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FRM Exam Part I · Regression Diagnostics

Serial Correlation and Autocorrelation of Regression Residuals

Updated 11 October 2026 · Fact-checked

Serial correlation means regression error terms are correlated across observations, usually over time. OLS coefficients stay unbiased, but standard errors are wrong, so t-tests mislead. Detect it with residual plots or the Durbin-Watson statistic (about 2 means none), and fix it with Newey-West standard errors or by modelling the dynamics.

Understand Serial Correlation and Autocorrelation of Residuals

A regression assumes the error term in one period tells you nothing about the error in another. That is the no-serial-correlation assumption: Cov(εt, εs) = 0 for t ≠ s. Serial correlation, also called autocorrelation, means this fails. A positive error today makes a positive error tomorrow more likely (positive serial correlation). Negative serial correlation means errors tend to flip sign.

It is common in time series data such as monthly returns, spreads or macro variables. It often appears when you leave out a variable that moves slowly over time, or when you pick the wrong functional form. The leftover pattern then shows up in the residuals.

What breaks? If the regressors are strictly exogenous, the OLS slope estimates remain unbiased and consistent. But OLS is no longer the best linear unbiased estimator, and the usual standard errors are wrong. With positive serial correlation, the usual standard errors are typically too small. That inflates t-statistics and makes you reject true null hypotheses too often (more Type I errors). R² and F-statistics are also unreliable. If a lagged dependent variable is a regressor and errors are autocorrelated, the OLS estimates themselves become inconsistent.

To detect it, plot the residuals over time and look for runs of the same sign. Then use the Durbin-Watson (DW) test, which checks for first-order autocorrelation. Another option is the Breusch-Godfrey test, which handles higher orders.

To correct it, keep the OLS coefficients but use Newey-West (HAC) standard errors, which are robust to both serial correlation and heteroskedasticity. You can also add missing variables or lags, or transform the model.

Do not confuse it with heteroskedasticity. Heteroskedasticity is non-constant error variance. Serial correlation is correlation between errors in different periods. They are different violations, though Newey-West handles both.

Key formulas to remember

Durbin-Watson statistic
DW = Σ(et − et−1)² ÷ Σet²
Numerator sums from t = 2 to n; denominator from t = 1 to n. et are OLS residuals.
DW and first-order autocorrelation
DW ≈ 2(1 − ρ)
Approximation for large samples. ρ = 0 gives DW ≈ 2; ρ near +1 gives DW near 0; ρ near −1 gives DW near 4.
Solving for ρ
ρ ≈ 1 − DW ÷ 2
Use to estimate the residual autocorrelation from a reported DW.
AR(1) error process
εt = ρεt−1 + ut
ut is white noise. |ρ| < 1 for stationary errors. Serial correlation exists if ρ ≠ 0.
DW decision rule
DW < dL: reject H0 (positive autocorrelation); DW > dU: do not reject; dL ≤ DW ≤ dU: inconclusive
H0: ρ = 0. dL and dU come from tables based on n and the number of regressors. For negative autocorrelation, compare 4 − DW with the same bounds.

How to solve Serial Correlation and Autocorrelation of Residuals questions

Use this sequence for any question on serial correlation, DW or its corrections.

  1. 1Identify the violation. Errors correlated across time means serial correlation. Non-constant variance means heteroskedasticity.
  2. 2If DW is given, compute or approximate ρ using ρ ≈ 1 − DW ÷ 2.
  3. 3Compare DW with 2. Well below 2 points to positive autocorrelation; well above 2 points to negative.
  4. 4If critical values dL and dU are given, apply the decision rule. Remember the inconclusive zone.
  5. 5State the consequence: coefficients unbiased (with exogenous regressors), standard errors biased, t-tests unreliable. Positive correlation usually understates standard errors.
  6. 6Pick the fix: Newey-West standard errors, added lags or variables, or a respecified model.
  7. 7Check that your final answer matches the direction asked (Type I error, standard errors too small or large, bias or no bias).

Quickest way: The DW-to-ρ shortcut

When to use it: Any question that gives a DW value or residual data and asks about autocorrelation.

  1. Compute ρ = 1 − DW ÷ 2 in your head.
  2. Near 0: no problem. Positive: standard errors too small, t-stats inflated. Negative: opposite.
  3. If the options include a fix, choose Newey-West for robust standard errors without changing coefficients.
  4. Eliminate any option claiming OLS coefficients are biased by serial correlation alone.

Common mistakes in Serial Correlation and Autocorrelation of Residuals

  • Saying serial correlation makes OLS coefficients biased.

    Students mix it up with omitted variable bias.

    Fix: With strictly exogenous regressors, OLS slopes stay unbiased and consistent. The standard errors are what fail.

  • Treating DW = 0 as no autocorrelation.

    Students assume zero means none.

    Fix: No autocorrelation gives DW ≈ 2. DW near 0 means strong positive autocorrelation; near 4 means strong negative.

  • Ignoring the inconclusive region of the DW test.

    Students use only one critical value.

    Fix: Use both dL and dU. Between them, the test cannot decide.

  • Confusing heteroskedasticity with serial correlation.

    Both are residual problems with similar fixes.

    Fix: Heteroskedasticity is about unequal variance. Serial correlation is about correlation between errors at different times.

  • Believing positive autocorrelation makes standard errors too large.

    Students guess the direction.

    Fix: Positive autocorrelation typically understates standard errors, inflating t-statistics and causing too many false rejections.

  • Using the DW test with a lagged dependent variable as a regressor.

    DW is applied mechanically.

    Fix: DW is biased toward 2 in that case. Use the Breusch-Godfrey test instead.

Worked examples

Example 1

A regression of monthly fund returns on a market factor yields residuals with a Durbin-Watson statistic of 0.80. Estimate the first-order autocorrelation of the residuals and state the likely effect on t-statistics.

Show the solution
  1. Use ρ ≈ 1 − DW ÷ 2.
  2. ρ ≈ 1 − 0.80 ÷ 2 = 1 − 0.40 = 0.60.
  3. ρ is clearly positive, so positive serial correlation is likely (DW is far below 2).
  4. With positive serial correlation, usual OLS standard errors are typically too small, so t-statistics are overstated.

Answer: ρ ≈ 0.60; positive autocorrelation; t-statistics are likely overstated and Type I errors more frequent.

Example 2

In a regression with n = 60 and one regressor, the DW statistic is 1.20. The critical values are dL = 1.55 and dU = 1.62. Test H0: no positive autocorrelation. Which action best addresses a rejection while keeping the coefficient estimates?

Show the solution
  1. Compare DW with dL: 1.20 < 1.55.
  2. Since DW < dL, reject H0.
  3. Conclude there is evidence of positive serial correlation.
  4. To fix inference while keeping OLS coefficients, compute Newey-West (HAC) standard errors.
  5. Estimated ρ ≈ 1 − 1.20 ÷ 2 = 0.40, consistent with positive autocorrelation.

Answer: Reject H0; positive serial correlation is present. Use Newey-West standard errors.

Exam tips

  • Memorize the DW anchors: about 0 is strong positive, about 2 is none, about 4 is strong negative.
  • Questions often ask what is affected: answer standard errors and t-tests, not coefficient bias.
  • When asked for a correction that handles both serial correlation and heteroskedasticity, think Newey-West.
  • Read carefully whether the question asks about positive or negative autocorrelation; the direction reverses the consequences.
  • Expect to be given DW and asked for ρ; practice 1 − DW ÷ 2 until automatic.

Practice questions from Regression Diagnostics

Serial Correlation and Autocorrelation of Residuals in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Serial Correlation and Autocorrelation of Residuals: frequently asked questions

How do I interpret the Durbin-Watson test?

A value near 2 suggests no first-order autocorrelation. Values well below 2 suggest positive autocorrelation and values well above 2 suggest negative. Use the dL and dU critical values for a formal decision, remembering the inconclusive range.

What is the difference between heteroskedasticity and serial correlation?

Heteroskedasticity means the error variance changes across observations. Serial correlation means errors in different periods are correlated with each other. Both distort standard errors, but they are different violations and are diagnosed differently.

How do you correct autocorrelation in regression residuals?

The common fix is Newey-West standard errors, which keep the OLS coefficients and correct the standard errors. You can also add omitted variables or lags, or respecify the model if the autocorrelation signals misspecification.

Does serial correlation bias OLS coefficients?

Not when regressors are strictly exogenous. The coefficients stay unbiased and consistent, but the standard errors are wrong and OLS is no longer efficient.