GARCH Errors Research Articles

The Efficient Market Hypothesis Re-Visited: New Evidence from 100 US Firms

In this paper, we test the efficient market hypothesis for 100 US firms listed on the New York Stock Exchange. To test the unit root null hypothesis, we develop a generalized autoregressive heteroskedasticity (GARCH) model that not only caters for the GARCH errors but also allows for two endogenous structural breaks in the data series. We study the size and power properties of the proposed GARCH structural break unit root test and find that it statistically performs well in finite samples. We find that only 22% of firms have a stationary stock price series.

INTERNATIONAL OUTPUT AND INFLATION UNCERTAINTY AND THEIR IMPACT ON COUNTRIES' MACROECONOMIC PERFORMANCE. EVIDENCE FROM A DYNAMIC FACTOR GARCH-IN-MEAN MODEL

We identify international output and inflation uncertainty and analyze their impact on individual countries' macroeconomic performance. Output and inflation uncertainty on an international level is measured through the conditional variances of common factors in inflation and output growth, estimated from a bivariate dynamic factor model with GARCH errors. The impact of international and country-specific uncertainty is analyzed by including the conditional variances as regressors. We find increases in uncertainty during the first and second oil crisis, the 1980s and 1990s recessions as well as the recent Great Recession to be confined to the international level. The effect of international uncertainty results to be highly significant and unambiguously negative on countries' output growth and inflation rates whereas the impact of country-specific uncertainty is very mixed.

Asymptotic properties of quasi-maximum likelihood estimators in observation-driven time series models

We study a general class of quasi-maximum likelihood estimators for observation-driven time series models. Our main focus is on models related to the exponential family of distributions like Poisson based models for count time series or duration models. However, the proposed approach is more general and covers a variety of time series models including the ordinary GARCH model which has been studied extensively in the literature. We provide general conditions under which quasi-maximum likelihood estimators can be analyzed for this class of time series models and we prove that these estimators are consistent and asymptotically normally distributed regardless of the true data generating process. We illustrate our results using classical examples of quasi-maximum likelihood estimation including standard GARCH models, duration models, Poisson type autoregressions and ARMA models with GARCH errors. Our contribution unifies the existing theory and gives conditions for proving consistency and asymptotic normality in a variety of situations.

Bootstrap-based unit root tests for higher order autoregressive models with GARCH(1, 1) errors

ABSTRACTBootstrap-based unit root tests are a viable alternative to asymptotic distribution-based procedures and, in some cases, are preferable because of the serious size distortions associated with the latter tests under certain situations. While several bootstrap-based unit root tests exist for autoregressive moving average processes with homoskedastic errors, only one such test is available when the innovations are conditionally heteroskedastic. The details for the exact implementation of this procedure are currently available only for the first order autoregressive processes. Monte-Carlo results are also published only for this limited case. In this paper we demonstrate how this procedure can be extended to higher order autoregressive processes through a transformed series used in augmented Dickey–Fuller unit root tests. We also investigate the finite sample properties for higher order processes through a Monte-Carlo study. Results show that the proposed tests have reasonable power and size properties.

Recent Developments in Recursive Estimation for Time Series Models

Recently there has been a growing interest in joint estimation of the location and scale parameters using combined estimation functions. Combined estimating functions had been studied in Liang et al. (2011) for models with finite variance errors and in Thavaneswaran et al. (2013) for models with infinite variance<br />stable errors. In this paper, first a theorem on recursive estimation based on estimating functions is extended to multi-parameter setup and it is shown that the unified approach can be used to estimate the location parameter recursively for models with finite variance/infinite variance errors. The method is applied for the joint estimation of the location and scale parameters for regression models with ARCH errors and RCA models with GARCH errors.

GEL estimation for heavy-tailed GARCH models with robust empirical likelihood inference

We construct a Generalized Empirical Likelihood estimator for a GARCH(1, 1) model with a possibly heavy tailed error. The estimator imbeds tail-trimmed estimating equations allowing for over-identifying conditions, asymptotic normality, efficiency and empirical likelihood based confidence regions for very heavy-tailed random volatility data. We show the implied probabilities from the tail-trimmed Continuously Updated Estimator elevate weight for usable large values, assign large but not maximum weight to extreme observations, and give the lowest weight to non-leverage points. We derive a higher order expansion for GEL with imbedded tail-trimming (GELITT), which reveals higher order bias and efficiency properties, available when the GARCH error has a finite second moment. Higher order asymptotics for GEL without tail-trimming requires the error to have moments of substantially higher order. We use first order asymptotics and higher order bias to justify the choice of the number of trimmed observations in any given sample. We also present robust versions of Generalized Empirical Likelihood Ratio, Wald, and Lagrange Multiplier tests, and an efficient and heavy tail robust moment estimator with an application to expected shortfall estimation. Finally, we present a broad simulation study for GEL and GELITT, and demonstrate profile weighted expected shortfall for the Russian Ruble–US Dollar exchange rate. We show that tail-trimmed CUE-GMM dominates other estimators in terms of bias, mse and approximate normality.

Change Point Analysis of Exchange Rates Using Bootstrapping Methods: An Application to the Indonesian Rupiah 2000–2008

In this paper, we investigate detecting single change point under time series regression model with GARCH errors using the cumulative sum of squares of the least squares residuals test and the log-likelihood ratio test. Furthermore we think it is important to calculate confidence interval for an estimated change point, for which we need to know the sampling distribution of the estimated change point. We obtain the sampling distribution to calculate confidence interval using Monte Carlo simulation based on a circular block bootstrap method and verify the performance of the above break point tests by Monte Carlo experiment. Then we detect a change point in the exchange rate of Indonesian Rupiah (IDR) using the above test to detect. The Government of Indonesia officially announced (de jure) to adopt a floating exchange rate regime in August 1997. However, from time to time, Bank Indonesia nevertheless maintains the stability of rupiah value in the market. Since there is no official information regarding on central bank’s intervention in the foreign exchange market, therefore detecting a structural change in the time series of the exchange market can be used as an indicator of exchange rate management. Our real data analysis shows that the IDR had been moving with the USD since 2000, but that the direction of the relationship changed in March 2002. This indicates that there was some control over the Rupiah’s movement.

Intraday Volatility: An Integer Autoregressive Model

Intraday and high frequency time series are mostly defined by a non-continuous prices process. This paper introduces an integer based ARMA model found to be a better predictor for absolute intraday price changes than continuous time estimators (such as GARCH or multiplicative error models). Using transactions data on the E-Mini S&P 500, we provide a forecasting model for absolute five and ten minutes price variations that over performs a number of other models.

A Bayesian Analysis of Autoregressive Models with Exogenous Variables and Power-Transformed and Threshold GARCH Errors

Consider a class of autoregressive models with exogenous variables and power transformed and threshold GARCH (ARX-PTTGARCH) errors, which is a natural generalization of the standard and special GARCH model. We propose a Bayesian method to show that combining Gibbs sampler and Metropolis-Hastings algorithm to give a Bayesian analysis can be applied to estimate parameters of ARX-PTTGARCH models with success.

Wild bootstrap testing for cointegration in an ESTAR error correction model

This paper proposes a bootstrap cointegration test in an exponential smooth transition autoregressive (ESTAR) error correction model (ECM). We introduce the t-type test that has a null hypothesis of no cointegration and an alternative hypothesis of the ESTAR-ECM and develop the test using a wild bootstrap. Monte Carlo simulation results show that the t-type test and conventional tests considerably overreject the null hypothesis of no cointegration in the presence of multivariate GARCH errors and multivariate stochastic volatility, whereas the proposed wild bootstrap test has correct empirical sizes, in contrast to other tests, and reasonable powers under cointegration with the ESTAR-ECM.

On Diagnostic Checking of Vector ARMA-GARCH Models with Gaussian and Student-t Innovations

This paper focuses on the diagnostic checking of vector ARMA (VARMA) models with multivariate GARCH errors. For a fitted VARMA-GARCH model with Gaussian or Student-t innovations, we derive the asymptotic distributions of autocorrelation matrices of the cross-product vector of standardized residuals. This is different from the traditional approach that employs only the squared series of standardized residuals. We then study two portmanteau statistics, called Q1(M) and Q2(M), for model checking. A residual-based bootstrap method is provided and demonstrated as an effective way to approximate the diagnostic checking statistics. Simulations are used to compare the performance of the proposed statistics with other methods available in the literature. In addition, we also investigate the effect of GARCH shocks on checking a fitted VARMA model. Empirical sizes and powers of the proposed statistics are investigated and the results suggest a procedure of using jointly Q1(M) and Q2(M) in diagnostic checking. The bivariate time series of FTSE 100 and DAX index returns is used to illustrate the performance of the proposed portmanteau statistics. The results show that it is important to consider the cross-product series of standardized residuals and GARCH effects in model checking.

Measuring time-varying financial market integration: An unobserved components approach

We measure the time-varying degree of world stock market integration of five developed countries (Germany, France, UK, US, and Japan) over the period 1970:1–2011:10. Time-varying financial market integration of each country is measured through the conditional variances of the country-specific and common international risk premiums in equity excess returns. The country-specific and common risk premiums and their conditional variances are estimated from a latent factor decomposition through the use of state space methods that allow for GARCH errors. Our empirical results suggest that stock market integration has increased over the period 1970:1–2011:10 in all countries but Japan. And while there is a structural increase in stock market integration in four out of five countries, all countries also exhibit several shorter periods of disintegration (reversals), i.e. periods in which country-specific shocks play a more dominant role. Hence, stock market integration is measured as a dynamic process that is fluctuating in the short run while gradually increasing in the long run.

Arbitrary initial values and random norm for explosive AR(1) processes generated by stationary errors

This article is concerned with a broad class of explosive AR(1) models. Allowing stationary dependence on the error process, we do not restrict ourselves to independent and identically distributed errors. The model accommodates, as special cases, GARCH errors, AR(1) errors and Gaussian ARMA errors. The error distribution is permitted to be non-normal. To circumvent the effect of initial values, the limit distribution of the least squares estimate using a random norm (rather than a constant norm) is derived. It is shown that the limit distribution using a random norm is free from the initial value provided the error is symmetrically distributed about zero.

Estimation of SEM with GARCH errors

The interdependence of financial markets combined with their volatility make the multivariate GARCH model a suitable econometric framework for analysing their behaviour. However, the non-availability of analytical derivatives in a general context and the computational heaviness resulting from a numerical calculation still represent a major hurdle for the use of such models in practical applications. In a general simultaneous equation model with multivariate GARCH errors, analytical expressions of the score, the Hessian and the information matrices are derived and used for implementing QML and GMM estimation procedures. The asymptotic variances of these estimators are obtained using the same expressions and the asymptotic superiority of GMM over QML is shown in the non-normal case. A simulation study comparing different gradient algorithms for ML as well as the finite sample behaviour of ML and GMM shows that using analytical results instead of numerical approximations in the optimisation procedure yields better results and reiterates the superiority of GMM over QML in finite samples under non-normality.

The influence of heteroskedastic variances on cointegration tests: A comparison using Monte Carlo simulations

This paper investigates the influence of heteroskedastic variances on cointegration tests. The Monte Carlo simulation results show that cointegration tests allowing for threshold adjustments or structural breaks overreject the null hypothesis of no cointegration in the presence of GARCH errors and variance breaks. In particular, multivariate GARCH and bivariate variance breaks cause severe size distortions in such cointegration tests. On the other hand, a variance ratio cointegration test yields reasonable empirical sizes under most cases of heteroskedastic variances, as compared to other tests including standard cointegration tests.

On the Oversized Problem of Dickey–Fuller-Type Tests with GARCH Errors

This article examines the finite-sample size of a class of Dickey–Fuller-type tests in the presence of GARCH errors, with and without the influence of initial conditions of the underlying simulated path. Oversizing is observed for all tests when the GARCH process is nearly degenerate and the volatility parameter is large, but the degree of size distortion varies across tests and is contingent on the initial condition. The result due to the initial effect is linked to the size distortion caused by a sequence of small downward variance breaks arising in the early stage of the underlying process.

Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA–GARCH/IGARCH models

This paper investigates the asymptotic theory of the quasi-maximum exponential likelihood estimators (QMELE) for ARMA--GARCH models. Under only a fractional moment condition, the strong consistency and the asymptotic normality of the global self-weighted QMELE are obtained. Based on this self-weighted QMELE, the local QMELE is showed to be asymptotically normal for the ARMA model with GARCH (finite variance) and IGARCH errors. A formal comparison of two estimators is given for some cases. A simulation study is carried out to assess the performance of these estimators, and a real example on the world crude oil price is given.

Unit root testing in the presence of ARFIMA–GARCH errors

AbstractWe consider asymptotic behavior of self‐normalized sums of autoregressive fractionally integrated moving average (ARFIMA) processes whose innovations are GARCH errors. The asymptotic distribution of the sums is derived under very mild conditions. Applications to unit root tests with ARFIMA–GARCH errors are discussed. It is shown that even when the errors exhibit both long‐range dependence and heavy‐tailed conditional heteroscedasticity, the asymptotic distributions of the Dickey–Fuller ρ‐type tests are functionals of standard Brownian motion rather than those of fractional Brownian motions. Some Monte Carlo simulations are provided to illustrate the finite sample properties of two of the tests. Copyright © 2010 John Wiley & Sons, Ltd.

PARAMETER ESTIMATION IN NONLINEAR AR–GARCH MODELS

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first-order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi-maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.

Parameter Estimation in Nonlinear AR-GARCH Models

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a functional coefficient autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. Strong consistency and asymptotic normality of the global Gaussian quasi maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.