You estimate an OLS regression on time-series data and diagnostic tests reveal AR(1) autocorrelation (ρ ≈ 0.75) in the residuals. What is the primary statistical consequence you should be concerned about?
AThe coefficient estimates are biased and do not represent the true population parameters
BThe standard errors are wrong (typically too small), making t-statistics and p-values unreliable
CThe R² statistic is inflated, overstating the model's explanatory power
DThe coefficient estimates are inefficient, but since they are unbiased, inference proceeds normally
Autocorrelation does not bias OLS coefficient estimates — they remain unbiased (and consistent). But the OLS standard error formula assumes uncorrelated errors; with autocorrelation, those formula-derived standard errors are incorrect, typically too small. This makes t-statistics appear larger than they should be and p-values smaller, leading to false confidence in statistical significance. Option D correctly identifies that coefficients are unbiased but wrongly concludes that inference proceeds normally — the standard errors are broken.
Question 2 Multiple Choice
A researcher fits a linear trend to GDP data that grows exponentially. Even if the true underlying shocks are independent white noise, what will the residuals likely show?
ANo autocorrelation, since the shocks are independent by assumption
BNegative autocorrelation, because the model alternates between over- and under-prediction
CPositive autocorrelation, because the misspecified model leaves a systematic curved pattern in residuals
DHeteroskedasticity but not autocorrelation, since the variance grows with the level
Model misspecification can generate apparent autocorrelation even when the true errors are white noise. A linear trend fitted to exponential growth will under-predict in the early and late periods and over-predict in the middle (or vice versa), leaving a systematic curved residual pattern that manifests as positive autocorrelation. This is a crucial diagnostic insight: residual autocorrelation may signal misspecification (wrong functional form, wrong trend specification) rather than a fundamentally autocorrelated error process.
Question 3 True / False
When OLS is applied to time-series data with AR(1) autocorrelation, the regression coefficients are biased.
TTrue
FFalse
Answer: False
Autocorrelation violates one of the classical OLS assumptions, but it does not cause bias in the coefficient estimates. OLS remains unbiased (and consistent) under autocorrelation. What breaks is the validity of the standard error formulas, which assume Cov(uₜ, uₛ) = 0 for t ≠ s. With correlated errors, the true sampling variance of the estimates is different from what the OLS formula computes. The practical consequence is invalid inference, not biased estimates.
Question 4 True / False
An ACF plot that decays slowly and geometrically across many lags is diagnostic of AR-type autocorrelation structure.
TTrue
FFalse
Answer: True
AR(p) processes produce an ACF that decays gradually (often geometrically for AR(1)) and a PACF that cuts off sharply after lag p. A slow, exponential decay in the ACF across many lags is the signature pattern of autoregressive structure. By contrast, an MA(q) process shows the opposite: the ACF cuts off sharply after lag q while the PACF decays slowly. Correctly reading these patterns from ACF/PACF plots is essential for diagnosing the type of autocorrelation and choosing the appropriate correction.
Question 5 Short Answer
Why does autocorrelation in OLS residuals lead to incorrect statistical inferences even though the coefficient estimates themselves are still correct?
Think about your answer, then reveal below.
Model answer: OLS coefficient estimates are unbiased and consistent under autocorrelation — the same conditional expectation formula applies. But the OLS standard error formula is derived under the assumption of uncorrelated errors (Cov(uₜ, uₛ) = 0). With positive autocorrelation, adjacent observations carry redundant information, and the effective sample size is smaller than the actual n. The standard errors that assume independent observations therefore understate the true uncertainty around the estimates, making the t-ratios too large and the p-values too small. Researchers who trust those standard errors will reject null hypotheses too often — a false precision problem.
A useful analogy: if you survey the same 100 people twice and pool the 200 responses as if they were independent, your standard errors will be too small because the two observations from each person are correlated. Autocorrelation in time-series data creates an analogous problem: observations close in time are not independent, so treating them as if they are overstates the information content of the sample.