Consider the sample regression Y i 1 2 X i u i Imposing the restrictions (i) u i 0 and (ii) u i X i 0, obtain the estimators 1 and 2 and show that they are identical with the least squares estimators given in Eqs (3 1 6) and (3 1 7) This method of obtaining estimators is called the analogy principl...

Imposing the first restriction we obtain u i Y i 1 2 X i 0 Simplifying this yields the first n ...

Consider the sample regression Y i = 1 + 2 X i + u i

Question:

Consider the sample regression

Y_i = β̂₁ + β̂₂X_i + û_i

Imposing the restrictions (i) Σû_i = 0 and (ii) Σ û_iX_i = 0, obtain the estimators β̂₁and β̂₂ and show that they are identical with the least-squares estimators given in Eqs. (3.1.6) and (3.1.7). This method of obtaining estimators is called the analogy principle. Give an intuitive justification for imposing restrictions (i) and (ii). (Hint: Recall the CLRM assumptions about u_i.) In passing, note that the analogy principle of estimating unknown parameters is also known as the method of moments in which sample moments (e.g., sample mean) are used to estimate population moments (e.g., the population mean). As noted in Appendix A, a moment is a summary statistic of a probability distribution, such as the expected value and variance.