View source for Durbin–Watson statistic

{{short description|Test statistic}}
{{more footnotes|date=December 2012}}
In [[statistics]], the '''Durbin–Watson statistic''' is a [[test statistic]] used to detect the presence of [[Autocorrelation#Regression_analysis|autocorrelation]] at lag 1 in the [[errors and residuals in statistics|residuals]] (prediction errors) from a [[regression analysis]]. It is named after [[James Durbin]] and [[Geoffrey Watson]]. The [[Sample size|small sample]] distribution of this ratio was derived by [[John von Neumann]] (von Neumann, 1941). Durbin and Watson (1950, 1951) applied this statistic to the residuals from [[Ordinary least squares|least squares]] regressions, and developed bounds tests for the [[null hypothesis]] that the errors are serially uncorrelated against the alternative that they follow a first order [[autoregressive]] process. Note that the distribution of this test statistic does not depend on the estimated regression coefficients and the variance of the errors.<ref name=pivotal>{{cite book|last=Chatterjee|first=Samprit|last2=Simonoff|first2=Jeffrey| title=Handbook of Regression Analysis|year=2013|publisher=John Wiley & Sons|isbn=1118532813}}</ref>

A similar assessment can be also carried out with the [[Breusch–Godfrey test]] and the [[Ljung–Box test]].

==Computing and interpreting the Durbin–Watson statistic==

If <math display="inline">e_t</math> is the [[errors and residuals in statistics|residual]] given by <math>e_t = \rho e_{t-1}+ \nu_t ,</math> the Durbin-Watson [[test statistic]] is

: <math>d = {\sum_{t=2}^T (e_t - e_{t-1})^2 \over {\sum_{t=1}^T e_t^2}},</math>

where <math display="inline">T</math> is the number of observations. For large <math display="inline">T</math>, ''<math display="inline">d</math>'' is approximately equal to <math display="inline">2(1- \hat{\rho})</math>, where <math>\hat \rho</math> is the sample autocorrelation of the residuals at lag 1.<ref name="Gujarati 2003">Gujarati (2003)  p. 469</ref> ''<math display="inline">d = 2</math>'' therefore indicates no autocorrelation. The value of ''<math display="inline">d</math>'' always lies between <math display="inline">0</math> and <math display="inline">4</math>. If the Durbin–Watson statistic is substantially less than 2, there is evidence of positive serial correlation. As a rough rule of thumb, if Durbin–Watson is less than 1.0, there may be cause for alarm. Small values of ''<math display="inline">d</math>'' indicate successive error terms are positively correlated. If ''<math display="inline">d>2</math>'', successive error terms are negatively correlated. In regressions, this can imply an underestimation of the level of [[statistical significance]].

To test for '''positive autocorrelation''' at significance <math display="inline">\alpha</math>, the test statistic ''<math display="inline">d</math>'' is compared to lower and upper critical values (''<math display="inline">d_{ L, \alpha}</math>'' and <math display="inline">d_{ U, \alpha}</math>):
:*If ''<math display="inline">d < d_{ L, \alpha}</math>'', there is statistical evidence that the error terms are positively autocorrelated.
:*If ''<math display="inline">d > d_{ U, \alpha}</math>'', there is '''no''' statistical evidence that the error terms are positively autocorrelated.
:*If <math>d_{ L, \alpha} < d < d_{ U, \alpha} </math>, the test is inconclusive.

Positive serial correlation is serial correlation in which a positive error for one observation increases the chances of a positive error for another observation.

To test for '''negative autocorrelation''' at significance ''<math display="inline">\alpha</math>'', the test statistic <math display="inline">(4 - d)</math> is compared to lower and upper critical values (''<math display="inline">d_{ L, \alpha}</math>'' and <math display="inline">d_{ U, \alpha}</math>):
:*If ''<math display="inline">(4 - d) < d_{ L, \alpha}</math>'', there is statistical evidence that the error terms are negatively autocorrelated.
:*If ''<math display="inline">(4 - d) > d_{ U, \alpha}</math>'', there is '''no''' statistical evidence that the error terms are negatively autocorrelated.
:*If <math>d_{ L, \alpha} < (4 - d) < d_{ U, \alpha} </math>, the test is inconclusive.

Negative serial correlation implies that a positive error for one observation increases the chance of a negative error for another observation and a negative error for one observation increases the chances of a positive error for another.

The critical values, ''<math display="inline">d_{ L, \alpha}</math>'' and ''<math display="inline">d_{ U, \alpha}</math>'', vary by level of significance (''<math display="inline">\alpha</math>'') and the [[Degrees_of_freedom_(statistics)#In_linear_models | degrees of freedom]] in the regression equation. Their derivation is complex—statisticians typically obtain them from the appendices of statistical texts.

If the [[design matrix]] <math>\mathbf{X}</math> of the regression is known, exact critical values for the distribution of <math>d</math> under the null hypothesis of no serial correlation can be calculated. Under the null hypothesis <math>d</math> is distributed as

: <math>
\frac
{\sum_{i=1}^{n-k} \nu_i \xi_i^2}
{\sum_{i=1}^{n-k} \xi_i^2},
 </math>

where <math display="inline">n</math> is the number of observations and <math display="inline">k</math> is number of regression variables; the <math> \xi_i </math> are independent standard normal random variables; and the <math> \nu_i </math> are the nonzero eigenvalues of <math>
( \mathbf{I} - \mathbf{X} ( \mathbf{X}^T \mathbf{X} ) ^{-1} \mathbf{X}^T ) \mathbf{A},
</math>
where <math>\mathbf{A}</math> is the matrix that transforms the residuals into the <math>d</math> statistic, i.e. <math>d = \mathbf{e}^T\mathbf{A}\mathbf{e}.</math><ref name="Durbin 1971">{{cite journal |last1=Durbin |first1=J. |last2=Watson |first2=G. S. |year=1971 |title=Testing for serial correlation in least squares regression.III |journal=Biometrika |volume=58 |issue=1 |pages=1–19 |doi=10.2307/2334313}}</ref> A number of computational algorithms for finding percentiles of this distribution are available.<ref name="Farebrother 1980">{{cite journal |last1=Farebrother |first1=R. W.|year=1980 |title=Algorithm AS 153: Pan's procedure for the tail probabilities of the Durbin-Watson statistic |journal=Journal of the Royal Statistical Society, Series C |volume=29 |issue=2 |pages=224–227}}</ref>

Although serial correlation does not affect the consistency of  the estimated regression coefficients, it does affect our ability to conduct valid statistical tests. First, the F-statistic to test for overall significance of the regression may be inflated under positive serial correlation because the mean squared error (MSE) will tend to underestimate the population error variance. Second, positive serial correlation typically causes the ordinary least squares (OLS) standard errors for the regression coefficients to underestimate the true standard errors. As a consequence, if positive serial correlation is present in the regression, standard linear regression analysis will typically lead us to compute artificially small standard errors for the regression coefficient. These small standard errors will cause the estimated t-statistic to be inflated, suggesting significance where perhaps there is none. The inflated t-statistic, may in turn, lead us to incorrectly reject null hypotheses, about population values of the parameters of the regression model more often than we would if the standard errors were correctly estimated.

If the Durbin–Watson statistic indicates the presence of serial correlation of the residuals, this can be remedied by using the [[Cochrane–Orcutt estimation|Cochrane–Orcutt procedure]].

The Durbin–Watson statistic, while displayed by many regression analysis programs, is not applicable in certain situations. For instance, when lagged dependent variables are included in the explanatory variables, then it is inappropriate to use this test. Durbin's h-test (see below) or likelihood ratio tests, that are valid in large samples, should be used.

==Durbin h-statistic==<!-- This section is linked from [[Autocorrelation]] -->

The Durbin–Watson statistic is [[bias of an estimator|biased]] for [[autoregressive moving average model]]s, so that autocorrelation is underestimated. But for large samples one can easily compute the unbiased [[Normal distribution|normally distributed]] h-statistic:

:<math>h = \left( 1 - \frac {1} {2} d \right) \sqrt{\frac {T}  {1-T \cdot \widehat {\operatorname{Var}}(\widehat\beta_1\,)}},</math>

using the Durbin–Watson statistic ''d'' and the estimated variance

: <math>\widehat{\operatorname{Var}} (\widehat\beta_1)</math>

of the regression coefficient of the lagged dependent variable, provided

: <math>T \cdot \widehat{\operatorname{Var}}(\widehat\beta_1)<1. \,</math>

==Implementations in statistics packages==
#[[R (programming language)|R]]: the <code>dwtest</code> function in the lmtest package, <code>durbinWatsonTest</code> (or dwt for short) function in the car package, and <code>pdwtest</code> and <code>pbnftest</code> for panel models in the plm package.<ref>{{cite book |last=Hateka |first=Neeraj R. |title=Principles of Econometrics: An Introduction (Using R) |publisher=SAGE Publications |year=2010 |isbn=978-81-321-0660-9 |chapter=Tests for Detecting Autocorrelation |pages=379–82 |chapter-url=https://books.google.com/books?id=SdaGAwAAQBAJ&pg=PA379 }}</ref>
#[[MATLAB]]: the dwtest function in the Statistics Toolbox.
#[[Mathematica]]: the Durbin–Watson (''d'') statistic is included as an option in the LinearModelFit function.
#[[SAS (software)|SAS]]: Is a standard output when using proc model and is an option (dw) when using proc reg.
#[[EViews]]: Automatically calculated when using OLS regression
#[[gretl]]: Automatically calculated when using OLS regression
#[[Stata]]: the command <code>estat dwatson</code>, following <code>regress</code> in time series data.<ref>{{cite web |title=regress postestimation time series — Postestimation tools for regress with time series |work=Stata Manual |url=https://www.stata.com/manuals15/rregresspostestimationtimeseries.pdf }}</ref> Engle's LM test for autoregressive conditional heteroskedasticity (ARCH), a test for time-dependent volatility, the Breusch–Godfrey test, and Durbin's alternative test for serial correlation are also available. All (except -dwatson-) tests separately for higher-order serial correlations. The Breusch–Godfrey test and Durbin's alternative test also allow regressors that are not strictly exogenous.
#[[Microsoft Excel|Excel]]: although Microsoft Excel 2007 does not have a specific Durbin–Watson function, the ''d''-statistic may be calculated using <code>=SUMXMY2(x_array,y_array)/SUMSQ(array)</code>
#[[Minitab]]: the option to report the statistic in the Session window can be found under the "Options" box under Regression and via the "Results" box under General Regression.
#[[Python (programming language)|Python]]: a durbin_watson function is included in the statsmodels package (<code>statsmodels.stats.stattools.durbin_watson</code>), but statistical tables for critical values are not available there.
#[[SPSS]]: Included as an option in the Regression function.
#[[Julia (programming language)|Julia]]: the ''DurbinWatsonTest'' function is available in the ''HypothesisTests'' package.<ref>{{cite web|url=https://juliastats.org/HypothesisTests.jl/latest/time_series/ |title= Time series tests |website=juliastats.org |access-date=2020-02-04}}</ref>
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==See also==
* [[Time-series regression]]
* [[Autocorrelation|ACF]] / [[Partial autocorrelation function|PACF]]
* [[Correlation dimension]]
* [[Breusch–Godfrey test]]
* [[Ljung–Box test]]

==References==
{{Reflist}}

==Further reading==
{{refbegin|60em}}
* {{cite journal |last=Durbin |first=J. |last2=Watson |first2=G. S. |year=1950 |title=Testing for Serial Correlation in Least Squares Regression, I |journal=[[Biometrika]] |volume=37 |issue=3–4 |pages=409–428 |jstor=2332391 |doi=10.1093/biomet/37.3-4.409}}
* {{cite journal |last=Durbin |first=J. |last2=Watson |first2=G. S. |year=1951 |title=Testing for Serial Correlation in Least Squares Regression, II |journal=Biometrika |volume=38 |issue=1–2 |pages=159–179 |jstor=2332325 |doi=10.1093/biomet/38.1-2.159}}
* {{cite book |last=Gujarati |first=Damodar N. |first2=Dawn C. |last2=Porter|author2-link=Dawn C. Porter |year=2009 |title=Basic Econometrics |edition=5th |location=Boston |publisher=McGraw-Hill Irwin |isbn=978-0-07-337577-9 |url=https://books.google.com/books?id=6l1CPgAACAAJ }}
* {{cite book |last=Kmenta |first=Jan |author-link=Jan Kmenta |title=Elements of Econometrics |location=New York |publisher=Macmillan |year=1986 |edition=Second |isbn=0-02-365070-2 |pages=[https://archive.org/details/elementsofeconom0003kmen/page/328 328]–332 |url=https://archive.org/details/elementsofeconom0003kmen |url-access=registration }}
* {{cite journal |last=Neumann |first=John von |year=1941 |title=Distribution of the ratio of the mean square successive difference to the variance |journal=[[Annals of Mathematical Statistics]] |volume=12 |issue=4 |pages=367–395 |jstor=2235951 |doi=10.1214/aoms/1177731677|doi-access=free }}
* {{cite book |last=Verbeek |first=Marno |author-link = Marno Verbeek|title=A Guide to Modern Econometrics |edition=4th |location=Chichester |publisher=John Wiley & Sons |year=2012 |pages=117–118 |isbn=978-1-119-95167-4 |url=https://books.google.com/books?id=4UTeOZDyoeUC&pg=PA117 }}
{{refend}}

==External links==
* [http://www.stanford.edu/~clint/bench/dwcrit.htm Table for high ''n'' and ''k''] {{Webarchive|url=https://web.archive.org/web/20110807130000/http://www.stanford.edu/~clint/bench/dwcrit.htm |date=2011-08-07 }}
* {{YouTube|id=3bipJW_vuBU&list=PLD15D38DC7AA3B737&index=7#t=38m51s|title=Econometrics lecture (topic: Durbin–Watson statistic)}} by [[Mark Thoma]]

{{Statistics|analysis}}

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[[Category:Time series statistical tests]]
[[Category:Autocorrelation]]