Enter Values
DW Quick Reference
DW ≈ 2(1 − ρ̂)
DW < dL = Reject H₀ (positive SC) |
dL–dU = Inconclusive |
dU < DW < 4−dU = Fail to reject |
4−dU–4−dL = Inconclusive |
DW > 4−dL = Reject H₀ (negative SC)
Test Results
DW Statistic
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dL Bound
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dU Bound
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Approx. ρ̂
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Bounds interpolated from nearest tabulated values. Conservative users should use the next lower tabulated n.
LM Statistic
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χ² Critical
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p-value
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DW Decision Zones
0
2
4
dL
dU
4-dU
4-dL
DW
Reject H₀ (positive SC)
Inconclusive
Fail to reject H₀
Reject H₀ (negative SC)
Formula Breakdown
DW ≈ 2(1 − ρ̂) | ρ̂ ≈ 1 − DW/2
Model Assumptions
- The DW test detects AR(1) serial correlation only and requires strictly exogenous regressors and a model with an intercept.
- DW is biased toward 2 when lagged dependent variables are present — use the Breusch-Godfrey test instead.
- DW critical bounds are from Savin-White tables (models with intercept). No-intercept models use different tables and are not supported here.
- The Breusch-Godfrey test handles AR(p) and is valid with lagged dependent variables.
- The standard BG LM test assumes homoskedastic errors. A heteroskedasticity-robust variant exists but is not implemented here.
- LM = n × R²aux is the large-sample form (standard in software). Wooldridge uses (n−q) × R² for finite-sample adjustment — results may differ slightly.
- R²aux must come from the auxiliary regression of OLS residuals on original regressors plus p lagged residuals.
- Newey-West standard errors are robust to both heteroskedasticity and autocorrelation. Bandwidth: g = ⌊4(n/100)2/9⌋ per Wooldridge; n1/4 is an alternative convention.
- This tool interprets externally computed DW/BG statistics — it does not compute them from raw residual data.
- For educational purposes. Not financial advice. Econometric assumptions simplified.
Understanding Serial Correlation Tests
What Is Serial Correlation?
Serial correlation (autocorrelation) occurs when error terms in a time series regression are correlated across time periods. The most common form is AR(1), where the error at time t is correlated with the error at time t−1. While serial correlation does not bias OLS coefficients under strict exogeneity, it invalidates the usual standard errors, making hypothesis tests unreliable.
The Durbin-Watson Test
The DW statistic tests for first-order (AR(1)) serial correlation. It ranges from 0 to 4, with DW ≈ 2 indicating no serial correlation. The test compares DW to critical lower (dL) and upper (dU) bounds from Savin-White tables. The five decision zones are:
- DW < dL: Reject H₀ — evidence of positive serial correlation
- dL ≤ DW ≤ dU: Inconclusive
- dU < DW < 4−dU: Fail to reject H₀ — no evidence of serial correlation
- 4−dU ≤ DW ≤ 4−dL: Inconclusive
- DW > 4−dL: Reject H₀ — evidence of negative serial correlation
Important: The DW test requires strictly exogenous regressors. It is biased toward 2 when lagged dependent variables are present. Use the Breusch-Godfrey test in that case.
The Breusch-Godfrey Test
The BG test is more general: it can detect higher-order AR(p) serial correlation and is valid even with lagged dependent variables. The procedure regresses the OLS residuals on the original regressors plus p lagged residuals, then computes LM = n × R²aux, which follows a χ²(p) distribution under the null hypothesis of no serial correlation.
Corrections for Serial Correlation
- Newey-West standard errors: Keep OLS coefficients, use HAC (heteroskedasticity and autocorrelation consistent) standard errors for valid inference.
- FGLS (Cochrane-Orcutt / Prais-Winsten): Transform the data using estimated ρ̂ for more efficient estimates.
- Add dynamics: Include lagged dependent variables or error terms if serial correlation reflects omitted dynamics.
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Frequently Asked Questions
Serial correlation (autocorrelation) occurs when error terms in a regression model are correlated across time periods. In time series data, this means the error at time t is correlated with the error at time t−1 (or earlier periods). While serial correlation does not bias OLS coefficient estimates under strict exogeneity, it invalidates the usual standard errors, making t-statistics and confidence intervals unreliable. This can lead to incorrect conclusions about statistical significance.
The DW statistic ranges from 0 to 4. A value near 2 indicates no serial correlation (ρ̂ ≈ 0). Values below 2 suggest positive serial correlation (successive errors tend to have the same sign), while values above 2 suggest negative serial correlation. To make a formal decision, compare DW to the lower (dL) and upper (dU) critical bounds from Savin-White tables. If DW < dL, reject H₀ for positive serial correlation; if DW > dU but < 4−dU, fail to reject; if between dL and dU, the test is inconclusive.
The BG test has three key advantages: (1) it can test for higher-order serial correlation (AR(p), not just AR(1)), (2) it is valid even when the regression includes lagged dependent variables (where DW is biased toward 2), and (3) it produces a definitive reject/fail-to-reject result with no inconclusive zone. Use BG when your model includes lagged Y values, when you suspect higher-order autocorrelation, or when the DW test falls in the inconclusive region.
Newey-West standard errors are heteroskedasticity and autocorrelation consistent (HAC) standard errors. They adjust for both non-constant variance and serial correlation in the error terms without requiring a specific model for either. The bandwidth parameter g = ⌊4(n/100)2/9⌋ determines how many lags of autocorrelation to account for (n1/4 is an alternative rule). Newey-West SEs are typically larger than OLS SEs when serial correlation is present, leading to more conservative inference.
Under strict exogeneity (static models with no lagged dependent variables), OLS coefficients remain unbiased even with serial correlation. However, the usual OLS standard errors become invalid, typically underestimating true variability, which inflates t-statistics and can lead to false rejections. With lagged dependent variables or other non-strictly exogenous regressors, serial correlation can make OLS inconsistent, meaning the estimates do not converge to the true values even in large samples. In either case, corrective measures such as Newey-West standard errors or FGLS are recommended.
Three main approaches: (1) Newey-West standard errors — keep OLS coefficients but use HAC standard errors for valid inference; simplest fix, widely used. (2) FGLS (Cochrane-Orcutt or Prais-Winsten) — transform the data using the estimated autocorrelation coefficient ρ̂ to produce efficient estimates. (3) Add dynamics — include lagged dependent variables or lagged error terms in the model if the serial correlation reflects omitted dynamics rather than a nuisance.
Disclaimer
This calculator is for educational purposes only and interprets externally computed serial correlation test statistics. It does not compute test statistics from raw data. Results should be cross-referenced with your statistical software output. Refer to Wooldridge, Introductory Econometrics, Chapter 12 for detailed methodology.
