(a) An analyst is using the regression model below to investigate the effects of the three independent variables, X₁, X₂, X₃ on the dependent variable Y:

Y_i = ₀ + ₁X_1i + ₂X_2i + ₃X_3i + u_i (1)

Based on relevant theory, all three of the independent variables should be relevant for explaining variation in the dependent variable. However, the analyst is concerned that there may be multicollinearity between some of the independent variables. To check this, they compute the sample correlation coefficients between each pair of independent variables:

Table 1: Correlation Coefficients Between Independent variables

X1 X2 X3

X1 1.00 0.29 0.86

X2 0.29 1.00 0.32

X3 0.86 0.32 1.00

They also estimate the following three auxiliary regressions:

X_1t = ₀ + ₁X_2t + ₂X_3t + _1t (2)

X_2t = ₀ + ₁X_1t + ₂X_3t + _2t (3)

X_3t = ₀ + ₁X_1t + ₂X_2t + _3t (4)

The R² values obtained for these three auxiliary regressions are 0.74 for model (2), 0.10 for model (3) and 0.75 for model (4).

(i) Based on the information above and any other relevant calculations, do you think the analyst is correct to be concerned about possible multicollinearity in model (1)? Clearly explain and justify your answer.

(ii) In situations in which multicollinearity is identified, in what ways could the model or analysis be changed to minimise the level of multicollinearity? Is it always correct to regard multicollinearity as a problem which needs to be solved in order to obtain reliable results from the model? Explain your answer.

(b) Briefly outline the principles or strategies that can be used when trying to select the most appropriate set of explanatory/independent variables for a particular regression model. Why is trying to identify the most suitable explanatory variables crucial in order to obtain reliable results?