Nonlinear dynamics in arbitrage of the S&P 500 index and futures: A threshold error-correction model

Economic Modelling 27 (2010) 566–573

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Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d

Nonlinear dynamics in arbitrage of the S&P 500 index and futures: A threshold error-correction model ☆ Bong-Han Kim a, Sun-Eae Chun b, Hong-Ghi Min c,⁎ a b c

Department of International Economics, Kongju National University, Republic of Korea Graduate School of International Studies, Chung-Ang University, Republic of Korea Department of Management Science, Korea Advanced Institute of Science and Technology, Daejun, Republic of Korea

a r t i c l e

i n f o

Article history: Accepted 20 November 2009 JEL classification: G13 Keywords: S&P 500 index and futures Three-regime threshold ECM SupLM test No-arbitrage band

a b s t r a c t Using a three-regime threshold error-correction model, we investigate the nonlinear dynamics of the S&P 500 index and futures. First, using the SupLM statistic, we report estimates of two thresholds for the three-regime model to explain the nonlinear dynamics in arbitrage of the S&P 500 index and futures. This provides empirical evidence of the no-arbitrage band predicted by the cost-of-carry model. Second, using quasi-maximum likelihood estimation, we demonstrate that those indexes that are located outside the no-arbitrage band are a nonlinear stationary process of mean-reversion to the no-arbitrage band. However, index and futures that are located within the no-arbitrage band are non-stationary. Third, we confirm an earlier finding that futures price leads the nonlinear mean-reverting behavior of the index but not vice versa. Impulse response function analysis and forecasting performance of three-regime error-correction model reinforce our findings and our estimation results are robust with different specifications of pricing error terms and endogenous variables. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Infrequent trading (Miller et al., 1994), market sentiment, and arbitrage trading (Kawaller et al., 1987; Martens et al., 1998; Dwyer et al., 1996)1 are major determinants that link stock index futures and the stock market. Kawaller, et al. (1987), Brennan and Schwartz (1990), Kawaller (1991), Stoll and Whaley (1990), Chan (1992), and Koch (1993) investigated the impact of index futures arbitrage on the dynamic adjustment of the S&P 500 index and futures using the concepts of cointegration and no-arbitrage relations. Other studies [Anderson (1997), Balk and Fomby (1997), Balk and Wohar (1998), Baum and Karasulu (1998), Enders and Falk (1998), Forbes et al. (1999)2, Lo and Zivot (2001), Obstfeld and Taylor (1997), and Martens et al. (1998)] explained the nonlinear adjustment process of asset prices using threshold cointegration. In particular, Martens et al. (1998), using Tsay's (1989) four-step method, provides estimates of four thresholds and five

☆ We are grateful to the editor and an anonymous referee for their invaluable comments but all remaining errors are our own. ⁎ Corresponding author. Tel.: +82 42 350 6306; fax: +82 42 350 6339. E-mail address: [email protected] (H.-G. Min). 1 Dwyer et al. (1996) present empirical results that the Threshold ECM which captures the nonlinear dynamic relationship between the S&P 500 index and futures better than the simple ECM model. However the methodology used in their study is not general, since they derive the result under the strong assumption of symmetric upper and lower limit for the no-arbitrage bound. 2 Forbes et al. (1999) estimated the Threshold ECM model with the prior threshold parameters assuming cointegration in the ECM without testing.

0264-9993/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2009.11.011

regimes to explain the no-arbitrage band and nonlinear adjustment process of stock market indexes and futures. However, they did not test whether the four thresholds provide a true data generating process using a proper statistic. As a result, while the model of five regimes (four thresholds) has the advantage of capturing infrequent trading, it does not provide econometric evidence that four thresholds (five regimes model) is a true data generating process. Building upon Martens et al. (1998) and Forbes et al. (1999), we provide econometric evidence that there are two thresholds (three regimes) in the S&P 500 index and futures using the SupLM test of Hansen and Seo (2002). Using the two-step quasi-maximum likelihood estimation, we estimate two thresholds for three regimes and provide evidence of a no-arbitrage band and nonlinearity in the adjustment of the S&P 500 index and futures. This paper is organized as follows. Section 2 presents a model of price determination for stock market index and futures, which is based on a cost-of-carry model3 and explains the equilibrium adjustment process with transaction costs. We then introduce a three-regime threshold error-correction model. Section 3 describes the data, reports on a nonlinearity test, estimates the three-regime threshold error-correction model (TRTECM), and analyzes the nonlinear impulse response functions for the S&P500 index and futures. Section 4 concludes the paper. 3 Cornell and French (1983) claim that differences in taxation on the indexes and futures cause a no-arbitrage band while Brennan and Schwartz (1990) claim that position restrictions are responsible. But Puttonen and Martikainen (1991) and Gay and Jung(1999) argue that short-sale restriction is the main reason.

B.-H. Kim et al. / Economic Modelling 27 (2010) 566–573

2. The Model 2.1. Three-regime threshold error-correction model Dwyer et al. (1996) and Martens et al. (1998)4 showed the existence of a nonlinear adjustment process in the form of a single threshold (two regimes) and four thresholds (five regimes), respectively. While the five regime models for nonlinear dynamics of futures and index has been explored by Martens et al. (1998), the three-regime transaction cost model of no-arbitrage band has been investigated much more extensively5 (Yadav et al., 1994; Anderson, 1997; Obstfeld and Taylor, 1997; Forbes et al., 1999; Baum and Karasulu, 1998). However, the dynamic adjustment process of price deviation can be modeled better if we build a model whose number of thresholds (number of regimes) can be decided by a proper econometric test. In this study, we find that the true data generating process has two thresholds using the SupLM test defined in Eq. (7) in Section 2.2. Let the price of the shares underlying the index be s, buyers are assumed to pay s(1 + v), where v the proportional transaction cost is for them. Similarly, we assume that the transaction cost is paid when the shares underlying the stock index are sold, receiving s(1 + μ), where μ is the proportional transaction cost for sellers. Under these assumptions, a futures price at time t of delivery date T should remain in the no-arbitrage band, as in Eq. (1).6 St ð1−μÞ expðrc ðT−tÞÞ≤ Ft ≤ St ð1 + μÞ expðrc ðT−tÞÞ

ð1Þ

where Ft is the future price at time t maturing at time T, St is the index at time t, r is the risk free interest rate, q is the dividend yield on the index futures and rc = r − q. The basis at time t is ln (Ft/St). In logarithmic deviation, the condition for a no-arbitrage opportunity can be rewritten as Eq. (2). lnð1−μÞ≤ ln Ft − ln St −ðrc ðT−tÞÞ≤ lnð1 + νÞ

ð2Þ

Eq. (2) states that the futures prices move within a certain range of no-arbitrage opportunity when the transaction cost is proportional to the existing price. Hansen and Seo (2002) extended the threshold cointegration model by taking into account the case of an unknown cointegrating vector and multivariate thresholds. In this paper, we use a SupLM test statistic to test the existence of two thresholds. We then use the quasi-maximum likelihood method to estimate the three-regime threshold error-correction model and explain the nonlinear dynamics of an index and futures price adjustment for the S&P500. The threshold error-correction model with three regimes can be modeled as follows.

where wt = lnFt − lnSt − rc(T − t) is the price deviation, γ1 is the threshold parameter for the lower limit of the no-arbitrage bound, and γ2 is the threshold parameter for the upper limit of the noarbitrage bound. In Regime I, where futures prices are undervalued, a profit can be made by selling the index and buying futures. In Regime III, where futures prices are overvalued, buying index and selling futures can create a profit. Regime II forms a no-arbitrage band and the arbitrage profit is smaller than the transaction costs in this regime. Combining the three regimes into one equation, we can propose a three-regime threshold error-correction model as expressed in Eq. (4). l

Δ xt = ðμ 1 + α1 wt−1 + ∑ Γ1j Δ xt−j Þ1ðwt−1 ≤γ1 Þ j=1

l

+ ðμ 2 + α2 wt−1 + ∑ Γ2j Δxt−j Þ1ðγ1 〈wt−1 ≤γ2 Þ

ð1Þ

ð1Þ

wt−1 ≤γ1

ð2Þ

ð2Þ

ð2Þ

γ1 〈wt−1 ≤γ2

ð3Þ

ð3Þ

ð3Þ

γ2 〈wt−1

wt = φ0 + φ1 wt−1 + εt wt = φ0 + φ1 wt−1 + εt

4

ðRegime IÞ ðRegime IIÞ

ðRegime IIIÞ;

ð3  1Þ ð3  2Þ ð3  3Þ

Existing literatures such as Martens et al. (1998) have provided the existence of thresholds using grid search methodology and found five regimes. 5 While Martens et al. (1998) claim that different arbitrageurs become active at different thresholds depending on their relative price advantage if arbitrage capital is constrained, Forbes et al. (1999) argue that large, sophisticated financial institutions will not leave any arbitrage opportunities to less equipped arbitrageurs. In particular, fully automated program trading will automatically trigger an appropriate arbitrage strategy based on present mispricing thresholds and it is difficult to imagine a shortage of arbitrage capital for least-cost arbitrageurs. 6 The economic meaning of Eq. (1) can be explained as follows. Assume that the transaction cost occur when investors take a long position of the futures contract and sell stocks simultaneously for the purpose of hedging. The difference is put in the bank as a deposit and the value of the future contract deposit maturing at time T is St(1 + μ) exp(rc(T − t)). This will not exceed Ft because of the arbitrage profit opportunity.

ð4Þ

j=1 l

+ ðμ 3 + α3 wt−1 + ∑ Γ3j Δ xt−j Þ1ðwt−1 〉γÞÞ + ut j=1

where xt = (lnFt,lnSt)′, μi is a (2 × 1) constant vector, αi is a (2 × 1) adjustment coefficient vector, and Γij are (2 × 2) coefficient matrices for the three regimes i = 1,2,3. Also, ut is a (2 × 1) error vector, and we assume that Et − 1(ut) = 0.1(·) is the indicator function that determines the three different regimes depending on the magnitude of the basis wt − 1. The adjustment coefficients, α1,α2, and α3 show different adjustment speeds in the three different regimes.7 When the size of the adjustment vector approaches zero the index and futures do not respond to the movement of wt − 1. However, if the adjustment vector is significantly different from zero, then the index and futures price are influenced by wt − 1 and this causes mean-reverting behavior of wt. 2.2. SupLM statistic for testing the threshold effects To test the existence of a threshold effect in the dynamic relationship between index and futures, we define the parameter vector as θ = (μ′,α′, vec(Γ1)′, ..., vec(Γl)′)′. The three-regime threshold ECM (TRTECM) can then be expressed as follows: ′

′

Δ xt = yt−1 θ1 1ðwt−1 ≤γ1 Þ + yt−1 θ2 1ðγ1 b ωt−1 ≤γ2 Þ +

′ yt−1 θ3 1ðwt−1 ≥ γ2 Þ

ð5Þ

+ ut ;

where yt − 1 = (1, wt − 1, Δxt′− 1, ..., Δxt′ − l )′, Δxt = (lnFt,lnSt)′. Eq. (6) follows from Eq. (5). ′

ð1Þ

wt = φ0 + φ1 wt−1 + εt

567

′

′

Δ xt = yt−1 θ + yt−1 δ1 1ðwt−1 ≤γ1 Þ + yt−1 δ2 1ðwt−1 ≥ γ2 Þ + ut ;

ð6Þ

where θ = θ2 = (μ′,α′,vec(Γ1)′, ..., vec(Γl)′)′, δ1 = θ1 − θ, and δ2 = θ3 − θ. Hypotheses for the presence of a threshold effect can be written as follows. Ho : δ = 0 vs: H1 : δ≠0

; where δ = ðδ′1 ; δ′2 Þ′ :

A null hypothesis states that the parameter vector for the three regimes is the same, while the alternative hypothesis states that the parameter vector is different. The null model is a linear ECM and the alternative model is a TRTECM. If we can reject the null hypothesis, the equilibrium adjustment process has nonlinear characteristics. In order to be able to estimate the threshold parameters that are unidentifiable under the null hypothesis, we follow the optimality arguments of Andrews and Ploberger (1994) and use the SupLM

7 This model is different from the conventional regime switching model, as the regime is determined by the equilibrium error and the threshold parameter.

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B.-H. Kim et al. / Economic Modelling 27 (2010) 566–573

statistic defined in Eq. (7), which is independent from the nuisance parameter. SupLMn = supγ∈Γ LMn ðγÞ;

ð7Þ

where Γ = [γ1,γ2], and γ1 satisfies P(wt ≤γ1) =p and γ2 satisfies P (wt ≤γ2) = 1 −p. Even though the SupLM statistic does not have a standard asymptotic distribution, Hansen and Seo (2002) have shown a methodology to calculate a bootstrap p-value for the hypothesis test.8 Since the maximum likelihood function is not differentiable in a threshold cointegration model we cannot apply the maximum likelihood method directly. However, Hansen and Seo (2002) propose a two-step methodology for the solution. Step-one: if the parameter vectors are divisible into two groups, that is, (γ1,γ2) and (θ1′,θ2′,θ3′), the threshold error-correction model becomes linear in the parameter and the maximum likelihood estimator is the same as the least squares estimator. Once the estimation of the first group of parameters (γ1,γ2) is completed, the second group of parameters (θ1′,θ2′,θ3′) can be estimated. The maximum likelihood function can be defined as follows: n n ⌢ −1 ⌢ Lðγ1 ; γ2 Þ = constant− log j∑j− μ∑ μ; 2 2 where ∑ =

1 n

Table 1 Descriptive statistics and unit root tests.

ð8Þ

n

⌢ ⌢′ ∑ μ t μ t , ut are defined in Eq. (4), and μ̑ = (μ1̑ ,μ2̑ ,…,μ̑ n).

t =1

Step-two: the second step is to estimate the model using a grid search methodology by creating a grid of Γ = [γ1,γ2] for (γ1,γ2). Here the lower and upper limits of the grids are set to satisfy P(wt ≤ γ1) = p and P(wt ≤ γ2) = 1 − p. Usually, p is set to be 0.05, 0.10, and 0.15, depending on the degrees of freedom, as suggested by Andrews and Ploberger (1994). In this paper, we use 0.10 for the value of p. When we compare the SupLM statistic of Hansen and Seo (2002) with the Bayesian approach of Forbes et al. (1999), the SupLM statistic has couple of advantages over the Bayesian approach. First of all, the Bayesian methodology assumes the number of regimes as three and a cointegrating relationship among variables without estimation. However, the SupLM statistic tests the number of thresholds and estimates the cointegrating relationship. Second, as a result, the Bayesian approach of Forbes et al. (1999) reports a significant drift term in index returns at the outer regime which is inconsistent with the economic theory, but the SupLM statistic does not.

ADF statisticsa Jarque–Bera statistic Ljung–Box Q(1)c Ljung–Box Q(10)c

lnFt

lnSt

lnHt

lnFt − lnSt

− 1.4243 357.25b

− 1.2692 327.951b

− 1.3003 378.502b

− 12.4603b − 14.8691b 74.937b 32.702b

7052.7b 7057.2b 7057.0b 6011.4b 70,133.0b 70,256.0b 70,247.0b 36,848.0b

Critical value of ADF statistics is − 2.8625. Denotes significance of the estimates at 1% critical level. c Ljung–Box Q statistics and figures in the parentheses shows p-value and lag length is selected based on the Schwartz criterion. b

This implies that there is a cointegrating relationship between the non-stationary time series of the S&P 500 index and futures. Fig. 1 shows the trends of the logarithmic value of the data for the sample period. The futures price moves closely with the index during the sample period, suggesting a cointegrating relationship between index and futures. Fig. 2 shows the basis trend, which is the logarithmic difference between the index and futures prices. The estimates of the basis lies between −0.004 and 0.004 and, on average, are smaller than zero in the first half while greater than zero in the second half. A Jarque–Bera test statistic of 74.9374 rejects the null hypothesis of a normal distribution and the estimated Ljung–Box Q statistics with a time lag of one and ten show high autocorrelations in the basis. 3.2. Testing the number of thresholds To check the robustness of the test results, we consider three different specifications of error-correction terms and dependent variables, as summarized in Table 2.

3.1. Data

8 Since the threshold parameter, γ, is unidentifiable under the null hypothesis, the SupLM statistics do not follow the standard distribution (e.g., χ2 distribution) under the null hypothesis. Therefore p-values are derived through simulation. In this paper, p-values of SupLM statistics are obtained by bootstrapping. A more detailed explanation on the SupLM can be found in Davies (1987), Andrews (1993), and Hansen and Seo (2002). 9 We use the daily U.S. discount rate as the interest rates, while the dividends data are taken from the S&P Information Bulletin, such as Martens et al. (1998). Estimation results for the November data are quite similar to that of May and are not reported here but estimation results are available upon request.

5855.8b 32,845.0b

a

3. Estimation

To make this research comparable to the previous studies, we use the same dataset used by Martens et al. (1998) and Forbes et al. (1999): minute-to-minute data of the S&P 500 index and futures traded in the Chicago Mercantile Exchange expiring after 6 months. Also, the dataset is comprised of trades made in May 1993 and the sample size is 7060.9 Descriptive statistics, an ADF test for the unit root, and cointegration for the variables used in this study are reported in Table 1. An ADF test of the unit root shows that the index (lnSt), futures (lnFt) and theoretical index (lnHt) are non-stationary while the price deviations (lnFt − lnSt − rc(T − t))and basis (lnFt− lnSt) are stationary.

lnFt − lnSt − rc (T − t)

Fig. 1. Trends of S&P 500 index and futures.

Fig. 2. Trends of S&P 500 basis.

B.-H. Kim et al. / Economic Modelling 27 (2010) 566–573 Table 2 Three different definitions of the variables for TRTECM.

Case I Case II Case III

Table 4 Estimation results of TRTECM (Case I).

Dependent variables

Error-correction terms

Regime I: wt − 1 ≤γ1

Regime II: γ1〈wt − 1 ≤γ2

Regime III: γ1〈wt − 1

lnFt,lnSt lnFt,lnSt lnFt,lnHt

wt = lnFt − lnSt − rc(T − t) wt = lnFt − lnSt wt = lnFt − lnSt − rc(T − t)

ΔlnF

ΔlnF

ΔlnF

ΔlnS

ΔlnS

γ1 = − 0.001027

ΔlnS

γ2 =0.001232

P(wt − 1 ≤γ1)=0.0533 P(γ1〈wt − 1 ≤γ2)=0.8800 P(γ1〈wt − 1)=0.0666

Note: Ht = Stexp{rc(T − t)}. Constant

First, for Case I, we use a basis (which is the logarithmic difference between futures and index) adjusted for the cost-of-carry as the errorcorrection term and use the logarithm of futures and index for the dependent variables. While Case I represents the economic implications of Eq. (2) very well and has been explored in previous studies (Martens et al., 1998; Forbes et al., 1999), this model is subject to the specification error. Second, for Case II, we use a basis as the error-correction term and logarithm of futures and index for the dependent variables. Finally, for Case III, we use a basis adjusted for the cost-of-carry as the errorcorrection term and use the logarithm of futures and theoretical futures price defined in Table 3 (note) as dependent variables. While Cases II and III are free from the econometric specification error, they do not represent the economic implications of Eq. (2) well. We use the SupLM statistic to test whether the equilibrium adjustment process between index and future prices is nonlinear. Estimation results of Hansen and Seo's (2002) two-step methodology are reported in Table 3. Table 3 reports the estimated SupLM statistic and Bootstrap p-value, which are obtained through a simulation repeated 1000 times. First, we can reject the null hypothesis of no threshold and confirm the alternative of two thresholds (three-regime model). Second, this implies that the S&P 500 index and futures have a nonlinear equilibrium adjustment property and this result is robust with different specifications of the model, i.e., there are two thresholds for all three cases.

α ΔlnFt − 1 ΔlnFt − 2 ΔlnFt − 3 ΔlnFt − 4 ΔlnFt − 5 ΔlnFt − 6 ΔlnFt − 7 ΔlnFt − 8 ΔlnSt − 1 ΔlnSt − 2 ΔlnSt − 3 ΔlnSt − 4 ΔlnSt − 5 ΔlnSt − 6 ΔlnSt − 7

3.3. Estimation of the three-regime threshold error-correction model In this section, we estimate the three-regime threshold errorcorrection model (TRTECM) to investigate the short-run and long-run adjustment processes of the index and futures toward equilibrium. In estimating TRTECM, the lag length is selected as eight, as in Martens et al. (1998) and Forbes et al. (1999). Estimation results of the TRTECM for the S&P 500 index and futures for Cases I–III are reported in Tables 4–6. 3.3.1. Case I First, Case I deals with a situation when the dependent variables are xt = (lnFt,lnSt)′ and the mispricing error is wt = lnFt − lnSt − γc(T − t). Most existing studies deal with Case I, including Martens et al. (1998) and Forbes et al. (1999). Estimation results for Case I are reported in Table 4. As we can see from the third row of Table 4, the threshold parameter estimates are {−0.1027% and 0.1232%}. Our estimates of thresholds are quite smaller than those of Forbes et al. (1999) and Martens et al. (1998) whose estimates of no-arbitrage thresholds are [−0.1039%, 0.1278%] and [− 0.072, 0.072], respectively. Also, when we convert no-arbitrage bandwidth in basis points, our estimates of Table 3 SupLM statistic for the number of thresholds. H0: number of threshold = 0, H1: number of threshold = 2

SupLM statistic 5% critical value Bootstrap p-value

569

Case I

Case II

Case III

149.569 109.703 0.000

135.851 112.307 0.000

149.569 113.107 0.000

ΔlnSt − 8 Log likelihood

0.0001 0.0002 (0.000) (0.000) 0.1030 0.2213a (0.087) (0.056) − 0.0297 0.1031b (0.067) (0.048) − 0.0453 0.1541a (0.059) (0.046) − 0.0232 0.1250a (0.058) (0.038) − 0.0472 0.0587 (0.073) (0.034) − 0.0732 0.1503a (0.066) (0.032) − 0.0636 0.0582 (0.063) (0.040) 0.0772 0.1463a (0.072) (0.042) − 0.1058 0.1209a (0.069) (0.044) 0.0361 0.0743 (0.065) (0.081) − 0.0513 −0.1555 a (0.065) (0.054) − 0.0248 − 0.1373a (0.066) (0.040) 0.0493 − 0.0465 (0.088) (0.040) 0.0817 − 0.1858a (0.084) (0.044) 0.0479 − 0.0863b (0.071) (0.044) − 0.1254 − 0.1121b (0.085) (0.049) − 0.0466 0.2314a (0.143) (0.078) 82,125.434

0.0000 (0.000) − 0.0065 (0.008) − 0.0326 (0.019) 0.0182 (0.018) 0.0302 (0.018) − 0.0004 (0.018) 0.0182 (0.017) − 0.0098 (0.015) − 0.0001 (0.017) − 0.0056 (0.018) 0.0031 (0.040) − 0.0488 (0.030) 0.0044 (0.033) − 0.0030 (0.029) 0.0076 (0.032) 0.0489 (0.027) − 0.0281 (0.028) 0.0093 (0.023)

− 0.0000 (0.000) 0.0198a (0.005) 0.0345a (0.009) 0.0945a (0.008) 0.1097a (0.009) 0.0784a (0.009) 0.0661a (0.009) 0.0404a (0.009) 0.0348a (0.008) 0.0187 (0.011) − 0.0510b (0.020) − 0.0269 (0.014) − 0.0228 (0.017) 0.0012 (0.015) 0.0138 (0.017) 0.0017 (0.014) 0.0082 (0.013) 0.0001 (0.015)

0.0001 (0.000) − 0.0910 (0.079) 0.0281 (0.065) 0.0074 (0.063) 0.0117 (0.072) − 0.0269 (0.071) 0.0031 (0.079) 0.0519 (0.071) 0.0920 (0.076) 0.0065 (0.067) 0.0832 (0.134) − 0.1415 (0.126) − 0.0762 (0.176) 0.0728 (0.117) 0.0294 (0.101) 0.0248 (0.129) − 0.2181 (0.125) 0.1132 (0.143)

− 0.0001 (0.000) 0.1491a (0.058) 0.0215 (0.043) 0.1304a (0.044) 0.1204a (0.054) 0.0560 (0.042) −.0121 (0.054) 0.0085 (0.035) 0.0213 (0.048) − 0.0270 (0.047) 0.1264 (0.109) − 0.0733 (0.076) 0.0423 (0.121) 0.0274 (0.075) 0.0030 (0.071) 0.0794 (0.080) − 0.0449 (0.111) 0.0439 (0.056)

Note: 1) xt = (lnFt,lnSt)′ and wt = lnFt − lnSt − rc(T − t). 2) Figures in parentheses are standard errors. a Denotes significance of the estimate at 1% critical level. b At 5% critical level.

1.00 (basis point) is smaller than that of Martens et al. (1998) [1.61 (basis point)] and Forbes et al. (1999) [1.03 basis point]. This implies that our estimates provide evidence of a much more efficient adjustment process for the S&P 500 index and futures than that of Martens et al. (1998) and Forbes et al. (1999), i.e., TRTECM provides much smaller opportunity of arbitrage between index and futures. This can be confirmed by the fact that the probability of data being in the no-arbitrage band (Regime II) is 88% in TRTECM while that of Martens et al. (1998) is 68 %.10 The coefficient of futures price (α) in regime I is 0.1030 but insignificant, while that of the index is 0.2213 and significant. This implies that the spot market is responding to futures market information but less so in the converse direction. The third and seventh columns (in the sixth row) in Table 4 show that the error-correction term in the index equations (α) is significant for Regime I and Regime III. This implies that the equilibrium adjustment is achieved through a decline of the index rather than through an increase of futures prices implying that futures lead the index (in the long-run).

10 Characteristics of our parameter estimates compared to those of Martens et al. (1998) and Forbes et al. (1999) for the Cases II and III are quite similar to that of Case I.

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B.-H. Kim et al. / Economic Modelling 27 (2010) 566–573

Table 5 Estimation results of threshold ECM model (Case II).

Table 6 Estimation results of threshold ECM model (Case III).

Regime I: wt − 1 ≤γ1

Regime II: γ1〈wt − 1 ≤γ2

Regime III: γ1〈wt − 1

Regime I: wt − 1 ≤γ1

Regime II: γ1〈wt − 1 ≤γ2

Regime III: γ1〈wt − 1

ΔlnF

ΔlnF

ΔlnF

ΔlnF

ΔlnF

ΔlnF

ΔlnS

ΔlnS

γ1 = − 0.001364

ΔlnS

γ2 = 0.001310

α ΔlnFt – 1 ΔlnFt − 2 ΔlnFt − 3 ΔlnFt − 4 ΔlnFt − 5 ΔlnFt − 6 ΔlnFt − 7 ΔlnFt − 8 ΔlnSt − 1 ΔlnSt − 2 ΔlnSt − 3 ΔlnSt − 4 ΔlnSt − 5 ΔlnSt − 6 ΔlnSt − 7 ΔlnSt − 8 Log likelihood

− 0.0000 (0.000) − 0.0561 (0.070) 0.0098 (0.068) − 0.0086 (0.063) 0.0106 (0.068) 0.0598 (0.050) − 0.0132 (0.073) − 0.0366 (0.065) 0.1052 (0.081) − 0.0527 (0.077) 0.0318 (0.123) − 0.0939 (0.077) − 0.0503 (0.092) − 0.0825 (0.060) 0.0086 (0.097) − 0.0097 (0.083) − 0.1229 (0.100) 0.2272 (0.077) 80876.999

0.0002 (0.000) 0.1526a (0.042) 0.1695a (0.038) 0.2343a (0.044) 0.1855a (0.041) 0.0540 (0.039) 0.1780a (0.043) 0.0328 (0.047) 0.1496a (0.053) 0.0853 (0.047) 0.2357 (0.080) − 0.3044a (0.069) − 0.2346a (0.051) − 0.541 (0.045) −0.2347 a (0.062) − 0.0752 (0.057) − 0.0977 (0.070) 0.1377 (0.080)

0.0000 (0.000) − 0.0044 (0.006) − 0.0400 (0.018) 0.0161 (0.017) 0.0253 (0.017) − 0.0045 (0.018) 0.0183 (0.016) − 0.0070 (0.015) − 0.0010 (0.016) − 0.0056 (0.017) 0.016 (0.036) − 0.0516 (0.027) − 0.0066 (0.031) 0.0028 (0.030) 0.0160 (0.030) 0.0599a (0.025) − 0.0446 (0.027) − 0.0041 (0.023)

0.0000 (0.000) 0.0188a (0.003) 0.0374a (0.008) 0.0998a (0.008) 0.1111a (0.008) 0.0839a (0.008) 0.0663a (0.009) 0.0475a (0.009) 0.0406a (0.008) 0.0205 (0.011) − 0.0443a (0.019) − 0.0278a (0.014) − 0.0221 (0.017) − 0.0053 (0.015) − 0.0002 (0.018) 0.0071 (0.014) 0.0038 (0.014) 0.0046 (0.015)

0.0002 (0.000) − 0.1196 (0.113) 0.0219 (0.099) − 0.0180 (0.094) − 0.0144 (0.014) − 0.0907 (0.108) − 0.0549 (0.132) − 0.0267 (0.099) 0.1002 (0.119) 0.0261 (0.095) 0.2209 (0.237) − 0.2681 (0.226) − 0.0083 (0.277) 0.2299 (0.170) − 0.0803 (0.181) − 0.0552 (0.180) − 0.1825 (0.185) 0.1948 (0.202)

− 0.0001 (0.000) 0.1386b (0.071) 0.0425 (0.071) 0.1257 (0.079) 0.1027 (0.076) 0.0551 (0.064) − 0.0314 (0.089) − 0.0346 (0.067) − 0.0453 (0.091) − 0.0372 (0.081) 0.2749b (0.152) − 0.1490 (0.141) − 0.0163 (0.190) 0.2377 (0.144) 0.0107 (0.117) − 0.0116 (0.096) 0.0476 (0.086) 0.0520 (0.0712)

ΔlnH

ΔlnH

γ2 = 0.001238

γ1 =−0.000955

P(wt − 1 ≤γ1)=0.0509 P(γ1〈wt − 1 ≤γ2)=0.9201 P(γ1〈wt − 1)=0.0289 Constant

ΔlnH

P(wt − 1 ≤γ1)=0.0658 P(γ1〈wt − 1 ≤γ2)=0.8690 P(γ1〈wt − 1)=0.0650 Constant α ΔlnFt − 1 ΔlnFt − 2 ΔlnFt − 3 ΔlnFt − 4 ΔlnFt − 5 ΔlnFt − 6 ΔlnFt − 7 ΔlnFt − 8 ΔlnSt − 1 ΔlnSt − 2 ΔlnSt − 3 ΔlnSt − 4 ΔlnSt − 5 ΔlnSt − 6 ΔlnSt − 7 ΔlnSt − 8 Log likelihood

0.0001 (0.000) 0.1220 (0.071) − 0.1114 (0.058) − 0.0467 (0.050) − 0.0426 (0.053) − 0.0576 (0.062) − 0.0614 (0.059) − 0.0554 (0.054) 0.0863 (0.066) − 0.0833 (0.062) 0.0329 (0.058) − 0.0458 (0.054) − 0.0046 (0.062) 0.0619 (0.077) 0.0669 (0.076) 0.0400 (0.065) − 0.1409 (0.080) − 0.0246 (0.126) 78,353.875

0.0002 (0.000) 0.2298a (0.044) 0.0790a (0.040) 0.1538a (0.042) 0.1181a (0.035) 0.0507 (0.031) 0.1411a (0.030) 0.0717 (0.041) 0.1385a (0.038) 0.1119a (0.039) 0.0845 (0.074) − 0.1443a (0.052) − 0.1309a (0.038) − 0.0282 (0.037) − 0.1690a (0.043) − 0.0884 (0.048) − 0.1025a (0.046) 0.2177 a (0.068)

0.000 (0.000) − 0.0051 (0.008) − 0.0310 (0.019) 0.0184 (0.018) 0.0323 (0.018) 0.0009 (0.018) 0.0180 (0.017) − 0.0096 (0.016) − 0.0010 (0.017) − 0.0065 (0.018) 0.0067 (0.039) − 0.0496 (0.031) 0.0014 (0.033) − 0.0015 (0.030) 0.0078 (0.032) 0.0499 (0.027) − 0.0266 (0.028) 0.0086 (0.023)

0.0000 (0.000) 0.0180a (0.005) 0.0357a (0.009) 0.0924a (0.008) 0.1093a (0.009) 0.0781a (0.009) 0.0661a (0.009) 0.0374a (0.009) 0.0336a (0.008) 0.0186 (0.011) − 0.0508a (0.020) − 0.0275a (0.014) − 0.0215 (0.017) − 0.0000 (0.015) 0.0134 (0.017) 0.0011 (0.013) 0.0084 (0.013) − 0.0004 (0.015)

0.0001 (0.000) − 0.0962 (0.082) 0.0271 (0.068) 0.0077 (0.065) 0.0096 (0.077) − 0.0305 (0.072) − 0.0004 (0.081) 0.0510 (0.070) 0.0931 (0.076) 0.0136 (0.067) 0.0899 (0.135) − 0.1327 (0.127) − 0.0428 (0.171) 0.0152 (0.116) 0.0444 (0.101) 0.0258 (0.131) − 0.2178 (0.126) 0.1130 (0.143)

− 0.0001 (0.000) 0.1558a (0.060) 0.0149 (0.045) 0.1311a (0.044) 0.1204a (0.057) 0.0572 (0.043) − 0.0132 (0.055) 0.0142 (0.035) 0.0271 (0.048) − 0.0277 (0.047) 0.1295 (0.111) − 0.0744 (0.075) 0.0404 (0.119) 0.0129 (0.084) 0.0059 (0.071) 0.0751 (0.081) − 0.0429 (0.111) 0.0431 (0.055)

Note: 1) xt = (lnFt,ln St)′ and wt = lnFt − lnSt. 2) Figures in parentheses are standard errors. a Denotes significance of the estimate at 1% critical level. b At 10% critical level.

Note: 1) xt = (lnFt,lnHt)′ and wt = ln Ft − lnSt − rc(T − t). 2) Figures in parentheses are standard errors. a Denotes significance of the estimate at 1% critical level.

Similarly, in the short-run, the lagged futures price leads the correction as we can be seen in the third and seventh column of Table 4. The estimated long-run adjustment coefficient α in Regime II is −0.0065 and 0.0198 for the futures and index respectively; they are very small compared to the estimates in Regimes I and III, implying that price deviation adjusts insignificantly while the price deviation persists within the no-arbitrage bound. However, in the short-run, lagged futures price leads the correction as can be seen in the fifth column of Table 4. In Regime III, where the futures price is overvalued, the adjustment coefficient α for the index is 0.1491 and is significant while that of the futures is −0.0910 but insignificant.

Since Regime II is the no-arbitrage band along with the extremely high estimated probability of 92.01% suggests that the future market is efficient and supports the cost-of-carry model. For Regime I, the estimated coefficient of the error-correction term for the futures price α is −0.0561 but is insignificant while that for the index is 0.1562 and significant indicating a lead of futures prices in the long-run as in the first case. In the short-run, futures prices mainly lead the adjustment process as can be seen from the significant estimated coefficients of the lagged futures price variables. In Regime II, a very small but significant estimated coefficient of α for the index, 0.0188, implies that a small and slow adjustment is at work. In Regime III, the estimated adjustment coefficient α for the index is 0.1386 and significant while that for the futures is −0.1196 but insignificant, implying that the adjustment is achieved through an increase of the index as in Case I. The fact that the equilibrium is achieved through the index supports the idea that the futures price and basis are weakly exogenous.

3.3.2. Case II Second, Case II deals with a situation where the basis (wt =lnFt − lnSt) is used as the error-correction term. The maximum likelihood estimation results are reported in Table 5. The third row of Table 5 provides the estimated threshold parameters for γ1 and γ2 as −.001364 and 0.001310, respectively. The fourth row of Table 5 shows that the probability of the basis being in Regime I is 5.09 % while that of being in Regime II and III are 92.01% and 2.89%, respectively.

3.3.3. Case III Finally, Case III considers a situation where the dependent variables are xt = (lnFt,lnHt)′ and the mispricing error is wt = lnFt − lnSt −rc(T −t).

B.-H. Kim et al. / Economic Modelling 27 (2010) 566–573 Table 7 Boundaries and number of observations for each regime.

Regime I Regime II Regime III

571

Table 8 Estimation results of AR (7) for each regime.

Threshold estimates

Number of observations

( −∞; − 0.001027] (− 0.001027; 0.001232] (0.001232; ∞)

377 6213 470

lnHt is a theoretical futures price defined in the Table 2. Generally, for the three different regimes, the estimation results in Table 6 show very similar long-run and short-run adjustment processes to those revealed in the other two cases. However, in Regime I, not only a lagged futures price but also a lagged index causes short-run adjustment of the index such that they can move into Regime II. We can summarize the estimation results of Cases I–III as follows. First, a no-arbitrage band exists, as suggested by the cost-of-carry model. Second, a thresholds effect suggests a nonlinear adjustment of index and futures price to the long-run equilibrium band. Finally, futures lead the index but not vice versa. In other words, the futures price is independent of the magnitude of mispricing error and the futures price does not respond to changes in the index. This confirms findings of earlier studies (Chan, 1992; Forbes et al., 1999; Kawaller et al., 1987; Martens et al., 1998; Stoll and Whaley, 1990) in which the futures price leads the index. For example, Martens et al. (1998), show (in Table 4 of their paper) that the error-correction term is mainly significant for the index returns, indicating a lead of the future prices. Also Forbes et al. (1999), from Bayesian estimates of adjustment coefficients, report that most of the equilibrium adjustment process appears to occur in the spot price changes and spot price changes are more time- and cross-dependent than futures price changes. These estimation results are robust with all three different specifications of the error-correction term and dependent variables. 3.4. Impulse response function and forecasting performance of TRTECM 3.4.1. Impulse response function analysis of TRTECM One useful way of understanding the dynamic behavior of a nonlinear process is to examine the impulse response function. As carried out in Martens et al. (1998), we check the dynamic properties of the S&P 500 index and futures by estimating the impulse response function of the AR model for each regime. The thresholds and the number of observation in every regime are given in Table 7. Fig. 3 shows the pricing errors. We can see that there are a small number of observations for the outer regimes (Regime I and Regime II). The estimation results for the AR model for each Regime are reported in Table 8. Table 8 shows that the estimates of AR(1) coefficient in the outer regimes are smaller than that of the inner regime, implying a band reverting property of the process of pricing error.

Constant ωt − 1 ωt − 2 ωt − 3 ωt − 4 ωt − 5 ωt− 6 ωt − 7 R2

Regime I

Regime II

Regime III

− 0.0008 (0.00) 0.4376 (0.00) − 0.071 (0.220) − 0.0244 (0.67) 0.1013 (0.08) − 0.0051 (0.93) 0.0538 (0.35) − 0.0645 (0.22) 0.1847

0.0001 (0.01) 0.8363 (0.00) 0.0277 (0.09) 0.0077 (0.64) − 0.0126 (0.44) 0.0016 (0.92) 0.0051 (0.76) 0.0333 (0.010) 0.7746

0.0005 (0.00) 0.4495 (0.00) − 0.0348 (0.50) 0.0252 (0.62) 0.0502 (0.33) 0.0716 (0.16) 0.0160 (0.75) 0.0840 (0.07) 0.2547

Note: 1) Figures in parentheses are p-values.

This band reverting property can be investigated more rigorously by analyzing the nonlinear impulse response functions. The impulse response function investigates the effect of the shocks at time t on the movement of the time series in the future. The impulse response function for a linear process can be defined as the difference between realization without shock and realization with shock. While the impulse response function of the linear model is proportional to the size of the shock and does not depend on the initial condition and the direction of the shock, the impulse response function of the nonlinear model has very different properties. The latter depends on the initial history of the time series and the size and direction of the shock as well as future shocks.11 Following Gallant et al. (1993), we obtain the nonlinear impulse response function by a Monte Carlo integration. The impulse response function is defined as the difference between the conditional expectation of the time series from (t + 1) to (t +N) with a shock q at time t and the conditional expectation of the process without a shock, where both have an identical initial history ht. The impulse response function after period j is given as Eq. (9). Iðj; qt ; ht Þ = E½ωt

+ j jqt ; ht −E½ωt + j jht 

ð9Þ

We generate simulated observations of ωt + j, j = 1,…,J, “with” shock at t and “without” shock using parameter estimates of AR(7) models to obtain the difference between two simulated observations. We do this 200 times and the impulse response function is the mean of the differences with the specific history and shock. We estimate the impulse response functions with ±1% and ±2% shocks to compare the impacts of small and large shocks on the series. Fig. 4 shows estimated values of the impulse response functions with four different shocks. Fig. 4(a) shows the impulse response function after a shock in Regime I, since the value of the pricing error at time t, ωt = −0.00122, belongs to the lower range, (−∞; −0.001027]. Fig. 4(b) and (c) shows the impulse response function after a shock in Regime II and III, respectively. The figures are only a representative example of many possible impulse response functions depending on the history. We can see that the degree of persistence in the inner regime is much longer than those of the outer regimes. When the size of shock is 0.02%, the half life of the inner regime is 6 days while those for the outer regimes are less than one day.12 The difference in the degree of persistence between the inner and outer regime supports a band reverting property implied by arbitrage with transaction costs. 3.4.2. Out-of-sample forecasting performance of the TRTECM We compare out-of-sample forecasting performance of TRTECM to those of random walk model and linear ECM. Using 6060 samples out

11

See Gallant et al. (1993) and Koop (1996) for details. The half life is defined as the expected time for a process to adjust halfway back to zero following a one-time shock. 12

Fig. 3. Pricing Errors.

572

B.-H. Kim et al. / Economic Modelling 27 (2010) 566–573 Table 9 Forecasting errors of TRTECM, Random Walk Model, and linear ECM. Horizon

1 2 3 4 5

TRTECM vs. random walk

TRTECM vs. linear ECM

TRTECM/RWa

DMb

P-valuec

TRTECM/ECMd

DMb

P-valuec

0.5894 0.4544 0.3879 0.3480 0.3222

− 4.3026 − 4.5331 − 4.7958 − 5.0992 − 5.3937

0.0000 0.0000 0.0000 0.0000 0.0000

0.9794 0.9759 0.9592 0.9272 0.8884

− 1.7143 − 1.9219 − 3.1071 − 4.5162 − 5.9437

0.0432 0.0273 0.0009 0.0000 0.0000

a

Ratio of RMSE of TRTECM to that of random walk model. Diebold and Mariano (1995) statistic: the long-run variances are estimated using Parzen window automatic bandwidth selection by Andrews. c P-value of Diebold-Mariano statistic. d Ratio of RMSE of TRTECM to that of linear ECM. b

Fig. 4. (a) Impulse response function after a shock in regime I. Observation t = 3800 and history (t − 6,…,t) − 0.00108, − 0.00119, − 0.00126, − 0.00126, − 0.00131, and − 0.00122 in Eq. (9). (b) Impulse response function after a shock in regime II. Observation t= 2500 and history (t − 6,…,t) − 0.00025, − 6.80E-06, − 6.80E-06, − 0.00025, 8.33E-05, 0.000376, and 0.000241 in Eq. (9). (c) Impulse response function after a shock in regime III. Observation t = 330 and history (t − 6,…,t) 0.001764, 0.002285, 0.002466, 0.002013, 0.001899, 0.001695, and 0.001673 in Eq. (9).

of the 7060, we estimate and calculate the out-of-sample forecast errors of 1000 samples for TRTECM, linear ECM, and random walk model. Table 9 reports RMSEs of three different models of S&P 500 index. From the p-values of DM statistics in Table 9, we can reject the null hypothesis that TRTECM and random walk (linear ECM) have equal forecasting errors against the alternative hypothesis that forecasting errors of TRTECM are smaller than those of random walk (linear ECM) at 1% critical level. Especially, Table 9 shows that forecasting errors of TRTECM relative to that of random walk model and linear ECM decrease as the forecasting horizons increase. From this forecasting performance, we can say that nonlinear three-regime model explains the dynamics of S&P 500 index better than other models. 4. Conclusions This paper estimates a no-arbitrage band and investigates the shortrun and long-run dynamic adjustment processes of index and futures prices using the S&P500 futures and index. First, using a SupLM statistic, we provide evidence of two thresholds for three regimes which supports the no-arbitrage band. Second, from quasi-maximum likelihood estimation of the three-regime error-correction model, we demonstrate that the estimated adjustment coefficients are different for three different regimes, implying that the adjustment speed changes depending on the size of the mispricing error. Third, this paper provides

evidence that futures price leads the index in the short-run and longrun, which is consistent with previous studies. This supports the idea that futures prices satisfy a weak exogeneity condition against price deviation, implying that futures prices are independent of the magnitude of the price deviation from the equilibrium value and that futures prices do not respond to movement of index prices. For these results, the estimation results are robust with different specifications of the error-correction term (mispricing error) and dependent variables. Above all, using the SupLM test, we report a single no-arbitrage band with three regimes, which has been utilized in the literatures but not tested in earlier studies. The existence of Regime II can be explained by the transaction costs, i.e., transaction fees, transaction taxes, position restrictions, and short selling restrictions. While price deviations located initially in Regimes I and III move into Regime II with a nonlinear equilibrium adjustment process in the long-run through arbitrage, the price deviations that are initially located in Regime II meander within the regime of the no-arbitrage band. A nonlinear impulse response function analysis and out-of-sample forecasting performance of TRTECM reinforce these findings.

References Anderson, H.M., 1997. Transaction costs and nonlinear adjustment towards equilibrium in the US Treasury bill market. Oxford Bulletin of Economics and Statistics 59, 465–484. Andrews, D., 1993. Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856. Andrews, D., Ploberger, W., 1994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414. Balk, N., Fomby, T., 1997. Threshold cointegration. International Economic Review 38, 627–645. Balk, N., Wohar, M.E., 1998. Nonlinear dynamics and covered interest rate parity. Empirical Economics 23, 535–559. Baum, C.F., Karasulu, M., 1998. Modelling federal reserve discounting policy. Computational Economics 11, 53–70. Brennan, M.J., Schwartz, E., 1990. Arbitrage in stock index futures. Journal of Business 63 (pt.2), s7–s31. Chan, K., 1992. Further analysis of the lead–lag relationship between the cash market and stock index futures markets. Review of Financial Studies 5, 123–152. Cornell, B., French, K.R., 1983. The pricing of stock index futures. Journal of Futures Markets 3, 1–14. Davies, R.B., 1987. Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43. Diebold, F.X., Mariano, R., 1995. Comparing predictive accuracy. Journal of Business and Economic Statistics 13, 253–263. Dwyer, G.P., Locke, P., Yu, W., 1996. Index arbitrage and nonlinear dynamics between the S&P500 futures and cash. Review of Financial Studies 9, 301–332. Enders, W., Falk, B., 1998. Threshold-autoregressive, median-unbiased, and cointegrating tests of purchasing power parity. International Journal of Forecasting 14, 171–186. Forbes, C.S., Kalb, G.R., Kofman, P., 1999. Bayesian arbitrage threshold analysis. Journal of Business and Economic Statistics 17, 364–372. Gallant, R., Rossi, P., Tauchen, G., 1993. Nonlinear dynamic structure. Econometrica 61, 871–907. Gay, G.D., Jung, D.Y., 1999. Further look at transaction costs, short sale restrictions, and futures market efficiency: the case of Korean stock index futures. Journal of Futures Markets 19, 153–174. Hansen, B., Seo, B., 2002. Testing for threshold cointegration. Journal of Econometrics 110, 293–318.

B.-H. Kim et al. / Economic Modelling 27 (2010) 566–573 Kawaller, I.G., 1991. Determining the relevant fair values of S&P500 futures: a case study approach. Journal of Futures Markets 11, 453–460. Kawaller, I.G., Koch, P.D., Koch, T.W., 1987. The temporal price relationship between S&P 500 futures and the S&P 500 index. Journal of Finance 42, 1309–1329. Koch, P.D., 1993. Reexamining intraday simultaneity in stock index futures markets. Journal of banking and Finance 17, 1191–1205. Koop, G., 1996. Parameter uncertainty and impulse response analysis. Journal of Econometrics 72, 135–149. Lo, M., Zivot, E., 2001. Threshold cointegration and nonlinear adjustment to law of one price. Macroeconomic Dynamics 5, 533–576. Martens, M., Kofman, P., Vorst, T.C.F., 1998. Threshold error-correction model for intraday futures and index returns. Journal of Applied Econometrics 13, 245–263. Miller, M.H., Muthuswamy, J., Whaley, R.E., 1994. Mean reversion of S&P500 index basis changes: arbitrage-induced or statistical illusion? Journal of Finance 49, 479–513.

573

Obstfeld, M., Taylor, A.M., 1997. Nonlinear aspects of goods market arbitrage and adjustment: Heckscher's commodity points revisited. Journal of the Japanese and International Economies 11, 441–479. Puttonen, V., Martikainen, T., 1991. Short sale restrictions: implication for stock index arbitrage. Economics Letters 37, 159–163. Stoll, H.R., Whaley, R.E., 1990. The dynamics of stock index and stock index futures returns. Journal of Financial and Quantitative Analysis 25, 441–468. Tsay, R.S., 1989. Testing and modeling threshold autoregressive processes. Journal of the American Statistical Association 84, 231–240. Yadav, P.K., Pope, P.F., Paudyal, K., 1994. Threshold autoregressive modeling in finance: the price differences of equivalent assets. Mathematical Finance 4, 205–221.