Figure 1. Research structure.
2. The Threshold Vector Error Correction Model
A natural approach to modeling economic variables is defining different states of the world/regimes in order to determine the likelihood of a time-dependent economic variable occurring at a particular time.
According to Aslanidis and Kouretas (2005), two main classes of regime-switching models have been proposed in the literature for modeling economic variables. The first class of models is the Tong and Lim (1980) initially proposed another approach which considers modeling explicitly the regime as a continuous function of an observable variable as in threshold autoregressive models. Although data have been gathered using this approach, statistical analysis techniques provide a better understanding of the regime. The second class of models is the Markov-switching type models, as originally employed in the business cycle context by Hamilton (1989). This approach assumes that the regime cannot actually be observed because the regime is determined by an underlying stochastic process.
The threshold vector error correction model (VECM) has been the major model used to analyze the macroeconomic dynamic or the causal relationships of stock prices. Examples of the applications of VECM include articles by Agrawal (2001), Calza, Gartner, and Sousa (2003), and Chen and Lin (2004).
2.1. Estimation of the Threshold Parameters
Let be a p-dimensional I (1) time series, with n observations, with l as the optimal lag length. A linear VECM of order l+ 1 can be written briefly as:
xt
∆
t 1
-t
t =A'X ( )+u
∆x β (1)
where
= p + 2. The error term is assumed to be a vector martingale difference sequence (MDS) with finite covariance matrix . Note that
l ut
Consider now an extension of equation (1), provided by:
where γ is the threshold parameter. Note that this dissertation uses the absolute value of error correction term as a threshold variable. In addition to being parsimonious in the modeling of threshold effect, use of this term is reasonable since transaction costs tend to be symmetric both long-term and short-term for arbitrage.
Alternatively, this may be written as:
,
and 1(．) denotes the indicator function. The existence of the threshold effect is confirmed if0< P
(
wt−1(β) ≤γ)
<1; otherwise the model simplifies to linear cointegration.The threshold VECM can be estimated using the maximum likelihood method proposed by Hansen and Seo (2002). Under the assumption that the errors are iid Gaussian, the likelihood function is: is just OLS regression:
( ) ( ) ( )
⎟0<π0< is a trimming parameter. Andrews (1993) suggests setting π0 between 0.05 and 0.15;
this dissertation sets π0= 0.05. This dissertation employs the grid-search algorithm developed by Hansen and Seo (2002) to obtain the parameter estimates, with the MLE (Aˆ1 ,Aˆ2) being
( )
βˆ,γˆAˆ
Aˆ1= 1 andAˆ2=Aˆ2
( )
βˆ,γˆ .2.2. Tests for Threshold Effects
Let represent the class of linear VECM in equation (1) and represent the class of two-regime threshold VECM in equation (2). These models are nested, with the constraint being the models in which gratify . The test used here will compare
(linear cointegration) with (threshold cointegration).
H0 H1
H0 H1 A1' = A2'
H0 H1
Hansen and Seo (2002) consider LM statistics for the threshold parameter. They do this for two reasons. First, the LM statistic is computationally quick, enabling feasible implementation of the bootstrap. Second, a likelihood ratio (LR) or Wald-type test would require a distribution theory for the parameter estimates for the unrestricted model, which they do not yet have. They now derive the LM test statistic.
Assume for the moment that (β,γ) are known and fixed. The model under is equation (1), and the model under is equation (2). Given
H0
H1 (β,γ), the models are linear, so the MLE is least squares. As equation (1) is nested in equation (2) and the models are linear, an LM-like statistic with robust heteroskedasticity can be calculated from a linear regression on equation (2).
The robust heteroskedasticity LM-like statistic is as follows:
( ) ( )
1 Section 2.1, but there is no estimate of γ under , so there is no conventionally defined LM statistic. Davies (1987) proposes the statistic
H0 0.15 is typically a good choice.
If the true cointegrating vector β0 is known a priori, the test takes form (10), except that β is fixed at the known valueβ0. Hansen and Seo (2002) denote this test statistic as
SupLM0 sup LM
(
β0,γ)
U L r r r≤≤
= (11)
Hansen and Seo (2002) believe that it is important to know that the values of γ that maximize the expressions in (10) and (11) will be different from the MLE γˆ presented in Section 2.1. Two separate reasons explain why these values are different. First, (10) and (11) were LM tests that were based on parameter estimates obtained under the null rather than the
alternative. Second, the LM statistics were computed with heteroskedasticity-consistent covariance matrix estimates. For this case, even the maximum of the SupWald statistics were different from the MLE (the latter equal only when homoskedastic covariance matrix estimates are used). This difference is generic in threshold testing and estimation for regression models but not specific to threshold cointegration.
Finally, this dissertation follows Hansen and Seo (2002) in developing two bootstrap methods to calculate critical values and P-values.
3. An Application to ADRs and Their Underlying Stocks
3.1. Literature Review
Over the past decade, several researchers have examined the direct and indirect causal transmissions among American Depository Receipts (ADRs) and their underlying stocks (UNDs). Among others, Alaganar and Bhar (2001) have examined whether arbitrage opportunities exist between ADRs and their UNDs within the developed markets, while Rabinovitch, Silva, and Susmel (2003) have investigated this issue within the emerging markets. However, these studies generally found that the prices of both the ADRs and UNDs were the same, leaving little, if any, opportunities for arbitrage.
Under perfect market assumptions, the ADRs and UNDs are closely related according 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 noise trader risk. Using the VECM, Kim, Szakmary, and Schwarz (2000) examined the dynamic price relationship of ADRs to exchange rates and UNDs. As arbitrage activities occur only when the spread between ADRs and UNDs is large enough to cover transaction costs, the use of threshold VECM could be potentially more meaningful in characterizing the price dynamics.
Ely and Salehizadeh (2001) employed cointegration techniques and estimated error correction models to examine the degree of integration between the United States and three foreign equity markets: UK, Japan, and Germany. They found that ADRs were cointegrated with ordinary shares trading between the three foreign markets, which implied that for long-term investors, they are a substitute for ordinary shares. Their analysis of the dynamic
relationships between ADRs and foreign equities suggested that information arising during trading hours from all the markets in the study affected portfolio valuation.
Chen, Chou, and Yang (2002) examined the price transmission effect between ADRs/
global depository receipts (GDRs) and their respective UNDs. An error-correction model was used to analyze the long-run causal relations where the stock returns data was nonstationary.
They also discussed the impact of premium and/or discount prices for overseas-listed stocks on the price transmission effect. Their results revealed a unidirectional causality from Taiwan’s capital market to other foreign markets. This asymmetry suggested that the domestic market plays a dominant role in price transmission relative to the foreign markets. Besides, both markets’ prices will adjust to establish a long-run cointegrated equilibrium.
Wang and Lin (2005) investigated the price interaction and arbitrage opportunities provided by the dual listing between the ADRs and their foreign UNDs. To inspect the linkage between the Taiwanese ADRs and their underlying shares, they applied the threshold cointegration model, which allowed for asymmetric adjustment towards a long-run equilibrium. They also examined the short-term adjustments by employing the threshold error correction model. Since some evidence of asymmetric adjustments was found in the results of the data, they implemented a complete multivariate threshold cointegration model instead of the univariate model to test for these asymmetries and determine the maximum likelihood estimation.
To the best of my knowledge, no study has yet been published characterizing the price dynamics between ADRs and UNDs through the use of the threshold VECM. Therefore, this dissertation explores two areas: the existence of arbitrage regimes and causal linkages
between the prices of ADRs and UNDs. First of all, it identifies the location of possible thresholds and explores the relationship leading to the determination of the error correction term in a two-regime strategy. Second, it estimates a threshold cointegration framework 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 UNDs.
3.2. Linear and Threshold Modes of VECM for ADR and Its UND
For ADR and UND, transaction costs and other market imperfection factors might cause the error correction effects on the price adjustment to be significant only when the deviation of price between ADR and UND is larger than a certain threshold. While previous studies, such as that of Enders and Chumrusphonlert (2004), employed a univariate threshold model to explore the properties of purchasing power parity, this research follows Hansen and Seo’s (2002) model to develop a multivariate threshold VECM. The model is employed to estimate the threshold parameters, to construct confidence intervals for the threshold parameters, and to develop new tests for the threshold effects of ADR and UND prices.
3.2.1. Estimation of the Threshold Parameters
This research applied the linear and the threshold VECM model to ADR and UND. Let ∆ xt be a p-dimensional I (1) time series, with n observations, with l as the optimal lag length. A linear VECM (12) of order l+ 1 can be written briefly as:
,
where ∆ is the first-order difference operator. The error term is assumed to be a vector martingale difference sequence with finite covariance matrix
ut
The linear VECM model explains the price changes for short-term as well as long-term adjustment (Figure 2). If the deviation from the long-term equilibrium is greater than the threshold γ, the price transmission process is defined by a different regime (regime 2) than in the case of smaller deviations from the long-term equilibrium (regime 1). As a variant and in line with approaches by Balke and Fomby (1997), the following specification of a threshold VECM (13) is proposed:
Regime 1
where γ is the threshold parameter. Note that this research uses the absolute value of error correction term as a threshold variable, as explained earlier.
pt
∆
−1
ECTt
Figure 2. Impact of the error correction term on the price adjustment
(linear error correction model).
This specification excludes the possibility of smaller deviations from a long-term equilibrium inside a regime of adjustment to larger deviations (Figure 3). Specification (13) allows this and is therefore economically more meaningful. Using threshold VECM (13), two regimes of price adjustment are used: one defined by absolute deviations from the long-term equilibrium that are below the threshold r (regime 1) and another defined by deviations that exceed the threshold r in absolute values (regime 2). Because of this regime definition, the threshold VECM is based on only one threshold and therefore is testable regarding threshold significance but also potentially allows for the economically meaningful regime 1 inside a regime of price adjustment to greater deviations from the long-term equilibrium (regime 2).
Note that threshold VECM (13) is essentially a restricted version of the general two-threshold model depicted; this is restricted in the sense that no asymmetric price transmission is
possible in (13), as the same price reaction occurs regardless of whether ECTt-1 is larger than γ or smaller than r− .
Figure 3. Impact of the error correction term on the price adjustment (threshold error correction model).
The threshold VECM of ADR and UND can be estimated using the maximum likelihood method proposed by Hansen and Seo (2002).
3.2.2. Tests for Threshold Effects
In order to assess the evidence, both the linear and the threshold VECM were tested by using the Lagrange Multiplier test developed by Hansen and Seo (2002). The test is used when the true cointegrating vector is unknown a priori and is denoted as:
( )
β~,γsup LM SupLM
U L r r r≤ ≤
= (14) pt
∆
Regime 1
Regime 2 Regime 2
−1
ECTt
−r r
where β~is the null estimate ofβ. For this test,
[
rL,rU]
is the search region set so that is Hansen and Seo (2002) developed two bootstrap methods to calculate critical values and P-values.3.3. Data and Empirical Results
The ADR and UND series were tested for stationarity in this study using unit root tests;
followed by an examination of the cointegration test between the two series. If they were cointegrated, the threshold VECM was 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 provided the data for analysis in this study, with Table 1 providing the basic description of their respective New York Stock Exchange-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.
Table 1. Data description
Symbol Company Industry Shares per DR
TGS TRANSPORTADORA DE GAS DEL SUR, S.A.
Oil and gas
operator 5 2 Jan 95
- 31 Jul 04 2,500
The log-price of the ADRs and the UNDs was used to carry out this 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 used 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 hypothesis of a unit root; the variables in the levels were I (1) for each ADR price and UND price. The variables in the first difference were integrated of order zero;
the null hypothesis of unit root was rejected at the 5% level for the price difference series.
These results indicate that the two price series are integrated in the first difference and thus validate the use of the cointegration test.
Table 2. Unit root and cointegration tests for log-prices of ADRs and UNDs
Augmented Dickey-Fuller Test Phillips-Perron Test Unit Root Test
Levels First
Differences Levels First Differences ADR -0.112758 -51.53653 ** -0.091492 -51.49286 **
YPF UND -0.138284 -48.78652 ** -0.126952 -48.83657 **
ADR -1.679652 -45.80010 ** -1.635612 45.39878 **
TEO UND -1.624543 -45.71221 ** -1.579939 -45.34922 **
ADR -2.256933 -38.23152 ** -1.811293 -51.83980 **
TGS UND -1.898783 -47.40127 ** -1.897981 -47.33906 **
Cointegration Tests Trace Test 5% CV Max-Eigenvalue Test 5% CV
None 78.15789 ** 15.41 78.15465 ** 14.07
YPF One at most 0.003231 3.76 0.003231 3.76
None 77.81962 ** 15.41 77.81962 ** 14.07
TEO One at most 2.827981 3.76 2.827981 3.76
None 111.4459 ** 15.41 107.8217 ** 14.07
TGS One at most 3.624222 3.76 3.624222 3.76
Notes:
1Total number of sample observations is 2,888 for YPF, 2,516 for TEO, and 2,500 for TGS. UND represents price of underlying stock.
** P < 0.05.
Given that all the variables of the same order were integrated, this study used two Johansen multivariate cointegration tests to determine whether the variables in each series were cointegrated. The maximum likelihood estimation procedure provided a likelihood ratio test, referred to as a trace test, with the likelihood ratio test being the test for maximum eigenvalue. The likelihood ratio statistic rejected the null hypothesis of no cointegration at the 5% level. A feature of this approach is that the VECM contains an error correction term that 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 that has been proven to be more effective than Granger causality.
Table 3 provides the estimates of the linear model. To address the issue of linear, or nonlinear, adjustment to the long-run equilibrium, this study estimated a linear VECM, given by equation (11), with the selection of the lag length being based upon the AIC and BIC criteria. As a comparison, this study first of all estimated the linear VECM for the price series of the ADRs and UNDs, reporting the results of the linear VECM estimation in Table 3. The estimated coefficients of the error correction term on the equations of the UND were all significant at the 5% level.
Table 3. Linear VECM estimations for log-prices of ADRs and UNDs
AIC -22529.2 -4510.15 -18063.0
BIC -22505.9 -4487.76 -18046.2
Notes:
1Values in parentheses are Eicker-White standard errors.
***P < 0.01; **P < 0.05; * P < 0.10.
The estimation results of the threshold VECM, and the test for the hypothesis of linearity versus the threshold effect of nonlinearity, provided by equation (13), 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 threshold VECM was provided by equation (12), with the 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 heteroskedasticity-robust covariance estimator, with the parameter estimates being calculated by the minimization of equation (8) over a 300 × 300 grid on the parameters (
βˆ
γ β, ).
Table 4 reports the threshold VECM results for ADR with ticker symbol ‘YPF’ along with UND. In this study, a lag length of l = 3 was selected, with the estimated cointegrating relationship being wt-1 = ADRt-1 −1.00123UNDt-1, quite close to a unit coefficient. This study also conducted analyses for the case where a unit coefficient is imposed, with the results being very similar. The estimated threshold parameter wasγ = 0.000368, indicating that the first regime corresponded to |ADRt-1 −1.00123UNDt-1|≤ 0.000368. This first regime, which comprised 78% of all of the observations in the sample, is referred to in this study as the
‘typical’ regime. Conversely, the second regime, which was |ADRt-1 −1.00123UNDt-1| >
0.000368, comprised 22% of all of the observations in the sample and is referred to here as the ‘extreme’ regime.
In the ‘typical’ regime specifically, both ∆ADRt and ∆UNDt had statistically insignificant error correction effects and minimal dynamics. They were close to white noise, which indicates that in this regime, ADRt and UNDt were close to random walks. In contrast, in the
‘extreme’ regime, the asymmetry of ∆ADRt and ∆UNDt was implied, in the sense that there was an error correction effect in the ADR and UND equation being statistically significant with dynamic coefficients. All in all, ADRt and UNDt were statistically significant in the error correction effects in the ‘extreme’ regime, but not in the ‘typical’ regime.
Table 4. Threshold VECM estimations of YPF for log-prices of ADR and UND
First Regime: |wt-1 |≤ 0.000368 Percentage of Obs = 0.783634
Second Regime: |wt-1 |> 0.000368 Percentage of Obs = 0.216366
Dep ∆ADRt ∆UNDt ∆ADRt ∆UNDt
Ind. Estimate Std Error Estimate Std Error Estimate Std Error Estimate Std Error wt-1 -0.032 0.027 0.015 0.016 -0.395 ** 0.200 0.442 *** 0.131 Constant (×10-3) 0.579 0.643 -0.774 0.478 -3.324 ** 1.572 2.064 1.563
∆ADRt-1 -0.005 0.039 0.144 *** 0.043 0.427 *** 0.138 0.217 ** 0.109
∆ADRt-2 0.078 0.049 -0.052 0.044 -0.257 * 0.141 0.106 0.115
∆ADRt-3 -0.017 0.056 0.057 * 0.034 0.241 * 0.133 0.054 0.113
∆UND t-1 -0.018 0.045 -0.016 0.037 -0.274 ** 0.127 -0.112 0.098
∆UND t-2 -0.015 0.038 0.018 0.027 0.197 *** 0.055 0.018 0.081
∆UND t-3 -0.018 0.036 0.009 0.037 -0.238 *** 0.086 -0.061 0.076 Threshold Estimate = 0.000368; Cointegrating Vector Estimate = 1.00123; AIC= -22653.1; BIC = -22606.4 Lagrange Multiplier Threshold Test
Fixed Regressor bootstrap = 84.114*** (P < 0.001) Residual bootstrap = 28.306*** (P < 0.001 ) Wald Test
Equality of Dynamic Coefficients = 34.188*** (P < 0.001) Equality of EC Coefficients = 24.911*** (P = 0.008) Note: ***P < 0.01; **P < 0.05; *P < 0.10.
The evidence of nonlinearity appeared to gain strength from the results of the Wald test diagnostics; thus, the null hypothesis of linearity in error correction terms was 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 adjustment, a result consistent with that 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 was negative; this result is consistent
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 adjustment, a result consistent with that 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 was negative; this result is consistent