ARIMA - Parameter Estimation

The Time Series module of STATISTICA includes different methods for estimating the parameters. All of them should produce very similar estimates, but may be more or less efficient for any given model. In general, during the parameter estimation phase a function minimization algorithm is used (the so-called quasi-Newton method; refer to the description of the Nonlinear Estimation method) to maximize the likelihood (probability) of the observed series, given the parameter values. In practice, this requires the calculation of the (conditional) sums of squares (SS) of the residuals, given the respective parameters. Different methods have been proposed to compute the SS for the residuals; in STATISTICA you can choose among: 1) the approximate maximum likelihood method according to McLeod and Sales (1983), 2) the approximate maximum likelihood method with backcasting, and 3) the exact maximum likelihood method according to Melard (1984).

Comparison of methods. In general, all methods should yield very similar parameter estimates. Also, all methods are about equally efficient in most real-world time series applications. However, method 1 above, (approximate maximum likelihood, no backcasts) is the fastest, and should be used in particular for very long time series (e.g., with more than 30,000 observations; note that the Time Series module is unique in that it will take full advantage of your computer's memory, and it will not impose fixed limitations on the lengths of time series that can be analyzed). Melard's exact maximum likelihood method (number 3 above) may also become inefficient when used to estimate parameters for seasonal models with long seasonal lags (e.g., with yearly lags of 365 days). On the other hand, the Time Series module will always use the approximate maximum likelihood method first in order to establish initial parameter estimates that are very close to the actual final values; thus, usually only a few iterations with the exact maximum likelihood method (3, above) are necessary to finalize the parameter estimates.

Parameter standard errors. For all parameter estimates, the Time Series module computes so-called asymptotic standard errors. These are computed from the matrix of second-order partial derivatives that is approximated via finite differencing (see also the respective discussion in Nonlinear Estimation).

Penalty value. As mentioned above, the estimation procedure requires that the (conditional) sums of squares of the ARIMA residuals be minimized. If the model is inappropriate, it may happen during the iterative estimation process that the parameter estimates become very large, and, in fact, invalid. In that case, the program will assign a very large value (a so-called penalty value) to the SS. This usually "entices" the iteration process to move the parameters away from invalid ranges. However, in some cases even this strategy fails, and you may see on the screen (during the Estimation procedure) very large values for the SS in consecutive iterations. In that case, carefully evaluate the appropriateness of your model. If your model contains many parameters, and perhaps an intervention component (see below), you may try again with different parameter start values.