3.3 Dynamic Price Transmission and Market Integration Models
3.3.3 Cointegration Analysis
The cointegration of a pair of markets means that the dynamics of the price relationships in the two markets may not obey the LOP in the short-run, but in the long run converge towards the LOP. If two price series, and , in two spatially separated markets contain stochastic trends and are integrated of the same order, say I (d), the markets are said to be cointegrated if there is a linear relationship between the price series. In other words, if the test of stationarity proves that I (1) and I (1), then and are cointegrated if cointegrated, their response adopts a long run equilibrium relationship (Prakash, 1997).
Cointegration also implies Granger causality.
Two commonly employed approaches to obtaining cointegration vectors and determining market integration exist. They are the two-step approach of Engel and Granger (1987) used for bivariate models and the Johansen (1990) variance autoregressive (VAR) approach, which is used in multivariate analyses. The first step in employing any of the two approaches is confirming that all the price series for the analysis are non-stationary and integrated of the same order. This involves testing the stationarity properties and the order of integration of the price series individually under a null hypothesis of non-stationarity of the series (unit roots) using the Dickey-Fuller (DF), augmented Dickey-Fuller (ADF), Phillips-Perron (PP) procedures or a host of other approaches to unit root tests.
An observed price series, for the market i is tested for stationarity using the DF (1979) by running a regression of equations (21) or (22) below.
i
Where εtis identically and independently distributed with a zero mean and constant variance (εt ∼iid(0,σ2). The null hypothesis of non-stationarity is β= 1.
Similarly, the augmented Dickey-Fuller (ADF) test estimates the price series, , as an autoregressive (AR) pattern in equation (20) below:
i
and n is the number of lags needed to eliminate serial correlation in the series. A failure to reject the null hypothesis, H0:β = 1, implies that the series exhibits a unit root (I [1]), otherwise the series is stationary. In the first case, the procedure is repeated for the first difference of the price series.
Once the price series are shown to be I[1] but their differences are I[0], they are candidates for cointegration analysis. The test for cointegration typically evaluates an equation such as (24) using OLS:
...(24)
i j
t t t
P = +α βP +ε
Where β measures the long run linear relationship between the individual price series, and are as earlier defined, and
i
Pt j
Pt εtis the estimated residual error of the long run relationship. Cointegration simply measures whether the above prices and move together; that is whether the differential given by
i
Pt Ptj εtis stationary.
The Engel and Granger (E-G) Cointegration Approach
The first step of the Engle and Granger procedure involves estimating equation (24) above using OLS. A Dickey-Fuller (DF) test on the residuals is then performed to determine their order of integration by running the regression below:
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Δ =ε α εˆt 1ˆt−1+et... (25) Where is a white noise term. et
If we cannot reject the null hypothesis of α1 = 1, we can conclude that the residual series contains unit root, and the two series are not cointegrated. Conversely, a rejection of the null hypothesis means that the residual sequence is stationary, and the series are cointegrated.
Given that two series are I(1) and the residuals are stationary, then the series are integrated of order (1, 1).
and
j
Pt Pti
The hypothesis of no cointegration may also be tested by verifying the unit root properties of εt using an improved version of the DF called the augmented Dickey-Fuller (ADF) test.
This test is conducted by estimating the equation:
1 1
2
ˆt ˆt t n
j t j
ε δ α ε − γ φ ε
Δ = + + + ∑= Δ − ... (26)
Where Δ denotes a first-order difference in the estimated residual term (i.e.Δ = −ε ε εt t t−1), and n = 1, 2 …n, are lag lengths. If the null hypothesis of α1 = 0 is rejected, then the residual series from this equation are stationary [I (1)], and since the two price series are integrated of the same order, then they are cointegrated of the order (1, 1).
The E-G approach to testing for cointegration is limited in a multi-variable case because it lacks a systematic approach for the separate estimation of multiple cointegration vectors. In addition, the reliance of E-G on two steps i.e. generating a residual series εˆtin the first step and then using εˆtto estimate an equilibrium equation in the second step means estimation errors from step 1 could be carried into step 2. In addition, where more than one variable is involved, the test of cointegration using the E-G yields results that are fairly sensitive to the variable selected for normalization.
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Johansen and Juselius Technique
In a multivariate case, the test for cointegration using the E-G technique yields results that are sensitive to the variable selected for normalization. In this case, a multivariate generalization of the Engel-Granger method proposed by Johansen and Juselius (1990) is used. This method is especially efficient in dealing with multivariate price series obtained from markets in which the direction of causality of price transmission among the markets is unknown (Johansen, 1995, and Ln and Inder, 1997 in Motamed et al, 2008). This is because of the technique’s ability to treat all the markets as endogenous and handle the response of the different variables to market shocks simultaneously. In addition, the Johansen Method overcomes the problem of normalization encountered in the E-G method, and allows the use of variables with different orders of integration i.e. I (0) and I (1) for the analysis.
The following vector autoregressive equations are estimated under the Johansen’s approach:
1
Where is a vector of time-ordered prices, Pt ηtis the p-dimensional vector of random errors, is the first difference operator,
Δ θ (monthly intercepts to account for seasonality), and Γ p-dimensional vectors and matrices of coefficients to be estimated, respectively. The vectors of random errors,η0t and η1t, are used to construct likelihood ratio test statistics that determine the number of unique cointegration vectors in . Pt
Two test statistics are used for the null hypothesis of no cointegration. The first test statistic, known as the trace test, evaluates the null hypothesis that there are at most r-cointegrating vectors in represented by: Pt
Where λ∧ denotes p – r smallest correlations of η0t with respect to η1tand T is the number of observations.
The second maximum likelihood ratio test, known as the maximal Eigen value test, evaluates the null hypothesis that there are exactly r cointegrating vectors in and is given by:
Pt
...(3
max ( , 1) ln(1 ... 0) 1
)
r r T
r λ = + = − −
λ
∧ +Because ln (1) = 0, the expression ln (1-λi) will be equal to zero if the prices are not integrated. The accuracy of the λtrace and λmaxdepend on the sample size, number of lags and the data series used (Cheung and Lai, 1993 in MacKinnon et al, 1998). Generally however, the further the estimated characteristic roots are from zero, the larger theλtraceandλmax statistics.
The null and alternative hypothesis for the λtraceand λmaxfor a case of k ( k = 1, 2, …k) price series are:
λtrace λmax
Null Hypothesis Alternative Hypothesis Null Hypothesis Alternative Hypothesis r = 0 r > 0 r = 0 r = 1
r ≤ 1 r > 1 r = 1 r = 2 r ≤ 2.. r > 2… r = 2… r = 3…
…r ≤ k …r > k … r = k …r = k+1
If none of the k null hypotheses above are rejected under the trace and maximum Eigen value tests, it is concluded that either there is no cointegrating vectors (i.e. no evidence of the LOP) or the null hypotheses are falsely rejected due to failure to account for other variables. If the null hypothesis r = 0 is rejected and the second and third null hypotheses are not rejected; it is concluded that there are two cointegrating vectors. If all k null hypotheses are rejected; then it is concluded that there are k cointegrating vectors and thus a
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confirmation of cointegration among the data series2. Critical values for the trace and maximal Eigen value test statistics are obtained from the Johansen and Juselius Table.
Ardeni (1989) and Baffes (1991) applied cointegration analysis to verifying the LOP in international commodity markets. The former revealed that prices would be generally cointegrated if non-stationarity in price series did not exist. Though this revelation is supported by Engel and Granger (1969), Ardeni did not find evidence of cointegration in international commodity markets and concluded that the LOP was not supported in international markets for the basic commodities that he considered. Baffes on the other hand pointed out that cointegration tests alone are insufficient in establishing the LOP. Alderman (1993) also used cointegration test to evaluate the integration of grain markets in Ghana.
Even though, cointegration has an advantage over the earlier methods because it deals with the problem of non-stationarity of time series data in the levels, the standard approach implicitly assumes a linear relationship between price series and stationary transactions costs. This assumption is misplaced because non-linearity in market relationships arises from arbitrage conditions, unsynchronized price cycles, discontinuous trade and non-stationary transactions costs (Baulch, 1997; McNew, 1996; McNew and Fackler, 1997;
Fackler and Goodwin, 2002; and Barrett and Li, 2002 reported by Rapsomanikis, 2003).
This means the use of linear cointegration analysis is justified only when the long run equilibrium relationship between prices is expected to remain fixed throughout the period of the study. According to Alexander and Wyeth, (1994) in Baulch (1997), cointegration is neither necessary nor sufficient for testing market integration, but is only a pre-test for other econometric techniques of market integration analysis.
2 This condition however does not make economic sense since for a number of N series; the number of cointegrating vectors should be N-1.
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