Conditional Heteroskedasticity Of Unknown Form Research Articles

Deconstructing the Yield Curve

We investigate the factor structure of the term structure of interest rates and argue that characterizing the minimal dimension of the data generating process is more challenging than currently appreciated. As a result, inference procedures for yield curve models that commit to a parsimoniously parameterized factor structure may be omitting important information about the underlying true factor space. To circumvent these difficulties, we introduce a novel nonparametric bootstrap that is robust to general forms of time and cross-sectional dependence and conditional heteroskedasticity of unknown form. We show that our bootstrap procedure is asymptotically valid and exhibits excellent finite-sample properties in simulations. We demonstrate the applicability of our results in two empirical exercises: first, we show that measures of equity market tail risk and the state of the macroeconomy predict bond returns beyond the level or slope of the yield curve; second, we provide a bootstrap-based bias correction and confidence intervals for the probability of recession based on the shape of the yield curve. Our results apply more generally to all assets with a finite maturity structure.

Testing for Non-Fundamentalness

Non-fundamentalness arises when observed variables do not contain enough information to recover structural shocks. This paper propose a new test to empirically detect non-fundamentalness, which is robust to the conditional heteroskedasticity of unknown form, does not need information outside of the specified model and could be accomplished with a standard F-test. A Monte Carlo study based on a DSGE model is conducted to examine the finite sample performance of the test. I apply the proposed test to the U.S. quarterly data to identify the dynamic effects of supply and demand disturbances on real GNP and unemployment.

Wild bootstrap tests for autocorrelation in vector autoregressive models

Tests for error autocorrelation (AC) are derived under the assumption of independent and identically distributed errors. The tests are not asymptotically valid if the errors are conditionally heteroskedastic. In this article we propose wild bootstrap (WB) Lagrange multiplier tests for error AC in vector autoregressive (VAR) models. We show that the WB tests are asymptotically valid under conditional heteroskedasticity of unknown form. WB tests based on a version of the heteroskedasticity-consistent covariance matrix estimator are found to have the smallest error in rejection probability under the null and high power under the alternative. We apply the tests to VAR models for credit default swap prices and Euribor interest rates. An important result that we find is that the WB tests lead to parsimonious models while the asymptotic tests suggest that a long lag length is required to get white noise residuals.

Inference in VARs with conditional heteroskedasticity of unknown form

We consider a framework for asymptotically valid inference in stable vector autoregressive (VAR) models with conditional heteroskedasticity of unknown form. A joint central limit theorem for the LS estimators of both the VAR slope parameters as well as the unconditional innovation variance parameters is obtained from a weak vector autoregressive moving average model set-up recently proposed in the literature. Our results are important for correct inference on VAR statistics that depend both on the VAR slope and the variance parameters as e.g. in structural impulse responses. We also show that wild and pairwise bootstrap schemes fail in the presence of conditional heteroskedasticity if inference on (functions) of the unconditional variance parameters is of interest because they do not correctly replicate the relevant fourth moments’ structure of the innovations. In contrast, the residual-based moving block bootstrap results in asymptotically valid inference. We illustrate the practical implications of our theoretical results by providing simulation evidence on the finite sample properties of different inference methods for impulse response coefficients. Our results point out that estimation uncertainty may increase dramatically in the presence of conditional heteroskedasticity. Moreover, most inference methods are likely to understate the true estimation uncertainty substantially in finite samples.

Specification test for panel data models with interactive fixed effects

In this paper, we propose a consistent nonparametric test for linearity in a large dimensional panel data model with interactive fixed effects. Both lagged dependent variables and conditional heteroskedasticity of unknown form are allowed in the model. We estimate the model under the null hypothesis of linearity to obtain the restricted residuals which are then used to construct the test statistic. We show that after being appropriately centered and standardized, the test statistic is asymptotically normally distributed under both the null hypothesis and a sequence of Pitman local alternatives by using the concept of conditional strong mixing that was recently introduced by Prakasa Rao (2009). To improve the finite sample performance, we propose a bootstrap procedure to obtain the bootstrap p-value. A small set of Monte Carlo simulations illustrates that our test performs well in finite samples. An application to an economic growth panel dataset indicates significant nonlinear relationships between economic growth, initial income level and capital accumulation.

Specification Test for Panel Data Models with Interactive Fixed Effects

In this paper, we propose a consistent nonparametric test for linearity in a large dimensional panel data model with interactive fixed effects. Both lagged dependent variables and conditional heteroskedasticity of unknown form are allowed in the model. We estimate the model under the null hypothesis of linearity to obtain the restricted residuals which are then used to construct the test statistic. We show that after being appropriately centered and standardized, the test statistic is asymptotically normally distributed under both the null hypothesis and a sequence of Pitman local alternatives by using the concept of conditional strong mixing that was recently introduced by Prakasa Rao (2009). To improve the finite sample performance, we propose a bootstrap procedure to obtain the bootstrap p-value. A small set of Monte Carlo simulations illustrates that our test performs well in finite samples. An application to an economic growth panel dataset indicates significant nonlinear relationships between economic growth, initial income level and capital accumulation.

TESTING HOMOGENEITY IN PANEL DATA MODELS WITH INTERACTIVE FIXED EFFECTS

This paper proposes a residual-based Lagrange Multiplier (LM) test for slope homogeneity in large-dimensional panel data models with interactive fixed effects. We first run the panel regression under the null to obtain the restricted residuals and then use them to construct our LM test statistic. We show that after being appropriately centered and scaled, our test statistic is asymptotically normally distributed under the null and a sequence of Pitman local alternatives. The asymptotic distributional theories are established under fairly general conditions that allow for both lagged dependent variables and conditional heteroskedasticity of unknown form by relying on the concept of conditional strong mixing. To improve the finite-sample performance of the test, we also propose a bootstrap procedure to obtain the bootstrap p-values and justify its validity. Monte Carlo simulations suggest that the test has correct size and satisfactory power. We apply our test to study the Organization for Economic Cooperation and Development economic growth model.

Testing the Martingale Difference Hypothesis in the European Emerging Unit-linked Insurance Markets

This study examines the martingale difference hypothesis (MDH) for the European emerging unit-linked insurance markets, using the automatic portmanteau (AQ) test of Escanciano and Lobato, 2009 for the three sub-periods of pre-crisis, crisis, and post-crisis. The martingale difference sequence is called conditional mean independence in the statistical literature, implying that the asset return is purely non-predictable from its own past. This study proposes a data-driven Box-Pierce for serial correlation. The automatic Portmanteau test for serial correlation test presents higher power in simulations than the existing ones for models commonly employed in empirical finance: the researcher does not need to specify the order of the autocorrelation tested, since the test automatically chooses this number; its asymptotic null distribution is chi-square with one degree of freedom, so there is no need of using a bootstrap procedure to estimate the critical values and the test is robust to the presence of conditional heteroskedasticity of unknown form. This paper examines return predictability of the daily ING unit-linked funds prices and aims at monitoring any improvement in the degree of efficiency in time.

An automatic Portmanteau test for serial correlation

This article introduces a data-driven Box–Pierce test for serial correlation. The proposed test is very attractive compared to the existing ones. In particular, implementation of this test is extremely simple for two reasons: first, the researcher does not need to specify the order of the autocorrelation tested, since the test automatically chooses this number; second, its asymptotic null distribution is chi-square with one degree of freedom, so there is no need of using a bootstrap procedure to estimate the critical values. In addition, the test is robust to the presence of conditional heteroskedasticity of unknown form. Finally, the proposed test presents higher power in simulations than the existing ones for models commonly employed in empirical finance.

Hybrid and Size-Corrected Subsampling Methods

This paper considers inference in a broad class of nonregular models. The models considered are nonregular in the sense that standard test statistics have asymptotic distributions that are discontinuous in some parameters. It is shown in Andrews and Guggenberger (2009a) that standard fixed critical value, subsampling, and m out of n bootstrap methods often have incorrect asymptotic size in such models. This paper introduces general methods of constructing tests and confidence intervals that have correct asymptotic size. In particular, we consider a hybrid subsampling/fixed-critical-value method and size-correction methods. The paper discusses two examples in detail. They are (i) confidence intervals in an autoregressive model with a root that may be close to unity and conditional heteroskedasticity of unknown form and (ii) tests and confidence intervals based on a post-conservative model selection estimator.

Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.

ROBUST ASYMPTOTIC INFERENCE IN AUTOREGRESSIVE MODELS WITH MARTINGALE DIFFERENCE ERRORS

This paper proposes a GMM-based method for asymptotic confidence interval construction in stationary autoregressive models, which is robust to the presence of conditional heteroskedasticity of unknown form. The confidence regions are obtained by inverting the asymptotic acceptance region of the distance metric test for the continuously updated GMM (CU-GMM) estimator. Unlike the predetermined symmetric shape of the Wald confidence intervals, the shape of the proposed confidence intervals is data-driven owing an estimated sequence of nonuniform weights. It appears that the flexibility of the CU-GMM estimator in downweighting certain observations proves advantageous for confidence interval construction. This stands in contrast to some other generalized empirical likelihood estimators with appealing optimality properties such as the empirical likelihood estimator whose objective function prevents such downweighting. A Monte Carlo simulation study illustrates the excellent small-sample properties of the method for AR models with ARCH errors. The procedure is applied to study the dynamics of the federal funds rate.

Bootstrapping autoregressions with conditional heteroskedasticity of unknown form

Conditional heteroskedasticity is an important feature of many macroeconomic and financial time series. Standard residual-based bootstrap procedures for dynamic regression models treat the regression error as i.i.d. These procedures are invalid in the presence of conditional heteroskedasticity. We establish the asymptotic validity of three easy-to-implement alternative bootstrap proposals for stationary autoregressive processes with martingale difference errors subject to possible conditional heteroskedasticity of unknown form. These proposals are the fixed-design wild bootstrap, the recursive-design wild bootstrap and the pairwise bootstrap. In a simulation study all three procedures tend to be more accurate in small samples than the conventional large-sample approximation based on robust standard errors. In contrast, standard residual-based bootstrap methods for models with i.i.d. errors may be very inaccurate if the i.i.d. assumption is violated. We conclude that in many empirical applications the proposed robust bootstrap procedures should routinely replace conventional bootstrap procedures for autoregressions based on the i.i.d. error assumption.

Testing for Autocorrelation and Unit Roots in the Presence of Conditional Heteroskedasticity of Unknown Form

The presence of conditional heteroskedasticity invalidates standard autocorrelation tests such as the Durbin-Watson statistic and its many variants, and reduces the power of standard unit root tests like the Dickey-Fuller test. This paper addresses the problem of testing for AR(1) and AR(p) alternatives in the presence of conditional heteroskedasticity, constructs new autocorrelation tests (called g- and gl- tests), and derives an asymptotic theory for the new statistics. The g test is shown to be robust to any form of conditional heteroskedasticity, while the gl test is robust to the joint presence of conditional heteroskedasticity and lagged dependent variables. The paper also develops a GLS-type unit root test that allows for conditional heteroskedasticity. Its limit distribution turns out to be a convex mixture of the Dickey-Fuller distribution and the standard normal distribution. It is shown that the GLS statistic improves the power of the Dickey-Fuller test under condictional heteroskedasticity. Some Monte Carlo simulations with the new tests are reported.

Bootstrapping Autoregressions with Conditional Heteroskedasticity of Unknown Form

Conditional heteroskedasticity is an important feature of many macroeconomic and financial time series. Standard residual-based bootstrap procedures for dynamic regression models treat the regression error as i.i.d. These procedures are invalid in the presence of conditional heteroskedasticity. We establish the asymptotic validity of three easy-to-implement alternative bootstrap proposals for stationary autoregressive processes with m.d.s. errors subject to possible conditional heteroskedasticity of unknown form. These proposals are the fixed-design wild bootstrap, the recursive-design wild bootstrap and the pairwise bootstrap. In a simulation study all three procedures tend to be more accurate in small samples than the conventional large-sample approximation based on robust standard errors. In contrast, standard residual-based bootstrap methods for models with i.i.d. errors may be very inaccurate if the i.i.d. assumption is violated. We conclude that in many empirical applications the proposed robust bootstrap procedures should routinely replace conventional bootstrap procedures based on the i.i.d. error assumption.

Bootstrapping Autoregressions with Conditional Heteroskedasticity of Unknown Form

Conditional heteroskedasticity is an important feature of many macroeconomic and financial time series. Standard residual-based bootstrap procedures for dynamic regression models treat the regression eroor as i.i.d. These procedures are invalid in the presence of conditional heteroskedasticity. We establish the asymptotic validity of three easy-to-implement alternative bootstrap proposals for stationary autoregressive processes with m.d.s. errors subject to possible conditional heteroskedasticity of unknown form. These proposals are the fixed-design wild bootstrap, the recursive design wild bootstrap and the pairwise bootstrap. In a simulation study all three procedures tend to be more accurate in small samples than the conventional large-sample approximation based on robust standard errors. In contrast, standard residual-based bootstrap methods for models with i.i.d. errors may be very inaccurate if the i.i.d. assumption is violated. We conclude that in many empirical applications the proposed robust bootstrap procedures should routinely replace conventional bootstrap procedures based on the i.i.d. error assumption. JEL Classification: C15, C22, C52

Testing for Zeros in the Spectrum of an Univariate Stationary Process: Part I

Non parametric and parametric estimation for the spectral density of a stationary process is a well-known topic, except when the spectrum vanishes for some frequency. Indeed, for this frequency, the limit law degenerates, and traditional inference no longer applies. The paper introduces non parametric tests of this hypothesis, which exploit the asymptotic behavior of the periodogram for some well-chosen sequence of frequencies. In particular, statistics free from nuisance parameters are derived, and conditional heteroskedasticity of unknown form is allowed. As an application, stationarity tests against seasonal unit-root alternatives are developed.

Adaptive estimation of heteroskedastic econometric models

We focus on the linear model with conditional heteroskedasticity of unknown form. Adaptive estimators of the coefficients of the linear model, based on no rigid parameterisation of the heteroskedasticity, but having the same asymptotic efficiency as estimators which do use such information, are surveyed. A small Monte Carlo study of their performance is reported. We describe a modification of the popular paradigm in which the variance is a function of the mean, allowing this function to be of unknown form. We describe a modification also of the autoregressive conditional heteroskedasticity (ARCH) model, in which the heteroskedasticity function is of unknown form.