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Estimation and Inference in Threshold Predictive Regression Models with Locally Explosive Processes
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Consider the linear predictive regression model with the following form
where $y_t \in \mathbb{R}$, $\boldsymbol{X}_{t-1} \in \mathbb{R}^{p+1} = \big( 1, x_{t-1}^{\prime} \big)^{\prime}$ includes both the intercept and the $p-$dimensional regressor vector $x_{t-1}$, with the parameter vector being $\theta_0 = \left( \alpha_0, \beta_0 \right)$ such that $\beta_0$ consists of $p$ coefficients. Moreover, throughout the paper, we assume that regressors are generated via the following locally persistent autoregression process studied by lieberman2017multivariate, lieberman2018iv, lieberman2020hybrid
where $\varphi$ is an unknown coefficient such that $\boldsymbol{\varphi} := \big( \varphi_1,..., \varphi_d \big)$ with $d \equiv p$, that is, we assume that the parameter $\varphi$ has the same dimension as the number of regressors included in the model. Notice that the inner product in expression (ref) implies that $\langle \varphi , u_{\varphi t} \rangle = \sum_{j=1}^d \varphi_j u_{\varphi, jt} \equiv \widetilde{\boldsymbol{\varphi}}_{( 1 \times n )}$. Although the persistence coefficients, $c_i \ \forall \ i \in \left\{ 1,..., p \right\}$ remain unchanged across $t$, the autocorrelation coefficient is estimable separately for each $t \in \left\{ 1,..., n \right\}$ due to the terms $\langle \varphi , u_{\varphi t} \rangle \otimes I_p$ and $\langle \varphi , u_{\varphi t} \rangle^2 \otimes I_p$. Furthermore, the consistent and robust estimation (to the presence of the unknown persistence) of the true parameter vector $\boldsymbol{\theta}_0$ is the main interest of our study. Additionally, the estimation of the parameter vector $\boldsymbol{\varphi}$ is a relevant but challenging aspect due to its stochastic nature. We focus on estimation and inference aspects in threshold predictive regression models with locally explosive regressors. In particular, the non time-invariant property of the autocorrelation coefficient in the nonstationary autoregressive model implies a time-specific estimation of these matrices
for all $j \in \left\{ 1,..., n \right\}$. Furthermore, in this paper we focus in the case of locally explosive processes which implies that for the diagonal matrix $\boldsymbol{C}_p = \mathsf{diag} \left( c_1,..., c_p \right)$, it holds that $c_i > 0 \ \forall \ i \in \left\{ 1,..., p \right\}$. Regardless of the additional terms in the particular representation of a local-to-unity process, it has been proved that the partial-sum process of $\boldsymbol{x}_t$ weakly convergence to the following limit process
Consider the stochastic process given by
where $c \in \mathbb{R}_{+}$ and $\varphi \in \mathbb{R}^d$. The particular specification implies that the autoregressive coefficient corresponds to a stochastic time varying parameter that fluctuates in the vicinity of unity according to the proprerties of the exogenous covariate $v_t$, the degree of persistence $c$, the sample size $n$. Based on these settings one can then estimate the unknown parameter pair $( c, \varphi )$ using a NLLS estimation approach (see, lieberman2020hybrid) such that
Overall, in this paper similar to the spirit of koul1990weakly, we consider the construction of estimators that are asymptotically efficient over a range of nuisance parameters, known as adaptive. Thus in our modeling environment, we consider two nuisance parameters: one being the degree of persistence, $c$, and the second being the unknown coefficient, $\boldsymbol{\varphi}_t := \big( \varphi_{1},..., \varphi_{d} \big)$, that corresponds to the exogenous covariate $\boldsymbol{v}_t$. Furthermore, we are concerned with the least squares estimators and endogenously instrumental generated estimators of the predictive and threshold predictive regression models when the nonstationary process that generate the regressors have an additional exogenous variation. In particular, our theory relies on making use of the Ornstein-Uhlenbeck processes within an uncertain environment, since the coefficients $\left\{ \beta_{nt} \right\}_{t=1}^n$ are random processes.
Thus in relation to the process (ref), we define the Ornstein-Uhlenbeck process in a random external environment $\left\{ G_{c, \varphi}(t) \right\}_{ t \in [0,1]}$ driven by the process $G$ such that
Furthermore, consider the autoregressive regression model with stochastic local unit roots (ref) and assume that the coefficient $\varphi$ is known. Then, lieberman2020hybrid proved that
Consequently, in the case when the parameter vector $\varphi = 0$, the above limit result becomes
which corresponds to the standard asymptotic theory result under the LUR specification of the autocorrelation coefficient without the additional exogenous variation term.
Our next concern is the estimation of the threshold effects in the linear predictive regression model with locally persistence processes (i.e., or locally explosive processes in our settings). A key contribution of our study is that we focus on comparing the asymptotic behaviour of both the classical least squares estimator as well as the instrumental variable based estimator (IVX). Although the study of chen2022estimation considers a similar setting of a threshold regression model with stochastic local unit root regressors and then develop a statistical testing procedure for the presence of threshold effects using the sup OLS-Wald test statistic, the limiting distribution is not free of any nuisance parameters due to its dependence on the unknown degree of persistence. On the other hand, in this paper we implement the IVX implementation of the sup-Wald test in the threshold predictive regression model that has been recently proposed by liu2022robust, in the case of the linear conditional mean and conditional quantile predictive regressions models.
Notice that the proposed data generating process that allows to consider departures from unit root behaviour involve temporary departures from unity at any sample point that can move the process in stationary or explosive directions. An alternative approach than the one we focus in our work, is to consider the unknown localizing coefficient of persistence to be time-varying such that $c_n \equiv c_n (t/n)$. Such type of functional coefficient representation is examined in the studies of bykhovskaya2020point and bykhovskaya2018boundary. Although, in this scenario the autocorrelation coefficient varies with time and can be arbitrarily close to unity the limit process of partial sum processes converge to the standard OU process rather than the OU process in a random environment as we have in this paper. The particular limit appears regardless of the presence of the threshold variable, which indeed complicates the asymptotic theory when considering the convergence results of estimators and test statistics under the assumption of diminishing threshold effects.
The predictive regression model is a popular model widely used the past two decades for investigating the predictability puzzle in stock returns. Specifically, the stock return predictability literature examines various statistical and empirical aspects; seminal studies include those of pesaran1995predictability and stambaugh1999predictive. A particular phenomenon discussed in the more recent literature is the detection of the so-called "pockets of predictability", which means that the presence of predictability is characterized by threshold effects and nonlinearities in relation to macroeconomic events and financial stability. On the other hand, a more realistic feature is to assume that the persistence properties of predictors can be stochastic, alternating along with the aforementioned features. In this paper, we study aspects of estimation and inference in threshold predictive regression models with locally explosive regressors. In particular, we develop the asymptotic theory for the OLS based estimator as well as the instrumental variable (IVX) based estimator proposed by magdalinos2009limit in threshold predictive regression models.
Various studies proposed inference methodologies for threshold models under the assumption of time series stationarity such as in hansen1996inference, hansen2000sample, caner2001threshold, gonzalo2002estimation, pitarakis2008comment, galvao2011thresholdt, galvao2014testing, kourtellos2014structural, kourtellos2017endogeneity and chiou2018nonparametric. Furthermore, in cointegrated and predictive regression models, gonzalo2006threshold, gonzalo2012regime, gonzalo2017inferring and chen2015robust proposed predictability tests under the presence of threshold effects for persistent regressors modelled by the local-unit-root specification. The particular predictability tests inspired by the IVX estimator of phillipsmagdal2009econometric are examined by kostakis2015Robust in linear predictive regressions with abstract degree of persistence.
In addition, the time series econometrics literature also focused on developing methodologies for modelling financial bubbles which include the study of explosive and nonstationary processes as presented in the papers of phillips2007limit, nielsen2010analysis, magdalinos2009limit, guo2019testing, pedersen2020testing and duffy2021estimation. Furthermore, another relevant aspect is the development of econometric methods for detecting and dating market exuberance, with some relevant studies being those of phillips2011explosive and phillips2011dating in which the focus is both estimation of such models under the presence of explosive bubbles as well as the dating of the exact appearance of this event. Moreover, farmer2019pockets, demetrescu2020testing and georgiev2021extensions consider the modeling and testing for the presence of time period specific predictability while bykhovskaya2020point consider modelling a time-varying degree of persistence as a way to capture these effects.
Therefore, motivated by these stylized facts in many financial and macroeconomic data, lieberman2017multivariate, lieberman2020hybrid acknowledge the shifting of persistence in time series and develop a more general specification of nonstationary processes which includes both the unknown persistence in the time series properties of regressors as well as a stochastic departure from local to unity behaviour. In this direction, the study of chen2022estimation propose a framework for estimation and inference in threshold regression with hybrid stochastic local unit root regressors. These authors develop their own asymptotic theory results which are useful when considering estimation in a threshold environment with stochastic unit root regressors (see also liu2022robust).
In terms of the estimation and identification of threshold effects and nonlinearities in predictability (e.g., maasoumi2002entropy, gonzalo2012regime and michaelides2016non) via autoregressions\footnote{A seminal work on the threshold model for autoregression processes is proposed by tsay1989testing, tsay1998testing while staiger1997instrumental introduce the framework of IV regression with weak instruments.} is crucial for correctly modelling economic phenomena such as price asymmetries and herd behaviour in financial markets (see, lux1995herd, brunnermeier2009deciphering and kapetanios2014nonlinear) as well as in valuing macroeconomic fundamentals (see, bohn1998behavior and hatchondo2016debt). When these features are not correctly captured they can manifest as structural breaks resulting to biased parameter estimates. On the other hand, the structural break literature emphasized the presence of shifts in persistence as in horvath2020sequential.
Therefore, the inclusion of hybrid stochastic unit roots to the threshold predictive regression model is a novel feature which can accommodate more realistic aspects when modelling return predictability. In terms of the econometric challenges appeared within our setting these include the problem of a nuisance parameter identification, only under the alternative hypothesis. To deal with this aspect, we follow similar techniques as the ones proposed by davies1977hypothesis, davies1987hypothesis, andrews1993tests andrews1994optimal, and hansen1996inference. Our contributions in this paper are threefold. Firstly, we study relevant aspects to the estimation and inference in threshold predictive regression models with a more general persistence properties, and specifically focusing on locally explosive processes contributing this way to the recent predictability literature. Secondly, we study the applied and theoretical implementation of the IVX estimator proposed by phillipsmagdal2009econometric within our setting. Thirdly, an empirical application examines the regime-specific predictability hypothesis using the economic uncertainty index as the switching threshold effect.
\paragraph{Structure.}
The rest of the paper is organized as follows. In Section (ref), we introduce the model estimation procedure and main assumptions of the proposed econometric environment. In Section (ref), we develop the asymptotic theory for the two proposed estimators under the presence of a threshold effect with locally explosive regressors. In Section (ref), we examine the finite-sample performance of the sup-Wald type test statistics and summarize our main findings based on an extensive Monte Carlo simulation study. Section (ref) provides empirical evidence of threshold effects and regime-specific predictability using financial ratios of equity indices. Section (ref), summarizes our main conclusions. The proofs of the main results in the paper can be found in the Appendix.
\paragraph{Notation.} We denote with $\left( B_t \right)_{ t \in [0,1] }$ the Brownian motion defined on the probability space $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$ which is independent of the $\sigma-$field $\mathcal{F}$. The limiting process $G$ is an $\mathcal{F}-$conditional Gaussian martinagle with $\mathcal{F}-$conditional mean. In particular, for each $t > 0$, $G_t$ or equivalently $G(t)$ has a mixed normal distribution. Moreover, we denote with $\left\lVert . \right\rVert_F$ and $\left\lVert . \right\rVert_1$ the spectral norm, $L_2$ (Frobenius) and the $L_1$ norm respectively. The symbol$=_d$ denotes equivalence in distribution and the notation $\to_p$ convergence in probability, while $\Rightarrow$ denotes weak convergence in the function space $C([0,1])$ equipped with a suitable topology.
Consider the following threshold predictive regression model
where $y_t \in \mathbb{R}$ and $x_t$ is a $p-$dimensional vector of predictors parametrized by a vector of locally explosive processes (a special case of locally stochastic unit roots) given by expression (ref). In the special case that the model has a single regressor then its assumed to be generated by
where $c$ is the localizing coefficient of persistence such that $c_j > 0 \ \forall \ j \in \left\{1,...,p \right\}$ and $\upepsilon_{k,t}$ is $p \times 1$ vector with the initial condition given by $x_{1} = u_{x1}$. The particular hybrid local unit root specification allows for general types of dependence to be modelled by the threshold predictive regression given by (ref) and (ref) with $t \in \left\{ 1,...,n \right\}$ but we focus on locally explosive processes.
Consider the vector $\xi_t = \left( u_{yt}, u^{\prime}_{xt}, u^{\prime}_{ \varphi t} \right)^{\prime}$, where both both $u^{\prime}_{xt}$ and $ u^{\prime}_{ \varphi t}$ are $p-$dimensional time series vectors, such that $\xi_t$ is a strictly stationary martingale difference sequence. The partial sums process satisfy the invariance principle (phillips1987time, phillips1987towards, phillips1988testing) such that
where the covariance matrix $\mathbf{\Sigma}_{\xi \xi}$ is defined as below
where $\mathbf{B}_{\xi }(r)$ is a vector Brownian motion with a positive-definite matrix $\mathbf{\Sigma}_{\xi \xi}$ such that all these covariance matrices are positive-definite (for the innovation structure, see phillips1992asymptotics).
The econometric identification of the threshold variable is implemented via the conditional mean specification function given by expressions (ref)-(ref), which implies that the presence of a threshold effect is represented by a fixed threshold parameter which is data-driven such that $\gamma \in \Gamma := \left[ \gamma_1, \gamma_2 \right]$. The lower threshold and upper threshold bounds are estimated such that $\mathbb{P} \left( q_t \leq \gamma_1 \right) = \pi_1 > 0$ and $\mathbb{P} \left( q_t > \gamma_1 \right) = \pi_2 < 1$. Moreover, we define the indicator variables to determine the threshold regime by $I_{1t} \equiv \mathbb{P} \left( q_t \leq \gamma \right)$ and $I_{2t} \equiv \mathbb{P} \left( q_t > \gamma \right)$. Then, by replacing the threshold variable with a uniformly distributed random variable, $U \sim Unif[0,1]$, and using the transformation principle\footnote{In practice this property allows us to estimate the threshold variable using the uniform distribution without having to impose additional regularity conditions on the CDF of the unknown threshold.} which allows us to transform the CDF of any random variable to that of uniformly distributed random variables such that $I \left( q_t \leq \gamma \right) = I \left( F(q_t) \leq F(\gamma) \right) \equiv I \left( U_t \leq \lambda \right)$.
For notation convenience, we express the threshold predictive regression model in a matrix form. To do this, we denote with $y$ the vector stacking $y_t$ and $X_i$ the matrix stacking $\big( I_{it} \ x_t I_{it} \big)$ for $i = 1,2$ such that $\boldsymbol{y} = \boldsymbol{X}_1 \boldsymbol{\theta}_1 + \boldsymbol{X}_2 \boldsymbol{\theta}_2 + \boldsymbol{u}$, with $\boldsymbol{X} = \big( \boldsymbol{X}_1 \ \boldsymbol{X}_2 \big)$, $\boldsymbol{\theta} = \big( \theta_1, \theta_2 \big)$ and $\theta_i = \left( \alpha_i, \beta_i \right)$ for $i =1,2$. Alternatively, in the case when the model has only slopes, we can define $y = X(\gamma) \theta + u$ with $\theta = ( \beta^{\prime}, \delta^{\prime} )$ and $X_t(\gamma ) = \left[ x_t^{\prime} , x_t^{\prime} I \left( q_t \leq \gamma \right) \right]$. For the remaining of the paper we use the symbol $\top$ to denote the transpose of a matrix or a vector.
Hence, given a nonstochastic threshold parameter $\gamma \in \left[ \gamma_1, \gamma_2 \right]$ we can estimate $\theta ( \gamma )$ with
Then, the least squares estimator of $\gamma$ is given by the following optimization function
where the objective function is defined as
and $\text{SSR} \left( \gamma \right)$ represents the concentrated sum of squared errors. To have consistent and efficient statistical inferences, we consider that the following general assumptions hold. In particular, for the consistent estimation of the threshold effects, we impose related regulatory conditions which are given by Assumptions (ref)-(ref) below.
Assumption (ref) provides a condition for the presence of strictly stationary threshold variable $q_t$. Moreover, Assumption (ref) states that the threshold variable $q_t$ is contemporaneously exogenous in the model and ensures that the multivariate invariance principle for the partial sum of the martingale difference array (ref) holds. Then, with Assumption (ref) we impose the existence of a time-invariant continuous distribution function for the threshold variable, which ensures the existence of dense true threshold levels as the sample size increases. Furthermore, we can impose additional assumptions which ensure that the threshold variable induces diminishing effects which vanish asymptotically. We denote with $W(r, \lambda )$ to be a two-parameter Brownian motion on the topological space $(r,\lambda) \in [0,1]^2$ in a similar manner as in the paper of caner2001threshold.
We define the following stochastic process\footnote{Notice that the particular stochastic process (i.e., notanionwise) can be seen as the solution of the stochastic differential equation: $dX_t = X_t d \widetilde{Z}_t + d Z_t$ such that $$ X_t = \mathsf{exp} \left\{ \widetilde{Z}_t - \frac{1}{2} \langle \widetilde{Z} \rangle_t \right\} . \left( X_0 + \int_0^t \mathsf{exp} \left\{ ... \right\} \right)$$} (see, lieberman2020hybrid)
such that
The nonlinear stochastic process above along with Assumption (ref) is instrumental when developing the asymptotic theory of our test statistics. Notice that since our proposed framework allows for the existence of hybrid stochastic local unit roots the diffusion processes have a nonlinear structure to capture the additional features. When all $a_j$ and $c_j$ are zero, then we have a threshold model with unit root regressors. Therefore, our proposed framework is more general and encompasses cases such as near-unit root or integrated regressors, which can be modelled in the case of LUR specification. Notice that the threshold indicator has the same lag as the predictor in the predictive regression model (which is one lag less than the regressand).
All random elements are defined with a suitable probability space, denoted by $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$. Throughout the paper, all limits are taken as $n \to \infty$, where $n$ is the sample size. The symbol $"\Rightarrow"$ is used to denote the weak convergence of the associated probability measures as $n \to \infty$. The symbol $\overset{d}{\to}$ denotes convergence in distribution and $\overset{\text{plim}}{\to}$ denotes convergence in probability, within the probability space (see, billingsley1968convergence). For further details on the martingale approximation results see hall1981martingale.
In this section, we examine the asymptotic theory of the OLS based estimator and the corresponding IVX based estimator for the threshold predictive regression as well as for the corresponding estimator of the threshold effect\footnote{Notice that the threshold effect can be identified within a range of values say, $-1/2 < \uptau < 1/2$. For $\uptau = 1/2$ then, the nuisance parameter is at the bound of this neighbourhood and thus there is weakly identification of the threshold variable.}. The following results provide the convergence rates for the consistent identification of the true threshold effect.
The proof of Lemma (ref) can be found in the Appendix. For the case of the stationary threshold model, consistency is proved by chan1993consistency. A stronger large sample result for the identification of the threshold effect is given by Lemma (ref) below.
and $\mathcal{L} := \underset{ r \in ( - \infty, + \infty ) }{ \text{arg max} } \ \Lambda(r)$ with $\Lambda(r)$ denotes a two-sided Brownian motion (see, the paper of khoshnevisan1996uniform for more details) such that
where $W_1(r)$ and $W_2(r)$ are two independent standard Brownian motions on $[0, \infty )$.
The novelty of our framework is that we consider the instrumental variable $Z_{tn}$ which is based on the IVX methodology proposed by phillipsmagdal2009econometric. The IVX instrument is constructed as below
The philosophy of the IVX instrumentation is that it induces a mildly integrated regressor which can control the unknown degree of persistence in the original regressor. In particular, in the case of hybrid stochastic local unit roots by allowing $\upgamma_z \neq 1$ which cover cases such as $\upgamma_z > 1$, $\upgamma_z \in (0,1)$ or $\upgamma_z < 1$, the IVX filter can achieve this property which is found to ensure weakly convergence to a mixed Gaussian distribution of the IVX estimator (see, kostakis2015Robust). It remains to verify that this asymptotic property still holds in our setting. Then, this will allow us to derive standard Brownian functionals for the asymptotic results of the tests of the next section even under the presence of threshold effects and nonlinear stochastic terms. The convenience of deriving an analytical known form for the corresponding limiting distributions of the tests is that we can easily obtain critical values and conduct statistical inference.
Next, we shall consider the IVX instrumentation within our modeling environment. Specifically, in the settings of SLUR IVX instrumentation has to be applied in a different manner than in the classical nonstationary autoregressive model, due the fact that the specification of the autocorrelation coefficient has a more complicated form. This is expressed in the following form
In practise the above instrumentation procedure is applied and we simplify it to the following terms
where the three terms above are defined as below
Another example, is to consider the case in which $p = d = 1$ (single SLUR regressor in the model). Then, it holds that
Notice that the above representation doesn't imply that the autocorrelation coefficient is estimated dynamically within a rolling window but that its value when the DGP is constructed is estimated at each time series observation of the full sample. Furthermore, an IVX instrumentation is also applied to the error term $\boldsymbol{u}_{\varphi t}$ such that
Then, the main result regarding the IVX estimator implies that a Mixed Gaussianity assumption holds in large samples such that
where $\mathbb{V}^{ivx} \in \mathbb{R}^{ p \times p }$.
In this Section, we focus on developing two tests: (i) testing for linearity and (ii) testing for Joint Nonlinearity and Predictability based on the threshold predictive regression model described above. Therefore, we can reformulate the threshold model as below
where $\alpha = \alpha_1$, $\beta = \beta_1$ and $\eta = \left( \alpha^{*}, \delta^{*} \right)^{\top}$ with $\alpha^{*} = \alpha_2 - \alpha_1$ and $\delta^{*} = \beta_1 - \beta_2$. Then, we can observe that the threshold effect diminishes when $\eta = 0$. Therefore, under the null hypothesis for a fixed $\gamma \in \Gamma = \left[ \gamma_1, \gamma_2 \right]$ the Wald statistic has the following form
where $P_x$ the projection matrix of $x_t$ and $I_n$ an $n \times n$ identity matrix.
We define the corresponding supremum functional as in caner2001threshold, pitarakis2008comment and gonzalo2012regime, gonzalo2017inferring such that
We aim to derive the asymptotic distribution of the test statistic given by (ref), under the null hypothesis $\mathbb{H}_0^{(1)} : \eta = 0$ and show that in the special case when $c > 0$ and $\phi = 0$, then the limit theory reduced to the asymptotic result proved by Proposition 1 of gonzalo2012regime. Due to the presence of nuisance parameter identified only under the alternative hypothesis, we follow davies1977hypothesis and hansen1996inference.
Therefore, in order to test for both non-linearity and predictability we employ the supremum functional with the unknown threshold variable being within the range of the values $\gamma_1$ and $\gamma_2$. Therefore, the estimation procedure involves scanning within this window and applying the maximizing to obtain the unknown threshold variable. When we obtain an estimate for the threshold variable, then the estimation procedure requires to substitute this estimate and then estimate the model parameters\footnote{A detailed description of the procedure can be found in the book of terasvirta2010modelling. Moreover, tong1980thresholdAR}.
Furthermore, related to the assumptions of the model as also argued in gonzalo2012regime the dependence assumptions such as for example how the correlation structure between the threshold variable and the innovations of the predictive regression can affect the asymptotic theory of the predictability tests. For instance, imposing the assumption that there is no correlation\footnote{In particular, assuming certain correlation structure between the threshold variable and the innovations of the predictive regression model implies that predictability is conditioned on whether in that certain period of time, there is a high correlation between these two random quantities. } between these two quantities, provides an equivalent assumption of having an exogenous determination of the threshold variable.
Notice that testing for $\beta_1 = \beta_2 = 0$ for a given $\lambda \in (0,1)$ then induces the Wald statistic with the estimated threshold parameter $\hat{\lambda}$ with a limiting distribution given by
regardless of whether $\alpha_1 = \alpha_2$ or $\alpha_1 \neq \alpha_2$.
In other words, the Wald statistic above is useful for conducting inferences regarding the presence of regime specific slopes in the predictive regression model without prior knowledge on whether the model intercepts are regime dependent or not. In other words, the limiting distribution of the sup-Wald statistic evaluated at the estimated threshold parameter, $\hat{\lambda}$, is the same regardless of whether $\alpha_1 = \alpha_2$ or $\alpha_1 \neq \alpha_2$. Moreover, due to the presence of the nuisance parameter of persistence as well as the presence of endogeneity in the system, such that, the allowed correlation between the Brownian motions $B_u$ and $B_v$, then the analytical expression of its limit process has its second component depending on $\sigma_{uv}$. On the other hand, the limiting distribution above simplifies further under the requirement that $\sigma_{un} = 0$ (exogeneity assumption), such that $W_n ( \hat{\lambda} ) \Rightarrow \chi^2(2)$, as $n \to \infty$.
Next, we focus on the null hypothesis that jointly tests the absence of linearity and no predictive power of the threshold predictive regression model which implies that $\mathbb{H}_0^{(2)}: \alpha_1 = \alpha_2 , \beta_1 = \beta_2 = 0$. Equivalently, based on the formulation given by expression (ref) the null hypothesis becomes $\mathbb{H}_0^{(2)}: \eta = 0 , \beta = 0$ and the limiting distribution\footnote{A related proof for the instrumental variable case is presented by lieberman2018iv.} of the corresponding sup Wald-OLS and sup Wald-IVX tests are given by Theorem (ref) below.
In this Section, we examine the finite sample performance of the proposed estimators and predictability tests via an extensive Monte Carlo simulation study.
We consider the following data generating process
where $q_t$ and $\xi_t = \left( u_t, v_t, \upepsilon_t \right)$ are independently normally distributed with mean zero and variance one and $\gamma_0 = 0.25$. Furthermore, we need to choose appropriate values for the unknown parameter $\varphi$ and the nuisance parameter of persistence. We consider that the coefficient of persistence takes values such that $c \in \left\{ 1, 2, 5, 10 \right\}$ and $\phi \in \left\{ 0, 0.05 , 0.25, 0.50 \right\}$. Each data generating process is replicated based on $B = 5,000$ and we consider sample sizes such as $n \in \left\{ 250, 500 \right\}$.
We obtain the threshold estimator based on both the OLS and the IVX estimation and compare the empirical size results. In summary we observe that the performance of the threshold estimator improves as the sample size increases. We also observe the diminishing threshold effect for the threshold estimator across different values of $(c , \phi)$ and $n$. However, when we observe the threshold estimators based on the OLS versus the IVX procedure in relation to the coefficient of persistence, we can clearly see that the IVX estimator produces smoother convergence rates and empirical sizes closer to the nominal size especially under the assumption of high persistence in the regressors.
\
Under the null hypothesis of no threshold effects, the model reduces to the standard predictive regression model, which implies that there is absence of nonlinearity. Thus, the empirical size is obtained with the use of the sup-Wald statistics by replicating the DGP and counting the frequency of rejecting the null hypothesis with respect to the replications. Similarly, we can obtain the empirical power of the tests under the alternative hypothesis of nonlinearity and predictability.
For the experimental design we consider $B = 1000$ and $n = \left\{ 250, 500 \right\}$ to simplify the computational time. Moreover, we consider a predictive regression model with three types of regressors, such that (i) mildly integrated regressors (where the degree of persistence implies that the LUR component is close to the unit boundary), (ii) mildly explosive regressors (where the degree of persistence implies that the LUR component is on the explosive side of the unit boundary) and (iii) near nonstationary regressors (where the degree of persistence implies that the LUR component is well below the unit boundary). Furthermore, to access the performance of the proposed test statistics with respect to the relative efficiency of the IVX to the OLS estimator, we only consider the case in which all the regressors are locally explosive.
In addition, to exclude other effects, we assume that the degree of endogeneity in the system remains the same across all variables and is given by the a fixed covariance matrix, which is unchanged throughout the simulation step. Then, under the null hypothesis of no threshold effect both test statistics are constructed under the assumption that the parameter vector across the two regimes remains the same, such that, $\beta_1 = \beta_2$ (e.g., in the case when we assume that the model includes no intercepts) and $\theta_1 = \theta_2$, where $\theta_j = ( \alpha_j, \beta_j )$ for $j \in \left\{ 1,2 \right\}$ when the model includes both a slope and an intercept. Furthermore, we use a pre-specified threshold cut-off point which also remains fixed through the replication steps. Another used specific parameter which is the coefficient of persistence for the instrumentation procedure, this is also kept fixed to avoid additional complexitiy when comparing the finite sample performance of the test statistics under the null hypothesis.
In this Section, we examine an empirical implementation of the proposed framework. Our goal is to uncover regime-specific predictability and threshold effects in financial markers, focusing on the pre-pandemic and post-pandemic sampling periods. More specifically, our aim is to assess whether the data support the presence of regime-specific predictability due to the socio-economic events resulted from the 2019 pandemic.
The stock return predictability is a major puzzle in financial economics. The literature goes back several decades; we briefly highlight important studies. The seminal paper of perron1989great discuss the unit root hypothesis around periods of financial turbulence. dejong1991temporal study the debate whether divided are trend-stationary or integrated processes while stein1991stock propose a framework for stock price distributions and stochastic volatility (see, also menzly2004understanding). Moreover, timmermann1993learning examines the excess volatility and predictability puzzle in stock prices. Further studies related to predictability testing include campbell2006efficient, kostakis2015Robust kasparis2015nonparametric, demetrescu2020testing. In this paper, we focus on the regime-specific\footnote{Notice that for instance formal econometric methodologies for testing for regime switching are proposed by cho2007testing, however in this paper our focus is to examine whether we find statistical significant evidence of regime-specific predictability rather than to test for regime-switching.} predictability hypothesis as proposed by the studies of gonzalo2012regime, gonzalo2017inferring.
We are also inspired by the work of hatchondo2016debt\footnote{The particular framework even though is driven from the perspective of sovereign debt it includes the main idea that an endogenous threshold variable such as the price at which long-term bonds would trade without current-period borrowing, drives the identification strategy of the model.} who introduce the idea of a threshold macroeconomic variable driving economic policy making as well as the paper of atanasov2020consumption who examine consumption fluctuations and predictability of expected returns. Therefore, we are motivated to study an additional aspect not previously examined in the predictability literature such as the effect of economic policy uncertainty as a potential threshold variable. baker2016measuring introduce a measure for economic policy uncertainty and examine the level of predictable policy responses\footnote{A different aspect but of possible interest is how systemic risk can affect stock return predictability. For instance, the aspects of uncertainty and systemic risk are examined by dicks2019uncertainty. Moreover, a discussion on the aspects of fluctuations in uncertainty is provided by bloom2014fluctuations.}.
In particular, economic policy uncertainty can be considered as a set of shocks which affects the predictability of returns in relation to the commonly used predictors in the literature. Therefore, we aim to use the indicator of economic policy uncertainty introduced by Baker et al. (2016) as an exogenous threshold variable to assess the existence of regime-specific predictability.
In this paper, we study a special special class of persistence which is the class of hybrid stochastic local unit roots, for the threshold predictive regression model, filling the gap in the literature of predictability tests. Our study extends the current methodologies of identifying regime specific predictability with persistence or unit root predictors. In particular, by incorporating such persistence shifts with the framework proposed by lieberman2020hybrid it allows us to reformulate the predictability tests proposed by gonzalo2012regime, gonzalo2017inferring and kostakis2015Robust generalizing this way the particular asymptotic results. We hope that our proposed framework will be helpful to practitioners who are interested in detecting predictability and threshold effects under the presence of potential hybrid stochastic local unit roots in equity indices and financial variables capturing economic conditions and market sentiment.
The asymptotic theory of this paper confirms that the estimation and inference of the threshold predictive regression model produces consistent parameter estimates. Moreover, the simulation experiment demonstrate good empirical size and power properties across different values of the unknown degree of persistence. The additional persistence properties we consider in this paper, allows to capture how the effect of shifting persistence as modelled via the hybrid stochastic local unit root affects the presence of the threshold effect. In terms of the IVX instrumentation procedure, the IVX instrument can filter out the abstract persistence occurred from the unknown coefficient of persistence as well as the component which comes from the additional term in the LUR specification. The additional component is considered as an extra source of innovation appeared in the system by a set of exogenous covariates. Thus, we can consider these being for example, the effect of economic shocks, such as the economic policy uncertainty to the persistence properties of regressors and the predictability in the model. Notice that a key assumption for the development of the asymptotic theory is the assumption of diminishing threshold effects. This allows us to obtain the limiting distribution of the threshold estimator as well as the asymptotic behaviour of the sup Wald statistic for testing the existence of the threshold effect under the null hypothesis.
Finally, some important extensions to the present work are worth mentioning. Firstly, time-varying predictability can be modelled in parallel to the hybrid stochastic local to unity specification. Secondly, one can consider extending our proposed threshold predictive regression model to allow for multiple regimes and thus the presence of multiple threshold effects (see, for example chiou2018nonparametric). This can increase significantly the complexity of the framework but nevertheless methods such as the ones proposed by gonzalo2002estimation and gonzalo2005subsampling could be utilized. Thirdly, considering the presence of an endogenous threshold variable or smooth transitions between regimes such as in luukkonen1988testing. Last but not least, examining the implementation of our proposed tests in forecasting and predictive accuracy environments such as in the studies of emilianouncovering and pitarakis2020novel is another possibility; all these being interesting applications for future research.