3.1. Marginal and Conditional Prior Distributions

Throughout this section, we will assume that k = 1 and w = 0 for notational convenience. The results will still be valid for k > 1 and w > 0 as long as prior independence between Ψ and the other parameter matrices is assumed. All probability distributions in this section will be conditional on a given cointegration rank, though this will not be written out explicitly.

The space of β is not euclidean because of the nonidentification of the cointegration vectors. It is deceptive to think in terms of the free parameter space of β under some arbitrarily chosen normalization, e.g., the linear normalization in Section 2, without regard to the fact that actual parameter space is non-euclidean. In the following paragraphs we shall describe the true parameter space of β and show that the prior in (3.1) implies a uniform distribution over this abstract space.

Let

denote the set of p × r real matrices of rank r(≤ p) and define the group of transformations X → XL, where

and L is any nonsingular r × r matrix. This group defines an equivalence relation

in

such that for any

if and only if sp(X) = sp(Y). Thus, the points of the resulting coset space of equivalence classes, usually denoted by

, stand in a 1-1 correspondence with the r-dimensional subspaces of

. The set of r-dimensional subspaces of

is an analytic manifold of dimension (p − r)r (James, 1954), which has been termed the Grassman manifold and is denoted by

.

The uniform distribution on

is naturally defined as the (unique) invariant distribution under the group of transformations of

induced by the group of orthonormal transformations of

(James, 1954).

As a result of the nonidentification of the cointegration vectors explained in Section 2, the actual parameter space of β is the Grassman manifold. We shall now prove that the distribution in (3.1) implies that β is marginally uniformly distributed over

. First we need a definition and a few lemmas.

DEFINITION 3.1. An m × s matrix D follows the matrix t distribution,

, if its density is given by

See Box and Tiao (1973) and Bauwens, Lubrano, and Richard (1999) for properties of the matrix t distribution.

LEMMA 3.2. Let R be a p × r matrix of independent N(0,1) variables. Then sp(R) is uniformly distributed over

.

Proof. See James (1954).

LEMMA 3.3. If N₁ and N₂ are independent s × s and m × s matrices of independent N(0,1) variables, then

Proof. See Phillips (1989) and Dickey (1967).

LEMMA 3.4. If β = (I_r B′)′ and B ∼ t_(p−r)×r(0,I_p−r,I_r,1), then sp(β) is uniformly distributed over

.

With the preceding definitions and lemmas out of the way, we are now prepared to state an important property of the distribution in (3.1).

THEOREM 3.5. β is marginally uniformly distributed over

.

To illustrate this rather abstract uniform distribution, let us consider the bivariate case with a single cointegration vector β = (1,B)′. According to the proof of Theorem 3.5 in the Appendix, the distribution in (3.1) implies a Cauchy(0,1) distribution on B. This is not surprising given that B is a ratio of two independent N(0,1) variates under the uniform distribution over

(see Lemmas 3.2–3.4). A more natural, but computationally inconvenient, parametrization of β is the polar parametrization

where θ is the angle of the cointegration vector. In this parametrization the distribution in Theorem 3.5 reduces to a constant density for θ (James, 1954). Slightly more generally, in the p-dimensional case with a single cointegration vector, the distribution in Theorem 3.5 reduces to the conventional uniform distribution over the p-dimensional hemisphere with unit radius (Mardia and Jupp, 2000). In the general case, we may say that the prior in (3.1) assigns equal probability to every possible cointegration space of dimension r. Although more informative prior information on the cointegration vectors may be available, the marginal prior on β implied by the prior in (3.1) satisfies all four of the desiderata stated in the Introduction and should therefore be a suitable reference prior.

It should be noted that the prior in (3.1) is by no means the only distribution on α and β that implies a uniform distribution on the Grassman manifold. The prior in (3.1) is especially interesting, however, in that it is both conceptually relevant and, as will be shown later, very convenient from a computational viewpoint.

THEOREM 3.6. The marginal prior of Σ is

where IW denotes the inverted Wishart distribution (Zellner, 1971).

Proof. This follows directly from the proof of Theorem 3.5. █

From (3.1) we immediately obtain

where A ∼ N_m×s(μ,Ω₁,Ω₂) means that vec A ∼ N_ms(vec μ,Ω₁ [otimes ] Ω₂). The linear normalization of β makes α difficult to interpret, however, and the conditional prior in (3.3) may not shed much light on the prior in (3.1).

Consider instead the prior of α conditional on β and Σ when β is orthonormal. Restricting β to be orthonormal is not sufficient to identify the model, however, as any orthonormal version of β can be rotated to a new one by postmultiplying it with an r × r orthonormal matrix. This need not concern us here as β only enters p(α|β,Σ) in the form β′β and p(α|β,Σ) is therefore invariant under these rotations. Define

and note that

is orthonormal. For Π = αβ′ to remain unchanged by the transformation

, we must make the corresponding transformation of the adjustment matrix from

. In the following theorem, let

denote the ith column of

and note that

describes how the p response variables are affected by the ith cointegrating relation under the orthonormal normalization.

THEOREM 3.7.

.

The rather restrictive form of the prior in Theorem 3.7 must be motivated. First, the restriction to conditional normal priors on α (and thereby also on

) is necessary for an efficient numerical evaluation of the posterior; see Sections 4 and 5. Second, nonidentical priors on the columns of

do not make sense unless overidentifying restrictions on the columns of β are used to give a unique meaning to each cointegration vector. Another way to see this is that within the class of matrix normal priors

, only the priors with μ = 0, Ω₁ = I_r are invariant to rotations of

. Third, the scale matrix in the conditional prior may be any positive definite matrix; the posterior computations remain nearly the same. By making the conditional covariance matrix proportional to Σ we are taking the possibly differing scales of the time series into account. Finally, the reason for centering the conditional prior over zero is motivated by the invariance requirement just stated. It has the effect of centering the prior over Π = 0, which is often a good starting point in an analysis; see the discussion of the “sum of coefficients” prior in Doan, Litterman, and Sims (1984) and Section 3.2.

In his influential development of Bayesian reference tests of sharp null hypotheses Jeffreys (1961, Ch. 5) argued that the prior on the parameters under the alternative hypothesis should be centered over the point in the null and that the prior spread around this point should be an increasing function of the model's scale parameter; see also Berger (1985, Sect. 4.3.3). Although the situation is quite a bit more complex here, the prior in Theorem 3.7, which is centered over the hypothesis Π = 0, or r = 0, with a prior scale depending on Σ, has the same flavor and should therefore be appropriate for inference on the cointegration rank; see Section 5.

Further clarification of the hyperparameters A, q, and v is obtained from the marginal prior of

. By multiplying

with the marginal inverted Wishart prior of Σ and integrating with respect to Σ, we obtain

Results in Box and Tiao (1973, pp. 446–447) then give

where E(Σ) = A/(q − p − 1) is the expected value of Σ a priori; see, e.g., Bauwens et al. (1999, p. 306).

The hyperparameter A is determined from E(Σ) and q, and the investigator thus faces subjective specification of (i) the expected value of Σ, (ii) the degree of certainty regarding Σ (large values of q imply large certainty), and (iii) the tightness around the point zero for

(large values of v give high concentration of probability mass around zero). Note that whether a value for v is large or not depends on E(Σ), which should therefore be specified before v.

The main difficulty for the investigator is likely to be the specification of E(Σ). If interest only centers on the posterior of α,β,Ψ,Σ conditional on a given cointegration rank, then A may be set equal to the zero matrix and q = 0. This corresponds to using the usual improper prior p(Σ) ∝ |Σ|^−(p+1)/2. If we also aim at analyzing the cointegration rank, but are either unable or unwilling to state our beliefs about Σ, then

may be used, where

is the maximum likelihood estimate of Σ in the full rank model; note that this implies that

. This suggestion is of course not a proper Bayesian solution as the prior then becomes dependent on the observed data. The consequences of this side step are minimized by the choice of the smallest possible q (maximum uncertainty) subject to a finite expected value of Σ.

3.2. Prior Stability

Define

The assumption of rank(Π) = r implies that r of the eigenvalues of Π_C are equal to one. A cointegrated process is stable if all the remaining eigenvalues of Π_C are smaller than one in modulus. It is clearly of interest to know what prior probability is implicitly being placed on the set of stable processes if the prior in (3.1) is used. This could be investigated either by analytical approximation or by simulation methods for different models, i.e., by varying p, r, and k. We shall here be content with simulating the special case p = 2 and r = k = 1. Table 1 displays the prior probability that the process is stable for A = I₂ as a function of q and σ = v^−1/2 (note that σ is on a standard deviation scale). Experiments with other choices of A with strong positive and negative correlation structure did not have a large impact on the probability. Note also that it is unnecessary to increase the magnitude of the diagonal elements in A as this has the same effect as increasing σ.

Implied prior probability that the process is stable

Densities of the unrestricted eigenvalue (λ) are displayed in Figure 1 for different values of σ. The densities are symmetric around the modal value λ = 1. A nonsymmetric density for λ that places more mass to the left of λ = 1 than to the right of this point would perhaps better represent actual beliefs. The gain from a nonsymmetric prior is probably less than the loss in computational efficiency in the posterior calculations, however.

Implied prior distribution on the unrestricted eigenvalue for A = I2 and q = 4. Here σ = 0.25 (- · -), σ = 0.5 (——), and σ = 1 (- - -).

A crude way to obtain a nonsymmetric prior is to simply exclude explosive processes a priori (or “too explosive” processes, e.g., with eigenvalues larger than 1.1 in modulus) by restricting the domain of the prior in (3.1) to the space of α, β, and Ψ where the process is stable. This is neatly handled in the posterior calculations for a given cointegration rank by simply rejecting the draws from the posterior corresponding to nonstable processes; see Section 4. Note that the latter region will be small if the process actually is stable and data informative and most draws will then be accepted. The posterior distribution of the rank will require heavier numerical computations, however.

4.1. Normalization Issues

The choice of variables used for normalizing β may be somewhat arbitrary, and it is important to show that the posterior distribution corresponding to the prior in (3.1) is invariant to this choice. Let

denote the set of indices for the r variables used to normalize β. Consider the change in normalization

, where

equals

with jth variable in the normalized set replaced by the kth variable in the nonnormalized set. This change in normalizing variables is accomplished by the transformation T_U : (α,β,Σ) → (α,β,Σ), where α = αU′, β = βU⁻¹, and U is an r × r invertible transformation matrix whose elements are functions of the kth row of B. The exact form of U need not concern us for the moment; it is sufficient to note that such a matrix always exists, and is unique, if the kth variable in the nonnormalized set has a nonzero coefficient in the jth cointegrating vector; see the proof of Theorem 4.2 in the Appendix. Such a change of normalizing variables will be termed valid. It is important to note that Π = αβ′ is unchanged by the transformation.

The next definition, adapted from Drèze and Richard (1983), formalizes the idea that the inference should not depend on whether we (i) work directly with

or (ii) start with

and then transform to

.

DEFINITION 4.1. A density p(α,β,Σ) is said to be invariant with respect to normalization if and only if its functional form is invariant with respect to the valid parameter transformation T_U : (α,β,Σ) → (α,β,Σ).

THEOREM 4.2. The posterior distribution corresponding to the prior (3.1) is invariant with respect to normalization.

The main advantages of the linear normalization are that the prior that assigns the same probability to every cointegration space is of rather simple form and that easily implemented numerical methods (see Sections 4.3 and 5) can be used to compute the posterior results. Note also that we are free to transform the posterior distribution of α and β as long as the space spanned by the columns of β and the matrix of long-run multipliers Π = αβ′ remain unchanged, i.e., the class of allowable transformations is (α,β) → (αV′,βV⁻¹), for any invertible r × r matrix V. For example, an orthonormal β is obtained with V = (β′β)^1/2. The transformation is conveniently performed directly on the posterior draws of α and β. Thus, as long as the initial linear normalization is valid (dubious normalizations may be excluded with the test of Luukkonen et al., 1999), the restriction to the linear normalization is no restriction at all as the final results may be transformed to any desired normalization.

4.2. Marginal Posterior Distribution of β

The next result gives the marginal posterior of the cointegration vectors.

THEOREM 4.3. The marginal posterior distribution of β is

where C₁ = X′M_Z X + vI_p, C₂ = vI_p + X′Q[I_T − Z(Z′QZ)⁻¹Z′Q]X, and Q = I_T − Y(A + Y′Y)⁻¹Y′.

The expression

in Theorem 4.3 is a 1-1 poly-matrix-t density (Bauwens and van Dijk, 1990). Theorem 3.1 in Bauwens and Lubrano (1996) is the limiting special case of Theorem 4.3 with A = 0 and q = v = 0 (which corresponds to a constant prior on α and β). Contrary to the family of multivariate poly-t densities (see, e.g., Dickey, 1968; Drèze, 1977; Bauwens et al., 1999), poly-matrix-t densities have remained largely unexplored. The following result can be shown, however.

THEOREM 4.4. The marginal posterior of B is integrable but possesses no finite integer moments.

Proof. The result follows from a trivial modification of the proof of Corollary 3.2 in Bauwens and Lubrano (1996).

The nonexistence of integer moments is not a consequence of the prior distribution in (3.1) but rather of the linear normalization of β, where each element of B is a matrix quotient with the upper r × r submatrix of β in the denominator. Phillips (1994) makes the same point about the distribution of the maximum likelihood estimator in the linear normalization, which he shows has Cauchy-like tails.

It is also possible to derive the marginal posterior distribution of α as in Kleibergen and van Dijk (1994, eq. (29)) in closed form. It is a complicated nonstandard distribution (see Section 6 for further discussion) and is not conveniently used in the numerical posterior evaluations discussed in the next section.

4.3. Numerical Posterior Evaluation

The marginal posterior distribution of the cointegration vectors in Theorem 4.3 is of the same 1-1 poly-matrix-t form as the distribution in Theorem 3.1 in Bauwens and Lubrano (1996). Bauwens and Lubrano discuss both importance sampling (Kloek and van Dijk, 1978) and Gibbs sampling (Smith and Roberts, 1993) approaches to evaluating such a density; Bauwens and Giot (1998) implement the Gibbs sampling approach and give details on convergence issues. The key properties used in those exercises are (i) the conditional distribution of one of the cointegration vectors conditional on all other cointegration vectors is a vector 1-1 poly-t, (ii) the 1-1 poly-t is amenable to direct simulation using the algorithm of Bauwens and Richard (1985), and (iii) the posteriors of α, Ψ, and Σ conditional on β are all standard. Once the marginal posterior of β has been evaluated by sampling methods the marginal posteriors of α, Ψ, and Σ may therefore be computed by averaging their posteriors conditional on β over the posterior sample of β. We refer the reader to Bauwens and Lubrano (1996) and Bauwens and Giot (1998) for details.

A major disadvantage of building the numerical posterior evaluations on the analytical form of

is the inability to handle posterior distributions of quantities with intractable posterior distribution conditional on β, such as impulse response functions or forecasts. The Gibbs sampler is a convenient algorithm for sampling from the joint posterior distribution of α, β, Ψ, and Σ and may thus be used in such situations; Geweke (1996) seems to have been the first to use Gibbs sampling in cointegration models. It turns out that the posterior distribution for the prior in (3.1) is amenable to an algorithm similar to the one in Geweke (1996). The Gibbs sample may also be used to efficiently compute the posterior distribution of the cointegration rank (Section 5 and Theorem 4.6, which follows).

The Gibbs sampler is an easily implemented method for generating observations from complex multidimensional distributions by sampling iteratively from the so-called full conditional posterior distributions. The full conditional posterior distribution of a subset of parameters in a model is the posterior distribution of the subset conditional on all other parameters. Initial values for all parameters are needed to start up the Gibbs sampler. The maximum likelihood estimates in Johansen (1995) are natural candidates. The sampled parameter values are not independent but can be shown to converge in distribution to the target posterior distribution independently of the choice of initial values (Tierney, 1994). Furthermore, the expected value of any well-behaved transformation of the parameters may be consistently estimated by sampling averages.

The full conditional posteriors of α, β, Ψ, and Σ are given in the next theorem.

Most of the model parameters are located in Ψ and Σ, and the Gibbs updating steps for these two matrices usually dominate the total computing time. The time to convergence of the Gibbs sampler also increases as the dimensions of Ψ and Σ grow. The next theorem gives the conditional posteriors necessary to perform a (marginal) Gibbs sampler to generate samples directly from

. This Gibbs sampler is also used in Section 5 to calculate the posterior distribution of the rank.

THEOREM 4.6.

The posterior densities of Ψ and Σ are obtainable by marginalizing their densities conditional on α and β, which belong to the matrix t and inverted Wishart family, respectively, using draws from the marginal Gibbs sampler in Theorem 4.6; Bauwens and Lubrano (1996) and Bauwens and Giot (1998) provide the details.

5.1. Monte Carlo Integration

The integrals in (5.2) with respect to α, Ψ, and Σ may be computed analytically, leading to the 1-1 poly-matrix-t density in Theorem 4.3, which is repeated here (along with its proportionality constant)

The final integral with respect to B must be computed numerically. A Monte Carlo integration approach is suggested by the following lemma, which is proved by expanding β′C₂ β in B and completing the square (see the proof of Corollary 3.2 in Bauwens and Lubrano, 1996).

LEMMA 5.2. For 1 ≤ r ≤ p − 1

where the expectation is taken with respect to the

distribution, C₂ (see Theorem 4.3) is partitioned as

The expected value in Lemma 5.2 may be computed by generating variates from the

distribution, computing |β′C₁ β|^{(T+q−d−p)/2} for each draw, and averaging over all draws.

5.2. Importance Sampling

Another method that may be used to approximate the integral with respect to β in (4.1) is importance sampling (Kloek and van Dijk, 1978; Geweke, 1989). In cases where the importance function well approximates the target integrand, importance sampling can be quite efficient as it produces independent draws without wasting an initial burn-in sample. The fact that the draws are independent makes a central limit theorem directly applicable, and the precision of the estimates is easily assessed (Geweke, 1989).

Given the heavy tails of the marginal posterior of β (Theorem 4.4), a natural suggestion for an importance function is the matrix Cauchy density. The maximum likelihood estimate of B and an estimate of its asymptotic covariance matrix (Johansen, 1995, Theorem 13.4) may be used as location and scale matrix, respectively. That is, we suggest the density

as an importance function. Further fine tuning may be introduced by multiplying (X₂′ X₂)⁻¹ by a scale factor.

Poly-t densities may be substantially skew and even bimodal. In such cases the matrix Cauchy may not perform well as an importance function. An alternative may be to generate each of the r cointegration vectors conditional on the maximum likelihood estimates of the remaining r − 1 vectors. These conditional posteriors are 1-1 poly-t (Bauwens and Lubrano, 1996) and may be generated by one of the algorithms in Bauwens and Richard (1985).

5.3. Marginal Likelihood Identity Approach

By a slight rearrangement of Bayes' theorem we obtain what Chib (1995) has termed the basic marginal likelihood identity:

Chib (1995) suggested using this identity in combination with a Gibbs sampler to estimate the marginal likelihood. The expression for

in (5.5) clearly holds for any α and B. Let

be the point where

is evaluated. As explained in Chib (1995), this point should preferably be of high posterior density; the posterior mode and median are good candidates (the posterior mean does not exist; see Theorem 4.4). The term

in (5.5) is given in the second part of Theorem 4.6, and the next result gives the expression for the numerator of (5.5).

LEMMA 5.3.

where W = Y − Xβα′.

The final term of the marginal likelihood identity

is not available in closed form, but its value in a point

, which is all we need, can be computed from a posterior sample B⁽¹⁾,…,B⁽ⁿ⁾ of B by

where

is given in the first part of Theorem 4.6. From the ergodic theorem (Tierney, 1994),

almost surely. This procedure for computing

will be named the marginal likelihood identity (MLI) algorithm.

The posterior sample from

needed in the MLI approach can be obtained from (i) a Gibbs sampler for the 1-1 poly-matrix-t density in (4.1) as described in Bauwens and Lubrano (1996) and Bauwens and Giot (1998), (ii) the marginal Gibbs sampler in Theorem 4.6, which samples from

, and (iii) the full Gibbs sampler in Theorem 4.5, which samples from

.

The matrix t conditional posteriors

in Theorem 4.6 are easily sampled using, e.g., the algorithm in Bauwens et al. (1999). Even though the second approach samples α in addition to β it is likely to be faster than the first approach, which requires draws from a 1-1 poly-t distribution for each of the cointegration vectors (for an algorithm, see Bauwens and Richard, 1985). The third approach is clearly not as fast as the second but has the advantage of yielding both the posterior distribution of the cointegration rank and the joint posterior

at the same time.