Linear Models

Abstract

Given the empirical evidence and theoretical arguments for some degree of predictability in financial markets (see Chapter 3), any departures in asset returns from the random walk model would provide a theoretical basis for a more general form for the expectations of asset returns. Under this assumption the naïve estimate, based purely on statistics of historical data, is replaced by a general estimate conditioned on the most recent past returns and/or fundamental (or market) factors that influence, to some degree, the future behaviour of the financial time series. The unconditional estimates of expected returns and risks, are substituted by statistical time series forecasting models defined by the conditional mean, conditional variance and conditional correlation and, for two time series x and y, take the form:

$$ \begin{array}{l} \hat y_c = E\left[ {y_t \left| {F_{t - \tau } } \right.} \right] \\ \sigma _c^2 = E\left[ {\left( {y_t - \hat y_c } \right)^2 \left| {F_{t - \tau } } \right.} \right] \\ \rho _c = E\left[ {\left( {y_{t - } \hat y_c } \right)\left( {x_{t - } \hat x_c } \right)\left| {F_{t - \tau } } \right.} \right] \\ \end{array} $$

(9.1)

where the expectations are conditional on some vector of lagged time series variables, F _t-1.