4. Nonlinear Models
ECO374H1
Department of Economics
Summer 2025
Linearity vs Nonlinearity
I Consider a stochastic process {Yt } and an information set
It -1 = {Yt -1 1 Yt -2 1 ...1 Xt -1 1 Xt -2 1 ...}
I We say that {Yt } is linear in mean if the conditional mean of Yt is a linear function of It -1 1 that is
E [Yt jIt -1] = φ1 Yt -1 + φ2 Yt -2 + ... + β1 Xt -1 + β2 Xt -2 + ...
I Under this definition, the ARMA(p1 q) models we have analyzed are linear models
I We say that {Yt } is nonlinear if the conditional mean of Yt is a nonlinear function of It -1
I By nonlinear we mean any functional form that is not linear
Threshold Nonlinearity
I As an example of a nonlinear model, suppose that {Yt } behaves di§erently for Yt > 0 than for Yt ≤ 0 :
where φ11 ≠ φ21
I Equivalently
Yt = φ11 Yt -1I(Yt -1 > 0) + φ21 Yt -1I(Yt -1 ≤ 0) + ε t (1)
where the indicator function I(A) = 1 if statement A is true and I(A) = 0 otherwise
I This model is called the self-exciting threshold autoregressive (SETAR) model with threshold value 0
I Here {Yt } is a mixture of two separate linear processes, which together constitutes a nonlinear process
I For a simulation from the SETAR process, see code file 4a. SETAR Simulation
Nonlinear Dynamics in Data: U.S. Industrial Production
I U.S. industrial production (year-to-year changes) exhibits threshold nonlinearity between recessions (shaded) and expansions
Nonlinear Dynamics in Data: 3-month T-bills
I The U.S. 3-month Treasury bills rate can be viewed as a mixture of two
processes: random walk for low rates and a stationary process for high rates
Linearity vs Nonlinearity
I ACF and PACF only measure linear dependence in {Yt}
I ARMA type models only exploit linear dependence in {Yt} for forecasting
I As such, they can be viewed as first-order linear approximations to nonlinear processes
I For detecting and modeling nonlinear dependence we mainly rely on statistical testing and forecasting performance measures
Nonlinear Models
I We will analyze the following types of nonlinear models:
1. Threshold models
2. Smooth transition models
Threshold Autoregressive Process (TAR)
I The Threshold Autoregressive Process (TAR) takes the general form.
I The model contains r regimes
I There is a separate AR(p) process in each regime
I The threshold variable xt can be any variable in or outside of the model
I When xt is any lag of Yt 1 the TAR process is called self-exciting (SETAR)
SETAR
I In (1), we have introduced SETAR(1) with xt = Yt -11 c1 = 01 and c2 = ∞
I Using the TAR notation of (2) with intercepts, the SETAR(1) model becomes
I Using D = I(Yt -1 ≥ c1) = 1 - I(Yt -1 < c1) rewrite the model (3) as
where φ-0 = φ20 - φ10 and φ-1 = φ21 - φ11
SETAR
I The model (4) corresponds to a regression model
I Assuming ε t is iid 1 we can use OLS to regress Yt on Yt -1 , D 1 and Yt -1 D
I We can test for linearity with H0 : 0 = 0 and φ-1 = 01 which is equivalent to H0 : φ20 = φ 10 and φ21 = φ11
I For application to 3-month U.S. Treasury bill rates, see code file 4b. SETAR Application
Smooth Transition
I The TAR model assumes that the shift from one regime to the next happens abruptly, due to the binary nature of the indicator function I(Yt -1 ≥ c)
I In many cases we would expect a smooth transition from one state to the next, e.g. in macroeconomic variables such as GNP
I The Smooth Transition Autoregressive Model (STAR) can be speciÖed as
Yt = φ0 + φ11 Yt -1 + φ12 Yt -2 + ... + φ1p Yt -p
+ (φ 1 + φ21 Yt -1 + φ22 Yt -2 + ... + φ2p Yt -p ) G (st, g , c) + ε t
where G (st, g , c) is a transition function that is bounded, and continuous in a transition variable st (typically lags of Yt )
I The parameter g captures the speed of transition and c is a threshold parameter
I A popular special case is the STAR(1) model
Yt = φ0 + φ11 Yt -1 + (φ1 + φ21 Yt -1)G (st, g , c) + ε t