Testing for Smooth Structural Changes in Time Series Models via Nonparametric Regression Bin Chen and Yongmiao Hong forthcoming in Econometrica

November 27, 2011

1.Ideas of the paper

Design a consistent test for a broad range of structural instabilities by using a special estimator. Usually there is no prior information about the alternative — Power vs Generality. We need a general estimator of the coefficients — a local linear estimator. (Robinson 1989, 1991, Cai 2007).

2.Hypothesis—The model

DGP: Yt = Xt′ at + ǫt for t = 1, . . . , T A simple linear model E(ǫt |Xt ) = 0

Xt can be exogenous or predetermined

2.Hypothesis—Null Hypothesis

H0 : at = a for some constant vector and for all t. Under H0 , a can be consistently estimated by OLS: a ˆ = arg min a

T X t=1

(Yt − Xt′ a)2

2.Hypothesis—Alternative Hypothesis

HA : at is time varing. Single break: Chow (1960), Andrews (1993). Multiple breaks: Bai (1998). Deterministic trend model: Hall and Hart (1990). Smooth transition regression (STR): Lin and Terasvirta (1994). To cover a wide range of alternatives, consider: Yt = Xt′ a(t/T ) + ǫt for t = 1, . . . , T where a : [0, 1] → Rd , is a unknown smooth function except for a finite number of points on [0, 1].

3.Local linear estimation For at , consider the following local liner estimator (Robinson 1991, Cai 2007). t+[T h]

min

β∈R2d

X

s=t−[T h]

h s − t
i2 kst Ys − a′o Xs − a′1 Xs T

t+[T h]

=

X

s=t−[T h]

where

i2 h kst Ys − β ′ Zst

s − t
, and k(·) is a density function. T h = h(T ) is a bandwidth: h → 0 and T h → ∞ s − t
′ ′ Xs ] β ′ = [a′0 , a′1 ] and Zst = [Xs′ , T a ˆt = a ˆo kst = k

4.Test statistics Define:

T X ˆ= 1 Q (X ′ a ˆt − Xt′ a ˆ)2 T t=1 t

Then the statistics are defined as: q √
ˆH ˆ − AˆH / B ˆ = T hQ H where:

ˆM ˆ −1 ) AˆH = h−1/2 CA trace(Ω ˆH = 4CB trace(M ˆ −1 Ω ˆM ˆ −1 Ω) ˆ B ˆ = T −1 M

T X

Xt Xt′

t=1

CA and CB depend only on k(·).

ˆ = T −1 Ω

T X t=1

ǫˆ2t Xt Xt′

5.Asymptotic distributions—Assumptions

Assumptions: 1 2 3 4 5

{Xt , ǫt } is a vector a stationary β−mixing processes . {ǫt } is a martingale difference sequence and E(ǫ2t ) = σ 2 .

E(Xt Xt′ ) is finite and positive definite; E(Xit8 ) < ∞; E(Yt8 ) < ∞. k(·) is a symmetric bounded probability density function. h = cT −λ for 0 < λ < 1.

5.Asymptotic distributions

Suppose above Assumptions and H0 hold, then ˆ → N (0, 1) as T → ∞. 1 H 2 Suppose E(ǫ2t |Xt ) = σ 2 , then AˆH = h−1/2 dCA σ ˆ 2 and 4 ˆH = 4dCA σ B ˆ