Testing for unit roots in time series models with non-stationary volatility

Many of the key macro-economic and financial variables in developed economies are characterized by permanent volatility shifts. It is known that conventional unit root tests are potentially unreliable in the presence of such behaviour, depending on a particular function (the variance profile) of the underlying volatility process. Somewhat surprisingly then, very little work has been undertaken to develop unit root tests which are robust to the presence of permanent volatility shifts. In this paper we fill this gap in the literature by proposing tests which are valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth-transition) volatility change processes as special cases. Our solution uses numerical methods to simulate the asymptotic null distribution of the statistics based on a consistent estimate of the variance profile which we also develop. The practitioner is not required to specify a parametric model for volatility. An empirical illustration using producer price inflation series from the Stock–Watson database is reported.     

Non-stationarity in garch models: A bayesian analysis

First, the non-stationarity properties of the conditional variances in the GARCH(1, 1) model are analysed using the concept of infinite persistence of shocks. Given a time sequence of probabilities for increasing/decreasing conditional variances, a theoretical formula for quasi-strict non-stationarity is defined. The resulting conditions for the GARCH(1,1) model are shown to differ from the weak stationarity conditions mainly used in the literature. Bayesian statistical analysis using Monte Carlo integration is applied to analyse both stationarity concepts for the conditional variances of the US 3-month treasury bill rate. Interest rates are known for their weakly non-stationary conditional variances but, using a quasi-strict stationarity measure, it is shown that the conditional variances are likely to be stationary. Second, the level of the treasury bill rate is analysed for non-stationarity using Bayesian unit root methods. The disturbances of the GARCH model for the treasury bill rate are t-distributed. It is shown that the unit root parameter is negatively correlated with the degrees-of-freedom parameter. Imposing normally distributed disturbances leads therefore to underestimation of the non-stationarity in the level of the treasury bill rate.     

Detection of Structural Change in the Long‐run Persistence in a Univariate Time Series*

In this paper, we investigate a test for structural change in the long‐run persistence in a univariate time series. Our model has a unit root with no structural change under the null hypothesis, while under the alternative it changes from a unit‐root process to a stationary one or vice versa. We propose a Lagrange multiplier‐type test, a test with the quasi‐differencing method, and ‘demeaned versions’ of these tests. We find that the demeaned versions of these tests have better finite‐sample properties, although they are not necessarily superior in asymptotics to the other tests.     

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.     

Cointegration tests of present value models with a time-varying discount factor

The paper analyses the impact of persistence and volatility in the discount rate in present-value models on cointegration tests in levels and in logarithms. In simulations we find that the probability of not rejecting the null of no cointegration depends on the persistence of the discount rate process and can be very high when the expected returns process is highly persistent. In contrast, the cointegration tests are very robust with respect to the level of volatility in the discount rate. We discuss the relevance of our findings for the US stock market where standard ADF tests do not reject the null of no cointegration between stock prices and dividends. Based on estimates of persistence in four asset pricing models, we find that a model which links expected returns to the dividend yield is sufficiently persistent to explain the failure of rejecting the null that stock prices and dividends are not cointegrated.     

Trend‐stationary GNP: evidence from a new exact pointwise most powerful invariant unit root test

There has been a substantial debate whether GNP has a unit root. However, statistical tests have had little success in distinguishing between unit‐root and trend‐reverting specifications because of poor statistical properties. This paper develops a new exact small‐sample, pointwise most powerful unit root test that is invariant to the unknown mean and scale of the time series tested, that generates exact small‐sample critical values, powers and p‐values, that has power which approximates the maximum possible power, and that is highly robust to conditional heteroscedasticity. This test decisively rejects the unit root null hypothesis when applied to annual US real GNP and US real per capita GNP series. This paper also develops a modified version of the test to address whether a time series contains a permanent, unit root process in addition to a temporary, stationary process. It shows that if these GNP series contain a unit root process in addition to the stationary process, then it is most likely very small. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Random walks,cointegration, and the transmission of shocks across global real estate and equity markets

This article presents tests of the random walk hypothesis for the U.S. and world commercial real estate markets along with the world stock market through utilizing appropriate market indices. The augmented Dickey-Fuller and Phillips-Perron unit root tests and Cochrane variance ratio test find each of these markets to exhibit random walk behavior. In addition, Johansen-Juselius cointegration tests reveal that the three markets are not cointegrated. The vector autoregressive model shows little or no predictive power in explaining the variation in monthly returns. The generalized impulse response functions suggest that shocks stemming from one market are quickly disseminated to the other markets within two months. (JEL G14, G15)     

OPEC news and predictability of oil futures returns and volatility: Evidence from a nonparametric causality-in-quantiles approach

This paper provides a novel perspective to the predictive ability of OPEC meeting dates and production announcements for (Brent Crude and West Texas Intermediate) oil futures market returns and GARCH-based volatility using a nonparametric quantile-based methodology. We show a nonlinear relationship between oil futures returns and OPEC-based predictors; hence, linear Granger causality tests are misspecified and the linear model results of non-predictability are unreliable. When the quantile-causality test is implemented, we observe that the impact of OPEC variables is restricted to Brent Crude futures only (with no effect observed for the WTI market). Specifically, OPEC production announcements, and meeting dates predict only lower quantiles of the conditional distribution of Brent futures market returns. While, predictability of volatility covers the majority of the quantile distribution, barring extreme ends.     

On economic uncertainty,stock market predictability and nonlinear spillover effects

This paper uses a k-th order nonparametric Granger causality test to analyze whether firm-level, economic policy and macroeconomic uncertainty indicators predict movements in real stock returns and their volatility. Linear Granger causality tests show that whilst economic policy and macroeconomic uncertainty indices can predict stock returns, firm-level uncertainty measures possess no predictability. However, given the existence of structural breaks and inherent nonlinearities in the series, we employ a nonparametric causality methodology, as linear modeling leads to misspecifications thus the results cannot be considered reliable. The nonparametric test reveals that in fact no predictability can be observed for the various measures of uncertainty i.e., firm-level, macroeconomic and economic policy uncertainty, vis-à-vis real stock returns. In turn, a profound causal predictability is demonstrated for the volatility series, with the exception of firm-level uncertainty. Overall our results not only emphasize the role of economic and firm-level uncertainty measures in predicting the volatility of stock returns, but also presage against using linear models which are likely to suffer from misspecification in the presence of parameter instability and nonlinear spillover effects.     

Using an artificial financial market for studying a cryptocurrency market

This paper presents an agent-based artificial cryptocurrency market in which heterogeneous agents buy or sell cryptocurrencies, in particular Bitcoins. In this market, there are two typologies of agents, Random Traders and Chartists, which interact with each other by trading Bitcoins. Each agent is initially endowed with a finite amount of crypto and/or fiat cash and issues buy and sell orders, according to her strategy and resources. The number of Bitcoins increases over time with a rate proportional to the real one, even if the mining process is not explicitly modelled. The model proposed is able to reproduce some of the real statistical properties of the price returns observed in the Bitcoin real market. In particular, it is able to reproduce the unit root property, the fat tail phenomenon and the volatility clustering. The simulator has been implemented using object-oriented technology, and could be considered a valid starting point to study and analyse the cryptocurrency market and its future evolutions.     

Stock market volatility and the business cycle

This paper investigates the joint time series behavior of monthly stock returns and growth in industrial production. We find that stock returns are well characterized by year-long episodes of high volatility, separated by longer quiet periods. Real output growth, on the other hand, is subject to abrupt changes in the mean associated with economic recessions. We study a bivariate model in which these two changes are driven by related unobserved variables, and conclude that economic recessions are the primary factor that drives fluctuations in the volatility of stock returns. This framework proves useful both for forecasting stock volatility and for identifying and forecasting economic turning points.     

A comparative goodness-of-fit analysis of distributions of some Lévy processes and Heston model to stock index returns

In this paper, we investigate the goodness-of-fit of three Lévy processes, namely Variance-Gamma (VG), Normal-Inverse Gaussian (NIG) and Generalized Hyperbolic (GH) distributions, and probability distribution of the Heston model to index returns of twenty developed and emerging stock markets. Furthermore, we extend our analysis by applying a Markov regime switching model to identify normal and turbulent periods. Our findings indicate that the probability distribution of the Heston model performs well for emerging markets under full sample estimation and retains goodness of fit for high volatility periods, as it explicitly accounts for the volatility process. On the other hand, the distributions of the Lévy processes, especially the VG and NIG distributions, generally improves upon the fit of the Heston model, particularly for developed markets and low volatility periods. Furthermore, some distributions yield to significantly large test statistics for some countries, even though they fit well to other markets, which suggest that properties of the stock markets are crucial in identifying the best distribution representing empirical returns.     

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.     

Powerful tests for structural changes in volatility

Detecting structural changes in volatility is important for understanding volatility dynamics and stylized facts observed for financial returns such as volatility persistence. We propose modified CUSUM and LM tests that are built on a robust estimator of the long-run variance of squared series. We establish conditions under which the new tests have standard null distributions and diverge faster than standard tests under the alternative. The theory allows smooth and abrupt structural changes that can be small. The smoothing parameter is automatically selected such that the proposed test has good finite-sample size and meanwhile achieves decent power gain.     

Bootstrapping Financial Time Series

It is well known that time series of returns are characterized by volatility clustering and excess kurtosis. Therefore, when modelling the dynamic behavior of returns, inference and prediction methods, based on independent and/or Gaussian observations may be inadequate. As bootstrap methods are not, in general, based on any particular assumption on the distribution of the data, they are well suited for the analysis of returns. This paper reviews the application of bootstrap procedures for inference and prediction of financial time series. In relation to inference, bootstrap techniques have been applied to obtain the sample distribution of statistics for testing, for example, autoregressive dynamics in the conditional mean and variance, unit roots in the mean, fractional integration in volatility and the predictive ability of technical trading rules. On the other hand, bootstrap procedures have been used to estimate the distribution of returns which is of interest, for example, for Value at Risk (VaR) models or for prediction purposes. Although the application of bootstrap techniques to the empirical analysis of financial time series is very broad, there are few analytical results on the statistical properties of these techniques when applied to heteroscedastic time series. Furthermore, there are quite a few papers where the bootstrap procedures used are not adequate.     

Priced risk and asymmetric volatility in the cross section of skewness

We investigate the sources of skewness in aggregate risk factors and the cross section of stock returns. In an ICAPM setting with conditional volatility, we find theoretical time series predictions on the relationships among volatility, returns, and skewness for priced risk factors. Market returns resemble these predictions; however, size, book-to-market, and momentum factor returns are not always consistent with our predictions. We find evidence that size and book-to-market may be priced post-crisis but not in the decade before. Momentum does not appear priced by our test. We link aggregate risk and skewness to individual stocks and find empirically that the risk aversion effect manifests in individual stock skewness. Additionally, we find several firm characteristics that explain stock skewness. Smaller firms, value firms, highly levered firms, and firms with poor credit ratings have more positive skewness.     

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.     

The stochastic volatility in mean model: empirical evidence from international stock markets

In this paper we present an exact maximum likelihood treatment for the estimation of a Stochastic Volatility in Mean (SVM) model based on Monte Carlo simulation methods. The SVM model incorporates the unobserved volatility as an explanatory variable in the mean equation. The same extension is developed elsewhere for Autoregressive Conditional Heteroscedastic (ARCH) models, known as the ARCH in Mean (ARCH‐M) model. The estimation of ARCH models is relatively easy compared with that of the Stochastic Volatility (SV) model. However, efficient Monte Carlo simulation methods for SV models have been developed to overcome some of these problems. The details of modifications required for estimating the volatility‐in‐mean effect are presented in this paper together with a Monte Carlo study to investigate the finite sample properties of the SVM estimators. Taking these developments of estimation methods into account, we regard SV and SVM models as practical alternatives to their ARCH counterparts and therefore it is of interest to study and compare the two classes of volatility models. We present an empirical study of the intertemporal relationship between stock index returns and their volatility for the United Kingdom, the United States and Japan. This phenomenon has been discussed in the financial economic literature but has proved hard to find empirically. We provide evidence of a negative but weak relationship between returns and contemporaneous volatility which is indirect evidence of a positive relation between the expected components of the return and the volatility process. Copyright © 2002 John Wiley & Sons, Ltd.