Heteroskedastic Innovations Research Articles

Determination of the Number of Common Stochastic Trends under Conditional Heteroskedasticity

Permanent-transitory decompositions and the analysis of the time series properties of economic variables at the business cycle frequencies strongly rely on the correct detection of the number of common stochastic trends (co-integration). Standard techniques for the determination of the number of common trends, such as the well-known sequential procedure proposed in Johansen (1996), are based on the assumption that shocks are homoskedastic. This contrasts with empirical evidence which documents that many of the key macro-economic and financial variables are driven by heteroskedastic shocks. In a recent paper, Cavaliere et al., (2010, Econometric Theory) demonstrate that Johansen's (LR) trace statistic for co-integration rank and both its i.i.d. and wild bootstrap analogues are asymptotically valid in non-stationary systems driven by heteroskedastic (martingale difference) innovations, but that the wild bootstrap performs substantially better than the other two tests in finite samples. In this paper we analyse the behaviour of sequential procedures to determine the number of common stochastic trends present based on these tests. Numerical evidence suggests that the procedure based on the wild bootstrap tests performs best in small samples under a variety of heteroskedastic innovation processes.

Predictive quantile regressions under persistence and conditional heteroskedasticity

This paper provides an improved inference for predictive quantile regressions with persistent predictors and conditionally heteroskedastic errors. The confidence intervals based on conventional quantile regression techniques are not valid when predictors are highly persistent. Moreover, the conditional heteroskedasticity introduces rather complicated nuisance parameters in the limit theory, whose estimation errors can be another source of distortion. We propose a size-corrected bootstrap inference thereby avoiding the nuisance parameter estimation. The bootstrap consistency is shown even with the nonstationary predictors and conditionally heteroskedastic innovations. Monte Carlo simulation confirms the significantly better test size performances of the new methods. The empirical exercises on stock return quantile predictability are revisited.

On the Evolution of U.S. Temperature Dynamics

Climate change is a multidimensional shift. While much research has documented rising mean temperature levels, we also examine range-based measures of daily temperature volatility. Specifically, using data for select U.S. cities over the past half-century, we compare the evolving time series dynamics of the average temperature level, AVG, and the diurnal temperature range, DTR (the difference between the daily maximum and minimum temperatures at a given location). We characterize trend and seasonality in these two series using linear models with time-varying coefficients. These straightforward yet flexible approximations provide evidence of evolving DTR seasonality, stable AVG seasonality, and conditionally Gaussian but heteroskedastic innovations for both DTR and AVG.

Mildly Explosive Autoregression Under Stationary Conditional Heteroskedasticity

A limit theory is developed for mildly explosive autoregressions under stationary (weakly or strongly dependent) conditionally heteroskedastic errors. The conditional variance process is allowed to be stationary, integrable and mixingale, thus encompassing general classes of generalized autoregressive conditional heteroskedasticity‐type or stochastic volatility models. No mixing conditions or moments of higher order than 2 are assumed for the innovation process. As in Magdalinos (), we find that the asymptotic behaviour of the sample moments is affected by the memory of the innovation process both in the form of the limiting distribution and, in the case of long range dependence, the rate of convergence, while conditional heteroskedasticity affects only the asymptotic variance. These effects are cancelled out in least squares regression theory, and thereby, the Cauchy limit theory of Phillips and Magdalinos () remains invariant to a wide class of stationary conditionally heteroskedastic innovations processes.

Mildly Explosive Autoregression Under Stationary Conditional Heteroskedasticity

A limit theory is developed for mildly explosive autoregressions under stationary (weakly or strongly dependent) conditionally heteroskedastic errors. The conditional variance process is allowed to be stationary, integrable and mixingale, thus encompassing general classes of generalized autoregressive conditional heteroskedasticity‐type or stochastic volatility models. No mixing conditions or moments of higher order than 2 are assumed for the innovation process. As in Magdalinos ([Magdalinos T, 2012]), we find that the asymptotic behaviour of the sample moments is affected by the memory of the innovation process both in the form of the limiting distribution and, in the case of long range dependence, the rate of convergence, while conditional heteroskedasticity affects only the asymptotic variance. These effects are cancelled out in least squares regression theory, and thereby, the Cauchy limit theory of Phillips and Magdalinos ([Phillips PCB, 2007a]) remains invariant to a wide class of stationary conditionally heteroskedastic innovations processes.

Limit theory for explosive autoregression under conditional heteroskedasticity

This paper studies an explosive autoregression with conditionally heteroskedastic innovations. The asymptotic distributions of LS, GLS, t-statistics, heteroskedasticity-consistent t-statistics and GLS t-statistics are derived for nonstationary local-to-unity and mildly explosive roots, in which ρn satisfies n1−ρn→−∞,0. Combined with the existing results, we can cover the range of ρn such that n1−ρn→−∞,∞. Some results on pure explosive cases are also discussed.

Limit Theory for Explosive Autoregression Under Conditional Heteroskedasticity

This paper studies an explosive autoregression with conditionally heteroskedastic innovations. The asymptotic distributions of LS, GLS, t-statitiscs, heteroskedasticity-consistent t-statistics and GLS t-statistics are derived for nonstationary local-to-unity and mildly explosive roots, in which ρ_{n} satisfies n(1-ρ_{n})→[-∞,0). Combined with the existing results, we can cover the range of ρ_{n} such that n(1-ρ_{n})→[-∞,∞]. Some results on pure explosive cases are also discussed.

Predictive Quantile Regressions Under Persistence and Conditional Heteroskedasticity

This paper provides an improved inference for predictive quantile regressions with persistent predictors and conditionally heteroskedastic errors. The confidence intervals based on conventional quantile regression techniques are not valid when predictors are highly persistent. Moreover, the conditional heteroskedasticity introduces rather complicated nuisance parameters in the limit theory, whose estimation errors can be another source of distortion. We propose a size-corrected bootstrap inference thereby avoiding the nuisance parameter estimation. The bootstrap consistency is shown even with the nonstationary predictors and conditionally heteroskedastic innovations. Monte Carlo simulation confirms the significantly better test size performances of the new methods. The empirical exercises on stock return quantile predictability are revisited.

A Comparison of Sequential and Information‐based Methods for Determining the Co‐integration Rank in Heteroskedastic VAR Models

AbstractIn this article, we investigate the behaviour of a number of methods for estimating the co‐integration rank in VAR systems characterized by heteroskedastic innovation processes. In particular, we compare the efficacy of the most widely used information criteria, such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) , with the commonly used sequential approach of Johansen [Likelihood‐based Inference in Cointegrated Vector Autoregressive Models(1996)] based around the use of either asymptotic or wild bootstrap‐based likelihood ratio type tests. Complementing recent work done for the latter in Cavaliere, Rahbek and Taylor [Econometric Reviews(2014) forthcoming], we establish the asymptotic properties of the procedures based on information criteria in the presence of heteroskedasticity (conditional or unconditional) of a quite general and unknown form. The relative finite‐sample properties of the different methods are investigated by means of a Monte Carlo simulation study. For the simulation DGPs considered in the analysis, we find that the BIC‐based procedure and the bootstrap sequential test procedure deliver the best overall performance in terms of their frequency of selecting the correct co‐integration rank across different values of the co‐integration rank, sample size, stationary dynamics and models of heteroskedasticity. Of these, the wild bootstrap procedure is perhaps the more reliable overall as it avoids a significant tendency seen in the BIC‐based method to over‐estimate the co‐integration rank in relatively small sample sizes.

A Comparison of Sequential and Information-based Methods for Determining the Co-integration Rank in Heteroskedastic VAR Models

In this paper we investigate the behaviour of a number of methods for estimating the co-integration rank in VAR systems characterized by heteroskedastic innovation processes. In particular we compare the efficacy of the most widely used information criteria, such as AIC and BIC, with the commonly used sequential approach of Johansen (1996) based around the use of either asymptotic or wild bootstrap-based likelihood ratio type tests. Complementing recent work done for the latter in Cavaliere, Rahbek and Taylor (2013, Econometric Reviews, forthcoming), we establish the asymptotic properties of the procedures based on information criteria in the presence of heteroskedasticity (conditional or unconditional) of a quite general and unknown form. The relative finite-sample properties of the different methods are investigated by means of a Monte Carlo simulation study. For the simulation DGPs considered in the analysis, we find that the BIC-based procedure and the bootstrap sequential test procedure deliver the best overall performance in terms of their frequency of selecting the correct co-integration rank across different values of the co-integration rank, sample size, stationary dynamics and models of heteroskedasticity. Of these the wild bootstrap procedure is perhaps the more reliable overall since it avoids a significant tendency seen in the BIC-based method to over-estimate the co-integration rank in relatively small sample sizes.

Bootstrap Determination of the Co-Integration Rank in Heteroskedastic VAR Models

In a recent paper Cavaliere et al. (2012) develop bootstrap implementations of the (pseudo-) likelihood ratio (PLR) co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying vector autoregressive (VAR) model which obtain under the reduced rank null hypothesis. They propose methods based on an independent and individual distributed (i.i.d.) bootstrap resampling scheme and establish the validity of their proposed bootstrap procedures in the context of a co-integrated VAR model with i.i.d. innovations. In this paper we investigate the properties of their bootstrap procedures, together with analogous procedures based on a wild bootstrap resampling scheme, when time-varying behavior is present in either the conditional or unconditional variance of the innovations. We show that the bootstrap PLR tests are asymptotically correctly sized and, moreover, that the probability that the associated bootstrap sequential procedures select a rank smaller than the true rank converges to zero. This result is shown to hold for both the i.i.d. and wild bootstrap variants under conditional heteroskedasticity but only for the latter under unconditional heteroskedasticity. Monte Carlo evidence is reported which suggests that the bootstrap approach of Cavaliere et al. (2012) significantly improves upon the finite sample performance of corresponding procedures based on either the asymptotic PLR test or an alternative bootstrap method (where the short run dynamics in the VAR model are estimated unrestrictedly) for a variety of conditionally and unconditionally heteroskedastic innovation processes.

Bootstrap Determination of the Co-Integration Rank in Heteroskedastic VAR Models

In a recent paper Cavaliere et al. (2012) develop bootstrap implementations of the (pseudo-) likelihood ratio [PLR] co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying VAR model which obtain under the reduced rank null hypothesis. They propose methods based on an i.i.d. bootstrap re-sampling scheme and establish the validity of their proposed bootstrap procedures in the context of a co-integrated VAR model with i.i.d. innovations.In this paper we investigate the properties of their bootstrap procedures, together with analogous procedures based on a wild bootstrap re-sampling scheme, when time-varying behaviour is present in either the conditional or unconditional variance of the innovations. We show that the bootstrap PLR tests are asymptotically correctly sized and, moreover, that the probability that the associated bootstrap sequential procedures select a rank smaller than the true rank converges to zero. This result is shown to hold for both the i.i.d. and wild bootstrap variants under conditional heteroskedasticity but only for the latter under unconditional heteroskedasticity. Monte Carlo evidence is reported which suggests that the bootstrap approach of Cavaliere et al. (2012) significantly improves upon the nite sample performance of corresponding procedures based on either the asymptotic PLR test or an alternative bootstrap method (where the short run dynamics in the VAR model are estimated unrestrictedly) for a variety of conditionally and unconditionally heteroskedastic innovation processes.

Asymptotics for LS, GLS, and Feasible GLS Statistics in an AR(1) Model with Conditional Heteroskedasticity

This paper considers a first-order autoregressive model with conditionally heteroskedastic innovations. The asymptotic distributions of least squares (LS), infeasible generalized least squares (GLS), and feasible GLS estimators and t statistics are determined. The GLS procedures allow for misspecification of the form of the conditional heteroskedasticity and, hence, are referred to as quasi-GLS procedures. The asymptotic results are established for drifting sequences of the autoregressive parameter and the distribution of the time series of innovations. In particular, we consider the full range of cases in which the autoregressive parameter rho_{n} satisfies (i) n(1 - rho_{n}) approaching infinity and (ii) n(1 - rho_{n}) approaching h in [0,infinity) as n approaching infinity, where n is the sample size. Results of this type are needed to establish the uniform asymptotic properties of the LS and quasi-GLS statistics.

Asymptotics for LS, GLS, and feasible GLS statistics in an AR(1) model with conditional heteroskedasticity

We consider a first-order autoregressive model with conditionally heteroskedastic innovations. The asymptotic distributions of least squares (LS), infeasible generalized least squares (GLS), and feasible GLS estimators and t statistics are determined. The GLS procedures allow for misspecification of the form of the conditional heteroskedasticity and, hence, are referred to as quasi-GLS procedures. The asymptotic results are established for drifting sequences of the autoregressive parameter ρn and the distribution of the time series of innovations. In particular, we consider the full range of cases in which ρn satisfies n(1−ρn)→∞ and n(1−ρn)→h1∈[0,∞) as n→∞, where n is the sample size. Results of this type are needed to establish the uniform asymptotic properties of the LS and quasi-GLS statistics.

Exact likelihood computation for nonlinear DSGE models with heteroskedastic innovations

Phenomena such as the Great Moderation have increased the attention of macroeconomists towards models where shock processes are not (log-)normal. This paper studies a class of discrete-time rational expectations models where the variance of exogenous innovations is subject to stochastic regime shifts. We first show that, up to a second-order approximation using perturbation methods, regime switching in the variances has an impact only on the intercept coefficients of the decision rules. We then demonstrate how to derive the exact model likelihood for the second-order approximation of the solution when there are as many shocks as observable variables. We illustrate the applicability of the proposed solution and estimation methods in the case of a small DSGE model.

COINTEGRATION RANK TESTING UNDER CONDITIONAL HETEROSKEDASTICITY

We analyze the properties of the conventional Gaussian-based cointegrating rank tests of Johansen (1996, Likelihood-Based Inference in Cointegrated Vector Autoregressive Models) in the case where the vector of series under test is driven by globally stationary, conditionally heteroskedastic (martingale difference) innovations. We first demonstrate that the limiting null distributions of the rank statistics coincide with those derived by previous authors who assume either independent and identically distributed (i.i.d.) or (strict and covariance) stationary martingale difference innovations. We then propose wild bootstrap implementations of the cointegrating rank tests and demonstrate that the associated bootstrap rank statistics replicate the first-order asymptotic null distributions of the rank statistics. We show that the same is also true of the corresponding rank tests based on the i.i.d. bootstrap of Swensen (2006, Econometrica 74, 1699–1714). The wild bootstrap, however, has the important property that, unlike the i.i.d. bootstrap, it preserves in the resampled data the pattern of heteroskedasticity present in the original shocks. Consistent with this, numerical evidence suggests that, relative to tests based on the asymptotic critical values or the i.i.d. bootstrap, the wild bootstrap rank tests perform very well in small samples under a variety of conditionally heteroskedastic innovation processes. An empirical application to the term structure of interest rates is given.

Sphet: Spatial Models with Heteroskedastic Innovations inR

<b>sphet</b> is a package for estimating and testing spatial models with heteroskedastic innovations. We implement recent generalized moments estimators and semiparametric methods for the estimation of the coefficients variance-covariance matrix. This paper is a general description of <b>sphet</b> and all functionalities are illustrated by application to the popular Boston housing dataset. The package in its current version is limited to the estimators based on Arraiz, Drukker, Kelejian, and Prucha (2010); Kelejian and Prucha (2007, 2010). The estimation functions implemented in <b>sphet</b> are able to deal with virtually any sample size.

Nonlinear principal components and long-run implications of multivariate diffusions

We investigate a method for extracting nonlinear principal components (NPCs). These NPCs maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and multivariate probability densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these NPCs. By exploiting the theory of continuous-time, reversible Markov diffusion processes, we give a different interpretation of these NPCs and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the NPCs maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the NPCs behave as scalar autoregressions with heteroskedastic innovations; this supports semiparametric identification and estimation of a multivariate reversible diffusion process and tests of the overidentifying restrictions implied by such a process from low-frequency data. We also explore implications for stationary, possibly nonreversible diffusion processes. Finally, we suggest a sieve method to estimate the NPCs from discretely-sampled data.

Co-Integration Rank Testing under Conditional Heteroskedasticity

We analyse the properties of the conventional Gaussian-based co-integrating rank tests of Johansen (1996) in the case where the vector of series under test is driven by globally stationary, conditionally heteroskedastic (martingale difference) innovations. We first demonstrate that the limiting null distributions of the rank statistics coincide with those derived by previous authors who assume either i.i.d. or (strict and covariance) stationary martingale difference innovations. We then propose wild bootstrap implementations of the co-integrating rank tests and demonstrate that the associated bootstrap rank statistics replicate the first-order asymptotic null distributions of the rank statistics. We show the same is also true of the corresponding rank tests based on the i.i.d. bootstrap of Swensen (2006). The wild bootstrap, however, has the important property that, unlike the i.i.d. bootstrap, it preserves in the re-sampled data the pattern of heteroskedasticity present in the original shocks. Consistent with this, numerical evidence suggests that, relative to tests based on the asymptotic critical values or the i.i.d. bootstrap, the wild bootstrap rank tests perform very well in small samples under a variety of conditionally heteroskedastic innovation processes. An empirical application to the term structure of interest rates is given.

USING SUBSPACE METHODS FOR ESTIMATING ARMA MODELS FOR MULTIVARIATE TIME SERIES WITH CONDITIONALLY HETEROSKEDASTIC INNOVATIONS

This paper deals with the estimation of linear dynamic models of the autoregressive moving average type for the conditional mean for stationary time series with conditionally heteroskedastic innovation process. Estimation is performed using a particular class of subspace methods that are known to have computational advantages as compared to estimation based on criterion minimization. These advantages are especially strong for high-dimensional time series. Conditions to ensure consistency and asymptotic normality of the subspace estimators are derived in this paper. Moreover asymptotic equivalence to quasi maximum likelihood estimators based on the Gaussian likelihood in terms of the asymptotic distribution is proved under mild assumptions on the innovations. Furthermore order estimation techniques are proposed and analyzed.