Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
60,904 characters · 8 sections · 90 citation commands
Bootstrap Adaptive Lasso Solution Path Unit Root Tests
\setcounter{footnote}{0}
In Arnold2024, we propose the $\tau$ and $\Breve{\tau}$ adaptive Lasso unit root tests, which exploit distinct stochastic orders of the activation knots of the lagged level regressor on the Lasso solution path in stationary and non-stationary autoregressions. Simulations reveal a decline in the tests’ precision when adjusting for deterministic components in the presence of higher-order serial correlation. As is well-documented for established unit root tests NgPerron2001, $\tau$ and $\Breve{\tau}$ also are vulnerable to large negative moving average (MA) coefficients, which cause significant upward size distortions. Time-varying variance in the error process can exacerbate such distortions. The latter invalidates the tests' homoskedastic limiting null distributions, further promoting spurious rejections.
A well-established strand of the literature on time series regressions addresses inference on model parameters when the generating process affects a test's distribution with nuisance terms. Several contributions to the unit root literature employ the bootstrap for first-order approximation of the null distribution to improve the finite-sample precision and ensure the asymptotic validity of a test. For autoregressive (AR) models, examples are recursive bootstraps in AR(1) models FerrettiRomo1996,Swensen2003, sieve bootstraps ChangPark2003,Park2003 and the residual-based block bootstrap PaparoditisPolitis2003. CavaliereTaylor2008,CavaliereTaylor2009heteroskedastic,CavaliereTaylor2009 devise wild bootstrap Liu1988 tests for asymptotically valid inference under conditionally heteroskedastic innovations as well as general forms of a time-varying unconditional variance. They propose analogues of the $M$ unit root tests suggested by Stock1999,NgPerron2001,PerronNg1996, the tests of Phillips1987 and PhillipsPerron1988 and the augmented Dickey-Fuller (ADF) tests of SaidDickey1984,Elliottetal1996.
Bootstraps have also been extensively applied in the recent literature on inference for penalised estimators. Chatterjee2011 utilise a residual-based bootstrap to reliably estimate both the distribution and bias of adaptive Lasso estimators in a cross-sectional setting. Similarly, Audrino2017 use a residual-based block bootstrap for heteroskedasticity-robust inference in adaptively $\ell_1$-penalised time series regressions. Chernozhukov2013 sparked active research by applying the multiplier (wild) bootstrap to penalised estimation in a high-dimensional heteroskedastic setup. Notably, they resample factors that shape the asymptotic distribution, excluding terms that drive asymptotically negligible nuisance parameters by not recalculating the estimator CandesTao2007 during each bootstrap cycle. Hansen2018 point out that this procedure is computationally efficient to obtain asymptotically valid inference or confidence intervals at the cost of failing to \enquote{capture any finite sample uncertainty introduced in the lasso selection}. Contrary to this minimalistic approach, Dezeure2017 estimate the distribution of a de-sparsified Lasso estimator for high-dimensional regression by computing the Lasso solutions in every iteration of the wild bootstrap. They assess the computational burden as feasible due to the advances in parallel computing.
In this paper, we address the tests' reliability issues outlined above and propose bootstrap analogues to $\tau$ and $\Breve{\tau}$, using a wild bootstrap scheme for resampling Lasso solution paths in ADF regressions. The method exploits the computational efficiency of the LARS algorithm to compute the Lasso solution path in every bootstrap iteration, similar to the method of Dezeure2017. Our wild bootstrap algorithm follows Cavaliereetal2015 and is grounded on the distribution theory, in particular the bootstrap invariance principle, for unconditional heteroskedastic autoregressions of CavaliereTaylor2008,CavaliereTaylor2009. We provide numerical evidence that the wild bootstrap attenuates finite sample size distortions from challenging AR and MA error processes, especially under detrending. Simulations further indicate that---contrary to the unadjusted tests---the wild bootstrap permits valid inference when there is non-stationary volatility in the innovations, mirroring the results of CavaliereTaylor2008,CavaliereTaylor2009heteroskedastic for bootstrap variants of established unit root tests.
The remainder of this article is structured as follows. In (ref), we discuss the theoretical setup and recap the activation knot unit root tests of Arnold2024. (ref) discusses the wild bootstrap algorithm and its implementation. Monte Carlo studies in (ref) investigate finite sample properties of the bootstrap tests. We illustrate applying the methods using real residential property prices for selected OECD countries in (ref). (ref) concludes and motivates avenues for further research.
We will use the following conventions throughout the manuscript. $\mathbb{I}(\cdot)$ is the indicator function and $\lfloor\cdot\rfloor$ denotes the integer part of its argument. We define $\lVert\bm x\rVert_q$ as the $\ell_q$ norm of a vector $\bm x$. Coefficients of the true ADF model have a superscript $\star$. A superscript $*$ signifies bootstrap quantities conditional on the observed data. $\xrightarrow[]{d^*}_p$ means bootstrap weak convergence in probability. Convergence in probability and distribution are denoted by $\xrightarrow{p}$ and $\xrightarrow{d}$, respectively.
We consider time series $y_t$ generated by an AR data-generating process (DGP)
where $\bm z_t := (1,t,\dots,t^m)'$ is an $m^\text{th}$ order deterministic component with coefficient vector $\bm \theta$ and $x_t$ is an AR(1) processes with errors $u_t$. The stochastic component $x_t$ satisfies $\varrho\in(-1, 1]$, i.e., $y_t$ is stationary when $\lvert\varrho\rvert<1$ and has a unit root when $\varrho=1$. We make the following assumptions on the error term $u_t$, cf. Assumption $\mathcal{A}$ in CavaliereTaylor2008.
(ref) comprises a set of standard conditions ChangPark2002, ensuring that $u_t$ is a weakly-stationary and invertible moving average (MA) process in the $\varepsilon_t$ with finite fourth-order moments. Together with (ref), which requires the error process $u_t$ to be unconditionally homoskedastic, (ref) imposes the same conditions on $u_t$ as we do in Assumption 1 of Arnold2024. We note that the MDS condition in part (ref) allows for conditionally heteroskedastic $\varepsilon_t$, e.g., weakly stationary generalised autoregressive conditionally heteroskedastic (GARCH) and Markov-switching processes, but excludes unconditionally heteroskedastic error processes. The latter is allowed for by the following generalisation of the volatility function in (ref).
(ref) is a relaxation of (ref) and allows $\sigma_t$ to exhibit non-stochastic time-varying volatility of a quite general form, including (a countable number of) abrupt jumps, polynomial trends and smooth transitions in the unconditional variance. It also generalises the setup of Arnold2024 towards a broader class of empirical processes relevant to macroeconomic applications. We refer to Remarks 1 and 2 in CavaliereTaylor2008 and Section 2 in CavaliereTaylor2009 for a discussion of heteroskedastic error processes covered by allowing for MDS innovations $\epsilon_t$ as in part (ref) of (ref) and deterministic non-stationary volatility implied by (ref).
Next, we will review the adaptive Lasso activation knot tests of Arnold2024.
Based on the ADF($\infty$) representation of (ref) if $\bm z_t'\bm \theta = 0$,
with $ \rho^\star\in(-2,0]$, $\sum_{j=1}^\infty \delta_j^\star < 1$, we consider the approximating ADF($p$) regression model
The AR lag order $p$ in the model (ref) meets the following assumption.
In Arnold2024, we investigate testing for a unit root using the adaptive Lasso solution path to (ref). Consider the adaptively penalised loss function
with adaptive weights $w_1 := 1/\lvert\widehat{\rho}\rvert$, $w_{2,\,j} := 1/\left\lvert\widehat{\delta}_j\right\rvert$ determined by the initial OLS estimates $\widehat{\rho}$ and $\widehat{\delta}_j$ in (ref), and adjustment parameters $\gamma_1,\gamma_2\in\mathbb{R}^+$. The adaptive Lasso solution path to (ref),
is a collection of solutions $\widehat{\bm \beta}_\lambda := (\widehat{\rho}_\lambda, \widehat\bm \delta_\lambda')'$ subject to the $\ell_1$ penalty parameter $\lambda$. A solution path $\mathcal{L}$ is characterised by knots at which variables are activated or deactivated. Our testing principle leverages that a unit root causes the activation knots of $y_{t-1}$ to be of different stochastic order than under stationarity. We propose a right-sided test of $H_0:\rho^\star=0$ against $H_1:\rho^\star\in(-2,0)$ via the statistic $\tau_{\gamma_1} := T^{\gamma_1-1} \lambda_{0,\,\rho^\star}$ with $\lambda_{0,\,\rho^\star}$, the first activation knot of $y_{t-1}$, satisfying
The knot $\lambda_{0,\,\rho^\star}$ is a standard output of algorithms for calculating (ref) such as LARS Efronetal2004, making $\tau_{\gamma_1}$ straightforward to compute.\footnote{We refer to Section 3.1 in Arnold2024 for the formal definition of an activation knot and a discussion of the stochastic properties of $\lambda_{0,\,\rho^\star}$ in particular.}
Given consistency of the estimator $\widehat{\bm \beta}_\lambda$ at $\lambda = \lambda_{0,\,\rho^\star\sim c/T}$, the activation knot of $y_{t-1}$ under local-to-unity roots with non-centrality parameter $c\in(-\infty,0]$, the (local) distribution of $\tau_{\gamma_1}$ can be identified. Specifically, for $\varrho^\star=1+c/T$ and $\gamma_1>1/2,\ \gamma_2>0$, Theorem 1 in Arnold2024 states that for unconditionally homoskedastic $u_t$ satisfying (ref),
where $W_c(r) := \int_0^r \mathrm{exp}[c(r-s)]\mathrm{d}W(s)$ is an Ornstein-Uhlenbeck process driven by the standard Wiener process $W(s)$ on $s\in[0,1]$.
A modification of $\tau_{\gamma_1}$ proposed in Arnold2024 derives from enhancing the penalty weight for $y_{t-1}$ with additional information on whether $\rho^\star=0$. This information enrichment of $w_1$ proceeds as $\Breve{w}_1 := w_1 \cdot J_\alpha$, using the statistic $J_\alpha$ HerwartzSiedenburg2010 which exploits different stochastic orders of the OLS estimator in time series regressions when the degree of integration differs. Analogous to $\tau_{\gamma_1}$, the modified statistic is calculated as $\Breve{\tau}_{\gamma_1} := T^{\gamma_1-1}\Breve{\lambda}_{0,\,\rho^\star}$, where $\Breve{\lambda}_{0,\,\rho^\star}$ denotes the first activation knot of $y_{t-1}$ on a solution path $\Breve{\mathcal{L}}$ based on $\Breve{w}_1$.\footnote{See Algorithm 1 in Arnold2024 for details on the computation of $J_\alpha$.} By Corollary 1 of Arnold2024,
where $J_{\alpha,\,c}$ is the $c$-dependent limit of $J_\alpha$.
For implementation we propose the natural choice $\gamma_1=\gamma_2=1$ which avoids an adjustment for $\phi(1)\neq1$ and yields tests with limit distributions
where $\widehat\sigma^2$ estimates the error variance based on OLS residuals from the penalty weights ADF regression (ref).
To accommodate for $\bm z_t\neq\bm 0$, standard detrending ideas can be applied before computing the Lasso solution. In the remainder, we follow Arnold2024 and consider first-difference (FD) detrending SchmidtPhillips1992 before computing the Lasso solutions and calculating $J_{\alpha}$ on OLS-adjusted data. Under detrending, the $W_c(r)$ in (ref) are replaced by the corresponding projection, resulting in limit distributions deviating from the case without adjustment for deterministic components. The $\Breve{\tau}$ limit is further affected by the distribution of $J_{\alpha,\,c}$, which is OLS-adjusted. Critical values for $\tau$ and $\breve{\tau}$ under FD adjustment for deterministic components are reported in Table D1 in Arnold2024.
In the next section, we summarise the implications of (ref) and discuss a wild bootstrap correction that addresses the ramifications for the activation knot tests.
While Arnold2024 find that $\tau$ and $\Breve{\tau}$ have mostly good size, simulation studies indicate some downward or upward size distortions in small samples for AR error processes with high autocorrelation or processes with MA roots close to $-1$. These distortions are somewhat more pronounced under detrending.
As we demonstrate in (ref), heteroskedastic error processes may amplify these undesirable finite sample properties. Unconditional heteroskedasticity invalidates inference using the homoskedastic null distributions in (ref). CavaliereTaylor2007 show that non-stationary volatility as permitted under (ref) alters the limit distributions of common (unpenalised) regression-based unit root tests in (ref) as non-stationary volatility introduces nuisance parameters to the tests' homoskedastic distributions in (ref) that do not vanish asymptotically, invalidating the critical values based on the limits for $c=0$. For DGP (ref) in the local-to-unity case $\varrho^\star = 1+c/T$ with $-\infty<c\leq0$, this can be traced back to the invariance principle
depending on the volatility function $\omega(\cdot)$, cf. the discussion of Theorem 1 in CavaliereTaylor2007. Here, $\overline{\omega}^2 := \int_0^1 \omega(s)^2\mathrm{d}s$ is the limit of $T^{-1}\sum_{t=1}^T\sigma_t^2$ and
is a diffusion process driven by a time-transformed Wiener process $W_{\eta}(\cdot) := W(\eta(\cdot))$ with directing process $\eta(\cdot)$. A key quantity is the functional
the so-called variance profile of the series, cf. Section 3 in CavaliereTaylor2007. A non-constant unconditional volatility function $\omega(\cdot)$ thus alters the tests' (asymptotic) null distributions and local power functions, invalidating the local asymptotic results for $\tau$ and $\Breve{\tau}$ in (ref). Under constant volatility, $\overline{\omega}^2=\sigma^2\in(0,\infty)$. Therefore, $\eta(s)=s,\, s\in[0,1]$ so that $W_{\eta}(\cdot) = W(\cdot)$ and the limiting r.v. in (ref) reduces to the same (scaled) Ornstein-Uhlenbeck process underlying the limiting functionals in (ref).
The wild bootstrap proposed in CavaliereTaylor2008 samples bootstrap innovations as $\varepsilon_t^* := \xi_t \cdot \check{\varepsilon}_{p,\,t}^{d}$ with $\check{\varepsilon}_{p,\,t}^{d}$ the OLS residuals from the ADF regression (ref) based on the detrended data and the $\xi_t$ are i.i.d. with $\operatorname{E}(\xi_t) = 0$ and $\operatorname{Var}(\xi_t) = 1$. The resampled data $y_t^*$ generated by the partial sum process
initialised at $y_0^*=0$, then satisfy the bootstrap invariance principle
cf. Eq. (4) in the proof of Theorem 2 of CavaliereTaylor2008.
Since the $\xi_t$ are i.i.d., the device $\varepsilon^*_t = \xi_t \cdot\check{\varepsilon}_{q,\,t}$ anihilates any serial correlation from the original shocks. While this does not impact the asymptotic validity of the bootstrap tests, as follows from the exposition in CavaliereTaylor2008,CavaliereTaylor2009, neglecting correlation in the original shocks may reduce the finite-$T$ precision of the bootstrap tests. We thus follow CavaliereTaylor2009,SmeekesTaylor2012 and build higher-order stationary dynamics estimated from the data into the bootstrap errors,
and construct the bootstrap sample as
The $\check\delta_{q,\,j}$, $j=1,\dots,q$ are estimated coefficients of the $\Delta y_{t-j}$ in a sieve ADF regression (ref) with lag order $q$, similarly as in the residual-based bootstrap schemes of FerrettiRomo1996, ChangPark2003, Park2003. Setting $q = p$, the recolouring recursion (ref) ensures that\footnote{Other choices for $q$ are discussed in (ref). A proof of (ref) under Assumptions equivalent to ours is given in the proof of Theorem 2 of SmeekesTaylor2012.}
(ref) states that the bootstrap correctly replicates effects from stationary serial correlation, cf. (ref), and thus may better reproduce nuisance terms from adjusting for $\phi(1)\neq 1$ in the finite-$T$ null distributions of $\tau$ and $\Breve{\tau}$. The invariance principles (ref) and (ref), and continuous mapping arguments are the theoretical foundation for applying the (sieve) wild bootstrap to the activation knot tests $\tau$ and $\Breve{\tau}$ for improving finite sample precision and valid asymptotic inference in the present heteroskedastic framework.
We next present the wild bootstrap algorithm, which adapts the wild bootstrap algorithms for the popular ADF and $M$ tests proposed by CavaliereTaylor2008,CavaliereTaylor2009,Cavaliereetal2015 to our activation knot tests.
To investigate sample properties of the tests, we generate time series as
with starting value $y_0=0$ for sample sizes $T\in\{75,100,150,250,500,1000\}$. We let $\varrho = 1+c/T$ so that setting $c=0$ obtains data under the unit root null. As (local) stationary alternatives, we set $c=-7$ when testing based on data adjusted for a constant and $c=-13.5$ for detrending. Pure AR or MA errors are generated with the recursion
with coefficients $\varphi,\vartheta\in\{-.8, -.4,\allowbreak 0, .4, .8\}$.
Following CavaliereTaylor2009heteroskedastic, we model deterministic smooth transitions in the unconditional volatility parameter $\sigma_{T,\,t}$ between two regimes with variances $s_1^2,s_2^2>0$ via a logistic function $\mathbb{S}_{T,\,t}$,
with transition midpoint $\lfloor \kappa T\rfloor$, $\kappa\in(0,1)$. The parameter $\tilde{\gamma}_T$ determines the transition speed between $s_1^2$ and $s_2^2$, yielding and abrupt regime switch at $t=\lfloor \kappa T\rfloor$ in the sense that $\mathbb{S}_{T,\,t}\to\mathbb{I}(t\geq \lfloor\kappa T\rfloor)$ as $\tilde{\gamma}_T\to\infty$. We adapt the local drift $\tilde{\gamma}_T := 25/T$ from CavaliereTaylor2009heteroskedastic. Setting $s_1^2 = 1$ throughout, we model negative shifts with early transition midpoints ($\kappa = .2$, $s_2^2 = .25$) as well as positive shifts with late midpoints ($\kappa = .8$, $s_2^2 = 4$).
The bootstrap variants of $\tau$ and $\Breve{\tau}$, denoted $\tau^*$ and $\Breve{\tau}^*$, are implemented as detailed in (ref) at the 5% level and computed with $B=499$ bootstrap replications. The $\xi_t$ in Step 3 of (ref) are standard Gaussian r.v.s, and the lag orders $p,q$ and $p^*$ are estimated using the RSMAIC, if not indicated otherwise. We compute the Lasso solution paths using the implementation of the LARS algorithm Efronetal2004 in the R (R) package lars pkg-lars.
We first examine the ability of the wild bootstrap to approximate the tests' finite sample distributions and assess the bootstrap tests' precision and local power in the baseline scenario with uncorrelated homoskedastic errors ($s_2^2=1$). The sieve regression lag order $q$ is zero in this experiment. The results presented in (ref) indicate that $\tau^*$ and $\Breve{\tau}^*$ have excellent precision (top panel), improving on the empirical size of $\tau$ and $\Breve{\tau}$, which may be somewhat conservative for small $T$. Furthermore, the bootstrap tests seem to replicate the (size-adjusted) local power function of the standard tests quite well, with only minor deviations for small $T$ notable for $\Breve{\tau}$ (bottom panel).
In a second experiment, we examine the precision of the sieve bootstrap tests under homoskedastic correlated errors. (ref) presents the results. The bootstrap tests perform well for AR errors with $\varphi\in(-.8, .8)$ and under MA errors with $\vartheta=.8$. The most challenging scenario is MA errors with $\vartheta = -.8$, which lead to upward size distortions across $T$ that are prohibitive for small samples and only decay slowly with the sample size. Although $\tau^*$ and $\Breve{\tau}^*$ are oversized, resampling with recolouring yields significant improvements over $\tau$ and $\Breve{\tau}$. Recolouring is particularly helpful under detrending, where the empirical size of the bootstrap tests converges quickly towards the $5\%$ level, undercutting the rejection rates of the unadjusted tests by over a third in larger samples. Furthermore, comparing the size of the SWB tests with the size of the WB tests with $q=0$ reported in (ref) of (ref) corroborates better precision through the sieve step under MA errors in particular.
Next, we investigate the impact of unconditional heteroscedasticity and its interplay with the effects of autocorrelated errors on the precision of the tests. Our focus is MA processes for which $\tau$ and $\Breve{\tau}$ show the largest size distortions, and bootstrap inference with better precision is the most needed. (ref) presents the size of the tests for MA errors with non-stationary volatility. Consistent with the theory, we see (upward) size distortions for $\tau$ and $\Breve{\tau}$ under smoothly trending variances across all scenarios. These distortions vary in magnitude with the correlation structure, the type of variance shift and the adjustment for $\bm z_t$. The smallest extent occurs for an early variance reduction and MA coefficient $\vartheta = .8$, where both tests mostly remain below 10%. There are differences between $\tau$ and $\Breve{\tau}$, e.g., for a late variance increase with $\vartheta = .8$, which is likely due to idiosyncracies of the $J_\alpha$ statistic under heteroskedasticity. As before, we observe the most pronounced distortions at $\vartheta=-.8$ for small $T$, which are worse than under homoscedasticity, cf. (ref). The bootstrap analogues consistently perform better, having size close to the 5% level for positive MA coefficients for upward and downward variance shifts and independently of the adjustment for a deterministic component. For $\vartheta=-.8$, $\tau^*$ and $\Breve{\tau}^*$ tend to be somewhat more oversized than under homoscedasticity and also require large $T$ for rejection rates to ameliorate. As in the homoskedastic settings, comparing with additional simulation results for the WB tests with $q=0$ in (ref) indicates better precision of the SWB tests. (ref) report qualitatively similar results for heteroskedastic AR errors.
Local power estimates for the SWB tests under heteroskedastic AR and MA processes are provided in (ref). With a few exceptions for error processes with negative coefficients and small $T$, we find that the bootstrap tests effectively approximate the local power functions of the infeasible size-adjusted tests $\tau$ and $\Breve{\tau}$. Furthermore, the simulation results show that information enrichment yields power gains even under non-stationary volatility and that the bootstrap retains these gains.
The housing price rally sparked by a sustained low-interest environment after the 2008 financial crisis has received much public and scientific attention Mianetal2015,Jordaetal2016. While monetary policies fueled price dynamics during the COVID-19 pandemic FranckeKorevaar2021, the recent trend reversal due to surges in mortgage and construction interest rates highlights the importance of understanding the dynamics in the market relevant to macroeconomic policy.
We use the tests to assess real residential property price inflation (RRPPI) rates of the six largest Eurozone economies for stochastic trends, covering 207 quarterly observations from Q1-1972 to Q2-2023 for Belgium, France, Germany, Italy, Spain and the Netherlands. The data RRPPdata obtained from CPI deflated price indices summarise all types of new and existing dwellings, except for Germany, where the underlying index considers owner-occupied houses only.
Besides the abovementioned events, the sample period covers further economic turmoil, such as OPEC oil shortages (1973, 1979) and recessions in the 1980s and 1990s, reflecting different variance regimes. We find evidence of such regimes in data displayed in the left panel of (ref), showcasing a high degree of co-movement. The estimated volatility profiles in the right panel further indicate the instability of the unconditional variances, showing considerable downward shifts for Belgium (BE), Italy (IT), Spain (ES), and the Netherlands (NL).
We consider ADF regressions with a non-zero mean,
where the $k_i$ in the baseline regression, as well as lag orders in (ref), are selected by RSMAIC. We compute the bootstrap tests using $B=4999$ bootstrap iterations.
(ref) presents the outcomes of the tests for three different periods: the entire data range (Q1-1972--Q2-2023) and subsamples before (Q1-1972--Q4-1998) and for the Euro-era (Q1-1999--Q2-2023) which is characterised by macroeconomic convergence, e.g., due to a mutual primary refinancing interest rate applying to the member economies.
Outcomes for the bootstrap tests mostly agree with the standard inference, giving mixed conclusions. For the entire period, the tests indicate mean reversion behaviour for all economies except Germany, where only $\breve{\tau}$ rejects at 5%. The pre-Euro subsample shows similar outcomes. Notably, $\breve{\tau}^*$ tends to have smaller p-values than $\tau^*$, likely due to power gains from information enrichment—a feature of $\breve{\tau}$ that the simulation outcomes indicate is preserved by the bootstrap. None of the tests rejects the null hypothesis of a stochastic trend for the Euro-era at $5\%$. The reduced sample size or possible trend changes in the generating process could contribute to the mixed evidence, especially since periods of exuberance in the housing market characterise the Euro-era subsample.
In this paper, we assessed the reliability of the adaptive Lasso solution path-based unit root tests $\tau$ and $\Breve{\tau}$ proposed in Arnold2024 under weaker assumptions and considered whether resampling offers robust alternatives. Drawing on the theoretical results in CavaliereTaylor2008,CavaliereTaylor2009, we propose the wild bootstrap analogues $\tau^*$ and $\Breve{\tau}^*$, implementing the computation of resampled Lasso solution paths efficiently using the LARS algorithm Efronetal2004, pkg-lars. Numerical evidence shows that the bootstrap yields tests with higher precision, allowing more robust inference under correlated error processes of a general form.
Consistent with the theory on heteroskedastic autoregressions in CavaliereTaylor2007, $\tau$ and $\Breve{\tau}$ do not attain their homoscedastic limits derived in Arnold2024 when the errors are unconditionally heteroscedastic, so that valid inference is not guaranteed, even asymptotically. Our simulations confirm that $\tau$ and $\Breve{\tau}$---like conventional unit root tests---have null distributions and local power functions affected by nuisance parameters for unconditionally heteroskedastic innovations. A consequence is size distortions, with the strength of the effect depending on the adjustment for deterministic components. Correlated errors, such as MA(1) processes with negative coefficients, exacerbate this effect. Our sieve wild bootstrap tests display higher precision than $\tau$ and $\Breve{\tau}$ in these scenarios, indicating their ability to accurately recover the first-order null distributions. In addition, our simulation results show the wild bootstrap variants to approximate the finite-sample (local) power functions of the infeasible size-adjusted implementations of the standard tests under variance shifts.
To illustrate the bootstrap tests, we consider real residential property price inflation rates for selected Eurozone economies. This data set seems representative of our setup, given signs of persistence and non-stationary volatility. Both bootstrap tests yield the same conclusions, pointing to stationarity for the entire period from 1972 to 2023 and the period before the introduction of the Euro. We find no evidence for mean-reversion for the Euro-era subsample.
There are various avenues for further research. Our simulation evidence suggests that the wild bootstrap preserves the benefits of information enrichment, motivating its application to inference for penalised regression when the (asymptotic) distribution is unknown. Given the positive results for $\tau$ and $\Breve{\tau}$, expanding the underlying testing principle to other (adaptively) penalised regression estimators, for example, the fused Lasso or the group Lasso seems worthwhile. Good starting points are Qian2016 and Schweikert2021, which employ adaptive variants of the group Lasso and fused Lasso to detect structural breaks in panel and cointegrating regressions.
Our empirical application raises the question of whether the advent of the Euro affected the heterogeneity in the housing price dynamics across the Eurozone countries. To address this question, contemplating additional measures of heterogeneity for information enrichment can improve the discriminatory power of the aforementioned penalised estimators. This topic is closely related to the heterogeneous treatment effect inference literature, which could inspire further extensions.
Another promising direction is inference in high-dimensional regressions for which the double Lasso of Bellonietal2014 has become a standard method. It would be appealing to investigate whether information enrichment furthermore improves high-dimensional inference for which the wild bootstrap has become a cornerstone Chernozhukov2023. To this end, one could apply our testing principle to causal inference problems or multivariate time series, potentially using adaptive penalty weights derived using the de-sparsified Lasso. We are currently investigating this approach.