A statistical method which is used to examine the relationship between two or more variables, where a dependent variable is affected or dependent on one or more independent variables, is a regression analysis.
A regression analysis does not necessarily test for causality between variables or the direction of causality between them. To be able to analyze the direction of the causality, one can use a Granger causality test.
Granger causality analysis is a statistical hypothesis test which investigate the direction of causality between variables to find the direction of a potential causal relationship. The null hypothesis of the Granger causality test states that there is no causality between two variables while the alternative hypothesis says that there is a causality.
The basic idea of a Granger causality test is to determine whether future values of a time series X can be predicted by the use of past values of an additional variable, for example, time series Y and identify which variables occur first. If time series Y can be predicted using past values of both Y and X, the result can be expressed as variable X is Granger causing variable Y. Granger causality tests observe two time series to be able to discern whether series Y occurs earlier than series X or if the movements of the variables occur simultaneously. (Studenmund 2017) Granger causality does not identify direct causality and does not analyzing if there are other influencing factors, the causality test only determines if one variable precedes another or not. By using annual data this thesis is not analyzing direct causality, instead the purpose of this work is to analyze in which order the different variables occurs, therefore a Granger causality analysis is used. An example of the Granger causality analysis is how meteorologists predict the weather through weather forecasts, these forecasts precede the actual weather. This does not mean that the meteorologist's forecast is causing the current weather, this is an example of Granger causality, how to interpret it and how it can be misleading. Granger causality only shows which variable that occurs first, but one cannot determine if the variable that occurs first is the cause of the change. However, one can determine that itβs not the other way around (Leamer 1985).
Granger causality tests can be applied through the technique of VAR. Equation 1.1 and 1.2 states that π
!-and π(- are related to past values of the other as well as the past value of itself. By running the Granger-test, conclusions about causality can be established. That is, if π!- Granger causes π(-or if π(- Granger causes π!-. If the result of the Granger test states that both π(- and π!- causes each other there is a bidirectional causality running between them and one can not identify which variables precedes another. If only one variable causes the other, there is a so-called unidirectional causality. There can also be a case of independence, that is, when the two variables coefficients are not statistically significant in either of the
In this section, focus is to determine a number of directions of Granger causality between selected variables.
The direction of causality that are investigated is presented in table 2. The study investigates the causal relationship between OMXSPI and GDP and the causal relationship between Bank development and GDP.
A unidirectional causal relationship is examined from St.dev, Turnover ratio and Turnover MSEK to GDP, jointly and separately.
Table 2. The directions of causality being investigated
OMXSPIβ GDP GDP β OMXSPI OMXSPI β GDP Bank β GDP GDP β Bank Bank β GDP Stock market variables β
GDP St.dev β GDP
Turnoverratio β GDP
TurnoverMSEK β GDP
In order to conduct a Granger causality test some steps need to be implemented in advance. First: each variableβs lagged terms are computed for regressing the unrestricted and restricted models. In the unrestricted models, each variable is regressed on lagged terms of itself and on the other variables in the system. The unrestricted model contains of all parameters. In the restricted model the variables are also regressed on its past values, but it excludes the lagged value of the investigated variable in the model. Both these models are needed to perform an F-test. (Gujarati & Porter 2009)
A restricted and an unrestricted model is created to be able to conduct the F-tests, to test for Granger causality. The unrestricted and restricted model can be determined from the equation 1.1 and 1.2. When testing for the direction of causality from π!- (GDP) to π(-(OMXSPI) equation 1.1 is the unrestricted model. By exclude, the variable π(- and all the lagged values connected to π(-, from equation 1.1, one get the restricted model which means that all π½-values related to π(- is equal to zero. By exclude several variables simultaneously, one can analyze the jointly causality between more than two variables. For example, by exclude the variables π*-, π+- and π,- in equation 1.1 and all lagged values connected to each variable, the π½-values related to this three variables is equal to zero and the outcome will be a restricted model. These steps, to create restricted and unrestricted models, are applied to analyze direction of causality between several variables.
The null hypothesis states that one variable does not Granger cause another. To test whether the null hypothesis is true or false, a F-test is conducted. In the tests both the restricted and unrestricted regression models are required to receive the restricted and unrestricted residual sum of squares. The residual sum of
squares, both restricted and unrestricted, is needed to be able to calculate the F value used in the F-test.
(Gujarati & Porter 2009)
πΉ = π ππ/β π ππ0/ / π π ππ0/ (π β π)
(1.6)
π ππ0/ : residual sum of squares from the unrestricted regression model π ππ/ : residual sum of squares from the restricted regression model k: number of parameters estimated in the unrestricted regression n: number of observations
m: number of terms that is lagged n-k: the degree of freedom
From the F-test a computed F-value is received, at a chosen level of significance, and is then compared with the critical F-value. The critical value of F is obtained from the F-distribution which is based the degrees of freedom. The null hypothesis can be rejected if the observed F-value exceed the critical F-value, at the chosen level of significance. If the computed F-value does not exceed the critical F-value the null hypothesis can not be rejected. Alternatively, the p-value can also be used which determines whether the null hypothesis can be rejected or not depending on the specified level of significance. If the p-value of the computed F-value is lower than the chosen level of significance the null hypothesis can be rejected. To check different directions of Granger causality all the steps above are applied in all cases. (Gujarati & Porter 2009)
4.4 T-test
In addition to an F-test, this thesis uses a t-test to answer some of the hypotheses. The use of a t-test is needed for testing individual significance, i.e. how one variable affect another given that the other variables in the regressions are kept constant. The t-test uses a certain level of significance to see if a hypothesis can be rejected or not by comparing an observed t-value with a critical t-value. The critical t-value is determined from the t-table where the difference between the number of observations and parameters determines which degree of freedom to use to find the correct critical value. Another way to interpret the result of the regression without using the observed t-value is to analyze the p-value. The p-value determines whether the null hypothesis can be rejected or not depending on the specified level of significance. A high p-value indicates that the null hypothesis can not be rejected, and a low p-value indicates that the null hypothesis can be rejected i.e. the p-value tells the probability that the null hypothesis can be true. (Gujarati & Porter 2009)