# Documentation

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# dwtest

Class: LinearModel

Durbin-Watson test of linear model

## Syntax

```P = dwtest(mdl) [P,DW] = dwtest(mdl) [P,DW] = dwtest(mdl,method) [P,DW] = dwtest(mdl,method,tail) ```

## Description

`P = dwtest(mdl)` returns the p-value of the Durbin-Watson test on the `mdl` linear model.

```[P,DW] = dwtest(mdl)``` also returns the Durbin-Watson statistic, `DW`.

```[P,DW] = dwtest(mdl,method)``` specifies the method `dwtest` uses to compute the p-value.

```[P,DW] = dwtest(mdl,method,tail)``` specifies the alternative hypothesis.

## Input Arguments

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Full, fitted linear regression model, specified as a `LinearModel` object constructed using `fitlm` or `stepwiselm`.

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

• `'exact'` — Calculates an exact p-value using Pan’s algorithm.

• `'approximate'` — Calculates the p-value using a normal approximation.

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

Alternative hypothesis to test, specified as one of the following:

TailAlternative 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 hypotheses.

## Output Arguments

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p-value of the test, returned as a numeric value. `dwtest` tests if the residuals are uncorrelated, against the alternative that there is autocorrelation among them. Small values of `P` indicate that the residuals are correlated.

Durbin-Watson statistic, returned as a numeric value.

## Examples

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Examine whether the residuals from a fitted model of census data over time have autocorrelated residuals.

Load the census data and create a linear model.

```load census mdl = fitlm(cdate,pop); ```

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

```P = dwtest(mdl) ```
```P = 0 ```

There is significant autocorrelation in the residuals.

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## Algorithms

Approximate calculation of the p-value uses a normal approximation [1]. Exact calculation uses Pan’s algorithm [2].

## 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.