The estimate method for arima models uses fmincon from Optimization Toolbox™ to perform maximum likelihood estimation. This optimization function requires initial (or, starting) values to begin the optimization process.
If you want to specify your own initial values, then use name-value arguments. For example, specify initial values for nonseasonal AR coefficients using the name-value argument AR0.
Alternatively, you can let estimate choose default initial values. Default initial values are generated 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.
When you generate initial values, estimate enforces stability and invertibility for all AR and MA lag operator polynomials. 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 keeps the user-specified initial values, and sets the remaining initial coefficient values to 0.
This table summarizes the techniques estimate uses to generate default initial values. The software uses the methods in this table and the main data set to generate initial values. If you specify seasonal or nonseasonal integration in the model, then estimate differences the response series before initial values are generated. Here, AR coefficients and MA coefficients include both nonseasonal and seasonal AR and MA coefficients.
| Technique to Generate Initial Values | |||
|---|---|---|---|
| Parameter | Regression Coefficients Present | Regression Coefficient Not Present | |
| MA Terms Not in Model | AR coefficients | Ordinary least squares (OLS) | OLS |
| Constant | OLS constant | OLS constant | |
| Regression coefficients | OLS | N/A | |
| Constant variance | Population variance of OLS residuals | Population variance of OLS residuals | |
| MA Terms in Model | AR coefficients | OLS | Solve Yule-Walker equations, as described in Box, Jenkins, and Reinsel [1]. |
| Constant | OLS constant | Mean of AR-filtered series (using initial AR coefficients) | |
| Regression coefficients | OLS | N/A | |
| Constant variance | Population variance of OLS residuals | Variance of inferred innovation process (using initial MA coefficients) | |
| MA coefficients | Solve modified Yule-Walker equations, as described in Box, Jenkins, and Reinsel [1]. | ||
For details about how estimate initializes conditional variance model parameters, see Initial Values for Conditional Variance Model Estimation.
[1] 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.