Fórmula para determinar el punto de equilibrio en porcentaje

Cointegration measures the long-run common stochastic trend among variables. In other words, it examines the long term behaviour of market prices. According to Dolado (1999), cointegration is defined as the co-movements among trending variables with the capability of testing for the existence of equilibrium relationships within a wholly dynamic specification framework. It can be deduced therefore that if two stock prices are cointegrated, then viable arbitrage profits can be explored when there is deviation from equilibrium.

This chapter adopts the following cointegration tests; Engle-Granger (EG) residual-based test, fully-modified OLS estimator, canonical correlation regression estimator, Johansen technique, and Gregory-Hansen (GH) regime shift test. Cointegration tests that do not account for structural breaks in the time series may lead to low power and bias result. However, the GH regime shift test accounts for structural changes that may shift the long-run relationship of the underlying variables.

A. Engle-Granger Methodology

The necessary condition for cointegration is that the variables should be integrated of the same order. According to Enders (2004). it is possible to find equilibrium relationships among variables that are integrated of different order. Floros (2005) further argues that stock markets are interdependent if the stock indices of two or more countries are cointegrated (that is, they exhibit long-run relationship).

The linear relationship between UK and US stock indices are specified as;

𝑙𝑛𝑃𝑈𝐾,𝑡 = 𝛼 + 𝛽𝑙𝑛𝑃𝑈𝑆,𝑡+ 𝑒𝑡 (2.1)

According to Engle and Granger (1987), if the two price series are non-stationary, but the deviations are stationary, then 𝑙𝑛𝑃𝑈𝐾,𝑡 and 𝑙𝑛𝑃𝑈𝑆,𝑡 are cointegrated of order (1,1). The

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estimated value of the departure from the long-run relationship is detected by the stationary disturbance term denoted as ê𝑡. The Augmented Dickey Fuller (ADF) is used to test the disturbance term and is expressed as;

∆ê_𝑡 = 𝛽₁ê_𝑡−1+ ∑𝑛 𝛽_𝑖+1∆ê_𝑡−𝑖

𝑖=1 + 𝜀𝑡 (2.2)

Having selected the lag lengths, if we fail to reject the null hypothesis 𝛽₁ = 0, then we cannot reject the null of no cointegration. On the other hand, if the null hypothesis is rejected against the alternative 𝛽₁ < 0, then it is concluded that the series are cointegrated of order (1,1), thus a form of long-run stock market integration is established.

This residual-based Engle-Granger static long-run regression has been challenged especially for its inefficiency in multivariate cases. According to Banerjee et al. (1986), bias in the estimated parameters is likely to be created when lagged terms in small samples are neglected. Blough (1988) further expresses concern about the low power of the cointegration test in relatively small samples.

B. FMOLS and CCR Cointegration Regressions

The fully modified ordinary least square (FMOLS) estimator was proposed by Phillips and Hansen (1990) while the canonical correlation regression (CCR) estimator was proposed by Park (1992). These cointegration regression models use a semiparametric correction to eliminate asymptotic bias and have fully efficient normal asymptotics. They allow the use of standard Wald tests based on asymptotic chi-squared statistical inference. These cointegration regression estimators eliminate asymptotically the endogeneity bias caused by the long-run correlation of 𝑦𝑡 and 𝑥𝑡′, the second-order bias (i.e. regression errors are serially correlated) of the OLS estimator.

Both FMOLS and CCR estimators can be derived by transforming the regressors and regressand and subsequently applying the OLS procedure (Wang and Wu, 2012).

We start by expressing the time series vector process (𝑦_𝑡, 𝑥_𝑡′)′ with cointegrating relationships as;

𝑦_𝑡 = 𝑥_𝑡′𝛽 + 𝑑_1𝑡′ 𝛾₁+ 𝑢_1𝑡 (2.3)

𝑦𝑡 = 𝛤1𝑑1𝑡 + 𝛤2𝑑2𝑡+ 𝜀𝑡 (2.4)

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where 𝑑1𝑡 and 𝑑2𝑡 are deterministic trend regressors; 𝑢2𝑡 are regressors innovations. The innovations 𝑢_𝑡 = (𝑢_1𝑡, 𝑢_2𝑡′ )′ are assumed to be strictly stationary and ergodic with zero means, finite covariance matrix, one-sided long-run covariance matrix Ʌ, and nonsingular long-run covariance matrix Ω.

The regressand is being transformed under FMOLS as follows;

𝑦𝑡+ = 𝑦𝑡 - 𝑤̂12Ω̂22−1𝑢̂2𝑡 (2.6)

where 𝑢̂_1𝑡 is the residual of the cointegration equation estimated by OLS, and 𝑢̂_2𝑡 are differenced residuals of regressor equations or the residuals of the difference regressor equations.

The FMOLS estimator proposed by Phillips and Hansen (1990) is given by;

𝜃̂𝐹𝑀𝑅 = [ 𝛽

̂

𝛾̂₁] = [∑𝑇𝑡=1𝑧𝑡𝑧𝑡′] [∑𝑇𝑡−1𝑧𝑡𝑦𝑡+− 𝑇 (𝜆

̂₁₂+′

0 )] (2.7)

where 𝜆̂₁₂+ = 𝜆̂₁₂ - 𝑤̂₁₂Ω̂₂₂−1Λ̂₂₂ are called bias-correction terms. 𝑧_𝑡= (𝑥_𝑡′, 𝑑_1𝑡′ )′. 𝑤̂_1,2 is the estimate of the long-run covariance of 𝑢1𝑡 conditional on 𝑢2𝑡.

In this study, a constant and a time trend are included in the equation for the FMOLS estimator;

𝑙𝑛𝑃_{𝑈𝐾,𝑡} = 𝛽₀+ 𝛽₁𝑡 + 𝛽₂𝑙𝑛𝑃_{𝑈𝑆,𝑡}+ 𝑢_𝑡 (2.8)

The quadratic spectral kernel and the Andrews automatic bandwidth selection method are adopted.

The CCR is constructed by adjusting the data using stationary components of a given model. The CCR estimation transforms both the regressand and the regressors as;

𝑦_𝑡+ _{= {Σ̂}−1_Λ̂ 2𝛽̃ + (_Ω̂ 0 22 −1_𝜔̂ 21)} ′ 𝑢̂_𝑡 and 𝓏_𝑡+ = (1, 𝓏_𝑡+′)′ with 𝑥_𝑡+ = 𝑥_𝑡 - (Σ̂−1Λ̂₂)′𝑢̂_𝑡

where Λ̂₂ = (Λ̂₁₂, Λ̂′₂₂)′. 𝛽̃ is the OLS estimator of 𝛽; 𝑢̂_𝑡 = [Λ̂_1𝑡, ∆𝑥_𝑡′]′comprises of the OLS residuals and the first difference of the I(1) regressors. Therefore, the CCR estimator is defined as;

𝜃̂𝐶𝐶𝑅 = (∑𝑇𝑡=1𝓏𝑡+𝓏𝑡+

′

)−1(∑𝑇 𝓏_𝑡+

𝑡=1 𝑦𝑡+) (2.9)

The FMOLS and CCR models transform the data such that OLS eventually give an asymptotically efficient estimators. According to Montalvo (1995), the CCR estimator shows

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lower bias than the OLS and the FMOLS. The drawback of these methods is the ambiguity created if the system contains more than one cointegrating relation. In this situation, the Johansen technique performs better than other tests.

C. Johansen Technique

The Johansen technique builds cointegrated variables directly on maximum likelihood estimation rather than OLS procedures (Johansen and Juselius, 1988).

The technique is specified by a VAR(p) model as follows;

∆𝑦_𝑡 = Π𝑦_𝑡−1+ ∑𝑝−1_𝑖=1 Γ_𝑖∆𝑦_𝑡−1+ 𝜀_𝑡 (2.10)

where 𝛱 = αβ', α and β are n x r matrices (r is the number of cointegrating vectors). The error correction parameters contained in α measure the degree to which the variable react to disturbances in the long-run equilibrium; the parameter (𝛤_𝑖,…., 𝛤_𝑝−1) of dimension n x n define the short-run adjustment to changes in the variables, which implies the presence of p – r common trends (Gonzalo and Granger, 1995). The variables in 𝑦_𝑡 are not cointegrated, as long as the rank of 𝛱 is zero, thus the characteristic roots will equal zero and no stationary linear combination can be identified.

Johansen techniques provide two statistics to test for the null hypothesis of no cointegration, which are the trace statistic and the maximum eigenvalue statistic. They are specified as;

𝜆𝑡𝑟𝑎𝑐𝑒(𝑟) = −𝑇 ∑𝑛𝑖=𝑟+1ln (1 − 𝜆𝑖) (2.11)

𝜆_𝑚𝑎𝑥(𝑟, 𝑟 + 1) = −𝑇ln(1 − λ_r+1) (2.12)

where T is the number of usable observations; 𝜆𝑖is the estimated values of the characteristic roots derived from the estimated 𝛱 matrix. If the test statistics is greater than the critical value, then the null hypothesis that there are r cointegrating vectors is rejected against the alternative that there are r + 1 (for trace) or more than r for maximum eigenvalue (Enders, 2010). This model is useful for determining the number of cointegrating relationships (that is, long-run relationships) among the variables.

Indeed, the Johansen technique is capable of testing long-term relationship. A fundamental drawback of the Johansen method is the symmetrical treatment of all variables in a VAR system, hence makes no clear distinction between endogenous and exogenous variables.

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The Gregory-Hansen (GH) test has the capability of detecting cointegrating relations when there is a break in the intercept and/or slope coefficients. The power of standard test for the null hypothesis of no cointegration can be significantly reduced if structural changes that manifest through changes in the long-run relationship whether by changes in the intercept or changes in the cointegrating vectors are not accounted for (Gregory and Hansen, 1996). The Gregory- Hansen test for the null of no cointegration against the alternative hypothesis of cointegration while allowing for trend and one-time regime shift of unknown timing.34 The test has the capability to detect cointegration relationship among variables of interest when there is a break in the intercept and/or slope coefficient. The limitation of Johansen technique and Engle- Granger test to falsely conclude on absence of cointegrating relationship has been overcome by the Gregory-Hansen’s test inclusion of a one-time regime shift in the cointegrating vector. The structural changes of the regime shift model is captured by a shift in the intercept or slope of the cointegrating relationship. Gregory-Hansen (1996) specifies the model as;

𝑦_1𝑡 = 𝜇₁+ 𝜇₂𝜓_𝑡𝜏+ 𝛼₁т_𝜇

2𝑡+ 𝛼2т𝑦2𝑡𝜓𝑡𝜏+ 𝑒𝑡 t = 1,…,n (2.13) where, 𝜇₁ denotes the intercept before the shift; 𝜇₂ is the change in the intercept at the time of the shift; 𝛼1 represents the cointegrating slope coefficients before the regime shift; 𝛼1 and 𝛼2 denote the change in the slope coefficient; 𝜓_𝑡 if the dummy variable that captures the structural change (if t > τ, dummy variable is 1; t < τ, dummy variable is 0); τ ϵ (0,1) is a relative timing of the change point.

In summary, applying these cointegration tests will help to explore robustly the long-run relationship between UK and US stock markets. Particularly, using cointegration test that identifies structural changes will influence the results of the long-run relationship between variables under scrutiny. However, we give more credence to models that make a clear distinction between endogenous and exogenous variables.