Questions: Time Series Data: Structure and Concepts

Question 1

Two trending variables — global average temperature and the number of pirates worldwide since 1800 — show a high R² and a statistically significant coefficient when regressed on each other. What is the most likely explanation?

• A. Pirates causally affect climate through ocean disruption
• B. The regression suffers from omitted variable bias
• C. Both variables share a time trend, producing a spurious regression
• D. The OLS estimator is consistent when applied to time series

Answer: (C)

This is a classic spurious regression: two unrelated non-stationary series that both trend over time will appear strongly correlated because their shared trend dominates. High R² and significant t-statistics do not indicate a real relationship — they reflect the shared trend. Standard OLS inference is invalid for non-stationary series.

Question 2

A time series with a stochastic trend is stationary after first-differencing.

• T. True
• F. False

Answer: True

A random walk (the canonical stochastic trend) has the form Yₜ = Yₜ₋₁ + εₜ. First-differencing yields ΔYₜ = εₜ, which is white noise — stationary with constant mean and variance. This is why differencing is a standard preprocessing step before applying OLS to time series data. A series that becomes stationary after one difference is called 'integrated of order 1' or I(1).

Question 3

Why does autocorrelation in the error term cause problems for standard OLS inference, even if OLS coefficient estimates remain unbiased?

Model answer: OLS standard errors are derived under the assumption that errors are uncorrelated. When errors are autocorrelated, the OLS formula for standard errors underestimates the true sampling variability, making t-statistics too large and p-values too small — leading to false positives.

Autocorrelation violates the Gauss-Markov assumption of uncorrelated errors. While the OLS coefficient estimator remains unbiased (errors average out), the estimated variance of those coefficients is biased downward because the standard OLS formula ignores the covariance between error terms across time. The result is overconfident inference — rejecting null hypotheses too often.