Normal mixture GARCH(1,1): applications to exchange rate modelling

Some recent specifications for GARCH error processes explicitly assume a conditional variance that is generated by a mixture of normal components, albeit with some parameter restrictions. This paper analyses the general normal mixture GARCH(1,1) model which can capture time variation in both conditional skewness and kurtosis. A main focus of the paper is to provide evidence that, for modelling exchange rates, generalized two‐component normal mixture GARCH(1,1) models perform better than those with three or more components, and better than symmetric and skewed Student's t‐GARCH models. In addition to the extensive empirical results based on simulation and on historical data on three US dollar foreign exchange rates (British pound, euro and Japanese yen), we derive: expressions for the conditional and unconditional moments of all models; parameter conditions to ensure that the second and fourth conditional and unconditional moments are positive and finite; and analytic derivatives for the maximum likelihood estimation of the model parameters and standard errors of the estimates. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Forecasting in GARCH models with polynomially modified innovations

Orthogonal polynomials can be used to modify the moments of the distribution of a random variable. In this paper, polynomially adjusted distributions are employed to model the skewness and kurtosis of the conditional distributions of GARCH models. To flexibly capture the skewness and kurtosis of data, the distributions of the innovations that are polynomially reshaped include, besides the Gaussian, also leptokurtic laws such as the logistic and the hyperbolic secant. Modeling GARCH innovations with polynomially adjusted distributions can effectively improve the precision of the forecasts. This strategy is analyzed in GARCH models with different specifications for the conditional variance, such as the APARCH, the EGARCH, the Realized GARCH, and APARCH with time-varying skewness and kurtosis. An empirical application on different types of asset returns shows the good performance of these models in providing accurate forecasts according to several criteria based on density forecasting, downside risk, and volatility prediction.     

Robust Likelihood Methods Based on the Skew-t and Related Distributions

The robustness problem is tackled by adopting a parametric class of distributions flexible enough to match the behaviour of the observed data. In a variety of practical cases, one reasonable option is to consider distributions which include parameters to regulate their skewness and kurtosis. As a specific representative of this approach, the skew‐t distribution is explored in more detail and reasons are given to adopt this option as a sensible general‐purpose compromise between robustness and simplicity, both of treatment and of interpretation of the outcome. Some theoretical arguments, outcomes of a few simulation experiments and various wide‐ranging examples with real data are provided in support of the claim.     

Skewness Risk and Bond Prices

This paper uses extreme value theory to study the implications of skewness risk for nominal loan contracts in a production economy. Productivity and inflation innovations are drawn from generalized extreme value distributions. The model is solved using a third‐order perturbation and estimated by the simulated method of moments. Results show that the data reject the hypothesis that innovations are drawn from normal distributions and favor instead the alternative that they are drawn from asymmetric distributions. Estimates indicate that skewness risk accounts for 12% of the risk premia and reduces bond yields by approximately 55 basis points. For a bond that pays 1 dollar at maturity, the adjustment factor associated with skewness risk ranges from 0.15 cents for a 3‐month bond to 2.05 cents for a 5‐year bond. Copyright © 2016 John Wiley & Sons, Ltd.     

Measuring Spot Variance Spillovers when (Co)variances are Time‐varying – The Case of Multivariate GARCH Models

We propose global and disaggregated spillover indices that allow us to assess variance and covariance spillovers, locally in time and conditionally on time‐t information. Key to our approach is the vector moving average representation of the half‐vectorized ‘squared’ multivariate GARCH process of the popular BEKK model. In an empirical application to a four‐dimensional system of broad asset classes (equity, fixed income, foreign exchange and commodities), we illustrate the new spillover indices at various levels of (dis)aggregation. Moreover, we demonstrate that they are informative of the value‐at‐risk violations of portfolios composed of the considered asset classes.     

Exploiting tail shape biases to discriminate between stable and student t alternatives

The nonnormal stable laws and Student t distributions are used to model the unconditional distribution of financial asset returns, as both models display heavy tails. The relevance of the two models is subject to debate because empirical estimates of the tail shape conditional on either model give conflicting signals. This stems from opposing bias terms. We exploit the biases to discriminate between the two distributions. A sign estimator for the second‐order scale parameter strengthens our results. Tail estimates based on asset return data match the bias induced by finite‐variance unconditional Student t data and the generalized autoregressive conditional heteroscedasticity process.     

Autoregresive conditional volatility, skewness and kurtosis

This paper proposes a GARCH-type model allowing for time-varying volatility, skewness and kurtosis. The model is estimated assuming a Gram–Charlier (GC) series expansion of the normal density function for the error term, which is easier to estimate than the non-central t distribution proposed by [Harvey, C. R. & Siddique, A. (1999). Autorregresive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–487). Moreover, this approach accounts for time-varying skewness and kurtosis while the approach by Harvey and Siddique [Harvey, C. R. & Siddique, A. (1999). Autorregresive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–487] only accounts for non-normal skewness. We apply this method to daily returns of a variety of stock indices and exchange rates. Our results indicate a significant presence of conditional skewness and kurtosis. It is also found that specifications allowing for time-varying skewness and kurtosis outperform specifications with constant third and fourth moments.     

International foreign exchange agreements and nominal exchange rate volatility: a GARCH application

For eight major national currencies, this study estimates, and tests several hypotheses with, a t-distribution GARCH model of daily spot nominal exchange rate changes. The sample period covered is June 1, 1982 through September 30, 1992. By using likelihood ratio and parameter stability tests, it finds that for most of the currencies considered, both the conditional means and variances of unexpected exchange rate changes experienced statistically significant structural breaks across the five subperiods that are associated with four episodes of international foreign-exchange policy coordination. The study also finds that the orderings of the GARCH-estimated unconditional standard deviations roughly match the orderings of the sample standard deviations across the five subperiods. An explanation is provided for what underlying factors contributed to these structural shifts.     

A Portfolio Index GARCH model

This paper develops the structure of a parsimonious Portfolio Index (PI) GARCH model. Unlike the conventional approach to Portfolio Index returns, which employs the univariate ARCH class, the PI-GARCH approach incorporates the effects on individual assets, leading to a better understanding of portfolio risk management, and achieves greater accuracy in forecasting Value-at-Risk (VaR) thresholds. For various asymmetric GARCH models, a Portfolio Index Composite News Impact Surface (PI-CNIS) is developed to measure the effects of news on the conditional variances. The paper also investigates the finite sample properties of the PI-GARCH model. The empirical example shows that the asymmetric PI-GARCH-t model outperforms the GJR-t model and the filtered historical simulation with a t distribution in forecasting VaR thresholds.     

Forecasting realized volatility: a Bayesian model‐averaging approach

How to measure and model volatility is an important issue in finance. Recent research uses high‐frequency intraday data to construct ex post measures of daily volatility. This paper uses a Bayesian model‐averaging approach to forecast realized volatility. Candidate models include autoregressive and heterogeneous autoregressive specifications based on the logarithm of realized volatility, realized power variation, realized bipower variation, a jump and an asymmetric term. Applied to equity and exchange rate volatility over several forecast horizons, Bayesian model averaging provides very competitive density forecasts and modest improvements in point forecasts compared to benchmark models. We discuss the reasons for this, including the importance of using realized power variation as a predictor. Bayesian model averaging provides further improvements to density forecasts when we move away from linear models and average over specifications that allow for GARCH effects in the innovations to log‐volatility. Copyright © 2009 John Wiley & Sons, Ltd.     

GARCH models,tail indexes and error distributions: An empirical investigation

We perform a large simulation study to examine the extent to which various generalized autoregressive conditional heteroskedasticity (GARCH) models capture extreme events in stock market returns. We estimate Hill's tail indexes for individual S&P 500 stock market returns and compare these to the tail indexes produced by simulating GARCH models. Our results suggest that actual and simulated values differ greatly for GARCH models with normal conditional distributions, which underestimate the tail risk. By contrast, the GARCH models with Student's t conditional distributions capture the tail shape more accurately, with GARCH and GJR-GARCH being the top performers.     

HEGY Tests in the Presence of Moving Averages*

We analyze the asymptotic distributions associated with the seasonal unit root tests of Hylleberg et al. (1990) for quarterly data when the innovations follow a moving average process. Although both the t‐ and F‐type tests suffer from scale and shift effects compared with the presumed null distributions when a fixed order of autoregressive augmentation is applied, these effects disappear when the order of augmentation is sufficiently large. However, as found by Burridge and Taylor (2001) for the autoregressive case, individual t‐ratio tests at the semi‐annual frequency are not pivotal even with high orders of augmentation, although the corresponding joint F‐type statistic is pivotal. Monte Carlo simulations verify the importance of the order of augmentation for finite samples generated by seasonally integrated moving average processes.     

Parametric Value-at-Risk analysis: Evidence from stock indices

We evaluate the performance of several volatility models in estimating one-day-ahead Value-at-Risk (VaR) of seven stock market indices using a number of distributional assumptions. Because all returns series exhibit volatility clustering and long range memory, we examine GARCH-type models including fractionary integrated models under normal, Student-t and skewed Student-t distributions. Consistent with the idea that the accuracy of VaR estimates is sensitive to the adequacy of the volatility model used, we find that AR (1)-FIAPARCH (1,d,1) model, under a skewed Student-t distribution, outperforms all the models that we have considered including widely used ones such as GARCH (1,1) or HYGARCH (1,d,1). The superior performance of the skewed Student-t FIAPARCH model holds for all stock market indices, and for both long and short trading positions. Our findings can be explained by the fact that the skewed Student-t FIAPARCH model can jointly accounts for the salient features of financial time series: fat tails, asymmetry, volatility clustering and long memory. In the same vein, because it fails to account for most of these stylized facts, the RiskMetrics model provides the least accurate VaR estimation. Our results corroborate the calls for the use of more realistic assumptions in financial modeling.     

Stochastic volatility models: conditional normality versus heavy‐tailed distributions

Most of the empirical applications of the stochastic volatility (SV) model are based on the assumption that the conditional distribution of returns, given the latent volatility process, is normal. In this paper, the SV model based on a conditional normal distribution is compared with SV specifications using conditional heavy‐tailed distributions, especially Student's t‐distribution and the generalized error distribution. To estimate the SV specifications, a simulated maximum likelihood approach is applied. The results based on daily data on exchange rates and stock returns reveal that the SV model with a conditional normal distribution does not adequately account for the two following empirical facts simultaneously: the leptokurtic distribution of the returns and the low but slowly decaying autocorrelation functions of the squared returns. It is shown that these empirical facts are more adequately captured by an SV model with a conditional heavy‐tailed distribution. It also turns out that the choice of the conditional distribution has systematic effects on the parameter estimates of the volatility process. Copyright © 2000 John Wiley & Sons, Ltd.     

Diagnostic analysis for a vector autoregressive model under Student′s t‐distributions

In this paper, we use the local influence method to study a vector autoregressive model under Student^′s t‐distributions. We present the maximum likelihood estimators and the information matrix. We establish the normal curvature diagnostics for the vector autoregressive model under three usual perturbation schemes for identifying possible influential observations. The effectiveness of the proposed diagnostics is examined by a simulation study, followed by our data analysis using the model to fit the weekly log returns of Chevron stock and the Standard & Poor's 500 Index as an application.     

Daily House Price Indices: Construction,Modeling, and Longer‐run Predictions

We construct daily house price indices for 10 major US metropolitan areas. Our calculations are based on a comprehensive database of several million residential property transactions and a standard repeat‐sales method that closely mimics the methodology of the popular monthly Case–Shiller house price indices. Our new daily house price indices exhibit dynamic features similar to those of other daily asset prices, with mild autocorrelation and strong conditional heteroskedasticity of the corresponding daily returns. A relatively simple multivariate time series model for the daily house price index returns, explicitly allowing for commonalities across cities and GARCH effects, produces forecasts of longer‐run monthly house price changes that are superior to various alternative forecast procedures based on lower‐frequency data. Copyright © 2015 John Wiley & Sons, Ltd.     

Modelling dependence using skew t copulas: Bayesian inference and applications

We construct a copula from the skew t distribution of Sahu et al. ( 2003 ). This copula can capture asymmetric and extreme dependence between variables, and is one of the few copulas that can do so and still be used in high dimensions effectively. However, it is difficult to estimate the copula model by maximum likelihood when the multivariate dimension is high, or when some or all of the marginal distributions are discrete‐valued, or when the parameters in the marginal distributions and copula are estimated jointly. We therefore propose a Bayesian approach that overcomes all these problems. The computations are undertaken using a Markov chain Monte Carlo simulation method which exploits the conditionally Gaussian representation of the skew t distribution. We employ the approach in two contemporary econometric studies. The first is the modelling of regional spot prices in the Australian electricity market. Here, we observe complex non‐Gaussian margins and nonlinear inter‐regional dependence. Accurate characterization of this dependence is important for the study of market integration and risk management purposes. The second is the modelling of ordinal exposure measures for 15 major websites. Dependence between websites is important when measuring the impact of multi‐site advertising campaigns. In both cases the skew t copula substantially outperforms symmetric elliptical copula alternatives, demonstrating that the skew t copula is a powerful modelling tool when coupled with Bayesian inference. Copyright © 2010 John Wiley & Sons, Ltd.     

A dynamic measure of inaccuracy between two past lifetime distributions

In the present communication we introduce a dynamic measure of inaccuracy between two past lifetime distributions over the interval (0, t). Based on proportional reversed hazard rate model (PRHRM), a characterization problem for this dynamic inaccuracy measure has been studied. An upper bound to the dynamic measure of inaccuracy H*(f, g; t) has also been derived.     

Structural breaks and GARCH models of exchange rate volatility

We investigate the empirical relevance of structural breaks for GARCH models of exchange rate volatility using both in‐sample and out‐of‐sample tests. We find significant evidence of structural breaks in the unconditional variance of seven of eight US dollar exchange rate return series over the 1980–2005 period—implying unstable GARCH processes for these exchange rates—and GARCH(1,1) parameter estimates often vary substantially across the subsamples defined by the structural breaks. We also find that it almost always pays to allow for structural breaks when forecasting exchange rate return volatility in real time. Combining forecasts from different models that accommodate structural breaks in volatility in various ways appears to offer a reliable method for improving volatility forecast accuracy given the uncertainty surrounding the timing and size of the structural breaks. Copyright © 2008 John Wiley & Sons, Ltd.