A radial basis function artificial neural network test for ARCH

Economics Letters 69 (000) 5 3 www.elsevier.com/ locate/ econbase A radial basis function artificial neural network test for ARCH * Andrew P. Blake, George Kapetanios National Institute of Economic and Social Research, Dean Trench Street, Smith Square, London SWP 3HE, UK Received July 999; accepted 3 March 000 Abstract We propose a test for ARCH that uses a radial basis function artificial neural network. It outperforms alternative neural network tests in a variety of Monte Carlo experiments. 000 Elsevier Science S.A. All rights reserved. Keywords: Artificial neural networks; Conditional heteroscedasticity; Hypothesis testing; Bootstrap JEL classification: C; C; C45. Introduction Following the introduction of autoregressive conditional heteroscedasticity (ARCH) models by Engle (98), a rapidly increasing amount of empirical and theoretical work has investigated their properties and extensions such as GARCH and M-GARCH. Comprehensive surveys may be found in, for example, Bollerslev et al., 99; Bera and Higgins (993) and Bollerslev et al., 994. Testing procedures for ARCH effects have also received considerable attention, beginning with the LM procedure proposed by Engle (98). Recently, Peguin-Feissolle (999) has proposed a neural network test for ARCH based on the neglected nonlinearity testing procedure of Lee et al. (993). In this paper we provide a general test for ARCH, based on an alternative neural network specification. This is shown to have superior power properties and is less subject to ad hoc parameter choices. We do find, however, that the test suffers from size distortions. We use the bootstrap to correct these distortions and find that although the power of the test is reduced it remains above that of alternative neural network procedures. The paper is organised as follow. Section provides the alternative testing procedures that will be examined. Section 3 provides details on the bootstrap procedure that corrects the size distortions of the original test. Section 4 provides the design of the *Corresponding author. Tel.: 44-0-7654-94; fax: 44-0-7654-900. E-mail address: ablake@niesr.ac.uk (A.P. Blake). 065-765/ 00/ $ see front matter 000 Elsevier Science S.A. All rights reserved. PII: S065-765(00)0067-6

6 A.P. Blake, G. Kapetanios / Economics Letters 69 (000) 5 3 Monte Carlo experiments. Section 5 presents the results of the Monte Carlo study. Finally, Section 6 concludes.. Testing procedures We consider the following regression model y 5 x9b e t t t where hetj is a sequence of disturbances and we denote by ^t the set of available information (s-field) at time t. Under the null hypothesis hetj is an i.i.d. sequence with variance s. Under the alternative hypothesis E(e u^ ) 5 h 5 f(e, e,...). t t t t t The ability of neural networks to approximate arbitrary functions has been used in a number of contexts in econometrics including tests for neglected nonlinearity (Lee et al., 993) and tests for ARCH (Peguin-Feissolle, 999). The underlying idea in ARCH testing is that the function f(? ) may be approximated by a neural network. A single hidden layer feedforward neural network model is defined as q f(e, e,...). a O ac(e,...,e, g,...,g ) () t t 0 j j t tp j0 jp j5 There are p inputs used to activate q hidden units regulated by the functions c (? ). Following Lee et j al. (993), Peguin-Feissolle (999) specified that logistic functions be used for c (? ) giving j q f(e t, e t,...). a0o]]]]]]]]] (g0jgjet...gpje tp) j5 e a j Then the null of no ARCH effects is equivalent to a test of a 5???5aq50where a j, j 5,...,q are the coefficients of the hidden units in a regression of the squared residuals of () on a constant and the hidden units. There are q( p ) q parameters to estimate in (3), a difficult and expensive exercise. Peguin-Feissolle (999) follows Lee et al. (993) to avoid estimation issues by assuming the weights g ij, i 5,..., p; j 5 0,..., q, are generated randomly from a uniform distribution on [, ]. q and p are chosen by the researcher. To avoid multicollinearity between the hidden units, the q largest principle components of the hidden units are used as regressors instead. Again the choice of q rests with the researcher. The actual test amounts to constructing the statistic ] e 9W(W9W ) W9e where e 5 (e,..., e ) is the vector of demeaned squared residuals from (). W is a matrix containing the T observations on the hidden units and a constant. The test has asymptotically a x distribution according to Peguin-Feissolle (999). We will refer to this as the ANN test. Our test amounts to two important departures from the ANN test. Firstly, we suggest using radial basis functions (RBFs) instead of logistic functions. These are widely used in single hidden layer neural network models. For a nontechnical introduction see Campbell et al. (993). The inspiration for q () (3)

A.P. Blake, G. Kapetanios / Economics Letters 69 (000) 5 3 7 the use of similar functions was originally to solve the exact interpolation problem. The hidden neural network unit arising out of RBFs is generally of the form c(uuvt cuu) where vt is a vector of inputs, c is a vector of constants referred to as centres, uu? uu denotes a norm and c(? ) is a scalar function. z Usually the function used is Gaussian, given by e t. Specifically, we use the following (v c), z 5 zz. 9 t t t t t z 5 R R is a diagonal matrix of radii. z is then the hidden unit output for a given c and R. t Our second departure from the ANN test is to use a strategy to construct the test statistic which avoids many of the ad hoc parameter choices associated with the White test. Following Peguin- Feissolle (999) and Kamstra (993) we consider both the residuals and the squared residuals of () at given lags as possible neural network inputs for the construction of hidden units. The strategy may be described as follows:. Form T potential hidden units by using each observation of the inputs, v t, t 5,..., T, asa possible center. Following common practice (Orr, 995) we use twice the maximum change from period t to period t, t 5,..., T of each input as the radius for all potential hidden units.. We regress the squared residuals of () on a constant and each hidden unit and obtain the sum of squared residuals (SSR t) from each regression. T 3. We compare the SSR with SSR 5 o t 0 t5 et and sort the hidden units according to the magnitude of SSR0 SSRt in descending order. 4. Starting with the hidden unit which provides maximum reduction in the sum of squared residuals, we regress the residuals, ˆ e t, on this unit and a constant, and successively add units to the regression until an information criterion (in this case BIC) is minimised. 5. For the chosen regression, we test the significance of the coefficients of the hidden units using the same LM test used for the ANN test. This test will be referred to as the RBF test. 3. Bootstrap approximation Preliminary investigation of the small sample properties of the RBF test indicates that the size properties of the test are not very good. In particular the test overrejects under the null hypothesis in a number of occasions. Such an outcome indicates that the x asymptotic approximation is not particularly accurate. A common method used to overcome this problem involves use of the bootstrap. The bootstrap is an improvement to the asymptotic approximation since, under given conditions, the / approximation error is smaller that T which is the error of the asymptotic approximation. In particular if a statistic is asymptotically pivotal (i.e. does not depend asymptotically on unknown parameters) then the bootstrap estimate of its distribution has statistical error of order T. Of course, the simulation error arising out of the need to estimate the bootstrap distribution using a finite number of replications has to be controlled. As Brown (999) points out T bootstrap replications are needed to take advantage of the improved statistical approximation. The above results apply to samples with

8 A.P. Blake, G. Kapetanios / Economics Letters 69 (000) 5 3 i.i.d. observations. Results for sequences of dependent data exist for some cases such as AR or MA models (see Bose, 988 and Bose, 990) indicating that the improvement is of order o p() rather than / O p(t ). In our framework the bootstrap can be applied as follows. Once the original test statistic, denoted by S has been obtained we retrieve the set of residuals, he ˆ,..., ˆ et j from (). We then resample randomly with replacement from the set of the residuals to obtain a bootstrap sample of residuals (e ˆ *,..., ˆ e * T ) where each ˆ e t* has been drawn with replacement from he ˆ,..., ˆ et j and stars denote generic bootstrap quantities. We then carry out the artificial neural network test on the bootstrap sample of residuals, (e ˆ *,..., ˆ e T* ). Repeating this process N times where N is the number of bootstrap replications we obtain a set of bootstrap test statistics, S *,..., S * N. We then use these samples to construct the bootstrap distribution of our test statistic. More specifically the estimated P-value of a given test statistic S is given by N On5 (S n* # S) pˆ 5]]]]] N where (? ) is the indicator function taking the value when its argument is true and zero otherwise. We will refer to the bootstrap test as RBF-B. Under the null hypothesis, the error terms in () are i.i.d. and therefore the resampling scheme described above is justified. Under an ARCH alternative the random resampling should ensure that the dependence between the resampled residuals is negligible asymptotically thereby providing a consistent testing procedure under certain conditions. Establishing that the error sequence in () is either mixing or near epoque dependent (see e.g. Davidson, 994, pp. 6 77) under a generalised ARCH alternative should be sufficient for the testing procedure to be consistent. Nevertheless the sufficiency of mixing or near epoque dependence is conjectured and a rigorous proof remains to be provided. 4. Monte Carlo study As we mentioned earlier, both the residuals and the squared residuals of (), at given lags, are used as possible inputs to the neural network. We refer to the tests using residuals in levels as ANN-L, RBF-L and RBF-BL tests respectively, and similarly the tests using squared residuals as ANN-S, RBF-S and RBF-BS. Following Peguin-Feissolle (999) we set q 5 0 and q 5 3 for ANN-L and ANN-S. We carry out Monte Carlo experiments for three widely used ARCH-type specifications. The first is a standard ARCH model. The second is a generalised ARCH (GARCH) model and the third is an exponential GARCH (E-GARCH) model. The first case coincides with that of Peguin-Feissolle (999) for comparative purposes. The other setups are considered because of their popularity in the literature. For all cases the regression model () is given by y 5 0? 5 0? 5x e t t t x 5 0? 7x t t t t n, n NID(0,) The data yt and xt have been normalised to lie between 0 and for the ANN-L and ANN-S tests.

A.P. Blake, G. Kapetanios / Economics Letters 69 (000) 5 3 9 The specification of h for the three experiments we will consider together with the lag order of the t residuals used as inputs to the neural network are: Case 0: (p5) e NID(0, ) t Case : (p53) h 5 0? 0? e 0? 4e 0? 3e, t t t t3 ] e 5 h v, v NID(0, ) t œ t t t Case : (p53) t t t t3 t h 5 0? 0? e 0? 3e 0? e 0? h 0? ht 0? h t3, ] e 5 h v, v NID(0, ) t œ t t t Fig.. P-discrepancy.

0 A.P. Blake, G. Kapetanios / Economics Letters 69 (000) 5 3 Case 3: (p53) log ht5 0? 0? zt 0? 3zt 0? zt3 0? log ht 0? log ht 0? log h t3, ] e 5 h v, v NID(0, ) z 5 uv u Euv u nv, n 50? 5 t œ t t t t t t t The negative value given to n for the E-GARCH setup reflects the so-called leverage effect commonly found in stock prices denoting the asymmetric responses to positive and negative shocks. For all cases four different sample sizes were considered to investigate the effect of the number of observations on the performance of the tests. The sample sizes considered were 50, 00, 50 and 50. For each sample size, T samples of size T 00 were constructed with 00 initial observations discarded to minimise the effect of starting values, which are set to zero. For each experiment 000 replications were carried out. Fig.. Size-power curve: Case.

5. Monte Carlo results A.P. Blake, G. Kapetanios / Economics Letters 69 (000) 5 3 In Figs. 4 we present the P-value discrepancy and size-power curves for the Monte Carlo study (see Davidson and MacKinnon (998) for details on the construction of the graphs). The P-value discrepancy graph (Fig. ) indicates that the RBF-S and RBF-L tests overreject the null hypothesis. However, when the bootstrap approximation is used in the RBF-BS and RBF-BL tests, the overrejection disappears. In fact these tests have much better size properties than the ANN-S and ANN-L tests which underreject. Thus the bootstrap approximation has the desired effect by correcting the size distortion apparent in the graphs. We need to establish the power of the new test and establish that the size correction is achieved with (at worst) acceptable power loss. The size-power curves (Figs. 4) indicate that for most sample sizes and most experiments the RBF tests significantly outperform the ANN tests. For the RBF tests the use of residuals in levels provides more powerful tests than the use of squared residuals. The bootstrap procedures (RBF-BS and RBF-BL) are dominated by the asymptotic procedures (RBF-S Fig. 3. Size-power curve: Case.

A.P. Blake, G. Kapetanios / Economics Letters 69 (000) 5 3 Fig. 4. Size-power curve: Case 3. and RBF-L) in terms of power but in all cases they dominate the ANN tests. The reduction in power is, however, quite modest for the bootstrap versions. We conclude that the RBF-BL is preferred overall. 6. Conclusions Neural networks have been used in a number of contexts in econometrics due to their ability to approximate arbitrary functions. This paper has investigated a new neural network test for ARCH effects based on radial basis functions, using both the level and square of the estimation residuals. We compared their performance to that of existing neural network tests for ARCH. It was found that although the new tests slightly overreject the null hypothesis they have far superior power properties. In an earlier version of this paper the experimental setup analysed by Peguin-Feissolle (999) was considered. There it was found that the RBF-S and RBF-L tests dominate the ANN tests in terms of power. When we apply the bootstrap procedures, they dominate the ANN tests in some cases but perform less well in others. However, the setups considered in Peguin-Feissolle (999) are less common in the empirical literature than the setups considered in the present paper. All results are available from the authors on request.

A.P. Blake, G. Kapetanios / Economics Letters 69 (000) 5 3 3 A bootstrap procedure has been utilised to remove the tendency of the tests to overreject. The bootstrap tests retain the superior power performance of the RBF tests while enjoying good size properties. References Bera, A.K., Higgins, M.L., 993. ARCH models: properties, estimation and testing. Journal of Economic Surveys 7 (4), 305 366. Bollerslev, T., Chou, R.Y., Kroner, K.F., 99. ARCH modelling in finance: a review of the theory and empirical evidence. Econometrics 5 (-), 5 60. Bollerslev, T., Engle, R.F., Nelson, D.B., 994. ARCH Models. In: Engle, R.F., McFadden, D. (Eds.), Handbook of Econometrics, Vol. 4, North Holland. Bose, A., 988. Edgeworth correction by bootstrap in autoregressions. Annals of Statistics 6, 709 7. Bose, A., 990. Bootstrap in moving average models. Annals of the Institute of Statistical Mathematics 4, 75 768. Brown, B.W., 999. Simulation variance reduction for bootstrapping. In: Mariano, R., Weeks, M., Schuermann, T. (Eds.), Simulation Based Inference: Theory and Application, Cambridge University Press. Campbell, J.Y., Lo, A.W., MacKinlay, A.C., 993. The Econometrics of Financial Markets, Princeton University Press. Davidson, J., 994. Stochastic limit theory. Oxford University Press. Davidson, R., MacKinnon, J.G., 998. Graphical methods for investigating the size and power of hypothesis tests. The Manchester School 66, 6. Engle, R.F., 98. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 504, 987 007. Kamstra, M., 993. A neural network test for heteroscedasticity, Working Paper, Simon Fraser University, Burnaby, Canada. Lee, T.H., White, H., Granger, C.W.J., 993. Testing for neglected nonlinearity in time series models: a comparison of neural network methods and alternative tests. Journal of Econometrics 56, 69 90. Orr, M.J., 995. Regularisation in the selection of the power of some tests for conditional heteroscedasticity. Economics Letters 63, 5 7. Peguin-Feissolle, A., 999. A comparison of the power of some tests for conditional heteroscedasticity. Economics Letters 63, 5 7.