The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 287 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 287

Amongst the most important stylized facts from the point of view of

finance theory, we can mention the following:

■ Returns of individual stocks exhibit nearly zero autocorrelation at

every lag.

■ Returns of some equity portfolios exhibit significant autocorrelation.

■ The volatility of returns exhibits hyperbolic decay with significant

autocorrelation.

■ The distribution of stock returns is not normal for time horizons from

a few minutes to a few days. The exact shape is difficult to ascertain

but power law decay cannot be rejected.

■ The distribution of stock returns is close to a log-normal after a few

days.

■ There are large stock price drops (that is, market crashes) that seem to

be outliers with respect to both normal distributions and power law

distributions.

■ Stock return time series exhibit significant cross-correlation.

These findings are, in a sense, model-dependent. For instance, the

distribution of returns, a subject that has received a lot of attention, can

be fitted by different distributions. There is no firm evidence on the

exact value of the power exponent, with alternative proposals based on

variable exponents. The autocorrelation is model-dependent while the

exponential decay of return autocorrelation can be interpreted only as

absence of linear dependence.

It is fair to say that these stylized facts set the stage for financial model-

ing but leave ample room for model selection. Financial time series seem to

be nearly random processes that exhibit significant cross correlations and,

in some instances, cross autocorrelations. The global structure of auto and

cross correlations, if it exists at all, must be fairly complex and there is no

immediate evidence that financial time series admit a simple DGP.

One more important feature of financial time series is the presence

of trends. Prima facie trends of economic and financial variables are

exponential trends. Trends are not quantities that can be independently

measured. Trends characterize an entire stochastic model. Therefore

there is no way to arrive at an assessment of trends independent from

the model. We will see later in this chapter that a number of models

reject the assumption of exponential trends. Exponential trends are,

however, a reasonable first approximation.

Given the finite nature of world resources, exponential trends are

not sustainable in the long run. However, they might still be a good

approximation over limited time horizons. An additional insight into

financial time series comes from the consideration of investors’ behav-