Conditional Homoskedasticity Research Articles

A POWERFUL SUBVECTOR ANDERSON–RUBIN TEST IN LINEAR INSTRUMENTAL VARIABLES REGRESSION WITH CONDITIONAL HETEROSKEDASTICITY

We introduce a new test for a two-sided hypothesis involving a subset of the structural parameter vector in the linear instrumental variables (IVs) model. Guggenberger, Kleibergen, and Mavroeidis (2019, Quantitative Economics, 10, 487–526; hereafter GKM19) introduce a subvector Anderson–Rubin (AR) test with data-dependent critical values that has asymptotic size equal to nominal size for a parameter space that allows for arbitrary strength or weakness of the IVs and has uniformly nonsmaller power than the projected AR test studied in Guggenberger et al. (2012, Econometrica, 80(6), 2649–2666). However, GKM19 imposes the restrictive assumption of conditional homoskedasticity (CHOM). The main contribution here is to robustify the procedure in GKM19 to arbitrary forms of conditional heteroskedasticity. We first adapt the method in GKM19 to a setup where a certain covariance matrix has an approximate Kronecker product (AKP) structure which nests CHOM. The new test equals this adaptation when the data are consistent with AKP structure as decided by a model selection procedure. Otherwise, the test equals the AR/AR test in Andrews (2017, Identification-Robust Subvector Inference, Cowles Foundation Discussion Papers 3005, Yale University) that is fully robust to conditional heteroskedasticity but less powerful than the adapted method. We show theoretically that the new test has asymptotic size bounded by the nominal size and document improved power relative to the AR/AR test in a wide array of Monte Carlo simulations when the covariance matrix is not too far from AKP.

To keep faith with homoskedasticity or to go back to heteroskedasticity? The case of FATANG stocks.

Did the pattern of US stock market volatility change due to COVID-19 or have the US stock markets been less volatile despite the pandemic shock? And as for tech stocks, are they even less volatile than the market overall? In this paper, we provide evidence in favor of a “quietness” in the stock markets, interrupted by COVID-19, by analyzing dispersion, skewness and kurtosis characteristics of the empirical distribution of nine returns series that include individual FATANG stocks (FAANG: Facebook, Amazon, Apple, Netflix and Google; plus Tesla) and US indices (S&P 500, DJIA and NASDAQ). In comparison with the years before, the daily average return after COVID-19 was 6.48, 2.58 and 2.34 times higher for Tesla, Apple and NASDAQ, respectively. In terms of volatility, the increase was more pronounced in the three stock indices when compared to the individual FATANG stocks. This paper also puts forward a new methodology based on semi-variance and semi-kurtosis. While the value of the ratio between semi-kurtosis and kurtosis is always higher than 70% for the three US stock indices, in the case of stocks the opposite is true, which highlights the importance of large positive returns when compared to negative ones. Structural breaks and conditional heteroskedasticity are also analyzed by considering the traditional symmetrical and asymmetrical GARCH models. We show that in the most recent past, despite the COVID-19 pandemic, the FATANG tech stocks are characterized mostly by conditional homoskedasticity, while the returns of US stock indices are characterized mainly by conditional heteroskedasticity.

On the inconsistency of nonparametric bootstraps for the subvector Anderson–Rubin test

Bootstrap procedures based on instrumental variable (IV) estimates or t-statistics are generally invalid when the instruments are weak. The bootstrap may even fail when applied to identification-robust test statistics. For subvector inference based on the Anderson–Rubin (AR) statistic, Wang and Doko Tchatoka (2018) show that the residual bootstrap is inconsistent under weak IVs. In particular, the residual bootstrap depends on certain estimator of structural parameters to generate bootstrap pseudo-data, while the estimator is inconsistent under weak IVs. It is thus tempting to consider nonparametric bootstrap. In this note, under the assumptions of conditional homoskedasticity and one nuisance structural parameter, we investigate the bootstrap consistency for the subvector AR statistic based on the nonparametric i.i.d. bootstrap and its recentered version proposed by Hall and Horowitz (1996). We find that both procedures are inconsistent under weak IVs: although able to mimic the weak-identification situation in the data, both procedures result in approximation errors, which leads to the discrepancy between the bootstrap world and the original sample. In particular, both bootstrap tests can be very conservative under weak IVs. The pairs bootstrap test can be very conservative even under strong IVs.

Testing a Class of Semi- or Nonparametric Conditional Moment Restriction Models using Series Methods

This paper proposes a new test for a class of conditional moment restrictions whose parameterization involves unknown, unrestricted conditional expectation functions. Examples of such conditional moment restrictions are conditional mean independence (leading to a nonparametric significance test) and conditional homoskedasticity (with an otherwise unrestricted conditional mean) and also arise from models of single-agent discrete choice under uncertainty and static games of incomplete information. The proposed test may be viewed as a semi-/nonparametric extension of the Bierens (1982) goodness-of-fit test of a parametric model for the conditional mean. Estimating conditional expectations using series methods and employing a Gaussian multiplier bootstrap to obtain critical values, the resulting test is shown to be asymptotically correctly sized and consistent. A simulation study applies the procedure to test the specification of a two-player, binary-action static game of incomplete information, treating equilibrium beliefs as nonparametric conditional expectations.

On the Inconsistency of Nonparametric Bootstraps for the Subvector Anderson-Rubin Test

Bootstrap procedures based on instrumental variable (IV) estimates or t-statistics are generally invalid when the instruments are weak. The bootstrap may even fail when applied to identification-robust test statistics. For subvector inference based on the Anderson-Rubin (AR) statistic, Wang and Doko Tchatoka (2018) show that the residual bootstrap is inconsistent under weak IVs. In particular, the residual bootstrap depends on certain estimator of structural parameters to generate bootstrap pseudo-data, while the estimator is inconsistent under weak IVs. It is thus tempting to consider nonparametric bootstrap. In this note, under the assumptions of conditional homoskedasticity and one nuisance structural parameter, we investigate the bootstrap consistency for the subvector AR statistic based on the nonparametric i.i.d. bootstap and its recentered version proposed by Hall and Horowitz (1996). We find that both procedures are inconsistent under weak IVs: although able to mimic the weak-identification situation in the data, both procedures result in approximation errors, which leads to the discrepancy between the bootstrap world and the original sample. In particular, both bootstrap tests can be very conservative under weak IVs. The pairs bootstrap can be very conservative even under strong IVs.

Two-stage least squares as minimum distance

SummaryThe two-stage least-squares (2SLS) instrumental-variables (IV) estimator for the parameters in linear models with a single endogenous variable is shown to be identical to an optimal minimum-distance (MD) estimator based on the individual instrument-specific IV estimators. The 2SLS estimator is a linear combination of the individual estimators, with the weights determined by their variances and covariances under conditional homoskedasticity. It is further shown that the Sargan test statistic for overidentifying restrictions is the same as the MD criterion test statistic. This provides an intuitive interpretation of the Sargan test. The equivalence results also apply to the efficient two-step generalized method of moments and robust optimal MD estimators and criterion functions, allowing for general forms of heteroskedasticity. It is further shown how these results extend to the linear overidentified IV model with multiple endogenous variables.

On Bootstrap inconsistency and Bonferroni-based size-correction for the subset Anderson–Rubin test under conditional homoskedasticity

We focus on the linear instrumental variable model with two endogenous regressors under conditional homoskedasticity, and study the subset Anderson and Rubin (1949, AR) test when the nuisance structural parameter, the unrestricted slope coefficient of endogenous regressor, may be weakly identified. Weak identification leads to nonstandard null limiting distributions, and alternative to the usual chi-squared critical value is needed. We first investigate the bootstrap validity for the subset AR test based on various plug-in estimators, and show that the bootstrap provides asymptotic refinement when the nuisance structural parameter is strongly identified, but is inconsistent when it is weakly identified. This is in contrast to the result of bootstrap validity in Moreira et al. (2009). Then, we propose a Bonferroni-based size-correction method that yields correct asymptotic size for all the test statistics considered. The power performance of size-corrected tests can be further improved by applying the mapping between structural and endogenous parameters in the model. Monte Carlo experiments confirm the bootstrap inconsistency and demonstrate that all the subset tests based on our correction technique control the size.

The causality link between energy electricity consumption, CO2 emissions, R&D stocks and economic growth in Mediterranean countries (MCs)

This study examined the four-way linkages between electricity consumption, CO2 emissions, R&D stocks and economic growth for the MCs over the 1990–2016 periods, by implementing several estimators: SUR, 3SLS and GMM. The specification of these econometric techniques is that the four sets of equations (electricity, CO2, R&D and growth models) are estimated jointly through the system of equations. These techniques make it possible to estimate all the parameters of the models at the same time and solve the problem of endogeneity. Under conditional homoskedasticity, GMM reduces to 3SLS if the set of instrumental variables is common to all equations. If we further assume that all the regressors are predetermined, then 3SLS reduces to SUR. Empirical results confirm the existence of strong feedback effects between electricity, CO2 emissions, R&D stocks and economic growth (especially in SUR estimator). However, the results support the occurrence of unidirectional causality between electricity and R&D stocks (in 3SLS) while the other relationships exist. On the one hand, here is unidirectional causality among R&D stocks and economic growth as well as unidirectional causality between R&D stocks and CO2 emissions, on the other hand (GMM estimator). The study consolidates the policy insights to control the environment from degradation by using energy-efficient technologies. It is important for policymakers in formulating a conclusive, economic, energy and environmental policies to meet the bourgeoning energy demand in MCs. They should concentrate on allocating funds to adopt environmentally friendly technologies and encourage using more energy-efficient technologies for production purposes and thus preserving the environment from degradation.

Robust inference for the Two-Sample 2SLS estimator

The Two-Sample Two-Stage Least Squares (TS2SLS) data combination estimator is a popular estimator for the parameters in linear models when not all variables are observed jointly in one single data set. Although the limiting normal distribution has been established, the asymptotic variance formula has only been stated explicitly in the literature for the case of conditional homoskedasticity. By using the fact that the TS2SLS estimator is a function of reduced form and first-stage OLS estimators, we derive the variance of the limiting normal distribution under conditional heteroskedasticity. A robust variance estimator is obtained, which generalises to cases with more general patterns of variable (non-)availability. Stata code and some Monte Carlo results are provided in an Appendix. Stata code for a nonlinear GMM estimator that is identical to the TS2SLS estimator in just identified models and asymptotically equivalent to the TS2SLS estimator in overidentified models is also provided there.

On the Asymptotic Size of Subvector Tests in the Linear Instrumental Variables Model

We calculate the asymptotic sizes of the subvector Anderson and Rubin (1949, AR) and Lagrange Multiplier (LM) tests in a linear instrumental variables model with two right hand side endogenous variables when the reduced form coefficient matrix is unrestricted. Under the assumption of conditional homoskedasticity we show that the subvector AR test has correct asymptotic size but that the asymptotic size of the subvector LM test is generally distorted. We provide size-corrected critical values for the subvector LM test.

Accurately sized test statistics with misspecified conditional homoskedasticity

We study the finite-sample performance of test statistics in linear regression models where the error dependence is of unknown form. With an unknown dependence structure, there is traditionally a trade-off between the maximum lag over which the correlation is estimated (the bandwidth) and the amount of heterogeneity in the process. When allowing for heterogeneity, through conditional heteroskedasticity, the correlation at far lags is generally omitted and the resultant inflation of the empirical size of test statistics has long been recognized. To allow for correlation at far lags, we study the test statistics constructed under the possibly misspecified assumption of conditional homoskedasticity. To improve the accuracy of the test statistics, we employ the second-order asymptotic refinement in Rothenberg [Approximate power functions for some robust tests of regression coefficients, Econometrica 56 (1988), pp. 997–1019] to determine the critical values. The simulation results of this paper suggest that when sample sizes are small, modelling the heterogeneity of a process is secondary to accounting for dependence. We find that a conditionally homoskedastic covariance matrix estimator (when used in conjunction with Rothenberg's second-order critical value adjustment) improves test size with only a minimal loss in test power, even when the data manifest significant amounts of heteroskedasticity. In some specifications, the size inflation was cut by nearly 40% over the traditional heteroskedasticity and autocorrelation consistent (HAC) test. Finally, we note that the proposed test statistics do not require that the researcher specify the bandwidth or the kernel.

The robustness of asset pricing models: Coskewness and cokurtosis

We consider the robustness of the least-squares inference of linear asset pricing models. We evaluate the asymptotic covariance matrices of the least-squares estimator of alphas and betas when the joint distribution of factors and error terms is independently and identically distributed without making any specific distributional assumptions. When the standard assumption of conditional homoskedasticity for the conditional covariance matrix of error terms given factors does not hold, we show the asymptotic covariance matrix for betas depends only on cokurtosis of factors and error terms while the asymptotic covariance matrix for alphas depends on cokurtosis as well as coskewness of factors and error terms. This implies the inference of betas is not affected by skewness of the underlying joint distribution. Numerical examples are provided using Fama and French's benchmark portfolio returns and factors.

A New Test of the Martingale Difference Hypothesis

In this paper we propose a new class of tests for the martingale difference hypothesis based on the moment conditions derived by Bierens (1982). In contrast with the existing consistent tests, the proposed test has a standard limiting distribution and is easy to implement. Comparing with many commonly used autocorrelation- and spectrum-based tests, it has better power against a larger class of alternatives that may be serially correlated or uncorrelated. Moreover, this test does not rely on the assumption of conditional homoskedasticity and requires a weaker moment condition. Our simulations confirm that the proposed test is powerful against various linear and nonlinear alternatives and is quite robust to the failure of higher-order moments. Our empirical study on exchange rate returns also shows that the conclusion resulted from the proposed test is different from that of the conventional tests.

A CONSISTENT TEST FOR CONDITIONAL HETEROSKEDASTICITY IN TIME-SERIES REGRESSION MODELS

We show that the standard consistent test for testing the null of conditional homoskedasticity (against conditional heteroskedasticity) can be generalized to a time-series regression model with weakly dependent data and with generated regressors. The test statistic is shown to have an asymptotic normal distribution under the null hypothesis of conditional homoskedastic error. We also discuss extension of our test to the case of testing the null of a parametrically specified conditional variance. We advocate using a bootstrap method to overcome the issue of slow convergence of this test statistic to its limiting distribution.

Testing for ARCH in the presence of a possibly misspecified conditional mean

Ever since the development of the Autoregressive Conditional Heteroskedasticity (ARCH) model ( Engle, 1982), testing for the presence of ARCH has become a routine diagnostic. One popular method of testing for ARCH is T times the R 2 from a regression of the squared residuals on p of its lags. This test has been shown to have a lagrange multiplier interpretation and is asymptotically distributed as a χ 2( p) random variable. Underlying this test is the assumption of a correctly specified conditional mean. In this paper, we consider the properties of the ARCH test when there is a possibly misspecified conditional mean. Examples of misspecification include omitted variables, structural change, and parameter instability. We show that, in general, misspecification will lead to overrejection of the null hypothesis of conditional homoskedasticity. We demonstrate the use of recursive residuals to improve the fit of a first-stage conditional mean regression. We illustrate these results via Monte Carlo simulations and consider two empirical examples.

LONG AND SHORT MEMORY CONDITIONAL HETEROSKEDASTICITY IN ESTIMATING THE MEMORY PARAMETER OF LEVELS

Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader conditions than parametric estimates. However, recent large sample theory for semiparametric estimates forbids conditional heteroskedasticity. We show that a leading semiparametric estimate, the Gaussian or local Whittle one, can be consistent and have the same limiting distribution under conditional heteroskedasticity as under the conditional homoskedasticity assumed by Robinson (1995, Annals of Statistics 23, 1630–61). Indeed, noting that long memory has been observed in the squares of financial time series, we allow, under regularity conditions, for conditional heteroskedasticity of the general form introduced by Robinson (1991, Journal of Econometrics 47, 67–84), which may include long memory behavior for the squares, such as the fractional noise and autoregressive fractionally integrated moving average form, and also standard short memory ARCH and GARCH specifications.

Testing for GARCH effects: a one-sided approach

ARCH models often lie at the boundary of the parameter space under conditional homoskedasticity, which invalidates the usual χ 2 distribution of LR and Wald tests. Although LM tests are not affected, the one-sided nature of the alternative hypothesis should result in more powerful tests. We propose a simple one-sided version of the LM test, which is closely related to the Kuhn–Tucker multiplier test. We also present critical values for LR, Wald and one-sided LM tests. The results of a Monte Carlo comparison suggest that one-sided tests are indeed more powerful than their two-sided counterparts.

An Asymptotic Theory for Estimating Beta‐Pricing Models Using Cross‐Sectional Regression

ABSTRACTWithout the assumption of conditional homoskedasticity, a general asymptotic distribution theory for the two‐stage cross‐sectional regression method shows that the standard errors produced by the Fama–MacBeth procedure do not necessarily overstate the precision of the risk premium estimates. When factors are misspecified, estimators for risk premiums can be biased, and the t‐value of a premium may converge to infinity in probability even when the true premium is zero. However, when a beta‐pricing model is misspecified, the t‐values for firm characteristics generally converge to infinity in probability, which supports the use of firm characteristics in cross‐sectional regressions for detecting model misspecification.