# naginterfaces.library.correg.linregm_​stat_​durbwat¶

naginterfaces.library.correg.linregm_stat_durbwat(ip, res)[source]

linregm_stat_durbwat calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.

For full information please refer to the NAG Library document for g02fc

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/g02/g02fcf.html

Parameters
ipint

, the number of independent variables in the regression model, including the mean.

resfloat, array-like, shape

The residuals, .

Returns
dfloat

The Durbin–Watson statistic, .

pdlfloat

Lower bound for the significance of the Durbin–Watson statistic, .

pdufloat

Upper bound for the significance of the Durbin–Watson statistic, .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, mean of .

Constraint: the mean of the residuals , where .

(errno )

On entry, all residuals are identical.

Notes

For the general linear regression model

 where y is a vector of length n of the dependent variable, X is an n×p matrix of the independent variables, β is a vector of length p of unknown parameters, and ϵ is a vector of length n of unknown random errors.

The residuals are given by

and the fitted values, , can be written as for an matrix . Note that when a mean term is included in the model the sum of the residuals is zero. If the observations have been taken serially, that is can be considered as a time series, the Durbin–Watson test can be used to test for serial correlation in the , see Durbin and Watson (1950), Durbin and Watson (1951) and Durbin and Watson (1971).

The Durbin–Watson statistic is

Positive serial correlation in the will lead to a small value of while for independent errors will be close to . Durbin and Watson show that the exact distribution of depends on the eigenvalues of the matrix where the matrix is such that can be written as

and the eigenvalues of the matrix are , for .

However bounds on the distribution can be obtained, the lower bound being

and the upper bound being

where the are independent standard Normal variables. The lower tail probabilities associated with these bounds, and , are computed by stat.prob_durbin_watson. The interpretation of the bounds is that, for a test of size (significance) , if the test is significant, if the test is not significant, while if and no conclusion can be reached.

The above probabilities are for the usual test of positive auto-correlation. If the alternative of negative auto-correlation is required, then a call to stat.prob_durbin_watson should be made with the argument taking the value of ; see Newbold (1988).

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

Granger, C W J and Newbold, P, 1986, Forecasting Economic Time Series, (2nd Edition), Academic Press

Newbold, P, 1988, Statistics for Business and Economics, Prentice–Hall