General GARCH Research Articles

COVID-19 and Seasonality in Monthly Returns: a Firm Level Analysis of PSX

Abstract The current study scrutinizes the calendar anomalies in the context of the local market by analyzing the Pakistan Stock Exchange (PSX). For this purpose, closing prices of KSE-100, KSE-30 and KSE-All share Index from January, 2009 to June, 2021 have been used as well as a thorough individual firm level analysis is done, taking average log-returns of selected sample firms returns using OLS regression, general GARCH (1,1), asymmetric TGARCH and PGARCH models. The results indicate monthly seasonality, with significant April, July, and September effect in PSX indices returns. The findings of the study reveal that weak form inefficiency exists in Pakistan Stock Market, which implies the possibility of earning abnormal returns by investors using timing strategies. Due to global pandemic conditions, investor psychology investors turned circumspect. Consequently, the individual firms’ trading has also reduced.

Testing Daily Seasonality Using Value Premium Portfolios Returns: The Case of Pakistan

Calendar anomalies are a well-documented phenomena in financial markets. The current study scrutinized calendar anomalies in the context of the local market by analyzing the Pakistan stock exchange. The data from the listed companies of PSX have been considered to test for seasonality in value premium portfolios using OLS regression, general GARCH (1,1), TGARCH, GJR-GARCH, PGARCH, and EGARCH models. The findings suggest that return seasonality among different value premium portfolios explains the economically and statistically significant magnitude of small firm effect. The current research also analyzed average stock returns with cross-sectional variation. It confirmed the relationship of size with daily seasonality. Furthermore, it also determined that a weak form of efficiency exists in the Pakistani stock market. Thus, the findings indicate that the investors are able to earn abnormal returns on their investments with the help of timing strategies.

Nearly nonstationary processes under infinite variance GARCH noises

Let Yt be an autoregressive process with order one, i.e., Yt = μ + ϕnYt−1 + εt, where [εt] is a heavy tailed general GARCH noise with tail index α. Let {{hat phi }_n} be the least squares estimator (LSE) of ϕn For μ = 0 and α < 2, it is shown by Zhang and Ling (2015) that {{hat phi }_n} is inconsistent when Yt is stationary (i.e., ϕn ≡ ϕ < 1), however, Chan and Zhang (2010) showed that {{hat phi }_n} is still consistent with convergence rate n when Yt is a unit-root process (i.e., ϕn = 1) and [εt] is a GARCH(1, 1) noise. There is a gap between the stationary and nonstationary cases. In this paper, two important issues will be considered: (1) what about the nearly unit root case? (2) When can ϕ be estimated consistently by the LSE? We show that when ϕn = 1 − c/n, then {{hat phi }_n} converges to a functional of stable process with convergence rate n. Further, we show that if limn→∞kn(1 − ϕn) = c for a positive constant c, then {k_n}({hat phi _n} - phi ) converges to a functional of two stable variables with tail index α/2, which means that ϕn can be estimated consistently only when kn → ∞.

Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations

This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient ρ=ρn is derived uniformly over stationary values in [0,1), focusing on ρn→1 as sample size n tends to infinity. For tail index α∈(0,4) of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on 1−ρn2, but no condition on the rate of ρn is required. It is shown that, for the tail index α∈(0,2), the LSE is inconsistent, for α=2, logn/(1−ρn2)-consistent, and for α∈(2,4), n1−2/α/(1−ρn2)-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index α∈(0,4); and no restriction on the rate of ρn is necessary.

Bitcoin returns and risk: A general GARCH and GAS analysis

This paper performs a general GARCH and GAS analysis for modelling and forecasting bitcoin returns and risk. Since Bitcoin trading exhibits excess volatility compared with other securities, it is important to model its risk and returns. We consider heavy-tailed GARCH models as well as GAS models based on the score function of the predictive conditional density of the bitcoin returns. We compare out-of-sample 1%-Value-at-Risk (VaR) forecasts under 45 different specifications using three backtesting procedures. We find that GAS models with heavy-tailed distributions provide the best out-of-sample forecast and goodness-of-fit properties to bitcoin returns and risk modelling. Normally-distributed GARCH models are always outperformed by heavy-tailed GARCH or GAS models. Besides, heavy-tailed GAS models provide the best conditional and unconditional coverage for 1%-VaR forecasts, illustrating the importance of modelling excess kurtosis for bitcoin returns. Hence, our findings have important implications for risk managers and investors for using bitcoin in optimal hedging or investment strategies.

On change point test for ARMA–GARCH models: Bootstrap approach

The problem of testing for a parameter change has been a core issue in time series analysis. It is well known that the estimates-based CUSUM test often suffers from severe size distortions in general GARCH type models. The residual-based CUSUM test has been used as an alternative, which, however, has a defect not to detect the ARMA parameter changes in ARMA–GARCH models. As a remedy, one can employ the score vector-based CUSUM test in ARMA–GARCH models as in Oh and Lee (0000). However, it shows some size distortions for relatively small samples. Hence, we consider the bootstrap counterpart for obtaining a more stable test. Focus is made on the verification of the weak consistency of the proposed test. An empirical study is illustrated for its evaluation.

HYBRID-GARCH: A Generic Class of Models for Volatility Predictions using High Frequency Data

We propose a general GARCH framework that allows the predict volatility using returns sampled at a higher frequency than the prediction horizon. We call the class of models High FrequencY Data-Based PRojectIon-Driven GARCH, or HYBRID-GARCH models, as the volatility dynamics are driven by what we call HYBRID processes. The HYBRID processes can involve data sampled at any frequency.

Macroeconomic News and the DM/$ Exchange Rate

Empirical confirmation that the effect of macroeconomic fundamentals on exchange rates is economically important has been scarce. This paper employs a general GARCH specification with asymmetric responses to investigate the effect of 35 U.S. and German macroeconomic news announcements on the daily DM/$ exchange rate over the 1980-1998 period. We conclude that FX rates are strongly connected to real and nominal sector developments in both countries, and that real sector announcements influence the exchange rate more strongly than money or inflation announcements. We find that surprisingly high real growth appreciates the exchange rate and raises yields.

Strict Stationarity Testing and Estimation of Explosive and Stationary Generalized Autoregressive Conditional Heteroscedasticity Models

This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of (generalized autoregressive conditional heteroscedasticity) GARCH(1, 1) models without strict stationarity constraints and considers applications to testing problems. The estimator is unrestricted in the sense that the value of the intercept, which cannot be consistently estimated in the explosive case, is not fixed. A specific behavior of the estimator of the GARCH coefficients is obtained at the boundary of the stationarity region, but, except for the intercept, this estimator remains consistent and asymptotically normal in every situation. The asymptotic variance is different in the stationary and nonstationary situations, but is consistently estimated with the same estimator in both cases. Tests of strict stationarity and nonstationarity are proposed. The tests developed for the classical GARCH(1, 1) model are able to detect nonstationarity in more general GARCH models. A numerical illustration based on stock indices and individual stock returns is proposed.

HYBRID-GARCH: A Generic Class of Models for Volatility Predictions Using Mixed Frequency Data

We propose a general GARCH framework that allows the predict volatility using returns sampled at a higher frequency than the prediction horizon. We call the class of models High FrequencY Data-Based PRojectIon-Driven GARCH, or HYBRID-GARCH models, as the volatility dynamics are driven by what we call HYBRID processes. The HYBRID processes can involve data sampled at any frequency. As far as empirical specifications go, we obtain some powerful findings that deviate substantially from the existing literature. Models featuring intra-daily asymmetries (news impact curves applied to intra-daily returns) dominate symmetric models up to weekly horizons, while the reverse is true for longer horizons. Models using daily realized volatility are less preferred than HYBRID involving intra-daily weighting scheme even for longer horizons.

Asymmetric GARCH and the Financial Crisis: A Preliminary Study

The paper deals with estimation of both general GARCH as well as asymmetric EGARCH and TGARCH models, used to model the leverage effect of good news and bad news on market volatility. We estimate the models using daily returns of S&P 500 stock index and describe the news impact curves (NICs) for these models. When estimating the crisis series, we show the possibility of using a news impact surface to describe the results from models of higher orders.

European Securitization

Asset-backed securitization is a highly flexible yet complex refinancing technique that involves the issuance of contingent claims with varying seniority on the cash flow performance of a designated pool of asset exposures. Efficient risk management of ABS obligations requires both investors and issuers to thoroughly understand the inherent spread dynamics in this growing segment of fixed income markets. We model the heteroskedasticity of weekly secondary market spreads of European CDO, MBS, and Pfandbrief transactions in a multi-factor GARCH process for valuation and forecasting purposes. We find that expected spread changes follow a positive trend with asymmetric mean reversion depending on the direction of past spread changes. A statistically and economically stronger contribution by negative first moments of past spreads on mean spread changes indicates stronger difference stationarity when spreads are tightning. However, we observe a random stochastic process if spreads widen over several consecutive period. Also most of the conditional spread volatility is informed by negative past innovations, which constitute most of the economic significance of a positive general GARCH effect. These spread dynamics over our sample period imply that a higher risk premium from a sudden contraction of spreads seems to induce higher spread volatility at times when widening spread levels testify to negative investor sentiment.

Conditional density and value-at-risk prediction of Asian currency exchange rates

We first demonstrate the simultaneous need for both more general GARCH structures and non-normal innovation distributions for modelling the returns on certain return series such as the highly volatile exchange rates on East Asian currencies against the US dollar. This is accomplished not only via in-sample goodness-of-fit criteria, but also in terms of the precision of Value-at-Risk calculations made on out-of-sample density predictions. Second, a forecasting strategy using weighted maximum likelihood estimation is proposed. We show that it gives rise to considerably improved forecast performance over longer horizons. Copyright © 2000 John Wiley & Sons, Ltd.

Efficient estimation in semiparametric GARCH models

It is well-known that financial data sets exhibit conditional hereroskedasticity. GARCH-type models are often used to model this phenomenon. Since the distribution of the rescaled innovations is generally far from a normal distribution, a semiparametric approach is advisable. Several publications observed that adaptive estimation of the Euclidean parameters is not possible in the usual parametrization when the distribution of the rescaled innovations is the efficient score functions in the parametric model of the autoregression parameters are orthogonal to the tangent space generated by the nuisance parameter, thus suggesting that adaptive estimation of the autoregression parameters is possible. Indeed, we construct adaptive and hence efficient estimators in a general GARCH in mean-type context including integrated GARCH models. Our analysis is based on a general LAN theorem for time-series models, published elsewhere. In contrast to recent literature about ARCH models we do not need moment condition.

Pricing Commodity Options when the Underlying Futures Price Exhibits Time‐Varying Volatility

AbstractStandard commodity option pricing models assume proportional changes in the underlying futures price are i.i.d. normal. Empirical evidence suggests commodity futures price movements exhibit excess kurtosis and time‐varying volatility. This paper presents option pricing models when time‐varying volatility and excess kurtosis in the underlying futures price can be modeled as a GARCH process. Empirical results suggest that the GARCH option pricing model outperforms the standard Black option‐pricing model, which uses historical volatilities. A closed‐form approximation model using a variance forecast generated from the estimated GARCH process also outperforms Black's model and in many cases the general GARCH model as well.