The Performance Of Nonlinearity Tests On Asymmetric Nonlinear Time Series

The Performance Of Nonlinearity Tests On Asymmetric Nonlinear Time Series Eun S. Ahn University of Hawaii - West Oahu Jin Man Lee1 DePaul University Abstract. This paper examines nonlinearity tests of asymmetric time series in a controlled Monte Carlo setting with the goal of exploring how well existing nonlinear test statistics performed in a variety of typical time series settings. The data generation processes and sample sizes were allowed to vary in a controlled fashion. The study confirmed that none of the test statistics were dominant relative to the others. Also, dependent upon the data generating process, the test statistics exhibited very different powers. Finally, our research showed that the test performance was heavily dependent upon sample size and the degree of asymmetric mechanism. JEL Classification: C1, C6 Keywords: Nonlinearity Test, Asymmetric Time Series, Monte Carol Simulation 1. Introduction Nonlinear econometric testing has received great interest from econometric empirical studies due to the implications of nonlinear modeling for economic theory. Ample theories have been introduced regarding the nonlinear dynamic nature of economic phenomena and many studies have reported evidence of nonlinearity in economic models. Linear models are still widely used as they continue to play an important role in forecasting through the application of simple parsimonious forms to explain various economic phenomena. However, linear models are limited in flexibility and explanation of nonlinear behavior. Among the unlimited alternative forms of nonlinearity, the asymmetric form has drawn considerable interest as a special case of nonlinearity in economic and financial research. Since Tong (1983) first introduced threshold autoregressive models, they have been subsequently extended to Markov regime switching and smooth transition autoregressive models to investigate the asymmetrical behavior of time series models. Nonlinear models of economic activity date as far back as the 1920s (Mitchell [1927], Keynes [1936], and Hicks [1950]) and emphasize that adjustments of the economy during business cycles may be asymmetric. Keynes (1936) argued that “the conditions in an economy are more violent but also more short lived than expansions and that, therefore, GNP follows an asymmetric cyclical process with 11

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upswings lasting longer than the downturns.” Recent empirical studies suggest economic variables such as output and unemployment behave asymmetrically during stages of a business cycle (i.e. expansions are more persistent but less sharp than recessions) as Neftçi (1984), Delong and Summers (1986), Falk (1986), and Sichel (1989) suggest. Neftçi (1984) found evidence of the US employment related series displaying two different regimes during the business cycle with the unemployment rate being characterized by sudden jumps and slower declines. Further evidence in this vein was also provided by Delong and Summers (1986), Falk (1986), and Sichel (1989). Asymmetric monetary effects have also been addressed. Cover (1992) examined whether positive and negative money-supply shocks have symmetric effects on output. The study found positive money supply shocks do not have an effect on output, while negative money supply shocks do have an effect on output. These findings were confirmed by Karras (1996), Karras and Stokes (1996), and Choi (1999). Ball and Mankiw (1994) investigated the asymmetric adjustment of nominal prices with menu cost models in which positive trend inflation causes firms’ relative prices to decline automatically between price adjustments. Brännäs and Ohlsson (1999) found asymmetric monthly unemployment series may become symmetric when aggregated to quarterly or annual frequencies by using the nonlinear autoregressive asymmetric moving average (ArasMA) model on Swedish unemployment rates. Speight and McMillan (1998) found strong corroborative evidence of asymmetric steepness relative to the trend in UK durable consumption, total investment, investment in plant and machinery, exports and unemployment based on the coefficient of skewness. Neumann (1995) investigated the real effects of exchange rate volatility on trade balance and found the long-run effect of increased exchange rate volatility on the trade balance cannot be determined unequivocally because of differences in domestic capital between the debtor country and creditor country. GARCH class asymmetric models have been considered to cover the shortcoming of the traditional GARCH model because the conditional variance is not influenced by the sign of the forecast errors. The asymmetric or leverage volatility models based on asymmetric GARCH models trace their development back to the work of Black (1976) and Christie (1982). Engle and Ng (1993) found Japanese daily stock returns displayed asymmetric volatility effects. Fornari and Mele (1997) developed sign and volatility switching ARCH models and unraveled evidence of asymmetric volatility from stock market returns in six countries. Clements and Smith (1999) demonstrated the usefulness of nonlinear models in forecasting due to the asymmetric behavior of time series data. Their study revealed that if the presence of nonlinearity is ignored in time series data, estimates may behave poorly and the fitted linear model may leave out key dependencies and lead to erroneous conclusions and inferences.

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Although economic theories support asymmetric behavior or nonlinearity of various time series data, there still exists considerable controversy relating to the empirical testing with economic data. The major concern relates to the choice of a test statistic that lacks robustness to detect nonlinearity. The second concern lies with the minimum sample size necessary to detect the nonlinearities with sufficient power. The lack of information regarding these two issues with the asymmetrical branch of nonlinearity, in particular, is the genesis for this study. This study will conduct a Monte Carlo simulation of various asymmetric time series models with the goal of exploring how well existing nonlinear test statistics performed in a variety of typical time series settings where the data generation processes and sample sizes were allowed to change in a controlled fashion. The power of the test statistics will assist researchers in the diagnosis of asymmetric models and may subsequently lead to future development of more robust test statistics. 1.1 Nonlinearity Tests on Asymmetric Time Series Models Barnett et al. (1997) proposed a single-blind controlled competition among five test statistics using five nonlinear models. Their results displayed the lack of a dominant test statistic across the different types of nonlinear models and also that no individual test can serve as a diagnostic for their specific form of nonlinearity. They discovered the strength of test statistics across various tests to provide stylized facts about time series process. Their study focused on nonlinear models to test the power of available test statistics. This paper extends on their study by focusing on asymmetric time series models which have developed significantly in applied economic and financial literature recently. Asymmetric time series models are respected for their ability to test switching regimes in economic models and unequal volatility dependent upon news from financial markets. Most of these tests are based on models with specified nulls and alternatives. The central criticisms have been addressed at sample sizes, the degree of asymmetry, and the performance and power of the different nonlinear test statistics. The size of sample is the most frequently discussed in literature. Davies and Petruccelli (1986) reported poor power in identifying nonlinearity using small samples with the Keenan (1985) and Mcleod and Li (1983) tests. They also discovered both tests only have power when the time series are near nonstationary with small sample sizes (i.e. less than 200). Ashly, Patterson and Hinich (1986) examined the Hinich test with a self-exiting threshold autoregressive model (SETAR) and found the power of tests was dependent upon the sample size. In their study, they were able to reject 33, 55, and 88 percent for sample sizes of 256, 512, and 1024, respectively. Hsich and LeBaron (1998) tested the SETAR using the BDS test, and concluded it had very little power with small sample sizes, but performed well in larger sample sizes (e.g. 1000). Tsay (1989) implemented the Tsay test and found it rejected the null of linearity 37.4% of the time while a Keenan type of

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portmanteau test rejected only 19.5% with a threshold model. Cook et al. (1999) tested the nonlinearity of simulated series using different asymmetric adjustment parameters with different variance. They performed nonlinear tests using a typical F type test. Their results showed it was difficult to detect nonlinearity with small sample sizes using this type of test; it was only capable of detecting nonlinearity of the higher asymmetric adjustment parameters with larger sample sizes (e.g. 500). Another difficulty in testing the nonlinearity of asymmetric models is the degree of the asymmetric mechanism. For example, if the difference between the positive and negative coefficients in a TAR model is very large, then the implied nonlinear process can be very easily detected. However, if the difference is relatively small, it would be difficult to detect the nonlinearity of the series despite the fact that the underlying data generation process is nonlinear asymmetric. Tsay (1989) tested the threshold autoregressive models and found the Tsay test rejected the null of linearity 37.4% of the time when the asymmetric parameter was 0.5 distance, and 70%, 80.4%, 85.6% when the distance increased to 1.0, 1.5, 2.5, respectively. Cook et. al. (1999) also experienced similar problems with the asymmetric error correction models. The performance of the nonlinearity tests could be varied by the degree of asymmetric mechanism with the coefficients and variance. In addition to the sample size and the degree of asymmetry, test statistics can have varied performance depending on the true data generating process. For example, Barnett et al. (1997) reported that the bispectrum tests have no power against flat bispectrum and nonflat higher order polyspectra. They recommended using the tests as complements. In summary, a number of studies have shown the difficulties in using nonlinearity tests in regards to the size of sample, the degree of asymmetry, and the various test statistics being applied. However, not one of these prior studies provide a consistent and uniform examination across the various factors influencing asymmetric models to provide researchers with any stylized facts and guidelines for using these types of models. 1.2 Empirical Studies on Asymmetric Affects There is controversy in the detection of nonlinearities in economic data despite the fact that economic theory provides little support for the assumption of linearity. Also, there is a lack of literature supporting robust measures of asymmetric models. This study will attempt to provide some insight in this regard by examining the effects and interactions of sample sizes and the parameters for the degree of asymmetry with the power of different nonlinear test statistics in typical asymmetric models. The results will attempt to provide researchers with guidance in the choice of test statistics in various empirical settings and environments. The study is organized as follows. First, we will discuss asymmetric time series models including asymmetric autoregressive moving average models and volatility switching models, and introduce nonlinear test statistics. Second, we will find the size and performance of each nonlinear test with simple true null models.

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Third, we will apply nonlinear tests on five asymmetric models: asymmetric autoregressive model (asAR), asymmetric moving average (asMA), smooth transition models, asymmetric error correction models, and asymmetric GARCH models. We will then discuss the power of the nonlinear tests of the different asymmetric time series. Finally, we will summarize the findings and discuss the potential implications and avenues for future work. 2. Asymmetric Time Series Models and Nonlinear Test Statistics 2.1 Asymmetric Time Series Kim and Mittnik (1996) discussed the asymmetric data generating process (DGP): yt  f ( zt 1, zt 2 , )   t . (1) They included a number of possible cases: 1) A symmetric innovation process (t) with a linear transmission mechanism (f (.)) generating symmetric output (yt); the case of conventional linear econometric models. For this kind of model, a Box and Jenkins (1970) ARIMA type of model with white noise process may be used. 2) The innovation process (t) is symmetric but the transmission mechanism (f (.)) is nonlinear and thereby generates asymmetric output (yt). We can see this type of data generating process from a threshold autoregressive model, a Markovian regime shifting model, a smooth transition autoregressive model for the univariate case, and an asymmetric error correction model as a structural model. 3) The innovation process (t) is asymmetric with a linear transmission mechanism (f (.)) which generates asymmetric output (yt). Since the transmission mechanism is linear we can use the asymmetric generalized autoregressive conditional heteroskcedastic (asymmetric GARCH) class models like volatility switching GARCH, threshold GARCH, GJR GARCH, logistic smooth transition GARCH, and sign switching GARCH. 4) The innovation process (t) is asymmetric and transmission mechanism (f (.)) is nonlinear and thereby generates asymmetric output (yt). We can include this class of model combined with the second and third cases; a threshold autoregressive asymmetric moving average model, and an asymmetric threshold GARCH model are examples which fall in this category. For the second and third cases, if we use an incorrectly specified transmission mechanism (e.g. linear instead of proper nonlinear using linear model) then the error term must be an asymmetric process. If we ignore this process, then the resulting estimation and inference are invalid due to misspecification The traditional Box and Jenkins’s (1976) ARIMA type of linear model belongs to the first category: the linear transmission with symmetric innovation. The autoregressive model (AR (p)), (2) xt  a0  a1 xt 1  ...  a p xt  p  ut the moving average model (MA (q)),

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xt  b0  ut  b1ut 1  ...  bqut q

(3)

and the mixed autoregressive moving average model (ARMA (p,q)), xt  a0  a1 xt 1  ...  a p xt  p  ut  b1ut 1  ...  bqut q .

(4)

All transform the symmetric innovations sequence

ut according to the linear rule of

filter to produce the observed sequences xt . The other three categories are asymmetric DGP. The process with nonlinear transformation and symmetric innovations can generate asymmetric observed output. Among the nonlinear mechanism processes, we are interested in the asymmetric transformation mechanism, which generate asymmetric time series; the transmission mechanism can respond to innovation with more than two different rules depending upon the stage of the input series. For example, for a sign asymmetric time series model, it encompasses two different innovations, positive and negative. A sign asymmetric series xt will exhibit different behavior as a system of dynamic changes depending upon positive or negative innovation. The examples of models that belong to this category are the threshold autoregressive (TAR) models, the self-exiting threshold autoregressive (SETAR) models, the asymmetric moving average (asMA) models, the smooth transition autoregressive (STAR) models, and the asymmetric error correction models (asymmetric ECM). The TAR, asMA and STAR series are all generated by the asymmetric transmission mechanism with symmetric innovations. Another DGP category of an asymmetric process is a linear transformation mechanism with asymmetric innovations. In this category, we include the asymmetric generalized autoregressive conditional heteroskedasticity class models. Engle (1982) introduced the autoregressive conditional heteroskedasticity model (ARCH) and Bollerslev (1986) developed it to the generalized autoregressive conditional heteroskedasticity model (GARCH). The GARCH process can be interpreted as a discrete time approximation of a diffusion model with stochastic volatility. In this study, the fourth case (asymmetric linear transmission with nonlinear innovation) was excluded. Including two nonlinear components would have made it difficult to identify the strength of the nonlinearity tests. Although it is ideal to conduct complex nonlinear testing, this is not easy to achieve as all nonlinearity tests have some weaknesses and strengths. 2.2 Nonlinearity Tests From the DGP, the resulting output series may be an asymmetric or a symmetric series. If the series are generated by category two or three, then they exhibit nonlinear behavior and we must be able to detect the nonlinearity of the series. Detection of nonlinearity requires the choice of an appropriate nonlinearity test.

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Since our models are generated from asymmetric DGP, the output series must show the asymmetric behavior as a special case of nonlinearity. Granger and Teräsvirta (1993) divided nonlinearity tests into two broad categories. The first category contains all the tests derived without specific nonlinear alternatives taken into consideration as the portmanteau tests, while the second category consists of tests against a specific nonlinear model. The study showed that the application of the Lagrange Multiplier (LM) principle is not always necessary. Barnet et al. (1997) used five highly regarded tests for nonlinearity or chaos without specific alternatives. Many linearity tests may be considered as diagnostic tests based on residuals of a linear model and their distribution. Some tests without specific nonlinear alternatives can be interpreted as a score or LM tests against a nonlinear alternative. 2.2.1 Bispectrum Test Hinich (1982) developed tests based on the bispectrum to determine whether a time series is Gaussian and linear. Hinich’s bispectrum is based on the bispectrum in the frequency domain as the double Fourier transformation of the third-order moments function. The Hinich bispectrum test directly examines a nonlinear generating mechanism, so prewhitening is not necessary to use this test. Ashely, Patterson, and Hinich (1986) showed that the test statistics for the Hinich bispectrum test are invariant to linear filtering of the data. So, if prewhitening is done using a nonlinear filter, the adequacy of the prewhitening model is irrelevant to the validity of this test. This aspect of the test is in contrast to other tests which require the tested series to be first order white (i.e. no ACF spikes). 2.2.2 Hinich’s Bicorrelation Test Hinich (1986) introduced a test for dependence in the residuals of a linear parametric time series model fitted to Gaussian data. The test statistic is a third order extension of the standard correlation test for whiteness. He proposed a third order portmanteau statistic computed from the estimated values of a bicorrelation of the residuals. The test is to detect nonlinear structure in the innovations if it has a number of nonzero bicorrelations. He proposed this as a complement to the portmanteau test of McLeod and Li (1983), which is based on a normalized sum of squared autocorrelations of squared residuals, and thus uses a subset of the sample normalized fourth order cumulates. 2.2.3 BDS Test The BDS test of Brock, Dechert, and Scheinkman (1996) is based on the idea of a correlation dimension to test whether the residuals of a linear regression are from an independent and identically distributed process with the null hypothesis being whiteness. If the data are generated by a nonlinear process, then the remaining process must be nonlinear after removing the linear structure.

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The BDS test statistic is a transformation of the correlation function and uses the correlation integral function. The sample distribution of the BDS test statistic is not known under nulls of nonlinearity or linearity. The transformed test statistic asymptotically becomes a standard Z statistics under null hypothesis of whiteness. To apply the BDS test we need two arguments, embedded M and epsilon. M is the embedding dimension and the metric bound, epsilon is the maximum difference between the pairs of observations counted in computing the correlation function. Hsich and LeBaron (1988) found the power of the tests is maximized when et is between 0.5 to 1.5 times the standard deviation of the data. In the nonlinear tests, we selected from 3 to 10, and tested the individual setting with epsilon being the sample standard deviation. 2.2.4 Engle Test Engle (1982) introduced the autoregressive conditional heteroskedasticity (ARCH) model, and developed a second moment test to examine the ARCH effects. Engle’s ARCH LM test has been used to test no ARCH effect, and it is very useful to test the time series with second moment tests as a form of nonlinearity. To use the test, we have to set the autoregressive order p in squared estimated residuals. In our test, we chose the range from 1 to 5 to evaluate the test performance. 2.2.5 Keenan Test Keenan (1985) assumed the series can be adequately approximated by a secondorder Volterra expansion (Wiener, 1956). The Keenan test is focused on a univariable restricted by its past information. The test is based on F test for of linearity. 2.2.6 Tsay Test Tsay (1986) proposed a nonlinear test for a stationary time series based on Tukey’s (1949) one degree of freedom for nonadditivity test. Tsay (1989) applied the test to threshold autoregressive models to measure the power of the tests. The Tsay test is a more general nonlinear form than the Keenan test. The Keenan test is the same as the Tsay test when {xt} is strictly stationary AR (1) process. However, if {xt} is not AR (1), then the Keenan test is a special case of the Tsay test with the difference being the auxiliary regression. 2.2.7 McLeod-Li Test The McLeod-Li test, a portmanteau test proposed by McLeod and Li (1983), is based on a large-sample estimation theory that requires the error from the ARIMA process to be independent and identically distributed with finite variance. The autocorrelation function of the square of a time series could be useful in identifying nonlinear time series. Based on the autocorrelations of squared residuals from a linear fit to detect nonlinearity in time series data, similar to Box and Pierce (1970) and Ljung and Box

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(1978), if the residuals follow whiteness, then their cross product correlation should exhibit the same correlation structure as the correlation of the squared cross product. 2.2.8 RESET (Regression Specification Error Test) Ramsey (1969) considered the standard linear regression model Y  X   u , where Y is a T x 1 vector of observations on the dependent variable; X is a T x K matrix on K explanation variables; u is a T x 1 vector of disturbances and  is a K x 1 vector of parameters. His model specification of the null hypothesis is E (u|X) =0, where u is normally distributed with the covariance matrix proportional to the identity matrix. The alternative hypothesis is that a specification error has occurred (e.g. omitted variables, incorrect functional form, and correlation between X and u). Thursby and Schmidt (1977) extended the RESET using the power of the explanatory variables, a Taylor series expansion in the explanatory variables, and powers of principle components of explanatory variables. We included the power of the explanatory variables for a modified RESET since their Monte Carlo test revealed it had superior power for both omitted variable models and incorrect functional form models. 2.3 Nonlinearity Tests of Asymmetric Time Series The BDS test’s asymptotic distribution is known under the null of independence, but the sampling distribution of the test statistic is not known under the nonlinear or linear null hypothesis. Therefore, we need to generate the sample critical values and use them along with asymptotic critical values from a null of independence. The Hinich bispectrum test is a test in frequency domain of flatness of bispectrum. The null hypothesis is linear and it can directly test nonlinearity as an alternative hypothesis. The test detects the nonflat bispectrum, but has little power to detect high order nonflat polyspectra. Rothman (1992) compared the power of the BDS and Hinich’s bispectrum tests against some simple self-exciting threshold autoregressive (SETAR) models and applied both tests to identify the presence of varying forms of nonlinear structures which are undetected by conventional time series techniques. He emphasized the two tests as a general unspecified alternative hypothesis. Ashley, Patterson and Hinich (1986) reported the Hinich test on the SETAR model and they discovered you could reject 33, 55, and 88 percent for sample sizes of 256, 512, and 1024, respectively. Hsich and LeBaron (1988) tested SETAR using the BDS test and concluded the BDS test has very little power on small sample sizes, but showed reasonable results for larger sample sizes(e.g. 1000). The McLeod-Li, Keenan and Tsay tests have all been utilized as threshold autoregressive models and considered effective test statistics in the presence of uncertainty about regime shifting. Petruccellli and Davies (1986) applied the threshold autoregressive models using a portmanteau test. They compared the power of detecting general nonlinearity using the Keenan test as an F test and the portmanteau test proposed by McLeod and Li (1983) based on autocorrelation of

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squared residuals from a linear fit. The portmanteau test displayed better performance than the Keenan test. Therefore, the study recommended the application of portmanteau test to detect and identify self-exciting threshold autoregressive type nonlinear models. However, it has strength on the independence of the optimal choice of pre-filtering, and using cumulative sums proves the correct choice in identifying threshold type models. A number of studies have examined the issue of sample sizes. Davies and Petruccelli (1986) reported the poor power of identifying small samples using Keenan (1985) and Mcleod and Li (1983) tests. They found both tests only have power with small sample sizes (i.e. less than 200) when the time series are near nonstationary. Tsay (1989) applied a generalized version of the tests by Petruccelli and Davies (1986). He found the new Tsay test had higher power than the portmanteau test proposed by Mcleod and Li (1983) on threshold autoregressive models. The threshold-disturbance models by Elwood (1998) are asymmetric moving average models (asMA) which can be used to discriminate between the effects of positive and negative shocks of output with apparent asymmetry in their persistence. Elmwood applied this technique to investigate GNP and industrial production data. He found both GNP and industrial products (IP) series were wellfitted by asymmetric moving average models. Brännäs and Ohlsson (1999) applied the autoregressive asymmetric moving average (ARasMA) models. ARasMA models nest the linear autoregressive moving average (ARMA) model as a special case and thereby enables a straightforward linearity test. Their results showed the detection of nonlinearities depended on sample frequency. They also found symmetry and linearity for yearly data, but not on monthly or quarterly data using Swedish unemployment rates. Wecker (1981) applied the asMA models on stock market and six industrial price series data and confirmed that the series exhibited asymmetric behavior. The Barnett et al. (1997) test applied ARCH and GARCH models to compare the power of nonlinearity tests using the Hinich bispectrum, BDS, Lyapunov exponent, White, and Kaplan test statistics. They found the BDS test to have high power on large sample cases, but marginal power with small samples. It was also discovered that the Hinich’s bispectrum test weakly accepted linearity with small samples, but rejected linearity in larger sample cases. For structural asymmetric and asymmetric error correction models, Granger and Lee (1989) proposed a cointegrated relationship between a set of variables and allowed the speed of adjustment of the endogenous variables to depend on whether the current deviation of the cointegrated vector was positive or negative. Using the asymmetric error correction model, Cook et al. (1999) tested the nonlinearity of simulated series using different asymmetric adjustment parameters with different variance. They performed the nonlinear tests using a typical F type test and found it was difficult to reject linearity in small sample sizes and nonlinearity can only be

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detected with the higher asymmetric adjustment parameter with larger sample sizes (e.g. 500). Detection of the nonlinearity of asymmetric models encounters several issues regarding the power of the tests. First, each nonlinearity test may have more power depending on the data generating process. Classifying the power of tests with different models enabled us to choose appropriate nonlinearity tests in the presence of uncertainty regarding model specification. The nonlinearity test itself does not give the underlying process when the null hypothesis of linearity is rejected. However, the test results can yield valuable model identification information even though it conveys little information about the kind of appropriate nonlinear model. The size of sample is one of the main issues with the performance of tests. Economic data often do not provide large sample sizes; therefore, identifying the boundary of the appropriate size of sample for each nonlinearity test is an integral part of this study. Economic data are potentially weak in the detection of nonlinearity where the available sample size does not provide an adequate representation of the feasible range of behavior implied by the true data generating process. In this study, we considered sample sizes of 100, 250, 500, and 1000 to examine this issue. The power sensitivity of tests may depend on the degree of asymmetry. Tests may not be able to detect the nonlinearity of asymmetric time series if the asymmetry is not high enough to distinguish them from linear processes even though the true data generating process is nonlinear. Taking this difficulty into consideration, our study included a flexible range of data generating processes to investigate this issue. 3. Size and Performance of Nonlinear Tests We discussed various forms of asymmetric time series models and proposed nonlinear tests to detect the nonlinearity without alternative specification. To detect the nonlinearity on time series data it is necessary to check each nonlinear test’s appropriate size and performance. We tested their behavior on both small and large samples to capture the continuum of sample sizes encountered with various economic and financial applications. To conduct nonlinearity tests on asymmetric models, we calculated the size of tests using asymptotic and sample critical values for each nonlinearity test. We generated AR (1) series using coefficients 0.75, 0.50, 0.25, and 0, which allowed us to see how the behavior of the test statistics impacts different data generation processes. The AR (1) process is from the following form: (5) xt   xt 1  et where et ~ iid (0,1). To consider the sample size effect, we included sample sizes of 100, 250, 500, and 1000. The AR (1) series is generated by IMSL routines of GGUBS and GGNML in the GENARMA command of the B34S statistical analysis program® developed by Stokes (1997). Each test statistic is obtained from 5000 replications of each AR (1) series2. To remove the starting value effects on the autoregressive

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process, we generated 200 additional observations and removed them before we applied the remainder of the series. The BDS and bicorrelation tests need ARMA prefiltered residuals from a linear model. To generate the residuals, we used an AR (1) model to estimate the original series using a least squared estimator. In this section, we first conducted the Monte Carlo study to obtain the small sample critical values of the tests, and performed the size tests on the null models using both asymptotic critical values and sample critical values. Tables for critical values for all tests were omitted due to space concerns (they are available upon request). 3.1 Size of Nonlinearity Tests Using the calculated sample critical values, we applied the nonlinear tests to discover how the critical values affected the size of each test. To test the sample data, we generated AR (1) models: xt   xt 1  et , where t=1,…,n; n=100, 250, 500, 1000, and =  5 and et is iid (0,1) random variable. Random numbers were generated using the RNDN command in the GAUSS statistical program. The process was replicated 1000 times for each model. To remove the starting value problem we discarded the first 200 observations as before. Table 1 shows the results of the size tests for each nonlinear test. The numbers in parentheses show the percent of rejection of the true null model. As we discussed earlier, the Engle, Keenan, Tsay, RESET69 and Mcleod-Li tests all showed no significant difference between sample critical values and asymptotic values. When we applied both critical values for our generated AR (1) data, the size test results were very similar except for the RESET77, BDS, bispectrum, and bicorrelation tests which showed the different size of tests between sample critical values and asymptotic values. In Hinich’s bispectrum tests, the tests of M=M showed the sample critical values as a good statistic. The size of tests indicated it is possible that if we use the normal value, the test may be biased toward linear. The RESET77 test showed that asymptotic values are not good statistics to use. The sample critical values showed fewer problems, but still they were in the 95% confidence interval (3.6-6.4 for 1000 iterations). Overall, the size of the nonlinearity tests indicated that sample critical values were preferred to asymptotic ones for the proper size of the null of linearity. To be cautious, we used the sample critical values with asymptotic values throughout this study. 4. Nonlinearity Tests of Asymmetric Time Series Models 4.1 Threshold Autoregressive (TAR) Model We generated TAR (1) models using

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xt      xt1    xt1  ut

23

(6)

 t i

where xti  max( xt i ,0) , x  min( xt i ,0) , and ut ~ iid (0,1). To examine the effects of coefficient distance between positive and negative coefficients, we generated 8 different coefficient values with =0. We considered the coefficients for ( +,  - ) from -0.95 to 0.953. We selected coefficients to show past negative shocks have greater effects than positive shocks. In business cycle applications, the research is focused on the asymmetry with steepness; moderate positive slope during expansions in contrast with the relatively steep negative slope during contractions. For example, we considered the cyclical movement to replicate asymmetry in the direction of movement of the business cycle for negative values. The TAR model has been applied to regime shifting effects on various time series like business cycles, monetary policy, and exchange rates. Detecting nonlinearity of the series is the first step in the analysis for this type of data. To prevent any distortion from different errors with different tests, we generated the sample data using the same asymmetric mechanism. We included eight different degrees of asymmetric series with sample sizes of 100, 250, 500, and 1000. We attempted to include all possible settings of each test with 1000 replications. From this procedure, we drew summary graph 11. The X axis is the different nonlinear test statistics, the Y axis is the TAR models by degree of asymmetry, and the Z axis is the rejection percent rate from 1000 replications. At the 5% significance level, the RESET, Keenan, and Tsay tests have superior power to test the nonlinear process of the TAR models. When the size of sample and the degree of asymmetry are sufficiently large, these are very good test statistics to reveal a potential asymmetric nonlinear process of autoregressive models. The test performance depends on sample size and degree of asymmetry of the series. Although the successful tests displayed very strong power, they still have limited power to detect nonlinearity when sample size and the degree of asymmetry are small. Despite the limitation to detect the nonlinearity of TAR models, the results can be applied to most time series data if the underlying process of the data displays doubt regarding regime shifting or threshold like process. 4.2 Smooth Transition Autoregressive (STAR) Models We applied our general nonlinearity tests with the ESTAR and LSTAR series. The ESTAR models are symmetric processes while LSTAR are asymmetric processes. For the STAR model,

xt  .2 xt 1  1.1xt 1F ( xt 1; , c)  ut where

(7)

ut ~nid (0,1).

For the ESTAR model, we used the smooth function: MODEL 1 : F ( xt 1 ,1000,0)  1  exp1000( xt 1  0)2  ,

(8)

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MODEL 2 : F ( xt 1 ,10,0)  1  exp10( xt 1  0)2  .

(9)

For the LSTAR model, we used the smooth function: MODEL 1 : F ( xt 1 ,1000,0)  [1  exp1000( yt 1  0)]1 MODEL 2 : F ( xt 1 ,10,0)  [1  exp10( yt 1  0)]

1

(10) (11)

As previously mentioned, F ( yt d ,  , c) becomes a Heaviside function if the smoothness parameter  goes to ∞ and becomes an AR (p) model if  approaches 0. Our model considered  = 10 and 1000 to examine nonlinearity test properties depending on the smoothness parameter. For the BDS test and the bicorrelation tests, we obtained residuals from the AR(1) model. With higher  and with the exponential property of (4-2), it can easily become an autoregressive process. The coefficients of exponential modes are dependent upon the absolute values of past values, which indicate symmetric behaviors; however, the logistic STAR always has unique coefficients corresponding to the past values as an asymmetric process. The nonlinear test results of the ESTAR model with different  : The process can be very close to a TAR process with the nonlinear tests displaying strong rejection of linearity for the models. The nonlinear test results of the LSTAR model with different  : The test results revealed nonlinearity tests have good power for TAR tests and also have higher power than the other models. For large samples, the BDS and bicorrelation tests performed well on LSTAR models. The RESET, Keenan, and Tsay tests successfully rejected the linearity on both LSTAR models which was consistent with the TAR models. For small sample sizes, those tests also rejected the null at high rates, especially on the lower  model (LSTAR Model 2) where the tests rejected all the linearity at higher than 95 % except for the Tsay test (93.9%). The ESTAR model had difficulty detecting nonlinearity using the 10 general nonlinear tests regardless of the setting of  and the sample size. Since the ESTAR with higher  approaches a symmetric autoregressive (AR) process, it may be a linear process. For lower  , the smooth parameter may make the process smoother, but it can also make it more difficult to distinguish between a linear and nonlinear process. LSTAR models exhibit highly nonlinear processes. The RESET, Keenan, and Tsay tests are able to detect the nonlinear property at high probability for all sample sizes except N=100. The LSTAR with higher  approaches a threshold autoregressive model, which means MODEL1 is similar to the TAR process (linearity test results confirm this finding). All the test statistics strongly rejected the linearity of the TAR series as well as the LSTAR process. If the  approaches 0, the nonlinearity power tends to be enhanced, and other inferior test statistics of TAR (1) tend to increase the rejection rates (see Hinich’s bispectrum and bicorrelation test.)

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4.3 Asymmetric MA1 (asMA) Model The asymmetric moving average models (asMA) (also referred to as the thresholddisturbance models by Elwood [1998]) can be used to discriminate between the effects of positive and negative shocks to output with apparent asymmetry in their persistence. Using this type of model, he investigated the GNP and industrial production data series and found both GNP and industrial products (IP) series are well-fitted by asymmetric moving average models. Brännäs and Ohlsson (1999) used autoregressive asymmetric moving average (ARasMA) models. These models nest the linear autoregressive moving average (ARMA) model as a special case, thereby enabling a straightforward linearity test. They found that the detection of nonlinearities could depend on the sample frequency. They also found symmetry and linearity for yearly data, but not on monthly or quarterly data using Swedish unemployment rates. Wecker (1981) used stock market and six industrial price data series and confirmed that the series exhibited asymmetric behavior using asMA models. We tested nonlinearity to see how we can use available nonlinear tests to find the nonlinearity of an asMA series. To generate the asymmetric MA1 model, we used

xt     ut1   ut1  ut

(12)

where xti  max( xt i ,0) , xti  min( xt i ,0) , and ut ~ iid (0,1). To diagnose the effects of the degree of asymmetry between two parameters, data were generated using 8 different coefficient values with =0. We considered coefficients for (+,-) from -09.95 to 0.95. Bispectrum tests of asMA (1) models showed it is difficult to detect the nonlinearity of the asymmetric moving average process. We found the test had very weak power on the asMA models, but showed better performance than the TAR and LSTAR models. For small samples, it is difficult to detect the nonlinearity from the asymmetric series. The power of bispectrum tests improves as sample size increases but it is still weak. The average of the test statistics from the range of upper and lower M performed better than the tests which set M as a square root of the number of observations. Overall, it is very difficult to detect nonlinearity with asymmetric MA models using bispectrum tests (only able to detect nonlinearity in large data sets with high degree of asymmetry). RESET77, Keenan, and Tsay tests displayed superior power to test the nonlinear process of asMA models at the 5% significance level. When the size of samples and the degree of asymmetry are sufficiently large, these tests proved to be very good test statistics to detect the asymmetric nonlinear process of the asMA models. The test performance depends on the sample size and degree of asymmetry of the series. Even though the successful tests showed very strong power, they still revealed limited power when sample size and the degree of asymmetry were small.

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Despite the limitation to detect the nonlinearity of asMA models, the results can be applied to most time series data if the underlyig process of the data displays doubt about regime shifting or a threshold like process. 4.4 Asymmetric Error Correction Model To test the Granger-Lee type error correction model, we followed the same data generation process as Cook et al. (1999) We generated xt using AR(1) with coefficient of 0.5, and used different values for symmetric and asymmetric coefficients of the error correction model. We generated 1000 replicated data sets with 4 different sample sizes using the RNDN command in the GAUSS statistics program to get standard N (0,1) residuals. The asymmetric error correction series were generated , (13) y  1 ( y  x)t1  1 ( y  x)t1  t where

xt  0.5xt 1  et ,

et ~ (0,1) ,

( y  x)t 1 if ( y  x)t 1  0 ( y  x)t1 =  otherwise 0 ( y  x)t 1 if ( y  x)t 1  0 ( y  x)t1 =  otherwise 0 and t ~ N (0,1) . Cook et al. (1999) used sample sizes of 50, 100, and 500 while we applied 100, 250, 500, and 1000. We also included the symmetric and asymmetric coefficients (+,-) of (-.25, -.25), (-.5, -.25), (-.75, -.25), and (-.95, -.25). To consider the relative size of the tests for asymmetric error correction models, a symmetric series (-.25,.25) was also considered. We also extended the asymmetric adjustment parameter to (-.95, .25) to examine the relative size of the tests based upon the degree of asymmetry. The ECM model needs different settings for nonlinear tests. As previously discussed, BDS and bicorrelation tests need to use residuals obtained from the initial regression. We regressed the following model and obtained the residuals (t), (14) y  1 ( y  x )t 1  t and used  yt for the other tests. Model 1, as a symmetric ECM series, displayed the reasonable size of tests for each nonlinear test. All the tests (except bispectrum) used the mean test statistic and exhibited reasonable sizes of tests for the asymmetric ECM model. The statistics using the mean of upper and lower M tests were very conservative in testing for nonlinearity of the ECM model. The size of tests revealed that less than 1.3 and 0.4 percent were rejected at 5% significance level. The BDS test showed relatively high power with the EMC model (especially for sample sizes of 500 and 1000). The results of our simulations can be summarized as follows:

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1) Rejection frequencies increased as sample size increased; however, they remained low for smaller sample sizes (N=100 and N=250), which are typical of macroeconomic data sets. 2) Rejection frequencies increased as the difference in the asymmetric adjustment parameter increased, reflecting the increasing degree of nonlinearity generated. 3) The RESET, Hinich’s bispectrum, BDS, and Tsay tests all exhibited better power than the ECM models; the performance of the RESET, in particular, stood out. Typical F type tests (e.g. Keenan and Engle tests) displayed less power to detect the nonlinearity of ECM, supporting the findings of Cook et al. (1999). 4.5 GARCH Class Models Since the introduction of generalized autoregressive heteroskedasticity (GARCH), a family of GARCH models has evolved. In particular, the asymmetric GARCH model was developed in order to offset some of the shortcomings of the original GARCH model. In the GARCH model, the sign of the forecast errors does not influence the conditional variance; therefore, it may contradict the observed dynamics of asset returns. The asymmetric GARCH model was developed to addresses this issue. The Monte Carlo experiment for generating GARCH series used the following basic model for all asymmetric models. yt   t (15)

 t  zt t ,  t2  w  0.75 t2i  0.09 t2 j , where w= (1-0.75-0.09)s2, and the s2 is unconditional variance. We set all other coefficients for the asymmetric GARCH models to ensure the series does not explode. Based on (51), we set the other coefficients for the other asymmetric coefficients as follows: TGARCH : (16)  t  w  0.1 t1  0.2 t1  0.75 t i We used a GARCH data generating procedure applying the GAUSS statistical program and extended it to generate the other asymmetric GARCH series. We generated 1000 samples of 100, 250, 500, and 1000 observations and used 0.25 and 0.95 as unconditional variances, s2. The BDS and the bicorrelation tests need the autocorrelation to be removed from the data, so we filtered the series using ARIMA and used the residuals from the regression for nonlinearity tests. We also included the BDS tests without pre-filtering the data (BDS1). See tables 1 & 2. Overall findings from nonlinear tests of the GARCH and asymmetric GARCH are as follows: 1) The BDS, all bispectrum with average of the test statistics between higher M to lower M, Mcleod-Li, and Engle’s LM test displayed high power. Consistent with Barnett et al. (1997), the BDS test showed outstanding results on all asymmetric GARCH models. The power of the bispectrum test was weak in rejecting the nonlinearity for the models as Barnet et al. (1997) also found.

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2) The power of nonlinearity tests was very weak with small sample sizes (e.g. 100 and 250). None of the test statistics performed well in detecting the nonlinearity with sample size = 100 and only the BDS test displayed marginal power with sample size = 250 (rejecting linearity approximately 80% of the time at the 5% significance level). 4.6 Stylized Facts of the Performance by Test Statistics We performed nonlinearity tests on various asymmetric models using different data generating processes (DGP). We used two categories of DGP; the nonlinear transmission mechanism with symmetric innovation and the linear transmission mechanism with asymmetric innovation. We found no dominant nonlinearity test across the asymmetric models. The performances of test statistics were heavily dependent upon the DGP. The BDS test performed very weakly when the nonlinear transmission mechanism with symmetric innovations was applied (TAR, LSTAR, asMA). See table 2. However, the BDS test showed strong power when the linear transformation with asymmetric innovations were used (asymmetric GARCH and GARCH models). The test showed very weak power with small sample sizes (N=100 and 250) except for the asymmetric ECM and GARCH models. Our results showed that the BDS test is not a very strong test statistics for detecting nonlinearity for threshold type models. The bispectrum test revealed very weak power for the asymmetric time series. The performance improves when we used the average of different M. It displayed strong power on GARCH type of asymmetric innovation cases (GARCH, TGARCH, and asymmetric ECM). Similar to the BDS test, this test also exhibited very weak power with the TAR, asMA, and LSTAR models. The Hinich’s bicorrelation test performed poorly with the asymmetric time series. It showed relatively good power with the GARCH class of models, while it failed to reject the nonlinearity of asMA, TAR, and asymmetric ECM. The main distinctions between the bispectrum and bicorrelation tests were: the bicorrelation test showed much better performance than the bispectrum test on the LSTAR models but weaker performance on asymmetric ECM. The McLeod-Li, and Engle tests successfully detected the nonlinearity of GARCH and asymmetric GARCH models. They also showed very weak power on TAR, asMA, and LSTAR models. Ramsey’s RESET69 and RESET77, Tsay, and Keenan tests successfully detected the nonlinearity of the TAR, asMA and LSTAR models, but exhibited very weak power on GARCH and asymmetric GARCH. These tests were very good tests when the DGP applied the nonlinear mechanism with symmetric innovation, but not for the linear mechanism with asymmetric innovation. Table 2 shows the overview results of our tests and the asymmetric models for comparison purposes. We displayed the linearity rejection rates using the sample critical values, and selected the higher asymmetric cases for each model. We selected

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models 4 and 5 from TAR, model 5 from asMA, LSTAR, model 2 from STAR, model 3 from asymmetric ECM, and higher conditional variance for asymmetric GARCH and GARCH models. The power of tests in small samples was very weak in rejecting the linearity even when the true DGP was nonlinear. As we moved to sample size = 250, there was at least one good test statistic to detect the asymmetric models in each scenario. The threshold type of asymmetric models showed that the F type test of test statistics displayed very strong power, although they performed poorly with the asymmetric volatility type models. For the later models, the bispectrum, BDS, and Engle tests showed good power; however, they lacked power for the threshold type models. 5. Summary and Conclusion Asymmetric time series models are respected for their ability to test switching regimes in economic models and unequal volatility dependent upon news from financial markets. Although economic theories support asymmetric behavior or nonlinearity of various time series data, there still exists considerable controversy relating to empirical testing with economic data. The major concerns relate to the choice of a test statistic to detect nonlinearity, the lack of robustness of each of the various statistics, and the minimum sample size necessary to detect the nonlinearities with sufficient power. We conducted a Monte Carlo simulation of various asymmetric time series models with the goal of exploring how well existing nonlinear test statistics performed in a variety of typical time series settings wherein the data generation processes and sample sizes were allowed to vary in a controlled fashion. Most of the tests were based on models with specified nulls and alternatives, thereby addressing the central criticisms which have been raised (sample sizes, the degree of asymmetry, and the performance of the different nonlinear test statistics). We performed nonlinearity tests of asymmetric time series in a controlled Monte Carlo setting. We considered two possible asymmetric data generating processes (DGP), the nonlinear transmission mechanism with symmetric errors and the linear transmission with asymmetric innovation. For the nonlinear transmission mechanism with symmetric errors, we included the threshold autoregressive (TAR) models, smooth transition autoregressive (STAR) models, asymmetric moving average (asMA) models, and the asymmetric error correction (asECM) models as a structural asymmetric model. For the linear transmission with asymmetric innovation, we included asymmetric GARCH class models such as the threshold GARCH models. To examine the asymmetric data, we generated the above data with different possible variations, various asymmetric coefficients, and various unconditional or conditional errors. We compared the power of eight nonlinear tests which are widely used in economic and financial literatures. Our null model was a simple linear process for the first nonlinear data generation process. We simulated the test as an initial data diagnostic test without any alternative. To consider limited data sizes in practical

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applications, we included sample sizes of 100, 250, 500, and 1000. We calculated the sample critical values, and used them in conjunction with the asymptotic values to perform the nonlinearity tests to prevent the small sample bias of the test statistics. The size of test showed the calculated sample critical values have reasonable size of test to accept null true models with small sample sizes and it approaches asymptotic values when the sample size increases (except Hinich’s bispectrum with M =M and RESET77 tests). The Hinich’s bispectrum tests do not follow typical bispectrum tests while original setting of M follows the normal distribution. Therefore, we used the sample critical values. The RESET77 critical values are different between the sample and asymptotic. To be cautious in the evaluation of the nonlinearity tests, we used both critical values to compare the findings. The power of tests in small samples was very weak in rejecting the linearity even when the true DGP was nonlinear. As we moved sample size from 100 to 250, at least one good test statistic to detect the asymmetric models in each scenario surfaced. The threshold type asymmetric models showed that the F test type of test statistics (the Keenan, RESET, and Tsay), have very strong power (although not powerful for the asymmetric volatility type of models). For asymmetric volatility type models, the bispectrum, BDS, and Engle tests displayed good power (although they did not exhibit good power with the threshold type of models). From the nonlinearity tests we summarize as follows: First, our study confirmed none of the test statistics were dominant relative to others. Also, dependent upon the data generating process, the test statistics exhibited very different powers. The RESET, Keenan, and Tsay tests displayed very strong powers to detect the nonlinearity of the asymmetric models generated by nonlinear transmission mechanism such as the TAR, STAR, asMA and asymmetric ECM. The BDS test weakly rejected the linearity of the LSTAR and asMA models. The BDS, bispectrum, bicorrelation, McLeod-Li, and Engle tests all showed superior power to detect the nonlinearity of asymmetric GARCH class models generated by asymmetric errors (ECM being an exception). The bicorrelation and Keenan tests performed poorly, but all other tests strongly rejected the linearity of the models. Second, the test performance was heavily dependent upon sample size. No test statistic was able to reject linearity for sample sizes of 100, but performance improved as the sample size increased. For sample size of 250, the test results were marginally strong; however, the performance was very strong when the sample size increased to 500. This explains the difficulty in applying nonlinearity tests in empirical economic studies since, typically, macroeconomic datasets are small. Third, the test is also dependent on the degree of the asymmetric mechanism. The performance improved as the degree of the asymmetry increased (i.e. difference between the positive and the negative coefficients of TAR, asMA and ECM increased). The performance improved when the conditional variance of the asymmetric GARCH models was increased.

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These findings have illustrated the difficulties encountered with empirical studies of these types of asymmetric models. Since all of our findings are comparable, other researchers may perform their own test diagnostics and compare their findings with the results herein. Although this does not give any specific evidence as to what type of model may or may not be used, it will provide a nice guideline and framework to build upon. Appendix. Details of Nonlinearity Tests 1. Bispectrum Test The skewness function 2 ( f1 , f 2 ) is defined in terms of the bispectrum Bxxx ( f1 , f 2 ) and power spectrum S x ( f ) for frequencies f1 and f 2 , where

2 ( f1 , f 2 ) | Bxxx ( f1 , f 2 ) |2 / S xx ( f1 ) S xx ( f 2 ) S xx ( f1  f 2 )

Bxxx ( f1 , f 2 ) 





 c

r  S 

xxx

( r, s)exp[i 2 ( f1r  f 2 s )] , and

(A1)

cxxx ( r, s) is the

multiplicity of third-order moments where s  r , r  0,1,2,... . The skewness function ( f1 , f 2 ) is constant over all frequencies ( f1 , f 2 ) when the time series

xt is linear, whereas ( f1, f 2 ) is zero over all frequencies if xt is

Gaussian. To compute the test statistics we estimated the bispectrum of pair

xt at the frequency

( f i , f j ) for the sample {x(0), x(1),..., x( N  1)} where f k  k / N . The

estimate of the bispectrum Fxxx ( f j , f k ) : Fxxx ( f j , f k )  N 1 X ( f j ) X ( f k ) X ( f j*k ) ,

(A2)

n 1

where X ( f j )   x(t )exp( i 2 f j t ) , and n=0,1,…,N-1 and * denotes complex 0

conjugate. Fxxx ( f j , f k ) must be smoothed to form a consistent estimator. Hinich (1982) obtained an estimator of Bxxx (m, n) by averaging over adjacent frequency pair of Fxxx ( f j , f k ) . Let the Bˆ xxx (m, n) denotes the estimated Bxxx (m, n) ; we have Bˆ xxx (m, n)  M 2

mM 1



mM 1



j ( m 1) M k ( n 1) M

Fxxx ( f j , f k )

(A3)

This estimated bispectrum illustrates that the test involves using an estimator of bispectrum, which is the average value of the Fxxx ( f j , f k ) over a square of M2 points and thus is critically dependent on the assumption of the appropriate M value. Stokes (1997) showed that the larger the M value, the less the finite sample variance and the larger the sample bias. Because of this trade-off, there is no one unique M that is

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appropriate to use in performing nonlinearity and Gaussianity tests. Hinich (1982) suggested that a good value for M is the square root of observations. Ashely, Patterson, and Hinich (1986) found that empirically larger (smaller) M values converge slower (faster) to the asymptotic distribution for the linearity test but faster (slower) for the Gaussianity test. The B34S® implementation of the Hinich test automatically does a grid search over these admissible M values to test the sensitivity of the results to the M value chosen. 2. Hinich’s Bicorrelation Test Let u(tk ) denote the residuals from the fitted linear model scaled by sample standard deviation so that its sample variance is one. To standardize its variance the r,s sample bicorrelation is multiplied by (N-s) where N is the number of observations, N s

G( r, s)  ( N  s) 1/ 2  u(tk )u(tk  r )u(tk  s)

(A4)

k 1

Under the null hypothesis, E[G(r, s)]  0 and E[G 2 (r, s)]  1 , which follows a pure white noise. In his theorem 1, the test statistic

H N is the following normalized sum

2

of the ( L  1) L / 2 values G ( r, s ) for 1
s 1

H N  L1 [G 2 ( r, s)  1]

(A5)

s 2 r 1

HN

is

asymptotically

N (0,1)

as

N

approaches

.

For

large

N,

2 Var( H N )  1  O(v(k ) N 2c1 ) where 0  c  0.5 and v(k )  k  6k  6 .

3. BDS Test The correlation integral for a set of data is C n ( ) 

1 n2

n

n

 H (  | x s 1 t 1

s

 xt |) ,

(A6)

where H is the Heaviside function 1 H (r )   0

if r > 0 otherwise

If the data are generated by a stationary stochastic process that is weakly dependent, or by a deterministic process with an invariant measure, the correlation integral converges to (A7) C ( )   H (  | x  y |)d  ( x)d  ( y) where  is understood to be the univariate distribution function in the stochastic case, or the invariant measure in the deterministic case. m-histories of the data are by the theory of state-space reconstruction,

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xtm  ( xt , xt 1 ,

, xt m1 )

33

(A8)

and a correlation integral is calculated on the basis of these constructed vectors, 1 n n (A9) C n ( )  2  H (  || xsm  xtm || ) , n s 1 t 1 where ||u|| = maxi |ui|| for u  Rm . The value of m is called the embedding dimension. If the data are deterministically generated by an n-dimensional system (A10) zt 1  f ( zt ) with observations

xt  h( zt )

(A11)

then it is a theorem that for m  2n+1 the map

J m ( z)  [h( z), h f ( z),

, h f m1 ( z)]

(A12)

is an embedding. 4. Engle Test Engle (1982) introduced the autoregressive conditional heteroskedasticity (ARCH) model, and developed a second moment test to examine the ARCH effects. In his ARCH form, (A13) yt  xt'    t where  t ~ N (o,  t2 ) , p

 t2  w  i t2i . i 1

Engle tested the null hypothesis that a1  a2  ...  a p  0 , which implies no ARCH effect. His test procedure is to obtain the estimated residual from linear model, regress yt on xt , and save the estimated residual eˆt . Using the residual, regress

eˆt2  a0  a1eˆt21  a2eˆt22  ...  a peˆt2 p   t ,

(A14)

R 2 from the regression. The test of the null hypothesis of a1  a2  ...  a p  0 follows a  2 ( p) distribution under the null hypothesis

and calculate

of no ARCH dependence. Engle’s ARCH LM test has been used to test no ARCH effect, and it is very useful to test the time series with second moment tests as a form of nonlinearity. To use the test, we have to set the autoregressive order p in squared estimated residuals. In our test, we chose the range from 1 to 5 to evaluate the test performance.

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5. Keenan Test Keenan (1985) assumed the series can be adequately approximated by a secondBased on truncated Volterra expansions to polynomials, a strictly stationary process, xt can be estimated by AR (p) model, p

xt  a1   1i xt i   t .

(A15)

i 1

p

xˆt  aˆ1   ˆ1i xt i

(A16)

i 1

From the estimate, we get the estimated residual,

ˆt

and square of fitted value xˆt2 .

Regress xˆt2 to the same dependent variable, p

2 ˆ We get eˆt , xˆt  aˆ2    2i xt i  et and regress i 1 ˆt   eˆ  vt

(A17)

,

and get

vˆt . Using ˆt , eˆt , and vˆt , we calculate the test statistics

ˆ ' eˆ(eˆ ' eˆ)1 eˆ ' ˆ . Fˆ1,n-2p-2  v ' v /(n  2 p  2) of linearity we can use the test statistics to test linearity.

(A18) (A19)

6. Tsay Test Using a strictly stationary process, xt , Tukey’s (1949) nonadditivity is as follows: (1) Regress xt on Wt, denote (A20) xt  Wt '   t , where Wt= (1, xt-1,… xt-p )’, and save estimated residuals ˆt . (2) Construct the vector Zt by Zt'  vech(U t'U t ) , where (.) denotes the half-staking vector operator and Ut= (xt-1,… xt-p )’. Regress Zt on the explanatory variable

Zt  Wt '  et

and obtain the estimated residual products of xt. The residual

eˆt

(A21) . The vector Zt contains the unique cross

eˆt includes p (p+1)/2 vectors since Zt has p (p+1)/2

vectors. (3) Regress the residual ˆt on

eˆt , and get vˆt ,

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 t  et '   v t .

35

(A22)

(4) Define

(eˆt ' ˆt )(eˆt ' eˆt ) 1 (ˆt ' eˆt ) / m (A23) (vˆt ' vˆt ) /(n  M  m  1) The Tsay test is a more general nonlinear form than the Keenan test. The Keenan test is the same as the Tsay test when {xt} is strictly stationary AR (1) process. However, if {xt} is not AR (1), then the Keenan test is a special case of the Tsay test with the difference being the auxiliary regression. If {xt} follows as AR (2) process, the Keenan test is Fm,n  M m1 

2

xˆt 2  aˆ   ˆi xt i  et

i 1 . However, the Tsay test uses the following auxiliary regression,

(A24)

2

xˆt 12  aˆ1   ˆ1i xt i  e1t i 1

(A25)

2

xˆt 1 xˆt 2  aˆ2   ˆ2i xt i  e2t i 1

(A26)

2

xˆt 2 2  aˆ3   ˆ3i xt i  e3t i 1

,

(A27)

and et = {e1t,e2t,e3t}. This procedure reduces to Keenan if one aggretates xt with weights determined by the initial regression, to become a scalar variable. 7. McLeod-Li Test Suppose that n observations, x1,…,xn, of a time series are generated by an ARIMA (p,q) model in which the innovation et are is independent and identically distributed p

q

i 1

i 1

xt  w   i xt i   j et  j  et ,

(A28)

where w, i, and j are coefficients. Regress the above model to get the estimated residuals, eˆt . A significance test is provided by the portmanteau statistic for fixed M using the autocorrelation function of squared residuals, M Q*  n (n  2) [rˆ(i )]2 (n  1) ,

 i 1

n

where

 (eˆ

rˆ(k )  t  k 1

2 t

 ˆ 2 )(eˆt2 k  ˆ 2 )

n

 (eˆ t 1

2 t

 ˆ 2 ) 2

, and

ˆ 2   eˆt2 / n .

(A29)

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DECEMBER 2012

Q* follows asymptotically 2 (M) if the residuals are independent. We can test the null hypothesis of linearity using the statistics with different M on the application. 8. RESET (Regression Specification Error Test) The approximate linear combination form of the model can be, Y  X   Z  u .

(A30)

The null for the RESET examines whether  = 0. For Z, Ramsey (1969) proposed square of the estimated fitted values of Y. 1) A strictly stationary process, xt can be estimated by AR (p) model, p

xt  a1   1i xt i   t .

(A31)

i 1

2) From the estimate, we get the estimated residual,

ˆt

and fitted value xˆt . Regress

xt to the power of fitted value xˆt , p

m

i 1

j 2

xt  a2   1i xt i   j xˆtj  vt . 3) We get

vˆt , and calculate (ˆ ' ˆ  vˆ ' vˆ) / m  1

Fˆm-1,n-m 

v ' v /(n  m)

.

(A32)

(A33)

The null hypothesis of RESET is Ho : 2 = 3 …= m Thursby and Schmidt (1977) extended the RESET using the power of the explanatory variables, a Taylor series expansion in the explanatory variables, and powers of principle components of explanatory variables. We included the power of the explanatory variables for a modified RESET since their Monte Carlo test revealed it had superior power for both omitted variable models and incorrect functional form models. The modified RESET substitutes the Z in the equation to the power of X. After first regression, the auxiliary regression is: regress ˆt on AR (p) and the m power of the explanatory variables. p

p

m

ˆt  a2   1i xt i  ij xˆtji  vt

, save the residuals, vˆt . The modified RESET statistic is i 1

i 1 j 1

(ˆ ' ˆ  vˆ ' vˆ) / m  1 Fˆm-1,n-mp- p-m  v ' v /(n  mp  p  m) .

(A34)

(A35)

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Notes 1 Eun S. Ahn is an associate professor of University of Hawaii - West Oahu, and Jin Man Lee is a research director and adjunct professor at DePaul University and University of Chicago. Corresponding author: Eun S. Ahn, University of Hawaii West Oahu, [email protected] 2 10,000 replications for Hinich’s bispectrum and bicorrelation tests were applied as the asymptotic critical values were unknown. 3 The considered (0.25, 0.35), (0.25, 0.50), (0.25, 0.75), (0.25, 0.95), (-0.35, -0.25), (0.50, -0.25), (-0.75, -0.25), and (-0.95, -0.25). The sample sizes for all models were 100, 250, 500, and 1000, respectfully. References Ashley, R., Douglas Patterson, and Melvin J. Hinich (1986), “A Diagnostic Test for Nonlinear Serial Dependence in Time Series Fitting Errors,” Journal of Time Series Analysis, Vol. 7, No.3, 165-177. Ball, Laurence and N. Gregory Mankiw (1994), “Asymmetric Price Adjustment and Economic Fluctuations,” Economic Journal, Vol. 104, 247-261. Barnett, William A., A. Ronald Gallant, Melvin J Hinich, Jochen A. Jungeilges, Daniel T. Kaplan, and Mark J. Jensen (1997), “ A Single-Blind Controlled Competition Among Tests for Nonlinearity and Chaos,” Journal of Econometrics, Vol. 82(1), 157-192. Black, Fischer (1976), “Studies of Stock Price Volatility Changes,” Journal of the American Statistical Association, Business Economics, 177-181. Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, Vol. 31, 307-327. Box, G. E. P. and G. M. Jenkins (1976), Time Series Analysis, Forecasting and Control , 2nd ed, Holden-Day, San Francisco. Box, G. E. P. and Pierce, D. A., "Distribution of the Autocorrelations in Autoregressive Moving Average Time Series Models", Journal of American Statistical Association, 65 (1970): 1509-1526. Brännäs, Kurt and Henry Ohlsson (1999), “Asymmetric Time Series and Temporal Aggregation,” The Review of Economics and Statistics, Vol. 81, 341-344. Brock, W. A., Dechert, W. D., and Scheinkman, J. (1996), “A Test for Independence Based on the Correlation Dimension,” Econometric Reviews, Vol. 15, 197235. Choi, Woon Gyu (1999), “Asymmetric Monetary Effects on Interest Rates across Monetary Policy Stances,” Journal of Money, Credit, and Banking, Vol. 31, No. 3, 386-416. Christie, A (1982), “The Stochastic Behavior of Common Stock Variance: Value, leverage and interest rate effects,” Journal of Financial Economics, Vol. 10, 407-432.

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Clements, Michael P., and Jeremy Smith (1999), “A Monte Carlo Study of the Forecasting Performance of Empirical SETAR Models,” Journal of Applied Econometrics, Vol. 14, 123-141. Cook, Steven, Sean Holly, and Paul Turner (1999), “The Power of Tests for NonLinearity: The Case of Granger-Lee Asymmetry,” Economics Letters, Vol. 62, 2, 155-159. Cover, James Peery (1992), “Asymmetric Effects of Positive and Negative MoneySupply Shocks,” Quarterly Journal of Economics, Vol. 107(4), 1261-1282. Davies, Neville and Joseph D. Petruccelli (1986), “Detecting Non-Linearity in Time Series,” Statistician, Vol. 35, No. 2, 271-280. DeLong, J. B. and L. H. Summers (1986), “Detecting Non-Linearity in Time Series,” The American Business Cycle: Continuity and Changes, Chicago, IL, Chicago University Press. Elwood, S. Kirk (1998), “Is the Persistence of Shocks to Output Asymmetric?,” Journal of Monetary Economics, Vol. 41, 411-426. Engle, Robert R (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U. K. Inflation, Econometrica, 50, 987-1008. Engle, Robert F and Victor K. Ng (1993), “Measuring and Testing the Impact of News on Volatility,” Journal of Finance, Vol. 48(5), 1749-1778. Falk, B. (1986), “Further Evidence on the Asymmetric Behavior of Economic Time Series over the Business Cycle,” Journal of Political Economy, Vol. 94, 1096-1109. Fornari, Fabio and Antonio Mele (1997), “Sign and Volatility-Switching ARCH Models: Theory and Applications to International Stock Markets,” Journal of Applied Econometrics, Vol. 12, 49-65. Granger, C. W. J. and T. H. Lee (1989), “Investigation of Production, Sales and Inventory Relationships using Multicointegration and Non-Symmetric Error Correction Models,” Journal of Applied Econometrics, Vol. 4, S145-S159. Granger, C. W. J and Timo Teräsvirta (1993), Modeling Nonlinear Economic Relationship, Oxford University Press, New York. Hinich, Melvin (1982), “Testing for Gaussianity and Linearity of a Stationary Time Series,” Journal of Time Series Analysis, Vol. 3, 169-176. Hinich, Melvin (1986), “Testing for Dependence in the Input to a Linear Time Series,” Nonparametric Statistics, Vol. 6, 205-221. Hicks, J. (1950), The Contribution to the theory of the trade cycle, Oxford: Clarendon Press. Hsich, D, and LeBaron B. (1988), “Small Sample Properties of the BDS statistics, I, II and III,” Working paper, University of Chicago and University of Wisconsin-Madison. Karras, Georgios (1996), “Are the Output Effects of Monetary Policy Asymmetric? Evidence from a Sample of European Countries,” Oxford Bulletin of Economics and Statistics, Vol. 58, No. 2, 267-278.

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Karras, Georgios and Houston H. Stokes (1996), “Why are the Effects of MoneySupply Shocks Asymmetric? Evidence from Prices, Consumption, and Investment,” Working Paper, University of Illinois at Chicago. Keenan, Daniel MacRae (1985), “A Tukey Nonadditivity-Type Test for Time Series Nonlinearity,” Biometrika, Vol. 72(1), 39-44. Keynes, J. M. (1936), The General Theory of Employment, Interest and Money, Macmillan, London. Kim, Jeong-Ryeol and Stefan Mittnik (1996), “Detecting Asymmetries in Observed Linear Time Series and Unobserved Disturbances,” Studies in Nonlinear Dynamics and Econometrics, Vol. 1(3), 131-143. Ljung, G. M. and G. E. P. Box (1978), “One a measure of Lack of Fit in Time Series Modles,” Biometrika, Vol. 65, 297-303. McLeod, A. I. and W. K. Li (1983), “Diagnostic Checking ARMA Time Series Models Using Squared-Residual Autocorrelations,” Journal of Time Series Analysis, Vol. 4, 269-273. Mitchell, W. (1927), Business Cycle, New York: National Bureau of Economic Research. Neftci, Salih N. (1984), “Are Economic Time Series Asymmetric over the Business Cycle?,” Journal of Political Economy, Vol. 92, No. 2, 307-328. Neumann, Manfred (1995), “Real Effects of Exchange rate Volatility,” Journal of International Money and Finance, Vol. 14, 417-426. Petruccelli, Joseph D. , and Neville Davies (1986), “A Portmanteau Test for SelfExciting Threshold Autoregressive-Type Nonlinearity in Time Series,” Biometrika, Vol. 73(3), 687-694. Ramsey, J. B. (1969), “Tests for Specification Errors in Classical Linear LeastSquares Regression Analysis,” Journal of the Royal Statistical Society, Vol. B, 21(2), 350-371. Rothman, Phillp (1992), “The Comparative Power of the TR Test Against Simple Threshold Models,” Journal of Applied Econometrics, Vol. 7, S187-S195. Sichel, Daniel E. (1989), “Are business Cycles Asymmetric? A Correction,” Journal of Political Economy, Vol. 87, 5, 1255-1260. Speight, A.E.H., and D.G. McMillan (1998), “Testing for Asymmetries in UK Macroeconomic Time Series,” Scottish Journal of Political Economy, Vol. 45, No.2, 158-170. Stokes, Houston H. (1997), Specifying and Diagnostically Testing Econometric Models, Second Edition, Quorum Books. T. Teräsvirta and H.M. Anderson (1992), “Characterizing Nonlinearities in Business Cycles Using Smooth Transition Autoregressive Models ,” Journal of Applied Econometrics, Vol. 7, S119-S136. Thursby, Jerry G. and Peter Schmidt (1977), “Some Properties of Tests for Specification Error in a Linear Regression Model,” Journal of the American Statistical Association, Vol. 72(359), 635-641.

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Tong, H (1983), Threshold Models in Nonlinear Time Series Analysis (Lecture Notes in Statistics No. 21), New York, Spinger-Verlag. Tsay, Ruey S. (1989), “Testing and Modeling Threshold Autoregressive Process,” Journal of the American Statistical Association, Vol. 84(405), 231-240. Tsay, Ruey S. (1986), “Nonlinearity Tests for Time Series,” Biometrika, Vol. 73(2), 461-466. Tukey, J. W. (1949), “One Degree of Freedom for Non-Additivity,” Biometrics, 5, 232-242. Wecker, William E. (1981), “Asymmetric Time Series,” Journal of the American Statistical Association, Vol. 76, No. 373 , 16-21. Wiener, N. (1956), “Nonlinear Prediction and Dynamics,” Proceedings of the third Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, California, University of California Press, 247-251.

Mean M

6.1(6.1)

7.7(3.6) 5.5(10.2) 5.5(5.5)

2.7(1.2) 3.3(7.7) 3.8(5.0)

4.9(.5) 4.8(5.7) 4.5(5.8)

4.0(.4) 5.3(6.0) 5.6(6.8)

6.6(.5) 7.0(7.3)

4.8(6.2)

5.0(.5) 5.4(5.8)

4.2(5.3)

5.2(.5) 5.8(5.5)

3.9(4.5)

3.0(.0) 4.3(4.2)

=0.5 4.4(4.0) 4.6(3.8) 4.7(4.0) 4.2(3.5) 4.1(3.2) 4.5(3.0) 4.5(2.6) 4.6(2.7)

69(2) 6.1(5.6) 4.1(3.9) 6.3(6.1) 3.6(3.3) 5.9(5.7) 4.1(3.9) 5.8(5.6) 6.0(5.8) 69(3) 4.9(4.1) 5.0(3.5) 6.4(5.2) 4.7(4.0) 5.1(4.4) 5.3(4.1) 5.4(5.2) 5.8(5.6) RESET 77(2) 7.5(23.6) 7.4(22.1) 7.3(23.8) 6.5(21.1) 10.1(25.9) 7.1(21.2) 7.9(25.5) 7.8(21.4) 77(3) 6.4(12.7) 6.0(11.4) 5.1(13.0) 5.0(11.2) 5.9(15.0) 5.1(12.2) 6.5(14.0) 4.7(12.5) 77(4) 5.5(10.3) 5.7(10.1) 5.6(10.5) 4.6(9.4) 6.8(11.7) 5.5(11.2) 6.0(11.5) 5.4(10.7) Number shows the percent of rejection at 5% significant Level from 1000 replication. ( ) shows the percent of rejection from 1000 replication using asymptotic value at 5%. Confident interval is (3.6-6.4)

BICORR

BISPEC

Table 1: Empirical Frequencies of Rejecting a Linear Model Based on 1000 Replications Of AR(1) Models the Empirical Frequencies Of Rejecting A Linear Model Based On 1000 Replications Of Ar (1) Models N=100 N=250 N=500 N=1000 Test Order =-0.5 =0.5 =-0.5 =0.5 =-0.5 =0.5 =-0.5 3 5.6((7.0) 5.0((6.5) 5.0(5.1) 5.3(5.4) 4.9(4.9) 5.9(6.0) 4.1(3.4) 4 6.0(6.7) 5.4(6.7) 5.8(5.5) 5.3(5.1) 5.3(4.7) 6.6(5.8) 4.1(3.0) 5 6.1(7.2) 5.5(6.6) 5.0(4.6) 5.3(4.7) 5.2(4.3) 6.5(5.8) 3.5(2.6) 6 6.8(7.0) 5.1(5.8) 5.6(5.0) 4.7(4.3) 5.8(4.3) 6.8(5.7) 3.6(2.9) BDS 7 6.6(6.4) 4.9(5.4) 5.5(4.7) 4.5(3.8) 5.2(4.2) 6.9(5.8) 4.3(2.8) 8 7.2(6.5) 4.7(4.5) 5.3(4.4) 4.5(4.1) 5.5(3.7) 6.9(6.0) 4.4(2.6) 9 6.8(6.3) 4.7(4.2) 5.5(4.6) 4.9(3.7) 5.9(3.9) 7.0(5.4) 4.4(2.6) 10 6.5(5.8) 4.9(3.7) 5.9(4.7) 4.6(3.7) 6.0(3.9) 6.5(4.4) 4.3(2.9)

VOL.9 NO.2 AHN AND LEE: NONLINEARITY TESTS 41

1 2 3 4 5 1 2 3

Order =-0.5 4.3(3.7) 5.1(4.6) 4.4(4.0) 4.3(3.9) 4.1(3.9) 6.2(5.9) 4.8(4.4) 5.0(4.9)

=-0.5 3.8(2.2) 4.0(2.8) 4.1(2.3) 3.9(2.8) 4.4(2.6) 4.6(4.3) 4.8(4.4) 4.7(4.2)

=0.5 4.6(3.0) 4.9(4.0) 3.7(2.3) 4.3(2.9) 3.9(2.8) 5.0(4.2) 5.1(5.1) 5.1(4.7)

N=250

N=100 =0.5 5.2(4.3) 4.6(3.7) 4.3(3.5) 4.3(3.8) 4.7(3.5) 5.3(5.2) 5.4(5.3) 3.6(3.5)

=-0.5 4.1(3.8) 5.2(5.0) 5.8(5.3) 5.6(5.5) 5.3(4.7) 5.6(5.6) 3.9(4.1) 4.3(4.3)

N=500 =0.5 5.6(4.9) 4.6(4.5) 4.6(4.6) 4.9(4.7) 5.6(5.0) 4.7(4.8) 5.3(5.4) 5.3(5.2)

=-0.5 4.4(4.4) 4.6(4.6) 3.8(3.8) 3.3(3.3) 3.6(3.6) 5.4(5.3) 6.7(6.6) 6.3(6.3)

N=1000 =0.5 5.1(5.1) 3.7(3.8) 4.4(4.4) 4.6(4.4) 4.3(4.3) 4.9(4.7) 5.2(5.1) 4.2(4.2)

KEENAN

1 6.3(5.6) 4.6(3.8) 6.3(6.1) 3.6(3.3) 5.9(5.7) 4.1(3.9) 5.8(5.6) 6.0(5.8) 2 6.6(5.3) 5.2(4.6) 6.1(5.9) 3.6(3.3) 5.4(5.2) 3.9(3.5) 5.4(5.3) 5.9(5.6) 4 6.5(6.0) 5.9(5.3) 5.5(5.4) 4.2(3.9) 5.5(5.4) 3.7(3.4) 5.7(5.6) 6.0(5.7) 1 4.2(2.9) 4.9(3.2) 5.1(3.9) 4.4(4.0) 5.1(4.7) 4.7(4.5) 4.2(4.2) 3.7(3.7) 2 4.4(3.1) 4.7(3.2) 4.4(3.9) 4.7(4.3) 5.6(5.5) 5.1(5.1) 3.4(3.4) 4.6(4.6) MCLEOD 3 5.7(4.7) 4.2(2.6) 4.2(4.1) 4.9(4.2) 6.2(6.3) 4.4(4.6) 4.1(4.1) 4.3(4.3) 4 5.8(3.7) 4.1(2.9) 5.3(4.8) 5.6(4.9) 5.9(6.0) 5.1(5.1) 4.5(4.4) 5.3(5.3) Number shows the percent of rejection at 5% significant Level from 1000 replication. ( ) shows the percent of rejection from 1000 replication using asymptotic value at 5%. Confident interval is (3.6-6.4)

TSAY

ENGLE

Test

Table 1: Empirical Frequencies Of Rejecting a Linear Model Based On 1000 Replications Of AR (1) Models (continued)

42 THE JOURNAL OF ECONOMIC ASYMMETRIES DECEMBER 2012

Tests TAR(4) TAR(5) LSTAR asMA ECM GARCH T GARCH A. n = 100 BDS 9.2 11.0 16.6 19.6 67.8 90.6 43.7 Bispec1 10.1 14.9 8.4 6.9 68.7 69.2 33.1 Bispec2 11.7 10.9 8.6 8.7 38.1 41.9 17.1 Bicor 7.6 11.5 15.0 7.3 19.1 69.4 35.2 McLi 5.8 7.1 8.6 8.1 19.4 75.4 32.0 Engle 6.4 7.4 10.3 8.0 26.1 82.9 32.4 RESET69 29.1 55.8 54.6 7.0 90.0 18.4 ----RESET77 27.3 60.2 63.5 77.8 ----28.5 ----Tsay 9.5 30.1 43.3 34.0 51.6 49.5 16.3 Keenan 18.8 57.4 73.6 32.0 25.2 60.9 44.4 B. n = 250 BDS 11.3 21.1 37.3 44.3 98.7 99.9 82.7 Bispec1a 6.2 25.8 24.1 16.8 97.6 94.2 69.8 Bispec2b 6.8 13.0 16.8 14.0 73.9 70.5 37.2 Bicor 10.2 16.9 40.7 7.4 23.3 87.0 60.4 McLi 7.7 9.6 21.2 16.2 45.7 97.8 68.9 Engle 9.1 11.7 27.1 15.7 55.6 99.4 67.0 RESET69 73.4 93.5 82.2 22.3 99.7 32.0 ----RESET77 54.6 94.1 99.0 99.0 ----46.1 ----Tsay 18.9 73.0 93.9 82.4 90.1 69.0 24.1 Keenan 60.4 94.5 99.1 64.1 44.3 69.9 44.4 Each number means the percent of rejection for each asymmetric model with different size of samples. For example BDS test found nonlinearity form 10% of TAR(4) model even though the true data generation process is nonlinear. BDS test show very strong detection rate on GARCH model at 90.6%, respectfully.

Table 2: Summary of Nonlinear Test Results

VOL.9 NO.2 AHN AND LEE: NONLINEARITY TESTS 43

Tests TAR(4) TAR(5) LSTAR asMA ECM GARCH T GARCH C. n = 500 BDS 15.7 30.6 67.5 70.3 100.0 100.0 98.3 Bispec1 4.3 44.7 42.8 34.2 100.0 99.3 87.1 Bispec2 6.3 28.4 25.1 20.8 86.8 86.2 56.5 Bicor 19.0 27.0 73.8 7.7 28.4 100.0 76.9 McLi 9.6 12.5 39.9 25.6 74.8 100.0 93.9 Engle 11.7 14.8 49.2 25.0 84.0 99.9 93.0 RESET69 97.4 100 94.6 49.4 ----41.5 ----RESET77 68.0 99.8 100.0 100.0 100.0 55.5 ----Tsay 38.3 97.1 99.9 99.3 99.9 80.8 26.2 Keenan 93.9 100.0 100.0 82.9 67.9 75.2 43.9 D. n = 1000 BDS 22.7 46.3 92.1 93.1 100.0 100.0 100.0 Bispec1a 4.1 72.8 77.0 77.2 100.0 99.9 97.4 Bispec2b 9.1 51.2 57.1 55.1 91.7 95.5 80.7 Bicor 32.9 43.5 93.6 10.9 34.7 99.3 89.4 McLi 15.4 18.8 69.7 46.5 97.6 100.0 99.9 Engle 17.4 22.7 77.8 44.8 99.2 100.0 99.6 RESET69 100.0 100.0 99.5 83.6 ----47.0 ----RESET77 94.4 100.0 100.0 100.0 100.0 65.1 ----Tsay 73.9 100.0 100.0 100.0 100.0 88.8 31.9 Keenan 100.0 100.0 100.0 95.2 88.1 78.6 46.2 Each number means the percent of rejection for each asymmetric model with different size of samples. For example, BDS test found nonlinearity form 15.7% of TAR(4) model even though the true data generation process is nonlinear. BDS test show very strong detection rate on GARCH model at 100%, respectfully .

Table 2: Summary of Nonlinear Test Results (continued)

44 THE JOURNAL OF ECONOMIC ASYMMETRIES DECEMBER 2012