Documentation

## Initial Values for regARIMA Model Estimation

`estimate` uses `fmincon` from Optimization Toolbox™ to minimize the negative loglikelihood objective function. `fmincon` requires initial (i.e., starting) values to begin the optimization process.

If you want to specify your own initial values, then use name-value pair arguments. For example, to specify 0.1 for the initial value of a nonseasonal AR coefficient of the error model, pass the name-value pair argument `'AR0',0.1` into `estimate`.

By default, `estimate` generates initial values using standard time series techniques. If you partially specify initial values (that is, specify initial values for some parameters), `estimate` honors the initial values that you set, and generates default initial values for the remaining parameters.

`estimate` enforces stability and invertibility for all seasonal and nonseasonal AR and MA lag operator polynomials of the error model. When you specify initial values for the AR and MA coefficients, it is possible that `estimate` cannot find initial values for the remaining coefficients that satisfy stability and invertibility. In this case, `estimate` honors your initial values, and sets the remaining initial coefficient values to 0.

The way `estimate` generates default initial values depends on the model.

• If the model contains a regression component and intercept, then `estimate` performs ordinary least squares (OLS). `estimate` uses the estimates for `Beta0` and `Intercept0`. Then, `estimate` infers the unconditional disturbances using the regression model. `estimate` uses the inferred unconditional disturbances and the ARIMA error model to gather the other initial values.

• If the model does not contain a regression component and an intercept, then the unconditional disturbance series is the response series. `estimate` uses the unconditional disturbances and the ARIMA error model to gather the other initial values.

This table summarizes the techniques that `estimate` uses to gather the remaining initial values.

Technique to Generate Initial Values
ParameterError Model Does Not Contain MA TermsError Model Contains MA Terms
AROLSSolve the Yule-Walker equations .
MAN/ASolve the Yule-Walker equations .
VariancePopulation variance of OLS residualsVariance of inferred innovation process (using initial MA coefficients)

 Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.