GARCH Toolbox

Glossary

Akaike information criteria (AIC) - A model-order selection criterion based on parsimony. More complicated models are penalized for the inclusion of additional parameters. See also Bayesian information criteria.

AR - Autoregressive. AR models include past observations of the dependent variable in the forecast of future observations.

ARCH - Autoregressive Conditional Heteroscedasticity. A time-series technique in which past observations of the variance are used to forecast future variances. See also GARCH.

ARMA - Autoregressive Moving Average. A time-series model that includes both AR and MA components. See also AR and MA.

autocorrelation function (ACF) - Correlation sequence of a random time series with itself. See also crosscorrelation function.

autoregressive - See AR.

Bayesian information criteria (BIC) - A model-order selection criterion based on parsimony. More complicated models are penalized for the inclusion of additional parameters. Since BIC imposes a greater penalty for additional parameters than AIC, BIC always provides a model with a number of parameters no greater than that chosen by AIC. See also Akaike information criteria.

conditional - Time-series technique with explicit dependence on the past sequence of observations.

conditional mean - Time-series model for forecasting the expected value of the return series itself.

conditional variance - Time-series model for forecasting the expected value of the variance of the return series.

crosscorrelation function (XCF) - Correlation sequence between two random time series. See also autocorrelation function.

equality constraint - A constraint, imposed during parameter estimation, by which a parameter is held fixed at a user-specified value.

excess kurtosis - A characteristic, relative to a standard normal probability distribution, whereby an area under the probability density function is reallocated from the center of the distribution to the tails (fat tails). Samples obtained from distributions with excess kurtosis have a higher probability of containing outliers than samples drawn from a normal (Gaussian) density. Time series that exhibit a fat tail distribution are often referred to as leptokurtic.

explanatory variables - Time series used to explain the behavior of another observed series of interest. Explanatory variables are typically incorporated into a regression framework.

fat tails - See excess kurtosis.

GARCH - Generalized Autoregressive Conditional Heteroscedasticity. A time-series technique in which past observations of the variance and variance forecast are used to forecast future variances. See also ARCH.

heteroscedasticity - Time-varying, or time-dependent, variance.

homoscedasticity - Time-independent variance. The GARCH Toolbox also refers to homoscedasticity as constant conditional variance.

i.i.d. - Independent, identically distributed.

innovations - A sequence of unanticipated shocks, or disturbances. The GARCH Toolbox uses innovations and residuals interchangeably.

leptokurtic - See excess kurtosis.

MA - Moving average. MA models include past observations of the innovations noise process in the forecast of future observations of the dependent variable of interest.

MMSE - Minimum mean square error. A technique designed to minimize the variance of the estimation or forecast error. See also RMSE.

moving average - See MA.

objective function - The function to be numerically optimized. In the GARCH Toolbox, the objective function is the log-likelihood function of a random process.

partial autocorrelation function (PACF) - Correlation sequence estimated by fitting successive order autoregressive models to a random time series by least squares. The PACF is useful for identifying the order of an autoregressive model.

path - A random trial of a time-series process.

P-value - The lowest level of significance at which a test statistic is significant.

realization - See path.

residuals - See innovations.

RMSE - Root mean square error. The square root of the mean square error. See also MMSE.

standardized innovations - The innovations divided by the corresponding conditional standard deviation.

time series - Discrete-time sequence of observations of a random process. The type of time series of interest in the GARCH Toolbox is typically a series of returns, or relative changes of some underlying price series.

transient - A response, or behavior, of a time series that is heavily dependent on the initial conditions chosen to begin a recursive calculation. The transient response is typically undesirable, and initially masks the true steady-state behavior of the process of interest.

unconditional - Time-series technique in which explicit dependence on the past sequence of observations is ignored. Equivalently, the time stamp associated with any observation is ignored.

volatility - The risk, or uncertainty, measure associated with a financial time series. The GARCH Toolbox associates volatility with standard deviation.

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Bibliography