Impulse-Response Analysis in Markov Switching Vector Autoregressive Models

Impulse-Response Analysis in Markov Switching Vector Autoregressive Models Hans-Martin Krolzig Economics Department, University of Kent, Keynes College, Canterbury CT2 7NP October 16, 2006 Abstract By utilizing the state-space representation of Markov-switching vector autoregressive models, we develop impulse response functions with regard to shocks to variables of the system and shifts in regime The proposed analysis is related to the concept of generalized impulse responses introduced by Koop, Pesaran and Potter (1996) but characterizes the properties of the model dynamics in a more concise form In contrast to the impulse response functions proposed by Ehrmann, Ellison and Valla (2003) the analysis here fully reflects the Markov property of the switching regimes Empirical illustrations of the approach suggested here include the univariate Hamilton (1989) model of the US business cycle and two Markov-switching vector autoregressions of US output growth, employment growth and the term structure Keywords: Impulse Response Analysis; Markov Switching; Generalized impulse responses; Nonlinear Models; Time Series; Regime Shifts JEL classification: C22; C32; C50 1 Markov-Switching Vector Autoregressive models By allowing for changes in regime of the process generating the time series, the MS-VAR model has been proposed as an alternative to the constant-parameter, linear time-series models of the earlier Box and Jenkins (1970) modelling tradition The general idea behind this class of regime-switching models is that the parameters of a, say, K-dimensional vector time series process {y t } depend upon an unobservable regime variable s t {1,,M}, which represents the probability of being in a particular state of the world f(y t Y t 1,X t ;θ 1 ) if s t = 1 p(y t Y t 1,X t,s t ) = f(y t Y t 1,X t ;θ M ) if s t = M where Y t 1 = {y t j } j=0 denotes the history of y t and X t are strongly exogenous variables; θ m is the parameter vector associated with regime m The general form of a Markov-switching vector autoregressive (MS-VAR) process is given by y t = ν(s t ) + A 1 (s t )y t 1 + + A p (s t )y t p + u t, u t s t NID(0,Σ(s t )), (2) (1) 1

2 where the presample values y 0,,y 1 p are fixed The parameter shift functions ν(s t ), A 1 (s t ),,A p (s t ), and Σ(s t ) describe the dependence of the parameters on the realized regime s t, for example: ν 1 if s t = 1, ν(s t ) = ν M if s t = M The MS-VAR model provides a very flexible framework for modelling time series subject to regime shifts While all parameters of the conditional model can be made dependent on the state s t of the Markov chain, in practice, only some parameters of interest will be regime dependent while the others will be regime invariant In order to establish a unique notation for each model, we specify with the general MS(M) term the regime-dependent parameters: M Markov-switching mean, I Markov-switching intercept, A Markov-switching autoregressive parameters, H Markov-switching heteroscedasticity A complete description of the statistical model requires the formulation of a mechanism that governs the evolution of the stochastic and unobservable regimes on which the parameters of (1) depend Once a law has been specified for the states s t, the evolution of regimes can be inferred from the data In MS models the regime-generating process is an ergodic Markov chain with a finite number of states defined by the transition probabilities: p ij = Pr(s t+1 = j s t = i), (3) M p ij = 1 i,j {1,,M} (4) j=1 More precisely, it is assumed that s t follows an ergodic M-state Markov process with an irreducible transition matrix p 11 p 1M P = (5) p M1 p MM Thus, the probability which regime is in operation at time t conditional on the information at time t 1 only depends on the statistical inference on s t 1, Pr(s t Y t 1,X t,s t 1 ) = Pr(s t s t 1 ) It will be useful to collect all the information about the realization of s t in the regime vector ξ t : ξ t = I(s t = 1) I(s t = M), (6) where the indicator variables are defined as I(s t = m) = { 1 if s t = m 0 otherwise, (7) for m = 1,,M Using the regime vector ξ t, we can represent the Markov chain as a VAR(1) process (see Hamilton, 1994): ξ t+1 = Fξ t + v t, (8) where F = P and v t is a martingale difference sequence This provides a convenient way to derive predictions of the regime which in turn can utilized for the calculation of the impulse responses

3 2 Impulse-Response Analysis There has been some recent interest in impulse response in non-linear models Beaudry and Koop (1993) have investigated the persistence of output innovations when output has been modelled in a non-linear fashion They show that previous results by Campbell and Mankiw (1987) are biased Their results show that the persistence of positive innovations had been underestimated whereas the persistence of negative innovations has been overestimated Koop et al (1996) offer a more general analysis of impulse responses in non-linear models, and introduce the concept of generalized impulse response The generalized impulse response differs from the traditional impulse response because of the conditional information set used in the dynamic analysis (that is, the type of shocks and the history of the variables) We will re-interpret this concept in the framework of the Markov switching model, where history is represented by the current state of the Markov chain and/or the last p observations of y Recently, Ehrmann et al (2003) proposed to calculate the impulse response functions for the Gaussian innovations under the assumption that a particular regime m is prevailing: s t+h = m for all h 0 For each regime, these impulse responses identical to those of linear VAR with parameter matrices ν m,a m1,,a mp This approach not only ignores the Markov property of the switching regimes, but the probability that their impulses responses are representative for the dynamic system captured by the MS-VAR is rapidly converging to zero as the horizon increases This is illustrated in figure 1 which plots the probability of staying in regime after h periods, ie Pr[s t+h = s t+h 1 = = s t = m], as implied by the estimated Markov chains of the Hamilton model (p 11 = 076 and p 22 = 09) and the three-regime MS-VAR models of output growth, employment growth and the term structure that will be considered in section 6 Policy conclusions based on an uncritical interpretation of the produced impulse responses could have severe implications In constrast, the approach taken here is chosen to fully reflect the structure of the Markov-switching model In line with the generalized impulse responses, we could measure the responses of the system to shocks to the variables in h periods as: IR u (h) = E[y t+h ξ t,u t + u;y t 1 ] E[y t+h ξ t,u t ;Y t 1 ], (9) where u is the shock at time t However, as we will see in the following the responses are proportional to the size and sign of the shock, so we define the responses to shocks to the variables as in case of the linear VAR processes: IR uk (h) = E[y t+h ξ t,u t ;Y t 1 ] u kt (10) These previous analyses mainly focused on the response of the system to Gaussian innovations We introduce here a dynamic analysis in which the system is subjected to a regime shift The responses to shifts in regime are defined in spirit of the generalized-impulse-response concept: IR ξ (h) = E[y t+h ξ t + ξ,u t ;Y t 1 ] E[y t+h ξ t,u t ;Y t 1 ], (11) where ξ is the shift in regime at time t We assume here that the regime shift has an economic interpretation such as a turning point of the business cycle The next sessions discuss how such impulse response functions can be derived from a suitable state space representations of the MS-VAR process We first consider MS-VAR models with regime-invariant AR dynamics, where the regimes may reflect business cycle phaenomona, eg, booms vs recessions, or financial cycle, eg, bull markets vs bear markets Then we analyse MS-VAR models with regimedependent AR dynamics where the regimes may corresponds to hysteresis (unit) vs mean reversion (stable roots), bubbles (explosive roots) vs corrections (stable roots), or regime-dependent effects of monetary policy

4 09 Hamilton (1989) Regime 1 Regime 2 09 MSIH(3) VAR(2) Regime 1 Regime 2 Regime 3 09 MSIAH(3) VAR(2) Regime 1 Regime 2 Regime 3 08 08 08 07 07 07 06 06 06 04 04 04 03 03 03 02 02 02 01 01 01 Figure 1 Regime dynamics: Probability of staying in regime m after h periods 3 Analysis of MS-VAR models with regime-invariant VAR matrices An important subclass of the MS-VAR model constitutes the model where regime shifts affects the level of the endogenous variables via shifts in the mean (MSM) or shifts in intercept (MSI), but leaves the coefficients unaltered Commencing Hamilton s seminal contribution in 1989, this model has been successfully applied to the analysis of the business cycle In this context, the analysis of the intertemporal transmission of regime shifts is of particular interest The beauty of this case is that very neat closed-form solution can be derived This section presents the mathematical background for the impulse-response analysis in models with regime-invariant coefficients To derive the impulse-response functions we use the stacked MS(M)- VAR(1) representation of the MS(M)-VAR(p) process: y t = Mξ t + A 1 y t 1 + + A p y t p + u t, (12) where u t is a Gaussian white noise process Denote y t = (y t,,y t p+1 ), then equation (39) can be rewritten as: where A = A 1 A p 1 A p I K 0 0 0 I K 0 y t = Hξ t + Ay t 1 + u t, (13) M 0 is a (Kp Kp) matrix, H = = ι 1 M is a (Kp M) 0 matrix The state-space representation is completed by the VAR(1) representation of the Markov chain in (8)

5 Hence the expectation of y t+h conditional upon {u t,ξ t,y t 1 } is given by: where the conditional expectation of ξ t+h is y t+h t = Hξ t+h t + Ay t+h 1 t (14) ξ t+h t = F h ξ t (15) Based on this representation the following types of analysis are feasible: a) Corresponding to the impulse response analysis in linear Gaussian VARs we can look at the response of the system to shocks arising from the Gaussian innovations in each of the variables b) Impulse responses to changes in regimes can be also considered where we can distinguish between (i) the study of the path of the variables when there is a change in regime such as a shift from Regime 1 to Regime 2 or any other shift between the existing regimes, (ii) the dynamics when we move from the ergodic distribution to a sure state, say Regime 1 31 Responses to shocks arising from the Gaussian innovations to the variables According to the impulse response analysis in time-invariant linear VARs we have that y t+h = JA h ι j, (16) u jt [ ] J = I K 0 0 = ι 1 I K is a (K Kp) matrix and where ι j is the j th column of the identity matrix If the variance-covariance matrix Σ u is regime-dependent the standardized and orthogonalized impulse-responses also become regime-dependent: y t+h ε jt = JA h D(ξ t )ι j, (17) where u t = D(ξ t )ε t and D(ξ t ) is a lower triangular matrix resulting from the Choleski decomposition of Σ u (ξ t ) = D(ξ t )D(ξ t ) 32 Regime impulse function The effects of regime shifts can be measured as the reaction of y t+h to a change in regime at time t : ( h ) IR ξ (h) = J A k HF h k ξ (18) k=0 We think here of two different types of thought experiments: (i) The information that the system in regime m at t s t = m, considered as a shift from the unconditional distribution ξ, ( h ) (ιm IR ξ (h) = J A k HF h k ξ ) (19) k=0 (ii) When there is a shift from regime l to regime m at time t then the responses of the system are given by: ( h IR ξ (h) = J A k HF )(ι h k m ι l ) (20) k=0

6 In both cases, the dynamics are generated by (i) changes of the current state, (ii) changes to the conditional expectation of future regimes, and (iii) the autoregressive transmission of intercept shifts It is worthwhile emphasizing that in the literature there is some confusion between Markovswitching models and the existence of non-linearities The non-linearities in our model come from the fact that the mean and variance of the process are state dependent However, all regimes share the same autoregressive parameters The logical conclusion is that the response of output and employment to a change from Regime 2 to regime one is the mirror image (but with negative coefficients) of the response of the variables of a change from regime one to Regime 2 This would not have been the case if the autoregressive parameters were regime-dependent as well In the next section we will analyse this class of Markov-switching models We will show that the approach taken here, while being more complicated due to changes to the parameters in the autoregressive part of the system, is nevertheless feasible 4 Analysis of MS-VAR models with regime-dependent VAR matrices For clarity of exposition consider the MSA(M)-VAR(1) model y t = A(ξ t )y t 1 + u t, ξ t = Fξ t 1 + v t, where u t is NID(0,Σ) and v t is an MDS For the impulse response analysis in this model class, we can utilize the linear state space representation in ψ t = ξ t y t proposed by Karlsen (1990): or in short: ξ 1t y t ξ Mt y t = p 11 A 1 p M1 A 1 ξ 1t 1 y t 1 p 1M A M p MM A M ξ Mt 1 y t 1 + ε t, (21) ψ t = Πψ t 1 + ε t (22) where ε t+1 is an MDS Hence the moving average representation of ψ t+h given ψ t can be written as: The conditional expectation is given by: ψ t+h = Π h ψ t + h Π j ε t+j (23) j=1 E [ ψ t+h ψ t ] = Π h ψ t (24) As y t = M i=1 ξ ity t, we have that the conditional expectation y t+h can be derived based on the representation (21): E[y t+h y t,ξ t ] = M E [ ] ξ it+h y t+h y t,ξ t = (1 M I K )E [ ] ψ t+h ψ t i=1 Thus the impulse responses are given by: = (1 M I K)Π h (ξ t y t ) IR u (h) = ( 1 M I K) Π h (ξ t u), IR ξ (h) = ( 1 M I K) Π h ( ξ y t )

7 5 Generalizations 51 Switching Intercept: MSIA(M)-VAR(1) Models If the regime shifts affect the mean of the variables of the system by regime dependent intercepts, y t = ν(s t ) + A(s t )y t 1 + u t, u t s t NID(0,Σ(s t )), (25) the state space representation changes as follows: ξ 1t y t p 11 A 1 p M1 A 1 ξ 1t 1 y t 1 p 11 ν 1 p M1 ν 1 = + ξ Mt y t p 1M A M p MM A M ξ Mt 1 y t 1 p 1M ν M p MM ν M (26) where ψ t = ξ t y t This has to be complemented by the VAR(1) representation of the Markov chain: ξ 1t ξ Mt p 11 p M1 = p 1M p MM ξ 1t 1 ξ Mt 1 ξ 1t 1 ξ Mt 1 + η t (27) ε t and η t are martingale difference sequences Collecting both equations using the notation introduced in 4 gives: +ε t, ψ t ξ t = Mξ t 1 + Πψ t 1 + ε t = Fξ t 1 + η t or in short: [ ψ t ξ t }{{} ψ t ] [ ][ ] [ Π M ψ = t 1 + 0 F ξ t 1 }{{}}{{} Π ψ t 1 ε t η t ] }{{} ε t (28) Hence the moving average representation of ψ t+h given ψ t can be written as: The conditional expectation is given by: h ψ t+h = Π h ψ t + Π j ε t+j (29) j=1 E [ ψ t+h ψ t] = Π h ψ t (30) As y t = M i=1 ξ ity t we have that the conditional expectation y t+h can be derived based on the representation (21): E[y t+h y t,ξ t ] = = M i=1 E [ ] ξ it+h y t+h y t,ξ t = (1 M I K : 0 K,M )E [ ψ ] t+h ψ t [ ] [ ] h [ 1 M I Π M K 0 K,M 0 F ξ t y t ξ t ]

8 Thus the impulse responses are given by: IR u (h) = IR ξ (h) = [ ] [ ] h [ 1 M I Π M K 0 K,M 0 F [ 1 M I K 0 K,M ] [ Π M 0 F ] h [ ξ t u 0 M,1 ξ t y t ξ t ] ], 52 Switching Mean: MSMA(M)-VAR(1) models Similarly, if the regime shifts affect the mean of the variables of the system via regime-dependent means, which can be rewritten as: y t µ(s t ) = A(s t )[y t 1 µ(s t )] + u t, (31) y t = µ(s t ) + z t (32) z t = A(s t )z t 1 + u t, (33) we can utilize the approach above to derive the impulse response functions The state space representation is then given by the measurement equation, y t = [ ] 1 M I K ξ 1t z t ξ Mt z t + ] [ν 1 ν M ξ 1t ξ Mt, (34) and the transition equations ξ 1t z t ξ 1t ξ Mt ξ Mt z t = = p 11 A 1 p M1 A 1 ξ 1t 1 z t 1 + ε t, p 1M A M p MM A M ξ Mt 1 z t 1 p 11 p M1 ξ 1t 1 + η t, p 1M p MM ξ Mt 1 where ε t and η t are martingale difference sequences The conditional expectation of y t+h given y t and ξ t can be written as: E[y t+h y t,ξ t ] = ( 1 M I K) Π h (ξ t y t ) + MF h ξ t (35) where the matrices M,F, and Π are defined as in 4 Thus the impulse responses are given by: IR u (h) = ( 1 M I K) Π h (ξ t u), IR ξ (h) = ( 1 M I K) Π h ( ξ y t ) + MF h ξ

9 53 Higher order vector autoregressions When the process is an MSA(M)-VAR(p) process, we use the stacked VAR(1) representation by defining: y t ε t A 1 A 2 A p ν m y y t = t 1, ε 0 t =, A I m = K 0 0, ν 0 m = (36) 0 0 I K 0 0 y t p+1 Then, the impulse responses are given by: where y t+h is calculated as before IR(h) = ( ι 1 I K) yt+h (37) 54 Cointegration Consider the MSI(M)-VECM(p 1) model, y t = Mξ t + αβ y t 1 + Γ 1 y t 1 + + Γ p 1 y t p+1 + u t, (38) where M =[ν 1 : : ν M ] and ξ t is the unobservable (M 1) state vector consisting of the indicator variables I(s t = m) for m = 1,,M The corresponding MS(M) VAR(p) representation is given by y t = Mξ t + A 1 y t 1 + + A p y t p + u t, (39) where A 1 = I K + αβ + Γ 1 and A j = Γ j Γ j 1 for 1 < j p with Γ p = 0 K The IRFs can be derived straightforwardly using our results for MS-VAR models 6 Empirical illustrations 61 Hamilton s Model of the US Business Cycle The Hamilton (1989) model of the US business cycle fostered a great deal of interest in the MS AR model as an empirical vehicle for characterizing macroeconomic fluctuations, and there have been a number of subsequent extensions and refinements (see the literature discussed in Krolzig, 1997) The Hamilton (1989) model of the US business cycle is an MSM(2)-AR(4) of the quarterly percentage change in US real GNP from 1953 to 1984: y t µ(s t ) = α 1 ( y t 1 µ(s t 1 )) + + α 4 ( y t p µ(s t 4 )) + u t, (40) where u t NID(0,σ 2 ), and the conditional mean µ(s t ) switches between two states: { µ µ(s t ) = 1 < 0 if s t = 1 ( contraction or recession ), µ 2 > 0 if s t = 2 ( expansion or boom ) (41) The variance of the disturbance term, σ 2, is assumed to be the same in both regimes Thus, contractions and expansions are modeled as switching regimes of the stochastic process generating the growth rate of real GNP The transition probabilities are constant: p 21 = Pr( contraction in t expansion in t 1), p 12 = Pr( expansion in t contraction in t 1)

10 For a given parametric specification, probabilities are assigned to the unobserved regimes expansion and contraction conditional on the available information set which constitute an optimal inference on the latent state of the economy Regimes reconstructed in this way are crucial for predicting the probability of future recessions Figure 2 shows the regime-invariant response following a Gaussian shock and the effect of a transition from an expansion to a recession This is complemented by the accumulated responses which can be calculated as in the linear case Note that the responses to the regime shift are proportional to the probability of staying in a recession shown in figure 1 Standard Gaussian shock USGNP Standard Gaussian shock (acc) USGNP 00 00 00 0 4 8 12 16 20 Transition: Expansion to Recession USGNP Pr[s t+h s t ] 0 0 4 8 12 16 20 Transition: Expansion to Recession (acc) USGNP 1 2 3 15 s t+h = Recession h>0 4 0 4 8 12 16 20 0 4 8 12 16 20 Figure 2 Hamilton model: Impulse Responses 62 Output growth, employment growth and the term structure Following Krolzig and Toro (1998), we extend the Hamilton s analysis by modelling jointly output growth, y t and employment growth, n t, with a single latent variable describing the business cycle phases As there has been some interest in the relation of the term structure of interest rates and the business cycle (see, inter alia, Labadie, 1994), and Osborn, Simpson and Sensier, 2001), we add to their system the term structure rt l rs t as measured by the difference between the yield on 10Y government bonds and the Fed Funds rate As in Krolzig and Toro (1998), we consider a three-regime model: n t y t = rt l rs t ν 1 (s t ) n t 1 n t 2 ν 2 (s t ) + A 1 (s t ) y t 1 + A 2 (s t ) y t 2 + ν 3 (s t ) rt 1 l rs t 1 rt 2 l rs t 2 u 1t u 2t u 3t, (42) where u t NID(0,Σ(s t )) We consider two different model specifications: An MSIH(3)-VAR(2) model with regime-invariant VAR matrices, A i (s t ) = A i, and an MSIAH(3)-VAR(2) model with regime-dependent VAR matrices

11 Note that in contrast to their data set, cointegration between y t and n t could not be established for our data vintage, where y t and n t are, respectively, the quarterly seasonally-adjusted US GNP and employment (total nonfarm payrolls: all employees) The Maximum Likelihood estimates using an EM algorithm (see Hamilton (1990) and Krolzig (1997)) as implemented in Krolzig (1998) for data from 1960(i) to 2002(ii) are reported in tables 1 and 2 Table 1 ML estimates of the MSIAH(3)-VAR(2) model (1960Q1 to 2002Q2) Regime 1 Regime 2 Regime 3 n t y t r l t r s t n t y t r l t r s t n t y t r l t r s t Intercept -0121 0223 0117 0098 0745 0452 0846 2338 16 n t 1 0855 0664-1335 13 7-0432 69 1798-0496 n t 2-0059 -06 0121 0171-0124 0119-0334 -0818-0045 y t 1-0074 -0316 0068 0084 0036-0121 -0250-0701 -0089 y t 2-0017 0037-0051 0018 0138-0045 -6 1 0183 rt 1 l rt 1 s -8-0094 0610 0075 0072 1294 2 0147 0800 rt 2 l rt 2 s 0065 0270 3 0072-0042 -0426-0113 -26 0086 Standard error 0382 0761 0012 0151 0493 0354 0207 0496 0460 Table 2 ML estimates of the MSIH(3)-VAR(2) model (1960Q1 to 2002Q2) n t y t r l t r s t Intercepts Recession -0115 0072 0222 Expansion 0254 0859 0376 High growth 0399 1257 0479 Coefficients n t 1 0677 0475-0248 n t 2-0067 -0381-0084 y t 1-0010 -0153-0119 y t 2-1 7-1 rt 1 l rt 1 s -0013-0039 20 rt 2 l rt 2 s 4 0099-0141 Standard errors Recession 0391 0786 1342 Expansion 0125 0430 0397 High growth 0286 0755 0452 The time series are plotted together with their fit in figure 3 Plotting the contribution of the Markov chain to the series (as based on the MSIH model) reveals that the inferred regimes represent the common cycle in output and employment growth

12 Employment Growth mean (MSIH) series fit (MSIH) fit (MSIAH) 001 001 1960 1965 1970 1975 1980 1985 1990 1995 2000 Output Growth 5 0 001 1960 1965 1970 1975 1980 1985 1990 1995 2000 Term Structure 001 1960 1965 1970 1975 1980 1985 1990 1995 2000 Figure 3 Time series and their fit Figure 4 displays the regime probabilities of the three regimes in the MSIH and the MSIAH model In both models, regime 1 tracks the NBER dated recessions of the US economy very precisely Regimes 2 and 3 both reflect economic expansions The MSIH-VAR model confirms the findings of Krolzig (2001): The 2nd regime represents the less volatile expansions of the post 1985 period, while the 3rd regime picks up the expansions until 1985 The separation is less pronounced in the model with regimevarying autoregressive coefficients Interestingly, the third regime emerges at troughs of the NBER cycle dating (up to the 1980-82 double-dip recession) reflecting the catching-up effects observed during economic recoveries in that time period Using the smoothed regime inferences, the matrix of transition probabilities and some regime properties can be calculated which are presented in tables 3 and 4 The regimes are persistent with an average recessions duration of seven quarters Table 3 Regime properties of the MSIAH(3)-VAR(2) model Transition probabilities Observations ErgProb Duration Recession 0856 0043 0101 422 0255 696 Expansion 0052 0904 0045 908 42 1040 High growth 0043 0203 0754 360 0203 406 Table 4 Regime properties of the MSIH(3)-VAR(2) model Transition probabilities Observations ErgProb Duration Recession 0845 0 0154 430 02588 647 Expansion 0045 0954 0 528 03102 2212 High growth 0060 0032 0907 732 04310 1077 Note that in figure 3 the common cycle can only explain a small fraction of the fluctuations of

13 MSIH: Probabilities of Regime 1 MSIAH: Probabilities of Regime 1 1960 1970 1980 1990 2000 MSIH: Probabilities of Regime 2 1960 1970 1980 1990 2000 MSIAH: Probabilities of Regime 2 1960 1970 1980 1990 2000 MSIH: Probabilities of Regime 3 1960 1970 1980 1990 2000 MSIAH: Probabilities of Regime 3 1960 1970 1980 1990 2000 1960 1970 1980 1990 2000 NBER Recession filtered smoothed Figure 4 Regime probabilities the term structure This observation is confirmed by a likelihood ratio test of the null hypothesis that ν 1 = ν 2 = ν 3 : Table 5 LR test for shifts in the mean H 0 : ν 1 = ν 2 = ν 3 MSIH(3)-VAR(2) MSIAH(3)-VAR(2) n χ 2 (2) = 25415 [00] χ 2 (2) = 62655 [00] y χ 2 (2) = 25096 [00] χ 2 (2) = 53184 [00] r l r s χ 2 (2) = 2025 [03632] χ 2 (2) = 1456 [04827] [ ] Marginal p value We, therefore, shall expect that term structure of interest is not significantly affected in its mean by regime shifts This is confirmed in the impulse response functions plotted in figure 5 Figure 5 illustrates the economic consequences of business cycle turning points in the form (i) of moves from the ergodic regime distribution ξ to a particular regime m ( ξ t = ι m ξ) (plots on the diagonal) and (ii) of transitions from regime m to regime n ( ξ t = ι n ι m ) (plots on the off-diagonal) The following observations deserve some interest: (i) the moves to a particular regime capture the informational value of knowing that the economy is in a particular state (eg, knowing that we are in regime 2 will not improve predictions greatly); (ii) the responses of a shift from regime m to regime n and vice versa are symmetric; (iii) some transitions are empirically irrelevant (eg, transitions from regime 1 to 2 and regime 2 to 3) as shown by the matrix of transition probabilities in table 3

14 Move to regime 1 Transition regime 2 to 1 Transition regime 3 to 1 0 0 0 5 5 5 0010 0010 0010 0 20 40 0 20 40 0 20 40 Transition regime 1 to 2 Move to regime 2 Transition regime 3 to 2 0010 00 00 5 0 25 50 25 50 0 20 40 0 20 40 0 20 40 Transition regime 1 to 3 Transition regime 2 to 3 Move to regime 3 0010 50 50 5 25 25 0 00 00 0 20 40 Figure 5 0 20 40 n y r l r s 0 20 40 Responses to regime shifts in the MSIH(3)-VAR(2) model Response of n to unit shock to n linear VAR(2) MSIH(3) VAR(2) Response of y to unit shock to n 00 Response of r l r s to unit shock to n 00 00 02 Response of n to unit shock to y Response of y to unit shock to y 02 Response of rl r s to unit shock to y 00 00 00 Response of n to unit shock to 0 rl r s Response of y to unit shock to 0 rl r s Response of r l r s to unit shock to r l r s 025 025 00 Figure 6 Responses to shocks in the MSIH(3)-VAR(2) model

15 Although the autoregressive coefficients are regime-invariant in the MSIH(3)-VAR(2) model, figure 6 reveals significant changes when compared to the linear VAR(2) In the linear model, a positive output shock increases employment growth and the term structure over the next three years However, if the analysis is conditioned on the state of the business cycle as in the Markov-switching model, output shocks are transitory and have no employment effects Its effect on the term structure is even inverted We also see that a loosening of monetary policy (positive shock to r l r s ) has no significant effect on economic growth when the analysis is conditioned on the state of the business cycle In contrast, we can find that in a linear world the same shock would increase economic growth after 4 quarters by 025% This supports the observation that the term structure of interest and the business cycle are highly correlated In figure 7, the response of the systems to regimes shifts are plotted for three different settings for the initial values of the variables: (i) x t = x t 1 = ˆµ = (26,0821,076) ; (ii) x t = (001,001,001) and x t 1 = 0 K,1 ; (iii) x t = x t 1 = 0 K,1 We can see that the economic circumstances of a business cycle transition have quite remarkable effects Figure 8 displays the responses of the system to unit shocks ( x kt = 001) to the variables of the system As the VAR matrices are regime dependent, the impulse response functions become regime dependent However, that the responses differ only marginally and are similar to those in figure 6 This indicates that the MSIH(3)-VAR(2) is the overall preferred model Move to regime 1 Transition regime 2 to 1 Transition regime 3 to 2 Transition regime 1 to 3 Move to regime 3 x t =x t 1 =µ x t =01, x t 1 =0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Move to regime 1 Transition regime 2 to 1 Transition regime 3 to 2 Transition regime 1 to 3 Move to regime 3 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Move to regime 1 Transition regime 2 to 1 Transition regime 3 to 2 Transition regime 1 to 3 Move to regime 3 x t =x t 1 =0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 n y r l r s Figure 7 Responses to regime shifts in the MSIAH(3)-VAR(2) model

16 Response to unit shock to n in regime 1 Response to unit shock to n in regime 2 Response to unit shock to n in regime 3 001 001 001 001 001 001 Response to unit shock to y in regime 1 001 Response to unit shock to y in regime 2 001 Response to unit shock to y in regime 3 001 001 001 Response to unit shock to rl r s in regime 1 Response to unit shock to rl r s in regime 2 001 Response to unit shock to rl r s in regime 3 001 001 001 001 001 001 7 Conclusions Figure 8 Responses to shocks in the MSIAH(3)-VAR(2) model n y r l r s In this paper we proposed a unified approach to the impulse response analysis in Markov-switching vector autoregressive models Based on the corresponding state-space representation of MS-VAR models, we derived the impulses response function IR(h) as the dynamic reaction (E t y t+h ) of the system (i) to shocks to the variables of the system ( u) and (ii) to regime shifts ( ξ) at time t The proposed analysis in related to the concept of generalized impulse responses introduced by Koop et al (1996) but characterizes the properties of the model dynamics only depending on the current values of y t and ξ t rather than the complete history of the process Y t = {y t 1,y t 2,} In contrast to the impulse response functions proposed by Ehrmann et al (2003) the analysis here fully reflects the Markov property of the switching regimes Empirical illustrations of the approach included Hamilton s US business cycle model and two Markov-switching vector autoregressions of output growth, employment growth and the term structure We found that explicitly modelling the business cycle with the hidden Markov chain significantly changes the results of the impulse response analysis, which could have important consequences for the judgment of the macroeconomic consequences of policy shocks References Beaudry, P, and Koop, G (1993) Do recessions permanently affect output? Journal of Monetary Economics, 31, 149 163 Box, G E P, and Jenkins, G M (1970) Time Series Analysis, Forecasting and Control San Francisco: Holden-Day Campbell, J, and Mankiw, N (1987) Are output fluctuations transitory? Quarterly Journal of Economics, 102, 857 880

17 Ehrmann, M, Ellison, M, and Valla, N (2003) Regime-dependent impulse response functions in a markov-switching vector autoregressive model Economic Letters, 78, 295 299 Hamilton, J D (1989) A new approach to the economic analysis of nonstationary time series and the business cycle Econometrica, 57, 357 384 Hamilton, J D (1990) Analysis of time series subject to changes in regime Journal of Econometrics, 45, 39 70 Hamilton, J D (1994) Time Series Analysis Princeton: Princeton University Press Karlsen, H (1990) A class of non-linear time series models PhD thesis, University of Bergen, Norway Koop, G, Pesaran, M, and Potter, S (1996) Journal of Econometrics, 74, 119 147 Impulse responses in nonlinear multivariate models Krolzig, H-M (1996) Statistical analysis of cointegrated VAR processes with Markovian regime shifts SFB 373 Discussion Paper 25/1996, Humboldt Universität zu Berlin Krolzig, H-M (1997) Markov-Switching Vector Autoregressions: Modelling, Statistical Inference and Application to Business Cycle Analysis Berlin: Springer Verlag Lecture Notes in Economics and Mathematical Systems, 454 Krolzig, H-M (1998) Econometric modelling of Markov-switching vector autoregressions using MSVAR for Ox Discussion Paper, Department of Economics, University of Oxford: http://wwweconoxacuk/research/hendry/krolzig Krolzig, H-M (2001) Business cycle measurement in the presence of structural change: International evidence International Journal of Forecasting, 17, 349 368 Krolzig, H-M, and Toro, J (1998) A new approach to the analysis of shocks and the cycle in a model of output and employment Working paper eco 99/30, EUI, Florence Labadie, P (1994) The term structure of interest rates over the business cycle Journal of Economic Dynamics and Control, 18(3-4), 671 697 Osborn, D, Simpson, P, and Sensier, M (2001) Forecasting UK industrial production over the business cycle Journal of Forecasting, 20, 405 424