Durbin–Watson statistic

Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation in the residuals from a regression analysis. It is named after James Durbin and Geoffrey Watson. The Durbin-Watson statistic ranges in value from 0 to 4. A value of 2 indicates no autocorrelation in the sample; values approaching 0 indicate positive autocorrelation; and values toward 4 indicate negative autocorrelation.

Overview

The Durbin-Watson statistic is especially useful in the field of econometrics and other disciplines that involve time series analysis to ensure that the residuals of a regression model are not autocorrelated. Autocorrelation can lead to inefficiencies in the ordinary least squares (OLS) estimation of regression coefficients, making the Durbin-Watson statistic a critical tool for diagnosing regression models.

Calculation

The Durbin-Watson statistic is calculated using the formula:

\[d = \frac{\sum_{t=2}^{T} (e_t - e_{t-1})^2}{\sum_{t=1}^{T} e_t^2}\]

where:

• \(e_t\) is the residual at time \(t\),
• \(T\) is the number of observations.

The numerator of the formula represents the squared differences between consecutive residuals, indicating the extent of autocorrelation. The denominator is the sum of squared residuals, a measure often used in the context of regression analysis.

Interpretation

Interpreting the Durbin-Watson statistic involves comparing the calculated value to tabulated values that depend on the level of significance (\(\alpha\)), the number of observations, and the number of explanatory variables in the regression model. If the Durbin-Watson statistic is substantially less than 2, it suggests positive autocorrelation. Conversely, a value significantly greater than 2 implies negative autocorrelation. However, determining the exact range of values that indicate autocorrelation requires consulting Durbin-Watson tables.

Limitations

While the Durbin-Watson statistic is widely used, it has limitations. It is most effective for detecting first-order autocorrelation. For higher-order autocorrelation, other tests, such as the Breusch-Godfrey test, may be more appropriate. Additionally, the Durbin-Watson statistic can be inconclusive when its value is close to 2, necessitating further testing or alternative methods to assess autocorrelation.

Applications

The Durbin-Watson statistic is applied in various fields that utilize regression analysis, including econometrics, finance, and the social sciences. It helps in validating the assumptions of linear regression models, ensuring that the estimated parameters are unbiased and efficient.

See also

• Autocorrelation
• Regression analysis
• Econometrics
• Time series analysis
• Breusch-Godfrey test