IV. Markov-switching models

Transcription

1 IV. Markov-switching models A. Introduction to Markov-switching models Many economic series exhibit dramatic breaks: - recessions - financial panics - currency crises Questions to be addressed: - how handle econometrically - how incorporate into economic theory

2 Economic recessions as changes in regime y t real GDP growth in quarter t s t 1 when economy is in expansion s t 2 when economy is in recesion y t m st t t N0, 2 Probs t j s t1 i, s t2 k,..., y t1, y t2,... p ij

3 y t m st t If s t is observed, m st AR(1) m st a m st1 v t a p 21 m 1 p 12 m 2 p 11 p 21 v t martingale difference sequence

4 If only t y t, y t1,..., y 1 is observed, Probs t 1 t is nonlinear in t. Given Probs t1 j t1, can calculate Probs t j t (and likelihood fy t t1 recursively:

5 Probs t j t1 p 1j Probs t1 1 t1 p 2j Probs t1 2 t1 2 fy t t1 Probs t i t1 fy t s t i, t1 i1 Probs t j t Probs tj t1 fy t s t j, t1 fy t t1

6 Could choose population parameters m 1, m 2,, p 11, p 22 by maximizing likelihood.

7 Plot of Prob(s t 2 t1, t1 with simulated real-time inference (historical real-time data sets from ALFRED) through Plot of actual real-time inference (announced publicly at each date t 1 since 2005.

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10 Another example of change in regime Dollar-denominated minus Peso-denominated (in logs)

11 Model of structural change: y t 1 y t1 1 t t t 0 y t 2 y t1 2 t t t 0

12 Questions: 1) How forecast with this model? 2) What caused change at t 0? 3) What is probability law for {y t }?

13 s t 1 t 1,2,..,t 0 s t 2 t t 0 1, t 0 2,... y t s y t t1 st1 t Need: probability law for s t

14 Markov chain: Ps t j s t1 i,s t2 k,... Ps t j s t1 i p ij Transition from 1 to 2 is permanent p 21 0

15 In general, if s t is a Markov chain taking on one of the values s t 1,2,...,N, let p ij Ps t j s t1 i. Collect in matrix P p ji p 11 p 21 p N1 P p 12 p 22 p N2 p 1N p 2N p NN

16 In general, if s t is a Markov chain taking on one of the values s t 1,2,...,N, let p ij Ps t j s t1 i. Collect in matrix P p ji p 11 p 21 p N1 P p 12 p 22 p N2 p 1N p 2N p NN

17 Let t e i (the ith column of I N when s t i. Then Ps t1 1 s t i E t1 t e i Ps t1 2 s t i Ps t1 N s t i Pe i P t

18 Suppose we had a set of observations t y t, y t1,..., y 1 that gave us an imperfect inference about s t summarized as Ps t 1 t t t E t t Ps t 2 t Ps t N t

19 Then t1 t E t1 t P t t (e.g., row j states that Ps t1 j t p 1j Ps t 1 t p 2j Ps t 2 t p Nj Ps t N t

20 Return to original example of interest: y t s y t t1 st1 t t ~ i.i.d. N0, 2 Ps t j s t1 i p ij i, j 1,2 s t T t1 independent of t T t1 t y t,y t1,...,y 1

21 Implication: y t t1,s t,s t1 ~ N y st t1 s, 2 t1

22 Convenient to summarize s t,s t1 with a single Markov chain: s t 1 s t 2 s t 3 s t 4 if s t 1 and s t1 1 if s t 2 and s t1 1 if s t 1 and s t1 2 if s t 2 and s t1 2

23 t e i (ith column of I 4 ) when s t i t1 t P t t p 11 0 p 11 0 P p 12 0 p p 21 0 p 21 0 p 22 0 p 22

24 py t s t 3, t1 py t s t 1, st1 2, t1 1 2 exp y t 1 y t

25 Collect the densities that might be associated with each of the N 4 states in an N1 vector py t s t 1, t1 t py t s t 2, t1 py t s t N, t1

26 Recall that Ps t 1 t1 P t1 t1 Ps t 2 t1 Ps t N t1

27 Thus t P t1 t1 py t s t 1, t1 Ps t 1 t1 py t s t 2, t1 Ps t 2 t1 py t s t N, t1 Ps t N t1

28 Summing the elements of this vector gives 1 t P t1 t1 N j1 py t t1, py t, s t j t1 the conditional likelihood of tth observation.

29 The result of dividing the jth element of t P t1 t1 by the conditional likelihood is py t, s t j t1 py t t1 Ps t j y t, t1 t P t1 t1 1 t P t1 t1 t t

30 t P t1 t1 1 t t t P t1 t1 Iterative algorithm similar to Kalman filter: Input for step t: t1 t1 (an N1vector whose jth element is Ps t j y t,y t1,...,y 1 ). Output for step t: t t

31 Options for initial value 0 0 : (1) If Markov chain is ergodic, use ergodic probabilities 0 0 A A 1 A e N1 A N1N I N P 1

32 (2) Set 0 0, a vector of free parameters to be estimated by maximum likelihood or Bayesian methods along with the other parameters.

33 (3) Set 0 0 N 1 1. (4) Set 0 0 based on prior beliefs.

34 Above assumed we knew parameters appearing in t py t s t j, t1 ;N j1 (in this case, 1, 2, 2 ) and p appearing in P (in this case p p 11,p 22 ).

35 However, as byproduct of step t of iteration we ended up calculating py t t ;,p and so we ve calculated log likelihood,p T t1 logpy t t ;,p which can be maximized numerically with respect to and p by numerical methods.

36 Note during numerical search we d want to be choosing 11 and 22 rather than p 11 and p 22 where p p

37 General case: py t s t 1, t1 t py t s t 2, t1 py t s t N, t1 py t t1 1 t P t1 t1 y,...,y ; T 1 T t1 log py t t1

38 IV. Markov-switching models A. Introduction to Markov-switching models B. Economic theory and changes in regime

39 B.1. Closed-form solution of DSGE s and asset-pricing implications Lucas tree model with CRRA utility: P t price of stock D t dividend coefficient of relative risk aversion P t D t k1 k 1 E t D tk

40 Cecchetti, Lam and Mark (1990): log D t log D t1 m st t P t st D t

41 B.2. Linear rational expectations models with changes in regime A st Ey t1 t, s t, s t1,..., s 1 d st B st y t C st x t A j n y n y matrix of parameters when s t j.

42 Davig and Leeper (2007): Let y jt value of y t when s t j Y t Nn y 1 y 1t n y 1 y Nt n y 1

43 N Ey t1 s t i, t j1 Hence when s t i, Ey t1 s t1 j, s t i, t p ij A st Ey t1 s t, t p i A i EY t1 Y t

44 p 1 1N A 1 n y n y d 1 n y 1 A Nn y Nn y p N 1N A N n y n y d Nn y 1 d N n y 1

45 B Nn y Nn y B B B N C Nn y n x C 1 n y n x C N n y n x

46 Consider non-regime-changing system AEY t1 Y t d BY t Cx t If we can find a stable solution of the form Y t h Nn y 1 Nn y 1 then the ith block y t h st n y 1 n y 1 H x t Nn y n x n x 1 H st x t n y n x n x 1 is a stable solution to our original equation of interest.

47 However, even if we find a unique stable solution to the invariant system, there may be other stable solutions to the original system - Farmer, Waggoner, and Zha (2010)

48 B.3. Multiple equilibria Multiplicity of stable equilibria could itself be of interest - Coordination externalities (Cooper and John, 1988; Cooper, 1994) - Equilibria indexed by expectations (Kurz and Motolese, 2001) Regime-switching model could describe transitions between equilibria - Kirman (1993); Chamley (1999)

49 B.4. Tipping points and financial crises In other models, there is a unique equilibrium, but small change in fundamentals can cause big change in outcome - Acemoglu and Scott (1997); Moore and Schaller (2002); Guo, Miao, and Morelle (2005); Veldkamp (2005); Startz (1998); Hong, Stein, and Yu (2007); Branch and Evans (2010) Financial crises - Brunnermeier and Sannikov (2014); Hamilton (2005); Asea and Blomberg (1998); Hubrich and Tetlow (2013)

50 B.5. Currency crises and sovereign debt crises Currency crises - Jeanne and Masson (2000); Peria (2002); Cerra and Saxena (2005) Sovereign debt crises - Greenlaw, et. al. (2013); Davig, Leeper and Walker (2011); Bi (2012)

51 B.6. Changes in policy as the source of changes in regime Monetary policy: hawks vs. doves Owyang and Ramey (2004); Schorfheide (2005); Liu, Waggoner, and Zha (2011); Bianchi (2013) Unsustainable fiscal policy and inflation Ruge-Murcia (1995, 1999)

52 IV. Markov-switching models A. Introduction to Markov-switching models B. Economic theory and changes in regime C. Extensions

53 C.1. Selecting the number of regimes Smith, Naik and Tsai (2006): y t x t st st t T i T t1 Probs t i T ; MLE MSC 2 N MLE i1 T i T i Nk T i Nk2

54 Calculate nonstandard properties of likelihood ratio test - Hansen (1992) - Garcia (1998) Use general specification tests of null of N regimes that have power against N +1 - Hamilton (1996) - Carrasco, Hu and Ploberger (2014)

55 C.2. Chib s multiple change-point model P p p 11 p p 22 p p N1,N p N1,N1 1

56 C.3. Allowing any parameters of distribution to change py t s t 1, t1 t py t s t 2, t1 py t s t N, t1 py t t1 1 t P t1 t1 y,...,y ; T 1 T t1 log py t t1

57 Example: Dueker (JBES, 1997). AR(1) with Student t innovations whose degrees of freedom change with regime: py t s t j, t1 ; j 1/2 j 1/2 j /2 1 y t c y t1 2 j c,,, 1, 2 j 1/2

58 Example: Krolzig (Markov-Switching Vector Autoregressions, Springer 1997): Gaussian VAR(1) with lag coefficients changing: py t s t j, t1, 2 n/2 1/2 exp1/2y t c j j y t1 1 c 1, c 2 y t c j j y t1,vec 1,vec 2,vech

59 Can also allow transition probabilities to be parametric function of exogenous or lagged dependent variables z t : P Ps t j s t1 i, z t ; p N i,j1. Example: P p, exp z t 1exp z t 1 1exp z t 1 1exp z t exp z t 1exp z t

60 What can t we do? Models where y t depends on a growing number of states: py t t1,s t,s t1,...,s 1 ; Example: ARMA process y t t st1 t1 y t t st1 y t1 st1 st2 y t2 y st1 st2 st3 t3

61 Example: GARCH process: y t h t t h t s y2 t t1 h t1 y2 st t1 y 2 st1 t2 2 s y 2 t2 t3

62 Solution: numerical Bayesian methods

63 IV. Markov-switching models A. Introduction to Markov-switching models B. Economic theory and changes in regime C. Extensions D. Bayesian analysis of Markov-switching models

64 Example: y t s x t t t t ~ i.i.d. N0, 2 Ps t j s t1 i p ij i,j 1, 2 (does not depend on x tk, tk, s tk1 for k 0,1, 2,...)

65 Gibbs sampler: 1, 2, 3, , 2 3 p 11,p 22 4 s 1,s 2,...,s T

66 (1) Generating 1 2 from p 1 2, 3, 4, Y, X. Prior: 2 ~N, Conditioning on 2, 3, 4, Y, X is equivalent to observing t T t1 t y t st Posterior: x t for 2 2, 3, 4,Y, X ~N T, S S T 2 t1 t

67 (2) Generating 2 1 p 2 1, 3, 4,Y, X. Priors:, 2 from i 2 ~ Nm i, 2 M i i 1,2 (independent of each other) Posterior: Conditioning on s t T t1, only those observations t for which s t 1 are relevant for posterior distribution of 1.

68 i 1, 3, 4,Y, X ~ Nm i, 2 M i M i M 1 T i t1 x t x t st i 1 m i M i M 1 i m i T t1 x t y t s ti

69 Label-switching problem: If switch 1 with 2 and p 11 with p 22, value of likelihood py X, 1, 2, 3 is identical. Implication: if priors for i and p ii are same for i 1,2, then true posterior distribution is bimodal and perfectly symmetric around the two modes.

70 Presume we have interpretive (as opposed to numerical) labels for regimes. E.g., regime 2 recession, should have faster GDP growth, so that, say, 1 1, first element of 1, should be bigger 2 1, the first element of 2.

71 Strategy (1): Intentionally use symmetric priors for regimes 1 and 2 and intentionally randomly perturb parameter draw j to switch across modes so as to get multimodal posterior distribution, and apply normalization rule to this.

72 Strategy (2): Impose normalization requirement at every draw. Drawback to (2): not clear it s same distribution as (1).

73 Drawback to either approach: Even though normalized posterior distribution has unique global mode, may still have local modes resulting from label switching. Recommendation: plot posterior distributions to check for this.

74 (3) Generating 3 p 11,p 22 from p 3 1, 2, 4,Y, X. Priors: A variable x is said to have a beta distribution with parameters 0 and 0, denoted x ~ Beta,, if px, x1 1x 1 for 0 x 1 and px, 0 elsewhere.

75 Ex Vx 2 1

76 Beta distribution p(x) ==0.5 ==1 ==6 == x

77 Beta distribution p(x) =1.5,=0.5 =3,=1 =6,=2 =18,= x

78 Priors: p ii ~ Beta i, i i 1, 2 (independent of each other)

79 Posterior: Observation of 1, 2, 4, Y, X only affects inference about p ii through 4 s 1, s 2,...,s T. Assume that initial probability Ps 1 1 does not depend on p ii (don t use ergodic probabilities).

80 Suppose that in the sequence 4 s 1,s 2,...,s T state s t 1 is observed to be followed by s t1 1 a total of n 11 times, whereas state s t 1 is followed by s t1 2 a total of n 12 times.

81 Then, for purposes of inference about p 11, can view the data 1, 2, 4, Y, X solely as a sample of n 11 n 12 observations from a Bernoulli variable with probability of success p 11 : n p 1, 2, 4,Y, X 3 p p 11 n 12

82 data: n p 1, 2, 4, Y, X 3 p p 11 n 12 prior: p ii ~ Beta i, i p 3 p p posterior: p ii ~ Beta i, i 1 1 n n n n 21

83 (4) Generating 4 s 1, s 2,...,s T from p 4 1, 2, 3, Y, X. Calculate Ps T 1 1, 2, 3,Y, X from first element of T T. Generate U T ~ U0,1 and set S T 1 if U T e 1 T T.

84 Consider PS t i S t1 j,s t2 k,...,s T z, 1, 2, 3, Y, X PS t i S t1 j, 1, 2, 3, t for t y t, y t1,...,y 1, x t, x t1,...,x 1

85 PS t i S t1 j, 1, 2, 3, t PS t i,s t1 j 1, 2, 3, t PS t1 j 1, 2, 3, t PS t1 j S t ips t i 1, 2, 3, t PS t1 j 1, 2, 3, t p ije i t t e j P t t

86 Iterating backwards t T 1, T 2,... we generate the sequence 4 s 1,s 2,...,s T from p 4 1, 2, 3, Y, X.

87 Generalization: N-state Markov chain. x ~ Beta 1, 2 px 1, x 11 1 x 21 0 x 1

88 x 1, x 2,..., x N ~ Dirichlet 1, 2,..., N px 1,..., N 1 N 1 N x x N 1 N 0 x i 1 x 1 x N 1 i 0

89 prior: p 11,p 12,...,p 1N ~ Dirichlet 11, 12,..., 1N data: n 1j number of times s t 1 is followed by s t1 j posterior: p 11,p 12,...,p 1N 1, 2, 3,Y, X ~ Dirichlet 11 n 11,..., 1N n 1N

90 Generalization: time-dependent transition probabilities: Ps t j s t1 i,x t x t predetermined at t

91 Convenient framework for Gibbs sampling: latent variable z t z t 0 s t1 x t u t u t ~ N0, 1 s t 1 if z t 0 2 if z t 0

92 Gibbs sampler: 1, 2, 3, , s 1,s 2,...,s T, z 1,z 2,...,z T 4 0,

93 Draws from p 1 2, 3, 4, Y,X and p 2 1, 3, 4, Y, X same as before (conditioning onz t adds no information beyond that ins t ).

94 Will draw 3 using p 3 1, 2, 4, Y, X ps 1,...,s T 1, 2, 4,Y,X pz 1,...,z T s 1,...,s T, 1, 2, 4, Y, X (a) To draw from ps 1,...,s T 1, 2, 4,Y,X use modification of filter for time-varying probabilities

95 Recall filter with constant probabilities: jth element of : t jt py t s t j, t1 jth element of t t1 P t1 t1 : ps t j t1 jth element of t t t1 : py t, s t j t1 t t t t t1 1 t t t1

96 Filter with time-varying probabilities: z t 0 s t1 x t u t u t N0, 1 s t 1 if z t 0

97 ps t 1 t1 ps t 1 s t1 1, t1 ps t1 1 t1 ps t 1 s t1 2, t1 ps t1 2 t1 p 0 x t u t 0ps t1 1 t1 p2 0 x t u t 0ps t1 2 t1 0 x t ps t1 1 t1 2 0 x t ps t1 2 t1

98 So we just replace t t1 P t1 t1 in the regular filter with t t1 ps t 1 t1 ps t 2 t1 t t t t t1 1 t t t1

99 (i) Run through filter to calculate t t, t t1 T t1 (ii) Generate s T 1 with probability e 1 T T and s T 2 with probability e 2 T T. (iii) To get s t for t T1,T2,... use modification of earlier smoothing algorithm:

100 PS t i S t1 j, 1, 2, 4, t PS t i,s t1 j 1, 2, 4, t PS t1 j 1, 2, 4, t PS t1 j S t ips t i 1, 2, 4, t PS t1 j 1, 2, 4, t 0 i x t e i t t e 1 t1 t if j i x t e i t t e 2 t1 t if j 2

101 p 3 1, 2, 4,Y,X ps 1,...,s T 1, 2, 4,Y,X pz 1,...,z T s 1,...,s T, 1, 2, 4,Y,X (b) To draw from pz 1,...,z T s 1,...,s T, 1, 2, 4,Y,X z t 0 s t1 x t u t Generate z t N 0 s t1 x t,1 keep if signz t signs t 1.5 otherwise draw new z t

102 Finally, to generate a value for 4 0, given 1, 2, 3, Y, X, notice that conditional on having observed s t1, z t T t1, generating 4 is standard regression problem: z t 0 s t1 x t u t

103 IV. Markov-switching models A. Introduction to Markov-switching models B. Economic theory and changes in regime C. Extensions D. Bayesian analysis of Markov-switching models E. State-space models with Markov switching

104 t1 F s t t v t1 y t A s t x t H st Ps t1 j s t i p ij Ev t1 v t1 Q st t w t Ew t w t R s t s t,v t,w t independent x t predetermined exogenous

105 1 unknown elements of Q 1, Q 2,R 1,R 2 2 unknown elements of F 1, F 2, A 1, A 2, H 1,H 2 3 p 11,p 22 4 s 0,s 1,s 2,...,s T 5 unknown elements of 0, 1,..., T

106 (1) Generating 1 Q 1, Q 2,R 1,R 2 given Y, X, 2, 3, 4, 5. t1 F s t t v t1 Ev t1 v t1 Q st

107 Notice for purposes of estimating Q 1, the likelihood satisfies py X, 2, 3, 4, 5 T Q t1 s 1/2 t1 exp 1/2 T t1 t F s t 1 t1 Q 1 st1 t F st 1 t1 T t1 Q 1 1/2 s t1 1 exp 1/2 T t1 t F s t 1 t1 Q 1 1 t F st 1 t1 st1 1

108 prior: Q 1 1 ~ WN Q 1, Q1 posterior: Q 1 1 2, 3, 4, 5, Y,X ~ WN Q1 T 1, Q1 S Q1 T 1 T t1 S T Q1 t1 st1 1 v t v t v t t F st 1 t1 s t1 1

109 (2) Generating 2 F 1, F 2,A 1, A 2,H 1, H 2 given Y,X, 1, 3, 4, 5. t1 F s t t v t1 Ev t1 v t1 Q st

110 prior: f 2 Q 2 ~ Nm F2, Q 2 M F2 posterior: f 2 Y, X, 1, 3, 4, 5 ~ Nm,Q F2 2 M F2 M F2 M 1 T F2 t1 t1 t1 s t1 2 1

111 m I F2 r M M 1 m F2 F2 F2 I r M T F2 t1 t1 t1 st1 2 f 2 f 2 vecf 2 F 2 T t1 t1 t1 st1 2 1 T t1 t1 t st1 2

112 (3) Generating 3 p 11, p 22 given Y, X, 1, 2, 4, 5. (4) Generating 4 s 0, s 1,s 2,...,s T given Y,X, 1, 2, 3, 5. Exactly same as for other Markovswitching models.

113 (5) Generating 5 0, 1,..., T given Y, X, 1, 2, 3, 4. t1 F s t t v t1 y t A s t x t H st Ev t1 v t1 Q st t w t Ew t w t R s t Conditional on s 0, s 1,...,s T, this is just a Kalman filter problem where we use different F, Q, A, H, R for different dates.

114 P t1 t F s tp t t F st Q s t P t1 t1 P t1 t P t1 t H st1 H st1 t1 t F s t t t P t1 t H R 1 H st1 st1 st1 P t1 t

115 t1 t y t1 A x st1 t1 H st1 t1 t t1 t1 t1 t P t1 t H st1 H st1 P t1 t H st1 R st1 1 t1 t

116 T Y,X, 1, 2, 3, 4 ~ N T T, P T T t t1, Y,X, 1, 2, 3, 4 ~ N t t,p t t J t P t t F P 1 st t1 t P t t P t t J t F P st t t