Twenty-four residuals that drift rather than scatter. d is 0.4162, far below the lower bound of 1.2728, so positive autocorrelation is rejected outright. Note the two autocorrelation figures: the familiar 1 − d/2 shorthand gives 0.7919 while the first-order autocorrelation computed directly is 0.5749 — overstated by 37.8%, because the shorthand quietly assumes the first and last residuals behave like the rest.
24 residuals · 1 regressor plus an intercept
d = 0.4162 — positive autocorrelation
The bounds at 5% are dL = 1.2728 and dU = 1.4457, mirrored at 2.5543 and 2.7272. The first-order autocorrelation implied by d is 0.7919; computed directly from the residuals it is 0.5749. A large-sample test on that direct figure gives z = 2.816, p = 0.00486.
Durbin-Watson d
0.4162
positive autocorrelation
Lower bound dL
1.2728
dU = 1.4457
ρ from d
0.7919
direct: 0.5749
Large-sample p
0.00486
z = 2.816
The five regions of the scale
The Durbin-Watson decision regions for this sample size and regressor count
Range of d
Conclusion
This sample
0 to 1.2728
positive autocorrelation
←
1.2728 to 1.4457
no conclusion available
1.4457 to 2.5543
no autocorrelation detected
2.5543 to 2.7272
no conclusion available
2.7272 to 4
negative autocorrelation
Two of these five regions are the test declining to answer, and together they cover 0.3460 of the scale — 8.6% of its whole range at 1 regressor and 24 observations. Add regressors and that share grows.
Two ways to read the autocorrelation
The implied and directly computed first-order autocorrelation
Estimate
Value
What it assumes
1 − d/2
0.79188
that the first and last residuals behave like the rest
Direct lag-1 correlation
0.57487
nothing beyond the residuals themselves
Difference
0.21702
large enough here to change how the result reads
Bounds computed, not tabled Both ρ estimates Any n and k Lag 1 only
What this tool shows
The bounds here are computed, not looked up. dL and dU come from inverting the bounding chi-square mixtures directly, which means they exist at any sample size and regressor count rather than only at the combinations a printed table happens to cover. At 24 observations the inconclusive band runs from 1.2728 to 1.4457 with one regressor and from 0.9249 to 1.9018 with five — 5.6× wider, and nearly a quarter of the statistic’s entire range.
The Durbin-Watson statistic from residuals in time order
dL and dU computed by Imhof inversion — no table lookup, any n and k
All five decision regions printed, including the two where the test declines to answer
Both first-order autocorrelation estimates: the 1 − d/2 shorthand and the direct one
A large-sample normal test as a fallback when the bounds are inconclusive
The negative-autocorrelation half of the test, which is usually ignored
Bounds computed Any n and k Both ρ estimates All five regions
This tests lag 1 only. Seasonal dependence passes it untouched.
Updated 13 September 2026 · Works in any browser, no installation
The Durbin-Watson statistic measures whether each residual resembles the one before it. It runs from 0 to 4: near 2 means independence, near 0 means consecutive residuals move together, near 4 means they alternate. Autocorrelated residuals do not bias the coefficients of a regression, but they make every standard error too small — so the model looks far more certain than it is, and the usual response is to widen the intervals rather than to change the fit.
At a glance
Formula shown
d = Σ(eₜ − eₜ₋₁)² / Σeₜ², over residuals in time order. Its exact null distribution depends on the regressor matrix, which is why the test comes with two bounds rather than one critical value: the distribution is bracketed by two weighted sums of chi-squares whose weights are the eigenvalues 4sin²(iπ/2n) of the differencing matrix, taken from either end of the spectrum. Those two quantiles are dL and dU, and they are obtained here by inverting Imhof’s integral rather than from a table.
Scenario support
Checking a time-series regression before trusting its standard errors, diagnosing a missing lagged variable or omitted trend, validating that residuals from a forecast model are white noise, and deciding whether to move to generalised least squares or a Newey-West correction.
Educational estimate
Planning support from the values you enter — not professional advice.
Why the test has two critical values instead of one
Every other test on this site has a single critical value. Durbin-Watson has two, and the reason is structural rather than a convention.
The null distribution of d depends on the regressor matrix, not only on n and k. Two regressions with the same shape and different predictors have different critical values.
Durbin and Watson’s answer was to bracket it. Whatever the regressors, the true distribution lies between two that depend only on n and k.
Those two are weighted sums of chi-squares with weights 4sin²(iπ/2n), the eigenvalues of the differencing matrix, taken from opposite ends of the spectrum.
So there are five regions, not three: reject positive, no conclusion, accept, no conclusion, reject negative. Software that prints d and a single p-value has quietly chosen one of the two bounds.
The inconclusive band is bigger than it looks
The two no-conclusion regions are usually dismissed as a technicality. At realistic sample sizes they are a large share of the scale.
At 24 observations with one regressor the band runs 1.2728 to 1.4457 — 0.1730 wide, and mirrored above 2, so 0.3460 of the 0-to-4 range is unanswerable.
With five regressors it runs 0.9249 to 1.9018. That is 0.9770 wide, 5.6× as much, and with its mirror it covers 48.9% of the scale.
The band shrinks with n and grows with k. At one regressor it is 0.2836 wide at n = 15, 0.1373 at n = 30, 0.0403 at n = 100 and 0.0201 at n = 200 — which is why the problem is mostly a small-sample one.
The shipped inconclusive preset lands there deliberately: d = 1.3471 between 1.2728 and 1.4457. The honest report is that the test has no answer, and the large-sample z is printed beside it as the usual fallback.
Two autocorrelations that do not agree
Almost every treatment of this statistic says d ≈ 2(1 − ρ). The approximation is usually presented without the word approximately doing any work.
On the shipped positive preset, 1 − d/2 gives 0.7919. The first-order autocorrelation computed directly from the same residuals is 0.5749.
The shorthand overstates it by 37.8%. Both figures are printed here so the gap cannot be missed.
The cause is the ends of the series. The numerator of d sums n − 1 squared differences while the denominator sums n squared residuals, so the first and last observations are treated differently from the rest — an edge effect that matters at n = 24 and fades by n = 200.
On the independent preset the two agree to within 0.003, which is what the approximation looks like when it is safe. Read the gap, not just the statistic.
It only ever looks one step back
d is built entirely from consecutive pairs, which makes it blind to any dependence that skips a step.
Monthly data with an annual cycle can pass cleanly. Each month resembles the same month a year earlier, and consecutive months need not resemble each other at all.
A quarterly series with a four-period pattern has the same problem. d sees nothing and the residuals are plainly not independent.
Which is what the Ljung-Box test is for: it pools many lags at once and catches dependence wherever it sits.
And the correlogram shows where. Durbin-Watson answers a yes-or-no question about lag 1; the other two answer the question that usually follows.
The other half of the scale
Almost every use of this test looks only for values below 2. The region above 2.5 is a real diagnostic and is usually skipped.
The shipped negative preset gives d = 3.1538, above the rejection point of 2.7272, with residuals that alternate rather than drift.
Negative residual autocorrelation usually means over-differencing. A series that was differenced when it did not need to be produces exactly this pattern.
It can also mean an over-fitted model chasing noise, or a moving-average error structure that the specification does not include.
The test is symmetric, so the bounds simply mirror: 4 − dU and 4 − dL. All five regions are printed here for that reason.
What to do when it rejects
A significant Durbin-Watson is not a reason to abandon the model. It is a reason to stop trusting its standard errors.
The coefficients stay unbiased. Autocorrelated errors do not pull the estimates; they misstate the uncertainty around them, almost always downward.
The first question is whether the model is missing something. An omitted trend, a missing lagged term or an unmodelled seasonal pattern all show up as autocorrelated residuals, and fixing the specification is better than patching the errors.
If the specification is right, correct the standard errors. A Newey-West covariance estimator is the usual route and needs no assumption about the form of the dependence.
Or model the error process directly with generalised least squares, which is more efficient when the structure really is first-order.
What not to do is read the test on a model that includes a lagged dependent variable. d is biased toward 2 there and will accept autocorrelation it should reject; Durbin’s h test exists for that case.
Reporting the statistic
Four items, and the second is what makes d interpretable at all.
Give d to four figures. Two is not enough to place a value inside a band 0.17 wide.
Give n and the number of regressors. Both bounds depend on them, and neither can be reconstructed without both.
Give the bounds, or say which convention you used. “d = 1.35, dL = 1.27, dU = 1.45, inconclusive” is a complete report; “d = 1.35, no autocorrelation” is a claim the test did not make.
And say what you did next. Whether the standard errors were corrected, the specification changed, or nothing done and why.
Sources and methodology
References for the statistic, its bounds and the inversion used here.
Method. d is computed from the residuals in the order given, and the bounds are not looked up. The bounding null distributions are weighted sums of chi-squares with weights 4sin²(iπ/2n), and their quantiles are obtained by bisection on Imhof’s numerical inversion — which means dL and dU exist at any n and k rather than only where a table happens to have a row. The suite checks that inversion where the answer is known in closed form, since equal weights make the mixture an ordinary chi-square and the two agree to seven decimals, and then checks the bounds themselves against Savin and White at eight (n, k) combinations: every one matches to 0.006, which is the table’s own rounding. Both first-order autocorrelation estimates are computed independently and printed, because the familiar 1 − d/2 is an approximation that is off by 37.8% on one of the shipped presets. The suite also asserts that d always lies between 0 and 4, that reversing or rescaling the residuals leaves it unchanged, that perfect alternation gives exactly 4(n − 1)/n rather than 4, and that the inconclusive band widens with every added regressor. That engine is verified on every change against 157 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Ljung-Box TestThe Ljung-Box portmanteau test with Box-Pierce beside it, the degrees-of-freedom correction for fitted models and a lag-by-lag table.
AutocorrelationACF and PACF at every lag with Bartlett bands that widen rather than a flat line, plus the shapes that identify a model.
Linear RegressionThe least-squares line with r and r² — and the regression of x on y beside it, because those are two different lines rather than one line rearranged.
ResidualResiduals, standardised and studentised residuals, leverage and Cook's distance, with Anscombe's quartet built in — including the dataset every residual rule scores cleanest and which one point controls entirely.
Moving AverageSimple, exponential, weighted and centred moving averages with the lag each one carries — exactly (n−1)/2 periods for a simple average, which is also the reason α = 2/(n+1) is in every EMA formula.
Kolmogorov-Smirnov TestKS goodness-of-fit with the Lilliefors correction when parameters are fitted, exact small-sample p-values and a two-sample mode.
An educational tool. The Durbin-Watson statistic tests first-order autocorrelation only, so seasonal or longer-lag dependence can pass it untouched, and it is biased toward 2 — accepting too readily — in any model containing a lagged dependent variable. Its exact critical value depends on the regressor matrix, which is why the result is bounded rather than exact and why two regions of the scale have no conclusion at all.
Launched Durbin-Watson with dL and dU computed by Imhof inversion of the bounding chi-square mixtures rather than read from a table, so the bounds exist at any n and k.
Checked those computed bounds against Savin and White at eight (n, k) pairs; every one matches to 0.006, the table rounding.
Printed all five decision regions including the two where the test declines to answer — 0.1730 wide at one regressor and 0.9770 at five, at 24 observations.
Reported both first-order autocorrelation estimates, since the familiar 1 - d/2 overstates the direct figure by 37.8% on the shipped preset.
Covered the negative-autocorrelation half of the scale, which is usually omitted and diagnoses over-differencing.
Add this calculator to your site
Responsive embed — and private: nothing your visitors type leaves their browser.