Durbin-Watson Test

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The Durbin-Watson Test is a measure of autocorrelation (also known as serial correlation) in the residuals resulting from regression analysis. Autocorrelation refers to the similarity of a time series across consecutive time intervals. This can lead to underestimating the standard error and may cause the false identification of significant predictors. The Durbin-Watson test specifically targets the AR(1) process for serial correlation.

There are several reasons why you might want to choose a Durbin-Watson test for your analysis:

  1. Simple and easy to interpret: The Durbin-Watson test is relatively straightforward to perform and interpret, with a test statistic ranging from 0 to 4. This simplicity can make it an attractive option for detecting first-order autocorrelation in regression residuals.
  2. Detecting first-order autocorrelation: If you suspect that there might be first-order autocorrelation in your regression model’s residuals, the Durbin-Watson test is specifically designed to identify this type of serial correlation.
  3. Assessing regression model assumptions: Autocorrelation in the residuals can violate the assumption of independent errors in linear regression models. The Durbin-Watson test can help you determine whether this assumption has been violated, allowing you to improve your model if necessary.
  4. Compatibility with statistical software: The Durbin-Watson test is widely available in most statistical software packages, making it a convenient choice for many researchers and analysts.
  5. Historical use: The Durbin-Watson test has been used extensively in the past to detect autocorrelation in regression residuals. As a result, it may be preferred in some cases due to familiarity or comparability with previous research.

However, it is essential to note that the Durbin-Watson test has limitations, such as being valid only for first-order autocorrelation and sensitivity to sample size. More modern tests like the Breusch-Godfrey test or the Ljung-Box test might be better alternatives depending on the specific dataset and research question.

Running the Durbin-Watson test

Technology options include:

  • Minitab: Select Stat > Regression > Regression > Fit Regression Model. Click “Results,” and look for the Durbin-Watson statistic.
  • MATLAB: The process is available on the Mathworks site.
  • SAS: Instructions can be found on the SAS website.
  • SPSS: From the primary regression dialog box, click Statistics. Check the Durbin-Watson box (found in the Residuals section under Linear Regression Statistics).

The null hypotheses for the Durbin-Watson test is H0  = no first-order autocorrelation. The alternate hypothesis is H1 = first-order correlation exists (for a first-order correlation, the lag is one time unit).

Assumptions for the Durbin-Watson test:

  1. Errors are normally distributed with a mean of 0.
  2. The errors are stationary.

The Durbin-Watson test statistic is calculated with the following formula:

Where Et are residuals from an ordinary least squares regression.

The Durbin-Watson test produces a test statistic, ranging from 0 to 4, where:

  • 2 indicates no autocorrelation.
  • 0 to <2 suggests positive autocorrelation (common in time series data).
  • 2 to 4 implies negative autocorrelation (less common in time series data).

A general guideline is that test statistic values between 1.5 and 2.5 are relatively normal. Values outside this range might be a cause for concern. Values below 1 or above 3 should definitely raise concerns.

Interpreting the p-value

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. A small p-value indicates the null hypothesis should be rejected. In a Durbin-Watson test, a small p-value typically indicates the presence of autocorrelation in the residuals of a regression model. The threshold for considering a p-value “small” depends on the chosen level of significance (α), which is often set at 0.05 or 5%. If the p-value is less than the chosen significance level (e.g., p < 0.05), the null hypothesis of no first-order autocorrelation is rejected, suggesting that there is evidence of autocorrelation in the residuals.

However, it’s important to note that the Durbin-Watson test itself doesn’t directly provide a p-value. Instead, the test generates a test statistic ranging from 0 to 4. To determine the statistical significance, you need to compare the test statistic with the critical values from the Durbin-Watson table or use additional calculations or software to obtain the corresponding p-value.

Alternatives to the Durbin-Watson test

The Durbin-Watson test is infrequently used and is considered outdated by some. It requires the use of tables, which are primarily found in older texts. Here’s an excerpt from one such table [2] at the 5% alpha level:

excerpt of a durbin-watson test table

The Durbin-Watson test has its limitations. For instance, it is only valid for first-order autocorrelation and can be sensitive to sample size. Consequently, some statisticians argue that the Durbin-Watson test is outdated and that better alternatives exist. Some of these alternatives include:

  1. Breusch-Godfrey test: This test is more general than the Durbin-Watson test for detecting autocorrelation. It can test for autocorrelation up to any order and is less sensitive to sample size.
  2. Ljung-Box test: This non-parametric test for autocorrelation does not make assumptions about data distribution, making it less sensitive to departures from normality.
  3. Portmanteau test: Another non-parametric test for autocorrelation, the Portmanteau test is similar to the Ljung-Box test but more powerful.

In addition to these tests, various other statistical methods can be used to examine autocorrelation. The choice of which test to use depends on the specific dataset and research question.

References

[1] Field, A. [2009], Discovering statistics using SPSS, third edition, Sage Publications (PDF download).

[2] Stanford. Table retrieved June 18, 2023 from: http://web.stanford.edu/~clint/bench/dwcrit.htm

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