ECON20003 · Quantitative Methods 2
Regression with Time-Series Data & Autocorrelation
Regression with Time-Series Data & Autocorrelation applies regression when observations are ordered in time, where errors in nearby periods tend to be correlated. You meet distributed-lag models (current and past values of a predictor) and autoregressive models (a variable on its own lags). The headline problem is autocorrelation: correlated errors leave OLS unbiased but inefficient with wrong standard errors, so the t and F tests are invalid. You detect it with the Durbin-Watson statistic (DW near 2 is clean, below 2 signals positive autocorrelation) or the Breusch-Godfrey LM test, and you remedy it with added lags, GLS/Cochrane-Orcutt, or Newey-West (HAC) standard errors.
What this chapter covers
- 01Cross-sectional vs time-series data; distributed-lag and autoregressive models
- 02Autocorrelation: OLS unbiased but inefficient, SEs wrong ⇒ invalid t/F
- 03Durbin-Watson: DW ≈ 2(1 − ρ̂); ≈ 2 clean, < 2 positive autocorrelation
- 04Breusch-Godfrey LM test (valid with higher orders and lagged dependent variables)
- 05Remedies: add lags, GLS/Cochrane-Orcutt, Newey-West (HAC) standard errors
Diagnosing autocorrelation with Durbin-Watson and Breusch-Godfrey
- 2 marksRead the Durbin-Watson: DW ≈ 2(1 − ρ̂), so ρ̂ ≈ 1 − DW/2 = 1 − 1.18/2 = 1 − 0.59 = 0.41 — a moderate positive first-order autocorrelation.
- 1 markApply the DW decision: DW = 1.18 is well below 2 with p = 0.004 < 0.05, indicating significant positive autocorrelation.
- 2 marksConfirm with Breusch-Godfrey: H₀ is no autocorrelation up to order 2; p = 0.002 < 0.05, so reject — serial correlation is present (BG stays valid even with a lagged dependent variable).
- 2 marksState the consequence: OLS coefficients remain unbiased, but the standard errors are wrong, so the t and F tests are invalid as reported.
- 1 markState a remedy: add appropriate lags, use GLS/Cochrane-Orcutt, or report Newey-West (HAC) standard errors before any inference.
Key terms
- Autocorrelation (serial correlation)
- Correlation between the regression errors at different time points, Corr(εₜ, εₜ₋ₛ) ≠ 0. It leaves OLS unbiased but inefficient and makes the standard errors wrong, invalidating the t and F tests.
- Durbin-Watson statistic
- A test for first-order autocorrelation with DW ≈ 2(1 − ρ̂). A value near 2 indicates no autocorrelation, below 2 positive autocorrelation, and above 2 negative autocorrelation.
- Breusch-Godfrey LM test
- A general test for autocorrelation up to a chosen order; H₀ is no serial correlation, so a small p rejects it. Unlike Durbin-Watson it handles higher orders and remains valid when a lagged dependent variable is included.
- Newey-West (HAC) standard errors
- Heteroskedasticity- and autocorrelation-consistent standard errors. They correct the standard errors for serial correlation so that valid t and F inference can proceed without re-specifying the model.
Regression with Time-Series Data & Autocorrelation FAQ
How do I read the Durbin-Watson statistic quickly?
Use ρ̂ ≈ 1 − DW/2. A DW near 2 means ρ̂ near 0 (no autocorrelation), a DW well below 2 means positive autocorrelation (ρ̂ > 0), and a DW well above 2 means negative autocorrelation. Confirm borderline cases with the Breusch-Godfrey test, which gives a clean p-value.
Does autocorrelation make my coefficients biased?
No — like heteroskedasticity, autocorrelation leaves the OLS coefficients unbiased; it only makes the standard errors wrong, so the t and F tests are unreliable. The fixes are adding lags, using GLS/Cochrane-Orcutt, or reporting Newey-West (HAC) standard errors.
Exam move
Practise the DW-to-ρ̂ conversion and a one-line reading of both DW and Breusch-Godfrey outputs. Memorise the shared refrain across heteroskedasticity and autocorrelation — coefficients unbiased, standard errors wrong, fix the SEs — because it earns marks in both chapters.
