Exploring Nonlinearity in Econometrics: Methods and Implications
This paper by Andrew P. Blake investigates nonlinearity in econometrics, covering crucial concepts like utility functions, production functions, the Phillips curve, and nonlinear time series. It discusses various methods for detecting nonlinearity, including Markov-switching models, threshold effects, and chaos theory, while addressing the challenges of testing for ARCH effects. By emphasizing the need for appropriate nonlinear functions and their estimation, Blake provides insights into how ignoring nonlinearity can mislead economic modeling and forecasting.
Exploring Nonlinearity in Econometrics: Methods and Implications
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Presentation Transcript
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 • 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
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:
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.
