Durbin-Watson Test

Performs the Durbin-Watson test for autocorrelation of disturbances.

DurbinWatsonTest(formula, order.by = NULL,
                 alternative = c("greater", "two.sided", "less"),
                 iterations = 15, exact = NULL, tol = 1e-10, data = list())

Arguments

formula: a symbolic description for the model to be tested (or a fitted "lm" object).
order.by: Either a vector z or a formula with a single explanatory variable like ~ z. The observations in the model are ordered by the size of z. If set to NULL (the default) the observations are assumed to be ordered (e.g., a time series).
alternative: a character string specifying the alternative hypothesis.
iterations: an integer specifying the number of iterations when calculating the p-value with the "pan" algorithm.
exact: logical. If set to FALSE a normal approximation will be used to compute the p value, if TRUE the "pan" algorithm is used. The default is to use "pan" if the sample size is < 100.
tol: tolerance. Eigenvalues computed have to be greater than tol to be treated as non-zero.
data: an optional data frame containing the variables in the model. By default the variables are taken from the environment which DurbinWatsonTest is called from.

Details

The Durbin-Watson test has the null hypothesis that the autocorrelation of the disturbances is 0. It is possible to test against the alternative that it is greater than, not equal to, or less than 0, respectively. This can be specified by the alternative argument.

Under the assumption of normally distributed disturbances, the null distribution of the Durbin-Watson statistic is the distribution of a linear combination of chi-squared variables. The p-value is computed using the Fortran version of Applied Statistics Algorithm AS 153 by Farebrother (1980, 1984). This algorithm is called "pan" or "gradsol". For large sample sizes the algorithm might fail to compute the p value; in that case a warning is printed and an approximate p value will be given; this p value is computed using a normal approximation with mean and variance of the Durbin-Watson test statistic.

Examples can not only be found on this page, but also on the help pages of the data sets bondyield, currencysubstitution, growthofmoney, moneydemand, unemployment, wages.

For an overview on R and econometrics see Racine & Hyndman (2002).

Value

An object of class "htest" containing:

statistic: the test statistic.
p.value: the corresponding p-value.
method: a character string with the method used.
data.name: a character string with the data name.

Note

This function was previously published as dwtest in the lmtest package and has been integrated here without logical changes.

Author

Torsten Hothorn, Achim Zeileis, Richard W. Farebrother (pan.f), Clint Cummins (pan.f), Giovanni Millo, David Mitchell

References

J. Durbin & G.S. Watson (1950), Testing for Serial Correlation in Least Squares Regression I. Biometrika 37, 409–428.

J. Durbin & G.S. Watson (1951), Testing for Serial Correlation in Least Squares Regression II. Biometrika 38, 159–178.

J. Durbin & G.S. Watson (1971), Testing for Serial Correlation in Least Squares Regression III. Biometrika 58, 1–19.

R.W. Farebrother (1980), Pan's Procedure for the Tail Probabilities of the Durbin-Watson Statistic (Corr: 81V30 p189; AS R52: 84V33 p363- 366; AS R53: 84V33 p366- 369). Applied Statistics 29, 224–227.

R. W. Farebrother (1984), [AS R53] A Remark on Algorithms AS 106 (77V26 p92-98), AS 153 (80V29 p224-227) and AS 155: The Distribution of a Linear Combination of \(\chi^2\) Random Variables (80V29 p323-333) Applied Statistics 33, 366–369.

W. Krämer & H. Sonnberger (1986), The Linear Regression Model under Test. Heidelberg: Physica.

J. Racine & R. Hyndman (2002), Using R To Teach Econometrics. Journal of Applied Econometrics 17, 175–189.

Examples


## generate two AR(1) error terms with parameter
## rho = 0 (white noise) and rho = 0.9 respectively
err1 <- rnorm(100)

## generate regressor and dependent variable
x <- rep(c(-1,1), 50)
y1 <- 1 + x + err1

## perform Durbin-Watson test
DurbinWatsonTest(y1 ~ x)
#> 
#> 	Durbin-Watson test
#> 
#> data:  y1 ~ x
#> DW = 2.1755, p-value = 0.8383
#> alternative hypothesis: true autocorrelation is greater than 0
#> 

err2 <- stats::filter(err1, 0.9, method="recursive")
y2 <- 1 + x + err2
DurbinWatsonTest(y2 ~ x)
#> 
#> 	Durbin-Watson test
#> 
#> data:  y2 ~ x
#> DW = 0.35099, p-value < 2.2e-16
#> alternative hypothesis: true autocorrelation is greater than 0
#> 

## for a simple vector use:
e_t <- c(-32.33, -26.603, 2.215, -16.967, -1.148, -2.512, -1.967, 11.669,
         -0.513, 27.032, -4.422, 40.032, 23.577, 33.94, -2.787, -8.606,
          0.575, 6.848, -18.971, -29.063)
DurbinWatsonTest(e_t ~ 1)
#> 
#> 	Durbin-Watson test
#> 
#> data:  e_t ~ 1
#> DW = 1.08, p-value = 0.01328
#> alternative hypothesis: true autocorrelation is greater than 0
#>