TESTING FOR COINTEGRATION RANK USING BAYES FACTORS. Katsuhiro Sugita WARWICK ECONOMIC RESEARCH PAPERS

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1 TESTING FOR COINTEGRATION RANK USING BAYES FACTORS Katsuhiro Sugita No 5 WARWICK ECONOMIC RESEARCH PAPERS DEPARTMENT OF ECONOMICS

2 Testing for Cointegration Rank Using Bayes Factors Katsuhiro Sugita Λ July 00 Department of Economics University of Warwick, UK Abstract This paper proposes Bayesian methods for estimating the cointegration rank using Bayes factors We consider natural conjugate priors for computing Bayes factors First, we estimate the cointegrating vectors for each possible rank Then, we compute the Bayes factors for each rank against 0 rank Monte Carlo simulations show that using Bayes factor with conjugate priors produces fairly good results We apply the method to demand for money in the US Key words: Cointegration; MCMC; Bayes factor JEL classification: C11; C1; C 1 Introduction This paper introduces Bayesian analysis of cointegrated VAR systems Several researchers have proposed Bayesian approach to cointegrated VAR systems These include Kleibergen and van Dijk (199), who proposed using a Jeffrey's prior instead of diffuse prior for the cointegrating vectors since the marginal posteriors may be nonintegrable with reduced rank of cointegrating vector Geweke(199) developed general methods for Bayesian inference with noninformative reference priors in the reduced rank regression model Kleibergen and Paap (1999) use a singular value decomposition of the unrestricted long-run multiplier matrix, Π, for identification of the cointegrating vectors and for Bayesian posterior odds analysis of the rank of Π Bauwens and Lubrano (199) reduce the ECM to the form of a multivariate regression model to identify the parameters The cointegrating rank is assumed to be known a priori, based on a theoretical economic model that defines equilibrium economic relations Λ The author thanks two anonymous referees, Luc Bauwens, Mike Clements, Richard Paap and audiences at the workshop Recent Advances in Bayesian Econometrics (001) in Marseilles for their useful comments on an earlier version of this paper 1

3 If we are interested in identifying the cointegration rank, they suggest checking the plot of the posterior density of the eigenvalues of generated sample Π 0 Π, which are equal to the square of the singular values of Π However, this informal visual inspection gives ambiguous results 1 Bauwens, et al (1999) suggest using the trace test of Johansen, since on the Bayesian side, the topic of selecting the cointegrating rank has not yet given very useful and convincing results (p8) In this paper we propose a simple method of determining of the cointegration rank by Bayes factors The method is very straightforward We consider using the conjugate priors for all parameters If there exist r cointegrating vectors in the system, the adjustment term ff has rank r Applying the Bayes factor to ff, which has r rank, against the null of ff, which has rank 0, for each rank gives the posterior probabilities for its rank The procedure for obtaining the posteriors has some similarities with Bauwens and Lubrano method However, instead of using diffuse priors for all parameters, the conjugate priors are chosen to be able to compute the Bayes factors The plan of this paper is as follows Section presents the prior specifications and derives the posterior densities for estimation of the cointegrated VAR systems In Section the Bayes factors for cointegration rank is introduced Section illustrates Monte Carlo simulations with DGPs of 1000 iterations for each rank to compare the performance of the proposed Bayesian methods with the classical Johansen test In Section 5, an illustrative example of the demand for money in the United Sates is presented Section concludes All computations in this paper are performed using code written by the author with Ox v0 for Linux (Doornik, 1998) Bayesian Inference in Cointegration Analysis In this section we present Bayesian analysis of cointegration Let X t denote an I(1) vector of n-dimensional time series with r linear cointegrating relations, then unrestricted VECM representation with deterministic trend is: X t = μ + ffflz + p 1 X Ψ i X t i + " t (1) h i 0, h i where z = X 0 t 1 t fl = fi 0 ffi P, t = p; p +1;:::;T, p is the number of lags in VAR, and the errors, " t, are assumed N (0; ) and independent over time μ, ", Ψ, ±, ff, andfi are parameters of dimensions n 1; n 1, n n, n n, n r, and n r, respectively Equation (1) can be rewritten in matrix format as: where Y = X +Zfl 0 ff 0 + E = WB + E () 1 Tsurumi and Wago (199) use a highest-posterior-density-region (HPDR) test to Π, then derive the posterior pdfs for singular values to see whether 99% highest-posterior-density-interval (HPDI) contains zero

4 Y = X = X 0 p X 0 p+1 X 0 T 5, Z = X 0 p 1 p X 0 p p +1 X 0 T 1 1 X 0 p 1 X X 0 p X 0 1 X 0 T 1 X 0 T p+1 T 5, E = h 5, W = " 0 p " 0 p+1 " 0 T 5, = X Zfl 0 i, B = μ 0 Ψ 0 1 Ψ 0 p 1 " 5, # ff 0 Let m be the number of columns of Y, so that m = T p +1, then X is m (1 + n(p 1)), ((1 + n(p 1)) n), W (m k), where k =1+n(p 1) + r, and B (k n) Thus, equation () represents the multivariate regression format of (1) This representation is a starting point We then describe the prior and likelihood specifications in order to derive posteriors First, we consider the case of applying the conjugate priors for some parameters strategy to select priors is to choose a conjugate prior only for the parameters that are used in computing Bayes factor For other parameters except for the cointegrating vector, we consider non-informative priors The conjugate prior density for B conditional on covariance ± follows a matrix-variate normal distribution with covariance matrix ± Ω A 1 of the form p(b j ±) /j±j k= jaj n= exp» Our 1 trφ ± 1 (B P ) 0 A (B P ) Ψ () where A is (k k) PDS and P (k n), k = n(p 1) + r +1 (the number of columns in W ) Choosing hyperparameters, P and A, in () should be careful since these have direct effect on the value of Bayes factor Kleibergen and Paap (199) choose The idea of using data in priors is similar to a g-prior proposed by Zellner (198) For the prior density for the covariance ± in (), we can assign an inverted Wishart ρ 1 tr ± S ff 1 p(±) /jsj h= j±j (h+n+1) exp () where h is a degree of freedom, S an n n PDS The prior for fi can be given as a matrix-variate normal ß (fi) /jqj n= jhj r= exp» 1 tr nq 1 fi fi 0 H fi fi o (5) where fi is a prior mean of fi, Q is r r PDS, H is n n PDS Note that r restrictions for identification are imposed on fi, for example, fi 0 = ( I r fi? 0 ), where fi? is (n r) r unrestricted matrix If we assign r restrictions on fi as I r, then only a part of fi, fi?, follows a matrix-variate normal If we assume that B, ± and fi are mutually independent, then the joint prior of the parameters in () is p(b;fi; ±) / p(bj±)p(fi)p(±) and thus can be derived as The restrictions imposed on fi need not to be I r but can be any r restrictions See Bauwens and Lubrano (199, page 1)

5 » p(b; ±;fi) / ß (fi) jaj n= j±j k+h+n+1 exp 1 trφ ΛΨ ± 1 S +(B P ) 0 A(B P ) () To derive the conditional posterior distributions, we need to derive the likelihood functions The likelihood function for B; ±; and fi is given by: n oio L (Y j B; ±;fi) / j±j t= exp n h± 1 tr 1 bs +(B B) b 0 W 0 W (B B) b () where b B =(W 0 W ) 1 W 0 Y, and b S =(Y W b B) 0 (Y W b B) Next we derive the posteriors from the priors and the likelihood function specified above The joint posterior distribution for the conjugate priors for ff is proportional to the joint prior () times the likelihood function (), thus we have p (B; ±;fi j Y ) / p (B; ±;fi) L (Y j B; ±;fi) / ß (fi) jaj n j±j (t+h+k+n+1)= exp / ß (fi) j±j c exp» 1 tr n± 1 hs +(B P ) 0 A(B P )+b S +(B b B) 0 W 0 W (B b B io h» 1 n± tr 1 S + S b +(P B) b 0 [A 1 +(W 0 W ) 1 ] 1 (P B) b ΛΨΛ +(B B? ) 0 A? (B B? )» 1 trφ ΛΨ ± 1 S? +(B B? ) 0 A? (B B? ) (8) = ß (fi) j±j c exp where c = t + k + h + n +1, A? = A + W 0 W, B? =(A + W 0 W ) 1 (AP + W 0 W b B), and S? = S + b S +(P b B) 0 [A 1 +(W 0 W ) 1 ] 1 (P b B) From (8), the conditional posterior of ± is derived as an inverted Wishart distribution, and the conditional posterior of B as a matrix-variate normal density with covariance, ± Ω A 1?, that is, p(± j fi;y ) /js? j t= j±j (t+h+n+1)= exp p(b j ±;fi;y) /ja? j n= j±j k= exp»» 1 tr ± 1 S? (9) 1 trφ ± 1 (B B? ) 0 A? (B B? ) Ψ (10) Thus, by multiplying (9) and (10), and integrating with respect to ±, we obtain the posterior density of B conditional on fi, which is a matrix-variate Student-t form, p(b j fi;y ) /js? j t= ja? j n= js? +(B B? ) 0 A? (B B? )j (t+k)= (11) The joint posterior of B and fi can be derived by integrating (8) with respect to ±,

6 p(b;fi j Y ) / ß (fi) js? +(B B? ) 0 A? (B B? ) j (t+h+k+1)= (1) By integrating (1) with respect to B we obtain the posterior density of the cointegrating vector fi, p(fi j Y ) / ß (fi) j S? j (t+h+1)= j A? j n= (1) The properties of (1) are not known, so that we have to resort to numerical integration techniques as Bauwens and Lubrano (199) use importance sampling to compute poly-t posterior results on parameters Other feasible methods are the Metropolis-Hastings algorithm and the Griddy-Gibbs sampling The Metropolis-Hastings algorithm requires assignment of a good approximating function, the candidate-generating function, to the posterior to draw random numbers, as importance sampling requires the importance function Since the Griddy- Gibbs sampling method does not require such an approximation, we employ the Griddy-Gibbs sampler for estimation of the cointegrating vector as Bauwens and Giot (1998) use the sampler for estimation of two cointegrating vectorsthe Griddy-Gibbs sampler that is proposed by Ritter and Tanner (199) approximates the true cdf of each conditional distribution by a piecewise linear function and then sample from the approximations The disadvantage of this sampling method is that we have to assign proper range and the number of the grid The range should be chosen so that the generated numbers are not truncated Bayes Factors for Cointegration Tests This section introduces the computation of the Bayes factors for cointegration rank Subsection 1 describes briefly the basic concept of the Bayes factors and some computation techniques Subsection presents the computation of the Bayes factors for cointegration rank 1 Bayes Factors and Determination of Rank The Bayes factor, which is defined as the ratio of marginal likelihood of null and alternative hypothesis, has been used for model selection The Bayes factors can be used to construct posterior probabilities for all models that seemed plausible In classical hypothesis test, one model represents the truth and the test is based on a pairwise comparison of the alternative For a detailed discussion of the advantages of Bayesian methods, see Koop and Potter (1999) Kass and Raftery (1995) provide an excellent survey of the Bayes factor Suppose, with data Y and the likelihood functions with the parameters, there are two hypotheses H 0 and H 1 The Bayes factor BF 01 is defined as follows: BF 01 = Pr(Y jh 0) Pr(Y jh 1 ) For more details, consult Chen, et al (000), Evans and Swartz (000) For tutorial for the M-H algorithm, see Chib and Greenberg (1995) 5

7 = R p( jh0 )L(Y j ;H 0 )d R p( jh1 )L(Y j ;H 1 )d (1) With the prior odds, defined as Pr(H 0 )=Pr(H 1 ), we can compute the posterior odds, which are PosteriorOdds 01 = Pr(H 0jY ) Pr(H 1 jy ) = Pr(Y jh 0) Pr(Y jh 1 ) Pr(H 0) Pr(H 1 ) When several models are being considered, the posterior odds yield the posterior probabilities Suppose q models with H 0 ;H 1 ;:::;H q 1 are being considered, and each hypotheses of H 1 ;H ;:::;H q 1 is compared with H 0 Then the posterior probability for model i under H i is (15) Pr(H i jy )= PosteriorOdds i0 P q j=0 PosteriorOdds j0 (1) where PosteriorOdds 00 is defined to be 1 These posterior probabilities are used to select the cointegrating rank, model selection, or as weights for forecasting There are several methods to compute the Bayes factors given in (1) For example, the Laplace approximation method (Tierney and Kadane, 198), or using numerical integration techniques such as importance sampling (Geweke, 1989) or the Metropolis-Hastings algorithm See Kass and Raftery (1995) for details Chib (1995) proposes a simple approach to compute the marginal likelihood from the Gibbs output In the case of nested models computation of the Bayes factor can be simplified by using the generalized Savage-Dickey density ratio, proposed by Verdinelli and Wasserman (1995), which is based on Dickey (191) Suppose we wish to test the null H 0 : ο = ο 0 versus H 1 : ο = ο 0, where ο can be scalar, vector, or matrix With the condition that p(φjο 0 ) = p 0 (Φ), where (Φ;ο) =, then the Bayes factor can be computed with the Savage-Dickey density ratio BF 01 = p(ο 0jY ) p(ο 0 ) The denominator, the marginal prior for ο evaluated at ο = ο 0, in (1) is trivial to calculate The numerator, the marginal posterior for ο evaluated at ο = ο 0, can be calculated by integrating out the other parameters, such as: (1) p(ο 0 jy ) = ' Z 1 N p(ο 0 jφ;y)p(φjy )d NX p(ο 0 jφ (i) ;Y) (18) where Φ (i), i =1;:::;N, are sample draws from the posterior

8 Bayes Factor for Cointegration Rank The Bayes factors are used for model selection, and thus can also be used for rank selection of the cointegration Kleibergen and Paap (1999) propose a cointegration rank test by Bayes factors using a decomposition derived from the singular value decomposition In this subsection, we propose a much simpler method for rank test using Bayes factors In a cointegrated system with n variables which arei(1), if there are r cointegrating vectors, then the error correction term Π has reduced rank of r: Π can be decomposed as products of ff and fi 0, both of which have reduced rank r Since ff is unrestricted and also is given a rank reduction when cointegration exists, we compute the Bayes factor for ff that is against the null ff = 0 for determining the number of the rank using inverted form of (1) for each possible rank (r =0; 1;:::;n) BF rj0 = = R R R R RRR p (ff; fi; ; ±) L (Y j ff; fi; ; ±) dffdfid d± (1=C r ) p (ff; fi; ; ±) j L (Y j ff; fi; ; rank(ff)=0 ±) j rank(ff)=0 dfid d± p(ff 0 =0 r n ) (19) (1=C r ) p(ff 0 =0 r n jy ) where C r = R R R p(ff; fi; ; ±) j rank(ff)=0 reduction of dimension dfid d± is the correction factor that is required for If there exists r cointegrating vectors, the Bayes factor for ff 0 in (19) is the most (r n) unlikely to be zero and thus should have the highest value in Bayes factors for other possible ranks Note that the Bayes factor for rank 0 equals to 1 In case of no cointegration, the Bayes factors for ff 0,wherer = 0, are less than 1 If we assign an equal prior probability toeach (r n) cointegration rank, the posterior probability for each rank can be computed as in (1) Pr(rjY )= BF rj0 P n j=0 BF jj0 (0) where BF 0j0 is defined as 1 The posterior probabilities given by (0) can be used for solutions of the prediction, decision making and inference problems that take account of model uncertainty Generally, the hypothesis that has the highest posterior probability can be selected as the 'best' model, only if it dominates the others Otherwise, analyses will fail to take uncertainty into account To compute the Bayes factors using (19), we use (18) with samples from the posteriors Since ff 0 is a partitioned element of B in () and thus the prior for alpha is a matrix-variate Student-t distribution as shown in (), so the numerator of (19) is: 8 p ff 0 =0 (n r) = ß nr h < jsj ja :1 j n= : where A :1 = A A 1 A 1 11 A 1, A = " ny j=1 h+r+1 j h+1 j 9 = h+r ; jsj (1) # A 11 A 1, A 11 is ((n(p 1) + 1) (n(p 1) + 1)) A 1 A

9 , A 1 ((n(p 1) + 1) r), A 1 (r (n(p 1) + 1)), A (r r) The posterior for alpha, which is a matrix-variate Student-t from (11), can be estimated by (18) as follows: p(ff 0 =0 (n r) jy ) = Z = 1 N p(ff 0 =0jfi;Y )p(fijy )dfi ' 1 N NX ß nr js (i)? j t+h ja (i)?;1 j n js (i)? + B (i)0? A(i)?:1 B(i)? ( ny NX p(ff 0 =0jfi (i) ;Y) t+h+r+1 i t+h+1 i ) j t+r () " # A?11 A?1, A?11 is ((n(p 1) + 1) (n(p 1) + 1)), A?1 A?1 A? ((n(p 1) + " 1) r) #, A?1 (r (n(p 1) + 1)), A? (r r), B? is obtained by partition of B?1 B? as B? =, where B?1 is ((n(p 1) + 1) n), B? (r n) B? where fi (i), S (i)?, and A (i)? are obtained from the i th iteration of the Gibbs sampler, A?:1 = A? A?1 A 1?11 A?1, A? = 1, To compute the value of C r in (19), Chen's method (199) can be used as R RRR p (ff; fi; ; ±) dffdfid d± = C r = Z Z Z p (ff; fi; ; ±) j ff=0 dfid d± = = R R R p (ff; fi; ; ±) jff=0 dfid d± R R R R R R R p (ff; fi; ; ±) dffdfid d± R p (ff; fi; R RRR ; ±) jff=0! (ff j fi; ; ±) dff dfid d± p (ff; fi; ; ±) dffdfid d± = ' R R RR p (ff; fi; ; ±) jff=0! (ff j fi; ; ±) dffdfid d± RRR R p (ff; fi; ; ±) dffdfid d± NX 1 N = 1 N = 1 N = 1 N NX NX NX p ff p (i) ff =0;fi (i) ; (i) j fi (i) ; (i) ; ± (i) (i) ; ± (i) p ff (i) ;fi (i) ; (i) ; ± p ff =0;fi (i) ; (i) ; ± (i) p fi (i) ; (i) ; ± (i) (i) (i) (i) p ff =0; (i) (i) j ± p (i) fi p (i) ± p (i) j ± (i) p fi p ± p ff =0; (i) j ± p (i) j ± (i) () where p (ff =0; j ±) and p ( j ±) in the last line of () are derived from () The Bayes factor for alpha can be obtained by dividing (1) by () and an inverse of () When the posterior probabilities are considered, we assign equal prior probabilities to 8

10 the possible cointegration ranks such that Pr(ff rank=r )=1=(n +1) for r =0; 1;:::;n With these n +1 Bayes factors, we can compute the posterior probabilities for each rank by using (1) Note that the method described above has the disadvantage over other methods such as Johansen's and Kelibergen and Paap's methods Their methods are based on eigenvalues or singular values of the long-run matrix Π, while our method is based on testing for zero restrictions on adjustment parameters The Bayes factors we obtain in our methodology depend upon r restrictions for identification imposed on fi Thus, different choice of the ordering of the variables generates different value of Bayes factors Monte Carlo Simulation To illustrate the performance of Bayesian tests for the rank of cointegration described in the previous section, we perform some Monte Carlo simulations The data generating processes (DGPs) consist of a four-variable VAR with an intercept term having various number of cointegrating vectors (0, 1,, and ) as following: 1 y t = μ + e t y t = μ + y t = μ + y t = μ + 5 y t = μ + where μ = h 0: 0: 0: 0: h 5 0: 0: 0: 0: 0: 0: 0: 0: " 5 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: i y t 1 + e t : 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0:1 0:1 0:1 0:1 # y t 1 + e t i 0, and et ο NID(0;I ) 5 y t 1 + e t 5 y t 1 + e t We demonstrate the performance of the test by varying the true rank Each simulation of DGPs for various ranks is repeated 1000 times The sample size t is 150, of which the first 50 are used for the first experiment and the rest of 100 are used for the second experiment We consider a VAR(1) model with a constant term throughout the experiments As noted 9

11 in the previous section, since the testing procedure depends upon the chosen ordering of the variables in the VAR, the ordering of the individual series in y is changed randomly during the simulation experiment The prior parameter specifications for the natural conjugate prior in () are given with P =0and A = (c W 0c W )=T, where c W = X Z b fi and b fi =0(n r) The specification for A is a g-prior proposed by Zellner (198) and used by Kleibergen and Paap In this experiments, we assign =1and 001 to see how this prior specification affects the results We also specify the prior parameters for fi in (5) with fi? = 0, Q = I n, H = fiz 0 Z, where fi = 1=T As for the prior specification for ± in (), we assign S = fix 0 X and h = n +1 For Johansen's LR trace test, the cointegrating rank for each iteration is chosen by p-value with the 5 per cent significance level and then the number of each selected rank is counted over iterations to obtain the rates of selection for each rank Table 1 summarizes the results of Monte Carlo simulation with the sample size t is 50 The values in the columns are the average posterior probabilities of 1,000 iterations for each true rank For each iteration, the Griddy-Gibbs sampling is performed with 5,000 draws and the first 1,000 discarded The column labelled as Pr(r Y) with =1shows the average posterior probabilities when =1 As shown in the table, the highest average posterior probability for each rank indicates the correct rank With the sample size is 50, the overall performances are slightly improved when =0:01, larger prior variance The results of the Johansen tests in this paper are obtained by using Pcfiml class of Ox v0 The source code of the class was modified and recompiled for the simulations 10

12 Table 1: Monte Carlo Results: The Average Posterior Probabilities with T =50 Pr(r Y) Pr(r Y) Johansen's DGP rank r with =1 with =0:01 trace test True rank r = True rank r = True rank r = True rank r = True rank r = The last column shows the results by Johansen's trace test The results show that the test apparently suffers from the shortage of samples especially for higher ranks To improve the finite sample properties for the likelihood ratio test, Johansen (000) proposed using Bartlett correction for a VAR with small sample Increasing the sample size to 100 improves the performances of all tests as shown in Table All highest average posterior probabilities indicate the true rank Larger the prior variance on ff (less value in ), it tends to choose a lower rank 11

13 Table : Monte Carlo Results: The Average Posterior Probabilities with T = 100 DGP rank r Pr(r Y) Pr(r Y) Johansen's with =1 with =0:01 trace test True rank r = True rank r = True rank r = True rank r = True rank r = Illustrative Example: The Demand for Money in the US In this section, we illustrate an example of cointegration analysis using the method that is presented in previous sections The focus is to show the usefulness of our method with a relatively small sample size and to compare two methods of computing Bayes factors and Johansen's likelihood ratio test The example is cointegration test for the demand for money in the United States There are many papers on the estimation of the money demand function, see Goldfeld and Sichel (1990) Atypical money demand equation is m t p t = μ + fl y y t + fl R R t + e t () 1

14 where m t is the log of money, p t is the log of price level, y t is log of income, R t is the nominal interest rate The data used for our analysis is based on the annual data provided by Lucas (1988) The money stock (m t ) is measured by M1 The income (y t ) is the net national product The interest rate (R t ) is the annual rate of the six-month commercial paper We use a part of the range of the original data - from 19 to 1989 With 0 observations, we test cointegration among three variables - (m p), y, and R Before analyzing the application, we briefly explain Bayesian hypothesis testing for the number of lags in VAR Since we do not know the actual lag length for the VAR and choice of the appropriate lag length affects the cointegration analysis, we apply our method that explained in Section to select the lag length Let's consider a VAR model, Y = XB + E, where B = μ 0 Φ 0 1 Φ 0 p, and X consists of vectors of lagged Y and 1s in the first column With conjugate and/or diffuse priors, we have the posteriors which are similar to our posteriors that are given in Section Then compute the Bayes factor for each Φ i =0to select the appropriate lag length Note that for this test we do not assign the correction factor C in Bayes factor The Bayes factor selects the appropriate lag length in the VAR is 1 dominantly, though the AIC indicates lags in the VAR We model a VAR(1) with a constant term for our cointegration analysis The prior specifications are the same as in the previous Monte Carlo experiments We assign equal prior probability to each rank Table illustrates the posterior probabilities with = 1 and 001 and p-values of Johansen's trace test For = 1 we see that the posterior probabilities indicate that there is one cointegration relation with 985 per cent For =0:01 our method supports one cointegration relation more strongly with 99 per cent Thus, both results by different prior specification select one cointegration relation, although Johansen's test results in no cointegration Table : Selection of the Cointegration Rank rank Pr(r Y) w/ =1 Pr(r Y) w/ =0:01 Johansen's p-value r = r = r = r = μ- 1 0:8 0:109 The posterior means of the cointegration relation among the variables Y t = m t p t ; y t ; R t is fi = Note that the first element offi is restricted to be 1 for identification The left column of Figure 1 shows the posterior densities of the second and the third element of fi The right column of Figure 1 shows the posterior densities of the three elements in ff 1

15 Figure 1: Posterior Densities of fi? (the left column) and ff (the right column) 0 15 Posterior density of b1 b1 Posterior density of alpha1 a Posterior density of alpha a Posterior density of b b Posterior density of alpha a Conclusion This paper shows simple methods of Bayesian cointegration analysis The Bayes factors are used for computing the posterior probabilities for each rank Monte Carlo experiments show that the methods proposed in this paper provide fairly good results The method that we propose requires to impose r restrictions on fi for each possible rank and then to estimate fi for each rank before computing Bayes factors for rank selection Thus, the choice of the restrictions on fi affects the values of Bayes factors in some degree Another disadvantage of the method is computing time Computing time depends upon the algorithm we choose for estimating the cointegrating vectors In this paper the Griddy- Gibbs sampler, which requires heavy computations, is chosen simply because we do not need to assign an approximation function that is needed in Metropolis-Hastings or importance sampling However, it will not be a problem in the future with much faster computers In this paper, a matrix-variate normal density for the cointegrating vector is chosen as a prior Instead, Jeffrey's or reference priors are also worth considering as Kleibergen and Paap used Jeffreys' priors for their cointegration analysis References [1] Bauwens, L, and Giot, P 1998, A Gibbs Sampling Approach to Cointegration, Computational Statistics, 1, 9-8 [] Bauwens, L, and Lubrano, M 199, Identification Restrictions and Posterior Densities in Cointegrated Gaussian VAR Systems, in TB Fomby (ed), Advances in Econometrics, vol 11B, JAI press, Greenwich, CT 1

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