# dwtest

Durbin-Watson test with linear regression model object

## Description

example

p = dwtest(mdl) returns the p-value of the Durbin-Watson Test on the residuals of the linear regression model mdl. The null hypothesis is that the residuals are uncorrelated, and the alternative hypothesis is that the residuals are autocorrelated.

p = dwtest(mdl,method) specifies the algorithm for computing the p-value.

p = dwtest(mdl,method,tail) specifies the alternative hypothesis.

[p,DW] = dwtest(___) also returns the Durbin-Watson statistic using any of the input argument combinations in the previous syntaxes.

## Examples

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Determine whether a fitted linear regression model has autocorrelated residuals.

Load the census data set and create a linear regression model.

mdl = fitlm(cdate,pop);

Find the p-value of the Durbin-Watson autocorrelation test.

p = dwtest(mdl)
p = 3.6190e-15

The small p-value indicates that the residuals are autocorrelated.

## Input Arguments

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Linear regression model, specified as a LinearModel object created using fitlm or stepwiselm.

Algorithm for computing the p-value, specified as one of these values:

• 'exact' — Calculate an exact p-value using Pan’s algorithm [2].

• 'approximate' — Calculate the p-value using a normal approximation [1].

The default is 'exact' when the sample size is less than 400, and 'approximate' otherwise.

Type of alternative hypothesis to test, specified as one of these values:

ValueAlternative Hypothesis
'both'

Serial correlation is not 0.

'right'

Serial correlation is greater than 0 (right-tailed test).

'left'

Serial correlation is less than 0 (left-tailed test).

dwtest tests whether mdl has no serial correlation, against the specified alternative hypothesis.

## Output Arguments

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p-value of the test, returned as a numeric value. dwtest tests whether the residuals are uncorrelated, against the alternative that autocorrelation exists among the residuals. A small p-value indicates that the residuals are autocorrelated.

Durbin-Watson statistic value, returned as a nonnegative numeric value.

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### Durbin-Watson Test

The Durbin-Watson test tests the null hypothesis that linear regression residuals of time series data are uncorrelated, against the alternative hypothesis that autocorrelation exists.

The test statistic for the Durbin-Watson test is

$DW=\frac{\sum _{i=1}^{n-1}{\left({r}_{i+1}-{r}_{i}\right)}^{2}}{\sum _{i=1}^{n}{r}_{i}^{2}},$

where ri is the ith raw residual, and n is the number of observations.

The p-value of the Durbin-Watson test is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. A significantly small p-value casts doubt on the validity of the null hypothesis and indicates autocorrelation among residuals.

## References

[1] Durbin, J., and G. S. Watson. "Testing for Serial Correlation in Least Squares Regression I." Biometrika 37, pp. 409–428, 1950.

[2] Farebrother, R. W. Pan's "Procedure for the Tail Probabilities of the Durbin-Watson Statistic." Applied Statistics 29, pp. 224–227, 1980.