Detecting parameter shift in garch models

This paper applies recent theories of testing for parameter constancy to the conditional variance in a GARCH model. The supremum Lagrange multiplier test for conditional Gaussian GARCH models and its robustified variants are discussed. The asymptotic null distribution of the test statistics are derived from the weak convergence of the scores, and the critical values from the hitting probability of squared Bessel process.

Monte Carlo studies on the finite sample size and power performance of the supremum LM tests are conducted. Applications of these tests to S&P 500 indicate that the hypothesis of stable conditional variance parameters can be rejected.     

Evaluating Multivariate GARCH Models in the Nordic Electricity Markets

ABSTRACT

This article considers a variety of specification tests for multivariate GARCH models that are used for dynamic hedging in electricity markets. The test statistics include the robust conditional moments tests for sign-size bias along with the recently introduced copula tests for an appropriate dependence structure. We consider this effort worthwhile, since quite often the tests of multivariate GARCH models are omitted and the models become selected ad hoc depending on the results they generate. Hedging performance comparisons, in terms of unconditional and conditional ex-post variance portfolio reduction, are conducted.     

Residual-based CUSUM of squares test for Poisson integer-valued GARCH models

This study considers the problem of testing for a parameter change in integer-valued time series models in which the conditional density of current observations is assumed to follow a Poisson distribution. As a test, we consider the CUSUM of the squares test based on the residuals from INGARCH models and find that the test converges weakly to the supremum of a Brownian bridge. A simulation study demonstrates its superiority to the residual and standardized residual-based CUSUM tests of Kang and Lee [Parameter change test for Poisson autoregressive models. Scand J Statist. 2014;41:1136–1152] and Lee and Lee [CUSUM tests for general nonlinear inter-valued GARCH models: comparison study. Ann Inst Stat Math. 2019;71:1033–1057.] as well as the CUSUM of squares test based on standardized residuals.     

A Lagrange multiplier test for testing the adequacy of constant conditional correlation GARCH model

ABSTRACT

A Lagrange multiplier test for testing the parametric structure of a constant conditional correlation-generalized autoregressive conditional heteroskedasticity (CCC-GARCH) model is proposed. The test is based on decomposing the CCC-GARCH model multiplicatively into two components, one of which represents the null model, whereas the other one describes the misspecification. A simulation study shows that the test has good finite sample properties. We compare the test with other tests for misspecification of multivariate GARCH models. The test has high power against alternatives where the misspecification is in the GARCH parameters and is superior to other tests. The test is not greatly affected by misspecification in the conditional correlations and is therefore well suited for considering misspecification of GARCH equations.     

On regression-based tests for persistence in logarithmic volatility models

Building on the work of Pantula (1986), this paper discusses how the hypothesis of conditional variance nonstationarity in the logarithmic family of generalized autoregressive conditional heteroskedasticity (GARCH) and stochastic volatility processes may be tested using regression-based tests. The latter are easy to implement, have well-defined large-sample distributions, and are less sensitive to structural changes than tests based on the quasimaximum likelihood estimator.     

Granger-causal analysis of GARCH models: A Bayesian approach

A multivariate GARCH model is used to investigate Granger causality in the conditional variance of time series. Parametric restrictions for the hypothesis of noncausality in conditional variances between two groups of variables, when there are other variables in the system as well, are derived. These novel conditions are convenient for the analysis of potentially large systems of economic variables. To evaluate hypotheses of noncausality, a Bayesian testing procedure is proposed. It avoids the singularity problem that may appear in the Wald test, and it relaxes the assumption of the existence of higher-order moments of the residuals required in classical tests.     

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.     

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.     

Modelling asymmetric volatility dynamics by multivariate BL-GARCH models

The class of Multivariate BiLinear GARCH (MBL-GARCH) models is proposed and its statistical properties are investigated. The model can be regarded as a generalization to a multivariate setting of the univariate BL-GARCH model proposed by Storti and Vitale (Stat Methods Appl 12:19–40, 2003a; Comput Stat 18:387–400, 2003b). It is shown how MBL-GARCH models allow to account for asymmetric effects in both conditional variances and correlations. An EM algorithm for the maximum likelihood estimation of the model parameters is derived. Furthermore, in order to test for the appropriateness of the conditional variance and covariance specifications, a set of robust conditional moments test statistics are defined. Finally, the effectiveness of MBL-GARCH models in a risk management setting is assessed by means of an application to the estimation of the optimal hedge ratio in futures hedging.     

Indirect estimation of randomized generalized autoregressive conditional heteroskedastic models

The class of generalized autoregressive conditional heteroskedastic (GARCH) models can be used to describe the volatility with less parameters than autoregressive conditional heteroskedastic (ARCH)-type models, their distributions are heavy-tailed, with time-dependent conditional variance, and are able to model clustering of volatility. Despite all these facts, the way that GARCH models are built imposes limits on the heaviness of the tails of their unconditional distribution. The class of randomized generalized autoregressive conditional heteroskedastic (R-GARCH) models includes the ARCH and GARCH models allowing the use of stable innovations. Estimation methods and empirical analysis of R-GARCH models are the focus of this work. We present the indirect inference method to estimate the R-GARCH models, some simulations and an empirical application.     

UNIT ROOT TESTING IN THE PRESENCE OF HEAVY‐TAILED GARCH ERRORS

We derive the asymptotic distributions of the Dickey–Fuller (DF) and augmented DF (ADF) tests for unit root processes with Generalized Autoregressive Conditional Heteroscedastic (GARCH) errors under fairly mild conditions. We show that the asymptotic distributions of the DF tests and ADF t‐type test are the same as those obtained in the independent and identically distributed Gaussian cases, regardless of whether the fourth moment of the underlying GARCH process is finite or not. Our results go beyond earlier ones by showing that the fourth moment condition on the scaled conditional errors is totally unnecessary. Some Monte Carlo simulations are provided to illustrate the finite‐sample‐size properties of the tests.     

Heteroscedasticity in non-stationary time series,some Monte Carlo evidence

A frequent question raised by practitioners doing unit root tests is whether these tests are sensitive to the presence of heteroscedasticity. Theoretically this is not the case for a wide range of heteroscedastic models. However, for some limiting cases such as degenerate and integrated heteroscedastic processes it is not obvious whether this will have an effect. In this paper we report a Monte Carlo study analyzing the implications of various types of heteroscedasticity on three types of unit root tests: The usual Dickey-Fuller test, Phillips' (1987) semi-parametric test and finally a Dickey-Fuller type test using White's (1980) heteroscedasticity consistent standard errors. The sorts of heteroscedasticity we examine are the GARCH model of Bollerslev (1986) and the Exponential ARCH model of Nelson (1991). In particular, we call attention to situations where the conditional variances exhibit a high degree of persistence as is frequently observed for returns of financial time series, and the case where, in fact, the variance process for the first class of models becomes degenerate.     

The effect of conditional heteroskedasticity on common statistical procedures for means and variances

Summary: Commonly used standard statistical procedures for means and variances (such as the t–test for means or the F–test for variances and related confidence procedures) require observations from independent and identically normally distributed variables. These procedures are often routinely applied to financial data, such as asset or currency returns, which do not share these properties. Instead, they are nonnormal and show conditional heteroskedasticity, hence they are dependent. We investigate the effect of conditional heteroskedasticity (as modelled by GARCH(1,1)) on the level of these tests and the coverage probability of the related confidence procedures. It can be seen that conditional heteroskedasticity has no effect on procedures for means (at least in large samples). There is, however, a strong effect of conditional heteroskedasticity on procedures for variances. These procedures should therefore not be used if conditional heteroskedasticity is prevalent in the data.*We are grateful to the referees for their useful and constructive comments.     

Specification and testing of multiplicative time-varying GARCH models with applications

In this article, we develop a specification technique for building multiplicative time-varying GARCH models of Amado and Teräsvirta (2008, 2013). The variance is decomposed into an unconditional and a conditional component such that the unconditional variance component is allowed to evolve smoothly over time. This nonstationary component is defined as a linear combination of logistic transition functions with time as the transition variable. The appropriate number of transition functions is determined by a sequence of specification tests. For that purpose, a coherent modelling strategy based on statistical inference is presented. It is heavily dependent on Lagrange multiplier type misspecification tests. The tests are easily implemented as they are entirely based on auxiliary regressions. Finite-sample properties of the strategy and tests are examined by simulation. The modelling strategy is illustrated in practice with two real examples: an empirical application to daily exchange rate returns and another one to daily coffee futures returns.     

A Wavelet-Based Approach of Testing for Granger Causality in the Presence of GARCH Effects

The size and power of the most commonly used tests and a new wavelet-based approach of testing for Granger causality is evaluated by means of a Monte Carlo study in which the error term follows a generalized autoregressive conditional heteroscedasticity consistent (GARCH) process. In the simulation study it is shown that the commonly used causality tests tend to overreject the true null hypothesis in the presence of GARCH errors and that the new wavelet-based approach improves the size properties of the Granger causality test for all of the different situations evaluated.     

Misspecification tests for periodic long memory GARCH models

Distributional theory for Quasi-Maximum Likelihood estimators in long memory conditional heteroskedastic models is not formally defined, even asymptotically. Because of that, this paper analyses the real size and power of the likelihood ratio and the Lagrange multiplier misspecification tests when periodic long memory GARCH models are involved. The performance of these tests is studied by means of Monte Carlo simulations with respect to the class of generalized long memory GARCH models. For this class of models, analytical derivatives are developed. An application to the USD/JPY exchange rate is also provided.     

BL-GARCH models and asymmetries in volatility

In this paper the class of Bilinear GARCH (BL-GARCH) models is proposed. BL-GARCH models allow to capture asymmetries in the conditional variance of financial and economic time series by means of interactions between past shocks and volatilities. The availability of likelihood based inference is an attractive feature of BL-GARCH models. Under the assumption of conditional normality, the log-likelihood function can be maximized by means of an EM type algorithm. The main reason for using the EM algorithm is that it allows to obtain parameter estimates which naturally guarantee the positive definiteness of the conditional variance with no need for additional parameter constraints. We also derive a robust LM test statistic which can be used for model identification. Finally, the effectiveness of BL-GARCH models in capturing asymmetric volatility patterns in financial time series is assessed by means of an application to a time series of daily returns on the NASDAQ Composite stock market index.     

Tests for the multivariate two-sample problem based on empirical probability measures

Nonparametric tests are proposed for the equality of two unknown p-variate distributions. Empirical probability measures are defined from samples from the two distributions and used to construct test statistics as the supremum of the absolute differences between empirical probabilities, the supremum being taken over all possible events. The test statistics are truly multivariate in not requiring the artificial ranking of multivariate observations, and they are distribution-free in the general p-variate case. Asymptotic null distributions are obtained. Powers of the proposed tests and a competitor are examined by Monte Carlo techniques.     

Modelling the persistence of conditional variances

This paper will discuss the current research in building models of conditional variances using the Autoregressive Conditional Heteroskedastic (ARCH) and Generalized ARCH (GARCH) formulations. The discussion will be motivated by a simple asset pricing theory which is particularly appropriate for examining futures contracts with risk averse agents. A new class of models defined to be integrated in variance is then introduced. This new class of models includes the variance analogue of a unit root in the mean as a special case. The models are argued to be both theoretically important for the asset pricing models and empirically relevant. The conditional density is then generalized from a normal to a Student-t with unknown degrees of freedom. By estimating the degrees of freedom, implications about the conditional kurtosis of these models and time aggregated models can be drawn. A further generalization allows the conditional variance to be a non-linear function of the squared innovations. Throughout empirical e imates of the logarithm of the exchange rate between the U.S. dollar and the Swiss franc are presented to illustrate the models.     

Impact study of volatility modelling of Bangladesh stock index using non-normal density

This article examines a wide variety of popular volatility models for stock index return, including the random walk (RW), autoregressive, generalized autoregressive conditional heteroscedasticity (GARCH), and asymmetric GARCH models with normal and non-normal (Student's t and generalized error) distributional assumption. Fitting these models to the Chittagong stock index return data from the period 2 January 1999 to 29 December 2005, we found that the asymmetric GARCH/GARCH model fits better under the assumption of non-normal distribution than under normal distribution. Non-parametric specification tests show that the RW-GARCH, RW-TGARCH, RW-EGARCH, and RW-APARCH models under the Student's t-distributional assumption are significant at the 5% level. Finally, the study suggests that these four models are suitable for the Chittagong Stock Exchange of Bangladesh. We believe that this study would be of great benefit to investors and policy makers at home and abroad.