Durbin Watson Test: Formula, Interpretation, R, Python, SPSS and Excel Guide 1

What Is the Durbin Watson Test?

The Durbin Watson Test is used after fitting a regression model. It examines whether the residual from one observation is related to the residual from the previous observation. In time-series regression, this matters because errors may follow a pattern across time. If residuals are positively autocorrelated, a positive error today may be followed by another positive error tomorrow. If residuals are negatively autocorrelated, a positive error may be followed by a negative error.

Most short online explanations stop at the statement “2 means no autocorrelation.” That is not enough for a serious data-analysis post. A better explanation must show the regression model, residuals, formula components, lagged residual plot, residual ACF, software code, and a careful warning about observation order. This guide does that using R, Python and SPSS verification.

If you are learning normality and variance-testing workflows alongside regression diagnostics, you may also find these related guides useful: DAgostino Pearson Test for skewness-kurtosis normality checking, Cramer von Mises Test for goodness-of-fit, Cochran C Test for checking the largest variance, and Brown Forsythe Test for robust equality-of-variance analysis.

Durbin Watson Test Formula

The Durbin Watson statistic is calculated from regression residuals. If the residual at observation t is written as e_t, the statistic is:

d = Σ(e[t] - e[t-1])² / Σe[t]²

The numerator measures how much consecutive residuals change. The denominator measures the total squared residual size. If residuals change randomly around zero, the statistic is usually close to 2. If consecutive residuals are too similar, the numerator becomes smaller and the statistic moves below 2. If consecutive residuals tend to alternate signs, the numerator becomes larger and the statistic moves above 2.

Rule-of-thumb interpretation

A common practical rule is that values between about 1.5 and 2.5 usually do not indicate a serious first-order autocorrelation problem. This rule is helpful for a quick diagnostic, but formal decisions should consider the number of observations, number of predictors, model structure and critical values.

Critical-value interpretation

The formal table method uses lower and upper bounds, often called dL and dU. For a positive autocorrelation test, if the statistic is below dL, there is evidence of positive autocorrelation. If it is above dU, the test does not reject no autocorrelation. If it falls between dL and dU, the result is inconclusive. For negative autocorrelation, the same logic is applied to 4 − d.

Durbin Watson Test Null Hypothesis and Alternative Hypothesis

Dataset and Regression Model Used

This worked example uses the student-por.csv student performance dataset. The main outcome variable is G3, the final grade. The regression model predicts G3 from earlier grades, study-related variables and family/academic background variables.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified Results in R, Python and SPSS

The analysis was reproduced in three environments. R used an OLS model and the lmtest workflow. Python used statsmodels OLS residuals and a permutation check. SPSS imported the clean CSV, saved predicted values and residuals, and manually computed the statistic from consecutive residual differences.

Main Model Result

SPSS Regression Model Summary

Main Regression Coefficients from SPSS

Model Comparison

Durbin Watson Test Charts and Interpretation

1. Actual vs Fitted G3

This chart shows whether the regression model predicts G3 reasonably well. The points follow the diagonal trend closely, which agrees with the high R Square value of about 0.851. Some low-grade outliers remain, but the model captures most of the final-grade pattern.

2. Regression Residuals by Observation Order

The residuals fluctuate around zero, which is what we want to see. There are some spikes, especially around outlier cases, but the plot does not show a strong smooth trend. This supports the conclusion that there is no serious first-order residual autocorrelation.

3. Consecutive Residual Differences

This chart shows how much each residual changes from the previous residual. Most changes remain near zero, with occasional large jumps. These large jumps contribute to the numerator of the statistic, but they do not create a severe autocorrelation pattern by themselves.

4. Durbin Watson Numerator Contributions

The tallest spikes identify observation positions where adjacent residuals changed sharply. This chart is useful because it opens the formula visually: the statistic is not a black-box number; it is built from these consecutive residual changes divided by total squared residuals.

5. Lagged Residual Scatter

This is one of the most important plots. If strong positive autocorrelation existed, the points would form a clear upward-sloping pattern. Here the fitted trend is only slightly upward, matching the small lag-1 residual correlation of about 0.0685.

6. Durbin Watson Statistic Across Regression Models

All three models produce statistics below 2 but still near 2. The background model has the lowest value, suggesting slightly more positive dependence, but all three remain within the common 1.5–2.5 range.

7. Permutation Null Distribution

The observed statistic is a little left of 2, which is consistent with weak positive autocorrelation. However, it is not extremely far from the null distribution center. The Python permutation two-sided p-value of about 0.0707 agrees with the R two-sided p-value of about 0.0672.

8. Residual Autocorrelation by Lag

The lag bars are mostly small. A few lags show mild positive values, but the chart does not show a very strong autocorrelation pattern. Since the statistic mainly targets lag-1 autocorrelation, the first bar is especially important and remains small.

How to Run the Durbin Watson Test in R, Python, SPSS and Excel

Durbin Watson Test in R

In R, the easiest method is to fit a linear model and use dwtest() from the lmtest package. You can also manually compute the statistic from residuals.

install.packages("lmtest")
library(lmtest)

student <- read.csv("student-por.csv", sep = ";", stringsAsFactors = FALSE)

student$G1 <- as.numeric(student$G1)
student$G2 <- as.numeric(student$G2)
student$G3 <- as.numeric(student$G3)
student$studytime <- as.numeric(student$studytime)
student$failures <- as.numeric(student$failures)
student$absences <- as.numeric(student$absences)
student$age <- as.numeric(student$age)
student$Medu <- as.numeric(student$Medu)
student$Fedu <- as.numeric(student$Fedu)

model <- lm(G3 ~ G1 + G2 + studytime + failures + absences + age + Medu + Fedu,
            data = student)

dwtest(model, alternative = "two.sided")
dwtest(model, alternative = "greater")

res <- resid(model)
dw_manual <- sum(diff(res)^2) / sum(res^2)
lag1_cor <- cor(res[-length(res)], res[-1])

dw_manual
lag1_cor

Durbin Watson Test in Python

In Python, use statsmodels to fit the regression and calculate the Durbin Watson statistic from the residuals.

import pandas as pd
import statsmodels.api as sm
from statsmodels.stats.stattools import durbin_watson

student = pd.read_csv("student-por.csv", sep=";")

cols = ["G1", "G2", "G3", "studytime", "failures", "absences", "age", "Medu", "Fedu"]
for col in cols:
    student[col] = pd.to_numeric(student[col], errors="coerce")

data = student[cols].dropna()

y = data["G3"]
X = data[["G1", "G2", "studytime", "failures", "absences", "age", "Medu", "Fedu"]]
X = sm.add_constant(X)

model = sm.OLS(y, X).fit()
residuals = model.resid

dw_statistic = durbin_watson(residuals)
lag1_correlation = residuals[:-1].corr(residuals[1:])

print(model.summary())
print("Durbin Watson statistic:", dw_statistic)
print("Lag-1 residual correlation:", lag1_correlation)

Durbin Watson Test in SPSS

SPSS can report the statistic through linear regression output, but the verified workflow below also saves residuals and calculates the statistic manually.

REGRESSION
  /MISSING LISTWISE
  /STATISTICS COEFF R ANOVA COLLIN
  /DEPENDENT G3
  /METHOD=ENTER G1 G2 studytime failures absences age Medu Fedu
  /SAVE PRED(prd_main) RESID(res_main).

SORT CASES BY caseid(A).
EXECUTE.

COMPUTE lag_res_main = LAG(res_main).
IF ($CASENUM = 1) lag_res_main = $SYSMIS.
COMPUTE res_diff_main = res_main - lag_res_main.
COMPUTE res_diff2_main = res_diff_main ** 2.
COMPUTE res2_main = res_main ** 2.
EXECUTE.

AGGREGATE
  /OUTFILE=* MODE=ADDVARIABLES
  /BREAK=
  /n_main = N(res_main)
  /sum_diff2_main = SUM(res_diff2_main)
  /sum_res2_main = SUM(res2_main).

COMPUTE dw_main = sum_diff2_main / sum_res2_main.
EXECUTE.

Durbin Watson Test in Excel

Excel does not need a special function for the statistic. You can compute it directly from a residual column.

Durbin Watson statistic = SUM(F3:F650) / SUM(G2:G650)

For this dataset, the Excel calculation should return approximately 1.8615 if the same residuals and same observation order are used.

How to Report the Result

A good report should mention the regression model, the statistic, the direction of possible autocorrelation, and the final interpretation. Avoid writing only “DW = 1.86” without explaining what it means.

Related Data Analysis Guides

After learning the Durbin Watson Test for autocorrelation, continue with related statistical testing guides on Online Internet Cafe: DAgostino Pearson Test for omnibus normality testing, Cramer von Mises Test for goodness-of-fit analysis, Cochran C Test for detecting the largest variance, and Brown Forsythe Test for robust variance comparison.

When Should You Use This Test?

Use this diagnostic when your regression residuals have a meaningful order. The most common case is time-series regression, where observations are arranged by day, month, year or another time unit. It can also be used when observations follow a production sequence, repeated measurement order, route order or another meaningful sequence.

Common Mistakes

1. Using row order without explaining it

The statistic depends on the order of residuals. If the row order has no real meaning, the result can be misleading. In this guide, the row order is clearly described as a demonstration order.

2. Treating 1.5 to 2.5 as a universal law

The 1.5–2.5 range is a practical rule of thumb, not a substitute for formal critical values or a p-value from a proper test implementation.

3. Ignoring visual diagnostics

A statistic alone is not enough. Residual sequence plots, lagged residual scatter plots and ACF charts help explain what the statistic is detecting.

4. Forgetting that the test targets first-order autocorrelation

The statistic mainly focuses on lag-1 residual autocorrelation. If you suspect higher-order serial correlation, use additional tests such as Breusch-Godfrey or Ljung-Box.

5. Confusing autocorrelation with model fit

A model can have a high R Square and still have autocorrelated residuals. Similarly, a model can have weak fit but no residual autocorrelation. These are different diagnostics.

Download SPSS Output and Verification Files

The SPSS PDF verifies the main model import, regression output, saved residuals and manual calculation of the statistic.