The researcher undertakes an Augmented Dickey-Fuller ADF test for the variables pce andpdi and obtains the following resultsDpce Coef Std ErrpceL1 -0632 0327LD 2249 1090L2D 2034 1099L4D 1754 0922L10D

(a) From the results of the ADF test, we can determine the order of integration for both pce and pdi. The estimated ADF test is:

For pce: Null hypothesis: The pce variable has a unit root (is non-stationary) Alternative hypothesis: The pce variable is stationary Critical values: There are different critical values depending on the number of observations and the inclusion of a trend term. We would need to know the number of observations and the presence of a trend term to determine the critical values.

Based on the coefficient estimates for the differenced variable (D.pce), we can see that the coefficient for L1 (lag 1) is negative and statistically significant. This suggests that the differenced pce variable is stationary, indicating that the pce variable itself is integrated of order 1 (pce is I(1)).

For pdi: Null hypothesis: The pdi variable has a unit root (is non-stationary) Alternative hypothesis: The pdi variable is stationary Critical values: There are different critical values depending on the number of observations and the inclusion of a trend term. We would need to know the number of observations and the presence of a trend term to determine the critical values.

Based on the coefficient estimates for the differenced variable (D.pdi), we can see that none of the coefficient estimates are statistically significant. This suggests that the differenced pdi variable is not stationary, indicating that the pdi variable itself is integrated of order 0 (pdi is I(0)).

It is important to note that the conclusion about the order of integration may be limited by the specific test used, the number of observations, and the presence of a trend term. Additional diagnostic tests and analysis should be conducted to further evaluate the results.

(b) To determine if there is evidence of a long-run cointegrating relationship between pce and pdi, we can examine the coefficient estimates for the residuals from the regression model. If the coefficient estimates are statistically significant and have the expected signs, it suggests the presence of a long-run relationship.

From the table of results for the residuals (vhat), we can see that the coefficient estimate for L1 (lag 1) is negative and statistically significant. This suggests that there is evidence of a long-run cointegrating relationship between pce and pdi.

(c) Assuming there is evidence of cointegration, the error correction model for pce can be written as:

D.pce = α + β1D.pdi + β2D.pdi(t-1) + β3D.pce(t-1) + γ(vhat(t-1)) + ε(t)

To determine the appropriateness of the error correction model, various diagnostic tests can be performed, such as checking for the significance and signs of the coefficient estimates, assessing the goodness of fit of the model, and conducting residual analysis. These tests can help evaluate the model's ability to capture the long-run relationship between pce and pdi and the dynamics of the adjustment process