ACF AIC using for model selection aicbic Akaike information criteria. See AIC AR model converting from ARMA model ARCH/GARCH effects hypothesis test archtest ARMA model converting to AR model converting to MA model asymptotic behavior for long-range forecast horizons autocorr auto-correlation function. See ACF auto-regressive model. See AR model Bayesian information criteria. See BIC BIC using for model selection compounding continuous and periodic conditional mean models with regression components conditional standard deviations inferred from observed return series of forecast errors simulating conditional variances constant of the innovations process constraints boundary boundary equality fixing model parameters stationarity and positivity conventions in GARCH Toolbox conventions in GARCH Toolbox convergence considerations determining status crosscorr cross-correlation function. See XCF default model estimation example forecasting example defining a model using a GARCH specification structure documentation conventions estimating initial parameters estimation count of coefficients count of coefficients incorporating a regression model of GARCH process parameters summary information estimation example estimating the model parameters post-estimation analysis pre-estimation analysis using the default model fixing model parameters forecast how to compute forecast errors conditional standard deviations conditional standard deviations forecasted explanatory data forecasting asymptotic behavior computing RMSE conditional mean conditional standard deviation incorporating a regression model minimum mean square error volatility. See MMSE volatility MMSE volatility MMSE volatility plotting results using the default model GARCH limitations overview uses for GARCH model default GARCH process forecasting inferring innovations objective function parameter estimation count of coefficients displaying results plotting results simulation GARCH specification structure contents creating and modifying parameters creating and modifying parameters definition of fields fixing model parameters parameters that affect convergence retrieving parameters use of parameters in simulation using as function input and output using to define a model GARCH Toolbox conventions and clarifications array definitions compounding precision of calculations row and column conventions stationarity overview recommendations and suggestions garchar garchcount garchdisp garchfit garchget garchinfer garchllfn garchma garchplot garchpred garchset garchsim homoskedasticity unconditional variance hypothesis tests ARCH/GARCH effects likelihood ratio Ljung-Box lack-of-fit inference using a regression model inferring conditional standard deviations GARCH innovations innovations distribution inferring from observed return series serial dependence simulating lack-of-fit hypothesis test lagged time series matrix lagmatrix lbqtest likelihood ratio hypothesis test likelihood ratio tests using for model selection Ljung-Box lack-of-fit hypothesis test log-likelihood objective function computing values gradient values maximization optimized value lratiotest MA model converting from ARMA model maximum likelihood estimation model parameters boundary constraints equality constraints estimating initial estimates model selection and analysis using AIC and BIC using likelihood ratio tests Monte Carlo simulation moving average model. See MA model PACF parameter estimation of GARCH process displaying results parcorr parsimonious parameterization parsimonious parameterization partial auto-correlation function See PACF plotting auto-correlation function cross-correlation function forecasting results parameter estimation results partial auto-correlation function simulation results prerequisites price series converting from return series converting to return series price2ret regression in a Monte Carlo framework regression components in estimation in forecasting in inference in simulation of conditional mean models ret2price return series converting from price series converting to price series data size and quality simulating RMSE computing for forecasted data root mean square error. See RMSE selecting a model shifted time series matrix simulating sample paths simulation of GARCH process plotting results using a regression model using ordinary least squares regression simulation example using a higher order model using the default model specification structure. See GARCH specification structure stationary and nonstationary time series time series correlation of observations stationary and nonstationary time series matrix lagged or shifted transient effects minimizing overview transients in the simulation process unconditional variances of the innovations process variances conditional and unconditional volatility clustering XCF