# dwtest: Durbin-Watson Test In lmtest: Testing Linear Regression Models

## Description

Performs the Durbin-Watson test for autocorrelation of disturbances.

## Usage

 ```1 2``` ```dwtest(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 `dwtest` 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`.

## Value

An object of class `"htest"` containing:

 `statistic` the test statistic. `method` a character string with the method used. `alternative` a character string describing the alternative hypothesis. `p.value` the corresponding p-value. `data.name` a character string with the data name.

## 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, 159–177.

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 χ^2 Random Variables (80V29 p323-333) Applied Statistics 33, 366–369.

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

`lm`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```## 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 dwtest(y1 ~ x) err2 <- filter(err1, 0.9, method="recursive") y2 <- 1 + x + err2 dwtest(y2 ~ x) ```