Package com.imsl.stat

Class LackOfFit

java.lang.Object
com.imsl.stat.LackOfFit
public class LackOfFit extends Object
Performs lack-of-fit test for a univariate time series or transfer function given the appropriate correlation function.

LackOfFit may be used to diagnose lack of fit in both ARMA and transfer function models. Typical arguments for these situations are:

Model lagMin lagMax npFree
ARMA (p, q) 1 \(\sqrt{\texttt{nObservations}} \) p + q
Transfer function 0 \(\sqrt{\texttt{nObservations}} \) r + s

LackOfFit performs a portmanteau lack of fit test for a time series or transfer function containing nObservations observations given the appropriate sample correlation function \(\hat{\rho}(k)\) for k = L, L+1,...,K where L = lagMin and K = lagMax.

The basic form of the test statistic Q is $$Q=n(n+2)\sum_{k=L}^{K}(n-k)^{-1}\hat{\rho}(k) $$

with L = 1 if \(\hat{\rho}(k)\) is an autocorrelation function. Given that the model is adequate, Q has a chi-squared distribution with \(K-L+1-m\) degrees of freedom where m = npFree is the number of parameters estimated in the model. If the mean of the time series is estimated, Woodfield (1990) recommends not including this in the count of the parameters estimated in the model. Thus, for an ARMA(p, q) model set npFree = p + q regardless of whether the mean is estimated or not. The original derivation for time series models is due to Box and Pierce (1970) with the above modified version discussed by Ljung and Box (1978). The extension of the test to transfer function models is discussed by Box and Jenkins (1976, pages 394-395).

See Also:

  • Method Summary

    Modifier and Type
    Method
    Description
    static double[]
    compute(int nObservations, double[] correlations, int npFree, int lagMax)
    Performs lack-of-fit test for a univariate time series or transfer function given the appropriate correlation function using a minimum lag of 1.
    static double[]
    compute(int nObservations, double[] correlations, int npFree, int lagMax, int lagMin)
    Performs lack-of-fit test for a univariate time series or transfer function given the appropriate correlation function.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Method Details

    • compute

      public static double[] compute(int nObservations, double[] correlations, int npFree, int lagMax)
      Performs lack-of-fit test for a univariate time series or transfer function given the appropriate correlation function using a minimum lag of 1.
      Parameters:
      nObservations - an int containing the number of observations of the stationary time series.
      correlations - a double array of length lagMax+1 containing the correlation function.
      npFree - an int scalar specifying the number of free parameters in the formulation of the time series model. npfree must be greater than or equal to zero and less than lagMax. Woodfield (1990) recommends npFree = p + q.
      lagMax - an int scalar specifying the maximum lag of the correlation function.
      Returns:
      a double array of length 2 with the test statistic, Q, and its p-value, p. Under the null hypothesis, Q has an approximate chi-squared distribution with lagMax-lagMin+1-npFree degrees of freedom.

    • compute

      public static double[] compute(int nObservations, double[] correlations, int npFree, int lagMax, int lagMin)
      Performs lack-of-fit test for a univariate time series or transfer function given the appropriate correlation function.
      Parameters:
      nObservations - an int containing the number of observations of the stationary time series.
      correlations - a double array of length lagMax+1 containing the correlation function.
      npFree - an int scalar specifying the number of free parameters in the formulation of the time series model. npfree must be greater than or equal to zero and less than lagMax. Woodfield (1990) recommends npFree = p + q.
      lagMax - an int scalar specifying the maximum lag of the correlation function.
      lagMin - an int scalar specifying the minimum lag of the correlation function. lagMin corresponds to the lower bound of summation in the lack of fit test statistic. Default value is 1.
      Returns:
      a double array of length 2 with the test statistic, Q, and its p-value, p. Under the null hypothesis, Q has an approximate chi-squared distribution with lagMax-lagMin+1-npFree degrees of freedom.