<s>
In	O
statistics	O
,	O
the	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
statistic	I-General_Concept
is	O
a	O
test	B-General_Concept
statistic	I-General_Concept
used	O
to	O
detect	O
the	O
presence	O
of	O
autocorrelation	O
at	O
lag	O
1	O
in	O
the	O
residuals	O
(	O
prediction	O
errors	O
)	O
from	O
a	O
regression	O
analysis	O
.	O
</s>
<s>
Durbin	O
and	O
Watson	O
(	O
1950	O
,	O
1951	O
)	O
applied	O
this	O
statistic	O
to	O
the	O
residuals	O
from	O
least	B-General_Concept
squares	I-General_Concept
regressions	O
,	O
and	O
developed	O
bounds	O
tests	O
for	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
the	O
errors	O
are	O
serially	O
uncorrelated	O
against	O
the	O
alternative	O
that	O
they	O
follow	O
a	O
first	O
order	O
autoregressive	B-Algorithm
process	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
the	O
distribution	O
of	O
this	O
test	B-General_Concept
statistic	I-General_Concept
does	O
not	O
depend	O
on	O
the	O
estimated	O
regression	O
coefficients	O
and	O
the	O
variance	O
of	O
the	O
errors	O
.	O
</s>
<s>
A	O
similar	O
assessment	O
can	O
be	O
also	O
carried	O
out	O
with	O
the	O
Breusch	O
–	O
Godfrey	O
test	O
and	O
the	O
Ljung	B-General_Concept
–	I-General_Concept
Box	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
If	O
the	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
statistic	I-General_Concept
is	O
substantially	O
less	O
than	O
2	O
,	O
there	O
is	O
evidence	O
of	O
positive	O
serial	O
correlation	O
.	O
</s>
<s>
As	O
a	O
rough	O
rule	O
of	O
thumb	O
,	O
if	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
is	O
less	O
than	O
1.0	O
,	O
there	O
may	O
be	O
cause	O
for	O
alarm	O
.	O
</s>
<s>
In	O
regressions	O
,	O
this	O
can	O
imply	O
an	O
underestimation	O
of	O
the	O
level	O
of	O
statistical	B-General_Concept
significance	I-General_Concept
.	O
</s>
<s>
To	O
test	O
for	O
positive	O
autocorrelation	O
at	O
significance	O
α	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
d	O
is	O
compared	O
to	O
lower	O
and	O
upper	O
critical	O
values	O
(	O
dL	O
,	O
α	O
and	O
dU	O
,	O
α	O
)	O
:	O
</s>
<s>
To	O
test	O
for	O
negative	O
autocorrelation	O
at	O
significance	O
α	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
(	O
4−d	O
)	O
is	O
compared	O
to	O
lower	O
and	O
upper	O
critical	O
values	O
(	O
dL	O
,	O
α	O
and	O
dU	O
,	O
α	O
)	O
:	O
</s>
<s>
If	O
the	O
design	B-Algorithm
matrix	I-Algorithm
of	O
the	O
regression	O
is	O
known	O
,	O
exact	O
critical	O
values	O
for	O
the	O
distribution	O
of	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
no	O
serial	O
correlation	O
can	O
be	O
calculated	O
.	O
</s>
<s>
Second	O
,	O
positive	O
serial	O
correlation	O
typically	O
causes	O
the	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
(	O
OLS	O
)	O
standard	O
errors	O
for	O
the	O
regression	O
coefficients	O
to	O
underestimate	O
the	O
true	O
standard	O
errors	O
.	O
</s>
<s>
The	O
inflated	O
t-statistic	O
,	O
may	O
in	O
turn	O
,	O
lead	O
us	O
to	O
incorrectly	O
reject	O
null	B-General_Concept
hypotheses	I-General_Concept
,	O
about	O
population	O
values	O
of	O
the	O
parameters	O
of	O
the	O
regression	O
model	O
more	O
often	O
than	O
we	O
would	O
if	O
the	O
standard	O
errors	O
were	O
correctly	O
estimated	O
.	O
</s>
<s>
If	O
the	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
statistic	I-General_Concept
indicates	O
the	O
presence	O
of	O
serial	O
correlation	O
of	O
the	O
residuals	O
,	O
this	O
can	O
be	O
remedied	O
by	O
using	O
the	O
Cochrane	O
–	O
Orcutt	O
procedure	O
.	O
</s>
<s>
The	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
statistic	I-General_Concept
,	O
while	O
displayed	O
by	O
many	O
regression	O
analysis	O
programs	O
,	O
is	O
not	O
applicable	O
in	O
certain	O
situations	O
.	O
</s>
<s>
The	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
statistic	I-General_Concept
is	O
biased	O
for	O
autoregressive	B-Algorithm
moving	O
average	O
models	O
,	O
so	O
that	O
autocorrelation	O
is	O
underestimated	O
.	O
</s>
<s>
R	B-Language
:	O
the	O
dwtest	O
function	O
in	O
the	O
lmtest	O
package	O
,	O
durbinWatsonTest	O
(	O
or	O
dwt	O
for	O
short	O
)	O
function	O
in	O
the	O
car	O
package	O
,	O
and	O
pdwtest	O
and	O
pbnftest	O
for	O
panel	O
models	O
in	O
the	O
plm	O
package	O
.	O
</s>
<s>
MATLAB	B-Language
:	O
the	O
dwtest	O
function	O
in	O
the	O
Statistics	O
Toolbox	O
.	O
</s>
<s>
Mathematica	B-Language
:	O
the	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
(	O
d	O
)	O
statistic	O
is	O
included	O
as	O
an	O
option	O
in	O
the	O
LinearModelFit	O
function	O
.	O
</s>
<s>
SAS	B-Language
:	O
Is	O
a	O
standard	O
output	O
when	O
using	O
proc	O
model	O
and	O
is	O
an	O
option	O
(	O
dw	O
)	O
when	O
using	O
proc	O
reg	O
.	O
</s>
<s>
Stata	B-Algorithm
:	O
the	O
command	O
estat	O
dwatson	O
,	O
following	O
regress	O
in	O
time	O
series	O
data	O
.	O
</s>
<s>
Engle	O
's	O
LM	O
test	O
for	O
autoregressive	B-Algorithm
conditional	O
heteroskedasticity	O
(	O
ARCH	O
)	O
,	O
a	O
test	O
for	O
time-dependent	O
volatility	O
,	O
the	O
Breusch	O
–	O
Godfrey	O
test	O
,	O
and	O
Durbin	O
's	O
alternative	O
test	O
for	O
serial	O
correlation	O
are	O
also	O
available	O
.	O
</s>
<s>
Minitab	B-Application
:	O
the	O
option	O
to	O
report	O
the	O
statistic	O
in	O
the	O
Session	O
window	O
can	O
be	O
found	O
under	O
the	O
"	O
Options	O
"	O
box	O
under	O
Regression	O
and	O
via	O
the	O
"	O
Results	O
"	O
box	O
under	O
General	O
Regression	O
.	O
</s>
<s>
Python	B-Language
:	O
a	O
durbin_watson	O
function	O
is	O
included	O
in	O
the	O
statsmodels	O
package	O
(	O
statsmodels.stats.stattools.durbin_watson	O
)	O
,	O
but	O
statistical	O
tables	O
for	O
critical	O
values	O
are	O
not	O
available	O
there	O
.	O
</s>
<s>
SPSS	B-Algorithm
:	O
Included	O
as	O
an	O
option	O
in	O
the	O
Regression	O
function	O
.	O
</s>
<s>
Julia	B-Application
:	O
the	O
DurbinWatsonTest	O
function	O
is	O
available	O
in	O
the	O
HypothesisTests	O
package	O
.	O
</s>
