Durbin-Watson Table

What Is a Durbin-Watson Table?

The Durbin-Watson table provides critical values for detecting autocorrelation in regression residuals. Autocorrelation occurs when residuals at one time point are correlated with residuals at adjacent time points—a common issue in time series data.

The Durbin-Watson statistic is:

$d = \frac{\sum_{t=2}^{n}(e_t - e_{t-1})^2}{\sum_{t=1}^{n}e_t^2}$

where e_t represents the residual at time t.

Understanding the DW Statistic

The statistic ranges from 0 to 4:

The Inconclusive Region

Unlike most statistical tests, the Durbin-Watson test has an inconclusive zone. The table provides two bounds:

• dL (lower bound)
• dU (upper bound)

Decision Rules for Positive Autocorrelation

Testing for Negative Autocorrelation

Use (4 - d) instead of d, then apply the same rules.

How to Read the Table

1. Identify n (sample size)
2. Identify k (number of regressors, excluding intercept)
3. Select your α level
4. Find dL and dU at the intersection

Example: Regression with 2 predictors and 50 observations at α = 0.05:

• k = 2, n = 50
• dL = 1.46, dU = 1.63
• If d = 1.35 < 1.46: Positive autocorrelation detected
• If d = 1.72 > 1.63: No positive autocorrelation

Why Autocorrelation Matters

When residuals are autocorrelated:

1. Standard errors are biased (usually underestimated)
2. t-tests and F-tests are unreliable
3. Confidence intervals are too narrow
4. R² may be inflated

The regression coefficients themselves remain unbiased, but statistical inference becomes invalid.

Common Causes of Autocorrelation

• Omitted variables: A missing predictor that varies over time
• Incorrect functional form: Linear model for a non-linear relationship
• Measurement error: Systematic bias in data collection
• True serial correlation: Genuine persistence in the error process

Remedies for Autocorrelation

If the Durbin-Watson test detects autocorrelation:

1. Add omitted variables that capture temporal patterns
2. Include lagged dependent variables
3. Use time series models (ARIMA, etc.)
4. Apply differencing to the data
5. Use generalized least squares (GLS)
6. Calculate robust standard errors (Newey-West)

Limitations

The Durbin-Watson test:

• Only detects first-order autocorrelation (lag 1)
• Requires the regressors to be non-stochastic
• May have low power in small samples
• Has an inconclusive region that grows with more regressors

For higher-order autocorrelation, consider the Breusch-Godfrey test.

Quick Reference Values

At α = 0.05 with k = 1 regressor:

Related Tools

• Linear Regression Calculator — Run regression
• Multiple Regression Calculator — Multiple predictors

Summary

The Durbin-Watson table is essential for validating regression assumptions in time series contexts. By comparing your calculated d statistic to the tabled dL and dU bounds, you can assess whether autocorrelation threatens the validity of your regression results. Remember that falling in the inconclusive region is common with smaller samples—in such cases, collecting more data or using alternative tests may help provide a definitive answer.