Unit Root Test For Real Price

Published Date: 02 Nov 2017

The null hypothesis is that the series has a unit root. As shown in the table above, the test statistics for industry demand for electricity is -3.032 and the p value is 0.0320. The absolute value of the critical value is taken (|-3.580|=3.580). As the 1st critical value is greater than the test statistics, the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Unit root test for Demand (Households)

The null hypothesis is that the series has a unit root. The above table shows that the test statistics for households demand for electricity is 0.326 and the p value is 0.9785. As the test statistics is positive, the null hypothesis is rejected and there is no unit root therefore it is stationary.

Unit root test for Real Price (Industry)

The null hypothesis is that the series has a unit root. The above table shows that the test statistics for industry real price is -0.586 and the p value is 0.8741. As the test statistic is smaller than the critical value of the greatest acceptable confidence level (10%), the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Unit root test for Real Price (Households)

The test statistics for households real price is -0.358 and the p value is 0.9169. The null hypothesis is that the series has a unit root. As the test statistic is smaller than the critical value of the greatest acceptable confidence level (10%), the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Unit root test for Income

The test statistics for income is -1.918 and the p value is 0.3236. The null hypothesis is that the series has a unit root. As the test statistic is smaller than the critical value of the greatest acceptable confidence level (10%), the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Co-integration is a relationship between two non stationary variables; it refers to the fact that these variables share a common stochastic drift and a long term equilibrium relationship. Co-integrated data are never expected to drift too far away from each other and tend to move together in the long run. Engle and Granger suggested a two step process to test for co- integration. The test procedure is to firstly regress one variable (demand) on another variable (income) using ordinary least squares. The second step is to then predict and test a residual for non stationary using the (augmented) Dickey-Fuller unit root test. If the series are co integrated, then the Dickey-Fuller test statistic will be statistically significant. The null hypothesis is that the residual is non stationary. If the residuals are stationary and co-integrated then the null hypothesis will be rejected.

A regression analysis is used to produce an equation that will predict a dependent variable using one or more independent variables and this equation has the form of ln(total demand)=β1 + β 2ln(Income) + ut

The above stata output shows the analysis of the variance (anova) results located on the top left along with the regression results which is located at the bottom. The dependent variable is demandindustry and the independent variable is income. The coefficients for income and constant are shown in the Coef. column. Std. Err. is Standard Error, t is test statistics, P>|t| are the p values, and 95% Confidence Interval. The results from the stata output can be written in regression equation as predicted ln(total demand) =9.782497+0.6714485Income. The coefficient for income is 0.6714485. This leads to a conclusion that an increase in one unit of income, a 0.67144851 unit increase in demand is predicted. The 95% confidence intervals for the coefficients are related to the p values such that the coefficient will not be statistically significant if the confidence interval includes 0. As the p value is less than 0.05, it is significant at 95% level. The ‘P>|t|’ column shows the two-tailed p-values used in testing the null hypothesis that each coefficient (parameter) is different from 0. Using an alpha of 0.05, the coefficient for income is statistically significant from 0 because its p-value of 0.000 is smaller than 0.05. The constant is an intercept of the regression line and is also significantly different from 0 at the 0.05 alpha level. As the p value is less than the significance level (0.05), there is sufficient evidence to reject the null hypothesis therefore the null hypothesis is rejected. The t-values shows the importance of a variable in the model and also tests whether a given coefficient is significantly different from zero. T-values are obtained by dividing the coefficients by its standard error. The t values in this table are relatively large. ‘Prob>F’ is the p-value associated with the above F-statistic.  It is used to test whether R-square is different from 0. As it is less than 0.05, there is a statistically significant relationship between demandindustry and income. R-squared is the proportion of variance in the dependent variable which can be explained by the independent variable. In this model, R-squared is 0.8318; meaning about 83% of the variance in demand is explained by this model. The adjusted R-squared is a modification of the value of R-squared that adjusts for the number of terms in a model. R-squared always increases when a new term is added to the model on the other hand the adjusted R-square only increases if the new term improves the model more than expected by chance. The adjusted R-squared in this model is 0.8283.

The above stata output shows the analysis of the variance (anova) results located on the top left along with the regression results which is located at the bottom. The dependent variable is demandhousehold and the independent variable is income. The coefficients for income and constant are shown in the Coef. column. Std. Err. is Standard Error, t is test statistics, P>|t| are the p values, and 95% Confidence Interval. The results from the stata output can be written in regression equation as predicted ln(total demand) =6.231413+1.361949Income. The coefficient for income is 1.361949. This leads to a conclusion that an increase in one unit of income, a 1.361949 unit increase in demand is predicted. As the p value is less than 0.05, it is significant at 95% level. Using an alpha of 0.05, the coefficient for income is statistically significant from 0 because its p-value of 0.000 is smaller than 0.05. The constant is an intercept of the regression line and is also significantly different from 0 at the 0.05 alpha level. As the p value is less than the significance level (0.05), there is sufficient evidence to reject the null hypothesis therefore the null hypothesis is rejected. T-values are obtained by dividing the coefficients by its standard error. The t values in this table are relatively large. ‘Prob>F’ is the p-value associated with the above F-statistic.  It is used to test whether R-square is different from 0. As it is less than 0.05, there is a statistically significant relationship between demandhousehold and income. In this model, R-squared 0.9888; meaning about 99% of the variance in demand is explained by this model. The adjusted R-squared for this model is 0.9886.

A Dickey Fuller unit root test is conducted on the residual. The null hypothesis is that the series has a unit root. The test statistics for the residual is -2.641 and the p value is 0.0849. As the test statistic is smaller than the 5% critical value, the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Error Correction Models are a group of multiple time series models that estimate the speed at which a dependent variable Y returns to equilibrium after a change in an independent variable X. The ECMs models are useful when dealing with integrated data but it can also be used with stationary data. It estimates both short and long term effects of one time series on another.

ECM

The 5% critical value for a cointegrating regression containing an intercept is -3.37 and the t-ratio is less than this. The null hypothesis of no cointegration is rejected when ct t ≤ , and not rejected when ct t > . The t-statistic in this case is −4.196< −3.37 and the null hypothesis that the least squares residuals are nonstationary is rejected; the residuals are stationary. This implies that the bond rate and the federal funds rate are cointegrated.

The t statisctic as shown in the table is -2.20. The 5% critical value for a co intergrating regression containing an intercept is – 0.6116589 and the t-statistic is