Autoregressive Moving Average Models 189
could be some Japanese daily stock index returns. These returns are not only
affected by their own past values but will also influence each other. Thus,
the advantage of VAR modeling is not only that one can estimate multiple
time series variables simultaneously but one can also study interrelation-
ships among variables.
The main drawback of VAR is that as the number of variables/lags
increases, the number of parameters to be estimated will increase signifi-
cantly. For example, to estimate a system of three variables with three lags,
we need to estimate a total of 30 parameters.^9 This may lead to having many
lags with statistically insignificant coefficients. In addition, the signs of the
coefficients may change across the lags making it difficult to interpret the
coefficient.
Thus, to study the interrelationship among variables, the estimated VAR
could be used to check for block significance tests, impulse responses, and
variance decompositions.^10 However, one significant benefit of VAR was
discovered by Granger and Engle when they introduced the concept of coin-
tegration, which is the subject of the next chapter.
Key Points
■ (^) Often financial time series exhibit trends where the current values are
related to the past or lagged values.
■ (^) The models that use past observations to predict the current value are
the autoregressive (AR) and moving average (MA) models.
■ (^) An AR model is appropriate when the current value is determined by
the values of variables in the recent past.
■ (^) An MA model is appropriate when the current value is influenced by a
recent shock and shocks in the recent past.
■ (^) Sometimes AR and MA models may require estimation of a large num-
ber of parameters to describe the data. In such circumstances, an autore-
gressive moving average (a mixture of AR and MA terms, or ARMA)
model is recommended.
■ (^) The ARMA model has the advantage of requiring fewer estimated
parameters.
■ (^) Regardless if it is an AR, MA, or ARMA model, it is important to select
the correct number of lags to describe the data.
(^9) The number of parameters to be estimated is determined by k + nk (^2) where k is
number of variables and n is number of lags.
(^10) For a further discussion of these topics, see Rachev et al., Financial Econometrics.