Regime-Switching Models

1 Regime-Switching ModelsMay 18, 2005 James D. HamiltonDepartment of Economics, 0508 University of California, San DiegoLa Jolla, CA for: Palgrave Dictionary of Economics0 Many economic time series occasionally exhibit dramatic breaks in their behavior, asso-ciated with events such asfinancial crises (Jeanne and Masson, 2000; Cerra, 2005; Hamilton,2005) or abrupt changes in government policy (Hamilton, 1988; Sims and Zha, 2004, Davig,2004). Of particular interest to economists is the apparent tendency of many economicvariables to behave quite differently during economic downturns, when underutilization offactors of production rather than their long-run tendency to grow governs economic dynam-ics (Hamilton, 1989, Chauvet and Hamilton, 2005). Abrupt changes are also a prevalentfeature offinancial data, and the approach described below is quite amenable to theoreticalcalculations for how such abrupt changes in fundamentals should show up in asset prices(Ang and Bekaert, 2003; Garcia, Luger, and Renault, 2003; Dai, Singleton, and Wei, 2003).

2 Consider how we might describe the consequences of a dramatic change in the behaviorof a single that the typical historical behavior could be described withafirst-order autoregression,yt=c1+ yt 1+ t,(1)with t N(0, 2),which seemed to adequately describe the observed data fort=1,2, .., that at datet0there was a significant change in the average level of the series, sothat we would instead wish to describe the data according toyt=c2+ yt 1+ t(2)fort=t0+1,t0+2, ..Thisfix of changing the value of the intercept fromc1toc2mighthelp the model to get back on track with better forecasts, but it is rather unsatisfactory as aprobability law that could have generated the data. We surely would not want to maintain1that the change fromc1toc2at datet0was a deterministic event that anyone would havebeen able to predict with certainty looking ahead from datet=1. Instead there must havebeen some imperfectly predictable forces that produced the change.

3 Hence, rather thanclaim that expression (1) governed the data up to datet0and (2) after that date, what wemust have in mind is that there is some larger model encompassing them both,yt=cst+ yt 1+ t,(3)wherestis a random variable that, as a result of institutional changes, happened in oursample to assume the valuest=1fort=1,2, .., t0andst=2fort=t0+1,t0+2, ..Acomplete description of the probability law governing the observed data would then requirea probabilistic model of what caused the change fromst=1tost=2. The simplest suchspecification is thatstis the realization of a two-state markov chain withPr(st=j|st 1=i, st 2=k, .., yt 1,yt 2, ..)=Pr(st=j|st 1=i)=pij.(4)Assuming that we do not observestdirectly, but only infer its operation through the observedbehavior ofyt, the parameters necessary to fully describe the probability law governingytare then the variance of the Gaussian innovation 2, the autoregressive coefficient ,thetwointerceptsc1andc2, and the two state transition probabilities, specification in (4) assumes that the probability of a change in regime depends on thepast only through the value of the most recent regime , though, as noted below, nothing in theapproach described below precludes looking at more general probabilistic specifications.

4 Butthe simple time-invariant markov chain (4) seems the natural starting point and is clearly2preferable to acting as if the shift fromc1toc2was a deterministic event. Permanence ofthe shift would be represented byp22=1, though the markov formulation invites the moregeneral possibility thatp22< in the case of business cycles orfinancial crises,we know that the situation, though dramatic, is not permanent. Furthermore, if the regimechange reflects a fundamental change in monetary orfiscal policy, the prudent assumptionwould seem to be to allow the possibility for it to change back again, suggesting thatp22<1is often a more natural formulation for thinking about changes in regime thanp22= model of the form of (3)-(4) with no autoregressive elements ( =0)appears to havebeenfirst analyzed by Lindgren (1978) and Baum, et. al. (1980). Specifications thatincorporate autoregressive elements date back in the speech recognition literature to Poritz(1982), Juang and Rabiner (1985), and Rabiner (1989), who described such processes as hidden markov Models .

5 markov - switching regressions were introduced in econometrics byGoldfeld and Quandt (1973), the likelihood function for which wasfirst correctly calculatedby Cosslett and Lee (1985). The formulation of the problem described here, in which allobjects of interest are calculated as a by-product of an iterative algorithm similar in spirit toaKalmanfilter, is due to Hamilton (1989, 1994). General characterizations of moment andstationarity conditions for such processes can be found in Tj stheim (1986), Yang (2000),Timmermann (2000), and Francq and Zako an (2001).Suppose that the econometrician observesytdirectly but can only make an inferenceabout the value ofstbased on what we see happening withyt. This inference will take the3form of two probabilities jt=Pr(st=j| t; )(5)forj=1,2, where these two probabilities sum to unity by construction. Here t={yt,yt 1, .., y1,y0}denotes the set of observations obtained as of datet,and is a vectorof population parameters, which for the above example would be =( , ,c1,c2,p11,p22)0,and which for now we presume to be known with certainty.

6 The inference is performediteratively fort=1,2, .., T,with steptaccepting as input the values i,t 1=Pr(st 1=i| t 1; )(6)fori=1,2and producing as output (5). The key magnitudes one needs in order to performthis iteration are the densities under the two regimes, jt=f(yt|st=j, t 1; )=1 2 exp (yt cj yt 1)22 2 ,(7)forj=1, , given the input (6) we can calculate the conditional density of thetth observation fromf(yt| t 1; )=2Xi=12Xj=1pij i,t 1 jt(8)and the desired output is then jt=P2i=1pij i,t 1 jtf(yt| t 1; ).(9)As a result of executing this iteration, we will have succeeded in evaluating the sampleconditional log likelihood of the observed datalogf(y1,y2, .., yT|y0; )=TXt=1logf(yt| t 1; )(10)4for the specified value of . Anestimateofthevalueof can then be obtained by maximizing(10) by numerical options are available for the value i0to use to start these iterations. If theMarkov chain is presumed to be ergodic, one can use the unconditional probabilities i0=Pr(s0=i)=1 pjj2 pii alternatives are simply to set i0=1/2or estimate i0itself by maximum calculations do not increase in complexity if we consider an(r 1)vector of obser-vationsytwhose density depends onNseparate regimes.

7 Let t={yt,yt 1, ..,y1}be theobservations through datet,Pbe an(N N)matrix whose rowj,columnielement is thetransition probabilitypij, tbe an(N 1)vector whosejth elementf(yt|st=j, t 1; )is the density in regimej,and t|tan(N 1)vector whosejth element isPr(st=j| t, ).Then (8) and (9) generalize tof(yt| t 1; )=10(P t t|t 1 t)(11) t|t=P t t|t 1 tf(yt| t 1; )(12)where1denotes an(N 1)vector all of whose elements are unity and denotes element-by-element multiplication. markov - switching vector autoregressions are discussed in detailin Krolzig (1997). Vector applications include describing the comovements between stockprices and economic output (Hamilton and Lin, 1996) and the tendency for some series tomove into recession before others (Hamilton and Perez-Quiros, 1996). There further is norequirement that the elements of tbe Gaussian densities or even from the same family of5densities. For example, Dueker (1997) studied a model in which the degrees of freedom ofa Student t distribution change depending on the economic is also often interested in forming an inference about what regime the economy wasin at datetbased on observations obtained through a later dateT,denoted t| referred to as smoothed probabilities, an efficient algorithm for whose calculation wasdeveloped by Kim (1994).

8 The calculations in (11) and (12) remain valid when the probabilities inPdepend onlagged values ofytor strictly exogenous explanatory variables, as in Diebold, Lee and Wein-bach (1994), Filardo (1994), and Peria (2002). However, often there are relatively fewtransitions among regimes, making it difficult to estimate such parameters accurately, andmost applications have assumed a time-invariant markov chain. For the same reason, mostapplications assume onlyN=2or3different regimes, though there is considerable promisein Models with a much larger number of regimes, either by tightly parameterizing the re-lation between the regimes (Calvet and Fisher, 2004), or with prior Bayesian information(Sims and Zha, 2004).In the Bayesian approach, both the parameters and the values of the statess=(s1,s2, .., sT)0are viewed as random variables. Bayesian inference turns out to be greatlyfacilitated by Monte Carlo markov chain methods, specifically, the Gibbs sampler.

9 This isachieved by sequentially (fork=1,2, ..)generating a realization (k)from the distributionof |s(k 1), Tfollowed by a realization ofs(k)from the distribution ofs| (k), , |s(k 1), T, treats the historical regimes generated at the previous iteration,6s(k 1)1,s(k 1)2, .., s(k 1)T,as iffixed known numbers. Often this conditional distribution takesthe form of a standard Bayesian inference problem whose solution is known analytically usingnatural conjugate priors. For example, the posterior distribution of given other parametersis a known function of easily calculated OLS coefficients. An algorithm for generating adraw from the second distribution,s| (k), T,was developed by Albert and Chib (1993).The Gibbs sampler turns out also to be a natural device for handling transition probabilitiesthat are functions of observable variables, as in Filardo and Gordon (1998).It is natural to want to test the null hypothesis that there areNregimes against thealternative ofN+1, for example, whenN=1, to test whether there are any changes inregime at all.

10 Unfortunately, the likelihood ratio test of this hypothesis fails to satisfythe usual regularity conditions, because under the null hypothesis, some of the parametersof the model would be unidentified. For example, if there is really only one regime , themaximum likelihood estimate p11does not converge to a well-defined population magnitude,meaning that the likelihood ratio test does not have the usual 2limiting distribution. Tointerpret a likelihood ratio statistic one instead needs to appeal to the methods of Hansen(1992) or Garcia (1998). An alternative is to rely on generic tests of the hypothesis that anN- regime model accurately describes the data (Hamilton, 1996), though these tests are notdesigned for optimal power against the specific alternative hypothesis ofN+1regimes. Atest recently proposed by Carrasco, Hu, and Ploberger (2004) that is easy to compute butnot based on the likelihood ratio statistic seems particularly promising.