Methodology

The mechanism. In the system \(r_{t+1} = \alpha + \beta x_t + u_{t+1}\), \(x_{t+1} = \theta + \rho x_t + v_{t+1}\), the AR(1) estimate of \(\rho\) is biased downward by roughly \((1+3\rho)/T\) (Kendall 1954), and that error transmits into the slope with its sign flipped: \(E[\hat\beta - \beta] \approx -(\sigma_{uv}/\sigma_{vv})(1+3\rho)/T\). With \(\sigma_{uv} < 0\) (a price shock moves return and yield oppositely) the slope bias is positive. Our data: \(\rho = 0.99\), \(\mathrm{corr}(u,v) = -0.95\).

Table 1 simulates the null (\(\beta = 0\)) at each subsample’s estimated \((\rho, T, \Sigma)\), 20,000 replications, vectorized over simulations with accumulated OLS cross-products.

Table 2 samples four posteriors. Specifications A and B (conditional likelihood) are conjugate: inverse-Wishart for \(\Sigma\), matrix-normal for the coefficients, with B truncating \(\rho\) to \((-1,1)\) by rejection. Specifications C and D (exact likelihood, with \(x_0\) drawn from the predictor’s stationary distribution) are not conjugate and are sampled by random-walk Metropolis-Hastings in an unconstrained parameterization, with the proposal covariance estimated from a pilot chain. We validated the sampler by running it on the conditional likelihood, where it reproduces the conjugate answer. An earlier importance-sampling approach is retained in the repository, documented as the attempt whose failure (weights concentrating on draws near \(\rho = 1\)) motivated the sampler.

Reproducibility. doit rebuilds everything: WRDS pulls, the tidy panel, every table and figure, the executed notebook, this site, and the test suite — 28 tests, split so simulation-based tests run in CI without credentials while data-dependent tests skip cleanly when the panel is absent.