Answer

Solved Step By Step with Explanation - LM Test for Specification

a) To perform the LM test for model specification, we need to compare the R^2 values of the quadratic form regression (Y = α1 + α2X + α3X^2 + u) and the auxiliary regression (Y = β1 + β2X + e).

Once we have the estimated coefficients, we can calculate the predicted values of Y based on the quadratic form regression equation.

Next, we can calculate the sum of squares of residuals (SSR_quad) and the total sum of squares (SST) for the quadratic form regression.

b) If we have determined that the quadratic form regression is the correct specification, estimating a linear regression (Y = β1 + β2X + e) would result in biased estimates of the slope coefficient.

The reason for this bias is that the quadratic form includes an additional term, α3X^2, which captures the non-linear relationship between CO2 emission and real GDP. If we exclude this term by estimating a linear regression, we are omitting an important component of the relationship, leading to biased results.