178 The Basics of financial economeTrics
Autoregressive Moving Average Models
Because in practical applications higher-order models may be required
to describe the dynamic structure of the data, AR and MA models may
require estimation of a large number of parameters. In 1938, Herman Wold
showed that a combined AR and MA process, referred to as an autore-
gressive moving average (ARMA) process, can effectively describe the time
series structure of the data as long as the appropriate number of AR and
MA terms are specified.^3 This means that any time series yt can be modeled
as a combination of past yt values and/or past εt errors. More formally, an
ARMA model can be expressed as
yt = c + ρ 1 yt− 1 + ρ 2 yt− 2 +... + ρnyt−n + εt + δ 1 εt− 1 +... + δqεt−q (9.5)
where n and q are the number of AR and MA terms, respectively.
Lags AIC BIC
11 2.035 2.107
12 2.035 2.113
13 2.038 2.122
14 2.041 2.130
15 2.034 2.131
16 2.036 2.138
17 2.039 2.147
18 2.041 2.155
19 2.044 2.164
20 2.046 2.172
21 2.049 2.181
22 2.051 2.189
23 2.053 2.198
24 2.055 2.205A model is selected based on the calculated
minimum of either AIC or BIC.
*Denotes minimum values.table 9.3 (continued)(^3) Herman Wold, A Study in the Analysis of Stationary Time Series (Stockholm,
Sweden: Almgrist & Wiksell, 1938).