Breusch-Godfrey Test for Detecting Higher-Order Serial Correlation in Regression Residuals

A test for autocorrelation in the residuals of regression models, generalizing the Durbin-Watson test to handle higher-order autocorrelation and models with lagged dependent variables.

Usage

breuschGodfreyTest(
  formula,
  data = list(),
  order = 1,
  orderBy = NULL,
  type = c("chisq", "f"),
  subset,
  na.action = na.omit,
  fill = 0
)

Arguments

formula: a symbolic description for the model to be tested (or a fitted "lm" object, in which case the model frame is taken from the fit and subset and na.action are ignored).
data: an optional data frame containing the variables in the model. By default the variables are taken from the environment which breuschGodfreyTest is called from. For a fitted "lm" object it is used for orderBy only, as the model itself carries its own model frame.
order: integer, the maximal order of serial correlation to be tested. Must be smaller than the residual degrees of freedom of the auxiliary regression.
orderBy: either a vector z or a formula with a single explanatory variable like ~ z. The observations in the model are ordered by the size of z; a formula with several terms is used as successive ordering keys. If set to NULL (the default) the observations are assumed to be ordered (e.g., a time series). z may be given at the length of the original data: rows dropped by subset or by na.action are then dropped from z as well. Missing values in z are ordered last.
type: the type of test statistic to be returned, either "chisq" (default) for the chi-squared test statistic or "f" for the F test statistic. Case-insensitive.
subset: an optional expression indicating which observations to use.
na.action: a function specifying how missing values are handled. Defaults to na.omit(): the auxiliary regression is fitted by lm.fit() and cannot carry missing values.
fill: a single value used as starting value for the lagged residuals in the auxiliary regression. By default 0 but can also be set to NA, in which case the leading incomplete rows are dropped.

Value

A list with class "breuschGodfreyTest" inheriting from "htest" containing the following components:

statistic: the value of the test statistic.
parameter: the degrees of freedom.
p.value: the p-value of the test.
method: a character string indicating what type of test was performed.
data.name: a character string giving the name(s) of the data.
coefficients: coefficient estimates from the auxiliary regression.
vcov: the corresponding covariance matrix estimate.
df.residual: the residual degrees of freedom of the auxiliary regression, for both types of test statistic.

Details

breuschGodfreyTest performs the Breusch-Godfrey test for higher-order serial correlation.

Under \(H_0\) the test statistic is asymptotically chi-squared with degrees of freedom as given in parameter. If type is set to "f" the function returns a finite sample version of the test statistic, employing an \(F\) distribution with degrees of freedom as given in parameter.

By default, the starting values for the lagged residuals in the auxiliary regression are chosen to be 0 (as in Godfrey 1978) but could also be set to NA to omit them.

breuschGodfreyTest also returns the coefficients and estimated covariance matrix from the auxiliary regression that includes the lagged residuals, accessible via coef() and vcov() on the result. (Note, however, that standard theory does not always apply to the standard errors and t-statistics in this regression.)

Note

Based on code by David Mitchell and Achim Zeileis previously published as bgtest() in the lmtest package, adapted to conform to package standards.

Unlike bgtest(), the residual degrees of freedom of the auxiliary regression are reported for both types of test statistic. They are a property of that regression and not of the statistic derived from it, whereas bgtest() hands back NULL for the chi-squared version. coeftest() therefore refers the coefficients to a \(t\) distribution in either case, where bgtest() switches to the normal one.

References

Breusch, T. S. (1978) Testing for autocorrelation in dynamic linear models. Australian Economic Papers, 17, 334-355.

Godfrey, L. G. (1978) Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica, 46, 1293-1301.

Examples

## Generate a stationary and an AR(1) series
set.seed(1)
x <- rep(c(1, -1), 50)

y1 <- 1 + x + rnorm(100)

## Perform Breusch-Godfrey test for first-order serial correlation:
breuschGodfreyTest(y1 ~ x)
#> 
#> 	Breusch-Godfrey test for serial correlation of order up to 1
#> 
#> data:  y1 ~ x
#> LM test = 0.0036887, df = 1, p-value = 0.9516
#> 

## or for fourth-order serial correlation
breuschGodfreyTest(y1 ~ x, order = 4)
#> 
#> 	Breusch-Godfrey test for serial correlation of order up to 4
#> 
#> data:  y1 ~ x
#> LM test = 3.0822, df = 4, p-value = 0.5442
#> 

## Compare with Durbin-Watson test results:
durbinWatsonTest(y1 ~ x)
#> 
#> 	Durbin-Watson test
#> 
#> data:  y1 ~ x
#> DW = 1.9762, p-value = 0.4924
#> alternative hypothesis: true autocorrelation is greater than 0
#> 

y2 <- stats::filter(y1, 0.5, method = "recursive")
breuschGodfreyTest(y2 ~ x)
#> 
#> 	Breusch-Godfrey test for serial correlation of order up to 1
#> 
#> data:  y2 ~ x
#> LM test = 19.907, df = 1, p-value = 8.128e-06
#> 

## finite sample F version, and dropping the leading lags instead of
## filling them with zeros
breuschGodfreyTest(y2 ~ x, order = 4, type = "f")
#> 
#> 	Breusch-Godfrey test for serial correlation of order up to 4
#> 
#> data:  y2 ~ x
#> LM test = 7.2942, df1 = 4, df2 = 94, p-value = 3.682e-05
#> 
breuschGodfreyTest(y2 ~ x, order = 4, fill = NA)
#> 
#> 	Breusch-Godfrey test for serial correlation of order up to 4
#> 
#> data:  y2 ~ x
#> LM test = 24.401, df = 4, p-value = 6.637e-05
#> 

## transformed terms and an explicit ordering variable
d <- data.frame(y = as.vector(y2), x = x, z = rnorm(100), tt = sample(100),
                grp = rep(c("A", "B"), each = 50))
breuschGodfreyTest(y ~ x + I(z^2), data = d, orderBy = ~ tt)
#> 
#> 	Breusch-Godfrey test for serial correlation of order up to 1
#> 
#> data:  y ~ x + I(z^2)
#> LM test = 1.5791, df = 1, p-value = 0.2089
#> 

## subset and orderBy combined: tt is given at the length of d and is
## reduced to the rows the model frame kept
breuschGodfreyTest(y ~ x, data = d, subset = grp == "A", orderBy = ~ tt)
#> 
#> 	Breusch-Godfrey test for serial correlation of order up to 1
#> 
#> data:  y ~ x
#> LM test = 0.83472, df = 1, p-value = 0.3609
#> 

## the test can also be applied to a fitted model
breuschGodfreyTest(lm(y1 ~ x))
#> 
#> 	Breusch-Godfrey test for serial correlation of order up to 1
#> 
#> data:  lm(y1 ~ x)
#> LM test = 0.0036887, df = 1, p-value = 0.9516
#>