Nonlinear Autoregressive Processes with Optimal Properties

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Nonlinear Autoregressive Processes with Optimal Properties F. Blasques S.J. Koopman A. Lucas VU University Amsterdam, Tinbergen Institute, CREATES OxMetrics User Conference, September 2014 Cass Business School, London 1 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Motivation Examples of nonlinear autoregressive (nonlinear AR) models : Treshold AR (TAR) : Tong (1983) y t = γ 1 y t 1 + γ 2 I(y t 2 < γ 3 )y t 1 + u t Smooth transition AR (STAR) : Chan & Tong (1986) and Teräsvirta (1994) y t = γ 4 x t 2 (γ 6 )y t 1 γ 5 [1 x t 2 (γ 6 )] y t 1 + u t where γ i is an unknown coecient, for i = 1,..., 6, I() is an indicator function and x t (γ) = 1 / [1 + exp( γ y t 2 )]. These are examples of nonlinear AR(2) models. 2 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Motivation Nonlinear AR models can also be written as linear AR models with an observation driven time-varying temporal dependence: Treshold AR (TAR) : Tong (1983) y t = ρ t y t 1 + u t, ρ t = γ 1 + γ 2 I(y t 2 < γ 3 ). Smooth transition AR (STAR) : Chan & Tong (1986) and Teräsvirta (1994) y t = ρ t y t 1 + u t, ρ t = γ 4 x t 2 (γ 6 ) + γ 5 [1 x t 2 (γ 6 )] where γ i is an unknown coecient, for i = 1,..., 6, I() is an indicator function and x t (γ) = 1 / [1 + exp( γ y t 2 )]. This is an interesting feature. 3 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Motivation More generally, we can consider a general nonlinear AR model y t = ϕ(y t 1 ; θ) + u t, u t p u (θ) for some "conveniently selected" function ϕ() of the innite past y t 1 := (y t 1, y t 2,...) and parameter vector θ. 4 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Motivation More generally, we can consider a general nonlinear AR model y t = ϕ(y t 1 ; θ) + u t, u t p u (θ) for some "conveniently selected" function ϕ() of the innite past y t 1 := (y t 1, y t 2,...) and parameter vector θ. The linear AR(1) model with time-varying dependence ρ t is y t = ρ t y t 1 + u t, u t p u (θ), ρ t = h(y t 1 ; θ) When h(y t 1 ; θ) is appropriately chosen, the two model equations are a.s. the same: h(y t 1 ; θ) = ϕ(y t 1 ; θ) / y t 1. 4 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Motivation More generally, we can consider a general nonlinear AR model y t = ϕ(y t 1 ; θ) + u t, u t p u (θ) for some "conveniently selected" function ϕ() of the innite past y t 1 := (y t 1, y t 2,...) and parameter vector θ. The linear AR(1) model with time-varying dependence ρ t is y t = ρ t y t 1 + u t, u t p u (θ), ρ t = h(y t 1 ; θ) When h(y t 1 ; θ) is appropriately chosen, the two model equations are a.s. the same: h(y t 1 ; θ) = ϕ(y t 1 ; θ) / y t 1. Alternative is to base the equivalence on (Taylor) expansions. 4 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Motivation What do we take from this equivalence? Rather than focussing on "some" general function ϕ(y t 1 ; θ) in the nonlinear AR model y t = ϕ(y t 1 ; θ) + u t, we consider the linear AR(1) model y t = ρ t y t 1 + u t where ρ t is an observation driven time-varying coecient. It is, perhaps, more practical and empirically more convenient! New nonlinear AR model formulations may arise with empirical relevance. 5 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Score updating How to let the coecient ρ t change in the AR(1) model? We use the local information from the likelihood function to adapt the value for ρ t. For this purpose, we use the score function of the predictive logdensity function at time t : log p(y t y t 1 ; θ) ρ t. For a given ρ t 1 value, score information is useful to determine new ρ t value, when new observation y t becomes available. It requires a specication for predictive density function p(y t y t 1 ; θ), we can depart from Gaussian assumptions. 6 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Our paper We present an observation-driven model specication for the time-varying dependency in autoregressive models. For the AR(1) case we have y t = h(f t ; θ)y t 1 + u t, u t p u (u t ; θ), f t = φ(y t 1, f t 1 ; θ), where h() and φ() are xed functions, both possibly depending on the xed parameter vector θ, with x t = {x t, x t 1, x t 2,...} for x = f, y. The AR(1) model is general and exible but need to specify φ(y t 1, f t 1 ; θ), p u (u t ; θ), h(f t ; θ). 7 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Time-varying temporal dependence in AR(1) model For the AR(1) model, y t = h(f t ; θ)y t 1 + u t, u t p u (u t ; θ), f t = φ(y t 1, f t 1 ; θ), we take the linear updating equation φ(y t 1, f t 1 ; θ) = ω + αs t 1 + βf t 1, s t = s(y t, f t ; θ), where ω, α and β are xed coecients and s t 1 is a deterministic function of past observations. We take s t as the score function of the conditional or predictive log-density function of y t, log p(y t f t, y t 1 ; θ) log p u (u t ; θ), as u t = y t h(f t ; θ)y t 1, with respect to f t. 8 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Time-varying temporal dependence in AR(1) model Our time-varying temporal dependence AR(1) model is given by with score function y t = h(f t ; θ)y t 1 + u t, u t p u (u t ; θ), f t = ω + αs(y t 1, f t 1 ; θ) + βf t 1, s t s(y t, f t ; θ) = log p(y t f t, y t 1 ; θ) f t. In spirit of score models : Creal, Koopman & Lucas (2011,2013) and Harvey (2013). Why the score? It provides optimality properties! In a Kullback-Leibler framework, see later. But what about the choice for p u (u t ; θ) and h(f t ; θ)? 9 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Time-varying temporal dependence in AR(1) model Our time-varying temporal dependence AR(1) model is given by y t = h(f t ; θ)y t 1 + u t, u t p u (u t ; θ), f t = ω + αs t 1 + βf t 1, where score function depends on choice s t = log p u(u t ; θ) f t h(f t ; θ) f t logit(f t ) p u (u t ; θ) Normal X Student's t 10 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Basic Model I The linear Gaussian updating case with score function y t = f t y t 1 + u t, u t N(0, σ 2 u), f t = ω + αs t 1 + βf t 1, s t = [c 0.5(y t f t y t 1 ) 2 /σ 2 u] f t = (y t f t y t 1 )(y t 1 /σ 2 u) = u t y t 1 /σ 2 u. The time-varying autoregressive parameter updating equation is f t = ω + α u t 1y t 2 σ 2 u + βf t 1. 11 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Basic time-varying temporal dependence We have the model y t = f t y t 1 + u t, Interesting interpretation : f t = ω + α u t 1y t 2 σ 2 u + βf t 1. update of f t reacts to error u t 1 multiplied by y t 2 and scaled by σ 2 u. role of y t 2 is to signal whether f t is below or above its mean. update distinguishes role of observed past data and of past parameter value. More interesting/intrinsic updating equations for other p u and h 12 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Models I, II and III Model I is based on Gaussian p u and unity function h(f) = f. Model II is based on Gaussian p u and logistic function for h(f). Model III is based on Student's t p u and unity function h(f). 13 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Basic time-varying temporal dependence Figure: Updating for f t : h(f) = f and p u (u) = N, t. 14 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Nonlinear AR specication Our AR(1) model implies that h(f t ) = y t u t y t 1. In the case of unity function for h(), we have f t = (y t u t )y 1 t 1 and the score-driven updating function becomes f t = ω + αs t 1 + β y t 1 u t 1 y t 2. By substituting this expression into y t = f t y t 1 + u t we obtain y t = ωy t 1 + αs t 1 y t 1 + β y t 1 u t 1 y t 2 y t 1 + u t, 15 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Nonlinear AR specication Our time-varying temporal dependence Model I y t = f t y t 1 + u t, f t = ω + α u t 1y t 2 σ 2 u + βf t 1, with f t 1 = (y t 1 u t 1 )yt 2 1, can be rewritten as y t = ωy t 1 + α y t 1y t 2 u t 1 σ 2 + β y t 1 u t 1 y t 2 y t 1 + u t. It is a nonlinear ARMA(2, 1)! Similar results can be obtained for Models II and III. But expressions become more intricate! 16 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Nonlinear ARMA models We have shown that our basic time-varying temporal dependence model is a nonlinear ARMA model. But what is new? Nonlinear ARMA models have been formulated! Threshold AR Tong (1991) y t = ϕ t y t 1 + u t, ϕ t = ϕ + ϕ I(y t 2 < γ), Smooth Transition Chan & Tong (1986), Teräsvirta (1994) y t = ϕ t y t 1 + u t, ϕ t = γ 1 x t 2 + γ 2 (1 x t 2 ), where x t = [1 + exp( γ 3 y t )] 1. 17 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Comparison with TAR and STAR 18 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Comparison with our basic model 19 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Properties Score function : Familiar entity in econometrics, has nice properties. Stationarity and Ergodicity : Conditions can be established, for both y t and f t. Maximum likelihood : Consistency and Asymptotic Normality, conditions can be established. Optimality : Updating using score provides a step closer to the true path of the time-varying parameter, optimality in the Kullback-Leibler sense. See Blasques, Koopman and Lucas (2014) 20 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Denition I: Realized KL Divergence KL divergence between p( f t ) and p ( f t+1 ; θ ) is given by D KL (p( f t ), p ( f t+1 ; θ )) = p(y t f t ) ln p(y t f t ) p ( y t f t+1 ; θ ) dy t. 21 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Denition I: Realized KL Divergence KL divergence between p( f t ) and p ( f t+1 ; θ ) is given by D KL (p( f t ), p ( f t+1 ; θ )) = p(y t f t ) ln p(y t f t ) p ( y t f t+1 ; θ ) dy t. The realized KL variation t 1 RKL of a parameter update from f t to f t+1 is dened as t 1 RKL = D KL (p( f t ), p ( f t+1 ; θ )) D KL (p( f t ), p ( f t ; θ )) 21 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Denition II: Conditionally Expected KL Divergence An optimal updating scheme, while subject to randomness, should have tendency to move in correct direction: On average, the KL divergence should reduce in expectation. 22 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Denition II: Conditionally Expected KL Divergence An optimal updating scheme, while subject to randomness, should have tendency to move in correct direction: On average, the KL divergence should reduce in expectation. The conditionally expected KL (CKL) variation of a parameter update from f t F to f t+1 F is given by [ t 1 CKL = q( f t+1 f t, f t ; θ) p(y f t ) ln p(y f ] t ; θ) p(y f t+1 ; θ) dy d f t+1, F Y where q( f t+1 f t, f t ; θ) denotes the density of f t+1 conditional on both f t and f t. For a given p t, an update is CKL optimal if and only if t 1 CKL 0. 22 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Condition for RKL Our basic time-varying temporal dependence model y t = f t y t 1 + u t, f t = ω + α u t 1y t 2 σ 2 u + βf t 1, we obtain RKL optimality under the condition α > σu 2 ω + (β 1) f t (y t 1 f t 1 y t 2 )y t 2, The new score information should have locally sucient impact on the updating for f t. A similar but dierent condition is derived for CKL optimality. 23 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

24 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Empirical illustration: Unemployment Insurance Claims We analyze the growth rate of US seasonally adjusted weekly Unemployment Insurance Claims (UIC) for roughly the last ve decades. Meyer (1995), Anderson & Meyer (1997, 2000), Hopenhayn & Nicolini (1997) and Ashenfelter (2005) have studied the UIC series. The importance of forecasting UIC has been highlighted by Gavin & Kliesen (2002): UIC is a leading indicator for several labor market conditions: how they can be used to forecasting GDP growth rates. Here we consider various models and do some comparisons amongst them. 25 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Empirical illustration Unemployment Insurance Claims: Model Comparison Model I TAR STAR AR(2) AR(5) LL 6744 6736 6737 6439 6968 AIC -13478-13462 -13464-12870 -13921 RMSE 0.750 0.752 0.752 0.848 1.20 26 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Empirical results 27 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

Conclusions We have introduced time-varying temporal dependence in the AR(1) model y t = f t y t 1 + u t, f t = ω + α u t 1y t 2 σ 2 u + βf t 1, an observation-driven approach to time-varying autoregressive coecient: GAS model is eective! reduced form : nonlinear ARMA models they can be compared with TAR and STAR models the ltered estimate f t has optimality properties in the KL sense when based on the score function! we provide some Monte Carlo evidence an empirical illustration for UIC is presented 28 / 28 Blasques, Koopman and Lucas Nonlinear Autoregressive Processes

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