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NAG Toolbox

NAG Toolbox: nag_stat_prob_durbin_watson (g01ep)

Purpose

nag_stat_prob_durbin_watson (g01ep) calculates upper and lower bounds for the significance of a Durbin–Watson statistic.

Syntax

[pdl, pdu, ifail] = g01ep(n, ip, d)
[pdl, pdu, ifail] = nag_stat_prob_durbin_watson(n, ip, d)

Description

Let r = (r1,r2,,rn)T r = (r1,r2,,rn)T  be the residuals from a linear regression of yy on pp independent variables, including the mean, where the yy values y1,y2,,yny1,y2,,yn 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 = (i = 1n1(ri + 1ri)2)/(i = 1nri2),
d=i=1 n-1 (ri+1-ri) 2 i=1nri2 ,
which can be written as
d = (rTAr)/(rTr) ,
d=rTAr rTr ,
where the nn by nn matrix AA is given by
A =
[ 1 − 1 0 … : − 1 2 − 1 … : 0 − 1 2 … : : 0 − 1 … : : : : … : : : : … − 1 0 0 0 … 1 ]
A=[ 1 -1 0 : -1 2 -1 : 0 -1 2 : : 0 -1 : : : : : : : : -1 0 0 0 1 ]
with the nonzero eigenvalues of the matrix AA being λj = (1cos(πj / n))λj=(1-cos(πj/n)), for j = 1,2,,n1j=1,2,,n-1.
Durbin and Watson show that the exact distribution of dd depends on the eigenvalues of a matrix HAHA, where HH is the hat matrix of independent variables, i.e., the matrix such that the vector of fitted values, y^, can be written as = Hyy^=Hy. However, bounds on the distribution can be obtained, the lower bound being
dl = (i = 1npλiui2)/(i = 1npui2)
dl=i=1 n-pλiui2 i=1 n-pui2
and the upper bound being
du = (i = 1npλi 1 + pui2)/(i = 1npui2),
du=i= 1 n-pλi- 1+pui2 i= 1 n-pui2 ,
where uiui are independent standard Normal variables.
Two algorithms are used to compute the lower tail (significance level) probabilities, plpl and pupu, associated with dldl and dudu. If n60n60 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 dd should be replaced by 4d4-d.

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
Farebrother R W (1980) Algorithm AS 153. Pan's procedure for the tail probabilities of the Durbin–Watson statistic Appl. Statist. 29 224–227
Imhof J P (1961) Computing the distribution of quadratic forms in Normal variables Biometrika 48 419–426
Newbold P (1988) Statistics for Business and Economics Prentice–Hall
Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan 7 328–337

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
nn, the number of observations used in calculating the Durbin–Watson statistic.
Constraint: n > ipn>ip.
2:     ip – int64int32nag_int scalar
pp, the number of independent variables in the regression model, including the mean.
Constraint: ip1ip1.
3:     d – double scalar
dd, the Durbin–Watson statistic.
Constraint: d0.0d0.0.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

work

Output Parameters

1:     pdl – double scalar
Lower bound for the significance of the Durbin–Watson statistic, plpl.
2:     pdu – double scalar
Upper bound for the significance of the Durbin–Watson statistic, pupu.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,nipnip,
orip < 1ip<1.
  ifail = 2ifail=2
On entry,d < 0.0d<0.0.

Accuracy

On successful exit at least 44 decimal places of accuracy are achieved.

Further Comments

If the exact probabilities are required, then the first npn-p eigenvalues of HAHA can be computed and nag_stat_prob_chisq_lincomb (g01jd) used to compute the required probabilities with c set to 0.00.0 and d to the Durbin–Watson statistic.

Example

function nag_stat_prob_durbin_watson_example
n = int64(10);
ip = int64(2);
d = 0.9238;
[pdl, pdu, ifail] = nag_stat_prob_durbin_watson(n, ip, d)
 

pdl =

    0.0610


pdu =

    0.0060


ifail =

                    0


function g01ep_example
n = int64(10);
ip = int64(2);
d = 0.9238;
[pdl, pdu, ifail] = g01ep(n, ip, d)
 

pdl =

    0.0610


pdu =

    0.0060


ifail =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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