# tailieunhanh - ECONOMETRICS phần 4

## có thể được sử dụng để tăng tốc độ tính toán, nhưng trong nhiều trường hợp có ít tính toán lợi thế để sử dụng các thuật toán hai bước. Thay vào đó, việc sử dụng chính là lý thuyết. Một ứng dụng phổ biến của định lý FWL, mà bạn có thể đã thấy trong một khóa học kinh tế giới thiệu là công thức hạ thấp phẩm giá cho hồi quy. Phân vùng | CHAPTER 4. THE ALGEBRA OF LEAST SQUARES 81 Theorem Frisch-Waugh-Lovell In the model the OLS estimator of @2 and the OLS residuals e may be equivalently computed by either the OLS regression or via the following algorithm 1. Regress y on X1 obtain residuals 1 2. Regress X2 on X1 obtain residuals X2 3. Regress 1 on X2 obtain OLS estimates @2 and residuals e. In some contexts the FWL theorem can be used to speed computation but in most cases there is little computational advantage to using the two-step algorithm. Rather the primary use is theoretical. A common application of the FWL theorem which you may have seen in an introductory econometrics course is the demeaning formula for regression. Partition X X1 X 2 where X1 is the vector of observed regressors and X2 b is a vector of ones . In this case M2 I - b b b 1 b . Observe that X1 yẽ M2X1 X1 - b b b 1 b X 1 X1 - X1 and M 2y y - b b b 1 b y y - y which are demeaned . The FWL theorem says that @1 is the OLS estimate from a regression of Vi - y on X1i - X1 @ 1 52 X1i X1 X1i X1 zL X1i - X1 yi - y . i 1 i 1 Thus the OLS estimator for the slope coefficients is a regression with demeaned data. Ragnar Frisch Ragnar Frisch 1895-1973 was co-winner with Jan Tinbergen of the first Nobel Memorial Prize in Economic Sciences in 1969 for their work in developing and applying dynamic models for the analysis of economic problems. Frisch made a number of foundational contributions to modern economics beyond the Frisch-Waugh-Lovell Theorem including formalizing consumer theory production theory and business cycle theory. CHAPTER 4. THE ALGEBRA OF LEAST SQUARES 82 Prediction Errors The least-squares residual êi are not true prediction errors as they are constructed based on the full sample including yi. A proper prediction for yi should be based on estimates constructed only using the other observations. We can do this by defining the leave-one-out OLS estimator of 3 as that obtained from the sample of n 1 .

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