172 The Basics of financial economeTrics
be inefficient and sometimes inappropriate because they may not take advan-
tage of the serial dependence in the most effective way. In this chapter, we
will introduce autoregressive moving average (ARMA) models that allow
for serial dependence in the observations.
Autoregressive Models
In finance, some asset returns show serial dependence. Such dependence can
be modeled as an autoregressive process. For example, a first-order autore-
gressive model can be represented as
yt = c + ρyt− 1 + εt (9.1)
where yt is the asset return c and ρ are parameters εt is assumed to be inde-
pendent and identically distributed (i.i.d). The i.i.d process is a white noise
process with mean zero and variance σε^2.
In words, equation (9.1) says that this period’s return depends on the prior
period’s return scaled by the value of ρ.
For example, an estimation of equation (9.1) using CRSP value-weighted
weekly index returns^2 for the period from January 1998 through October
2012 (774 observations) yields
yt^ = 0.15 − 0.07yt− 1
t-statistic (1.65) (1.97)The fact that last week’s return has a statistically significant coefficient shows
that lagged weekly returns might be useful in predicting weekly returns. In
other words, next period’s forecast is a weighted average of the mean of the
weekly return series and the current value of the return.
The first-order autoregressive model can be generalized to an nth order
autoregressive model and can be written as
yt = c + ρ 1 yt− 1 + ρ 2 yt− 2 +... + ρnyt−n + εt (9.2)
(^2) The CRSP value-weighted index, created by the Center for Research in Security
Prices, is a value-weighted index composed of all New York Stock Exchange (NYSE),
American Stock Exchange (AMEX), and NASDAQ stocks. By “value-weighted” it is
meant each stock in the index is weighted by its market capitalization (i.e., number
of common stock shares outstanding multiplied by the stock price).