Confidence Distributions for the Autoregressive Parameter

Figures & data

Fig. 1 Simulated and asymptotic confidence distributions, n = 100, 200, 400 and ${\widehat{\varphi }}_{\text{obs}}=0.95,0.975,0.9875$ (i.e., c = 5 in (12)) from left to right.

Fig. 2 Simulated and asymptotic confidence distributions, n = 100, 200, 400 and ${\widehat{\varphi }}_{\text{obs}}=0.90,0.95,0.975$ (i.e., c = 10 in (12)) from left to right.

Fig. 3 Simulated empirical confidence curve, ${\widehat{\varphi }}_{\text{obs}}=0.90$, n = 100.

Fig. 4 Log confidence density functions, for n = 100, 10⁵ replicates. In this simulation, ${\widehat{\varphi }}_{\text{obs}}=0.533$.

Fig. 5 Simulated logs of implied priors $c_{\text{emp}}(\varphi )/L(\varphi ,{\sigma }^2)$ for generating parameters ${\varphi }_0=0.5,{\sigma }^2=1$, known ${\sigma }^2$. We have 1000 replicates for each ${\widehat{\varphi }}_{\text{obs}}$, and 50,000 replicates of ${\widehat{\varphi }}_{\text{obs}}$.

Fig. 6 Simulated logs of implied priors $c_{\text{emp}}(\varphi )/L(\varphi ,{\sigma }^2)$ for generating parameters ${\varphi }_0=0.5,{\sigma }^2=1$, unknown ${\sigma }^2$. We have 1000 replicates for each ${\widehat{\varphi }}_{\text{obs}}$, and 50,000 replicates of ${\widehat{\varphi }}_{\text{obs}}$.

Fig. 7 Log exchange rate SKr/Euro, second half of 2021.

Fig. 8 Log exchange rate SKr/Euro, confidence curve.

Fig. 9 Log exchange rate SKr/Euro, “confidence curve” obtained via a flat Bayesian prior.

Fig. 10 Log exchange rate SKr/Euro, Bayesian and frequentist confidence curves.

Fig. 11 U.S. unemployment rate, ages 25–54, years 1948–2020.

Fig. 12 U.S. unemployment rate, Bayesian and frequentist confidence curves.