Aggregation and memory of models of changing volatility

In this paper we study the effect of contemporaneous aggregation of an arbitrarily large number of covariance stationary processes featuring short memory dynamic conditional heteroskedasticity, when heterogeneity is allowed for across units. We look at the memory properties of the limit aggregate. General conditions for long memory heteroskedasticity are obtained. More specific results relative to certain stochastic volatility models are also developed, providing some examples of how long memory heteroskedasticity can be obtained by aggregation.     

Testing for Linearity

The problem of testing for linearity and the number of regimes in the context of self‐exciting threshold autoregressive (SETAR) models is reviewed. We describe least‐squares methods of estimation and inference. The primary complication is that the testing problem is non‐standard, due to the presence of parameters which are only defined under the alternative, so the asymptotic distribution of the test statistics is non‐standard. Simulation methods to calculate asymptotic and bootstrap distributions are presented. As the sampling distributions are quite sensitive to conditional heteroskedasticity in the error, careful modeling of the conditional variance is necessary for accurate inference on the conditional mean. We illustrate these methods with two applications — annual sunspot means and monthly U.S. industrial production. We find that annual sunspots and monthly industrial production are SETAR(2) processes.     

Bootstrap inference in systems of single equation error correction models

We analyze OLS-based tests of long-run relationships, weak exogeneity and short-run dynamics in conditional error correction models. Unweighted sums of single equation test statistics are used for hypothesis testing in pooled systems. When model errors are (conditionally) heteroskedastic tests of weak exogeneity and short run dynamics are affected by nuisance parameters. Similarly, on the pooled level the advocated test statistics are no longer pivotal in presence of cross-sectional error correlation. We prove that the wild bootstrap provides asymptotically valid critical values under both conditional heteroskedasticity and cross-sectional error correlation. A Monte-Carlo study reveals that in small samples the bootstrap outperforms first-order asymptotic approximations in terms of the empirical size even if the asymptotic distribution of the test statistic does not depend on nuisance parameters. Opposite to feasible GLS methods the approach does not require any estimate of cross-sectional correlation and copes with time-varying patterns of contemporaneous error correlation.     

Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity

This paper explores the asymptotic distribution of the cointegrating vector estimator in error correction models with conditionally heteroskedastic errors. Asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector, which estimates the cointegrating vector and the multivariate GARCH process jointly, are provided. The MLE of the cointegrating vector follows mixture normal, and its asymptotic distribution depends on the conditional heteroskedasticity and the kurtosis of standardized innovations. The reduced rank regression (RRR) estimator and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity, and thus the efficiency gain of the MLE emerges as the magnitude of conditional heteroskedasticity increases. The simulation results indicate that the relative power of the t-statistics based on the MLE improves significantly as the GARCH effect increases.     

Nonlinearities and Fractional Integration in the US Unemployment Rate*

This paper proposes a model of the US unemployment rate which accounts for both its asymmetry and its long memory. Our approach introduces fractional integration and nonlinearities simultaneously into the same framework, using a Lagrange multiplier procedure with a standard null‐limit distribution. The empirical results suggest that the US unemployment rate can be specified in terms of a fractionally integrated process, which interacts with some nonlinear functions of labour‐demand variables such as real oil prices and real interest rates. We also find evidence of a long‐memory component. Our results are consistent with a hysteresis model with path dependency rather than a non‐accelerating inflation rate of unemployment (NAIRU) model with an underlying unemployment equilibrium rate, thereby giving support to more activist stabilization policies. However, any suitable model should also include business cycle asymmetries, with implications for both forecasting and policy‐making.     

Analysing inflation by the fractionally integrated ARFIMA–GARCH model

This paper considers the application of long-memory processes to describing inflation for ten countries. We implement a new procedure to obtain approximate maximum likelihood estimates of an ARFIMA—GARCH process; which is fractionally integrated I(d) with a superimposed stationary ARMA component in its conditional mean. Additionally, this long memory process is allowed to have GARCH type conditional heteroscedasticity. On analysing monthly post-World War II CPI inflation for ten different countries, we find strong evidence of long memory with mean reverting behaviour for all countries except Japan, which appears stationary. For three high inflation economies there is evidence that the mean and volatility of inflation interact in a way that is consistent with the Friedman hypothesis.     

Assessing DSGE model nonlinearities

We develop a new class of time series models to identify nonlinearities in the data and to evaluate DSGE models. U.S. output growth and the federal funds rate display nonlinear conditional mean dynamics, while inflation and nominal wage growth feature conditional heteroskedasticity. We estimate a DSGE model with asymmetric wage and price adjustment costs and use predictive checks to assess its ability to account for these nonlinearities. While it is able to match the nonlinear inflation and wage dynamics, thanks to the estimated downward wage and price rigidities, these do not spill over to output growth or the interest rate.     

A multivariate GARCH in mean approach to testing uncovered interest parity: evidence from Asia-Pacific foreign exchange markets

The existence of time-varying risk premia in deviations from uncovered interest parity (UIP) is investigated based on a conditional capital asset pricing model (CAPM) using data from four Asia-Pacific foreign exchange markets. A parsimonious multivariate generalized autoregressive conditional heteroskedasticity in mean (GARCH-M) parameterization is employed to model the conditional covariance matrix of excess returns. The empirical results indicate that when each currency is estimated separately with an univariate GARCH-M parameterization, no evidence of time-varying risk premia is found except Malaysian ringgit. However, when all currencies are estimated simultaneously with the multivariate GARCH-M parameterization, strong evidence of time-varying risk premia is detected. As a result, the evidence supports the idea that deviations from UIP are due to a risk premium and not to irrationality among market participants. In addition, the empirical evidence found in this study points out that simply modeling the conditional second moments is not sufficient enough to explain the dynamics of the risk premia. A time-varying price of risk is still needed in addition to the conditional volatility. Finally, significant asymmetric world market volatility shocks are found in Asia-Pacific foreign exchange markets.     

Asymptotic and bootstrap tests for linearity in a TAR-GARCH(1,1) model with a unit root

This paper derives the limiting distribution of the Lagrange Multiplier (LM) test for threshold nonlinearity in a TAR model with GARCH errors when one of the regimes contains a unit root. It is shown that the asymptotic distribution is nonstandard and depends on nuisance parameters that capture the degree of conditional heteroskedasticity and non-Gaussian nature of the process. We propose a bootstrap procedure for approximating the exact finite-sample distribution of the test for linearity and establish its asymptotic validity.     

Long memory and structural breaks in modeling the return and volatility dynamics of precious metals

We investigate the potential of structural changes and long memory (LM) properties in returns and volatility of the four major precious metal commodities traded on the COMEX markets (gold, silver, platinum and palladium). Broadly speaking, a random variable is said to exhibit long memory behavior if its autocorrelation function is not integrable, while structural changes can induce sudden and significant shifts in the time-series behavior of that variable. The results from implementing several parametric and semiparametric methods indicate strong evidence of long range dependence in the daily conditional return and volatility processes for the precious metals. Moreover, for most of the precious metals considered, this dual long memory is found to be adequately captured by an ARFIMA–FIGARCH model, which also provides better out-of-sample forecast accuracy than several popular volatility models. Finally, evidence shows that conditional volatility of precious metals is better explained by long memory than by structural breaks.     

Estimation of variance of housing prices using spatial conditional heteroskedasticity (SARCH) model with an application to Boston housing price data

In this paper we investigate housing price volatility within a spatial econometrics setting. We propose an extended spatial regression model of the real estate market that includes the effects of both conditional heteroskedasticity and spatial autocorrelation. Our suggested model has features similar to those of autoregressive conditional heteroskedasticity (ARCH) in the time-series context. We utilize the spatial ARCH (SARCH) model to analyze Boston housing price data used by Harrison and Rubinfeld (1978) and Gilley and Pace (1996). We show that measuring the variability of housing prices is an important issue and our SARCH model captures the conditional spatial variability of Boston housing prices. We argue that there is a different source of spatial variation, which is independent of traditional housing and neighborhood characteristics, and is captured by the SARCH model.     

Robust tests for heteroskedasticity in the one-way error components model

This paper constructs tests for heteroskedasticity in one-way error components models, in line with Baltagi et al. [Baltagi, B.H., Bresson, G., Pirotte, A., 2006. Joint LM test for homoskedasticity in a one-way error component model. Journal of Econometrics 134, 401–417]. Our tests have two additional robustness properties. First, standard tests for heteroskedasticity in the individual component are shown to be negatively affected by heteroskedasticity in the remainder component. We derive modified tests that are insensitive to heteroskedasticity in the component not being checked, and hence help identify the source of heteroskedasticity. Second, Gaussian-based LM tests are shown to reject too often in the presence of heavy-tailed (e.g. t$t$-Student) distributions. By using a conditional moment framework, we derive distribution-free tests that are robust to non-normalities. Our tests are computationally convenient since they are based on simple artificial regressions after pooled OLS estimation.     

GENERALIZED AUTOREGRESSIVE SCORE MODELS WITH APPLICATIONS

We propose a class of observation‐driven time series models referred to as generalized autoregressive score (GAS) models. The mechanism to update the parameters over time is the scaled score of the likelihood function. This new approach provides a unified and consistent framework for introducing time‐varying parameters in a wide class of nonlinear models. The GAS model encompasses other well‐known models such as the generalized autoregressive conditional heteroskedasticity, autoregressive conditional duration, autoregressive conditional intensity, and Poisson count models with time‐varying mean. In addition, our approach can lead to new formulations of observation‐driven models. We illustrate our framework by introducing new model specifications for time‐varying copula functions and for multivariate point processes with time‐varying parameters. We study the models in detail and provide simulation and empirical evidence. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonparametric simultaneous testing for structural breaks

In this paper we consider a regression model with errors that are martingale differences. This modeling includes the regression of both independent and time series data. The aim is to study the appearance of structural breaks in both the mean and the variance functions, assuming that such breaks may occur simultaneously in both the functions. We develop nonparametric testing procedures that simultaneously test for structural breaks in the conditional mean and the conditional variance. The asymptotic distribution of an adaptive test statistic is established, as well as its asymptotic consistency and efficiency. Simulations illustrate the performance of the adaptive testing procedure. An application to the analysis of financial time series also demonstrates the usefulness of the proposed adaptive test in practice.     

Volatility persistence,long memory and time-varying unconditional mean: Evidence from 10 equity indices

This paper re-examines evidence of volatility persistence and long memory in the light of potential time-variation in the unconditional mean of the volatility series. Daily equity volatility is generally regarded as exhibiting long memory, however, recent evidence has suggested that long memory may be a spurious finding arising from neglected breaks or time-variation in unconditional variance. The results presented here suggested that long memory is apparent when analysed on the assumption that unconditional variance is constant. However, both breakpoint tests and a moving average application suggest that unconditional variance exhibits substantial, although slow moving, time-variation. The apparent long-memory property largely disappears when this time-variation is taken into account. A modification of the GARCH model to allow for mean variation generates improved volatility forecasting performance, but only over long horizon. At the daily level the assumption of a constant unconditional variance does not seem to affect forecasts.     

Mean reversion and long memory in African stock market prices

We examine the behavior of stock market prices in several African countries by means of fractionally integrated techniques. In doing so, we can test for mean reversion in these markets. Our results can be summarized as follows: we cannot find evidence of mean reversion in any single market, and evidence of long memory returns (i.e., orders of integration above 1 in the logged stock prices) is obtained in the cases of Egypt and Nigeria, and, in a lesser extent in Tunisia, Morocco and Kenya. Permitting the existence of a structural change, the break dates take place in the earlier 2000s in the majority of the cases, and evidence of mean reversion seems to have taken place in the periods before the breaks in most of the countries. If we focus on the absolute and squared returns, evidence of long memory is obtained in Nigeria and Egypt. Thus, for these two countries, a long memory model incorporating positive fractional degrees of integration in both the level and the volatility process should be considered.     

Sources of volatility persistence: A case study of the U.K. pound/U.S. dollar exchange rate returns

This paper utilizes a new approach to examine the inherent nonlinear dynamics of the exchange rate returns volatility. Specifically, we utilize a regime switching threshold (i) generalized autoregressive conditional heteroskedasticity (RS-TGARCH) and (ii) a fractional generalized autoregressive conditional heteroskedasticity (RS-TFIGARCH) model. The RS-TGARCH model is found to be adequate in analyzing the first two moments of the U.K. pound/U.S. dollar monthly exchange rate returns series. The RS-TFIGARCH is found to be adequate for the daily returns series. The volatility persistence and leverage effects associated with exchange rate returns series are jointly tested by means of a Wald Chi-square test.     

Multimodality in GARCH regression models

It is shown empirically that mixed autoregressive moving average regression models with generalized autoregressive conditional heteroskedasticity (Reg-ARMA-GARCH models) can have multimodality in the likelihood that is caused by a dummy variable in the conditional mean. Maximum likelihood estimates at the local and global modes are investigated and turn out to be qualitatively different, leading to different model-based forecast intervals. In the simpler GARCH(p,q) regression model, we derive analytical conditions for bimodality of the corresponding likelihood. In that case, the likelihood is symmetrical around a local minimum. We propose a solution to avoid this bimodality.     

Dynamic volatility transmission and portfolio management across major cryptocurrencies: Evidence from hourly data

This study used hourly data to examine the dynamic conditional correlations and hedging strategies in the main cryptocurrency markets: Bitcoin (BTC), Ethereum (ETH), Litecoin (LTC), and Ripple (XRP). Multivariate generalized autoregressive conditional heteroskedasticity family models provided evidence of significant positive dynamic conditional correlations among these markets. A weaker conditional correlation was observed for the LCT–XRP portfolio than for the BTC–ETH portfolio, which had the highest correlation value. The dynamic correlations intensified after the cryptocurrency crisis. The results of a portfolio risk analysis suggested that investors should hold less BTC than LTC, ETH, and XRP to minimize risk while maintaining consistent expected portfolio returns. Investors should hold less BTC than the other cryptocurrencies during a crisis. In addition, the cheapest hedge strategy is to hold long BTC and short XRP regardless of the period. Holding long BTC and short LTC was found to be the most expensive hedge strategy. Finally, the study showed that an optimally weighted diversified portfolio provides the greatest reduction in risk and downside risk for ETH and LTC. For XRP, portfolio hedging is the best mechanism for reducing risk.     

Two estimators of the long-run variance: Beyond short memory

This paper deals with the estimation of the long-run variance of a stationary sequence. We extend the usual Bartlett-kernel heteroskedasticity and autocorrelation consistent (HAC) estimator to deal with long memory and antipersistence. We then derive asymptotic expansions for this estimator and the memory and autocorrelation consistent (MAC) estimator introduced by Robinson [Robinson, P. M., 2005. Robust covariance matrix estimation: HAC estimates with long memory/antipersistence correction. Econometric Theory 21, 171–180]. We offer a theoretical explanation for the sensitivity of HAC to the bandwidth choice, a feature which has been observed in the special case of short memory. Using these analytical results, we determine the MSE-optimal bandwidth rates for each estimator. We analyze by simulations the finite-sample performance of HAC and MAC estimators, and the coverage probabilities for the studentized sample mean, giving practical recommendations for the choice of bandwidths.