We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

The strong consistency and asymptotic normality of the Gaussian pseudo-maximum likelihood estimate of the parameters in a class of ARCH(∞) processes are established. The conditions imply that the process need not have finite variance, and allow for a wide range of rates of decay of the influence of observations in the remote past on the conditional variance, thereby covering a variety of parametric specifications of interest.

We consider the estimation of a random level shift model for which the series of interest is the sum of a short memory process and a jump or level shift component. For the latter component, we specify the commonly used simple mixture model such that the component is the cumulative sum of a process which is 0 with some probability (1−α) and is a random variable with probability α. Our estimation method transforms such a model into a linear state space with mixture of normal innovations, so that an extension of Kalman filter algorithm can be applied. We apply this random level shift model to the logarithm of absolute returns for the S&P 500, AMEX, Dow Jones and NASDAQ stock market return indices. Our point estimates imply few level shifts for all series. But once these are taken into account, there is little evidence of serial correlation in the remaining noise and, hence, no evidence of long-memory. Once the estimated shifts are introduced to a standard GARCH model applied to the returns series, any evidence of GARCH effects disappears. We also produce rolling out-ofsample forecasts of squared returns. In most cases, our simple random level shifts model clearly outperforms a standard GARCH(1,1) model and, in many cases, it also provides better forecasts than a fractionally integrated GARCH model. JEL Classification Number: C22.

Working Paper No. 8/2007On the macroeconomic causes of exchange rates volatility

In this paper we study the effect of contemporaneous aggregation of an arbitrarily large number of processes featuring dynamic conditional heteroskedasticity with short memory when heterogeneity across units is allowed for. We look at the memory properties of the limit aggregate. General, necessary, conditions for long memory are derived. More specific results relative to certain stochastic volatility models are also developed, providing some examples of how long memory volatility can be obtained by aggregation. JEL classification: C43

The strong consistency and asymptotic normality of the Whittle estimate of the parameters in a class of exponential volatility processes are established. Among many models of interest, this class includes one-shock models, such as the EGARCH model of Nelson (1991), and two-shock models, such as the SV model of Taylor (1986). The variable of interest might not have finite fractional moment of any order and so, in particular, finite variance is not imposed. We allow for a wide range of degrees of persistence of shocks to conditional variance, allowing for both short and long memory. A detailed Monte-Carlo exercise shows the small-sample properties of the estimator. We present an empirical application using the Standard & Poor’s 500 composite stock index.