Business cycles and changes in regime. 1. Motivating examples 2. Econometric approaches 3. Incorporating into theoretical models

Transcription

1 Business cycles and changes in regime 1. Motivating examples 2. Econometric approaches 3. Incorporating into theoretical models 1

2 1. Motivating examples Many economic series exhibit dramatic breaks: - recessions - financial panics - currency crises Questions: - how handle econometrically? - how incorporate into economic theory? 2

3 An example of change in regime Dollar-denominated minus Peso-denominated (in logs)

4 Model of structural change: y t 1 y t1 1 t t t 0 y t 2 y t1 2 t t t 0 4

5 Questions: 1) How forecast with this model? 2) What caused change at t 0? 3) What is probability law for {y t }? 5

6 s t 1 t 1, 2,..,t 0 s t 2 t t 0 1,t 0 2,... y t st y t1 st1 t Need: probability law for s t 6

7 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

8 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 8

9 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 Em st s t1 1 p 21 m 1 p 12 m 2 p 11 p 21 m 1 p 12 m 2 p 11 m 1 Em st s t1 2 p 21 m 1 p 12 m 2 p 11 p 21 m 2 p 21 m 1 p 12 p 11 p 21 m 2 p 21 m 1 1p 21 m 2 p 21 m 1 p 22 m 2 9

10 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: 10

11 Probs t j t1 p 1j Probs t1 1 t1 p 2j Probs t1 2 t1 fy t s t i, t exp y tm i 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 11

12 Could choose population parameters m 1, m 2,, p 11, p 22 by maximizing likelihood. 12

13 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

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16 2. Econometric treatment of changes in regime 2.1. Multivariate or non-gaussian processes and multiple regimes 2.2. Time-varying transition probabilities 2.3. Processes that depend on current and past regimes 2.4. Latent-variable models with changes in regime 2.5. Analysis using Bayesian methods 2.6. Selecting the number of regimes 2.7. Deterministic breaks 2.8. Chib s multiple change-point model 2.9. Smooth transition models 16

17 2.1. Multivariate or non-gaussian processes and multiple regimes Previous recursion used 2 fy t t1 Probs t i t1 fy t s t i, t1 i1 for fy t s t i the Nm i, 2 density. But works same for fy t s t i, t1 any multivariate, non-gaussian density. 17

18 Krolzig (1997) switching VAR: fy t t1 ; j 1 2 n/2 j 1/2 exp 1/2y t jt j 1 y t jt jt c j 1j y t1 2j y t2 rj y tr. Dueker (1997): degrees of freedom on Student t change with regime. 18

19 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 19

20 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 20

21 t t P t1 v t v t martingale difference sequence. In other words, N-state Markov chain can be represented using VAR(1). 21

22 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 22

23 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 23

24 Collect the densities that might be associated with each of the N states in ann 1 vector py t s t 1, t1 t py t s t 2, t1 py t s t N, t1 24

25 Recall that P t1 t1 Ps t 1 t1 Ps t 2 t1 Ps t N t1 25

26 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 26

27 Summing the elements of this vector gives 1 t P t1 t1 N j1 py t,s t j t1 py t t1, the conditional likelihood of tth observation. 27

28 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 28

29 t P t1 t1 1 t P t1 t1 t t Iterative algorithm similar to Kalman filter: Input for step t: t1 t1 (an N 1 vector whose jth element is Ps t j y t, y t1,...,y 1 ). Output for step t: t t 29

30 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 30

31 (2) Set 0 0, a vector of free parameters to be estimated by maximum likelihood or Bayesian methods along with the other parameters. 31

32 (3) Set 0 0 N 1 1. (4) Set 0 0 based on prior beliefs. 32

33 Above assumed we knew parameters N appearing in t py t s t j, t1 ; j1 (in first example, 1, 2, 2 ) and p appearing in P (in this case p p 11, p 22 ). 33

34 However, as byproduct of step t of iteration we ended up calculating py t t1 ;, p and so we ve calculated log likelihood, p T t1 log py t t1 ;, p which can be maximized numerically with respect to and p by numerical methods. 34

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

36 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 36

37 2.2. Time-varying transition probabilities Simple recursion used Probs t j t1 p 1j Probs t1 1 t1 p 2j Probs t1 2 t1. But works the same when p 1j is replaced by any known function p 1j t1. 37

38 2.3. Processes that depend on current and past regimes Simple example assumed density fy t s t, t1 depends only on current regime s t. If instead depends on current and past regimes can simply stack regimes as in companion form for VAR(p). 38

39 s t 1 when s t 1 and s t1 1 2 when s t 2 and s t1 1 3 when s t 1 and s t1 2 4 when s t 2 and s t1 2 Prob(s t j s t1 i p ij i, j 1,..., 4. 39

40 2.4. Latent variable models with changes in regime F t st F t1 t y t F t q t q jt j q j,t1 v jt 40

41 Can approximate likelihood function and optimal inference -Kim (1994) Useful for real-time inference - Chauvet and Hamilton (2006) - Chauvet and Piger(2008) - Camacho and Perez-Quiros(forthcoming) 41

42 2.5. Analysis using Bayesian methods Gibbs sampler -Albert and Chib(1993) -Kim and Nelson (1999) Time-varying transition probabilities - Filardo and Gordon (1998) Label-switching problem Frühwirth-Schnatter(2001) 42

43 2.6. 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 it ink T ink2 43

44 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) 44

45 2.7. Deterministic breaks If breaks are deterministic, test as in Bai and Perron(1998, 2003) How forecast? - Pesaran and Timmermann(2007) 45

46 2.8. Chib smultiple change-point model P p p 11 p p 22 p p N1,N p N1,N1 1 46

47 2.9. Smooth transition models Teräsvirta (2004) y t expz t1c x 1expz t1 c t1 1 1 x 1expz t1 c t1 2 u t By contrast, in Markov-switching regression the switching weights Prob(s t1 i t1 depend on y t1, y t2,..., y 1 not just z t1. 47

48 3. Economic theory and changes in regime 3.1. Closed-form solution of DSGE s and asset-pricing implications 3.2. Approximating the solution to DSGE s using perturbation methods 3.3. Linear rational expectations models with changes in regime 3.4. Multiple equilibria 3.5. Tipping points and financial crises 3.6. Currency crises and sovereign debt crises 3.7. Changes in policy as the source of changes in regime 48

49 3.1. 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 49

50 Cecchetti, Lam and Mark (1990): log D t log D t1 m st t P t st D t 50

51 Portfolio allocation -Angand Bekaert(2002a); Guidolinand Timmermann(2008) Financial implications of rare-event risk - Evans (1996); Barro(2006) Option pricing -Elliott, Chan and Siu (2005) Term structure of interest rates -Angand Bekaert(2002b); Bansal and Zhou (2002) 51

52 3.2. Approximating the solution to DSGE s using perturbation methods E t ay t1, y t, x t, x t1, t1, t, st1, st 0 y t control variables x t predetermined variables t innovations to exogenous variables s t follows an N-state Markov chain 52

53 Cecchetti, Lam and Mark: y t P t /D t x t lnd t /D t1 st m st, In general we seek solutions of the form y t st x t1, t x t h st x t1, t 53

54 Foerster, et. al. (2014): Sequence of economies indexed by 0 deterministic steady state 1 previous solution y t st x t1, t, x t h st x t1, t, 54

55 x, y are steady-state solution when 0, t 0, st ergodic value from Markov chain. y t y R x st x t1 x R st t R s t x t x H x st x t1 x H st t Hs t 55

56 3.3. 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. 56

57 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 57

58 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 58

59 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 59

60 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 60

61 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. 61

62 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) 62

63 3.4. Multiple equilibria Multiplicity of stable equilibria could itself be of interest -Coordination externalities (Cooper and John, 1988; Cooper, 1994) -Equilibria indexed by expectations (Kurzand Motolese, 2001) Regime-switching model could describe transitions between equilibria - Kirman(1993); Chamley(1999) 63

64 Stock market bubbles - Hall, Psaradakis and Sola (1999) Difficult to distinguish from unobserved fundamentals -Hamilton (1985); Driffilland Sola (1998); Gürkaynak(2008) 64

65 3.5. Tipping points and financial crises In other models, there is a unique equilibrium, but small change in fundamentals can cause big change in outcome -Acemogluand 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 -Brunnermeierand Sannikov(2014); Hamilton (2005); Aseaand Blomberg(1998); Hubrichand Tetlow(2013) 65

66 3.6. 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, Leeperand Walker (2011); Bi (2012) 66

67 3.7. 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) 67