hard · Quantitative Finance Probability, Statistics & Linear Algebra
Consider the AR(1) process X_t=φ X_t-1+varepsilon_t with varepsilon_t i.i.d. N(0,σ^2) and |φ|<1, in stationarity. You compute the OLS estimator hatφ from T observations.
Which statement about hatφ is correct?
- hatφ is biased downward in finite samples (toward zero / its sign), with bias of order -1/T, even though it is consistent and asymptotically normal
- hatφ is unbiased for every finite sample size T, because the regression error term is conditionally mean-zero given the entire observed past
- hatφ is biased upward in finite samples, overstating persistence, because positive autocorrelation in the residuals reinforces itself across the sample path
- hatφ is inconsistent as Tto∞ because the regressor X_t-1 is correlated with past shocks, which violates the strict-exogeneity assumption OLS needs
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