A Time-Varying Threshold STAR Model of Unemloyment michael dueker a michael owyang b martin sola c,d a Russell Investments b Federal Reserve Bank of St. Louis c Deartamento de Economia, Universidad Torcuato di Tella d School of Economics, Mathematics Statistics, Birkbeck College, University of London January 2009 Abstract Smooth-transition autoregressive (STAR) models have roven to be worthy cometitors of Markov switching models of regime shifts, but the assumtion of a time-invariant threshold level does not seem realistic it holds back this class of models from reaching their otential usefulness. Indeed, an estimate of a time-varying threshold level of unemloyment, for examle, might serve as a meaningful estimate of the natural rate of unemloyment. More recisely, within a STAR framework, one might call the time-varying threshold the tiing level rate of unemloyment, at which the mean dynamics of the unemloyment rate shift. In addition, once the threshold level is allowed to be time-varying, one can add an error-correction term between the lagged level of unemloyment the lagged threshold level to the autoregressive terms in the STAR model. In this way, the time-varying latent threshold level serves dual roles: as a demarcation between regimes as art of an error-correction term. Keywords: Regime switching, smooth-transition autoregressive model, unemloyment, nonlinear models. JEL Classification: C22; E31; G12. 1
1 Introduction Our starting oint in modeling the rate of unemloyment is the contemoraneous smooth-transition autoregressive model from Dueker, Sola Sagnolo (2007), which weights the two regimes by the ex ante robability that the unemloyment rate will be below/above the contemoraneous value of the threshold: P (y t < y t I t 1, Θ 0 ) P (y t y t I t 1, Θ 1 ), where Θ 0 Θ 1 are the arameters that ertain to the two regimes the resective regime weights are P 0t = P 1t = P (y t < yt I t 1, Θ 0 ) P (y t < yt I t 1, Θ 0 ) + P (y t yt I t 1, Θ 1 ) P (y t yt I t 1, Θ 1 ) P (y t < yt I t 1, Θ 0 ) + P (y t yt I t 1, Θ 1 ) (1) (2) Note that P 0t + P 1t = 1 this way. With the time-varying threshold comes the ossibility of adding an errorcorrection term to a STAR model: y t = P 0t µ 0 + P 1 tµ 1 + i=1 i=1 Φ (i) 0 y t i + Γ 0 (y t 1 y t 1) + Φ (i) 1 y t i + Γ 1 (y t 1 y t 1) + ɛ t We model the time-varying threshold level of the unemloyment rate as an autoregressive rocess, although the coefficient θ could be close to one to accommodate a high level of ersistence in the threshold level: (4) y t = λ + θy t 1 + w t (5) 2
The threshold is endogenous in that its innovation is correlated with the shocks to the observable data: Cov(ɛ t, w t ) 0. the regime-secific covariance matrices are denoted Ω 0 Ω 1. Estimation algorithm for the endogenous, timevarying threshold model Because the vector of latent thresholds enters the regime weights, P 0t P 1t, the model cannot be estimated by maximum likelihood. Nor is it ossible to derive exact conditional distributions for Gibbs samling. Instead, a Metroolis-Hastings algorithm is needed for Bayesian estimation of this model. The key is to have a good, yet tractable, roosal draw for the latent threshold series, {y t } T t=1. The arameter grouings for Markov Chain Monte Carlo estimation of the model are: {yt } T t=1 latent threshold series M-H draw, unscented Kalman filter roosal φ, ρ, µ, γ, λ, θ regression coeffs. M-H draw Normal roosal Ω 0, Ω 1 cov. matrices inverted Wishart Estimation results for U.S. unemloyment We alied the model to the U.S. unemloyment rate since 1968. Figure 1 shows the relationshi we found between the unemloyment rate the estimated time-varying threshold level. We also include out-of-samle forecasts of both the unemloyment rate the threshold level. Starting from November 2008, when the unemloyment rate was 6.7 ercent, the STAR model redicts that the unemloyment rate would reach 7.5 ercent at its eak in November 2009. This rojected eak seems low, but it is imortant to bear in mind that this forecast is coming from a univariate model of the unemloyment rate. With additional information about the current state of the business cycle, the model should be able to rovide accurate forecasts 3
of unemloyment kee the interesting self-referential unemloyment dynamics catured by the model. Aendix: analysis using the unscented Kalman filter We can rewrite the model in the receding section in a state sace reresentation y t = Hz t + ε t, z t = α + g(z t 1 ) + u t (6) where the nonlinear function g(.) contains the cumulative density weighting function P at. Given this state-sace reresentation, inferred values for the latent variable z t can be obtained from the unscented Kalman filter (UKF). The UKF is a nonlinear filter that serves as an alternative to the extended Kalman filter, which uses first-order Taylor-series aroximations to any nonlinear functions in the measurement transition equations. The UKF tracks the state variable by comuting its distribution across a set of deterministic oints called sigma oints. We begin with a set of initial values. We then augment the mean with the exectation of the transition noise z a t 1 t 1 = zt 1 t 1 E u t augment the state covariance P a t 1 t 1 = Pt 1 t 1 0 N,N 0 N,N I N, where s t 1 t 1 P t 1 t 1 are the estimates of the state its covariance matrix at time t 1. Our task is to construct a set of 2L + 1 sigma oints, 4
where L is the dimension of s a t 1 t 1. t 1 t 1 = Let z a t 1 t 1, for = 0 z a t 1 t 1 + ( ) (L + λ) Pt 1 t 1 a, for = 1,..., L i z a t 1 t 1 ( ) (L + λ) Pt 1 t 1 a, for = L + 1,..., 2L define the initial sigma oints, where λ = a 2 (L + κ) L, a κ are user-chosen ( ) arameters that govern the sread scale, resectively. Here, X is the ith column of the lower triangular Cholesky factorization of i the square marix X. Given the set of initial sigma oints, we can then roagate { t 1 t 1} through the transition function g (.) to recover i L t t 1 = g ( t 1 t 1), for = 0,..., 2L. The redicted states covariances can then be extraolated from a weighted sum of the roagated sigma oints P t t 1 = wc =0 The weights are defined as w c = w s = { λ { ẑ t t 1 = =0 w s t t 1 t t 1 ẑ t t 1 t t 1 ẑ t t 1. λ, for = 0 L+λ 1 2(L+λ), for = 1,..., 2L + (1 L+λ a2 + b), for = 0 1, for = 1,..., 2L, 2(L+λ) where a is defined as above b is a arameter that incororates rior information of the distribution. 5
Filter recursions The udating ste roceeds in a similar manner. We can augment the redicted state covariance with the exectation covariance of the measurement noise z a t t 1 = ẑt t 1 v t P a t t 1 = Pt t 1 0 n,n 0 n,n Ω, where Ω is the covariance of the measurement noise. We again comute 2L + 1 sigma oints from t t 1 = z a t t 1, for = 0 z a t t 1 + ( ) (L + λ) Pt t 1 a, for = 1,..., L i z a t t 1 ( ). (L + λ) Pt t 1 a, for = L + 1,..., 2L i L We then roagate the udate sigma oints through the measurement equation: γ t t = H t t 1, = 0,..., 2L. The udated roagation oints, γ t t, are used to construct the redicted measurement error ŷ t = =0 w sγ t t, where the weights are defined as before. We then form the udated state similar to the stard Kalman filter ẑ t t = ẑ t t 1 + K t (y t ŷ t ), where K t is the Kalman gain defined by K t = P yz P 1 yy. 6
Here, P yz defines the cross-covariance P yz = wc =0 P yy is the redicted covariance P yy = wc =0 The udated covariance is defined by Smoothing t t 1 ẑ t t 1 γ t t ŷ t γ t t ŷ t γ t t ŷ t. P t t = P t t 1 K t P yy K t. Multi-move samling of the latent attractor, {s } T t=1, requires backwards samling from one-eriod smoothed inferences of the state vector. Here the unscented Rauch-Tung-Striebel smoother comes into lay. The URTS smoother (Sarkka, 2007) begins by augmenting the unscented Kalman filter with a smoothing ste that recomutes the state estimate. The smoothed state is with covariance matrix P s t ẑ s t = ẑ t + D t ẑ s t+1 ẑ t+1 t = P t + D t P s t+1 P t+1 t D t, where D t is the smoother gain defined by where with P zt,z t+1 = D t = P zt,z t+1 P 1 t+1 t, wc =0 t+1 t ẑ t+1 t t t ẑ t t, t t = t t 1 + K t (y t ŷ t t 1 ). 7
References 1 Julier, Simon J. Jeffery K. Uhlmann. A New Extension of the Kalman Filter to nonlinear Systems, in The Proceedings of AeroSense: The 11th International Symosium on Aerosace/Defense Sensing, Simulation Controls, Multi Sensor Fusion, Tracking Resource Management II, SPIE, 1997. 2 Särkkä, Simo. Unscented Rauch-Tung-Striebel Smoothing, forthcoming IEEE, 2007. 3 Wan, Eric van der Merwe, Rudolh. The Unscented Kalman Filter, in S. Hakin, Kalman Filtering Neural Networks, Wiley, Setember 2001. 8
Unemloyment rate latent threshold level with out-ofsamle forecasts 12 10 8 6 4 2 0 2/1/1968 2/1/1972 2/1/1976 2/1/1980 2/1/1984 2/1/1988 2/1/1992 2/1/1996 2/1/2000 2/1/2004 2/1/2008 ercent Latent threshold Actual Forecasts