Time-series regressions have a habit their cross-sectional cousins mostly lack: this quarter’s error looks like last quarter’s. That is autocorrelation, and like heteroskedasticity it leaves your OLS coefficients standing while quietly wrecking every standard error printed beside them. The repair — Newey–West standard errors — is one option in any software package, but examiners want the mechanism, not the menu click. It is a topic students bring constantly to our online econometrics tutors; this page builds it from the residual plot up.
1 · What autocorrelation is
The classical model assumes the errors are uncorrelated across observations: Cov(uₜ, uₛ) = 0 for t ≠ s. Autocorrelation (or serial correlation) is the failure of that assumption — the error process remembers itself.
The workhorse description is the AR(1) error: uₜ = ρuₜ₋₁ + eₜ, where eₜ is fresh noise and ρ measures the memory. With ρ > 0 — much the most common case in economics — a positive shock today tends to be followed by another positive one: a supply shock that lifts inflation this quarter is still echoing next quarter. In the residuals it shows up as runs: long stretches above zero, then long stretches below, instead of noise bouncing across the line. The left panel of the diagram below is exactly that signature, and the right panel — the correlogram, the bar chart of the residuals’ correlation with their own lags — shows the memory decaying lag by lag.
2 · What it does to OLS
Keep the two consequences separate; the marks usually sit on the distinction.
The coefficients survive. With regressors that are exogenous, OLS remains unbiased and consistent under autocorrelated errors — the slope formula never uses the no-serial-correlation assumption. (One sharp exception: with a lagged dependent variable among the regressors, autocorrelated errors make OLS biased and inconsistent.)
The standard errors fail. The classical variance formula assumes each observation brings independent information. With ρ > 0, adjacent observations partly repeat each other, so the sample holds less information than the formula thinks: the classical standard errors are too small, and t-tests over-reject. OLS is also no longer efficient, but modern practice fixes the inference rather than reweighting the estimator.
3 · Detecting it
Plot first. Residuals against time: runs of same-signed residuals mean positive autocorrelation. The correlogram turns the impression into numbers — sample autocorrelations at each lag, judged against the ±2/√T significance band.
The Durbin–Watson statistic is the classic single number. It is tied to the lag-1 autocorrelation by the approximation DW ≈ 2(1 − ρ̂): no autocorrelation gives DW ≈ 2, strong positive autocorrelation drags it toward 0, negative pushes it toward 4. Its limits: it tests only lag 1, needs special tables, and is invalid with a lagged dependent variable.
The Breusch–Godfrey test is the general tool: regress the OLS residuals on the original regressors plus p lags of the residuals, and compare LM = n·R² from that auxiliary regression against a chi-squared critical value with p degrees of freedom. It handles higher-order autocorrelation and survives lagged dependent variables.
4 · The fix — Newey–West standard errors
You could model the error process and transform it away, but that requires knowing the process. Newey–West standard errors — also called HAC, heteroskedasticity- and autocorrelation-consistent — skip that step. Like White standard errors, they rebuild the variance estimate from the residuals themselves; unlike White, they also add in the sample covariances between observations up to a chosen lag length L, with nearer lags weighted more heavily. Pick L generously enough to capture the memory (with quarterly data, four lags is a natural start) and the result is valid whether the errors are autocorrelated, heteroskedastic, both or neither.
The crucial invariant carries over: Newey–West changes the uncertainty attached to β̂, never β̂ itself.
Worked example — a consumption regression with memory
You regress quarterly consumption growth on income growth over 15 years (T = 60) and get β̂₁ = 0.50 with a classical standard error of 0.10.
Step 1 — The naive inference. t = 0.50/0.10 = 5.0. The default output says the effect is overwhelming.
Step 2 — Look at the residuals. Plotted over time they run — several quarters above zero, then several below, like panel (a) of the diagram. Independent errors don’t queue up like that.
Step 3 — Measure the memory. Regress the residuals on their own first lag: ρ̂ = 0.6. Each quarter’s error carries 60% of the last one forward.
Step 4 — Durbin–Watson confirms. DW ≈ 2(1 − 0.6) = 0.8, far below 2 — strong positive autocorrelation.
Step 5 — How wrong is the classical SE? For AR(1) errors alongside an AR(1) regressor, the true sampling variance of the OLS slope exceeds the classical formula by roughly (1 + ρρₓ)/(1 − ρρₓ). Income growth is itself persistent, with ρₓ = 0.6, so the factor is (1 + 0.36)/(1 − 0.36) = 2.125: the classical formula understates the variance by more than half.
Step 6 — The honest standard error. Correcting inflates the SE by √2.125 ≈ 1.46, to 0.10 × 1.46 ≈ 0.15 — the order of magnitude a Newey–West estimate with a few quarterly lags delivers here, without assuming the AR(1) form.
Step 7 — Redo the inference. t = 0.50/0.146 = 3.43, down from 5.0. Still significant — but the evidence is meaningfully weaker than the default output claimed, and β̂₁ = 0.50 never moved. When a naive t sits nearer the critical value, this correction is what flips verdicts.
When autocorrelation strikes, can you say exactly what breaks — the coefficient, or the standard error? Keeping the estimate and the inference apart, and computing the Newey–West correction rather than just naming it, is what these questions actually test. Drilling that separation is what a one-on-one online econometrics tutor does with your own past papers. Book a trial session.
Practice
Q1. (a) Residuals have a lag-1 autocorrelation of ρ̂ = 0.35. Approximate the Durbin–Watson statistic. (b) A regression reports DW = 2.6. Approximate ρ̂ and interpret its sign.
Q2. A slope is estimated as β̂₁ = 0.24 with classical SE 0.06. The errors and the regressor are both AR(1) with ρ = 0.5 and ρₓ = 0.8. Compute the variance-inflation factor, the corrected standard error, and both t-statistics. Is the coefficient still significant at 5% (critical value 1.96)?
Q3. With n = 80, a Breusch–Godfrey auxiliary regression with one lag of the residuals gives R² = 0.06. Compute the LM statistic and test at 5% (χ²(1) critical value 3.84).
Answers. Q1: (a) DW ≈ 2(1 − 0.35) = 1.3. (b) ρ̂ ≈ 1 − 2.6/2 = −0.3 — negative autocorrelation: the errors tend to alternate sign. Q2: factor = (1 + 0.4)/(1 − 0.4) = 2.33; corrected SE = 0.06 × √2.33 ≈ 0.09; classical t = 0.24/0.06 = 4.0, corrected t = 0.24/0.092 ≈ 2.62. Still significant at 5%, but far less emphatic. Q3: LM = n·R² = 80 × 0.06 = 4.8 > 3.84, so reject the null of no autocorrelation — the errors are serially correlated and HAC standard errors are warranted.
Key takeaways
- Autocorrelation means the errors remember: Cov(uₜ, uₛ) ≠ 0, typically AR(1) with ρ > 0 in economic time series — visible as runs in the residual plot and a tall, decaying correlogram.
- OLS coefficients survive (absent a lagged dependent variable); the classical standard errors are too small with positive autocorrelation, so t-tests over-reject.
- Detect with the residual plot and correlogram, DW ≈ 2(1 − ρ̂) for a quick read, and the Breusch–Godfrey n·R² test as the general tool.
- Newey–West (HAC) standard errors fix the inference — robust to autocorrelation and heteroskedasticity at once, leaving the estimate untouched. Choose the lag length to cover the memory.
Why LSE, UCL and Oxbridge students choose our econometrics tutoring
- Matched to your course’s toolkit: whether your module leans on Durbin–Watson, Breusch–Godfrey, or goes straight to HAC estimation, sessions use your lecture notes’ notation and your package’s commands.
- The distinctions that carry marks, drilled: unbiased versus efficient, estimate versus standard error, when a lagged dependent variable changes the answer — rehearsed until they come out cleanly under exam pressure.
- Online and one-on-one: tutors work from your problem sets and past papers over a shared screen, so each session lands on the questions your paper will ask.
FAQ
Q: Does autocorrelation bias OLS estimates?
A: Not by itself. With exogenous regressors the coefficients remain unbiased; what breaks is the standard errors and everything built on them. The exception is a model with a lagged dependent variable, where autocorrelated errors do bias OLS.
Q: What do Newey–West standard errors do?
A: They estimate the coefficient variances allowing for both autocorrelation (up to a chosen lag length) and heteroskedasticity, using the residuals’ own covariances. Only the standard errors change — never the estimates.
Q: How do I choose the Newey–West lag length?
A: Long enough to cover the errors’ memory: rules of thumb grow the lag with the sample size, and four lags is a common quarterly starting point. If results are fragile to reasonable choices, say so and show both.
Q: What does a Durbin–Watson statistic of 2 mean?
A: Roughly zero first-order autocorrelation, since DW ≈ 2(1 − ρ̂). Values well below 2 indicate positive autocorrelation; values above 2, negative.
Q: Should I use Newey–West or model the error process directly?
A: For inference about the coefficients, HAC standard errors are the modern default — they take no stand on the error process. Model the dynamics instead when the dynamics themselves are the object of interest.
Book an online econometrics tutor for LSE, Kings, UCL, Oxford or Cambridge
Autocorrelation questions reward students who keep the estimate and the inference separate — and can compute the correction, not just name it. One-on-one online sessions build both, on your course’s own past papers. Tell us your module and exam date, and we’ll match you with the right tutor this week.
