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2004issue C041

Constructing first-order residual autocorrelation tests

A first-order residual autocorrelation test pairs each residual with the residual from the immediately preceding step, scores that pairing from 0 through 4, and uses tabled lower and upper critical values so serial correlation is treated as present, absent, or undetermined.

  • The test compares the residual at each time step with the residual from the immediately preceding step and reports a Durbin-Watson statistic from 0 through 4, with 2 as the reference for no serial correlation.
  • Lower and upper critical values are looked up from a table using sample size, independent-variable count, and significance level. The worked lookup uses a sample size of 60 and one independent variable.
  • A statistic below the lower critical value is treated as evidence of positive residual autocorrelation. A statistic above the upper critical value is treated as showing no evidence of residual autocorrelation.
  • A statistic inside the inconclusive interval, between the two critical values, leaves the presence of autocorrelation undetermined.
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Current trackAutocorrelation test23 readings

What the test compares

A first-order residual autocorrelation test compares the residual at each time step with the residual from the immediately preceding step.

That pairing is first-order autocorrelation: dependence between a regression residual at one sampling date and the residual from the immediately prior date.

How the statistic is scored

The Durbin-Watson statistic is a 0-to-4 residual serial-correlation score formed by pairing each residual with the residual one step earlier.

The resulting statistic takes values from 0 through 4, and 2 is the reference value that indicates no serial correlation.

Looking up the critical bounds

Decision thresholds are a lower critical value and an upper critical value looked up from a table using sample size, the independent-variable count, and the significance level.

Sample size is the number of ordered observations used both to fit the regression and to look up the critical bounds. Independent-variable count is the number of regressors that, together with sample size and significance level, selects the pair of critical bounds.

The worked lookup uses a sample size of 60 and one independent variable.

Reading the three outcomes

A statistic below the lower critical value is treated as evidence of positive autocorrelation among residuals.

A statistic above the upper critical value is treated as showing no evidence of residual autocorrelation.

A statistic that falls between the two critical values sits in the inconclusive interval and leaves the presence of autocorrelation undetermined.

5% Durbin–Watson bounds for one regressor

Independence of first-order residuals is rejected when the Durbin–Watson statistic sits below the lower curve and accepted only above the upper curve; the gap between them is inconclusive. The points are the tabled 5% dL and dU entries for p = 1, read from the sidebar critical-value sheet for sample sizes 15 through 35.
Independence of first-order residuals is rejected when the Durbin–Watson statistic sits below the lower curve and accepted only above the upper curve; the gap between them is inconclusive. The points are the tabled 5% dL and dU entries for p = 1, read from the sidebar critical-value sheet for sample sizes 15 through 35.

The source table is a 5% test. The sidebar example looks up n = 60 and p = 1; n = 60 is past the last fully visible row, so the plotted bounds stop at n = 35.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
10 of 15 in the Autocorrelation test track
20041-3 pp.Next on Autocorrelation testA two-stage Durbin-Watson residual screenPrice or return correlation can be high while residual autocorrelation after a linear fit remains low.
All readings on this track · 15 readings
  1. 1987Evaluating weekly price cycles with autocorrelation cutoffs
  2. 1988Serial correlation is not an independence test
  3. 1988Event-triggered chi-square tests for jittered serial links
  4. 1988Reverse the rule before you trust the system
  5. 1990Chi-square evaluation of lagged breadth correlations
  6. 1990Two-gate out-of-sample tests for trading rules
  7. 1990Sign-cell correlation, chi-square checks, and lag shifts
  8. 1998Test serial dependence before you optimize a mechanical system
  9. 2003How a one-rank rotation is constructed for an offset autocorrelation check
  10. 2004Constructing first-order residual autocorrelation tests
  11. 2004A two-stage Durbin-Watson residual screen
  12. 2011Constructing trend labels from increment correlation atoms
  13. 2012Overnight versus daytime return evaluation
  14. 2012Constructing a multi-horizon folded-correlation reversal filter
  15. 2016Tuning indicators to measured market cycles
By fieldStatistical tests95 readingsBy year2004180 readings
All 23 readings tagged Autocorrelation test
Also on Autocorrelation test5 readings
1982Construct ARIMA forecasts from lag diagnostics1986Evaluating daily price and volume serial independence windows1986Weekly resamples as a diagnostic filter for statistical windows1987Evaluating money-supply serial dependence before a forecast1987Evaluating weekly price cycles with autocorrelation cutoffs