Evaluating GARCH models

In this paper, a unified framework for testing the adequacy of an estimated GARCH model is presented. Parametric Lagrange multiplier (LM) or LM type tests of no ARCH in standardized errors, linearity, and parameter constancy are proposed. The asymptotic null distributions of the tests are standard, which makes application easy. Versions of the tests that are robust against nonnormal errors are provided. The finite sample properties of the test statistics are investigated by simulation. The robust tests prove superior to the nonrobust ones when the errors are nonnormal. They also compare favourably in terms of power with misspecification tests previously proposed in the literature.     

Multivariate residual‐based finite‐sample tests for serial dependence and ARCH effects with applications to asset pricing models

In this paper, we propose several finite‐sample specification tests for multivariate linear regressions (MLR). We focus on tests for serial dependence and ARCH effects with possibly non‐Gaussian errors. The tests are based on properly standardized multivariate residuals to ensure invariance to error covariances. The procedures proposed provide: (i) exact variants of standard multivariate portmanteau tests for serial correlation as well as ARCH effects, and (ii) exact versions of the diagnostics presented by Shanken ( 1990 ) which are based on combining univariate specification tests. Specifically, we combine tests across equations using a Monte Carlo (MC) test method so that Bonferroni‐type bounds can be avoided. The procedures considered are evaluated in a simulation experiment: the latter shows that standard asymptotic procedures suffer from serious size problems, while the MC tests suggested display excellent size and power properties, even when the sample size is small relative to the number of equations, with normal or Student‐t errors. The tests proposed are applied to the Fama–French three‐factor model. Our findings suggest that the i.i.d. error assumption provides an acceptable working framework once we allow for non‐Gaussian errors within 5‐year sub‐periods, whereas temporal instabilities clearly plague the full‐sample dataset. Copyright © 2009 John Wiley & Sons, Ltd.     

A forecast comparison of volatility models: does anything beat a GARCH(1,1)?

We compare 330 ARCH‐type models in terms of their ability to describe the conditional variance. The models are compared out‐of‐sample using DM–$ exchange rate data and IBM return data, where the latter is based on a new data set of realized variance. We find no evidence that a GARCH(1,1) is outperformed by more sophisticated models in our analysis of exchange rates, whereas the GARCH(1,1) is clearly inferior to models that can accommodate a leverage effect in our analysis of IBM returns. The models are compared with the test for superior predictive ability (SPA) and the reality check for data snooping (RC). Our empirical results show that the RC lacks power to an extent that makes it unable to distinguish ‘good’ and ‘bad’ models in our analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

金融时序数据建模的模型设定问题分析

W．J．Granger与D．F．Hendry（2004）关于建模思路的对话引起了国际计量经济学界关于模型设定问题的争论，本文就这一问题分析讨论了在金融时序数据实证研究中得以广泛应用的ARCH／GARCH模型的设定问题，认为在金融时序数据的建模中，ARMA族模型不宜作为数据生成过程的模型设定，其统计性质也不能直接扩展到ARMA-GARCH族数据生成过程。虽然ARCH／GARCH族模型作为金融时序数据的生成过程有着良好的统计性质，但不宜单纯采用一般到特殊的建模思路，而应是一般到特殊和特殊到一般两种建模思路的结合。ARCH／GARCH族模型的设定应当包含事前检验、事后检验等设定检验步骤。     

Exact and asymptotic tests for possibly non-regular hypotheses on stochastic volatility models

We study the problem of testing hypotheses on the parameters of one- and two-factor stochastic volatility models (SV), allowing for the possible presence of non-regularities such as singular moment conditions and unidentified parameters, which can lead to non-standard asymptotic distributions. We focus on the development of simulation-based exact procedures–whose level can be controlled in finite samples–as well as on large-sample procedures which remain valid under non-regular conditions. We consider Wald-type, score-type and likelihood-ratio-type tests based on a simple moment estimator, which can be easily simulated. We also propose a C(α)-type test which is very easy to implement and exhibits relatively good size and power properties. Besides usual linear restrictions on the SV model coefficients, the problems studied include testing homoskedasticity against a SV alternative (which involves singular moment conditions under the null hypothesis) and testing the null hypothesis of one factor driving the dynamics of the volatility process against two factors (which raises identification difficulties). Three ways of implementing the tests based on alternative statistics are compared: asymptotic critical values (when available), a local Monte Carlo (or parametric bootstrap) test procedure, and a maximized Monte Carlo (MMC) procedure. The size and power properties of the proposed tests are examined in a simulation experiment. The results indicate that the C(α)-based tests (built upon the simple moment estimator available in closed form) have good size and power properties for regular hypotheses, while Monte Carlo tests are much more reliable than those based on asymptotic critical values. Further, in cases where the parametric bootstrap appears to fail (for example, in the presence of identification problems), the MMC procedure easily controls the level of the tests. Moreover, MMC-based tests exhibit relatively good power performance despite the conservative feature of the procedure. Finally, we present an application to a time series of returns on the Standard and Poor’s Composite Price Index.     

Recent Theoretical Results for Time Series Models with GARCH Errors

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA–GARCH are summarized. Various new ARCH–type models, including double threshold ARCH and GARCH, ARFIMA–GARCH, CHARMA and vector ARMA–GARCH, are also reviewed.     

Modeling and forecasting of stock index volatility with APARCH models under ordered restriction

This article examines volatility models for modeling and forecasting the Standard & Poor 500 (S&P 500) daily stock index returns, including the autoregressive moving average, the Taylor and Schwert generalized autoregressive conditional heteroscedasticity (GARCH), the Glosten, Jagannathan and Runkle GARCH and asymmetric power ARCH (APARCH) with the following conditional distributions: normal, Student's t and skewed Student's t‐distributions. In addition, we undertake unit root (augmented Dickey–Fuller and Phillip–Perron) tests, co‐integration test and error correction model. We study the stationary APARCH (p) model with parameters, and the uniform convergence, strong consistency and asymptotic normality are prove under simple ordered restriction. In fitting these models to S&P 500 daily stock index return data over the period 1 January 2002 to 31 December 2012, we found that the APARCH model using a skewed Student's t‐distribution is the most effective and successful for modeling and forecasting the daily stock index returns series. The results of this study would be of great value to policy makers and investors in managing risk in stock markets trading.     

Robust and Powerful Tests for Nonlinear Deterministic Components

We develop a test for the presence of nonlinear deterministic components in a univariate time series, approximated using a Fourier series expansion, designed to be asymptotically robust to the order of integration of the process and to any weak dependence present. We show that our proposed test has uniformly greater local asymptotic power than the existing tests of Harvey, Leybourne and Xiao (2010) when the shocks are I(1), identical local asymptotic power when the shocks are I(0), and also improved finite sample properties. We also consider the issue of determining the number of Fourier frequencies used to specify any nonlinear deterministic components.     

Testing for ARCH in the presence of nonlinearity of unknown form in the conditional mean

Tests of ARCH are a routine diagnostic in empirical econometric and financial analysis. However, it is well known that misspecification of the conditional mean may lead to spurious rejection of the null hypothesis of no ARCH. Nonlinearity is a prime example of this phenomenon. There is little work on the extent of the effect of neglected nonlinearity on the properties of ARCH tests. We investigate this using new ARCH testing procedures that are robust to the presence of neglected nonlinearity. Monte Carlo evidence shows that the problem is serious and that the new methods alleviate this problem to a very large extent. We apply the new tests to exchange rate data and find substantial evidence of spurious rejection of the null hypothesis of no ARCH.     

Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations

ARCH and GARCH models are widely used to model financial market volatilities in risk management applications. Considering a GARCH model with heavy-tailed innovations, we characterize the limiting distribution of an estimator of the conditional value-at-risk (VaR), which corresponds to the extremal quantile of the conditional distribution of the GARCH process. We propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. Finally, we apply the proposed approach to an energy market data set.     

Testing for a break in trend when the order of integration is unknown

Harvey, Leybourne and Taylor [Harvey, D.I., Leybourne, S.J., Taylor, A.M.R. 2009. Simple, robust and powerful tests of the breaking trend hypothesis. Econometric Theory 25, 995–1029] develop a test for the presence of a broken linear trend at an unknown point in the sample whose size is asymptotically robust as to whether the (unknown) order of integration of the data is either zero or one. This test is not size controlled, however, when this order assumes fractional values; its asymptotic size can be either zero or one in such cases. In this paper we suggest a new test, based on a sup-Wald statistic, which is asymptotically size-robust across fractional values of the order of integration (including zero or one). We examine the asymptotic power of the test under a local trend break alternative. The finite sample properties of the test are also investigated.     

Power of Tests for Unit Roots in the Presence of a Linear Trend*

Dickey and Fuller [Econometrica (1981) Vol. 49, pp. 1057–1072] suggested unit‐root tests for an autoregressive model with a linear trend conditional on an initial observation. TPower of tests for unit roots in the presence of a linear trendightly different model with a random initial value in which nuisance parameters can easily be eliminated by an invariant reduction of the model. We show that invariance arguments can also be used when comparing power within a conditional model. In the context of the conditional model, the Dickey–Fuller test is shown to be more stringent than a number of unit‐root tests motivated by models with random initial value. The power of the Dickey–Fuller test can be improved by making assumptions to the initial value. The practitioner therefore has to trade‐off robustness and power, as assumptions about initial values are hard to test, but can give more power.     

A unified approach to standardized‐residuals‐based correlation tests for GARCH‐type models

In this paper, we propose a unified approach to generating standardized‐residuals‐based correlation tests for checking GARCH‐type models. This approach is valid in the presence of estimation uncertainty, is robust to various standardized error distributions, and is applicable to testing various types of misspecifications. By using this approach, we also propose a class of power‐transformed‐series (PTS) correlation tests that provides certain robustifications and power extensions to the Box–Pierce, McLeod–Li, Li–Mak, and Berkes–Horváth–Kokoszka tests in diagnosing GARCH‐type models. Our simulation and empirical example show that the PTS correlation tests outperform these existing autocorrelation tests in financial time series analysis. Copyright © 2008 John Wiley & Sons, Ltd.     

Evaluating interval forecasts of high‐frequency financial data

A number of methods of evaluating the validity of interval forecasts of financial data are analysed, and illustrated using intraday FTSE100 index futures returns. Some existing interval forecast evaluation techniques, such as the Markov chain approach of Christoffersen ( 1998 ), are shown to be inappropriate in the presence of periodic heteroscedasticity. Instead, we consider a regression‐based test, and a modified version of Christoffersen's Markov chain test for independence, and analyse their properties when the financial time series exhibit periodic volatility. These approaches lead to different conclusions when interval forecasts of FTSE100 index futures returns generated by various GARCH(1,1) and periodic GARCH(1,1) models are evaluated. Copyright © 2003 John Wiley & Sons, Ltd.     

A non-local perspective on the power properties of the CUSUM and CUSUM of squares tests for structural change

We consider the power properties of the CUSUM and CUSUM of squares (CUSQ) tests in the presence of a one-time change in the parameters of a linear regression model. A result due to Ploberger and Krämer [1990. The local power of the cusum and cusum of squares tests. Econometric Theory 6, 335–347.] is that the CUSQ test has only trivial asymptotic local power in this case, while the CUSUM test has non-trivial local asymptotic power unless the change is orthogonal to the mean regressor. The main theme of the paper is that such conclusions obtained from a local asymptotic framework are not reliable guides to what happens in finite samples. The approach we take is to derive expansions of the test statistics that retain terms related to the magnitude of the change under the alternative hypothesis. This enables us to analyze what happens for non-local to zero breaks. Our theoretical results are able to explain how the power function of the tests can be drastically different depending on whether one deals with a static regression with uncorrelated errors, a static regression with correlated errors, a dynamic regression with lagged dependent variables, or whether a correction for non-normality is applied in the case of the CUSQ. We discuss in which cases the tests are subject to a non-monotonic power function that goes to zero as the magnitude of the change increases, and uncover some curious properties. All theoretical results are verified to yield good guides to the finite sample power through simulation experiments. We finally highlight the practical importance of our results.     

Conditional heteroscedasticity of exchange rates: further results based on the fractionally integrated approach

A recent article (Tse, 1998 ) published in this journal analysed the conditional heteroscedasticity of the yen–dollar exchange rate based on the fractionally integrated asymmetric power ARCH model. In this paper, we present replication results using Tse's ( 1998 ) yen–dollar series. We also examine the robustness of Tse's ( 1998 ) findings across different currencies, sample periods and non‐nested GARCH‐type models. Unlike Tse ( 1998 ), we find some evidence of asymmetric conditional volatility for daily returns of currencies measured against the dollar or the yen. Copyright © 2004 John Wiley & Sons, Ltd.     

A Note on the Power of Money-Output Causality Tests

This study suggests that some empirical findings against money-output causality can be the consequence of ignoring autoregressive conditional heteroskedastic (ARCH) errors. Monte Carlo results confirm that ARCH effects drastically reduce the power of the standard causality test. The maximum likelihood approach allowing for ARCH effects, on the other hand, provides a good power performance. Using different specifications and sample period, Friedman and Kuttner (1993) and Thomas (1994) report limited evidence of money causing output. We detect significant ARCH effects in the models considered by these studies. Once ARCH effects are explicitly accounted for, we find that the monetary effect is significant though its magnitude is quite small.     

金融时间序列中加性异常值的鉴别与校正

借鉴Franses and Ghijsel[1](1999)和Charles and Darne[2](2005)提出的鉴别和校正金融序列加性异常值的方法,以GARCH模型为例,对我国的上证综合指数和深圳成分指数进行了加性异常值的鉴定与校正,并对校正后的残差进行了正态检验。结果表明该方法效果显著,进行异常值校正后的GARCH(1,1),更好地拟合金融时间序列中的尖峰厚尾和波动丛聚性的特性,纠正了正态分布的GARCH(1,1)对时间序列拟合的偏误。     

Time series properties of ARCH processes with persistent covariates

We investigate the time series properties of a volatility model, whose conditional variance is specified as in ARCH with an additional persistent covariate. The included covariate is assumed to be an integrated or nearly integrated process, with its effect on volatility given by a wide class of nonlinear volatility functions. In the paper, such a model is shown to generate many important characteristics that are commonly observed in financial time series. In particular, the model yields persistence in volatility, and also well predicts leptokurtosis. This is true for any type of volatility functions considered in the paper, as long as the covariate is integrated or nearly integrated. Stationary covariates cannot produce important characteristics observed in many financial time series. We present two empirical applications of the model, which show that the default premium (the yield spread between Baa and Aaa corporate bonds) affects stock return volatility and the interest rate differential between two countries accounts for exchange rate return volatility. The forecast evaluation shows that the model generally outperforms GARCH and FIGARCH at relatively lower frequencies.     

Testing for unit roots in the possible presence of multiple trend breaks using minimum Dickey–Fuller statistics

Trend breaks appear to be prevalent in macroeconomic time series, and unit root tests therefore need to make allowance for these if they are to avoid the serious effects that unmodelled trend breaks have on power. Carrion-i-Silvestre et al. (2009) propose a pre-test-based approach which delivers near asymptotically efficient unit root inference both when breaks do not occur and where multiple breaks occur, provided the break magnitudes are fixed. Unfortunately, however, the fixed magnitude trend break asymptotic theory does not predict well the finite sample power functions of these tests, and power can be very low for the magnitudes of trend breaks typically observed in practice. In response to this problem we propose a unit root test that allows for multiple breaks in trend, obtained by taking the infimum of the sequence (across all candidate break points in a trimmed range) of local GLS detrended augmented Dickey–Fuller-type statistics. We show that this procedure has power that is robust to the magnitude of any trend breaks, thereby retaining good finite sample power in the presence of plausibly-sized breaks. We also demonstrate that, unlike the OLS detrended infimum tests of Zivot and Andrews (1992), these tests display no tendency to spuriously reject in the limit when fixed magnitude trend breaks occur under the unit root null.