Estimating Markov-switching regression models in Stata
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1 Estimating Markov-switching regression models in Stata Ashish Rajbhandari Senior Econometrician StataCorp LP Stata Conference 2015 Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
2 ARMA models Time series data are autocorrelated due to the dependence with past values. Autoregressive moving average (ARMA) class of models is a popular tool to model such autocorrelations. The AR part models the current value as a weighted average of past values with some error. y t = φy t 1 + ε t where y t is the observed series φ is the autoregressive parameter ε t is an IID error with mean 0 and variance σ 2 Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
3 ARMA(1,1) model The MA part models the current value as a weighted average of past errors. y t = ε t + θε t 1 where θ is the moving average parameter. The AR and MA models generate completely different autocorrelations. Combining these lead to a flexible way to capture various correlation patterns observed in time series data. y t = φy t 1 + ε t + θε t 1 Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
4 Linear ARMA models Current value of the series is linearly dependent on past values The parameters do not change throughout the sample This precludes many interesting features observed in the data Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
5 Examples In economics, the average growth rate of gross domestic product (GDP) tend to be higher in expansions than in recessions. Furthermore, expansions tend to last longer than recessions In finance, stock returns display periods of high and low volatility over the course of years In public health, incidence of infectious disease tend be different under epidemic and non-epidemic states Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
6 Nonlinear models In all these examples, the dynamics are state-dependent. The states may be recession and expansion, high volatility and low volatility, or epidemic and non-epidemic states Parameters may be changing according to the states Nonlinear models aim to characterize such features observed in the data Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
7 Markov-switching model Hamilton (1989) Finite number of unobserved states Suppose there are two states 1 and 2 Let s t denote a random variable such that s t = 1 or s t = 2 at any time s t follows a first-order Markov process Current value of s t depends only on the immediate past value We do not know which state the process is in but can only estimate the probabilities The process can switch between states repeatedly over the sample Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
8 Features Estimate the state-dependent parameters Estimate transition probabilities P(s t = j s t 1 = i) = p ij Probability of transitioning from state i to state j Estimate the expected duration of a state Estimate state-specific predictions Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
9 Background Consider the following state-dependent AR(1) model y t = µ st + φ st y t 1 + ε t where ε t N(0, σ 2 s t ) s t is discrete and denotes the state at time t The parameters µ, φ, and σ 2 are state-dependent The number of states are imposed apriori For example, a two-state model can be expressed as { µ 1 + φ 1 y t 1 + ε t,1 if s t = 1 y t = µ 2 + φ 2 y t 1 + ε t,2 if s t = 2 Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
10 Assumptions on the state variable Recall the two-state model { µ 1 + φ 1 y t 1 + ε t,1 if s t = 1 y t = µ 2 + φ 2 y t 1 + ε t,2 if s t = 2 If the timing when the process switches states is known, we could Create indicator variables to estimate the parameters in different states. For example economic crisis may alter the dynamics of a macroeconomic variable. Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
11 States are unobserved s t is drawn randomly every period from a discrete probability distribution Switching regresssion model The realization of s t at each period are independent from that of the previous period s t follows a first-order Markov process The current realization of the state depends only on the immediate past s t is autocorrelated Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
12 mswitch regression command in Stata Markov-switching autoregression mswitch ar depvar [ nonswitch varlist ] [ if ] [ in ], ar(numlist) [ options ] Markov-switching dynamic regression mswitch dr depvar [ nonswitch varlist ] [ if ] [ in ] [, options ] Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
13 MSAR with 4 lags Hamilton (1989) models the quarterly growth rate of real GNP as a two state model The dataset spans the period 1951q1-1984q4 The states are expansion and recession rgnp t = µ st + φ 1 (rgnp t 1 µ st 1 ) + φ 2 (rgnp t 2 µ st 2 )+ φ 3 (rgnp t 3 µ st 3 ) + φ 4 (rgnp t 4 µ st 4 ) + ε t Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
14 Quarterly growth rate of US RGNP 1950q1 1951q2 1952q3 1953q4 1955q1 1956q2 1957q3 1958q4 1960q1 1961q2 1962q3 1963q4 1965q1 1966q2 1967q3 1968q4 1970q1 1971q2 1972q3 1973q4 1975q1 1976q2 1977q3 1978q4 1980q1 1981q2 1982q3 1983q4 1985q recession growth rate US RGNP Figure : Quarterly growth rate of US RGNP Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
15 Markov-switching autoregression. mswitch ar rgnp, ar(1/4) nolog Performing EM optimization: Performing gradient-based optimization: Markov-switching autoregression Sample: 1952q2-1984q4 No. of obs = 131 Number of states = 2 AIC = Unconditional probabilities: transition HQIC = SBIC = Log likelihood = rgnp rgnp Coef. Std. Err. z P> z [95% Conf. Interval] ar L L L L State1 _cons State2 _cons sigma p p Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
16 Transition probabilities State 1 is recession and State 2 is expansion. Let P denote a transition probability matrix for 2 states. The elements of P are [ ] [ ] p11 p P = = p 21 p such that j p ij = 1 for i,j = 1,2. p 11 denotes the probability of transitioning to recession in the next period given that the current state is in recession. Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
17 Predicting the probability of recession q1 1951q2 1952q3 1953q4 1955q1 1956q2 1957q3 1958q4 1960q1 1961q2 1962q3 1963q4 1965q1 1966q2 1967q3 1968q4 1970q1 1971q2 1972q3 1973q4 1975q1 1976q2 1977q3 1978q4 1980q1 1981q2 1982q3 1983q4 1985q1 date (quarters) recession probability of recession Figure : Probability of recession Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
18 Expected duration Compute the expected duration the series spends in a state Let D i denote the duration of state i D i follows a geometric distribution The expected duration is E[D i ] = 1 1 p ii The closer p ii is to 1, the higher is the expected duration of state i Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
19 Estimating duration of a state. estat duration Number of obs = 131 Expected Duration Estimate Std. Err. [95% Conf. Interval] State State Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
20 Equivalent AR specifications Consider the following equivalent AR(1) models: y t δ = φ(y t 1 δ) + ε t y t = µ + φy t 1 + ε t The unconditional means for the above models are related: δ = µ 1 φ Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
21 MSAR and MSDR specifications This equivalence is not possible if the mean is state-dependent y t = δ st + φ(y t 1 δ st 1 ) + ε t y t = µ st + φy t 1 + ε t (AR) (DR) A one time change in the state leads to an immediate shift in the mean level in the AR specification. A one time change in the state leads to the mean level changing smoothly over several time periods in the DR specification. Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
22 State vector of MSAR The observed series depends on the value of states at time t and t 1. A two-state Markov process becomes a four-state Markov process. In general, AR specification increases the state vector by the factor K p+1, where p is the number of lags. Used for modeling data with smaller frequency such as quarterly, annual, etc. Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
23 Markov-switching model of interest rates interest rate q1 1967q3 1980q1 1992q3 2005q1 date (quarters) Figure : Short term interest rate Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
24 Estimating interest rates Estimate using data for the period 1955q3-2005q4 Assume the following specification for interest rates where intrate is the interest rate e st N(0, σ 2 s t ) µ and σ 2 is state-dependent intrate t = µ st + e st Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
25 Estimate the model using mswitch dr. mswitch dr intrate, varswitch nolog Performing EM optimization: Performing gradient-based optimization: Markov-switching dynamic regression Sample: 1954q3-2005q4 No. of obs = 206 Number of states = 2 AIC = Unconditional probabilities: transition HQIC = SBIC = Log likelihood = intrate Coef. Std. Err. z P> z [95% Conf. Interval] State1 _cons State2 _cons sigma sigma p p Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
26 Predicted probability of State q1 1967q3 1980q1 1992q3 2005q1 date (quarters) stdintrate probability of State 2 Figure : Predicted probabilities using MSDR model Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
27 Dynamic forecasting with MSAR Estimate using data for the period 1955q3-1999q4 Assume the following specification for interest rates intrate t = µ st + ρ intrate t 1 + φ st inflation t + γ st ogap t + e t where intrate is the interest rate inflation is the inflation rate ogap is the output gap e t N(0, σ 2 ) ρ is constant µ, φ, and γ are state-dependent Out-of-sample forecasting from period 2000q1-2007q1 Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
28 Estimate the model using mswitch dr. mswitch dr intrate L.intrate if tin(,1999q4), switch(inflation ogap) nolog Performing EM optimization: Performing gradient-based optimization: Markov-switching dynamic regression Sample: 1955q3-1999q4 No. of obs = 178 Number of states = 2 AIC = Unconditional probabilities: transition HQIC = SBIC = Log likelihood = intrate Coef. Std. Err. z P> z [95% Conf. Interval] intrate intrate L State1 inflation ogap _cons State2 inflation ogap _cons sigma p p Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
29 Out-of-sample dynamic forecasts q3 2002q1 2003q3 2005q1 2006q3 date (quarters) interest rate forecasts in State 1 forecasts in State 2 weighted forecasts Figure : Forecasts using MSDR model Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
30 Thank you! Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
31 Hamilton, J. D. (1989), A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57(2), Ashish Rajbhandari (StataCorp LP) Markov-switching regression Stata Conference / 31
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