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Akaike information criteria (AIC) - A model order selection criteria based on parsimony. More complicated models are penalized for the inclusion of additional parameters. See also Bayesian information criteria (BIC).
AR - Auto-Regressive. AR models include past observations of the dependent variable in the forecast of future observations.
ARCH - Auto-Regressive Conditional Heteroscedasticity. A time series technique in which past observations of the variance are used to forecast future variances. See also GARCH.
ARMA - Auto-Regressive Moving Average. A time series model that includes both AR and MA components. See also AR and MA.
auto-correlation function (ACF) - Correlation sequence of a random time series with itself. See also cross-correlation function (XCF).
Bayesian information criteria (BIC) - A model order selection criteria 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 (AIC).
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.
cross-correlation function (XCF) - Correlation sequence between two random time series. See also auto-correlation function (ACF).
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 Auto-Regressive 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.
homoskedasticity - Time-independent variance. The GARCH Toolbox also refers to homoskedasticity 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. An technique designed to minimize the variance of the estimation or forecast error. See also RMSE.
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 auto-correlation function (PACF) - Correlation sequence estimated by fitting successive order auto-regressive models to a random time series by least squares. The PACF is useful for identifying the order of an auto-regressive model.
path - A random trial of a time series process.
P-value - The lowest level of significance at which a test statistic is significant.
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.
stationarity constraint - Constraint imposed during estimation such that the sum of the GARCH model conditional variance parameters is less than unity.
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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