Some Empirical Evidence on Models of the Fisher Relation: Post-Data Comparison

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1 HIAS-E-68 Some Empirical Evidence on Models of the Fisher Relation: Post-Data Comparison Jae-Young Kim Department of Economics, Seoul National University Hitotsubashi Institute for Advanced Study, Hitotsubashi University Woong Yong Park Department of Economics, Seoul National Univeristy April 08 Hitotsubashi Institute for Advanced Study, Hitotsubashi University -, Naka, Kunitachi, Tokyo , Japan tel: HIAS discussion papers can be downloaded without charge from: All rights reserved.

2 Some Empirical Evidence on Models of the Fisher Relation: Post-Data Comparison Jae-Young Kim Woong Yong Park April 08 Abstract The Fisher relation, describing a one-for-one relation between the nominal interest rate and the expected inflation, underlies many important results in economics and finance. Although it is a conceptually simple relation, the Fisher relation has more or less complicated with mixed results. There are several alternative models proposed in the empirical literature for the Fisher relation that have di erent implications. We evaluate those alternative models for the Fisher relation based on a post-data model determination method. Our results for data from the U.S. Japan and Korea show that models with both regimes/periods, a regime with nonstationary fluctuations and the other with stationary fluctuations, fit data best for the Fisher relation. Keywords: Fisher relation, nonlinear behavior, post-data model determination. JEL classification: C, C, C5. Department of Economics, Seoul National University and Hitotsubashi Institute for Advanced Studies (HIAS), Hitotsubashi University, Correspondence: Kwanak-ro Kwanak-gu, Seoul, 0886, Korea. jykim07@snu.ac.kr. The first author would like to thank HIAS for providing superior environment and support for finishing this work. Department of Economics, Seoul National University

3 Introduction The Fisher relation underlies many important results in economics and finance. The relation explains that the nominal interest rate is determined as the sum of the expected inflation and the real interest rate which is constant or stable. The Fisher relation, thus, signifies that nominal interest rate has a statistical one-for-one relation with the expected rate of inflation. Although the Fisher relation looks a simple relation, there are several alternative models proposed in the literature for the Fisher relation that have di erent implications. In this paper, we examine the Fisher relation by evaluating those alternative models based on a post-data model determination method. Empirical analysis on the Fisher relation was initiated by Fama (975). Constancy of the real interest was studied by Nelson and Schwert (977), Garbade and Wachtel (978), Mishkin (98, 984), and Fama and Gibbons (98). Correlation between the nominal interest rate and the inflation rate noted as the Fisher e ect was studied by Summers (98), Huizinga and Mishkin (986), and Mishkin (990). Also, Rose (988), Atkins (989), Mishkin (99), and Wallace and Warner (993) studied the real interest rate based on the concepts of a unit root and cointegration. On the other hand, it was studied by Clarida and Friedman (984), Huizinga and Mishkin (986), and Roley (986) that change in the U.S. monetary policy in late 970 s through mid 980 s might have a ected dynamics of interest rates and inflation. Evans and Lewis (995) and Garcia and Perron (996) used models of regime switch to analyze behavior of the U.S. real interest rate for post-war data including the period of policy regime change and oil shocks. Kim and Park (06) studied the possibility of short-run instability as well as long-memory properties of the Fisher relation. As described above there are several approaches and models for the Fisher relation proposed in the literature that have di erent implications. We, however, do not know which model is the most appropriate among the several models for the Fisher relation. This issue is an important one since di erent models have di erent implications for the Fisher relation, some of which are conflicting with each other. In this paper, we evaluate those alternative models for data from the U.S. Japan and Korea in the post war period before the world financial crisis. For this purpose we apply a post-data model determination method to get the model that best fits the data. The post-data model selection method evaluates relative probability of each model among the alternatives. We

4 use a Markov-Chain-Monte-Carlo (MCMC) method for computation of a criterion that quantifies the relative probability of each model. The model yielding the highest postdata probability is the one that best fits the data. We use Gibbs sampler to compute the relative post-data probabilities. Our results indicate that the best model is not the same for the three countries. However, models with both regimes/periods, a regime with nonstationary fluctuations and the other with stationary fluctuations, fit data best for the Fisher relation. The paper is organized as follows. Section introduces the Fisher relation and related issues. Section 3 explains several models for the Fisher relation proposed in the literature. In Section 4 we discuss how to select the model that is the most appropriate for the Fisher relation. Section 5 concludes the paper. The Fisher Relation and Related Issues The Fisher relation explains how the nominal interest rate is determined. Let t+ e be the expected inflation from period t to period t +. Also, let rt and i t,respectively,be the ex-ante real interest rate and the nominal interest rate at time t. Then, the Fisher relation defines that the nominal interest rate is equal to the real interest rate: (.) i t = r t + t+ e + " t allowing a temporary disturbance " t to the relation. As is well explained in Kim and Park (06) and others, the Fisher relation describes that the nominal interest rate has a one-for-one relation with the expected rate of inflation. In other words, the Fisher relation describes that there is a stable level of the real interest rate that is equal to the nominal interest rate minus expected inflation, allowing atemporarydisturbance.intermsofex-ante variables the relation writes as (.) rt = i t t+ e " t. We have an ex-post version of the Fisher relation as (.3) r t = i t t+ " t. 3

5 where t+ and r t are, respectively, ex-post inflation and ex-post real interest rate. Notice that we use the same notation for the disturbance " t in both cases of ex-ante and ex-post to save the notation. Denoting by v t the error of the inflation expectation: v t = t e t,itistruethat r t = rt + v t. If v t is stationary, which is the case under rational expectations, then the ex ante real interest rate rt and the ex post real interest rate r t have the same statistical properties. In this case one can analyze the Fisher relation based on the ex post interest rate as well as the ex ante rate. Existence of a stable Fisher relation is, in statistical sense, the same as that the real interest rate is a constant, or a stationary variable fluctuating around a constant mean. Thus, the Fisher relation is a simple relation in its concept. However, empirical analysis of the Fisher relation is more or less complicated with mixed results. There are several alternative models proposed in the literature for the Fisher relation. They have di erent implications on the stationary and nonstationary behavior of the real interest rate. Therefore, the issue of which model is the most appropriate for the empirical Fisher relation is a very important problem. We evaluate those alternative models for the Fisher relation based on a post-data model determination method. 3 Models for the Real Interest Rate In the following discussion we use the variable y t for the real interest rate. Also, let T T = {,,T} be the sample period. We have all four alternative models for the Fisher relation in the following, M i, i =0,,, An Autogression: M 0 The basic model is the p th order autoregression in y t : (3.) (y t µ)= px s= s(y t s µ)+" t, where " t iidn(0, ), and all roots of the characteristic equation z pz p =0 lie outside the unit circle. 4

6 Notice that we can rewrite the model (3.) as in the following:! px px (3.) y t = s µ + sy t s + " t, s= s= which is a more commonly used form in the usual time series analysis. We use the mean-deviated form (3.) instead of the more common one (3.) since the former is more convenient for adopting standard regime switching models in our study. 3. A Model with Partial-sample Instability: M We incorporate the possibility of partial-sample (or short-run) instability in M 0 following the suggestion in Kim (003), Andrews and Kim (008) and Kim and Park (06). Suppose that in a relatively short period T T B ( TT )theprocessy t becomes unstable, having properties of a nonstationary unit root or of higher volatility. In the model M we assume that T T B is identified a priori, which is di erent from the model M 3 below. Then, we have the following model for y t : px (3.3) (y t t )= [ s I(t T S )+ s I(t T B )](y t s t s )+" t s= where T S = T \ T B ; " t N(0, 0); I( ) istheindicatorfunction,and s and s are parameters. For t T S we assume that the mean of y t is t = µ 0 and for t T B t = µ + y t. This means that in the period T B the process y t has a unit root and its first di erence y t is a stationary autoregressive process. 3.3 Markov Regime Switching Model: M Assume that the variable y t follows regime switching dynamics across K states s t =,,,K: px (3.4) (y t t )= s(y t s t s )+" t. s= for " t N(0, t ), where t = µ st and t = s t in state s t.weassumethatµ < <µ K for identification. The state variable s t follows the first order Markov process with the transition probability from the state i to j, p ij =P[s t = j s t = i] fori, j =,,K. Garcia and Perron (996) used models of regime switch of the M type to anlyze behavior of the U.S. real interest rate for post-war data. 5

7 3.4 An Extended Markov Switching Model: M 3 We now consider a Markov switching model that contains a state of nonstationarity. It is an extension of M in the above subsection that only contains K stationary regimes. In this extended model M 3,thelastK th state is set to be a nonstationary state. Then, the extended model M 3 is (3.5) (y t t )= px [ s I(s t 6= K)+ s I(s t = K)](y t s t s )+" t s= where " t N(0, t ). The mean of y t in a stationary state is t = µ st and that in the nonstationary state is t = µ K + y t.thisimpliesthatthevariabley t has a unit root in the state s K. Remark: Thereisanalternativemodelingschemewiththeregimeswitchingduetoits own lagged variable, known as the self-exciting threshold regression. It was introduced by Tong and Lim (980) and studied by Seo (008), for instance, in relation to the unit root testing for the model. This alternative specification may be a relevant option for modeling the Fisher relation. We do not, however, consider this specification in this paper since our objective is to evaluate the existing models of the Fisher relation. This modeling scheme can be applied to the Fisher relation in any future work. 4 Model Selection For the Fisher Relation In this section we discuss how to compare di erent models and select the one that best fits data for the Fisher relation. Then, we provide the result of selecting a model among those explained in Section Post-data Model Selection In this subsection we explain how to determine the best model for the real interest rate out of several alternatives. Our approach is a post data model selection method developed in the Bayesian framework. Thus, it is a generalized version of the Bayesian information criterion. In the method we can evaluate relative merit of each model among several 6

8 alternatives and select a model that best fits the data. We use a Markov-Chain-Monte- Carlo (MCMC) method for the computation of the criterion that quantifies the relative merit of each model. Denote by Y T =(y,,y T ), a sample of T observations for the process y t.afamily M consists of candidate models for Y T in the presence of uncertainty of the true model. Amodelm i Mis associated with a parameter space i of dimension p i for i Iwhere I = {,...,I}. Assume that for each m i afamilyq i T ( i,y T )ofdistributionfunctions, with a density qt i ( i,y T ), is defined. Let Pr(m i Y T )bethepost-data(posterior)probabilitythatm i is true. By the Bayes rule, we have (4.) Pr(m i Y T )= q T (Y T m i )Pr(m i ) P ji q T (Y T m j )Pr(m j ) where Pr(m i )isthepriorprobabilitythatm i is true and q T (Y T m j )=q i T (Y T ). But, notice that Z (4.) q T (Y T m j )= q T (Y T j,m j )'( j m j )d i = E j [q T (Y T j )], where '( j m j ) is the prior density associated with the model m j. If we further assume that Pr(m j )isthesameforallj, themodelselectionruleistochoosem i for which E i [q T (Y T i )] is the largest. The quantity in (4.) can be alternatively interpreted as the marginal likelihood. The marginal likelihood q T (Y T m j )=E i [q T (Y T i )] can be rewritten as (4.3) q T (Y T )= q T (Y T )'( ) '( Y T ) where the script j and m j are omitted for convenience, and '( Y T )istheposteriordensity of. Theequation(4.3)isareversedversionoftheBayes rule.since(4.3)holdsforany, wemayevaluateq T (Y T )foraconvenient, say = the posterior mean. Taking the logarithm of (4.3) for =,wehave (4.4) ln q n (Y T )=lnq T (Y T )+ln'( ) ln '( Y T ) Our decision rule is to choose the model m j that yields the highest value of (4.4). Calculation of the log-likelihood and the log-prior at = is relatively easy. However, 7

9 calculation of the posterior '( Y T )isnoteasy. Tocompute'( Y T )wecanusean MCMC method such as Gibbs sampler as in Chib (995). Computation of the marginal likelihood or the posterior '( Y T )forthemodelsinsection3isaverydemandingwork with highly sophisticated programming. We use standard priors for the parameters that are used in the literature. That is, the regression coe cients and have normal priors. The error variance has an inverted gamma distribution. This normal-inverted gamma prior distribution is a conjugate prior and is standard one used in the literature. The mean µ i has a normal prior, and the transition probability {p ij } has a prior of Dirichlet distribution, which are also conjugate priors. The prior for the transition probability reflects information on duration of a state. For example, if it is reasonable to think that duration of the state i is four quarters, then /( p ii )=4,sothatp ii =0.75 for the prior. 4. Data and Empirical Results As in many existing works we use the 3-month Treasury bill rate or equivalence for the nominal interest rate and the consumer price index (CPI) for the price level to compute the inflation rate. We get the U.S. data of the T-bill rate and the CPI from the Federal Reserve Board and the Bureau of Labor Statistics, respectively. The Japan and Korean data are from the International Financial Statistics (IFS). All the data are seasonally adjusted. The data period is 957:Q-006:Q for Japan, 953:Q-006:Q for the U.S. and 976:Q3-006:Q for Korea. We consider most possible aspects of each model for the model selection: the number of states K up to 5 and the order of the autoregression up to 5 for each M i, i =0,,, 3. Table shows results of model selection by the method of post-data model selection. As shown in the Tables the best model selected is di erent for the three countries: M is the best model for the U.S. and M 3 for Japan and Korea. However, we can see that for data from the three countries the models with both regimes/periods, a regime with stationary fluctuations and the other with nonstationary/unstable fluctuations, (M and M 3 )areselectedasthemostappropriatemodels. Thisresultdoesnotapparentlyconfirm Good references for the MCMC method and the Gibbs sampler are Gelman, Carlin, Stern and Rubin (000), Chib and Greenberg (996), and Casella and George (99), among others. A detailed explanation of the MCMC used for our models is provided in Appendix A. 8

10 Table The result of model selection Selected model Number of states lag order Japan M Korea M U.S. M - 3 the existence of Fisher relation since the Fisher relation implies that the real interest rate is stable with stationary fluctuations around a constant. However, we can say that the Fisher relation prevails if the nonstationary fluctuations occur only temporarily. Kim and Park (06) have shown that the length of the period of nonstationary fluctuations in the real interest rate is relatively short for a data set similar to that used in this paper. 5 Concluding Remarks We have compared several alternative models of the Fisher relation for data from Japan, Korea and the U.S based on the Bayesian model determination method. Among four alternative models for the Fisher relation our result shows that models with both regimes/periods of nonstationarity and stationarity fit data best although the best model is not the same for the three countries. It is a new and interesting result about the Fisher relation and would motivate further investigation of related issues. References [] Andrews, D.W.K. and J.Y. Kim (006), Tests for Cointegration Breakdown over a Short Time Period, Journal of Business & Economic Statistics [] Atkins, F. (989), Co-integration, Error Correction and the Fisher E ect, Applied Economics, [3] Casella, G. and George, E. I. (99), Explaining the Gibbs Sampler, American Statistician 46,

11 [4] Chib, S. (995), Marginal Likelihood from the Gibbs Output, Journal of American Statistical Association 90, [5] Chib, S. and Greenberg, E. (996), Markov Chain Monte Carlo Simulation Methods in Econometrics, Econometric Theory, [6] Evans, M. D. D. and Lewis, K. K. (995), Do Expected Shifts in Inflation A ect Estimates of the Long-Run Fisher Relation?, Journal of Finance 50. [7] Fama, E. F. (975), Short-term Interest Rates as Predictors of Inflation, American Economic Review 65, [8] Fama, E. F. and Gibbons, M. R. (98), Inflation, Real Returns, and Capital Investment, Journal of Monetary Economics 9, [9] Garbade, K. and Wachtel, P. (978), Time Variation in the Relationship between Inflation and Interest Rates, Journal of Monetary Economics 4, [0] Garcia, R. and Perron, P. (996), An Analysis of the Real Interest Rate under Regime Shifts, Review of Economics and Statistics 78, 5. [] Gelman, A., Carlin, C. B., Stern, H. S. and Rubin, D. B. (000), Bayesian Data Analysis, Chapman & Hall/CRC, Boca Raton, FL. [] Huizinga, J. and Mishkin, F. S. (986), Monetary Policy Regime Shifts and the Unusual Behavior of Real Interest Rates, Carnegie-Rochester Conference Series on Public Policy 4, [3] Kim, J.Y. (003), Inference on Segmented Cointegration, Econometric Theory 9, [4] Kim, J.Y. and W.Y. Park (06), New Empirical Evidence for the Fisher Relation: Integration and Short-run Instability, Seoul Journal of Economics. [5] Mishkin, F. S. (98), The Real Interest Rate: An Empirical Investigation, Carnegie-Rochester Conference Series on Public Policy 5,

12 [6] Mishkin, F. S. (984), The Real Interest Rate: A Multi-country Empirical Study, Canadian Journal of Economics 7, [7] Mishkin, F. S. (990), What does the Term Structure of Interest Rates Tell Us about Future Inflation?, Journal of Monetary Economics 5, [8] Mishkin, F. S. (99), Is the Fisher E ect for Real?, Journal of Monetary Economics 30, [9] Nelson, C. R. and Schwert, G. W. (977), Short-term Interest Rates as Predictors of Inflation: On Testing the Hypothesis that the Real Rate of Interest Is Constant, American Economic Review 67, [0] Roley, V. V. (986), The Response of Interest Rates to Money Announcements under Alternative Operating Prosedures and Reserve Requirement Systems. NBER Working Paper No.8. [] Rose, A. K. (988), Is the Real Interest Rate Stable?, Journal of Finance 43, 095. [] Seo, M. H. (008), Unit root test in a threshold autoregression: asymptotic theory and residual-based block bootstrap. Econometric Theory, 4, 6, [3] Summers, L. H. (98), The Non-Adjustment of Nominal Interest Rates: A Study of the Fisher E ect, NBER Working Paper No [4] Tong, H. and K. S. Lim (980), Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society, Series B (Methodological), [5] Wallace, M. S. and Warner, J. T. (993), The Fisher E ect and the Term Structure of Interest Rates: Tests of Cointegration, Review of Economics and Statistics 75,

13 Appendix A: Inference on Posterior An Autoregression: M 0 Assume that the initial p observations y 0 = {y p+,,y 0 } are given. Let y T = {y,,y T }. Rewrite the given linear autoregression (y t µ)= px s= s(y t s µ)+" t, as in the following: px y t = 0 + sy t s + " t, where 0 =( s= P p s= s)µ, " t N(0, ). Writing it in a matrix form, we have y t = X T + " where X T = {x,, x T } 0 for x t = {,y t,,y t p } 0,and = { 0,,, p}. Also, " = {",," T } 0. The posterior of this model is well known. We divide prior of and as in the following p(, )=p( )p( ) (B.) Assume that, given prior of is normal N(, H )andpriorof is inverted-gamma IG( /,s /), that is, / has a gamma distribution. This prior is a conjugate prior whose family of distributions is the same as that of its posterior. Now, the prior (B.) can be written as p(, )=p( )p( ) p apple k/ = p H / exp s / s exp ( /) ( +k)/+ apple / exp [s +( ( )0 H( ) 0 H( )],

14 where k = dim( )=p +. The probabilitydensity functionof y T,thatis,thelikelihood of and is l(, T T/ apple y T, y 0 )= p exp (y T X T ) 0 (y T X T ) T T/ = p apple exp [s +( ˆ)0 X 0 T X T ( ˆ) where ˆ =(X 0 T X T ) X 0 T y T,ands =(y T X T ) 0 (y T X T ). The posterior density of and can be obtained as in the following: p(, y T, y 0 ) / l(, y T, y 0 )p(, ( +k)/+ ) / apple exp [s + s +( ) 0 H( )+( ˆ)0 X 0 T X T ( ˆ)] ( +k)/+ apple = exp [s +( ) 0 [H + X 0 T X T ]( ) (B.) where s = s + s +( ˆ)0 [H +(X 0 T X T ) ] ( ˆ)] and =[H + X 0 T X T ] [H +(X 0 T X T ) ˆ]. Expansion from the third row to the fourth row of (B.) is possible by the following fact: [A + BCD] = A A B[DA B + C ] DA. From (B.) we know that the posterior of and that of are, respectively,, y T, y 0 N(, [H + X 0 T X T ] ) y T, y 0 IG( /,s /) 3

15 where = + T.Weneedposteriorofµ and not that of 0. Werewritethemeanand variance of the conditional posterior of, and [H + X 0 T X T ],respectively,as =( 0; 0 ) 0 and [H + X 0 T X T ] = h H H 0 H where 0 =(,, p) 0, h = V ar( 0 ), and H = Cov( 0, 0), H = V ar( 0 ). From properties of multivariate normal distribution, we have (Poirier 995)! 0 0,, y T, y 0 N( 0 + H H ( 0 0),h H H H 0 ). Therefore, we get posterior of µ 0, as µ 0,, y T, y 0 N 0 + H H ( 0 0), h H H H P 0 ( p. s= s) P p s= s On the other hand, it is easy to show that posterior of 0 is 0, y T, y 0 N( 0,H ). A Model with Partial-sample Instability: M Posterior of model M can be directly obtained from the extended Markov switching regime change model M 3 by assuming that the state variable is known a priori. 3 Markov Regime Switching Model: M We explain how to simulate the posterior distribution of the Markov switching regime change model based on the Gibbs sampling. To get the posterior of the Markov switching regime change model M by the Gibbs sampler we have together the parameters of the model, vector of the state variables {s,,s T },meanµ,,µ K,variance,, K,autoregressioncoe cients,, p, 4

16 and = {,, p}. InGibbssamplingtheposteriorofparametersinapartisgenerated from a conditional distribution of the other part. ) Generation of S T : Given µ,,,p we can write the posterior of the state vector S T as in the following: p(s T y T )=p(s,,s T y T ) TY = p(s T y T ) p(s t s t+,,s T, y T ) t= TY = p(s T y T ) p(s t s t+, y T ). t= (B.3) In (B.3) derivation of the second line from the first line is based on a basic property of conditional probability, and the third line is obtained from the first order Markov properties of state variables. (B.3) shows that we generate s T,assumingthaty T is given, and then generate s t for t = T,T,, successivelyconditionedonthepre-generateds t+ and y T,bywhichS T can be generated. The problem is how to get p(s t s t+, y t ), which can be easily available by applying the Hamilton(989) s filter. Assuming that the other parameters are given, we can get the probability of current state p(s t y t ) by Hamilton filter if information up tp t is available. Since, for t = T,T,,, p(s t s t+, y t )= p(s t,s t+ y t ) s t+ y t ) = p(s t+ s t, y t )p(s t y t ) p(s t+ y t ) = p(s t+ s t )p(s t y t ) p(s t+ y t ) from the assumed transition probability and probability obtained from the Hamilton s filter, we can get p(s t s t+,y t ). From this result we get, for t = T,T,,, P [s t = i s t+,y t ]= p(s t+ s t = i)p(s t = i y t ) P M j= p(s t+ s t = i)p(s t = i y t ). (B.4) Given s t+, p(s t+ y t )isdeterminedindependentlyofs t, so that it is cancelled out in the numerator and denominator. Generation of the state variable from (B.4) can be done based on the uniform distribution. On the other hand, p(s T y T )canbeimmediatelyobtained from the Hamilton filter. 5

17 ) Generation of µ: GivenS T,,,P modify M as in the following: ỹ t s t = µ + + µ M t t s M,t t + " t t (B.5) Here ỹ t = y t ( y t + + p y t p )and s i,t = s i,t ( s i,t + + p s i,t p ). Also, s i,t = I(s t = i). Letting u t = " t / t,wehavethatu t N(0, ). Now, rewriting (B.5), ỹ = sµ + u (B.6) where ỹ = {ỹ / t,, ỹ t / t } 0, s i = { s i,,, s i,t } 0,i=,,M), and s = { s,, s M }. Also, u = {u,,u T } 0 and u N(0 T,I T ). Then, the likelihood of (B.6) becomes l(µ ỹ, S T, T apple,, P) = p exp (ỹ sµ)0 (ỹ sµ). (B.7) Now, assume that the prior distribution of µ is independent of the other parameters and is Gaussian with mean a 0 and variance m 0 : µ S T,,, P N(a 0,m 0 ) I(µ < <µ K ) (B.8) where I(µ < <µ K ) is a restriction for the identification of states. With this restriction the prior of µ is a truncated normal distribution. From (B.7) and (B.8) we get a conditional posterior of µ: p(µ S T,,, P, y T ) / l(µ ỹ,s T,,, P)p(µ S T,,, P) apple / m 0 / exp [(µ a 0) 0 m 0 (µ a 0 )+(ỹ sµ) 0 (ỹ sµ)] apple / m / exp [(µ µ ) 0 (µ µ ), where m =(m 0 + s 0 s), µ = m (m 0 a 0 + s 0 ỹ). Therefore, the conditional posterior of µ is a normal distribution: µ S T,,, P, y T N( µ,m ) I(µ < <µ K ) 6

18 where I(µ < <µ K ) is a restriction for the identification of states. After generation of µ we take only those satisfying this restriction. 3) Generation of :GivenS T, µ,, P,wegenerate,, K individually one by one since they are independent of each other. First, for let " t =(y t t ) LX p= " i = {" t s t = i} 0 p(y t p t p ) for i =,,K. With " t having a normal distribution with mean 0 and variance have the likelihood of " i as l( i " i,s T, µ,, P ) / i T i exp apple " i0 " i i st,we where T i is the sample size of the i th states (s t = i). Assuming that the prior of i is an inverted Gamma, IG( i /, i/), p( i S T, µ,, P ) / i we know that a conditional posterior of i is p( i S T, µ,, P, " i ) / i i / exp Ti + i / exp which is another inverted Gamma, Ti + i i S T, µ,, P, " i IG, i i, i + " i0 " i i + " i0 " i, i We can generated the posterior of i from the fact that the inverse of a Gamma has an inverted Gamma distribution. 4) Generation of :GivenS T and,µ,p,let! t = y t t and rewrite M as follows:! t! t! t p = + + p + " t t t t t (B.9) Letting u t = " t / t,wehaveu t N(0, ). Now, write (B.9) as z = X + u (B.0) 7

19 where z = {! /,,! T / T } 0,andforx t = {! t /,,! t / t } 0 X = {x,,x T } 0. Also, for u = {u,,u T } 0, u N(0 T,I T ). Then, the likelihood of (B.0) is T apple l( z,s T, µ,,p)= p exp (z X )0 (z X ). (B.) Now, assume that the prior of distribution with mean b 0 and variance B 0, is independent of the other parameters and has a normal S T, µ,,p N(b 0,B 0 ) I( S ). (B.) where I( S ) is a stability condition for where S is the space of such that roots of the characteristic equation z pz p =0lieoutsidetheunitcircle. From (B.) and (B.) we get a conditional posterior of : p( S T, µ,,p,z), / l( z,s T, µ,,p)p( S T, µ,,p) apple / B 0 / exp [( b 0) 0 B 0 ( b 0 )+(z X ) 0 (z X )] apple / B / exp [( ) 0 B 0 ( ) where B =(B 0 +X 0 X) and = B (B 0 b 0 +X 0 z). Therefore the conditional posterior of is a Gaussian as in the following: S T, µ,,p,z N(ˆ,B ) I( ) which is a truncated normal. As for µ we take only the generated values satisfying these restrictions. 5) Generation of P : Given S T,,µ,, S T being given conditionally, we can count the number of transitions between states in a sample. Thus, let n ij be the number of transitions from state i to state j. Then, P K i= P K j= n ij = T. Since for i =,,K the number of transitions to states j =,,K follows a multinomial distribution, the likelihood of p ij (j =,,K)is l(p i,,p ik y T )= KY j= 8 p n ij ij! KX j= p ij! nik.

20 Since p ik = (p i + p i(k ) )wedonotconsiderp im.fortheprioroftransitionprobability we use a Dirichlet distribution. The pdf of p ij having a Dirichlet distribution with parameters i,, i(k ) is as follows p(p i,,p i(k ) S T )= Therefore, a conditional posterior of p ij is p(p i,,p i(k ) S T, y T )= KY j= KY j= p ij ij! p n ij+ ij ij! KX j= KX j= p ij! ik p ij! nik+ ik That is, p ij has a Dirichlet distribution with parameters n i + i,,n i(k ) + i(k ) : p i,,p i(k ) S T, y T Dirichlet(n i + i,,n i(k ) + i(k ) )... We generate Dirichlet random numbers by a Gamma distribution. generated from Gamma( ij, ) we have p ij as x ij / P x ij. That is, with x ij 4 An Extended Markov Switching Model: M 3 Inclusion of instability state(s) in M does not substantially a ect estimation of posterior distribution by Gibbs sampler. However, given S T,,,P we need to modify the part of generation of µ. Since,inM 3, t = µ s,t + + µ K s K,t +(µ K + y t )s K,t,wecan rewrite M 3 as in the following with s i,t = I(s t = i): px y t y t s K,t s,t(y t s y t s s K,t s ) s= = µ [s,t px s= s,ts,t s ]+ + µ K [s K,t px s= s,ts K,t s ]+" t where s,t = [ s I(s t 6= K) + s I(s t = K)]. Thus, having ỹ t = y t y t s K,t P p s= s,t(y t s y t s s K,t s )in(b.5)wecangenerateµ by the same method. In this case the identifying restriction for the distribution of µ becomes I(µ < <µ K can generate all the other parameters except t by the same method as in M. ). We 9

21 Appendix B: Estimation of Marginal Likelihood Estimation of marginal likelihood can be done by Chib(995) s algorithm. This algorithm calculates a posterior density based on the fact that Gibbs sampler generates parameters of a posterior. For Gibbs sampling we need to know the exact form of a posterior density. Suppose that we consider computation of the following posterior density at = (,, B ): p( y T )=p( y T )p( y T, ) p( B y T,,, B ) (C.) where (r =,,B)canbeavector. Supposethatthereexistunobservedstatevector S T = {s,,s T } in this model. The first term can be computed by conditional densities of the other parameters Z p( y T )= p( y T, T,,, B,S T )dp(,, B,S T y) (C.) where the integral is taken with respect to parameters other than.theset{( (g),, (g) B,S(g) T ) g =,,G} is obtained as a generated sample from G times Gibbs sampling of the joint posterior of (,, B,S T ), p(,, B,S T y). Therefore, the integral (C.) is approximately computed as ˆp( y T )=G G X g= p( y T, (g),, (g) B,S(g) T )). Now for r =,,B we calculate consecutively p( r y T,,, r ), p( r y T,,, r ) Z = p( r y T,,, r, r+,, B,S T )dp( r+,, B,S T y), so that we can get X G p( r y T,,, r ) =G p( r y T,,, r g=, (g) r+,, (g) B,S(g) T ). 0

22 Here, ( (g) r+,, (g) B,S(g) T )canbeobtainedfromthepreviouslyappliedgibbssamplerby fixing (,, r )=(,, r ) andbyexecutinggibbssamplingagain. Thatis, we perform Gibbs sampling consecutively for a conditional posterior as in the following (g) (g ) (g ) r p( r y,,, r, r,, B,S (g ) T ) (g) r+ p( r+ y,,, r, (g) (g ) (g ) r, r+,, B,S. (g) B p( B y,,, r, (g) r,, (g) S (g) T p(s T y,,, r, (g) r,, (g) B ). ) B,S(g T ) (g ) T ) The value of r generated along with values of the other parameters in Gibbs sampling is used for posterior instead of the fixed value r. After computing posterior of (C.) we get the marginal likelihood ln ˆp(y T )=lnl(y T )+lnp( ) ln ˆp( y T ). We can easily calculate l(y T ) andp( ). Numerical standard deviations of a posterior density and a marginal likelihood can be calculated as is explained in Chib(995). For the model of Markov switching regime change we can partition parameters as µ,,,p and apply the above algorithm.