Durbin-Watson Test for Detecting First-Order Autocorrelation in Regression Residuals

Tests for first-order autocorrelation in the residuals of a linear regression model, based on the ratio of successive squared residual differences to the total residual sum of squares.

Usage

durbinWatsonTest(x, ...)

# S3 method for class 'formula'
durbinWatsonTest(
  x,
  data = list(),
  orderBy = NULL,
  alternative = c("greater", "two.sided", "less"),
  iterations = 15,
  exact = NULL,
  tol = 0.0000000001,
  subset,
  na.action = na.omit,
  ...
)

# S3 method for class 'lm'
durbinWatsonTest(
  x,
  data = list(),
  orderBy = NULL,
  alternative = c("greater", "two.sided", "less"),
  iterations = 15,
  exact = NULL,
  tol = 0.0000000001,
  ...
)

# S3 method for class 'numeric'
durbinWatsonTest(
  x,
  data = list(),
  orderBy = NULL,
  alternative = c("greater", "two.sided", "less"),
  iterations = 15,
  exact = NULL,
  tol = 0.0000000001,
  ...
)

# Default S3 method
durbinWatsonTest(x, ...)

Arguments

x: a symbolic description of the model to be tested (a formula), a fitted "lm" object, or a numeric vector of residuals.
...: further arguments passed to or from other methods.
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. For the lm and numeric methods it is used for orderBy only, as the model frame is already fixed there.
orderBy: either a vector z or a one-sided formula like ~ z. The observations in the model are ordered by the size of z; a formula with several terms is used as successive ordering keys. If set to NULL (the default) the observations are assumed to be ordered (e.g., a time series). z may be given at the length of the original data: rows dropped by subset or by na.action are then dropped from z as well. Missing values in z are ordered last.
alternative: a character string specifying the alternative hypothesis, must be one of "greater" (default), "two.sided" or "less".
iterations: an integer specifying the number of iterations used by the "pan" algorithm when computing the exact p-value.
exact: logical. If TRUE the exact p-value is computed via the "pan" algorithm; if FALSE a normal approximation is used. The default is TRUE for sample sizes below 100 and FALSE otherwise.
tol: numeric tolerance. Eigenvalues smaller than tol are treated as zero.
subset: an optional expression indicating which observations to use. Only used for the formula method.
na.action: a function specifying how missing values are handled. Defaults to na.omit(). Only used for the formula method.

Value

An object of class "htest" containing the following components:

statistic: the Durbin-Watson test statistic.
p.value: the p-value of the test.
alternative: a character string describing the alternative hypothesis.
method: a character string indicating the test performed.
data.name: a character string describing the data.

Details

The Durbin-Watson test has the null hypothesis that the autocorrelation of the disturbances is 0. The alternative hypothesis can be specified as greater than, not equal to, or less than 0 via 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 "pan" algorithm (Farebrother, 1980, 1984), implemented via Rcpp. For large sample sizes the algorithm might fail to compute the p-value; in that case a warning is issued and an approximate p-value is returned instead, computed via a normal approximation using the mean and variance of the Durbin-Watson statistic.

Three methods are dispatched:

formula: Fits the model from scratch using data.
lm: Extracts design matrix and response from a fitted "lm" object. The data.name field in the result reflects the model formula, not the name of the object.
numeric: Treats x as a vector of pre-computed residuals and fits an intercept-only design matrix. Note that this is not equivalent to testing residuals from a fitted model with predictors; use the lm or formula method when a model exists.

Note

Based on code by Torsten Hothorn, Achim Zeileis, Clint Cummins, Giovanni Millo and David Mitchell previously published in the lmtest package, with an Rcpp reimplementation of the "pan" algorithm, adapted to conform to package standards.

References

Durbin, J. and Watson, G. S. (1950) Testing for serial correlation in least squares regression I. Biometrika, 37, 409-428.

Durbin, J. and Watson, G. S. (1951) Testing for serial correlation in least squares regression II. Biometrika, 38, 159-178.

Durbin, J. and Watson, G. S. (1971) Testing for serial correlation in least squares regression III. Biometrika, 58, 1-19.

Farebrother, R. W. (1980) Pan's procedure for the tail probabilities of the Durbin-Watson statistic. Applied Statistics, 29, 224-227.

Farebrother, R. W. (1984) AS R53: A remark on algorithms AS 106, AS 153 and AS 155. Applied Statistics, 33, 366-369.

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

Examples

## formula method
set.seed(1)
x <- rep(c(-1, 1), 50)

err1 <- rnorm(100)
durbinWatsonTest(y ~ x, data = data.frame(y = 1 + x + err1, x = x))
#> 
#> 	Durbin-Watson test
#> 
#> data:  y ~ x
#> DW = 1.9762, p-value = 0.4924
#> alternative hypothesis: true autocorrelation is greater than 0
#> 

## autocorrelated errors (rho = 0.9)
err2 <- stats::filter(err1, 0.9, method = "recursive")
durbinWatsonTest(y ~ x, data = data.frame(y = 1 + x + err2, x = x))
#> 
#> 	Durbin-Watson test
#> 
#> data:  y ~ x
#> DW = 0.45961, p-value = 7.862e-15
#> alternative hypothesis: true autocorrelation is greater than 0
#> 

## subset and an ordering variable given at the length of the data
d <- data.frame(y = 1 + x + as.vector(err2), x = x, tt = sample(100),
                grp = rep(c("A", "B"), each = 50))
durbinWatsonTest(y ~ x, data = d, subset = grp == "A", orderBy = ~ tt)
#> 
#> 	Durbin-Watson test
#> 
#> data:  y ~ x
#> DW = 2.2901, p-value = 0.8511
#> alternative hypothesis: true autocorrelation is greater than 0
#> 

## lm method
fit <- lm(y ~ x, data = data.frame(y = 1 + x + err1, x = x))
durbinWatsonTest(fit)
#> 
#> 	Durbin-Watson test
#> 
#> data:  y ~ x
#> DW = 1.9762, p-value = 0.4924
#> alternative hypothesis: true autocorrelation is greater than 0
#> 

## numeric method (pre-computed residuals, intercept-only design assumed)
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)
#> 
#> 	Durbin-Watson test
#> 
#> data:  e_t
#> DW = 1.08, p-value = 0.01328
#> alternative hypothesis: true autocorrelation is greater than 0
#>