Half-life: estimate, bias, interval

From the monthly gold/silver log ratio (2001–2026, $n = 310$ regression observations), $\hat\varphi = 0.95033$. (a) Prove $h_{1/2} = -\ln 2/\ln\varphi$ and $h_{1/2}\Delta = \ln 2/\kappa$ for a sampled OU process. (b) Compute the raw half-life and the Kendall-corrected half-life. (c) Using $\text{se}(\hat\varphi) \approx \sqrt{(1-\varphi^2)/n}$, give a 95% interval for $\varphi$ and map it to a half-life interval. Also give the delta-method standard error of $\hat h$ and explain why it is misleading here. (d) What does this say about using the ratio as a trading signal?