Robust tests for ARCH in the presence of the misspecified conditional mean: A comparison of nonparametric approches

This study compares statistical properties of ARCH tests that are robust to the presence of the misspecified conditional mean. The approaches employed in this study are based on two nonparametric regressions for the conditional mean. First is the ARCH test using Nadayara-Watson kernel regression. Second is the ARCH test using the polynomial approximation regression. The two approaches do not require specification of the conditional mean and can adapt to various nonlinear models, which are unknown a priori. Accordingly, they are robust to misspecified conditional mean models. Simulation results show that ARCH tests based on the polynomial approximation regression approach have better statistical properties than ARCH tests using Nadayara-Watson kernel regression approach for various nonlinear models.


Introduction
The presence of heteroskedasticity has significant impacts on estimation and inference of time series analysis. For example, Becker and Hurn (2009) and Pavlidis, Paya, and Peel (2010) demonstrated that the presence of heteroskedasticity frequently leads to over-rejections of the null hypothesis with linear time series models when we test for linearity of the conditional mean model. In addition, Pavlidis, Paya, and Peel (2013) showed that causality tests of the conditional mean provide evidence of spurious causality relationships in the presence of multivariate heteroskedasticity. These facts indicate that tests for whether data generating processes (DGP) have heteroskedasticity play an important role in time series analysis.
Most representative model of heteroskedasticity is the autoregressive conditional heteroskedasticity (ARCH) introduced by Engle (1982). ARCH is one of simple and popular volatility models and is still used well. When we test for the presence of heteroskedasticity, we first estimate a regression model of the conditional mean. We next test for ARCH by statistics such as Lagrange multiplier test. If the conditional mean regression model is correctly specified, the ARCH test performs well. However, if the conditional mean is misspecified, it causes serious problems for ARCH tests. Lumsdaine and Ng (1999) examined properties of ARCH tests under the misspecified conditional mean. They showed that the misspecification of the conditional mean leads to the over-rejection of the null hypothesis of homoskedasticity. Similarly, Balke and Kapetanios (2007) clarified influence of neglected nonlinearity of the conditional mean on ARCH tests and found evidence of over-rejection of the no ARCH effect when nonlinearity of the conditional mean regression model is neglected. In order to test for ARCH appropriately, it is necessary to avoid the misspecified model of the conditional mean.
This study compares statistical properties of ARCH tests that do not depend on the conditional mean model. They are applicable for various nonlinear conditional mean models and are robust to the misspecified conditional mean model. We employ two nonparametric approaches to avoid the misspecification of the conditional mean model. First is the regression using Nadaraya-Watson kernel estimator, which is one of the representative nonparametric methods. The method was introduced by Nadaraya (1964) and Watson (1964) and used the kernel density function to do regression analysis that does not depend on the model. As can be seen in McMillan (2001) and Exterkate, Groenen, Heij, and van Dijk (2016), Nadaraya-Watson estimator is useful in various nonlinear models. Second is the regression analysis using a polynomial approximation of a general unknown nonlinear model. Stone (1977) and Katkovnik (1979) proposed the local polynomial estimator based on a polynomial approximation. Balke and Kapetanios (2007) developed the method to approximate unknown models by using neural network. Péguin-Feissolle, Strikholm, and Teräsvirta (2013) proposed a new causality test that is based on a Taylor approximation of a general nonlinear model and is applicable to various nonlinear models. These approaches are relevant methods from the viewpoint of the use of a polynominal approximation. This study introduces ARCH tests using these nonparametric regression approaches in order to avoid misspecification of the conditional mean and investigates statistical properties of the introduced tests in various linear and nonlinear models. If we were to perform erronous ARCH tests based on misspecified conditional mean models and fail to obtain reliability from the derived results, the correct model construction and statistical evaluation are extremely difficult. It is important to clarify the influence of tests on the misspecified models.
We examine rejection frequencies under the null and alternative hypotheses of introduced ARCH tests using Monte Carlo simulations. The simulation analyzes the influence of the lag length, the bandwidth selection for Nadaraya-Watson estimator, and the approximation order for the polynominal approximation method on the results. The conditional mean models ivestigated in this study are linear autoregressive, threshold autoregressive, smooth transition autoregressive, markov switching, and bilinear models. These nonlinear models are popular ones for empirical analysis, and tend to cause spurious ARCH effects. The results of Monte Carlo simulation provide evidence that ARCH tests based on the polynomial approximation regression approach have better statistical properties than ARCH tests using Nadayara-Watson kernel regression approach when DGP are various nonlinear models.
The remainder of this paper is organized as follows. Section 2 presents the influence of the misspecification of the conditional mean on ARCH tests and introduces ARCH tests using nonparametric regression approaches for the conditional mean. Section 3 provides statistical properties of the tests under various nonlinear models. Finally, the paper concludes in Section 4.

ARCH test using nonparametric regression approaches for the conditional mean
We consider the following DGP with order m.
where f (·; ·) is an unknown function and β is a parameter vector. u t is a disturbance term with mean zero and variance given by where ǫ t are independently and identically distributed (iid) random variables with mean zero and variance equal to one. Although the conditional variance could also have model misspecification similar to the conditional mean, standard heteroskedastic tests have the ability to find linear ARCH effects even if the true conditional variance is generalized ARCH (GARCH) with/without nonlinear parts. On the other hand, spurious ARCH effects tend to be observed when the conditional mean has model misspecification. The presence of misspecification of the conditional mean has clear impacts on inference of variance as shown by Lumsdaine and Ng (1999) and Balke and Kapetanios (2007).
Accordingly, we particularly investigate the influence of model misspecification of the conditional mean on ARCH effects.
For the test for ARCH effect, we have the null hypothesis of homoskedasticity denoted by and the alternative hypothesis Even if we assume a GARCH model as heteroskedasticity, the testing procedure is basically same as pointed out by Lee(1991) and Gel and Chen(2012). Therefore, we focus only on the ARCH test. The standard ARCH test introduced by Engle (1982) uses the auxiliary regression of squared residuals.
When true DGP are (1), suppose that we estimate a misspecified model where g(·; ·) is a misspecified function,m is a lag length, and α is a parameter vector for a misspecified model. It follows that the residual is denoted bŷ where e t = f (y t−1 , · · · , y t−m ; β) −ĝ(y t−1 , · · · , y t−m ; α). The squared residual forû t iŝ Equation (9) (7) is a linear autoregressive model, e t includes nonlinearity. As pointed out by Lumsdaine and Ng (1999) and Blake and Kapetanios (2007), such misspecification causes spurious ARCH effect.
Therefore, a regression approach that does not depend on a specific model is necessary to avoid model misspecification and spurious ARCH effect.
The first approach that is robust to model misspecification is a nonparametric regression based on Nadayara-Watson kernel estimator. We consider the following conditional mean regression model.
where m(·) is the unkown regression function without any parametric form. The regression function The most representative method to estimate the function is the Nadaraya-Watson estimator. The estimator is given byẑ where ) · · · K( yt−s−y hs ) is a product kernel function and h denotes the bandwidth to determine the smoothness of the kernel function. The kernel funcion satisfies the followings K(y)dy = 1, yK(y)dy = 0, y 2 K(y)dy > 0.
This study uses Gaussian kernel given by 2 We use two bandwidth selections for h. They are derived by minimizing the integrated mean squared error (IMSE). First is the plug-in method proposed by Silverman (1986). The bandwidth based on the plug-in method is based on the equation as follows where c 0 is a constant depending on the kernel function and p is the number of the regressor. When we use Gaussian kernel, the optimal bandwidth selection is given by where σ denotes the standard deviation of y t . The modified h opt that is robust to outliers is written whereQ is the estimate of the interquartile range of y t 3 .
Second is the cross-validation procedure. The procedure was developed by Rudemo (1982). When Gaussian kernel is used, we consider the following mean squeared error called the cross-validation criterion whereẑ(Y −i ) is leave-one-out estimator, which is an estimator that excludes ith observation. The optimal bandwidth h for the cross-validation procedure is determined by minimizing CV (h). Stone (1984) showed that the bandwidth h for the cross-validation can asymptotically select the optimal bandwidth from the viewpoint of IMSE and has probability convergence to the bandwidth for the plug-in method. While the bandwidth h for plug-in method depends on the assumed probability density function, the cross-validation does not need to assume the probability density function and can obtain consistent estimator of the bandwidth that minimizes IMSE. It is possible that the residuals obtained from Nadaraya-Watson estimator (12) with bandwidth selection (17) or (18) have similar properties. Accordingly, this means that the above nonparametric regression approach is robust to model misspecification of the conditional mean and enables the correct ARCH test 4 .
The next approach to avoid misspecification is the use of a polinomial approximation of a general unknown nonliner model. When we use k th-order Taylor approximation to the true model (1), the regression model is given by where q is a lag length and ǫ t is an error term that includes the remainder term of the Taylor series approximation. We assume q ≤ k. For example, (19) with p = 2 and k = 2 is written as The difference between the true model and the polynomial approximation regression model becomes smaller because the polynomial regression can approximate various nonlinear models including threshold autoregressive and markov switching models. When we want to test for ARCH effect under unkown (true) model, the use of residuals obtained from polinomial approximation regression (19) has advantages that the residuals have similar statistical properties as the true model. Therefore, the ARCH test using the residuals from the polinomial approximation regression does not appear to have influence of model misspecification.

Statistical properties of ARCH tests using nonparametric regression models
This section examines statistical properties of ARCH tests using nonparametric regression models of the conditional mean presented in section 2. We conduct Monte Carlo simulations to compare the rejection frequencies of the test statistics under various conditional mean models with/without ARCH effects. The simulations are based on 10,000 replications, the nominal level at 5%, and sample sizes with T = 100, 250 and 500. In order to avoid the effect of initial conditions, data with T +100 are generated.
The initial 100 samples are discarded and we use the data with sample size T . We compare ARCH tests (6) using the following regression models of the conditional mean: the AR model denoted AR(p), polynomial approximation model (19) with second and third order Taylor approximation denoted as T 2(p) and T 3(p), respectively, and nonparametric regression model (12) with plug-in method (17) and cross validation method (18) denoted as N P pl (p) and N P cv (p), respectively. We set the lag length p to p = 1 or p = 2 5 . The AR model is used as a benchmark for comparison.
First, we consider the following autoregressive processes to examine the influence of lag length on performance of the tests.
where u t ∼ i.i.d.N (0, 1). β 0 is set to β 0 = 0. Table 1 presents the rejection frequencies of the ARCH tests obtained from each regression model of the conditional mean. Data generating processes (DGP) from DGP1-1 to DGP1-4 have the followings.
These DGP have homoskedastic errors with a 0 = 1 and b 0 = 0 for (23). The rejection frequencies presented in Table 1 mean the empirical size of ARCH tests based on each regression model.
For DGP1-1 and DGP1-2, most of the tests have a small underrejection but reasonable size performance except for N P pl (2) and N P cv (2). N P pl (2) and N P cv (2) have overrejections for DGP1-1 and DGP1-2, respectively. The rejection frequencies of N P pl (2) for DGP1-1 with T = 500 and N P cv (2) for DGP1-2 with T = 500 are 0.143 and 0.101, respectively. Additional lag for nonparametric regression of the conditional mean using Nadaraya-Watson estimator leads to size distortions for ARCH tests. In contrast, AR(2), T 2(2), and T 3(2) does not have overrejections for DGP1-1 and DGP1-2. The results show that additional lag for AR and polynomial approximation regression does not have impact on size of ARCH tests. However, less lag length has clear impact on empirical size for all the tests. We can see that ARCH tests based on AR(1), T 2(1), T 3(1), N P pl (1), and N P cv (1)  Next, we examine the empirical size of ARCH tests under the following conditional mean generated by threshold autoregressive models.
As shown by Figures 1, 2, and 3, it is generally difficult to distinguish between nonlinear conditional mean with homoskedastic error and linear AR model with ARCH effect. Such similarlity between TAR model with homoskedastic errors and linear AR model with ARCH effect may cause spurious statistical properties.
The simulation results are tabulated in Table 2. AR(2) has overrejections of the null hypothesis of no ARCH. For DGP2-2 and DGP2-5 that have strong asymmetry, size distortions of AR(2) are huge. These results indicate that the use of the AR model for the conditional mean leads to spurious ARCH effect when true DGP are TAR or MTAR models. In additon, the overrejections increase with the large sample size. Unlike the performance of AR(2), polynomial approximation regression models T 2(2) and T 3(2) and nonparametric regression models N P pl (2) and N P cv (2) perform better.
We observe that AR(2), T 2(2) and N P pl (2) partially tend to reject the null hypothesis of no ARCH effect. The rejection frequencies of AR(2) is higher than other regression models for DGP3-2 and 3-6.
T 2(2) has size distortions for DGP3-2. N P pl (2) has a slight overrejection with T = 500. The shape of the transition function does not have clear impact on the empirical size of T 3(2) and N P cv (2). T 3 (2) and N P cv (2) can capture the properties of STAR models and conduct the ARCH test well.
Additionally, we show the results of each test for other nonlinear processes as follows.
The simulation results from Tables 1 to 4 show evidence that model misspecification of the conditional mean causes size distiortions of the null hypothesis of no ARCH effect. The ARCH tests using AR regression model are sensitive to the presence of nonlinear conditional mean and have high overrejections. This is caused by the neglected nonlinearity and the difficulty of distinguishment between nonlinearity of the conditional mean and the ARCH effect. Although noparametric regression models using Nadaraya-Watson estimator partially perform well, the rejection frequencies strongly depend on DGP and the bandwidth selection. By contrast, the size properties of T 3(2) outperform other models are close to the nominal size at 5%. Therefore, T 3(2) can approximate (unknown) linear and nonlinear conditional mean models well and lead to reliable ARCH tests.
Tables 5 and 6 report nominal power and size corrected power properties for ARCH tests. We use DGP1-3, DGP2-1, 2-4, 3-1, 3-4, 4-1, and 4-4 for power comparison. Each DGP have ARCH effect given by where a 0 and b 0 are set to a 0 = 1 and b 0 = (0.1, 0.3), respectively. The powers of AR(2) are clearly higher than other models in Table 5. We can relatively have reasonablely evaluation of power for DGP1-3 because the size properties of AR(2) and other tests are close to nominal level 0.05 shown in Table 1. However, the high nominal powers of AR(2) for other DGP are not correct interpretation.
The higher powers of AR(2) are influenced by size distortions presented in Tables from 2 to 4. We can evaluate the power properties of nonparametric models more appropriately because T 2(2) and T 3(3) do not have overjection for DGP in Table 5 and size distortions of N P pl (2) and N P cv(2) are smaller than those of AR(2). In their comparison, we can observe that polynomial approximation models T 2(2) and T 3(2) perform better than N P pl (2) and N P cv (2). Note that the powers of N P pl (2) are quite small when ARCH effect is b 0 = 0.1. For b 0 = 0.3, nonparametric regression models have sufficient power to find ARCH effect.
In order to compare power properties among the models without the influences of size distortions, Table 6 demonstrates size corrected power. The powers of AR(2) in Table 6 are lower than those in Table 5 because size distortions are corrected. AR(2) still performs well even if size is corrected.
Abilities to detect ARCH effect in nonlinear models for T 2(2) are high similar to those of AR(2).
Although the powers of T 3(2) is slightly smaller than those of T 2(2) because T 3(2) has additional regression parameters for the conditional mean, it has sufficient power to find ARCH effect. The rejection frequencies of N P pl (2) and N P cv (2) for b 0 = 0.1 are inferior to other models even in Table   6. While they relatively perform well for b 0 = 0.3 with T = 100, other models have better power properties especially for T = 250 and 500.
The comparison of ARCH tests using each regression model of the conditional mean indicates that the presence of the nonlinear conditional mean has influences of size and power properties on ARCH tests. AR regression models brings higher overrejection of the null hypothesis of no ARCH effect for nonlinear conditional mean. ARCH tests based on AR models for nonlinear conditional mean is not effective in the viewpoint of size and power because the size corrected tests are needed, and furthermore, the true model is generally unknown a priori. Nonparametric regression models using Nadaraya-Watson estimator tend to have slight size distortions and low power. Polynomial approximation model T 2(2) has slight overrejection depending on nonlinear conditional mean and sample size, whereas it has better power properties for ARCH effect with nonlinear conditional mean.
T 3(2) has reasonable size and power properties and yeilds reliable results of ARCH tests regardless of the conditional mean models.

Summary and conclusion
This study compared statistical properties of ARCH tests that are robust to misspecified conditional mean models. ARCH tests are important for statistical modeling because the presence affects statistical inference of the conditional mean regression model. However, it is difficult to know the correct specified conditional mean model and possible to employ misspecified conditonal mean model. This may lead to unreliable results. Therefore, it is neccesary to compare robust ARCH tests to unknown various conditional mean models and clarify its statistical properties. The approaches employed in this study are based on two nonparametric regressions. First is the ARCH test using Nadayara-Watson kernel regression. Second is the ARCH test using the polynomial approximation. The two approches can adapt to various nonlinear models, which are unknown a priori. Accordingly, they are robust to misspecfied models. Monte Carlo simulations provide evidence that ARCH tests based on the polynomial regression approach have better statistical properties than ARCH tests using Nadayara-Watson kernel regression approach for various nonlinear models. In particular, the test using the regression approach based on the third order Taylor approximation has reasonable and acceptable size and suf- 3. Sneather and Jones (1991) proposed another bandwidth selection based on the plug-in method.