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Nonlinearity in Econometrics

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

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Nonlinearity in Econometrics

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

  2. 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)

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

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

  5. 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.:

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

  7. 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)

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

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

  10. 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

  11. 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?

  12. 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

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

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

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

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

  17. Rejection probabilities of the TLG-2 test

  18. Rejection probabilities of the LWG test

  19. Rejection probabilities of the RBF test

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

  21. 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

  22. 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

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

  24. Size and power: ADF test, SETAR model

  25. Size and power: RBF test, SETAR model

  26. 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?

  27. 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.

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