Stock markets and their relationship
with the real economy in Latin America
Samuel Brugger *and Edgar Ortiz **

Econometric Modeling ( ...continuation )

where and ; β is the coefficient of the time trend and k is the order of the autoregressive progress. The coefficients α and β should be equal to zero indicating a random way model with deviation. The null and alternative hypotheses are ; . When the null hypothesis is accepted there is no stationarity. Finally, the order of the lags enables the test to be performed by a regressive process of the order n The number of lags should be determined. The results are obtained comparing the value obtained with the critical values. The bigger the negative number the greater the possibility for the existence of the unit root in the logarithmized series. Alternatively, p values from McKinnons test can be used; the smaller the probability, the stronger the rejection of the null hypothesis.

Once the presence of stationarity has been established in the series it can be proved whether the series of each country have different unit roots, which would mean the absence of co-integration. If, on the other hand, the series have the same unit root, this proves the existence of co-integration and therefore a balanced long term relationship.

The main idea of cointegration is that if a vector of [1 - β]T exists for two variables, making it possible to establish a lineal relationship between the two variables, of the order I (0), then the two variables are cointegrated of the order (1,1). The lineal relationship dominates the regression of the cointegration where "β" is a cointegration parameter and Z corresponds to an equilibrium error and measures the magnitude of the short term deviation in terms of long term equilibrium. The cointegration test therefore consists of an analysis of the integration equation residues so that the regression residue is I (0). The optimal number of lags should be determined in advance. A test that is often used for cointegration in financial series is the Johansen Test (1992, 1995). More specifically, a VAR in the order of p:

( 2 )

where yt is the vector k of the non-stationary monthly variables I(1); xt is a vector of deterministic variables; y εt is an innovations vector. This VAR can be expressed as shown in the following equations:

( 3 )

( 4 )

According to Grangers theorem (1969) if the matrix of coefficients π; has a reduced range of r < k, therefore k(r) matrices a and β exist, each with a range of r so that π; = α β’ and β’y t I(0). The number of cointegrating relationships is provided by r, i.e. the cointegration range. Finally, each β column is the cointegrating vector: if r = 0, there are no cointegrating vectors. The α elements are the adjustment parameters in the error correction vector. Johansens method estimates the matrix π in a non-restrictive way, and therefore rejects the restriction implied by the reduced range π.

Two tests are associated with Johansens model: trace analysis and maximum eigenvalue testing. Trace analysis is a joint test. The null hypothesis indicates that that the number of cointegrating vectors is less or equal to r, in that the alternative hypothesis shows that there are more cointegrating vectors than r. Maximum eigenvalue testing involves separate tests for each eigenvalue. The null hypothesis indicates that r cointegrating vectors are present while the alternative hypothesis indicates that there are r + 1 cointegrating vectors.

Although two variables are stock market returns and GDP and they maintain equilibrium long term, it is possible to detect short term deviations. To determine the magnitude of this imbalance and how to adjust from one period to the next, Sargans Error Correction Model Analysis (ECM) (1964; 2000) should be applied as well as Engle and Granger tests (1987). It is important to point out that a significant estimated lagged error indicates the presence of important relationships among the variables under study because in this case the series quickly adjusts to any long term deviation from imbalance that the variables share. In general this model only includes a lag of the difference of the independent variables. The current study includes additional lags, tested with serial correlation tests, heteroscedasticity correlogram, Q-tests and residual tests. The following generic ECM models can be applied both to the GDP and the stock exchange indices of each country to demonstrate relationships between the variables.