You have collected time series for various macroeconomic variables to test if there is a single cointegrating relationship among multiple variables. Formulate the null hypothesis and compare the EG–ADF statistic to its critical value

(a) Canadian unemployment rate, Canadian Inflation Rate, United States unemployment rate, United States inflation rate; t = (-3.374).
(b) Approval of United States presidents (Gallup poll), cyclical unemployment rate, inflation rate, Michigan Index of Consumer Sentiment; t = (-3.837).
(c) The log of real GDP, log of real government expenditures, log of real money supply (M2); t = (-2.23).
(d) Briefly explain how you could potentially improve on VAR(p) forecasts by using a cointegrating vector.
What will be an ideal response?

Answer:
(a) The null hypothesis of a unit root in the error correction term cannot be rejected even at the 10% level. Hence there is little support of a single cointegrating relationship between these four variables.
(b) The critical value is (-4.20) at the 10% significance level. Hence you cannot reject the null hypothesis of the error correction term having a unit root.
(c) Since the critical value for three variables is (-3.84) at the 10% significance level, there does not seem to be a cointegrating relationship between the three variables.
(d) Adding the error correction term from the cointegrating relationship between variables to the VAR(p) model results in a vector error correction model (VECM). The advantage of this model over a VAR model is that it incorporates both short-run and long-run information into the forecasting equation.