Nonlinear Econometrics

Introduction to Nonlinear Econometrics

Dirk Nachbar∗ March 1, 2006 ∗ [email protected]

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Outline 1. 2. 3. 4. 5.

Introduction Methodology Application Conclusion References

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1. Introduction Assumptions Since we all studied economics we touched on basic econometrics. Some of you might find this boring. Rationale As government economists we often make predictions about certain outcomes using some econometric relationships. These relationship are usually linear. But as we live in a complex world many variables are not linear (think of business cycles). Since there is a large literature on nonlinear models which can be scary for most of us, I restrict myself here to some simple nonlinear models, namely Threshold Autoregressive Models (TAR).

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2. Methodology We all know autoregressive models (AR) in which past values predict future values: yt = φ 0 +

p X

φi yt−i + ²t,

(1)

i=1

where E[²t ] = 0, E[²2t ] = σ 2 and E[²t ²s ] = 0

∀s 6= t.

If absolute values of the coefficients are above 1 then we have a random walk. We can integrate the series.

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Self-exciting TAR (SETAR)

½

yt−1 ≤ c, yt−1 > c,

(2)

yt = (φ0,1 + φ1,1yt−1 )(1 − I[yt−1 > c]) +(φ0,2 + φ1,2 yt−1 )I[yt−1 ≤ c] + ²t

(3) (4)

yt =

φ0,1 + φ1,1 yt−1 + ²t if φ0,2 + φ1,2 yt−1 + ²t if

This can also be written as

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Logistic STAR yt = (φ0,1 + φ1,1 yt−1 )(1 − G(yt−1;γ,c )) +(φ0,2 + φ1,2yt−1 )G(yt−1 ; γ, c) + ²t,

(5) (6)

where G(yt−1 ; γ, c) = (1 + exp(−γ[yt−1 − c]))−1 . When γ → 0 then G becomes a constant (0.5).

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Estimation We need to estimate the coefficients ρ and c. We can do this by sequential conditional least squares. Rewrite the SETAR as: yt = φ1 xI[yt−1 ≤ c] + φ2 xI[yt−1 > c] + ²t

(7)

where x = (1, yt−1, xt−2 , .., yt−p ). Then à n !−1 à n ! X X ˆ= φ x(c)x(c)0 x(c)yt , t=1

(8)

t=1

where x(c) = (x0 I[yt−1 ≤ c], x0 I[yt−1 > c])0 . This is similar to the standard OLS estimator: (XX 0 )−1 (XY ). The threshold can be obtained: ˆ c = arg min σ ˆ2

(9)

c∈C

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3. Application I generated a SETAR(2) model in Excel: Real Est.

a0 0.5 0.66

a1 0.8 0.67

a2 -0.5 -0.46

b0 0 -0.02

b1 0.8 0.78

b2 -0.2 -0.23

C 0 -0.01

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4. Conclusion Nonlinear is fun! Relevant in Policy??

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5. Reference Franses and van Dijk. Non-linear time series models in empirical finance. Cambridge UP. 2000

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