GARCH Toolbox™

lbqtest - Run Ljung-Box Q-statistic lack-of-fit hypothesis test

Syntax

[H,pValue,Qstat,CriticalValue] = ... lbqtest(Series,Lags,Alpha,DoF)

Description

[H,pValue,Qstat,CriticalValue] = ... lbqtest(Series,Lags,Alpha,DoF) performs the Ljung-Box lack-of-fit hypothesis test for model misspecification, which is based on the Q-statistic

where N = sample size, L = the number of autocorrelation lags included in the statistic, and is the squared sample autocorrelation at lag k.

Once you fit a univariate model to an observed time series, you can use the Q-statistic as a lack-of-fit test for a departure from randomness. Under the null hypothesis that the model fit is adequate, the test statistic is asymptotically chi-square distributed.

Input Arguments

`Series`	Vector of observations of a univariate time series for which `lbqtest` computes the sample Q-statistic. The last row of `Series` contains the most recent observation of the stochastic sequence. Typically, `Series` is either: The sample residuals derived from fitting a model to an observed time series, or The standardized residuals obtained by dividing the sample residuals by the conditional standard deviations.
`Lags`	Vector of positive integers indicating the lags of the sample autocorrelation function included in the Q-statistic. If specified, each lag must be less than the length of `Series`. If `Lags = []` or is unspecified, the default is `Lags =` `min([20,` `length(Series)-1])`.
`Alpha`	Significance levels. `Alpha` can be a scalar applied to all lags, or a vector the same length as `Lags`. If `Alpha = []` or is unspecified, the default is `0.05`. For all elements, α, of `Alpha`,0 < α < 1.
`DoF`	Degrees of freedom. `DoF` can be a scalar applied to all lags, or a vector the same length as `Lags`. If specified, all elements of `DoF` must be positive integers less than the corresponding element of `Lags`. If `DoF = []` or is unspecified, the elements of `Lags` serve as the default degrees of freedom for the chi-square distribution.

Output Arguments

`H`	Boolean decision vector. 0 indicates acceptance of the null hypothesis that the model fit is adequate (no serial correlation at the corresponding element of `Lags`). 1 indicates rejection of the null hypothesis. `H` is the same size as `Lags`.
`pValue`	Vector of p-values (significance levels) at which `lbqtest` rejects the null hypothesis of no serial correlation at each lag in `Lags`.
`Qstat`	Vector of Q-statistics for each lag in `Lags`.
`CriticalValue`	Vector of critical values of the chi-square distribution for comparison with the corresponding element of `Qstat`.

Examples

Example 1

Create a vector of 100 Gaussian random numbers:

randn('state', 100)       % Start from a known state.
Series = randn(100, 1);   % 100 Gaussian deviates ~ N(0, 1)

Compute the Q-statistic for autocorrelation lags 20 and 25 at the 10 percent significance level:

[H, P, Qstat, CV] = lbqtest(Series, [20 25]', 0.10);
[H, P, Qstat, CV]

ans =
         0    0.9615   10.3416   28.4120
         0    0.9857   12.1015   34.3816

Example 2

See Pre-Estimation Analysis.

References

Box, G.E.P., G.M. Jenkins, and G.C. Reinsel, Time Series Analysis: Forecasting and Control, Third edition, Prentice Hall, 1994.

Gourieroux, C., ARCH Models and Financial Applications, Springer-Verlag, 1997.

lagmatrix

lratiotest