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1 A Proof of Theorem 2 (i) If the likelihood ratio p(x θ i )/p(x θ ) is an absolutely continuous random variable for any i, then p(x θ ), = 1,..., m, have the same support and the posterior distribution is well defined for any prior π and ν-almost all x X. Moreover, ties in the posterior probabilities (p(θ i X) = p(θ X), i ) happen with probability zero under any θ Θ. An HPD credible set ϕ(, ; π) is uniquely defined and continuous in π whenever there are no ties in the posterior probabilities. The function z(π) defined in Theorem 1 is therefore continuous in π and Theorem 1 implies that there exists a prior π for which ϕ(, ; π ) has coverage of at least 1 α. (ii) Next, let us show that π > 0 for any and ϕ(, ; π ) is a similar 1 α confidence set. If π = 0 for some then θ is not contained in the 1 α HPD credible set for any x and ϕ(, ; π ) has zero coverage at θ. Thus, π > 0 for all. Since ϕ(, ; π ) is a 1 α credible set ϕ(θ, x; π )p(x θ )π = α p(x θ )π. Integration of the last display implies [ ] ϕ(θ, x; π )p(x θ )dν(x) π = α. (12) Since the coverage of ϕ(, ; π ) is at least 1 α, ϕ(θ, x; π )p(x θ )dν(x) α. Because π > 0 for all the equality in (12) can hold only if ϕ(θ, x; π )p(x θ )dν(x) = α for all or, in other words, ϕ(, ; π ) is similar. (iii) It suffices to show that if m ϕ (θ, x) m ϕ(θ, x; π ) for all x X and m ϕ (θ, x) > m ϕ(θ, x; π ) for all x X l with ν(x l ) > 0, then 25

2 ϕ (θ, x)p(x θ )dν(x) > α for some. The HPD set ϕ(θ, x; π ) can be defined for ν-almost all x by the minimum length property that for all ϕ with m ϕ (θ, x) = m ϕ(θ, x; π ), m ϕ(θ, x; π )p(θ x) m ϕ (θ, x)p(θ x). Thus, for ν-almost all x X l, ϕ(θ, x; π )p(x θ )π < ϕ (θ, x)p(x θ )π and for all x X the inequality holds weakly. Integrating this inequality with respect to ν yields m π (ϕ(θ, x; π ) ϕ (θ, x))p(x θ )dν(x) < 0. Since m π ϕ(θ, x; π )p(x θ )dν(x) = α by part (ii), this implies that there exists such that ϕ (θ, x)p(x θ )dν(x) > α. B Proof of Theorem 3 (i) By the assumed uniform continuity and max diam( Θ m ) 0, there exists M ɛ such that for any m M ɛ and θ 1, θ 2 Θ m, = 1,..., m, p(x θ 1 ) p(x θ 2 ) dν(x) < ɛ. (13) In order to obtain a contradiction, assume there exists and θ Θ m such that ϕ m ( θ, x; π m ) p(x θ )dν(x) < α ɛ. (14) For any θ 1 Θ m, ϕm ( θ, x; π m ) = ϕ m ( θ 1, x; π m ) as ϕ m is constant on Θ m by definition. Therefore, by (13) and (14), ϕ m ( θ 1, x; π m ) p(x θ 1 )dν(x) < α, θ 1 Θ m, which would make the equality in (6) impossible. A contradiction for ϕ m ( θ, x; π m ) p(x θ )dν(x) > α + ɛ can be obtained in the same way. 26

3 (ii) Suppose the claim does not hold. Then, there exists a subsequence {m k } with φ( θ, x)d θ (1 + ɛ) ϕ m k ( θ, x; π m k )d θ for ν-almost all x. Pick m k > M αɛ, with M ɛ defined in part (i) of this proof. For this m k, and any θ Θ m k, define φ (θ, x) = φ ( θ, x) = φ( θ1, x)d θ 1 /V and φ (θ, x) = φ ( θ, x) = φ ( θ, x)/(1 + ɛ), Θ where V = vol( Θ m k ), = 1,..., m k. Note that φ( θ, x)d θ = φ ( θ, x)d θ = (1 + ɛ) φ ( θ, x)d θ = (1 + ɛ) φ (θ, x)v. Thus, φ (θ, x)v ϕ m k ( θ, x; π m k )d θ = ϕm k (θ, x; π m k )V, and since ϕ m k (θ, x; π m k ) maximizes ϕ(θ, x)v subect to [α ϕ(θ, x)]p(x θ )π m k 0, φ (θ, x)p(x θ )π m k Integration of this inequality with respect to ν gives π m k ϕ m k (θ, x; π m k )p(x θ )π m k. φ (θ, x)p(x θ )dν(x) α. Thus, there exists such that φ (θ, x)p(x θ )dν(x) α and φ (θ, x)p(x θ )dν(x) (1 + ɛ)α. (15) At the same time, φ (θ, x)p(x θ )dν(x) = Θ m k Θ m k φ( θ 1, x)p(x θ 2 )dν(x)d θ 1 d θ 2 /V 2. Because φ( θ1, x)p(x θ 1 )dν(x) α for all θ 1 Θ by assumption on φ, φ (θ, x)p(x θ )dν(x) α+ sup θ 1, θ 2 Θ m k Combining (15) and (16) yields the desired contradiction. 27 p(x θ 1 ) p(x θ 2 ) dν(x) < (1+ɛ)α. (16)

4 C Computational Details for Applications C.1 Algorithm The fixed point iterations described in Section 3 require repeated evaluation of coverage probabilities. These may be computed using an importance sampling approach: Let p be a proposal density such that p(θ x) is absolutely continuous with respect to p, and let X i, i = 1,..., N be N i.i.d. draws from p. Then non-coverage probability of a set ϕ at θ can then be written as RP = ϕ(θ, x)p(θ x)dν(x) = ϕ(θ, x) p(θ x) p(x)dν(x), yielding the approximation p(x) RP (ϕ) = N 1 N i=1 ϕ( θ, X i ) p(θ X i ) p(x i ). Write ϕ π for the set ϕ(θ, x; π) of Theorem 1. We employ the following algorithm to obtain an approximate π such that the HPD set ϕ π has nearly coverage close to the nominal level: 1. Compute and store p(θ X i ) p(x i, i = 1,..., N, = 1,..., m. ) 2. Initialize π (0) at π (0) = 1/m, = 1,..., m. 3. For l = 0, 1,... (a) Compute z = RP (ϕ π (l)) α, = 1,..., m. (b) If max z min z < ε, set π = π (l) and end. (c) Otherwise, set π (l+1) = exp(ωz )π (l) / m k=1 exp(ωz k)π (l) k, = 1,..., m, and go to step 3a. 28

5 We set ε = , and found ω = 1.5 to yield reliable results as long as N is chosen large enough. In the context of obtaining a credible set with approximately uniform coverage in a bounded but continuous set Θ, we employ the above algorithm for a given partition m with ϕ π = ϕ m (θ, x; π) now defined as described in Section In addition, we evaluate the uniform coverage properties of the resulting set estimator on Θ, ϕ m ( θ, x; π m ), by computing the (approximate) non-coverage probabilities RP( θ) = N 1 N i=1 ϕm ( θ, X i ; π m ) p( θ X i ) p(x i ) over a fine grid of values of θ. If these uniform properties are unsatisfactory, then the algorithm is repeated using a finer partition. For an unbounded parameter space Θ, we implement the guess and verify approach described in Section We first choose an appropriate κ S by computing the coverage of the HPD set relative to a flat prior on Θ, and select κ S to be ust large enough for coverage to be sufficiently close to 1 α for all θ with ς( θ) > κ S. We then partition Θ NS into m 1 subsets, and set Θ m m = Θ S. As discussed in Section 2.3.4, it makes sense to rule out a large discontinuity of Π m ( θ, π m ) at the boundary between Θ NS and Θ S. Thus, in the above algorithm, we directly adust the m 1 values of Π m ( θ, π) on Θ m, = 1,..., m 1 without any scale normalization, and simply set Π m ( θ, π) on Θ S equal to the value of Π m ( θ, π) of a subset Θ m neighboring Θ S. This has the additional advantage of avoiding computation of the potentially ill-defined RP m. After the iterations have concluded, we evaluate the coverage properties of the resulting set estimator φ m ( θ, x; π m ) on a fine grid on Θ NS, and on a fine grid in the part of Θ S where φ m ( θ, x; π m ) is affected by the shape of the prior 29

6 Π m ( θ, π) on Θ NS. If these are unsatisfactory, we increase m and/or κ S. C.2 Details for Break Date and Magnitude The continuous process X is approximated with 800 steps. Uniform coverage is evaluated on the Cartesian grid with λ {0.15, , ,..., 0.85} and δ { 15.0, 14.99, 14.98,..., 15}. We set N = , and p to be uniform on {(λ, δ) : 0.13 λ 0.87, 16 δ 16}, which yields Monte Carlo standard deviations of coverage probabilities uniformly smaller than We impose symmetry with respect to the sign of δ, and around λ = 0.5, in the computation of π m. Computations for the finer partition take about 6 hours on a modern PC. In the application, we use Elliott and Müller s (2014) estimate of 2.6 for the long-run standard deviation of the quarterly data y t. C.3 Details for Autoregressive Root Near Unity The continuous process X is approximated with 800 steps. Uniform coverage is evaluated on 5001 values θ {120( 5000 )2 } 5000 =0. We set N = , and set p to be uniform on the 101 values θ {160( 100 )2 } 100 =0, which yields Monte Carlo standard deviations of coverage probabilities uniformly smaller than For the application, we rely on output of the DF-GLS regression also employed in Lopez, Murray, and Papell (2013) to obtain small sample analogues to 1 0 X(s)dX(s) and 1 0 X(s)2 ds. Specifically, let ˆρ, ˆσ ρ and ˆφ be the usual OLS estimate of ρ, its standard error and the additional coefficients in an aug- 30

7 mented Dickey-Fuller regression using GLS demeaned data (with lag length as determined by Lopez, Murray, and Papell (2013)). We then employ the analogue (T 1 ˆφ(1)(ˆρ 1)/ˆσ 2 ρ, T 2 ˆφ(1) 2 /ˆσ 2 ρ) ( 1 0 X(s)dX(s), 1 0 X(s)2 ds) for the empirical results in Table 1. 31

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