Documentation

## Initial Values for Conditional Variance Model Estimation

The `estimate` function for conditional variance 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, use name-value arguments. For example, specify initial values for GARCH coefficients using the name-value argument `GARCH0`.

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 you do specify, and generates default initial values for the remaining parameters.

When generating initial values, `estimate` enforces any stationarity and positivity constraints for the conditional variance model being estimated. The techniques `estimate` uses to generate default initial values are as follows:

• For the GARCH and GJR models, the model is transformed to an equivalent ARMA model for the squared, offset-adjusted response series. Note that the GJR model is treated like a GARCH model, with all leverage coefficients equal to zero. The initial ARMA values are solved for using the modified Yule-Walker equations as described in Box, Jenkins, and Reinsel . The initial GARCH and ARCH starting values are calculated by transforming the ARMA starting values back to the original GARCH (or GJR) representation.

• For the EGARCH model, the initial GARCH coefficient values are found by viewing the model as an equivalent ARMA model for the squared, offset-adjusted log response series. The initial GARCH values are solved for using Yule-Walker equations as described in Box, Jenkins, and Reinsel . For the other coefficients, the first nonzero ARCH coefficient is set to a small positive value, and the first nonzero leverage coefficient is set to a small negative value (consistent with the expected signs of these 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.