TURUN YLIOPISTO UNIVERSITY OF TURKU TALOUSTIEDE DEPARTMENT OF ECONOMICS RESEARCH REPORTS. A nonlinear moving average test as a robust test for ARCH

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1 TURUN YLIOPISTO UNIVERSITY OF TURKU TALOUSTIEDE DEPARTMENT OF ECONOMICS RESEARCH REPORTS ISSN ISBN A nonlinear moving average test as a robust test for ARCH Jussi Tolvi No 81 May 1999 Department of Economics FIN 0014 University of Turku Finland

2 Jussi Tolvi A nonlinear moving average test as a robust test for ARCH Abstract The possibility of using a nonlinear moving average (NMA) test as a test for ARCH is considered. A joint test of ARCH and NMA tests is derived. The asymptotic relative efficiency of the NMA test against ARCH is also computed, and is found to be equal to zero. The small sample properties of the tests are examined by some simulation experiments, along with the effects of additive outliers. Overall, both tests powers are low in small samples. The NMA test has even less power compared to the ARCH test, but it is more robust to outliers. In addition, often only one of the tests detects ARCH, whereas the other does not. An example is given (interest rate data, 75 observations), where due to an outlier the ARCH test can not detect ARCH, whereas the NMA test can. Keywords: GARCH, Lagrange multiplier test, outliers, asymptotic relative efficiency, ARE, joint test. Financial support from the Yrjö Jahnsson Foundation and comments from Hannu Koiranen, Pentti Saikkonen, Dick van Dijk and Matti Virén are gratefully acknowledged.

3 1. Introduction The commonly used Lagrange multiplier test for ARCH has two problems. First, in small samples the power of the test is low, especially against GARCH series. Secondly, if the data contains outliers the test suffers size and power distortions, even in larger samples. Not much can be done about the low power of the test in small samples. Robust tests may help in the problems caused by outliers, but they have some drawbacks as well. First, they are often difficult to use in practice. Second, their power is lower than the LM test s if the data do not contain outliers. A different approach will therefore be taken in this paper. Instead of robust ARCH tests, I will examine whether an LM test against a different alternative nonlinear model would be useful also in ARCH testing. The alternative considered here is a nonlinear moving average (NMA) model. This idea originates from my (unpublished) licentiate thesis, where it was noted that one form of an NMA test was quite robust to the effects of outliers, but had at least some power against ARCH. In this paper a slightly more powerful (against ARCH) form of the test will be examined. The two nonlinear models and LM tests for them will be presented in Section. A joint test for simultaneous testing of both ARCH and NMA will be derived, in the hope that it will have more power than the single tests. The asymptotic relative efficiency of the NMA test against ARCH is also computed. Section 3 contains the results of some simulation experiments, and Section 4 an empirical example.. Theory.1. Models and single tests The linear model for the observed series Y t to be considered is an autoregressive model with errors ε t, φ(l)y t = ε t, (.1) 1

4 where φ(l) is a lag polynomial of order P, with all roots outside the unit circle. ARCH models were introduced by Engle (198), and generalized by Bollerslev (1987). A generalized ARCH, or GARCH(p, q) process has a variance depending on past errors, such that ε t Ω t-1 ~ N(0, h t ), h t = σ + α(l)ε t + β(l)h t, (.) where α(l) and β(l) are lag polynomials of degrees q and p, respectively, and Ω t denotes information available at time t. If α(l) and β(l) are both zero, the variance h t is constant. The case of α(l) 0 and β(l) = 0 gives an ARCH(p) process. Engle (198) proposed a simple test for ARCH effects, based on an auxiliary regression using squared OLS residuals of equation (.1). The test is asymptotically equivalent to an exact LM test, and it is also a test for GARCH errors (see Lee 1991). Because this testing procedure is so simple it has become very popular, and is used routinely in many software packages as part of diagnostic testing of estimated models. GARCH models are also popular and successful in economics, especially in financial markets (see, e.g. Bollerslev, Engle and Nelson 1994). The nonlinear moving average (NMA) model has product terms of past errors as explanatory variables (see e.g. Granger and Teräsvirta 1993). A formulation using terms up to the third power will be considered here, such as u v,i>j φ(l)y t = γ ij ε t i ε t j + γ kmn ε t k ε t m ε t n + ε t, (.3) i=1 j= w x,m > k y,n >m k =1 m = where γs are the NMA parameters. Again, the error terms are not observed, but testing for nonlinear moving average effects is possible by an auxiliary regression using the OLS residuals of (.1). The ARCH test examined here is computed using four lags of squared residuals in the auxiliary regression, i.e. n =3

5 e t = a 0 + a 1 e t 1 + a e t and the NMA test uses an auxiliary regression of + a 3 e t 3 + a 4 e t 4 +ξ t, (.4) P e t = b 0 + b i Y t i + c 1 e t-1 e t- + c e t-1 e t-3 + c 3 e t- e t-3 + c 4 e t-1 e t- e t-3 + ξ t, (.5) i =1 where e t are the estimated residuals from the linear AR(1) model, and as, bs and cs are parameters. The test statistics are TR, where T is the sample size and R is the squared multiple coefficient of correlation from the auxiliary regressions. The tests are asymptotically distributed as χ with degrees of freedom 4, the number of nonlinear terms in the test regressions. The NMA model is close to ARCH, especially if, unlike here, terms where i = j (or k = m and so on) are allowed in the formulation (.3). It is nevertheless important to keep in mind that an ARCH model is nonlinear in the variance, whereas an NMA model is nonlinear in the mean. There is very little research on nonlinear moving average models in general (one of the few references is Robinson 1977, but see also Granger and Teräsvirta 1993), and they do not seem to have been applied in economics. One interesting exception and a potential justification for searching for NMA errors is the idea of using such models in forecasting white noise (Granger 1983 and 199). However, the closeness of NMA to ARCH may provide an exploitable advantage, which will be examined in this paper. If NMA tests have power against ARCH and are not too severely affected by outliers, they may provide a simple robust test for ARCH. The traditional LM test for ARCH has already been shown to have larger than nominal size and low power in the presence of outliers by van Dijk et al. (1999). These results will be replicated in Section 3 of this paper. I chose to use the same simulation design as van Dijk et al. (with the addition of a sample size of 50) for two reasons. First, to check the precision of the results, since van Dijk et al. based their results on only 1000 replications. And second, this choice facilitates a comparison of various tests for ARCH, including a robust ARCH test proposed by van Dijk et al. 3

6 .. The joint test A further possibility for testing for ARCH is provided by joint tests, originally proposed for ARCH and bilinearity testing by Higgins and Bera (1989). The idea is simply to add the test statistics of two LM tests together, and use this sum as a joint test statistic. It can be shown that under certain conditions the sum is also asymptotically distributed as χ, with degrees of freedom equal to the sum of the single tests degrees of freedom. Using the technique of the proof in Higgins and Bera (1989), it is easy to prove that tests for ARCH and NMA can also be combined as a joint test. The proof is as follows (see Higgins and Bera 1989 for a more extensive discussion, and note that Granger and Teräsvirta 1993 have a general discussion which could also be used as a basis for proving this). The required condition is the block diagonality of the information matrix. In this case the full model is (note that here only second-order terms will be considered in the NMA formulation; this is only to make the notation more tractable, since the results generalize immediately to higher-order cases) P Y t = φ i Y t 1 + γ ij ε t i ε t j +ε t, (.6) i =1 u i=1 v,j>i j= ε t Ω t 1 ~ N(0,h t ) and (.7) q h t =σ + α i ε t i. (.8) i =1 Partition first the parameter vectors C = (φ 1,..., φ P, γ 1,..., γ uv ) and A = (σ, α 1,..., α q ) to C1 = (φ 1,..., φ P ), C = (γ 1,..., γ uv ), A1 = (σ) and A = (α 1,..., α q ) to separate the NMA and ARCH parameters (γ and α). The joint null hypothesis of linearity is H 01 : C = 0 and H 0 : A = 0. The necessary and sufficient condition allowing these hypotheses to be tested separately and the test statistics to be added together is that H 01 I 1 H 0 = 0, or (.9) 4

7 [ 0 uv P 1 uv uv 0 uv 1 0 uv q ] I 1 0 P q 0 uv q 0 1 q 1 q q = 0, (.10) where 0 a b is a null matrix, 1 a b is an identity matrix, I is the information matrix and tildes ( ~ ) denote that these quantities are evaluated at the maximum likelihood (ML) estimates of parameters subject to both restrictions H 01 and H 0. Note that the subscripts do not agree with the dimensions of these matrices, they are only used to link the blocks with corresponding parameters. The required condition is obviously met, if the information matrix (and consequently also its inverse) is block diagonal of the form P+ uv P + uv 0 P +uv 1+ q 0 1 +q P + uv 1 + q 1+q, (.11) where denotes blocks of non-zero terms, and 0 denotes blocks of zero terms. This block diagonality can be used for other purposes as well, for example to show that in an ARCH regression model the estimation of regression parameters and ARCH parameters and their variances can be carried out separately (see Engle 198). To prove the required block diagonality of the information matrix for the case considered here, start with the log-likelihood for one observation (assuming normally distributed errors) l t = 1 log(h t ) 1 P ε t h t +constan t, (.1) ε t = Y t φ i Y t 1 γ ij ε t i ε t j. (.13) i=1 u v,j > i i= 1 j = l t is differentiated with respect to the mean and the variance parameters. Note that only the cross derivatives are needed to make sure that the required blocks of the information matrix are zero; the other (non-zero) elements do not matter here. We get first a differential with respect to a mean parameter C i 5

8 l t = 1 h t ε t h t h t ε t + ε t h t h t, (.14) and then with respect to a variance parameter A j l t A j = 1 h t h t h t A j 1 h t h t A j + ε t h t ε t h t ε t h t 3 A j h t h t + 1 ε t A j h t h t A j. (.15) The negative of the (i, j) element of the information matrix is formed by summing the expectations of these second derivatives over all observations (t = 1,..., T), i.e. T I (i,j) = E l t. (.16) t = 1 A j Taking these expectations iteratively, and noting that since ε t Ω t 1 follows a N(0,h t ) distribution, E(ε t Ω t 1 ) = 0 and E(ε t Ω t 1 ) = h t, the conditional expectations simplify to l I (i,j) = E E t Ω A t 1 = E 1 h t h t. (.17) j h t A j But since these are evaluated under the null hypothesis, i.e. at C = (γ 11,..., γ uv ) = 0 and A = (α 1,..., α q ) = 0, the above is equal to zero, because the middle term on the right hand side of (.17), h t = α 1 ε t 1 ε t α q ε t q ε t q, (.18) is now clearly equal to zero for all i and t. Therefore the information matrix and its inverse are block diagonal in the way required, and the tests for ARCH and NMA can be combined into a joint test. The joint test of ARCH and NMA used here is the sum of the two auxiliary regressions TR s, from equations (.4) and (.5), which is asymptotically distributed as χ with degrees of freedom 8. Whether it does in fact have more power than the single 6

9 tests (c.f. ARCH and bilinearity in Higgins and Bera 1989) will be examined by the simulations of Section The asymptotic relative efficiency of the NMA test against ARCH The powers of the single tests against right and wrong alternatives will first be briefly discussed. The optimal LM test for each model (or data generating process, DGP) is naturally given by the corresponding test. However, the powers of other tests can in some situations be quite considerable as well. Such results are found in, e.g. the simulations of Luukkonen, Saikkonen and Teräsvirta (1988). I will next show that the asymptotic relative efficiency (ARE, as defined later) of the NMA test against an ARCH alternative is equal to zero. For technical details of the calculations in this section, see Saikkonen (1989). Again, Granger and Teräsvirta (1993) has a general discussion of this topic, and a method for an alternative proof. To examine the ARE of the NMA test relative to the ARCH test, specify first the log-likelihood of the NMA model as T L 1T (φ, γ) = l 1t ( φ,γ ), (.19) t =1 and the log-likelihood of the ARCH model as T L T (φ,α) = l t ( φ,α), (.0) t =1 where φ is a p-dimensional parameter vector (AR parameters φ 1,..., φ p ), γ a w-dimensional parameter vector (NMA parameters γ 11,..., γ uv ), α a q-dimensional parameter vector (ARCH parameters α 1,..., α q ), and T the sample size. The hypothesis to be tested is H 0 : γ = 0 against H 1 : γ 0 (i.e. an NMA test), but the true model (ARCH) is given by L T, where α 0. Assume further that L 1T (φ, 0) = L T (φ, 0), or that the null models are the same (a linear AR model without ARCH, which is also the model considered earlier). The usual regularity conditions are also assumed to hold (see Saikkonen 1989). 7

10 The asymptotic distribution of the LM test for NMA, denoted LM γ, is examined in a sequence of local alternatives α = δ/t 1/, δ 0. Define the gradient l k t = 1t (φ,0) φ l 1t (φ,0) γ l t (φ,0) α (.1) with covariance matrix J = E[k t k t ] = [J ij ], i, j {φ, γ, α}. Define further the partial covariance matrices J γ φ = J γ J γφ J 1 φ J φγ and J γα φ = J γα J γφ J 1 φ J φα = J αγ φ. Now if the true data generating process is given by L T (φ,δ/t 1/ ), the NMA test statistic LM γ follows asymptotically a non-central chi-square distribution with degrees of freedom w, and noncentrality parameter λ c (δ) = δ J αγ φ J 1 γ φj γα φ δ. (.) The LM test for ARCH, LM α is in this situation an optimal test, since it is based on the true DGP. The ARCH test statistic LM α follows also asymptotically a non-central chisquare distribution, with degrees of freedom q and non-centrality parameter λ α (δ). The exact form of λ α (δ) does not matter here, as will become clear later. What is important, is that λ α (δ) > 0. The power of the wrong test can now be compared to that of the correct test. For a fixed δ the asymptotic relative efficiency of LM γ with respect to LM α is equal to (assuming for simplicity that w = q as well; this assumption does not matter to the main conclusion here, the case of w q is discussed in Saikkonen 1989) e γα = λ γ (δ) λ α (δ). (.3) The next step is to note that the relevant information matrices are in this case block diagonal, as discussed earlier. It follows therefore that J γα φ = J γα. It is then easy to show 8

11 that J γα = 0, which is the result we are looking for here. The required derivatives of the log-likelihood functions are, first for the incorrect alternative NMA l 1t (φ,0) γ and for the correct alternative ARCH = ε tε t 1 ε t h t,k, ε tε t uε t v h t, (.4) l t (φ,0) α = ε t 1 h t + ε t ε t 1 h t,k, ε t q h t + ε t ε t q h t. (.5) Elements of J γα are then equal to zero, since the expected values of the products E ε tε t i ε t j ε t k h t ε tε t iε t jε t ε t k 3 h t (.6) are equal to zero for all i, j, k. This means that J γα φ = 0 and that λ γ (δ) = 0 as well, and that the asymptotic relative efficiency e γα = 0. In other words, the asymptotic power of the NMA test against ARCH is not higher than the size of the test. Nevertheless, since this result is based only on local alternatives, there may still be ARCH models against which the NMA test has at least some, and possibly even quite notable power. This is noted in the simulations of Luukkonen, Saikkonen and Teräsvirta (1988) for bilinearity tests against ARCH. 3. Simulation experiments In all experiments of this section, a time series of T observations is simulated, in some simulations additive outliers are added to the data, an AR(1) model estimated, and test statistics calculated and compared to the nominal asymptotic 5% critical level. The percentage of the test statistics greater than the critical value is reported. The simulation design is as follows. Sample sizes considered are 50, 100 and, for a comparison with asymptotic results, 500 (also samples of 50 were simulated, but the results do not provide any additional information and will not be given here). Outliers of magnitudes 0 9

12 (i.e. no outliers), 3, 5 and 7 are added to the simulated series such that each observation is an outlier with probability If an outlier occurs, it is either positive or negative, with equal probabilities. Formally, we have the observed series X t, such that X t = Y t + I t (ρ)ω, where Y t is the underlying series, the indicator variable I t (ρ) = 0 with probability 0.95, I t (ρ) = 1 with probability 0.05 and I t (ρ) = 1 with probability 0.05, and ω is the outlier magnitude. The results here are based on 5000 replications, and the first 100 observations of every series were discarded to avoid dependence on initial values. The sizes of the tests are examined in an AR(1) series, with autoregressive parameter φ = 0.0, 0.1, 0.3, 0.5, 0.7, 0.9. The powers are computed for GARCH(1,1) series with parameter values representing those typically estimated for real financial data. These are (α, β) = (0.1,0.7), (0.1,0.8), (0.,0.6), (0.,0.7), (0.3,0.6). Apart from the differences in sample sizes, this is exactly the same setting as was used in van Dijk et al. (1999). Computed sizes for the two single tests are given in Table 1. In the absence of outliers, the ARCH test s size is usually lower than the nominal 5%, whereas the NMA test s sizes are almost always closer to the nominal. Outliers distort the sizes of both tests. In the smallest sample the NMA test seems to be slightly more distorted, but with larger samples the ARCH test becomes more distorted. The distortions are always worse for larger values of the autoregressive parameter φ, but when φ is close to or equal to zero, the distortions are relatively minor. Note also, that in some cases larger outliers (ω = 7) induce lesser distortions than smaller ones (ω = 5). The size of the joint test is usually between the two single tests sizes, and will not be tabulated here to save space. The results for sample sizes 100 and 500 are quite well in line with the results of van Dijk et al. (1999). Direct comparisons are not possible, since they computed the ARCH test using 1, 5 and 10 lags in the auxiliary regression. Comparing their results for the ARCH5 test with the ones obtained here (ARCH4), there are some cases where the difference in simulated sizes is quite noticeable. These occur where the distortions are major, i.e. with large values of φ and ω. These results are therefore not necessarily very 10

13 accurate, although they clearly provide a rough indication of the distortions. The same remarks apply also to the simulated powers of the ARCH test, which will be discussed next. Table 1. Sizes (in %) of the tests. See text for details. ARCH NMA ω T φ The powers of the single tests are given in Table. In the absence of outliers the tests are equally powerful when the sample size is 50, but in larger samples the ARCH test becomes clearly more powerful. When T is equal to 100, the NMA test s power is approximately half of the ARCH test s power, and this ratio decreases further as the sample size grows, to (on average) one third with T equal to 500. When outliers are present, however, the NMA test is more powerful than the ARCH test in smaller samples and with larger outliers. This is due to the severe power loss of the ARCH test. The NMA test, on the other hand, is more robust to outliers, although it suffers considerably as well. When comparing the powers, it should however be kept in mind, that the sizes of the NMA test are somewhat greater than those of the ARCH test. 11

14 Table. Powers (in %) of the tests. See text for details. ARCH NMA ω T α β On the whole, both of these tests have very little power in small samples. Similarly, the presence of outliers distorts the sizes and creates a power loss in both of the tests. Although the NMA test is slightly more powerful in these situations, the powers of both tests are rather dismal. Compared with the robust test of van Dijk et al. (1999) the NMA test is usually clearly less powerful, in a sample of 100 sometimes almost as powerful, and in one isolated case (T = 100, GARCH parameters equal to (0.1, 0.8) and outlier magnitude 5) slightly more powerful than the robust test. van Dijk et al. have not considered samples smaller than 100 observations, so it is possible that the NMA test would be more powerful than the robust test in a sample of, say 50 observations (since in the absence of outliers the robust test is clearly less powerful than the nonrobust LM test, and the NMA test is approximately as powerful as the nonrobust test when T = 50). Also of interest may be the fact, noted by Diebold and Pauly (1989), that in small samples the exact LM form of the ARCH test is somewhat more powerful than the TR form of the test used here. The difference is usually only a few percentage points, however. 1

15 Table 3 provides interesting additional information on the simulation results. On the left side of the table is the power of the joint test. For sample sizes 50 and 100, the joint test is always slightly more powerful than either of the single tests. The right side of Table 3 shows the total percentage of cases when at least one of the two single tests is significant. At least in small samples it seems that quite often only one of the tests is significant, whereas the other is not. These numbers are therefore somewhat higher than Table 3. Powers (in %) of the tests. See text for details. JOINT ARCH or NMA ω T α β either of the single tests powers alone, and also higher than the joint test s powers. This knowledge could be useful in empirical testing situations, since one could increase the power of the testing by computing both tests. Naturally, the effects of the simultaneity on the significance levels of the tests must then be taken into account. However, with a sample of 50 observations and no outliers, the single tests sizes are clearly below the nominal 5% level, whereas the frequency of at least one of them being significant is roughly between 5 and 6 per cent (these results are not tabulated here). As in the power simulations, only rarely are both tests significant in the same sample. In very small samples the combination of these tests is therefore a better alternative than using only one 13

16 of them with asymptotic critical values. Unfortunately, the spurious detections of nonlinearity caused by outliers are also often detected by only one of the tests at a time. This means that a combined testing procedure will also produce more spurious detections of ARCH which are really due to outliers. 4. An example In this section, an empirical time series is examined. The data are daily observations of the Finnish Helibor one month interest rate from July 3 to November 4, 1991, T = 75. Both the original data and the differences are plotted in Figure 1 (in the appendix). The differenced series is used in the analysis here. Based on the plot it seems that, as is usually the case for financial time series, ARCH is quite probably present in the data, but on the other hand it seems that the series may contain outliers as well. After a preliminary examination, an AR(4) model was chosen to remove linear autocorrelation from the series. Tests for nonlinearities were then computed from the residuals of this model, using the tests presented earlier, as well as some simpler forms of the tests. ARCH tests were computed with lags up to 4, and the four reported NMA tests are as follows. NMA1 has only one NMA term in the auxiliary regression, namely c 1 e t-1 e t- (or, in the notation of equation (.5), c = c 3 = c 4 = 0). NMA adds to this the term c e t-1 e t-3, (c 3 = c 4 = 0) NMA3 the term c 3 e t- e t-3 (c 4 = 0) and finally NMA4 the term c 4 e t-1 e t- e t-3. Note that ARCH4 and NMA4 are the tests used in the simulations of Section 3. The p-values of these tests are presented in the first column of Table 4. From there it can be seen that none of the ARCH tests can be considered significant at any reasonable level. Two of the NMA tests on the other hand are clearly significant, NMA1 with a p-value of 0.060, and NMA4 with a p-value as small as There is therefore at least some evidence of nonlinearity in the series. The AR(4) residuals are plotted in Figure (in the appendix), and seem to contain several observations that have noticeably larger values than the rest of the data. Most of these appear in clusters (possibly indicating ARCH), but one, at t = 8, seems to be isolated, and could be considered an outlier. The influence of this observation was 14

17 examined by adding a dummy variable to the AR(4) model for this observation. Residuals from this model were again tested for nonlinearities, and the tests p-values are Table 4. p-values of LM tests from the AR(4) residuals, with and without a dummy. Test No dummies Dummy at t = 8 ARCH1 0.7 < ARCH 0. < ARCH < ARCH4 0.6 < NMA NMA NMA NMA < presented in the second column of Table 4. All of the ARCH tests are now highly significant (all with p-values less than 0.001). Similarly, the p-value of NMA4 decreases further, whereas those of the first three NMA tests increase slightly. There is therefore very little doubt about the presence of ARCH in the series, it is only hidden by an outlier. This conclusion is reinforced by estimating ARCH models for the residual series. Table 5 provides the results for two models. An ARCH(1) model, in the upper part of the table, seems to be a reasonable specification. When there is no dummy in the AR(4) model, the estimated ARCH parameter α 1 is not significant at the 5% level, but when the dummy is added, it becomes clearly significant at that level. Note that the value of α 0 decreases to almost half of what it is in the absence of the dummy, whereas the value of α 1 increases slightly. Estimated ARCH(), ARCH(3) and ARCH(4) models all have either negative estimated parameter values or the sum of the parameter estimates is greater than one, even when a dummy is added to the model. A GARCH(1,1) model on the other hand, as can be seen from the lower part of Table 5, is not admissible without the dummy (α 1 < 0 and β 1 > 1). When a dummy is added to the AR model, however, the parameter estimates are acceptable (and significant). The sum α 1 + β 1 is close to being equal to one, implying perhaps an IGARCH model. Nevertheless, the main point of this example is that there are 15

18 real situations where ARCH tests can not detect ARCH, in this case because of outliers, whereas an NMA test can. Table 5. Estimated ARCH models for the AR(4) residuals, with and without the dummy (standard errors of the estimates are given in parentheses). Model/parameters No dummies Dummy at t = 8 ARCH(1) α (0.040) 0.15 (0.05) GARCH(1,1) α (0.33) 0.66 (0.6) α (0.0013) 0.03 (0.019) α (0.00) 0.38 (0.16) β (0.011) 0.57 (0.13) 5. Conclusions The simulation results show again the problems of ARCH testing in small samples. The power of the LM test is low, and outliers remove even what little power there is. The NMA test considered here is comparable to the ARCH test in the smallest sample, and is more robust to outliers. In very small samples it can therefore be superior to (literally) robust ARCH tests. In larger samples the ARCH test is still clearly superior. The joint test seems to do quite well in some situations, and the simultaneous use of both ARCH and NMA tests may also be of value. The practical relevance of these findings is, however, not likely to be great. Although the example of Section 4 shows that there may be data sets where an NMA test is notably better than the ARCH test, such data sets are probably rare in real life. 16

19 Appendix: Figures Figure 1. Helibor 1 month interest rate. Original data (a) and differences (b). (a) Original data obs. (b) Differences obs. 17

20 Figure. AR(4) residuals. 3 Residuals obs. 18

21 References Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, Bollerslev, T., R. F. Engle and D. B. Nelson (1994) ARCH models. In R. F. Engle and D. L. McFadden (Eds.) Handbook of econometrics IV. North-Holland, Amsterdam. Diebold, F. X. and P. Pauly (1989) Small sample properties of asymptotically equivalent tests for autoregressive conditional heteroskedasticity. Statistical Papers - Statistische Hefte, 30, Engle, R. F. (198) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, Granger, C. W. J. (1983) Forecasting white noise. In A. Zellner (ed.) Applied time series analysis of economic data. U. S. Department of Commerce, Bureau of the Census, Economic Research Report ER-5. Granger, C. W. J (199) Forecasting stock market prices: Lessons for forecasters. International Journal of Forecasting, Vol. 8, Granger, C. W. J. and T. Teräsvirta (1993) Modelling nonlinear economic relationships. Oxford University Press, Oxford. Higgins, M. L. and A. K. Bera (1989) A joint test for ARCH and bilinearity in the regression model. Econometric Reviews, 7, Lee, J. H. H. (1991) A Lagrange multiplier test for GARCH models. Economics Letters, 37, Robinson, P. M. (1977) The estimation of a nonlinear moving average model. Stochastic Processes and their Applications, 5, Saikkonen, P. (1989) Asymptotic relative efficiency of the classical test statistics under misspecification. Journal of Econometrics, 4, van Dijk, D., P. H. Franses and A. Lucas (1999) Testing for ARCH in the presence of additive outliers. Journal of Applied Econometrics, forthcoming. 19

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