ABSTRACT
Recall that for autoregressive moving average (ARMA) models with a nontrivial AR component time reversibility holds only for models driven by a Gaussian white noise; an alternative route of generalizing the ARMA model is to construct time series that are non-Gaussian distributed. This is motivated by potential applications in hydrology. See for example the reports by Quimpo (1967) and O’Connell and Jones (1979) where linear time series models driven by lognormal white noise is considered. Figure 8.1 gives the sample path of an AR(1) time series driven by a lognormal noise. It clearly exhibits the time irreversibility feature mentioned in Chapter 5. The modeling of ARMA models driven
Figure 8.1 Sample path of an autoregressive process with lognormal innovations
by non-Gaussian innovations was taken up by Li and McLeod (1988) and Li (1981, Chapter 5). Davies, Spedding, and Watson (1980) studied the skewness and kurtosis of ARMA models with non-Gaussian residuals. Under the assumptions in these references it can be shown that the residual autocorrelations
rˆk = ∑ (aˆt − aˆ)(aˆt−k − aˆ)∑
(aˆt − aˆ)2 , k = 1, . . . ,m (8.1)
where aˆt are residual from the fitted non-Gaussian ARMA model, a¯ =∑ aˆt/n, have an asymptotic multivariate normal distribution similar to
that of (2.8) albeit with a different information matrix I. Note that in (8.1) the at’s are centered so as to take into account the fact that at could have a nonzero mean which is the case with gamma or lognormal innovations. As an example consider the ARMA(1, 0) process
(1− φB)Zt = at , (8.2) where log at is N(0, 1). Note that the maximum likelihood estimator for σ2 is simply
∑ log a2t
/ n, thus, after maximizing over σ2, the concen-
trated conditional log-likelihood can be written
l(max) = constant− n∑
log at − (n− p)2 log (∑ log a2t
n
) (8.3)
A nonlinear optimization algorithm can then be used to find the maximum likelihood estimate φˆ. The three parameter lognormal situation is much more difficult. Hill (1963) has suggested maximum likelihood estimates which may be useful in this situation. Then straightforward calculation yields the information matrix
I =
( e(e− 1) 1− φ2 +
e
(1 − φ)2 ) 2e2 . (8.4)
This implies that the asymptotic variance of rˆ(1) is
1 n
[ 1−
( e(e− 1) 1− φ2 +
e
(1− φ)2 )−1
1 2e2
] ,
and the asymptotic variance of rˆ(k), k > 0, is using (2.8)
1 n
[ 1−
( e(e− 1) 1− φ2 +
e
(1 − φ)2 )−1
φ2(k−1)
2e2
] . (8.5)
Hence the asymptotic variance for rˆ(k) is much closer to 1/n than the corresponding Gaussian situation.
