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In [[statistics]], the '''Breusch–Godfrey-Bertolo test''' is used to assess the validity of some of the modelling assumptions inherent in applying [[regression analysis|regression-like]] models to observed data series. In particular, it [[statistical hypothesis testing|tests]] for the presence of [[serial dependence]] that has not been included in a proposed model structure and which, if present, would mean that incorrect conclusions would be drawn from other tests, or that sub-optimal estimates of model parameters are obtained if it is not taken into account. The regression models to which the test can be applied include cases where lagged values of the [[independent and dependent variables|dependent variables]] are used as [[independent and dependent variables|independent variables]] in the model's representation for later observations. This type of structure is common in [[econometric model]]s.


An alternative name for the test is the '''Breusch–Godfrey serial correlation Lagrange multiplier test''', where this indicates that the test is equivalent to one based on the idea of [[Lagrange multiplier test]]ing.


The test is named after [[Trevor S. Breusch]] and Leslie G. Godfrey.
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==Background==
The Breusch–Godfrey serial correlation LM test is a test for [[autocorrelation]] in the [[errors and residuals in statistics|errors]] in a regression model. It makes use of the [[errors and residuals in statistics|residuals]] from the model being considered in a [[regression analysis]], and a test statistic is derived from these.  The [[null hypothesis]] is that there is no [[serial correlation]] of any order up to ''p''.<ref>[http://www.macrodados.com.br/English/help/Econometrics_Part3.htm Macrodados 6.3 Help – Econometric Tools]</ref>
 
The test is more general than the [[Durbin–Watson statistic]] (or Durbin's ''h'' statistic), which is only valid for nonstochastic regressors and for testing the possibility of a first-order autoregressive model (e.g. AR(1)) for the regression errors.{{citation needed|date=November 2010}} The BG test has none of these restrictions, and is statistically more [[statistical power|powerful]] than Durbin's ''h'' statistic.{{citation needed|date=November 2010}}
 
== Procedure ==
Consider a [[linear regression]] of any form, for example
 
:<math>Y_t = \alpha_0+ \alpha_1 X_{t,1} + \alpha_2 X_{t,2} + u_t \,</math>
 
where the residuals might follow an AR(''p'') autoregressive scheme, as follows:
 
:<math>u_t = \rho_1 u_{t-1} + \rho_2 u_{t-2}  + \cdots + \rho_p u_{t-p} + \varepsilon_t. \, </math>
 
The simple regression model is first fitted by [[ordinary least squares]] to obtain a set of sample residuals <math>\hat{u}_t</math>.
 
Breusch and Godfrey{{citation needed|date=November 2010}} proved that, if the following auxiliary regression model is fitted
 
: <math> \hat{u}_t = \alpha_0 + \alpha_1 X_{t,1} + \alpha_2 X_{t,2} + \rho_1 \hat{u}_{t-1} + \rho_2 \hat{u}_{t-2} + \cdots + \rho_p \hat{u}_{t-p} + \varepsilon_t \, </math>
 
and if the usual <math>R^2</math> statistic is calculated for this model, then the following [[asymptotic distribution|asymptotic approximation]] can be used for the distribution of the test statistic
 
: <math>n R^2\,\sim\,\chi^2_p, \, </math>
 
when the null hypothesis <math>{H_0: \lbrace \rho_i = 0 \text{ for all } i \rbrace }</math> holds (that is, there is no serial correlation of any order up to&nbsp;''p''). Here ''n'' is the number of data-points available for the second regression, that for  <math>\hat{u}_t</math>,
:<math>n=T-p, \, </math>
where ''T'' is the number of observations in the basic series. Note that the value of n depends on the number of lags of the error term (''p'').
 
==Software==
* In [[R (programming language)|R]], this test is performed by function '''bgtest''', available in [[R (programming language)#Packages|package]] '''lmtest'''.
* In [[Stata]], this test is performed by the command '''bgodfrey'''.
* In [[SAS (software)|SAS]], the '''GODFREY''' option of the '''MODEL''' statement in '''PROC AUTOREG''' provides a version of this test.
* In Python, the acorr_breush_godfrey function in the module statsmodels.stats.diagnostic <ref>Breusch-Godfrey test in Python http://statsmodels.sourceforge.net/devel/generated/statsmodels.stats.diagnostic.acorr_breush_godfrey.html?highlight=autocorrelation</ref>
 
==See also==
*[[Breusch–Pagan test]]
 
{{more footnotes|date=November 2010}}
{{primary sources|date=January 2011}}
 
== Notes==
{{reflist}}
 
==References==
*Breusch, T.S. (1979) "Testing for Autocorrelation in Dynamic Linear Models", ''Australian Economic Papers'', 17, 334&ndash;355. {{doi|10.1111/j.1467-8454.1978.tb00635.x}}
*Godfrey, L.G. (1978) "Testing Against General Autoregressive and Moving Average Error Models when the Regressors Include Lagged Dependent Variables", ''[[Econometrica]]'', 46, 1293&ndash;1302. {{jstor|1913829}}
*Godfrey, L.G. (1988), ''Misspecification tests in econometrics'', Cambridge, UK: Cambridge ISBN 0-521-26616-5
*Godfrey, L.G. (1996), "Misspecification tests and their uses in econometrics", ''Journal of Statistical Planning and Inference'', 49 (2), (Econometric Methodology, Part II), 241&ndash;260 {{DOI| 10.1016/0378-3758(95)00039-9}}
 
{{DEFAULTSORT:Breusch-Godfrey Test}}
[[Category:Statistical tests]]
[[Category:Regression diagnostics]]
[[Category:Regression with time series structure]]

Latest revision as of 22:56, 9 December 2014


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