Details

SAS Macros and Functions

Details

Assume that the transformed series Y_t is a stationary pth order autoregressive process generated by independent normally distributed innovations.

$(1 - {\Theta}(B))(Y_{t} - {\mu}) = {\epsilon}_{t}$

${\epsilon}_{t} \sim iid \rm{N}(0,{\sigma}^2)$

Given these assumptions, the log likelihood function of the transformed data Y_t is

$l_{Y}({\cdot}) = &-& \frac{n}2{\ln}(2{\pi}) - \frac{1}2{\ln}(|{\Sigma}|) - \frac... ...ln}({\sigma}^2) \cr &-& \frac{1}{2{\sigma}^2}(Y-1{\mu})'{\Sigma}^{-1}(Y-1{\mu})$

In this equation, n is the number of observations, ${\mu}$ is the mean of Y_t, 1 is the n-dimensional column vector of 1s, ${{\sigma}^2}$ is the innovation variance, Y = (Y₁, ... ,Y_n)', and ${{\Sigma}}$ is the covariance matrix of Y.

The log likelihood function of the original data X₁, ... , X_n is

$l_{X}({\cdot}) = l_{Y}({\cdot}) + ({\lambda}-1) \sum_{t=1}^n{{\ln}(X_{t}+c)}$

where c is the value of the CONST= option.

For each value of ${\lambda}$ ,the maximum log likelihood of the original data is obtained from the maximum log likelihood of the transformed data given the maximum likelihood estimate of the autoregressive model.

The maximum log likelihood values are used to compute the Akaike Information Criterion (AIC) and Schwarz's Bayesian Criterion (SBC) for each ${\lambda}$ value. The residual mean square error based on the maximum likelihood estimator is also produced. To compute the mean square error, the predicted values from the model are re-transformed to the original scale (Pankratz 1983, pp. 256-258, and Taylor 1986).

After differencing as specified by the DIF= option, the process is assumed to be a stationary autoregressive process. You can check for stationarity of the series with 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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