Lecture 9: Markov Switching Models

Transcription

1 Lecture 9: Markov Switching Models Prof. Massimo Guidolin Financial Econometrics Winter/Spring 2018

2 Overview Defining a Markov Switching VAR model Structure and mechanics of Markov Switching: from univariate to multivariate models Understanding MS models through simulations MS models as normal mixtures The properties of Markov chain in MS models Filtered, smoothed, and predicted state probabilities 2

3 Motivation and Introduction One of the worst problems often plaguing econometric models regressions, ARMA, VAR, GARCH, etc. is their instability, the fact that the estimated parametric relations suddenly change over time o Famous to damage effectiveness of economic policy o Also worrisome in financial forecasting and risk management o D Four approaches/reactions: 1 Happy go lucky, ignore it and hope for the best (dashed red lines) 2 Test for breaks and shifts and just use data after most recent break 3

4 Definition of a Markov Switching VAR Model 3 Use rolling window estimation schemes o However, although popular, RW schemes are optimal only under specific assumptions on how instability occurs o RW scheme is optimal only when every period there is a break odd! 4 Model and forecast instability, when recurrent in the form of regimes In regime switching models (RSM), state variables govern how part or all parameters of a time series framework may change over time In a specific type of RSM Markov switching models (MSM) the state is latent and follows a simple (finite state) Markov chain o MC process = N-branch tree in which the probs. depend on finite history 4

5 Structure of Markov Switching VAR Models o When M = 1 (first-order MC) and we call the KxK matrix collecting the probabilities Prob. of switching from state i to state j the transition matrix of the K-state Markov process Regimes are unobservable (latent) even with unlimited time series information, estimation never reveal the actual, true state S t+1 o The same sample data concerning the N variables in y t are used also to produce inferences on the sample path followed by S t o Although intuition will be sought after, no attempt is made to provide a formal model of either the reason that regime changes occur or to explain the timing of such changes o Assume the absence of roots outside the unit circle in all regimes o In the definition, the Nx1 vector μ St+1 collects the N regime-dependent intercepts, while the p alternative NxN A j,st+1 (j = 1,, p) vector autoregressive matrices capture regime-dependent VAR effects o With p VAR lags and K regimes, there are a total of pk matrices to deal with, each potentially containing (unless restrictions) N 2 parameters 5

6 Structure of Markov Switching VAR Models o The (lower triangular) matrix represents the factor applicable to latent state S t+1 in a state-dependent Cholesky factorization of Conditioning information set o Conditionally on the state, MSIVARH(K, p) defines a standard Gaussian reduced form VAR(p); this is the case when we take S t+1 is treated as given and observable (we shall not of course) o In applications, K = 2 tends to be common, although not compelling o Especially with daily/weekly series common to support MSIH(K) (to be precise, MSIH(K,0)): o p = 0 may work at all frequencies because when K 2, possible that need of p 1 in single-state VARs arises from omission of regimes o The general model simplifies in univariate applications, when N = 1: o For instance, consider monthly international excess stock returns for the sample 1986: :12) of US and Japanese excess equity returns (denominated in US dollars) and rate of change in the VXO index, N = 3 6

7 One Application to International Equity Returns o Analyze for each series what type of first-order MSIARH(K, p) Constant Expected Returns Model ARMA Models The table for Japanese excess return is in Appendix A 7

8 One Application to International Equity Returns o The precise model favored by each information criterion in the case of each series may differ; in the end, simple heteroskedastic MSIH(2,0) model with no AR components always picked, for all three series, by BIC P-values Nonpersistent o For the transition probs, p-values possibile but trickier o Stock markets feature typical bull (== high-risk premia and low volatility) and bear (low or even 0, in the sense of not statistically significantly, risk premia and high volatility) phases o Both regimes are persistent for both US and Japan, Pr S t = bull S t-1 = bull is btw and 0.98, and Pr S t = bear S t-1 = bear btw and

9 One Application to International Equity Returns o In the case of VXO, state-dependent means are never precisely estimated; in the second regime, the volatility of implicit volatility is almost double than in the former state o When VIX-like volatility falls, it does so slowly and following a low variability path, while when it increases, it does so in an erratic way o Given that all the individual series contain regimes how many Markov states should we expect when the series are jointly modeled? o Naïve to expect K = 2, because the univariate state probability series above are not sufficiently synchronized o Their sample Spearman rank correlations are 0.46, 0.05, and

10 One Application to International Equity Returns o AIC and Hannan-Quinn (H-Q) converge on the choice of a rather richly parameterized MSIVARH(3,1) saturation ratio just above threshold o Unsurprisingly, the number of regimes equals three, an attempt to accommodate the different features of the state processes 10

11 One Application to International Equity Returns 11

12 Simulating from MS Models o While from any of the 3 regimes it is possible to switch to any other, this admits one exception as Pr S t = 1 S t-1 = 2 is estimated to be 0 For large N, MSVAR models are often richly parameterized, with a total number of parameters of: o K(K 1) is the elements that can be estimated from the transition matrix, when by-row summing up constraints are taken into account o For instance, for K = 2, N = 8, and p = 1 (not such an extreme case, see e.g., Guidolin and Ono, 2006), this implies the estimation of 218 parameters less than recommendable saturation ratios are possible o ML estimation may pose serious numerical as well as statistical problems: (i) the log-likelihood may present flat regions so that convergence of standard algorithms becomes impossible; (ii) identification issues may appear numerical algorithms may get confused Consider the simple case of We use sets of 1000 identical simulated shocks to better understand what a MSIARH model can do in terms of plausibility of the resulting time series o When possible, calibrate the selected parameters to US monthly data 12

13 Simulating from MS Models K = 1 K = 2 Simulated regimes o Simulated MSI(2) yields an unconditional mean of 6% and volatility of 19% per year as Gaussian IID o Most observers would detect the presence of more structure in the rightmost vs. leftmost plot, but could not exactly detect an MSI(2) o Some additional variability would be guessed, but this would be incorrect, as the 2 series are generated to have identical variance o Key driver of the appearance of simulations is persistence of Markov chain 13

14 Simulating from MS Models Leverage o On the right, appearance that may remind some readers of the occurrence of frequent (negative) jumps in returns 14

15 Simulating from MS Models o In the rightmost plot, eventually unit root is bent to stationarity by the mixing provided by the ergodic Markov chain o Leftmost plot shows highly visible intercept switches, around which we then find the typical no structure patterns of a white noise o In the rightmost plot, same nonlinear persistence (low), the presence of near-unit roots in each of the two regimes becomes visible and tends to cloud the fact that there are frequent regime shifts 15

16 MS Models Are Normal Mixtures A mixture of normal densities is a weighted sum of normal densities, in which the weights are themselves random variables and may change over time o In the case of MS, weights are random state probabilities over time o Mixtures of normal distributions provide a flexible family that can be used to approximate many distributions, capturing skewness and excess kurtosis as sources of non-normality (even multi-modality) o E.g., in an MS model, variance is not simply the average of the variances across the two regimes: differences in means also impart an effect because the switch to a new regime contributes to volatility Skewness > 0 Skewness < 0 Excess kurtosis >0 16

17 MS Regressions Useful to generalize framework, when specialized to N = 1, to reflect a more general form that also includes exogenous, fixed predictors and predictors whose coefficient does not follow an MS process: This is a MS regression o Let s forecast monthly Japanese excess aggregate stock returns using one lag of the same, one lag of US excess stock returns, and one lag of S&P 100 implied volatility 17

18 MS Regressions: One Example o All criteria unanimously select a MS regression in which all coefficients are time invariant but the standard error of regression is MS o Extensions to 3 regimes are rejected by the information criteria Standard regression 2-state MS regression o What makes the MS regression superior to a simple regression is the regime shifts in standard errors that as we expect when heteroskedasticity is dealt with allow us to obtain more precise estimates MS models are defined as driven by a hidden, discrete Markov state that is also latent, ergodic, and irreducible Although they can be generalized, most MS models are estimated assuming a homogeneous, first-order Markov chain 18

19 Markov Chain Processes in MS Models 1 Homogeneous, as, the prob. of transition to state j does not depend on past values of y 2 First-order, meaning that, or that all the memory of the past of series is retained by just one lag of S t o S t is latent because it cannot be extracted from the data with perfect precision, but at most the time series of the states may be inferred from the observed, available data o Ergodicity existence of a stationary Kx1 vector of probs satisfying called ergodic or long-run unconditional probabilities o All information needed to compute is in transposed transition matrix o If you start the system from a configuration of state probabilities equal to, then your prediction for the probabilities of the regimes oneperiod forward is identical MS model had reached a steady-state o It is the entire matrix P that matters to compute the ergodic probabilities and not only the values on its main diagonal o Given estimates of the stayer probs, the average estimated duration is: 19

20 Markov Chain Processes in MS Models o can also be interpreted as the average, long-run time of occupation of the different regimes by the MC: o Irreducibility of an MC implies that > 0, meaning that all regimes are possible and remain possible over time and no absorbing states or cycles among states exist o When K = 3, the transition matrix o implies that it is impossible to reach state 3 from the other two states o As soon as one leaves regime 3, which will occur almost surely if p 33 < 1, it becomes impossible to ever return again to state 3 o The third element of will have to be 0 as o The lecture notes show that is the eigenvector of P associated with the unit eigenvalue; there is always a unit eigenvalue as P rows sum to 1 o For instance, in the case of this P, the eigenvalues are 1, 0.87, and 0.74 and the first eigenvector is [ ] j-th element of 20

21 o As one would have expected from its persistence, the tri-variate system spends on average almost 60% of the time in the second regime o However, in spite of their very low persistence, regimes 1 and 3 also occur on average 16% and 25% of the time; these positive rates at which they are visited are helped by the fact that regimes 1 and 3 also communicate with each other o The average durations of the three regimes are 1.7, 6.3, and 1.4 months 21 Markov Chain Processes in MS Models o This eigenvector is not yet because it fails to have unit length o Now sufficient to scale the eigenvector to have unit length simply divide its entries by their sum , resulting in = [ ] o In the special case of K = 2, one obtains explicit solutions for the ergodic probabilities: o In our earlier international equity return application, the estimated transition matrix is:

22 Inference on the State Process in MS Models Several types of inferences on the state S t can be derived from MS o The fact that one needs to use and to extract inferences concerning the dynamics of regimes over time (technically, concerning ) derives from the latent nature of regimes in a MS model o The following notion is instead useful in forecasting problems 22

23 Inference on the State Process in MS Models o The filtered probs are the product of a limited information recursion o Once has been calculated, the lecture notes describe an algorithm by Kim to recover the sequence of smoothed probs o The difference btw. filtered and smoothed probs. is similar to asking (i) Given what I know about the weather in the past few weeks, what is chance of recording a high temperature today (also given observed conditions today)? This requires a real-time, recursive assessment, vs. (ii) Given the information on the weather in the past 12 months and up to today, what was the chance of a high temperature being recorded 4 months ago? This requires a full-information, but backward-looking assessment 23

24 Filtered and smoothed probs. The State Process in MS Models: One Example o In finance, we operate in real time and focus on forecasting so that we tend to care more for filtered probabilities than for smoothed ones o The two concepts coincide by construction at the end of the sample o In our example concerning monthly Japanese excess stock returns: Predicted probs. 24

25 Appendix A: Model Selection for Japanese Data 25