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
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
estimate honors the initial values
that you set, and generates default initial values for the remaining parameters.
When you generate initial values,
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
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
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|
|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 .|
|Constant||OLS constant||Mean of AR-filtered series (using initial AR coefficients)|
|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 .|
For details about how
conditional variance model parameters, see Initial Values for Conditional Variance Model Estimation.
 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.