Computes bounds for the significance of a Durbin–Watson statistic

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NAG[g01epc] NAG[nag_prob_durbin_watson] - Computes bounds for the significance of a Durbin–Watson statistic

Calling Sequence

g01epc(n, ip, d, pdl, pdu, 'fail'=fail)

nag_prob_durbin_watson(. . .)

Parameters

n - integer;

On entry: , the number of observations used in calculating the Durbin–Watson statistic.

Constraint: . .

ip - integer;

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

Constraint: . .

d - float;

On entry: , the Durbin–Watson statistic.

Constraint: . .

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, .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_prob_durbin_watson (g01epc) calculates upper and lower bounds for the significance of a Durbin–Watson statistic.

Description

Let be the residuals from a linear regression of on independent variables, including the mean, where the values can be considered as a time series. The Durbin–Watson test (see Durbin and Watson (1950), Durbin and Watson (1951) and Durbin and Watson (1971)) can be used to test for serial correlation in the error term in the regression.

The Durbin–Watson test statistic is:

d=((∑)(r[i+1],-,r[i])^2)/((∑)r[i]^2)

which can be written as

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

where the by matrix is given by

A=[[[1,-1,0,..,:],[-1,2,-1,..,:],[0,-1,2,..,:],[:,0,-1,..,:],[:,:,:,..,:],[:,:,:,..,-1],[0,0,0,..,1]]]

with the non-zero eigenvalues of the matrix being , for .

Durbin and Watson show that the exact distribution of depends on the eigenvalues of a matrix , where is the hat matrix of independent variables, i.e., the matrix such that the vector of fitted values, , can be written as . 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 are independent standard Normal variables.

Two algorithms are used to compute the lower tail (significance level) probabilities, and , associated with and . If the procedure due to Pan (1964) is used, see Farebrother (1980), otherwise Imhof's method (see Imhof (1961)) is used.

The bounds are for the usual test of positive correlation; if a test of negative correlation is required the value of should be replaced by .

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_REAL"

On entry, . Constraint: .

	Accuracy
	On successful exit at least 4 decimal places of accuracy are achieved.

	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 c set to and d to the Durbin–Watson statistic.