Breusch Godfrey Test

Unveiling the Mysteries of the Breusch-Godfrey Test for Autocorrelation

The accurate estimation of regression models hinges on the fundamental assumption of independent and identically distributed (i.i.d.) errors. Violation of this assumption, specifically the presence of autocorrelation (correlation between error terms), can severely bias parameter estimates and invalidate statistical inferences. This article delves into the Breusch-Godfrey (BG) test, a powerful tool used to detect autocorrelation of any order in regression models. We will explore its underlying principles, implementation, interpretation, and limitations, offering practical examples to solidify understanding.

Understanding Autocorrelation and its Implications

Autocorrelation, often denoted as serial correlation, occurs when the error terms in a time series or spatial data model are correlated across observations. This correlation can manifest in various forms, from positive autocorrelation (errors tend to have the same sign) to negative autocorrelation (errors alternate in sign). The presence of autocorrelation violates the classical linear regression model (CLRM) assumptions, leading to:

Inefficient estimates: Standard errors of regression coefficients are underestimated, leading to inflated t-statistics and spurious significance.
Inconsistent estimates: As the sample size increases, the estimated coefficients do not converge to their true values.
Invalid hypothesis tests: The calculated p-values are unreliable, making inferences about the model's parameters untrustworthy.

The Breusch-Godfrey Test: A Comprehensive Approach

Unlike earlier tests like the Durbin-Watson test which is primarily effective for detecting first-order autocorrelation, the Breusch-Godfrey test is a more general approach capable of detecting autocorrelation of any order (p). It employs an auxiliary regression to test the null hypothesis of no autocorrelation against the alternative hypothesis of autocorrelation of order p.

The test procedure involves the following steps:

1. Estimate the original regression model: Obtain the residuals (eᵢ) from the fitted regression model.
2. Run an auxiliary regression: Regress the residuals (eᵢ) on the original regressors and p lags of the residuals (eᵢ₋₁, eᵢ₋₂, ..., eᵢ₋ₚ). This regression is expressed as: eᵢ = α + β₁X₁ᵢ + β₂X₂ᵢ + ... + βₖXₖᵢ + γ₁eᵢ₋₁ + γ₂eᵢ₋₂ + ... + γₚeᵢ₋ₚ + υᵢ
3. Test for autocorrelation: The null hypothesis (H₀: no autocorrelation) is tested by examining the overall significance of the auxiliary regression. This is typically done using an F-test or, equivalently, by testing the joint significance of the lagged residual terms (γ₁, γ₂, ..., γₚ) using a chi-squared test. If the test statistic is significant (p-value < chosen significance level, e.g., 0.05), we reject H₀ and conclude that autocorrelation of order p is present.

Practical Example

Let's consider a model predicting house prices (Y) based on size (X₁) and location (X₂). After estimating the model, we suspect autocorrelation. To test this, we perform a Breusch-Godfrey test with p=2 (checking for autocorrelation up to the second order). We regress the residuals from the original model on X₁, X₂, eᵢ₋₁, and eᵢ₋₂. If the chi-squared test statistic for γ₁ and γ₂ is significant, it suggests the presence of autocorrelation.

Interpreting the Results

A significant BG test statistic indicates a rejection of the null hypothesis, suggesting the presence of autocorrelation in the model. The order of autocorrelation (p) used in the auxiliary regression determines the type of autocorrelation detected. However, it's crucial to remember that a non-significant BG test doesn't definitively prove the absence of autocorrelation; it simply lacks sufficient evidence to reject the null hypothesis.

Limitations of the Breusch-Godfrey Test

While powerful, the BG test is not without limitations:

Model specification: The test's validity relies on the correct specification of the original regression model. Omitted variables can lead to spurious detection of autocorrelation.
Small sample sizes: The test's performance may be less reliable with small sample sizes.
Non-linear autocorrelation: The BG test primarily detects linear autocorrelation. Non-linear patterns might not be effectively detected.

Conclusion

The Breusch-Godfrey test is a valuable tool for detecting autocorrelation in regression models. Its ability to test for higher-order autocorrelation provides a significant advantage over simpler tests. However, careful consideration of its limitations and proper interpretation of the results are crucial for drawing accurate conclusions about the presence and nature of autocorrelation in your data.

FAQs

1. What is the difference between the Breusch-Godfrey test and the Durbin-Watson test? The Durbin-Watson test primarily detects first-order autocorrelation, while the Breusch-Godfrey test can detect autocorrelation of any order. BG is also more robust to model misspecification.

2. How do I choose the value of 'p' in the Breusch-Godfrey test? 'p' represents the maximum order of autocorrelation you are testing. Start with a lower value (e.g., 1 or 2) and increase it if there's evidence of higher-order autocorrelation. Theoretical considerations or prior knowledge about the data can also guide the selection.

3. What should I do if the Breusch-Godfrey test reveals autocorrelation? If autocorrelation is detected, you need to address it. Methods include using autoregressive integrated moving average (ARIMA) models, generalized least squares (GLS) estimation, or including lagged dependent variables in the original regression model.

4. Can I use the Breusch-Godfrey test with non-time series data? Yes, the Breusch-Godfrey test can be applied to spatial data or any data where the error terms might exhibit correlation.

5. Is the Breusch-Godfrey test applicable to all types of regression models? While widely applicable, the test's assumptions should be considered. Its validity may be affected by severe heteroskedasticity or non-linear relationships in the model.

Search Results:

regression - Breusch–Godfrey test for the presence of serial ... 22 May 2017 · The question asks me to explain how I would run the Breusch-Godfrey test for the presence of serial ...

hypothesis testing - How to implement Breusch-Godfrey test for a ... 14 Oct 2020 · According to this R forum the Breusch-Godfrey test for an ARIMA model can be done by fitting a simple regression of the residuals from the fitted model on a constant and then perform a bgtest. But it only concerns a simple AR(1) model with no exogenous regressors.

Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey 24 Apr 2015 · The Breusch-Godfrey test is as Lagrange Multiplier test derived from the (correctly specified) likelihood function (and thus from first principles). The Ljung-Box test is based on second moments of the residuals of a stationary process (and thus of …

r - Breusch-Godfrey Test for autocorrelation - Cross Validated 6 Jun 2015 · Following the steps of Breusch–Godfrey test, I wrote my own R code which differs from the R function for bgtest under package 'lmtest' . Though both of them reject the null hypothesis that at least one $\rho$ is statistically significant .

hypothesis testing - Breusch–Godfrey test under … The Autocorrelation (AR) 1-2 test is defined as follows - often referred to as the Breusch–Godfrey test : 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).

regression - Durbin vs. Breusch-Godfrey test for autocorrelation: … 8 Apr 2021 · I'm being asked to justify why I use either the Durbin's alternative test for Serial Correlation or the Breusch-Godfrey test. It seems that both are relatively competent tests however there is little distinction between the two online and which test is better for a given scenario.

why Durbin Watson result could be so different from Ljung-box or ... 25 Jun 2019 · Same contradiction observed for Breusch-Godfrey test: bgtest(out.lm) LM test = 13.448, df =1, p-value = 0.0002452 I also tried different lags for Ljung-Box and BG test, it gave similar result as lag 1.

Breusch-Godfrey Test and the length of the lag, p 27 Oct 2015 · I'll use Breusch-Godfrey (BG) test to test correlation of an AR(1) model. In order to perform a BG test, the simple regression model is first fitted by ordinary least squares to obtain a set of sample residuals. Then the residuals are used the as the dependent variable and regressed over independent variables and its first p-lags.

Breusch-Godfrey autocorrelation test: bgtest for panel data yields ... 29 Jun 2017 · For panel models, the test needs to be run on the (quasi-)demeaned data and pbgtest() being a wrapper around lmtest::bgtest() does excatly that: extract the (quasi-)demeaned data and pass them on to lmtest::bgtest(). For a pooling model, you will get the same numbers as the data are not transformed.

R: ACF/PACF plots contradict Breusch-Godfrey test results 27 Aug 2021 · Breusch-Godfrey is a portmanteau-type test; it looks at all lags up to 20 (or whatever maximum lag order you choose). Now, the ACF shows that autocorrelation is statistically significant only for one lag among the first 20 (a single bar sticks out from the confidence bound).