Computes Durbin–Watson test statistic

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NAG[g02fcc] NAG[nag_durbin_watson_stat] - Computes Durbin–Watson test statistic

Calling Sequence

g02fcc(p, res, d, pdl, pdu, 'n'=n, 'fail'=fail)

nag_durbin_watson_stat(. . .)

Parameters

p - integer;

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

Constraint: . .

res - Vector(1..n, datatype=float[8]);

On entry: the residuals, .

Constraint: the mean of the residuals , where . .

d - assignable;

Note: On exit the variable d will have a value of type float.

On exit: the Durbin–Watson statistic, .

pdl - assignable;

Note: On exit the variable pdl will have a value of type float.

On exit: lower bound for the significance of the Durbin–Watson statistic, .

pdu - assignable;

Note: On exit the variable pdu will have a value of type float.

On exit: upper bound for the significance of the Durbin–Watson statistic, .

'n'=n - integer; (optional)

On entry: , the number of residuals.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_durbin_watson_stat (g02fcc) calculates the Durbin–Watson statistic, for a set of residuals, and the upper and lower bounds for its significance.

Description

For the general linear regression model

where is a vector of length of the dependent variable,

is a by matrix of the independent variables,

is a vector of length of unknown arguments,

and is a vector of length of unknown random errors.

The residuals are given by

and the fitted values, , can be written as for a by 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

d=((∑)(r[i+1],-,r[i])^2)/((∑)r[i]^2)

Positive serial correlation in the will lead to a small value of while for independent errors will be close to 2. 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

d = r^T*A*r*(1/(r^T*r))

and the eigenvalues of the matrix are , for .

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

d[l] = (sum(lambda[i]*u[i]^2, i = 1 .. n-p))/(sum(u[i]^2, i = 1 .. n-p))

and the upper bound being

d[u] = (sum(lambda[i-1+p]*u[i]^2, i = 1 .. n-p))/(sum(u[i]^2, i = 1 .. n-p))

where the are independent standard Normal variables. The lower tail probabilities associated with these bounds, and , are computed by g01epc (nag_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 g01epc (nag_prob_durbin_watson) should be made with the argument d taking the value of ; see Newbold (1988).

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_INT"

On entry, . Constraint: .

"NE_INT_2"

On entry, , . Constraint: .

"NE_INTERNAL_ERROR"

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.

"NE_RESID_IDEN"

On entry, all residuals are identical.

"NE_RESID_MEAN"

On entry, The mean of res is not approximately , .

	Accuracy
	The probabilities are computed to an accuracy of at least 4 decimal places.

	Further Comments
	If the exact probabilities are required, then the first eigenvalues of can be computed and g01jdc (nag_prob_lin_chi_sq) used to compute the required probabilities with the argument c set to and the argument d set to the Durbin–Watson statistic .

Examples

>	n := 10: p := 2: res := Vector([3.735719, 0.912755, 0.683626, 0.416693, 1.9902, -0.444816, -1.283088, -3.666035, -0.426357, -1.918697], datatype=float[8]): NAG:-g02fcc(p, res, d, pdl, pdu, 'n' = n):