice of $\theta_{0}^{*}$ leads to a different bootstrap. For example, if $\theta_{0}^{*}=\hat{\theta}$, then the bootstrap becomes the standard nonparametric bootstrap in Hall and Horowitz, (1996) because $\Delta y_{it}^{*}=\Delta y_{i^{*}t}$ holds true for $i=1,...,n$ and $t=t_{0},...,T$ in step 2. Note that, for $\theta_{0}^{*}$ not equal to $\hat{\theta}$, step 2 of Algorithm 1 generates $\Delta y_{it}^{*}$’s that are generally different from $\Delta y_{i^{*}t}$’s. The following subsections detail three different choices of $\theta_{0}^{*}$ for three different inference problems.