lbqtest - Ljung-Box Q-test for residual autocorrelation

Syntax

Description

[h,pValue,stat,cValue] = lbqtest(res) and [h,pValue,stat,cValue] = lbqtest(res,Name,Value) perform the Ljung-Box lack-of-fit hypothesis test for model misspecification.

The "portmanteau" test of Ljung and Box assesses the null hypothesis that a series of residuals exhibits no autocorrelation for a fixed number of lags L, against the alternative that some autocorrelation coefficient ρ(k), k = 1, ..., L, is nonzero. The test statistic is

where T is the sample size, L is the number of autocorrelation lags, and ρ(k) is the sample autocorrelation at lag k. Under the null, the asymptotic distribution of Q is chi-square with L degrees of freedom.

Input Arguments

res	Vector of residuals for which the test statistic is computed. The last element corresponds to the most recent observation. Typically, res contains (standardized) residuals obtained by fitting a model to an observed time series.

Name-Value Pair Arguments

'lags'	Scalar or vector of positive integers indicating the number of lags L used to compute the test statistic. Each element must be less than length(res−1). Default: min[20,T-1]
'alpha'	Scalar or vector of nominal significance levels for the tests. Elements must be greater than zero and less than one. Default: 0.05
'dof'	Scalar or vector of degree-of-freedom parameters for the asymptotic chi-square distributions of the test statistics. Elements must be positive integers less than the corresponding element of lags. Default: value of lags

Output Arguments

h	Vector of Boolean decisions for the tests, with length equal to the number of tests. Values of h equal to 1 indicate rejection of the null of no autocorrelation in favor of the alternative. Values of h equal to 0 indicate a failure to reject the null.
pValue	Vector of p-values of the test statistics, with length equal to the number of tests.
stat	Vector of test statistics, with length equal to the number of tests.
cValue	Vector of critical values for the tests, determined by alpha, with length equal to the number of tests.

Examples

Test exchange rates for autocorrelation, ARCH effects:

load Data_MarkPound
returns = price2ret(Data);
residuals = returns-mean(returns);
h1 = lbqtest(residuals)
h2 = lbqtest(residuals.^2)

Algorithms

• The input lags affects the power of the test. If L is too small, the test will not detect high-order autocorrelations; if it is too large, the test will lose power when significant correlation at one lag is washed out by insignificant correlations at other lags. The default value of min[20,T-1] is suggested by Box, Jenkins, and Reinsel [1]. Tsay [4] cites simulation evidence that a value approximating log(T) provides better power performance.

• When res is obtained by fitting a model to data, the degrees of freedom are reduced by the number of estimated coefficients, excluding constants. For example, if res is obtained by fitting an ARMA(p,q) model, dof should be L−p−q.

• lbqtest does not test directly for serial dependencies other than autocorrelation, but it can be used to identify conditional heteroscedasticity (ARCH effects) by testing squared residuals. See, e.g., McLeod and Li [3]. Engle's test, implemented by archtest, tests for ARCH effects directly.

References

[1] Box, G.E.P., G.M. Jenkins, and G.C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1994.

[2] Gourieroux, C. ARCH Models and Financial Applications. New York: Springer-Verlag, 1997.

[3] McLeod, A.I. and W.K. Li. "Diagnostic Checking ARMA Time Series Models Using Squared-Residual Autocorrelations." Journal of Time Series Analysis. Vol. 4, 1983, pp. 269–273.

[4] Tsay,R.S. Analysis of Financial Time Series. Hoboken, NJ: John Wiley & Sons, Inc., 2005.

See Also

archtest | autocorr

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