# Breusch Godfrey Hypothesis Statement

Testing for Autocorrelation |

The GODFREY= option in the FIT statement produces the Godfrey Lagrange multiplier test for serially correlated residuals for each equation (Godfrey 1978a and 1978b). is the maximum autoregressive order, and specifies that Godfrey’s tests be computed for lags 1 through . The default number of lags is four.

The tests are performed separately for each equation estimated by the FIT statement. When a nonlinear model is estimated, the test is computed by using a linearized model.

The following is an example of the output produced by the GODFREY=3 option:

** Figure 19.43 Autocorrelation Test Output**

The MODEL Procedure

1 | 6.63 | 0.0100 |
---|---|---|

2 | 6.89 | 0.0319 |

3 | 6.96 | 0.0732 |

The three variations of the test reported by the GODFREY=3 option are designed to have power against different alternative hypothesis. Thus, if the residuals in fact have only first-order autocorrelation, the lag 1 test has the most power for rejecting the null hypothesis of uncorrelated residuals. If the residuals have second- but not higher-order autocorrelation, the lag 2 test might be more likely to reject; the same is true for third-order autocorrelation and the lag 3 test.

The null hypothesis of Godfrey’s tests is that the equation residuals are white noise. However, if the equation includes autoregressive error model of order (AR(),) then the lag test, when considered in terms of the structural error, is for the null hypothesis that the structural errors are from an AR() process versus the alternative hypothesis that the errors are from an AR() process.

The alternative ARMA() process is locally equivalent to the alternative AR() process with respect to the null model AR(). Thus, the GODFREY= option results are also a test of AR() errors against the alternative hypothesis of ARMA() errors. See Godfrey (1978a and 1978b) for more detailed information.

Do I need to account for heteroskedasticity when performing the (vector) AR1-2 test?

The Autocorrelation (AR) 1-2 test is defined as follows - often referred to as the **Breusch–Godfrey test** (Wiki link):

The test is performed through the auxiliary regression of the residuals on the original variables and lagged residuals (missing lagged residuals at the start of the sample are replaced by zero, so no observations are lost). Unrestricted variables are included in the auxiliary regression. The null hypothesis is no autocorrelation, which would be rejected if the test statistic is too high. This LM test is valid for systems with lagged dependent variables and diagonal residual autocorrelation, whereas neither the Durbin--Watson nor the residual autocorrelations provide a valid test in that case.

I have a VAR model and I'm trying to determine the amount of lags to include. My model suffers from heteroskedasticity so I'm using the Wald test to take that into account when doing inference. There is a large difference between the normal standard errors and the heteroskedasticity-consistent standard errors in my model.

I'm using OxMetrics and it returns the same AR1-2 test statistic both when I estimate the model with normal errors and heteroskedasticity-consistent errors. Is this because the test on the auxiliary regression is not affected by the heteroskedasticity in the main model or is it just because OxMetrics doesn't perform the right test in this case?

hypothesis-testingautocorrelationheteroscedasticityvarf-test

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