EPISODIC NON-LINEARITY AND NON-STATIONARITY

							Labuan Bulletin

OF INTERNATIONAL BUSINESS & FINANCE

Labuan Bulletin of International Business & Finance 1(2), 2003, 79-93 ISSN 1675-7262

EPISODIC NON-LINEARITY AND NON-STATIONARITY IN ASEAN EXCHANGE RATES RETURNS SERIES
Kian-Ping Lima∗, Melvin J. Hinichb and Venus Khim-Sen Liewa
a b

Labuan School of International Business and Finance, Universiti Malaysia Sabah Applied Research Laboratories, University of Texas at Austin

Abstract A method proposed by Hinich and Patterson (1995) is employed in this study to examine the stability of the non-linear dependency structures underlying the exchange rates returns series of four ASEAN countries- Indonesia (IDR), the Philippines (PHP), Singapore (SGD) and Thailand (THB). The bicorrelation test results reveal the episodic and transient nature of these non-linear dependencies, which suggest that they are not persistent enough for investors to benefit from it. By transforming the returns into a set of binary data, the extended test procedure demonstrates that, while the GARCH-type models are commonly applied to financial time series such as exchange rates, they cannot provide an adequate characterization for the underlying process of IDR, PHP and THB bilateral exchange rates. Further investigation reveals that the violation of the covariance stationarity assumption as required by the GARCH process is due to the presence of episodic non-stationarity in the data. Given the prevalence of these episodic transient features across financial markets in the world, there is the need for researchers to take into account these salient features in their model construction. Keywords: GARCH; Non-linearity; Non-stationarity; Correlations; Bicorrelation; ASEAN foreign exchange markets.

∗

Corresponding author: Kian-Ping Lim, Labuan School of International Business and Finance, Universiti Malaysia Sabah, P.O. Box 80594, 87015 F.T. Labuan, Malaysia. E-mail: kianping@ums.edu.my.

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1. Introduction It is an accepted fact that financial economics has been dominated over the past decade by linear paradigm, which assumes economic time series conform to linear models or can be well approximated by a linear model. However, there is no theoretical reason to believe that economic systems must be intrinsically linear (see, for example, Pesaran and Potter, 1993; Campbell et al., 1997; Barnett and Serletis, 2000). This assumption of linearity, which has been made as an approximation of the real world, is now found to be inappropriate (Liew et al., 2003). With the recent breakthroughs pertaining to non-linear dynamics, coupled with the rapid acceleration in computer power, testing for non-linearity has become extremely popular in the financial econometrics literature, though the focus is on financial markets of developed countries. In principle, testing for non-linearity can be viewed as general test of model adequacy for linear models (Hinich and Patterson, 1989) and it has been argued that if the underlying generating process for a time series is nonlinear in nature, it would then be inappropriate to employ linear methods. For instance, most of the widely applied statistical tests like the unit root or stationary tests, the Granger causality test and the cointegration test are all build on the basis of linear autoregressive model. However, Sarno (2000) and Kapetanious et al. (2003), amongst others, illustrated that the adoption of linear stationarity tests are inappropriate in detecting mean reversion if the true data generating process (DGP) is in fact a stationary non-linear process. On the other hand, the Monte Carlo simulation evidence in Bierens (1997) indicated that the standard linear cointegration framework presents a mis-specification problem when the true nature of the adjustment process is non-linear and the speed of adjustment varies with the magnitude of the disequilibrium. Other related works provided by Pippenger and Goering (1993) and Balke and Fomby (1997) suggested a potential loss of power in standard unit root and cointegration tests under threshold autoregressive DGP. The major implication, as argued in Liew et al. (2003), is that estimating the linear model, implicitly disregarding any possible non-linearity in the series under consideration, will yield a mis-specified model and thereby provide wrong clues in policy matters. Thus, researchers can no longer take the linear assumption as granted. Linear model is valid only when formal linearity test result fails to provide evidence on the existence of non-linearity. To sum, testing for non-linearity is gaining popularity among researchers as a preliminary diagnostic tool to determine the nature of the DGP before any further empirical analysis. Though the presence of non-linearity suggests the inadequacy of linear models, it does not provide any insight into the appropriate functional form for the non-linear model. This is due to the portmanteau or general nature of most non-linearity tests in the literature, which do not have a specific alternative hypothesis (Brooks and Hinich, 2001). As such, even with substantial empirical evidence of non-linearity in financial time series (see, for example, Hsieh, 1989, 1991; Scheinkman and LeBaron, 1989; De Grauwe et al., 1993; Abhyankar et al., 1995; Steurer, 1995; Brooks, 1996; Barkoulas and Travlos, 1998; Opong et al., 1999), researchers have thus far failed to exploit the detected non-linearity in making improved point forecasts. This issue has been well posed by Diebold and Nason (1990: 317), who questioned: “Why is it that while statistically significant rejections of linearity in exchange rates and many other economic and financial series routinely occur, no nonlinear model has been found

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that can significantly outperform even the simplest linear model in out-of-sample forecasting?” The major challenge for researchers is thus to identify the relevant non-linear model from a vast class of potential alternatives that could characterize the returns generating process for financial time series. In this effort of model construction, it is imperative for researchers to examine the stability1 of the underlying data generating process. Specifically, if the DGP is found to be unstable, it would then be difficult to model the process accurately over a particular period of time, thus rendering prediction impossible. For instance, Brooks and Hinich (1998) found that the Sterling exchange rates were characterized by transient epochs of dependencies surrounded by long periods of white noise, which could not be captured by a GARCH model, or any of its variants. Ammermann and Patterson (2003) demonstrated that the non-linear dependencies show up at random intervals for a brief period of time but then seem to disappear again, which are not persistent enough for investors to benefit from it. These authors have suggested that such episodic dependencies are likely to be prevalent in many financial markets. Thus, it would be interesting to investigate whether the financial data of developing countries exhibit similar instability in the DGP, which would certainly pose a serious challenge to researchers in constructing a better econometric model. With this motivation, this study contributes to the current literature by broadening the existing evidence to include emerging ASEAN foreign exchange markets. The tool used to examine the issue of stability of the underlying DGP is the windowed test procedure of Hinich and Patterson (1995). This test is designed to detect episodes of transient serial dependencies within a data series, by breaking the full sample into smaller sub-samples or windows of data. In other words, this procedure examines whether the dependencies found in the full sample are in fact due to strong but episodic occurrences that appear only infrequently and fleetingly. Hinich and Patterson (1995), Brooks and Hinich (1998), Brooks et al. (2000) and Ammermann and Patterson (2003) have utilized this procedure to investigate the time series properties and stability of the underlying dynamics for financial data. The remaining structure of this paper is as follows: Section 2 discusses the windowed test procedure of Hinich and Patterson (1995) and section 3 gives a brief description of the data used in this study. Section 4 presents the empirical results as well as the analysis of the findings. Finally, conclusions are given at the end of the paper. 2. Hinich and Patterson’s Windowed Test Procedure In the windowed test procedure of Hinich and Patterson (1995), a correlation portmanteau test similar to the Box-Pierce Q-statistic is developed for the detection of correlation or linear serial dependencies within a window. For detecting non-linear serial dependencies within a window, the procedure uses a bicorrelation portmanteau
1

A stable process implies that there is one correct model that describes the underlying data generating process and the parameters of this correct model remains constant throughout the time period from which the data are drawn. For example, the model parameters of a linear autoregressive process will remain constant throughout the entire sample period. Even for non-linear processes like ARCH and GARCH, the evolution can be described by equations whose parameters are constant over time.

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test, which can be considered as a time-domain analog of the Hinich bispectrum test statistic (Hinich, 1982). In applying these tests, the full sample is broken down into smaller sub-samples or windows of data. If the full data sample does exhibit significant linear or non-linear serial dependencies, but there are only a few smaller windows that are significant, then this suggests that the data may instead be characterized by episodes of transient dependencies. In other words, it is the activity of these few windows that is actually driving the results of the overall sample. As demonstrated in the Monte Carlo simulations of Hinich and Patterson (1995), the test performed well even with small sample sizes. In this section, we provide a brief description of the test statistics used in this windowed test procedure2. Let the sequence {x(t)} denote the sampled data process, where the time unit, t, is an integer. The test procedure employs non-overlapped data window, thus if n is the window length, then k-th window is {x(tk), x(tk+1),….., x(tk+n-1)}. The next non-overlapped window is {x(tk+1), x(tk+1+1),….., x(tk+1+n-1)}, where tk+1 = tk+n. The null hypothesis for each window is that x{t} are realizations of a stationary pure noise process3 that has zero bicovariance. The alternative hypothesis is that the process in the window is random with some non-zero correlations Cxx(r) = E[x(t)x(t+r)] or non-zero bicorrelations Cxxx(r, s) = E[x(t)x(t+r)x(t+s)] in the set 0 < r < s < L, where L is the number of lags. Define Z(t) as the standardized observations obtained as follows:

Z (t ) =

x(t ) − mx sx

(1)

for each t = 1, 2,………, n where mx and sx are the sample mean and sample standard deviation of the window. The sample correlation is:
CZZ (r ) = (n − r )
− 1 n−r 2 t =1

∑ Z (t )Z (t + r )

(2)

The C statistic, which is developed for the detection of linear serial dependencies within a window, is defined as:
C = ∑ [CZZ (r ) ]
r =1 L 2

~ χ2 (L)

(3)

2

Interested readers can refer Hinich and Patterson (1995) and Hinich (1996) for a full theoretical derivation of the test statistics and some Monte Carlo evidence regarding the size and power of the test statistics. 3 A stationary time series is called pure-noise or pure white-noise if x(n1),…., nN. A white noise time series, by contrast is one for which the autocovariance function is zero for all lags. Whiteness does not imply that x(n) and x(m) are independent for m ≠ n unless the series is Gaussian.

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The (r, s) sample bicorrelation is: CZZZ (r , s ) = (n − s ) −1 ∑ Z (t ) Z (t + r ) Z (t + s )
t =1 n−s

for 0 < r < s

(4)

The H statistic, which is developed for the detection of non-linear serial dependencies within a window, is defined as: H = ∑∑ G 2 (r , s )
s = 2 r =1 L s −1

~ χ2 (L-1) (L/2)

(5)

where G (r , s ) = (n − s ) 2 CZZZ (r , s )

1

In both the C and H statistics, the number of lags L is specified as L = nb with 0 < b < 0.5, where b is a parameter under the choice of the user. Based on the results of Monte Carlo simulations, Hinich and Patterson (1995) recommended the use of b = 0.4 in order to maximize the power of the test while ensuring a valid approximation to the asymptotic theory. A window is significant if either the C or H statistic rejects the null of pure noise at the specified threshold level. This study uses a threshold of 0.01. In this case, the chance of obtaining a false rejection of the null is approximately one out of every 100 windows. With such a low-level threshold, it would minimize the chance of obtaining false rejections of the null hypothesis, indicating the presence of dependencies where these actually do not exist. Another element that must be decided upon is the choice of the window length. In fact, there is no unique value for the window length. According to Brooks and Hinich (1998), the window length should be sufficiently long to provide adequate statistical power and yet short enough for the test to be able to pinpoint the arrival and disappearance of transient dependencies, which is the main purpose of using a windowed test procedure. In this study, we follow the choice of Brooks and Hinich (1998) in which the data are split into a set of non-overlapping windows of 35 observations in length, approximately seven trading weeks. In fact, it was found that the choice of the window length did not alter much the results of both test statistics.
3. The Data

The Association of Southeast Asian Nations (ASEAN) comprises Brunei Darussalam, Cambodia, Indonesia, Laos, Malaysia, Myanmar, the Philippines, Singapore, Thailand and Vietnam. However, this study focuses only on Indonesia, the Philippines, Singapore, and Thailand (hereafter denoted as ASEAN-4) due to data availability of the selected member countries. Malaysia is excluded from this study because it adopted a fixed ringgit regime from September 1, 1998. At the time of writing, the Malaysian ringgit peg of 3.80 to the U.S. dollar has held firm.

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The daily spot exchange rates for the ASEAN-4 currencies (expressed as the price of a country’s currency in terms of the U.S. dollar) were obtained over the period January 2, 1990 to December 31, 2001. However, the daily data of Indonesia rupiah (IDR/USD) and the Philippines peso (PHP/USD) for the first half of 1990s were difficult to obtain. Thus, for these two currencies, the sample period covers November 16, 1995 to December 31, 2001. The daily data of Singapore dollar (SGD/USD) and Thai baht (THB/USD) used in this study were drawn from the Federal Reverse Statistical Release4. The source for the Indonesia rupiah (IDR/USD) and the Philippines peso (PHP/USD) daily rates came from the Pacific Exchange Rate Service5. The daily spot exchange rates obtained were expressed in logarithm form to compute a set of continuously compounded percentage daily returns, using the relationship: rt = 100 [ln (St) – ln (St-1)] (6)

where ln denotes the natural logarithm operator; St is the exchange rates at time t; and St-1 the rates on the previous trading day.
4. Empirical Results

Descriptive Statistics Table 1 provides summary statistics for all the ASEAN-4 exchange rates returns series. The means are quite small. The range of daily changes, however, is relatively high. Moving beyond the basic mean and standard deviation measurements to higherorder moments, the SGD and PHP exhibit some degree of negative or left-skewness while THB and IDR are right-skewed. On the other hand, the distribution of returns for all the series are highly leptokurtic, in which the tails of its distribution taper down to zero more gradually than do the tails of a normal distribution. Not surprisingly, given the non-zero skewness levels and excess kurtosis demonstrated within these series of returns, the Jarque-Bera (JB) test strongly rejects the null of normality for all the ASEAN-4 exchange rates returns series. These results conform to the consensus in the literature that the distributions of exchange rates returns series are non-normal (see, for example, Hsieh, 1988; Steurer, 1995; Brooks, 1996). Linear and Non-linear Serial Dependencies Table 2 presents the correlation (C) and bicorrelation (H) test statistics for the full data sets. Both the C and H statistics are highly significant, with extremely small p-

4

These daily data were obtained from the Federal Reserve Board’s official website at http://www.federalreserve.gov/releases/H10/hist on 6/2/2002. The H.10 release contains daily rates of exchange of major currencies against the U.S. dollar. 5 These daily data were obtained from the web location of Pacific Exchange Rate Service at http://pacific.commerce.ubc.ca/xr/data.html on 6/2/2002.

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values6. This indicates the presence of strong linear and non-linear serial dependencies within all the ASEAN-4 exchange rates returns series. These results corroborate those of Lim et al. (2002) which have also utilized similar data sets but with different methodologies- the popular BDS test (Brock et al., 1996) and Hinich bispectrum test (Hinich, 1982).
Table 1 Summary Statistics for ASEAN-4 Exchange Rates Returns Series
SGD Sample Period No. of observations Mean Median Maximum Minimum Standard deviation Skewness Kurtosis JB normality test statistic (p-value) 2/1/1990 31/12/2001 3016 -0.000990 0.000000 2.761759 -4.144408 0.362038 -0.953452 21.13454 41783.87 (0.000000)* THB 2/1/1990 31/12/2001 2959 0.018342 0.000000 20.76903 -6.353245 0.777282 6.071569 186.9490 4190031 (0.000000) IDR 16/11/1995 31/12/2001 1508 0.100374 0.020068 30.18815 -23.31619 2.696840 1.080621 30.52847 47909.65 (0.000000) PHP 16/11/1995 31/12/2001 1522 0.044510 0.007645 7.175615 -12.51779 0.828162 -1.271720 48.86232 133797.8 (0.000000)

* denotes extremely small p-value.

Table 2 C and H Statistics for ASEAN-4 Exchange Rates Returns Series
SGD Sample Period No. of observations No. of lags p-value - C Statistic - H Statistic 2/1/1990 31/12/2001 3016 25 0.0000* 0.0000 THB 2/1/1990 31/12/2001 2959 24 0.0000 0.0000 IDR 16/11/1995 31/12/2001 1508 19 0.0000 0.0000 PHP 16/11/1995 31/12/2001 1522 19 0.0000 0.0000

* denotes extremely small p-value.

Episodic Non-linearity In this section, we examine the stability of the detected non-linear dependency structures underlying the exchange rates returns series of these ASEAN-4 markets. One approach is to break the whole sample into relatively narrow sub-samples or
6 Instead of reporting the C and H statistics as chi-square variates, the T23 program written by the second author reports the statistics as p-values. Based on the appropriate chi square cumulative distribution value, the T23 program transforms the computed statistic to a p-value.

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windows. The window length should be sufficiently long to provide adequate statistical power and yet short enough for the data generating process to have remained roughly constant. In this study, the data are split into a set of nonoverlapping windows of 35 observations in length, approximately seven trading weeks. The windowed test of Hinich and Patterson (1995) can be applied either to the raw returns or to the residuals of an autoregressive fit of the data. Since our focus is to examine the stability of the underlying non-linear dependency structures, we apply the test to the residuals of an AR(p) fit for each series. The fitted AR(p) model serves to remove linear dependencies from the data so that a rejection of the null of pure noise at the specified threshold level is due to significant H statistic. It is found that an AR(2) model is sufficient to remove all the correlations from the ASEAN-4 exchange rates returns series under study. Table 3 presents the results for the windowed testing on residuals of an AR(2) model. The third column in Table 3 shows the number of windows where the null of pure noise is rejected by the H statistic. The fourth column is the percentage of the total number of windows where rejection occurs7. For example, for the SGD returns series, the null is rejected in 1 window by the H statistic, which is equivalent to 1.16%. By using a threshold of 0.01, we would expect the H statistic to reject 1% of the windows by random chance. The results show that the percentages of windows exhibiting significant non-linear serial dependencies are greater than the expected 1% in all the ASEAN-4 exchange rates returns series. Since the H statistic is highly significant for the full data series, one would expect these serial dependencies to be persistent throughout the data or at least many more of the windows to exhibit strong non-linear serial dependencies. Instead, these significant test results in the full data series are reflected in only a relatively few windows. In other words, it is the activity of these few windows that is actually driving the results of the overall sample. For example, out of eighty six windows for the SGD returns series, only one (1.16%) exhibit non-linear serial dependencies. It can partly be attributed to the high power of these portmanteau non-linearity tests (bicorrelation, bispectrum and BDS tests) employed in the earlier section and in Lim et al. (2002) that have masked the episodic non-linearity presence in the data. Table 3 also provides us with the dates when these episodic non-linearities occurred, which is potentially useful for our future investigation into the causes of these detected episodic behaviour. The episodic behaviour of the underlying non-linearity can be observed graphically. For example, the histograms in Figure 1 show the percentiles (i.e. one minus the pvalue) into which the H statistic falls in each window for the IDR returns series. Thus, a very significant window is plotted as a value near 1.0. It can be observed from these figures that the episodic occurrence of these non-linear dependencies appear within the data only infrequently. Another salient feature is the transient nature of these nonlinearities, in which some windows appear highly significant, but then quickly disappear, or become too weak to be detected, in subsequent windows. Similar
7

In this study, the threshold level was set at 0.01. The level of significance is the bootstrapped thresholds that correspond to 0.01.

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patterns are observed for PHP, SGD and THB8. Thus, these results demonstrate that the underlying non-linear process for all the ASEAN-4 exchange rates returns series are not stable.
Table 3 Windowed-Test Results for AR(2) Residuals
Percentage of significant H windows 1.16% 5.95%

Total number of windows SGD THB 86 84

Significant H windows 1 5

Dates of significant H windows 24/7/90-11/9/90 18/4/90-8/6/90 19/12/94-8/2/95 21/3/96-8/5/96 24/2/99-13/4/99 14/4/99-2/6/99 22/1/96-11/3/96 12/3/96-1/5/96 16/6/97-5/8/97 9/6/99-4/8/99 16/6/97-5/8/97

IDR

43

4

9.30%

PHP

43

1

2.33%

Figure 1 H Statistics for IDR Exchange Rates Returns Series
1.00 0.90 0.80 0.70

Percentile

0.60 0.50 0.40 0.30 0.20 0.10 0.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Window

8

Figures are available upon request from the authors.

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Episodic Non-Stationarity We extend the test procedure of Hinich and Patterson (1995) to search for deeper forms of non-linear dependencies. Figure 2 provides a plot of the normalized standard deviation for IDR returns series across windows9. Many empirical studies have used standard deviation or variance as well as the absolute change to measure the volatility of asset returns. As depicted, the volatility of IDR returns series is time variant. The ARCH model introduced by Engle (1982) or its extension GARCH by Bollerslev (1986) tends to be popular for it provides a parsimonious specification in capturing the time series properties of volatility. Thus, this section utilizes the test procedure of Hinich and Patterson (1995) to test whether a GARCH formulation represents an adequate characterization of the underlying non-linear process for all the ASEAN-4 exchange rates returns series.
Figure 2 Normalized Standard Deviation for IDR Exchange Rates Returns Series
4.50 4.00 3.50 3.00

Std Deviation

2.50 2.00 1.50 1.00 0.50 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Windows

The simple GARCH (1,1) model for {y(t)}can be written as: y(t) = εt ht½ εt |ψ t-1 ~ N (0, ht) ht = α 0 + α1 εt-12 + β1ht-1

(7)

To ensure a well-defined process in the context of GARCH (1, 1) model, the following customary constraints are applied to the parameters: α1 > 0, β1 > 0, and α1 +
9

Figures for PHP, SGD and THB are available upon request from the authors.

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focus of this section is to test the assumption of covariance stationarity of the unconditional variances, in which the legitimacy of any GARCH-type modelling relies heavily on.

β1< 1 suffices for covariance stationarity10. In Equation (7), εt is conditional on the information set ψ t-1 and is normally distributed with zero mean and variance ht. The

In this extended procedure, the original data series Z(t) have to be transformed into a set of binary data, {y(t)}, by setting all non-negative returns equal to 1 and all negative returns equal to –1. If Z(t) are generated by a GARCH process whose innovations {εt} are symmetrically distributed around a zero mean, then the binary set {y(t)} will be a stationary pure noise series. The justification for the binary transformation is that it turns a GARCH into pure noise11. Putting it differently, a GARCH process that has symmetric innovations produces independently distributed binary output. The binary transformed data has moments which are well-behaved with respect to asymptotic theory. Therefore, if the null of pure noise is rejected by the C or H statistic, then there are statistical structures present in the data that cannot be captured by GARCH-type models. The rejection may be due to serial dependence in the innovations but this violates a critical assumption for GARCH-type models. Table 4 presents the correlation (C) and bicorrelations (H) test statistics for the binary transformed data set {y(t)} covering the full sample period. The results show that the null of pure noise is strongly rejected by the C or H statistic, or both for returns series of THB, IDR and PHP. This implies that these returns series cannot be generated by a strongly stationary pure noise process as required by the GARCH-type models. However, for SGD, the GARCH-type models seem to be able to provide an adequate characterization of the underlying process.
Table 4 C and H Statistics for Whole Sample of Binary Transformed Data
SGD Sample Period No. of observations No. of lags p-value - C Statistic - H Statistic 2/1/1990 31/12/2001 3016 25 1.0000 1.0000 THB 2/1/1990 31/12/2001 2959 24 0.0120 0.0000* IDR 16/11/1995 31/12/2001 1508 19 0.0000* 0.0000* PHP 16/11/1995 31/12/2001 1522 19 0.0261 0.0000*

* denotes extremely small p-value.

A time series x(t) is covariance stationary if: (i) the mean of x(t) is constant over time; (ii) the variance of x(t) is also constant over time; and (iii) the covariance between x(t) and x(t+h) depends only on their time difference or lag, represented by h. 11 Though ARCH/GARCH is a martingale difference process, and thus white noise, it is not pure noise.

10

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It would be revealing to investigate the persistence of the underlying non-stationarity for these ASEAN-4 returns series, by breaking the whole sample into relatively narrow windows. Similarly, the data are split into a set of non-overlapping windows of 35 observations in length, approximately seven trading weeks. Table 5 presents the results for the windowed testing. The third column in Table 5 shows the number of windows where the null of pure noise is rejected by the C statistic. In parenthesis is the percentage of the total number of windows where rejection occurs. For example, for the THB returns series, the null is rejected in 6 windows by the C statistic, which is equivalent to 7.14%. By using a threshold of 0.01, we would expect the C statistic to reject 1% of the windows by random chance. Similarly, the fourth column provides the number and percentage of windows where the H statistic rejects the null of pure noise. The results in the fifth column of Table 5 reveal that the number of significant windows is larger than the 1% one would expect purely by chance for the THB, IDR and PHP returns series, either due to the rejection by C or H statistic, or both12. The findings from this windowed test procedure reveal that the violation of the covariance stationarity assumption as required by the GARCH process is due to the presence of episodic non-stationarity in the data. However, for SGD returns series, the null of pure noise is rejected only in 1 window by the H statistic. There is a possibility of obtaining a false rejection of the null since the percentage is marginally higher than the threshold level of 0.01. Even though the rejection by the H statistic is not by chance, the activity of this sole window is not strong enough to drive the results of the overall sample, which has been shown in Table 4 where C or H statistics cannot reject the null of pure noise for SGD covering the whole sample of binary data set.
Table 5 Windowed-Test Results for Binary Transformed Data
Total number of windows SGD THB IDR PHP 86 84 43 43 Significant C windows 0 6 (7.14%) 19 (44.19%) 5 (11.63%) Significant H windows 1 (1.16%) 8 (9.52%) 22 (51.16%) 8 (18.60%) Total number of significant C or H windows 1 (1.16%) 8 (9.52%) 23 (53.49%) 8 (18.60%)

12

One has to take note that in some windows the null of pure noise is rejected by both the C and H statistics.

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5. Conclusions

The outcome of our econometric investigation using the whole sample supports the presence of significant non-linear serial dependencies in the data generating process of all the ASEAN-4 exchange rates returns series. However, when the sample is broken down into relatively narrow sub-samples or windows, such significant nonlinear dependencies in the full data series are reflected in only a relatively few windows. In other words, it is the activity of these few windows that is actually driving the results of the overall sample, or triggers the detection of non-linearity by those portmanteau non-linearity tests. These results demonstrate that the underlying non-linear generating process for all the ASEAN-4 exchange rates returns series are episodic in nature in which the non-linear dependencies appear only infrequently. Another pertinent feature is the transient nature of these dependencies, in which some windows appear highly significant, but then quickly disappear, or become too weak to be detected, in subsequent windows. This provides a plausible explanation for the failure of researchers to exploit the detected non-linearity in making improved point forecasts. In particular, though the presence of non-linearity implies the potential of returns predictability, the dependency structures do not seem to be persistent enough for investors to benefit from it. Following the interpretation of Ammermann and Patterson (2003), these dependencies show up at random intervals for a brief period of time but then disappear again before they can be exploited. We extend the test procedure to examine whether the GARCH-type models are able to characterize the behaviour of the underlying DGP for all the exchange rates returns series under study. By transforming the returns into a set of binary data, the findings support the adequacy of GARCH-type models only for the case of SGD. However, for the returns series of THB, IDR and PHP, the null of pure noise is strongly rejected by the C and H statistics, suggesting the inadequacy of non-linear in variance models such as GARCH to describe the underlying returns series of these three markets. Further investigation using the windowed test procedure reveals that the violation of covariance stationarity assumption as required by the GARCH process is due to the presence of episodic non-stationarity in the data. These results are not surprising as Pagan and Schwert (1990a) and Loretan and Phillips (1994) have demonstrated that covariance stationary for stock market and foreign exchange data over even quite short periods of time is implausible. Thus, there should be continuous efforts to improve models that assume covariance stationarity such as GARCH-type models, as pointed out by Pagan and Schwert (1990b). May be one can consider the suggestion of Ramsey and Zhang (1997), who also found similar structures using waveform analysis which the authors described as ‘localized frequency bursts’. The authors suggested that the relevant model is one of oscillations induced by packets of information that leave the median of changes invariant to zero. The packets are characterized in time frequency space by short bursts of activity over narrow frequency ranges. This would certainly be a stimulating task. It would also be interesting to investigate the events that have contributed to these episodic dependencies in the data. This is possible because the windowed test procedure of Hinich and Patterson (1995) permits a closer examination of the precise time periods during which such non-linear dependencies are occurring, which has

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been provided in Table 3. For instance, in the work of Ammermann and Patterson (2003), the linear dependency is found to be directly attributable to changes in the Taiwan Stock Exchange’s price limits that were made during 1987 and 1988. Brooks et al. (2000) found that the non-linear dependency structures in their data were due to widespread upsets in the currency markets and a change in US accounting procedures. This line of inquiry is certainly worth investigating and will be included in future research agenda.
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