Power transformations of absolute returns and long memory estimation

Different power transformations of absolute returns of various financial assets have been found to display different magnitudes of sample autocorrelations, a property referred to as the Taylor effect. In this paper, we consider the long memory stochastic volatility model for the returns, under which, the asymptotic rate of decay of the autocorrelations of powers of absolute returns is governed by their long memory parameter. Although the true long memory parameter of powers of absolute returns is the same across different powers, we show that the local Whittle estimator of the long memory parameter has finite-sample bias that differs across the power transformations chosen. A Monte-Carlo experiment provides evidence in support of our theoretical finding that the reported variation of the estimates of the long memory parameter for power transformations of returns could be due to finite-sample bias of the estimator. The local Whittle estimates of powers of absolute returns for the S&P500 index and the DM/USD exchange rate are also examined.     

US stock market volatility persistence: evidence before and after the burst of the IT bubble

In this paper we test whether the US stock market volatility presents a different behavior before and after the burst of the IT bubble. Using long range dependence techniques we examine the order of integration in the absolute and squared returns in three daily stock market indices (DJIA, S&P and NASDAQ). The results indicate that both absolute and squared returns present long memory behavior. In general, the highest orders of integration in the volatility processes correspond to the NASDAQ index. The results also show that in most cases the volatility is more persistent in the bear market than in the bull market.     

MENA stock market volatility persistence: Evidence before and after the financial crisis of 2008

The global financial crisis began with a financial meltdown in the United States in early 2008 and then it had spread to the rest of the world. In this paper we test whether the MENA equity market volatility presents a different behavior before and after the financial crisis of 2008. Using long range dependence techniques we test for long memory in the returns, absolute and squared returns of the MENA equity markets. We subject the series to unit root tests that allow for structural breaks and use the Bai and Perron (1998, Econometrica, 66, 47; 2003a, J. Appl. Econometrics, 6, 72; 2003b, Econometrics J., 18, 1) to test for multiple breaks in the mean returns. The results indicate that the volatility measures represented by absolute and squared returns show evidence of long memory for the full and subsample periods, while the returns show a weak evidence of long memory. Considering the shift dates and corresponding to the 2008 financial crisis, the returns and volatility measures display less evidence of long memory in the after crisis period as opposed to the before crisis period. The change in the returns and volatility dynamics of these markets was due to financial and economic conditions that took place in the MENA region after the crisis.     

Realized volatility and transactions

This paper re-examines the impact of number of trades, trade size and order imbalance on daily stock returns volatility. In contrast to prior studies, we estimate daily volatility using realized volatility obtained by summing up intraday squared returns. Consistent with the theory of quadratic variation, realized volatility estimates are shown to be less noisy than standard volatility measures such as absolute returns used in previous studies. In general, our results confirm [Jones, C.M., Kaul, G., Lipson, M.L., 1994. Transactions, volume, and volatility. Review of Financial Studies 7, 631–651] that number of trades is the dominant factor behind the volume–volatility relation. Neither trade size nor order imbalance adds significantly more explanatory power to realized volatility beyond number of trades. This finding is robust to different time periods, firm sizes and regression specifications. The implications of our results for microstructure theory are discussed.     

Trading frequency and volatility clustering

Volatility clustering, with autocorrelations of the hyperbolic decay rate, is unquestionably one of the most important stylized facts of financial time series. This paper presents a market microstructure model that is able to generate volatility clustering with hyperbolically decaying autocorrelations via traders with multiple trading frequencies, using Bayesian information updates in an incomplete market. The model illustrates that signal extraction, which is induced by multiple trading frequencies, can increase the persistence of the volatility of returns. Furthermore, we show that the volatility of the underlying time series of returns varies greatly with the number of traders in the market.     

Testing of a market fraction model and power-law behaviour in the DAX 30

This paper tests a simple market fraction asset pricing model with heterogeneous agents. By selecting a set of structural parameters of the model through a systematic procedure, we show that the autocorrelations (of returns, absolute returns and squared returns) of the market fraction model share the same pattern as those of the DAX 30. By conducting econometric analysis via Monte Carlo simulations, we characterize these power-law behaviours and find that estimates of the power-law decay indices, the (FI)GARCH parameters, and the tail index of the selected market fraction model closely match those of the DAX 30. The results strongly support the explanatory power of the heterogeneous agent models.     

Why Do Absolute Returns Predict Volatility So Well?

Our objective is volatility forecasting, which is core to manyrisk management problems. We provide theoretical explanationsfor (i) the empirical stylized fact recognized at least sinceTaylor (1986) and Ding, Granger, and Engle (1993) that absolutereturns show more persistence than squared returns and (ii)the empirical finding reported in recent work by Ghysels, Santa-Clara,and Valkanov (2006) showing that realized absolute values outperformsquare return-based volatility measures in predicting futureincrements in quadratic variation. We start from a continuoustime stochastic volatility model for asset returns suggestedby Barndorff-Nielsen and Shephard (2001) and study the persistenceand linear regression properties of various volatility-relatedprocesses either observed directly or with sampling error. Wealso allow for jumps in the asset return processes and investigatetheir impact on persistence and linear regression. Extensiveempirical results complement the theoretical analysis.     

Forecasting financial market volatility: Sample frequency vis-à-vis forecast horizon

This paper explores the return volatility predictability inherent in high-frequency speculative returns. Our analysis focuses on a refinement of the more traditional volatility measures, the integrated volatility, which links the notion of volatility more directly to the return variance over the relevant horizon. In our empirical analysis of the foreign exchange market the integrated volatility is conveniently approximated by a cumulative sum of the squared intraday returns. Forecast horizons ranging from short intraday to 1-month intervals are investigated. We document that standard volatility models generally provide good forecasts of this economically relevant volatility measure. Moreover, the use of high-frequency returns significantly improves the longer run interdaily volatility forecasts, both in theory and practice. The results are thus directly relevant for general research methodology as well as industry applications.     

Box-Cox stochastic volatility models with heavy-tails and correlated errors

This paper presents a Markov chain Monte Carlo (MCMC) algorithm to estimate parameters and latent stochastic processes in the asymmetric stochastic volatility (SV) model, in which the Box-Cox transformation of the squared volatility follows an autoregressive Gaussian distribution and the marginal density of asset returns has heavy-tails. We employed the Bayes factor and the Bayesian information criterion (BIC) to examine whether the Box-Cox transformation of squared volatility is favored against the log-transformation. When applying the heavy-tailed asymmetric Box-Cox transformed SV model, three competing SV models and the t-GARCH(1,1) model to continuously compounded daily returns of the Australian stock index, we find that the Box-Cox transformation of squared volatility is strongly favored by Bayes factors and BIC against the log-transformation. While both criteria strongly favor the t-GARCH(1,1) model against the heavy-tailed asymmetric Box-Cox transformed SV model and the other three competing SV models, we find that SV models fit the data better than the t-GARCH(1,1) model based on a measure of closeness between the distribution of the fitted residuals and the distribution of the model disturbance. When our model and its competing models are applied to daily returns of another five stock indices, we find that in terms of SV models, the Box-Cox transformation of squared volatility is strongly favored against the log-transformation for the five data sets.     

Implications of market microstructure for realized variance measurement

Volatility measuring and estimation based on intra-day high-frequency data has grown in popularity during the last few years. A significant part of the research uses volatility and variance measures based on the sum of squared high-frequency returns. These volatility measures, introduced and mathematically justified in a series of papers by Andersen et al. [1999. (Understanding, optimizing, using and forecasting) realized volatility and correlation. Leonard N. Stern School Finance Department Working Paper Series, 99-061, New York University; 2000a. The distribution of realized exchange rate volatility. Journal of the American Statistical Association 96, no. 453: 42–55; 2000b. Exchange rate returns standardized by realized volatility are (nearly) Gaussian. Multinational Finance Journal 4, no. 3/4: 159–179; 2003. Modeling and forecasting realized volatility. NBER Working Paper Series 8160.] and Andersen et al. 2001a. Modeling and forecasting realized volatility. NBER Working Paper Series 8160., are referred to as ‘realized variance’. From the theory of quadratic variations of diffusions, it is possible to show that realized variance measures, based on sufficiently frequently sampled returns, are error-free volatility estimates. Our objective here is to examine realized variance measures, where well-documented market microstructure effects, such as return autocorrelation and volatility clustering, are included in the return generating process. Our findings are that the use of squared returns as a measure for realized variance will lead to estimation errors on sampling frequencies adopted in the literature. In the case of return autocorrelation, there will be systematic biases. Further, we establish increased standard deviation in the error between measured and real variance as sampling frequency decreases and when volatility is non-constant.     

Investor Psychology and Security Market Under- and Overreactions

We propose a theory of securities market under- and overreactions based on two well-known psychological biases: investor overconfidence about the precision of private information; and biased self-attribution, which causes asymmetric shifts in investors' confidence as a function of their investment outcomes. We show that overconfidence implies negative long-lag autocorrelations, excess volatility, and, when managerial actions are correlated with stock mispricing, public-event-based return predictability. Biased self-attribution adds positive short-lag autocorrelations ("momentum"), short-run earnings "drift," but negative correlation between future returns and long-term past stock market and accounting performance. The theory also offers several untested implications and implications for corporate financial policy.     

The one-trading-day-ahead forecast errors of intra-day realized volatility

Two volatility forecasting evaluation measures are considered; the squared one-day-ahead forecast error and its standardized version. The mean squared forecast error is the widely accepted evaluation function for the realized volatility forecasting accuracy. Additionally, we explore the forecasting accuracy based on the squared distance of the forecast error standardized with its volatility. The statistical properties of the forecast errors point the standardized version as a more appropriate metric for evaluating volatility forecasts.We highlight the importance of standardizing the forecast errors with their volatility. The predictive accuracy of the models is investigated for the FTSE100, DAX30 and CAC40 European stock indices and the exchange rates of Euro to British Pound, US Dollar and Japanese Yen. Additionally, a trading strategy defined by the standardized forecast errors provides higher returns compared to the strategy based on the simple forecast errors. The exploration of forecast errors is paving the way for rethinking the evaluation of ultra-high frequency realized volatility models.     

Leverage and Volatility Feedback Effects in High-Frequency Data

We examine the relationship between volatility and past andfuture returns using high-frequency aggregate equity index data.Consistent with a prolonged "leverage" effect, we find the correlationsbetween absolute high-frequency returns and current and pasthigh-frequency returns to be significantly negative for severaldays, whereas the reverse cross-correlations are generally negligible.We also find that high-frequency data may be used in more accuratelyassessing volatility asymmetries over longer daily return horizons.Furthermore, our analysis of several popular continuous-timestochastic volatility models clearly points to the importanceof allowing for multiple latent volatility factors for satisfactorilydescribing the observed volatility asymmetries.     

On the persistence and volatility in European,American and Asian stocks bull and bear markets

In this paper we examine the statistical properties of several stock market indices in Europe, the US and Asia by means of determining the degree of dependence in both the level and the volatility of the processes. In the latter case, we use the squared returns as a proxy for the volatility. We also investigate the cyclical pattern observed in the data and in particular, if the degree of dependence changes depending on whether there is a bull or a bear period. We use fractional integration and GARCH specifications. The results indicate that the indices are all nonstationary I(1) processes with the squared returns displaying a degree of long memory behaviour. With respect to the bull and bear periods, we do not observe a systematic pattern in terms of the degree of persistence though for some of the indices (FTSE, Dax, Hang Seng and STI) there is a higher degree of dependence in both the level and the volatility during the bull periods.     

S&P 500 trading strategies and stock betas

This paper shows that S&P 500 stock betas are overstatedand the non-S&P 500 stock betas are understated becauseof liquidity price effects caused by the S&P 500 tradingstrategies. The daily and weekly betas of stocks added to theS&P 500 index during 1985-1989 increase, on average, by0.211 and 0.130. The difference between monthly betas of otherwisesimilar S&P 500 and non-S&P 500 stocks also equals 0.125during this period. Some of these increases can be explainedby the reduced nonsynchroneity of S&P 500 stock prices,but the remaining increases are explained by the price pressureor excess volatility caused by the S&P 500 trading strategies.I estimate that the price pressures account for 8.5 percentof the total variance of daily returns of a value-weighted portfolioof NYSE/AMEX stocks. The negative own autocorrelations in S&P500 index returns and the negative cross autocorrelations betweenS&P 500 stock returns provide further evidence consistentwith the price pressure hypothesis.     

Robust estimation of intraweek periodicity in volatility and jump detection

Opening, lunch and closing of financial markets induce a periodic component in the volatility of high-frequency returns. We show that price jumps cause a large bias in the classical periodicity estimators and propose robust alternatives. We find that accounting for periodicity greatly improves the accuracy of intraday jump detection methods. It increases the power to detect the relatively small jumps occurring at times for which volatility is periodically low and reduces the number of spurious jump detections at times of periodically high volatility. We use the series of detected jumps to estimate robustly the long memory parameter of the squared EUR/USD, GBP/USD and YEN/USD returns.     

Aggregation of exponential smoothing processes with an application to portfolio risk evaluation

In this paper we propose a unified framework to analyse contemporaneous and temporal aggregation of a widely employed class of integrated moving average (IMA) models. We obtain a closed-form representation for the parameters of the contemporaneously and temporally aggregated process as a function of the parameters of the original one. These results are useful due to the close analogy between the integrated GARCH (1, 1) model for conditional volatility and the IMA (1, 1) model for squared returns, which share the same autocorrelation function. In this framework, we present an application dealing with Value-at-Risk (VaR) prediction at different sampling frequencies for an equally weighted portfolio composed of multiple indices. We apply the aggregation results by inferring the aggregate parameter in the portfolio volatility equation from the estimated vector IMA (1, 1) model of squared returns. Empirical results show that VaR predictions delivered using this suggested approach are at least as accurate as those obtained by applying standard univariate methodologies, such as RiskMetrics.     

The Dynamic Relation Between Stock Returns, Trading Volume, and Volatility

We examine the dynamic relation between returns, volume, and volatility of stock indexes. The data come from nine national markets and cover the period from 1973 to 2000. The results show a positive correlation between trading volume and the absolute value of the stock price change. Granger causality tests demonstrate that for some countries, returns cause volume and volume causes returns. Our results indicate that trading volume contributes some information to the returns process. The results also show persistence in volatility even after we incorporate contemporaneous and lagged volume effects. The results are robust across the nine national markets.     

A Comparison of Seasonal Adjustment Methods When Forecasting Intraday Volatility

In this article we compare volatility forecasts over a thirty‐minute horizon for the spot exchange rates of the Deutsche mark and the Japanese yen against the U.S. dollar. Explicitly modeling the intraday seasonal pattern improves the out‐of‐sample forecasting performance. We find that a seasonal estimated from the log of squared returns improves with the use of simple squared returns, and that the flexible Fourier form (FFF) is an efficient way of determining the seasonal. The two‐step approach that first estimates the seasonal using the FFF and then the parameters of the generalized autoregressive conditional heteroskedasticity (GARCH) model for the deseasonalized returns performs only marginally worse than the computationally expensive periodic GARCH model that includes the FFF.     

The generality of spurious predictability

Previous empirical work has documented significant predictability (non-zero cross autocorrelations) in short-term security returns. Extant theoretical papers have shown that these cross autocorrelations can arise due to partial impounding of information in securities whose returns are driven by a common factor. In this paper, we show that non-zero cross autocorrelation in security returns can arise under weaker conditions than is generally known. We demonstrate that the existence of cross autocorrelations crucially depends on the information structure of informed traders. Thus, a common factor in security returns is neither sufficient nor necessary. Any one of the following conditions on the information structure can generate non-zero cross autocorrelations:

(1) existence of an informed trader with information relevant to two securities;

(2) correlation in the signal of informed traders with information relevant to different securities; or

(3) correlation in uninformed trading.

These cross autocorrelations are then shown to be spurious. That is, traders without any private information cannot make positive trading profit by exploiting cross autocorrelations.