Nonlinearity in Econometrics Andrew P. Blake HKMA/CCBS May 2004

Nonlinearities in Economics ‘Functional’ nonlinearity Utility functions, production functions Zero-bound constraint, Phillips curves Nonlinear time series Thresholds - exchange rate bands Markov-switching behaviour Chaos Rare (and not usually very interesting)

Defining nonlinearity Nonlinear ‘in mean’ (Lee, White & Granger 1993) Null hypothesis Alternative

Are unit roots linear processes? Dickey-Fuller test usually conducted is: An alternative is an ergodic linear or nonlinear process:

Is ARCH a nonlinear process? Not nonlinear ‘in mean’ Forecast value unaffected Coefficient estimates unbiased but inefficient May be nonlinear in argument of conditional variance, i.e.:

A test for nonlinearity Estimate augmented model: Construct Wald test of significance of : where R is a selector matrix, W the regressors

Testing in practice Need to specify: Appropriate nonlinear function, Number of extra functions estimated, q Nonlinear function needs to be ‘general’ Has to capture a wide variety of potential nonlinearities Needs to be straightforward to implement (estimation procedure, parametric choices)

Choosing an appropriate function Power functions: RESET, TLG (1993) Logistic function: LWG (1993) Radial basis function: BK (2000, 2003a,b)

All you ever wanted to know about artificial neural networks…. …but were afraid to ask

Design problems for an ANN test Power functions Choose an expansion Logistic function Choose number of logistic functions Randomly generate coefficients Identifies under the null Radial basis function Use information criterion to choose RBFs by significance - Bootstrap problem

How good is a test? Evaluate the tests by Monte Carlo Size Power General problem, no analytic results Small sample distributions unknown Size What is the probability of Type 1 error? Are the nominal and actual sizes the same? Power What is the probability of Type 2 error? Is it powerful against different models?

1. Neglected nonlinearity (BK, 2003c) Evaluate the size/power characteristics Monte Carlo ‘design’ Set up linear model (size) Set up appropriate nonlinear models (power) Look at sample size effects ‘Bad’ Monte Carlo design can give misleading results

Self Exciting Threshold AR Models SETAR : Tong (1978) Model 1 Model 2

Smooth Transition AR models STAR models: Chan and Tong (1986) Model 1: Model 2: ESTAR LSTAR

Markov Switching Models Hamilton (1989): St is a Markov chain Model 1 Model 2

Bilinear Models Granger and Anderson (1978), common in the finance literature: Model 1 Model 2

Rejection probabilities of the TLG-2 test

Rejection probabilities of the LWG test

Rejection probabilities of the RBF test

2. Testing for ARCH (BK, 2000) Following Peguin-Feisolle (1999): No ARCH (size): ARCH (power): Other complex ARCH models tested

3. Does neglected nonlinearity look like ARCH? (BK, 2003c) Often assume that there is a linear model when testing for ARCH effects (we did!) Neglected nonlinearity might induce variation in the conditional variance ARCH ‘powerful’ against variety of mis-specified models Try to construct ‘nonlinearity robust’ ARCH test

Nonlinearity robust ARCH tests Complicated problem, as difficult to know what to do We propose a ‘nonlinear filter’, i.e. fit a neural network model and test the residuals Lots of options, possibilities, pitfalls Turns out we can find a good test: Filter using RBF, AIC Test using Engle’s LM test

4. Nonlinear unit root testing (BK, 2003a) SETAR model again: Nonlinear 6: Nonlinear 7: Nonlinear 8:

Size and power: ADF test, SETAR model

Size and power: RBF test, SETAR model

Conclusions on nonlinearity testing Nonlinearity testing is related to other forms of mis-specification Structural breaks are a type of nonlinearity Difficult to detect nonlinearity of the forms we often model - Markov switching, for example ‘Too many’ unit roots - need more power against nonlinear alternatives in general ‘Too much’ ARCH Neural networks weren’t that hard, were they?

References Blake, A.P. & G. Kapetanios (2000) ‘A radial basis function artificial neural network test for ARCH’, Economic Letters 69(1), 15-23. Blake, A.P. & G. Kapetanios (2003a) ‘Pure significance tests of the unit root hypothesis against nonlinear alternatives’, Journal of Time Series Analysis 24(3), 253-267. Blake, A.P. & G. Kapetanios (2003b) ‘A radial basis function artificial neural network test for neglected nonlinearity’, The Econometrics Journal 6(2), 357-373. Blake, A.P. & G. Kapetanios (2003c) ‘Testing for ARCH in the presence of nonlinearity of unknown form in the conditional mean’, Queen Mary, University of London, Department of Economics Working Paper No. 496. Blake, A.P. & G. Kapetanios (2004) ‘Testing for neglected nonlinearity in cointegrating relationships’, QMUL, Dept. Economics WP No. 508.