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 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?

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

22. 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

23. 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

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

25. Size and power: ADF test, SETAR model

26. Size and power: RBF test, SETAR model

27. 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?

28. References
Blake, A.P. & G. Kapetanios (2000) ‘A radial basis function artificial neural network test for ARCH’, Economic Letters 69(1),
Blake, A.P. & G. Kapetanios (2003a) ‘Pure significance tests of the unit root hypothesis against nonlinear alternatives’, Journal of Time Series Analysis 24(3),
Blake, A.P. & G. Kapetanios (2003b) ‘A radial basis function artificial neural network test for neglected nonlinearity’, The Econometrics Journal 6(2),
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.