Question

Consider the regression model with the heterogeneous coefficient. Yi​=β0i​+β1i​Xi​+vi​ where (vi​,Xi​,β0i​,β1i​) are i.i.d random variables with β0​=E(β0i​) and β1​=E(β11​) It is possible to show that the model can be written as Yi​=β0​+β1​Xi​+ui​ where ui​=(β0i​−β0​)+ (β1i​−β1​)Xi​+vi

Answer 1

The given model represents a linear regression with a heterogeneous coefficient, where the coefficients β0 and β1 vary across individuals (i). The ui term captures the deviations of the individual coefficients from their population means β0 and β1.

Step-by-step explanation:

In the given model, Yi = β0i + β1iXi + vi, where (vi, Xi, β0i, β1i) are i.i.d random variables with β0 = E(β0i) and β1 = E(β11). We can rewrite the model as Yi = β0 + β1Xi + ui, where ui = (β0i - β0) + (β1i - β1)Xi + vi.

This model represents a linear regression with a heterogeneous coefficient, where the coefficients β0 and β1 vary across individuals (i). The ui term captures the deviations of the individual coefficients from their population means β0 and β1. It is important to note that the ui term represents the individual-specific effects that are not captured by the average coefficients.