Durbin-Watson+tables

The Durbin-Watson Test for Serial Correlation with Extreme Sample Sizes or Many
Regressors
Author(s): N. E. Savin and Kenneth J. White
Source: Econometrica, Vol. 45, No. 8 (Nov., 1977), pp. 1989-1996
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1914122
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Econometrica
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Econometrica, Vol. 45, No. 8 (November, 1977)
THE DURBIN-WATSON TEST FOR SERIAL CORRELATION
WITH EXTREME SAMPLE SIZES OR MANY REGRESSORS'
BY N. E. SAVIN AND KENNETH J. WHITE
This paper presents extended tables for the Durbin and Watson [3 and 4] bounds test
The tables can be used for samples with 6 to 200 observations and for as many as 20
regressors.
1. INTRODUCTION
Recent studies by Durbin and Watson [5], L'Esperance and Taylor [10], Koerts
and Abrahamse [8], Tillman [15], Vinod [16], Savin and White [14] and others
have shown increasing interest in the test of autocorrelation based on the d
statistic proposed by Durbin and Watson [3 and 4]. The focus of these papers has
been the computation of the exact distribution of d and the power of the test based
on d. The exact distribution of d has been developed by Imhof [7] and Pan Jie-Jian
[12]. However, few of the generally available computer programs for regression
analysis incorporate these methods,2 possibly because of computational costs,
particularly for large samples. With the Durbin and Watson [4] tables the bounds
test is restricted to time series regressions with 15 to 100 observations and a
maximum of 5 regressors in addition to unity. Often regression studies do not
meet these restrictions since samples with less than 15 observations commonly
occur with annual time series and regressions with more than 5 regressors are
often found in the context of simultaneous equations and of distributed lags.3 In
this paper we present extended tables for the bounds test. Our tables can be used
for samples with 6 to 200 observations and for as many as 20 regressors.
2. THE DISTRIBUTION OF THE d STATISTIC
Consider the regression model
y = X3 + ?-
where y is an n x 1 vector of consecutive observations of the dependent variable,
X is an n x k matrix of observations of k fixed regressors, and s is an n x 1
1 This paper was completed while the authors were visiting at the University of British Columbia in
1975-1976. We would like to thank James Durbin, Geoffrey Watson, Gene Golub, Wilford
L'Esperance, Hrishikesh Vinod, Chinh Le, and Ernst Berndt for helpful discussions and assistance.
We would also like to thank the University of British Columbia for its generous support which made
this paper possible.
2 One such program is SHAZAM, written by White [17] which incorporates both the Imhof an
Pan Jie-Jian methods and uses a computational procedure suggested by Golub [6] which is outlined in
Savin and White [14].
3 Durbin [2] has shown that in the case of simultaneous equations the appropriate number o
regressors for the bounds test is the number of exogenous variables in the system and not the number of
right-hand variables in the structural equation being estimated. Clearly, the number of exogenous
variables in the system can easily exceed 5. This point, which is frequently overlooked, was noted by
Berndt and Christensen [1], but they were unable to carry out the bounds test with the existing Durbin
and Watson tables.
1989
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1990 N. E. SAVIN AND K. J. WHITE
normally distributed disturbance vector. It is assumed that X has full rank k and
that the elements of one of its columns are all unity. The d statistic is defined as
d = e'Ae/e'e
where
e = [I - X(XXf'X']s = Me
is an n x 1 vector of least squares residuals and where A is the usual successive
difference matrix. The distribution of d depends on X since it involves the roots of
MA. Durbin and Watson found a pair of bounding random variables dL and du
with distributions involving only the roots of A and tabulated their lower tail
significance points in [4].
Originally Durbin and Watson [4] computed the tables by hand using an
approximation based on the beta distribution and Jacobi polynomials. This
required interpolation in Karl Pearson's Tables of the Incomplete Beta Function
[13]. Imhof [7] presented an exact method for computing the distribution of
quadratic forms in normal variables. Koerts and Abrahamse [8] used the Imhof
method to recalculate the significance points of the bounding random variables dL
and du. Vinod [16] also used the Imhof method to generalize the bounds test for
higher order autoregressive processes. Pan Jie-Jian [12] used the results of his
earlier paper [11] to develop an alternative to the Imhof method. A proof of the
Pan Jie-Jian method is given in Durbin and Watson [5]. Recently, L'Esperance,
Chall, and Taylor [9] have suggested a method which employs complex integration.
Our extended tables are presented below. All computations were performed in
double precision Fortran on an IBM 370-168 computer. For sample sizes less than
15 we found the Pan Jie-Jian method to be computationally efficient compared to
the Imhof method. This confirms the experience reported by Durbin and Watson
[5]. For sample sizes 1 5 to 100 and up to 5 regressors in addition to unity our tables
are taken from Koerts and Abrahamse [8]. For 6 to 20 regressors and up to 80
observations we employed the Pan Jie-Jian method. All entries for more than 80
observations are computed using the Imhof method. While either method may be
used we recommend the switch to the Imhof method at 80 observations on the
basis of our computations.
The computer programs which implement the Imhof and Pan Jie-Jian methods
involve the numerical analysis of certain integrals. The accuracy of the computations is a function of the step size and truncation error. These were set in order to
obtain at least three digit accuracy past the decimal point. For values of n not
shown in the table, we suggest linear interpolation which should yield an accuracy
of two digits past the decimal point. It is worth remarking that in a few cases the
Imhof program failed to converge when the d statistic was equal to the value of
one of the roots of A. Finally, we note that the computations required for large
values of n were extremely heavy, thus making any further extension of the tables
impractical.
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DURBIN-WATSON
TEST
1991
In examining the critical values of the bounds test it is instructive to also
consider the minimum and maximum roots of the A matrix since these represent
the minimum and maximum values of the d statistic. The minimum and maximum
values of the d statistic for selected sample sizes are presented in Table I. For small
TABLE I
THEORETICAL RANGE OF d STATISTIC
n Minimum Maximum
2
3
2.0000
1.0000
3.0000
4 .5858
5 .3820
10 .0979
15 .0437
30 .0110
50 .0039
100 .0010
200 .0002
300 .0001
3.4142
3.6180
3.9021
3.9563
3.9890
3.9961
3.9990
3.9998
3.9999
500
.0000
2.0000
4.0000
sample sizes the inconclusive region is large relative to the range of the d statistic
which suggests that an exact test may often be required.
Trinity College, Cambridge
and
Rice University
Manuscript received June, 1976; revision received December, 1976.
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1992 N. E. SAVIN AND K. J. WHITE
TABLEI
90.58 (13P2790.8'U-
DURBIN-WATSOC:1PEGIFANOTSdLDW'
10.6539"27 8.03'51249-
130.78624950.31P628490.3182-
201.6893 701.6 239751.6082 71.59
150.63798 1.67593810.729315.867 10.52693810.26 8710.693718.56 951.02873691. 8250361.987531.6 901.86547321.89603 1.687 321.769
851.29 345781.60 32971.68 2731. 801.6594 1.687305942.8163 57.29183
751.48902371.689031.28697 1.85 701.4298 361.978 2361.58079241.6 651.407839 1.6823091.68720931.8 601.389547201.83945 1.637982410.78
51.3642708 91.U20 76381.490 781.3 501.324896 1.203857691.02378 50.9164
891.2P3764018.92 065183.97 021834.9 801.24639 871.05 94620.81759 6. 391.278 931.078 2690.317820.197 381.27069 81.720"968.1370 89.51
371.2 69,8 1.0 986520.17 839 0.21 361.20957821.0439)521.687903 1.278 391.078 4391.207861.9087 .6923 381.2960 78431.2950 67.83109 257.6 31.729803.619)58730.1698 750.612 321.60879541.20 795861.0 R7349681.502
31.87290 .6197083.2179068.5213 301.26791840.57R~6812.09 62.8105 291.843501.92 860719.23 58706.3219 281.043796"80.1327690.183 6290.4 271.0893 40.8719 6230.197 3620.89 261.07 39810.5763 90.41R726580.413 251.098360.15298670.1542930.6 241.037968240.19576820.;1"45798320.1 231.087Q940.15367280.1943 280.79
20.9718 340.79156 .849051237 6.80254 210.9768 30.719562 '0.478592310.685 20.9518763'04.951783 0.4621 38950.271
190.28356741C.0P896713.20957.16283 180.9265781.0u236495 .82013 .846721905 170.82 9561430.S78 9q20Q31.467928013.5 160.8 732596C.10837 92.r,160829.31
150.8729460.Q81739240.6518730.6 180.7692510.47938250.1672983-
1207.69354a019.728346059-.12 3O
io0.6"137290.6-
80.497135P2-
70.4351629-
60.39142-
ndLOIui]oUdL08JOt
k6=1I2'0 95786=qCkI10
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DURBIN-WATSON
TEST
1993
ak'isthenumbrofgxcldp.
201.567983 241.650798 413.62
150.47389 21.40863579 1
10.3479286 51.039248761
951.2073684 19.057362 1.0954
901.26478 5 1.69748 021.647
851.23604 891.32065 1.27804
801.2574 93061.825793
751.0894632 .094632157
701.3896 51.0974238 0.14927
651.0874392 0.57186230.97
601.3785942 0.1873652
50.9718426350.17842 90.63571
50.913287 640.52318764
450.831972 C65.01234785.61
40.7239615 0.8247615 380.
390.725614 382.05746938.12
380.726514 .082514637 .9
370.682915C40.7326895 0.
360.5821 4 0.9261378 4
350.6421879 30.52671980.3
340.6125 7890.3261 57
30.5821796,4 30.28754691
320.58164920.3761 8452
310.524876 930.2514,768 3
30.528479 60.5248317 60.9
290.4731568 2.4097136 8.450
280.436795 0.146392 85
270.413956 870.13245 96
260.3814 7590.143275 68
250.3481796 523.01748695.23
240.3158679 20.3651497-
230.81657 9 0.31745692-
20.46791835 0-
210.8763495C -
20.178943 60.89- -
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190.453276-
180.34675-
170.843265- -
160.34-
ndLUOt'I S
k'-1=234567890
1994
N.
E.
SAVIN
AND
K.
J.
WHITE
80.76312590.687
TABLEI
120.9738561.40273956.82017349
13.048652.71 049.5230869.25014736
DURBIN-WATSO C:5PERNTSIGFAOrdLNDUa
10.9273458691.2048365.20 - -
90.824136 52.18096- -
201.758 4931.7280 1.783694 1.527863 4 150.7246 193.74685102. 763812.40659187 10.6549371 .65928170.5328160.54871629 951.648723091.57 81.3502 74891.56 42903 901.63572 891.657421.80942761.85 4209 851.62470951.207451.8047291.5 86391 801.6258601.734 5721.80 453 21.86973 25 751.9862 05431.79 8701.45e 28391.67 0395 701.58364 251.70349 681.30247691.83 05948 651.7293 501.6473861.04537861.2093E64 601.549 2801.6947861.3205891.42603 98 51.286049 521.8473681. 429 531.029710 501.38462 1.738251.982467510.93 861024 451.7630 81.6 720 1.3859 1.802 3 401.2539681.52730861.5429061.7829451 391.45082731.65 28791.6504327.908532164 381.4275 9318.65270419.68 39102.7 89120 371.49506 371.5249Ilqo71.380 9412.05 891 361.452 8791.65423 791.480537912.40 8621 351.4029 831.65270831.94 67012.5984 236 341.95 80271.65 8410. 95702.6981 257 31.8502781.659302781.6904127.856 0921 321.7509421.65073981.04 72 904.18362790 31.64927501.6 73509182. 5018792. 60413 301.52489 671.504397180.3926485.1072 36 291.348705619.24730518.9402586.147320896 281.34765018. 4702815.9 7420198. 3065241 271.36490512.684753016.9274850362.194067 261.304 513.620759183. 2061735.4 2908153 251.8406 123.54876091.278410.2641950 214.7368510. 3759210.837521406.8 506213 231.57468 01.96785 20.46175802.34516270 21.39475031.64987 0.692 74058. 27143 21.40538126.907 8164.32079542.613086 201.4 5370981.6427910.625390.14627385 19.804753691.80 47523.69042 56.890372 4 18.539046 .190827. 603257.461027.38047 17.38056.971 0.64215380.27560.2971834 16.0379825.1703495.6217038.926408.390154 15.073694 81.75069 20.47 32 51.970326 48 14.0539 1.76 032 .5960827. 4031.276-
10.879326410.5 376240.8- - -
70.1356489- -_ -
60.14- _ _-
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dLU driLU l k'-1=2345k'6=78910
DURBIN-WATSON
TEST
1995
170.38 5-
ak'isthenumbrofg xcldintherp.
190.2356 407.2-
201.6583921.0869 31.508*176I.95 41 150.79826',153).90 4156.0972814.065231840 10.438926510.743-21.06351280.7 53129.68 951.0482 63701.98452C 0(1.96827 0.16 59728 901.3576 041.95382 01.6952 3.1087296 851.369042715.98206731. 520917. 2061 801.34957 283.0(15923.15 62 I3.01,28765 751.309 267.04315289.31562 09.315827 701.298P63 02.176390(8.152 73082.51 -97362 651.23094162.03 851.02967802.3941 601.8234579106.2 370.928513 .A20874 936 51.29067 (52.103 96f;.28103796.82045 17. 501.642390.725 89.306241(79 0.53261 .7 450.982163 7.908236 9.7051268 .905273 8 40.8962 3I0785.124367850. 1275.80 92437 390.875261 3.04729653.0714925.08319240. 380.5267913.048625 10a.7352904.8263705 370.8125 70.652 '04.8739520.397 510 360.827960.23157680.2 4172.96035 142 350.782 462.510 9(72.604q81392.0 539.210 340.758269a304.5726(18.'50249.5031270.634 30.71286940.58629A70.389 23.1089 23 320.71685 2.017359.026359.1023 8 310.672 8504.6 275980.3617 293.08 16 30.627591.0823970.526318.095613 290.615 3792.0(1P83592.0173928.016439528 280.57 16052.839 05.271368 0.32184950 270.56 3092.85471320.8 9413.602581 260.58937 2.1035679.2031 .5087362 4 250.7 4352.98071 35.27601983.65702419 280.43176 92.053 196270.4351 70.683- 230.9186 753.20 1(9.03576 8.- 20.398715 20.6319580.2371- 210.37964 82.301e9583.70-
20.63 41590.3267-
180.732654 -
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nLI=1dUk263L0=1dk5U6L7=1'k8dIL20
160.9835- -
1996 N. E. SAVIN AND K. J. WHITE
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