Analysis of Financial Time Series, Third Edition

2.4 Simple AR Models

The fact that the monthly return rt of CRSP value-weighted index has a statistically significant lag-1 autocorrelation indicates that the lagged return rt−1 might be useful in predicting rt. A simple model that makes use of such predictive power is

2.8

where {at} is assumed to be a white noise series with mean zero and variance . This model is in the same form as the well-known simple linear regression model in which rt is the dependent variable and rt−1 is the explanatory variable. In the time series literature, model (2.8) is referred to as an autoregressive (AR) model of order 1 or simply an AR(1) model. This simple model is also widely used in stochastic volatility modeling when rt is replaced by its log volatility; see Chapters 3 and 12.

The AR(1) model in Eq. (2.8) has several properties similar to those of the simple linear regression model. However, there are some significant differences between the two models, which we discuss later. Here it suffices to note that an AR(1) model implies that, conditional on the past return rt−1, we have

That is, given the past return rt−1, the current return is centered around ϕ0 + ϕ1rt−1 with standard deviation σa. This ...