The dynamic relationship between the prices of ADRs and their underlying stocks: evidence from the threshold vector error correction model

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Applied Economics

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The dynamic relationship between the prices of

ADRs and their underlying stocks: Evidence from the

threshold vector error correction model

Huimin Chung a , Tsung-Wu Ho b & Ling-Ju Wei c a

Graduate Institute of Finance , National Chiao Tung University , ShinChu, 300, Taiwan E-mail:

b

Department of Finance , Shih-Hsin University , Taiwan c

Department of Management Science , National Chiao Tung University , Taiwan Published online: 17 Feb 2007.

To cite this article: Huimin Chung , Tsung-Wu Ho & Ling-Ju Wei (2005) The dynamic relationship between the prices of ADRs and their underlying stocks: Evidence from the threshold vector error correction model, Applied Economics, 37:20, 2387-2394, DOI: 10.1080/00036840500218729

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The dynamic relationship between

the prices of ADRs and their

underlying stocks: evidence from

the threshold vector error correction

model

Huimin Chung

a,

*, Tsung-Wu Ho

b

and Ling-Ju Wei

c

a

Graduate Institute of Finance, National Chiao Tung University, ShinChu, 300, Taiwan

b

Department of Finance at Shih-Hsin University, Taiwan

c

Department of Management Science at the National Chiao Tung University, Taiwan

This paper sets out to estimate the dynamic relationship that exists between the prices of ADRs and their underlying stocks, in both the short run and the long run, using a number of recent developments of the threshold cointegration framework. The empirical results support the notion of nonlinear mean reversion of the prices of ADRs and their underlying stocks.

I. Introduction

The relationship between nonlinear error correction models and the concept of cointegration has attracted considerable attention in recent years. Applications of the threshold cointegration, introduced by Balke and Fomby (1997), are especially popular, evidenced by the many references reviewed in Hansen and Seo (2002) on multivariate threshold vector error correction model (hereafter, VECM). More recently, Peel and Taylor (2002) used univariate threshold autoregressive model and multivariate threshold VECM to investigate the covered interest rate arbitrage in the interwar period and found strong support for the Keynes–Einzig conjecture. Enders and Chumrusphonlert (2004) applied a threshold cointegration methodology to explore the properties

of long-run purchasing power parity in the Paciﬁc nations and found that asymmetric adjustments of nominal exchange rates play an important role in eliminating deviations from long-run PPP.

Most studies on price transmission using threshold models tend to use either one threshold to separate the adjustment process into two regimes (Balke and Fomby, 1997; Enders and Granger, 1998; Abdulai, 2002; Deidda and Fattouh, 2002; Escribano and Mira, 2002; Hansen and Seo, 2002; Cook, 2003; Cook and Manning, 2003; Sephton, 2003; Oscar et al., 2004) or two thresholds to separate the adjust-ment process into three regimes (Obstfeld and Taylor, 1997; Goodwin and Piggott, 2001; Serra and Goodwin, 2002; Seo, 2003). This paper aims to pro-pose a two-regime threshold VECM for ADR and its underlying stock price.

*Corresponding author. E-mail: chunghui@mail.nctu.edu.tw

Applied EconomicsISSN 0003–6846 print/ISSN 1466–4283 online # 2005 Taylor & Francis 2387 http://www.tandf.co.uk/journals

DOI: 10.1080/00036840500218729

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Given the increasing global competition, many companies have chosen to raise capital in the USA by issuing American Depositary Receipts (ADRs) in order to diversify their capital market risk, whilst also reducing the overall cost of capital and promot-ing the ﬁrm’s reputation in the global market. Through the purchase of ADRs, investors can also indirectly invest in foreign securities as a means of circumventing foreign exchange barriers and various investment regulations. Thus, for both foreign investors and issuing companies alike, ADRs have become one of the most popular ﬁnancial instruments currently in use.

Over the past decade several researchers have examined the direct and indirect causal transmissions among ADRs and their underlying stocks. Among others, Alaganar and Bhar (2001) have examined, within the developed markets, whether arbitrage opportunities exist between ADRs and their under-lying stocks, while Rabinovitch et al. (2003) have investigated this issue within the emerging markets. However, these studies generally found that the prices of both the ADRs and their underlying stocks were the same, leaving little, if any, opportunities for arbitrage.

Under perfect market assumptions, the ADR and its underlying stock price are closely related accord-ing to the law of one price. However, in practice, deviations from this no-arbitrage relation are usually observed because of market imperfections such as transaction costs and price uncertainty due to noisy trader risk. Using the VECM, Kim et al. (2000) examine the dynamic price relationship of American Depositary Receipt (ADR) price to exchange rate and underlying stock price. As arbitrage activities only occur when the spread between an ADR and its underlying stock price is larger enough to cover trading costs, the use of threshold VECM could be potentially more meaningful in characterizing their price dynamics.

To the best of our knowledge, no study has yet been published characterizing the price dynamics between ADRs and their underlying stocks through the use of the threshold VECM. Therefore, this paper sets out to explore in two parts, the existence of var-ious arbitrage regimes and causal linkages between the prices of ADRs and their underlying stocks. This paper begins, first of all, by identifying the loca-tion of possible thresholds and then exploring the relationship leading to the determination of the error correction term in a two regime strategy. This paper then estimates a threshold cointegration frame-work in both the short run and the long run, and finds that a significant threshold effect exists in the

error correction term of the prices of ADRs and their underlying stocks.

The remainder of this paper is organized as follows. Section II introduces the econometric models, followed, in Section III, by a description of the data and the empirical results. A brief summary, along with the conclusions drawn from this study, are provided in Section IV.

II. Econometric Methods

VECM has been the major model for the analysis of macroeconomic dynamic or the causal relation-ships of stock prices. Examples of the applications of VECM include Agrawal (2001) and Calza et al. (2003). For the case of ADR and its underlying stock (UND) price, the existence of transaction costs and other market imperfection factors might cause the error correction effects on the price adjust-ment be significant only when the deviation of price between ADR and UND is larger than a certain threshold. While previous studies, such as Enders and Chumrusphonlert (2004), employed a univariate threshold model to explore the properties of purchas-ing power parity, this paper follows the Hansen and Seo’s (2002) model to develop a multivariate threshold VECM. The model is employed to estimate the threshold parameters, to construct asymptotic confidence intervals for the threshold parameters, and to develop new tests for the threshold effects of ADRs and their underlying stocks (UNDs) prices.

Estimation of the threshold parameters

Let xt be a p-dimensional I(1) time series, with

n observations, with l as the maximum lag

length. A linear VECM of order l þ 1 can be written brieﬂy as

xt¼A0Xt1ðÞ þ ut ð1Þ

where

Xt1ðÞ ¼½1 wt1ðÞ xt1 xt2, . . . , xtl0

and is the ﬁrst-order diﬀerence operator; the repressor Xt1() is k 1; A is k p; and k ¼ pl þ 2.

The error term, ut is assumed to be a vector

Martingale diﬀerence sequence with ﬁnite covariance matrix ¼ Eðutu0tÞ. Note that wt1() ¼ 0xt1 is an

I(0) error correction term. For the bivariate case

of ADR and UND price, xt corresponds to

[ADRtUNDt].

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Consider now an extension of Equation 1, provided by: xt¼ A01Xt1ðÞ þ ut, if wt1ðÞ A02Xt1ðÞ þ ut, if wt1ðÞ >

where is the threshold parameter. Note that this paper uses the absolute value of error correction term as a threshold variable. In addition to the merit of parsimony in the modeling of threshold eﬀect, the assumption is reasonable since trans-action costs tend to be symmetric for either long or short position in the ADR for its arbitrage. Alternatively, this may be written as

x_t¼A01Xt1ðÞd1tð, Þ þA02Xt1ðÞd2tð, Þ þ ut, ð2Þ where d1tð, Þ ¼ 1ð wt1ðÞ Þ, d_2tð, Þ ¼ 1ð w _t1ðÞ> Þ,

and 1(.) denotes the indicator function. The existence of the threshold eﬀect is conﬁrmed if 0 < Pðjwt1ðÞj Þ <1, otherwise the model

simpliﬁes to linear cointegration.

The threshold VECM of ADRs and UNDs can be estimated using the maximum likelihood method pro-posed by Hansen and Seo (2002). Under the assump-tion that the errors utare iid Gaussian, the likelihood

function is Ln A1, A2, X , , ¼ n 2log j j 1 2 Xn t¼1 utðA1, A2, , Þ0 X1utðA1, A2, , Þ, ð3Þ where utðA1, A2, , Þ ¼ xtA 0 1Xt1ðÞd1tðÞ A02Xt1ðÞd2tðÞ:

MLEð ^AA1, ^AA2, ^, ^, ^Þ are the values which

maxi-mize L_nðA1, A2, , , Þ in order to maximize the

log-likelihood, to hold (, ) fixed, and to compute the constrained MLE for ðA₁, A₂, Þ: This is just OLS regression: ^ A A1ð, Þ ¼ Xn t¼1 Xt1ðÞXt1ðÞ0d1tð, Þ !1 X n t¼1 Xt1ðÞx0td1tð, Þ ! , ð4Þ ^ A A2ð, Þ ¼ Xn t¼1 Xt1ðÞXt1ðÞ0d2tð, Þ !1 X n t¼1 Xt1ðÞx0td2tð, Þ ! , ð5Þ ^ u utð, Þ ¼ utð ÂA1ð, Þ, ÂA2ð, Þ, , Þ, and ^ X X ð, Þ ¼1 n Xn t¼1 ^ u utð, Þ ûutð, Þ0: ð6Þ

Note that Equations 4 and 5 are the OLS regressions of xt on Xt1() for the samples of

which jwt1ðÞj and jwt1ðÞj > , respectively.

Lnð, Þ ¼ Lnð ^AA1ð, Þ, ^AA2ð, Þ, ^ X X ð, Þ, , Þ ¼ n 2log ^ X X ð, Þ np₂ : ð7Þ

From the grid search procedure, the model with the lowest value of log j ^PPð, Þj is used to provide the MLEð ^, ^Þ, while the limitation of is 0Pðjwt1ðÞj Þ 1 0, where 0 <0<1 is a

trimming parameter; this paper sets 0¼0.05. This

paper employs the grid-search algorithm developed by Hansen and Seo (2002) to obtain the parameter estimates, with the MLEð ÂA1, ÂA2Þbeing ÂA1¼ ÂA1ð ^, ^Þ

and ^AA2¼ ^AA2ð ^, ^Þ.

Tests for threshold effects

Let H0 represent the class of linear VECM in

Equation 1, and H1represent the class of two regime

threshold VECM in Equation 2. These models are nested, with the constraint H0being the models in H1

which gratify A01¼A02. Our test will compare

H0 (linear cointegration) with H1 (threshold

cointegration).

In order to assess the evidence, both linearity and the threshold VECM are tested by using the Lagrange Multiplier (SupLM) test developed by Hansen and Seo (2002). The LM statistic employed is:

LMð, Þ ¼ vecð ÂA1ð, Þ ÂA2ð, ÞÞ 0 ð ^VV1ð, Þ þ ^VV2ð, ÞÞ1 vecð ÂA₁ð, Þ ÂA₂ð, ÞÞ ð8Þ SupLM ¼ Sup rLrrU LMð ~, Þ ð9Þ

where ~ is the null estimate of . The bootstrap method proposed by Hansen and Seo’s (2002) is employed to calculate the asymptotic critical values and p-values.

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III. Data and Empirical Results

The ADRs and UNDs series are tested for stati-onarity in this paper using unit root tests; followed by an examination of the cointegration test between the two series. If they are cointegrated, the threshold VECM is then applied to determine the short-run dynamics and the long-run equilibrium between the ADR and the UND markets.

The daily returns of three locally-traded Argentinean ﬁrms provide the data for analysis in this study, with Table 1 providing the basic des-cription of their respective NYSE-traded ADRs. Although the ADRs are priced in US dollars, UNDs in the home stock market are priced in Argentinian pesos. The prices of ADRs are calculated into the Argentinian peso price using the daily closing exchange rate. ADRs prices, the prices of UNDs, and the exchange rates used in this study were obtained from Datastream.

The log-price of the ADRs and the UNDs are used to carry out our empirical analysis, with the returns of ADRs and UNDs being calculated, first of all, by taking the difference in the log-price. Table 2 presents the results of the unit root and cointegration tests; the unit root test uses the null hypothesis versus the alternative of stationarity in the variables for the results of the Augmented Dickey–Fuller (ADF) and Phillips–Perron (PP) tests. The results thus cannot reject the null hypo-thesis of a unit root; the variables in the levels are I(1) for each of the ADR price and for those of UND. The variables in the first difference are integrated of order zero; the null hypothesis of unit root is rejected at the 5% level for the price differe-nce series. These results indicate that the two price series are integrated in the first difference, and thus validates the use of the cointegration test.

Given that all the variables of the same order are integrated, this paper uses two Johansen multivariate

Table 1. Data description

Symbol Company Industry

Shares per DR Sample period Number of observations

YPF YPF, S.A. Oil and gas

operator

1 7 Jul 93–31 Jul 04 2888

TEO TELECOM ARGENTINA

STET-FRANCE TELECOM, S.A.

Telecoms 5 12 Dec 94–31 Jul 04 2516

TGS TRANSPORTADORA DE

GAS DEL SUR, S.A.

Oil and gas operator

5 2 Jan 95–31 Jul 04 2500

Table 2. Unit root and cointegration tests for log-prices of ADRs and their underlying stocks

Unit root test

Augmented Dickey–Fuller test Phillips–Perron test

Levels First diﬀerences Levels First diﬀerences

YPF ADR 0.112758 51.53653** 0.091492 51.49286** UND 0.138284 48.78652** 0.126952 48.83657** TEO ADR 1.679652 45.80010** 1.635612 45.39878** UND 1.624543 45.71221** 1.579939 45.34922** TGS ADR 2.256933 38.23152** 1.811293 51.83980** UND 1.898783 47.40127** 1.897981 47.33906**

Cointegration tests Trace test 5% CV Max-eigenvalue test 5% CV

YPF None 78.15789** 15.41 78.15465** 14.07 One at most 0.003231 3.76 0.003231 3.76 TEO None 77.81962** 15.41 77.81962** 14.07 One at most 2.827981 3.76 2.827981 3.76 TGS None 111.4459** 15.41 107.8217** 14.07 One at most 3.624222 3.76 3.624222 3.76

Notes: Total number of sample observations is 2888 for YPF, 2516 for TEO and 2500 for TGS. UND represents the price of underlying stock.

**Indicates signiﬁcance at the 5% level.

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cointegration tests to determine whether the variables in each respective series are cointegrated. The maximum likelihood estimation procedure provides a likelihood ratio test, referred to as a trace test, with the likelihood ratio test being the test for maximum eigenvalue. The likelihood ratio statistics reject the null hypothesis of no cointe-gration at the 5% level. A feature of this approach is that the VECM contains an error correction term which reﬂects the current error in achieving long-run equilibrium. Therefore, the VECM can be used to jointly estimate the long-run relationship with short-run dynamics, a process which has been proven to be more eﬀective than Granger causality.

Table 3 provides the estimates of the linear model. In order to address the issue of linear, or nonlinear, adjustment to the long-run equilibrium, this study estimates a linear VECM, given by Equation 1, with our selection of the lag length being based upon the AIC and BIC criteria. As a comparison, this paper first of all estimates the linear VECM for the price series of the ADR and under-lying stock, reporting the results of the linear VECM estimation in Table 3. The estimated coefficients of the error correction term on the equations of the underlying stock are all significant at the 5% level.

The estimation results of the threshold VECM, and the test for the hypothesis of linearity versus the threshold eﬀect of non-linearity, provided by

Equation 9, are presented in Tables 4, 5 and 6, under the application of the SupLM test for the complete bivariate speciﬁcation. The p-values of the results supporting the threshold cointegration hypothesis were calculated using both the ﬁxed repressor and a residual bootstrap experiment, with 1 000 simulation replications. The estimated thresh-old VECM was provided by Equation 2, with our selection of the lag length being based upon the AIC and BIC criteria; it was also considered in this study that the cointegrating vector ^ should be estimated. Standard errors were calculated from the heteroscedasticity-robust covariance estimator, with the parameter estimates being calculated by the minimization of Equation 7 over a 300 300 grid on the parameters (, ).

Table 4 reports the threshold VECM results for ADR with ticker symbol ‘YPF’ along with UND. In this study, this paper selected a lag length of l ¼3, with the estimated cointegrating relationship being wt1¼ADRt11.00123UNDt1, quite close

to a unity coefficient. This paper also conducted analyses for the case where a unity coefficient is imposed, with the results being very similar. The estimated threshold parameter was ¼ 0.000368, indicating that the first regime corresponded to |ADRt11.00123UNDt1| 0.000368. This first

regime, which comprised of 78% of all of the obser-vations in the sample, is referred to in this study as

Table 3. Linear VECM estimations for log-prices of ADRs and their underlying stocks

YPF TEO TGS

ADRt UNDt ADRt UNDt ADRt UNDt

wt1 0.044* 0.035** 0.037 0.150*** 0.082** 0.035** (0.026) (0.016) (0.028) (0.036) (0.039) (0.016) Constant (103) 0.242 0.781** 4.299 17.368 2.754** 1.335* (0.414) (0.380) (3.418) (14.260) (1.332) (0.736) ADRt1 0.022 0.214*** 0.671*** 0.117** 0.072 0.056** (0.038) (0.046) (0.130) (0.052) (0.044) (0.027) ADRt2 0.068 0.053 0.036 0.614*** 0.182*** 0.019 (0.045) (0.049) (0.027) (0.063) (0.054) (0.038) ADRt3 0.009 0.084** 0.513*** 0.091 (0.051) (0.038) (0.137) (0.062) UNDt1 0.030 0.063* 0.030 0.429*** 0.098** 0.061** (0.043) (0.038) (0.019) (0.064) (0.043) (0.027) UNDt2 0.018 0.044 0.245** 0.104 0.050 0.076** (0.040) (0.029) (0.113) (0.085) (0.050) (0.035) UNDt3 0.049 0.022 0.015 0.226*** (0.036) (0.035) (0.011) (0.054) Cointegration vector estimate 0.998549 1.19591 1.041 AIC 22529.2 4510.15 18063.0 BIC 22505.9 4487.76 18046.2

Notes: Values in parentheses are Eicker–White standard errors.

***, ** and * indicate signiﬁcance at the 1%, 5% and 10% levels, respectively.

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the ‘typical’ regime. Conversely, the second regime, which was |ADRt1 1.00123UNDt1| > 0.000368,

comprised of 22% of all of the observations in the sample, and is referred to in this study as the ‘extreme’ regime.

In the ‘typical’ regime speciﬁcally, both ADRt

and UNDt have statistically insigniﬁcant error

correction eﬀects and minimal dynamics. They are close to white noise, which indicates that in this regime, ADRt and UNDt are close to random

walks. In contrast, in the ‘extreme’ regime, the asymmetry of ADRt and UNDt is implied, in

the sense that there is an error correction eﬀect in the ADR and UND equation being statistically

Table 5. Threshold VECM estimations of TEO for log-prices of ADRs and their underlying stocks First regime: |wt1| 0.439982

Percentage of Obs ¼ 0.926693

Second regime: |wt1| > 0.439982

Dep ADRt UNDt ADRt UNDt

Ind. Estimate Std error Estimate Std error Estimate Std error Estimate Std error

wt1 0.138 0.109 0.006 0.045 0.031* 0.018 1.069*** 0.188 Constant (103₎ _28.461 _29.030 _21.562* _12.326 _71.085* _40.829 _139.86 _349.526 ADRt1 0.669*** 0.157 0.018 0.072 0.207*** 0.056 0.317*** 0.080 ADRt2 0.014* 0.008 0.748*** 0.079 0.011** 0.005 0.052 0.121 ADRt3 0.466*** 0.163 0.024 0.086 0.565*** 0.100 0.102 0.074 UNDt1 0.002 0.011 0.501*** 0.086 0.004 0.004 0.079 0.098 UNDt2 0.197* 0.118 0.073 0.104 0.970*** 0.117 0.369*** 0.142 UNDt3 0.001 0.010 0.353*** 0.078 0.003 0.002 0.001 0.069

Threshold estimate ¼ 0.439982; Cointegrating vector estimate ¼ 0.789472; AIC ¼ 4740.20; BIC ¼ 4695.41. Lagrange Multiplier threshold test

Fixed regressor (asymptotic) bootstrap ¼ 103.117*** ( p-value<0.001). Residual bootstrap ¼ 34.232*** ( p-value<0.001).

Wald test

Equality of dynamic coeﬃcients ¼ 24.806*** ( p-value<0.001). Equality of EC coeﬃcients ¼ 26.127*** ( p-value<0.001).

Notes: ***, ** and * indicate signiﬁcance at the 1%, 5% and 10% levels, respectively.

Table 4. Threshold VECM estimations of YPF for log-prices of ADRs and their underlying stocks First regime: |wt1| 0.000368

wt1 0.032 0.027 0.015 0.016 0.395** 0.200 0.442*** 0.131 Constant (103) 0.579 0.643 0.774 0.478 3.324** 1.572 2.064 1.563 ADRt1 0.005 0.039 0.144*** 0.043 0.427*** 0.138 0.217** 0.109 ADRt2 0.078 0.049 0.052 0.044 0.257* 0.141 0.106 0.115 ADRt3 0.017 0.056 0.057* 0.034 0.241* 0.133 0.054 0.113 UNDt1 0.018 0.045 0.016 0.037 0.274** 0.127 0.112 0.098 UNDt2 0.015 0.038 0.018 0.027 0.197*** 0.055 0.018 0.081 UNDt3 0.018 0.036 0.009 0.037 0.238*** 0.086 0.061 0.076

Threshold estimate ¼ 0.000368; Cointegrating vector estimate ¼ 1.00123; AIC ¼ 22653.1; BIC ¼ 22606.4. Lagrange Multiplier threshold test

Fixed regressor (asymptotic) bootstrap ¼ 84.114*** (p-value<0.001). Residual bootstrap ¼ 28.306*** ( p-value<0.001).

Wald Test

Equality of dynamic coeﬃcients ¼ 34.188*** ( p-value<0.001). Equality of EC coeﬃcients ¼ 24.911*** ( p-value ¼ 0.008).

Notes: ***, ** and *indicate signiﬁcance at the 1%, 5% and 10% levels, respectively.

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signiﬁcant with dynamic coeﬃcients. All in all, ADRt

and UNDt are statistically signiﬁcant in the error

correction eﬀects in the ‘extreme’ regime, but not in the ‘typical’ regime.

The evidence of non-linearity appears to gain strength from the results of the Wald test diagnostics, thus the null hypothesis of linearity in error correction terms is rejected. Comparing the estimated coefficients of the error correction terms in Tables 3 and 4 shows that the linear error correction models imply very slow speed of adjust-ment, a result consistent to those reported in Enders and Chumrusphonlert (2004). Since the null hypothesis is of equality of the coefficients on the error correction terms and of the dynamic coefficients across the two regimes, an important finding of the estimated linear VECM and threshold VECM is that the error correction term for the ADR is negative; this result is consistent with the error correction terms. This implies specifically, that from the long-run equilibrium, the ADR adjusts to any short-run deviations. Furthermore, the negative sign of the error correction term implies that if the ADR premium is above its equilibrium level, the ADR will decline. This is as predicted in the model when the ADR overshoots its long-run equili-brium; the result is therefore just as this paper would have expected to see in this study. Details of the procedures and analyses provided above are also presented in Tables 5 and 6. The error correction term appears to be significant only in the ‘extreme’ regime. The estimated coefficients of the error

correction terms in the extreme regime appear to be larger than those in the linear VECM. The short-run dynamic effects of ADR and UND price show sig-nificant differences between ‘typical’ and ‘extreme’ regimes.

IV. Summary and Conclusions

This paper employs the threshold VECM to investi-gate the dynamic price relationship between ADRs and their underlying stocks. The results provided by the LM test statistics reject the null hypothesis of no threshold effect, while the Wald test results reject the null hypothesis of the coefficients of the error correction term in the two regimes having the same value. This study therefore provides strong evidence to show that a threshold effect does exist in the prices of ADRs and their underlying stocks.

The main findings of our analyses can be sum-marized as follows. First of all, the results based on the threshold VECM demonstrate that linearity is rejected in favour of threshold effect nonlinearity and that the estimated two-regime threshold VECM forms a statistically sufficient representation of the data with separating regimes. Secondly, through the threshold parameters, this paper classifies the ‘typical’ regime and the ‘extreme’ regime, with only the error correction effect appearing in the ‘extreme’ regime being statistically significant, since it is not significant in the ‘typical’ regime. Finally, the negative sign of the error correction term in the

Table 6. Threshold VECM estimations of TGS for log-prices of ADRs and their underlying stocks First regime: |wt1| 0.000323

wt1 0.056 0.043 0.004 0.016 0.265*** 0.090 0.374*** 0.083 Constant (103) 3.095** 1.483 2.837*** 0.920 0.705 1.247 2.619** 1.075 ADRt1 0.009 0.054 0.029 0.034 0.095 0.070 0.046 0.057 ADRt2 0.167** 0.073 0.094* 0.051 0.148** 0.075 0.060 0.063 UNDt1 0.016 0.052 0.105*** 0.032 0.213*** 0.053 0.102** 0.043 UNDt2 0.009 0.065 0.081* 0.046 0.108* 0.063 0.018 0.051

Threshold estimate ¼ 0.000323; Cointegrating vector estimate ¼ 0.993680 AIC ¼ 18146.3; BIC ¼ 18112.8. Lagrange Multiplier threshold test

Fixed regressor (asymptotic) bootstrap ¼ 20.910*** ( p-value<0.001). Residual bootstrap ¼ 17.305*** ( p-value<0.001).

Wald test

Equality of dynamic coeﬃcients ¼ 20.772*** ( p-value ¼ 0.008). Equality of EC coeﬃcients ¼ 49.256*** ( p-value<0.001).

Notes: ***, ** and * indicate signiﬁcance at the 1%, 5% and 10% levels, respectively.

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‘extreme’ regime implies that if the ADR’s premium is above its equilibrium level, then the ADR price will decline; that is, nonlinear mean reversion is evident.

Last but not least, this study points to threshold VECM, which is consistent with the stylized fact of the error correction, and suggests that the eﬀectiveness of the threshold cointegration model surpasses that of the linear cointegration model. Further analytical studies, using the threshold VECM model, should be undertaken in the future, with its application being targeted at predicting the achievements of ADRs and their underlying stock prices.

References

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