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Assume that a time series *X*_{t} is a stationary *p*th order
autoregressive process with normally distributed white noise innovations.
That is,

where is the mean of *X*_{t}.

The log likelihood function of *X*_{t} is

where *n* is the number of observations,
**1** is the *n*-dimensional column vector of 1s,
is the variance of the white noise,
**X** = (*X _{1}*, ... ,

On the other hand, if the log transformed time series
*Y*_{t} = ln(*X*_{t}+*c*) is a stationary *p*th order
autoregressive process,
the log likelihood function of *X*_{t} is

where
is the mean of *Y*_{t},
**Y** = (*Y _{1}*, ... ,

The %LOGTEST macro compares the maximum values of
*l _{1}*(·) and

The %LOGTEST macro also computes the Akaike Information Criterion (AIC), Schwarz's Bayesian Criterion (SBC), and residual mean square error based on the maximum likelihood estimator for the autoregressive model. For the mean square error, retransformation of forecasts is based on Pankratz (1983, pp. 256-258).

After differencing as specified by the DIF= option, the process is assumed to be a stationary autoregressive process. You may wish to check for stationarity of the series using the %DFTEST macro. If the process is not stationary, differencing with the DIF= option is recommended. For a process with moving average terms, a large value for the AR= option may be appropriate.

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