TAILIEUCHUNG - ECONOMETRICS phần 3

Những người sáng tạo nội dung rất nhiều động lực để làm cho nó hoạt động duy trì các máy chủ, với nội dung mới, đoán thị hiếu của các cô gái, và tạo ra Thoái e1 trên X 2; có được ước lượng OLS b 2 và chất thải điện tử: ~ ^ Trong một số hoàn cảnh, định lý FWL | CHAPTER 3. CONDITIONAL EXPECTATION AND PROJECTION 52 and the second given by the dashed line is the linear projection on experience and its square P log wage I Experience Experience . It is fairly clear from an examination of Figure that the first linear projection is a poor approximation. It over-predicts wages for young and old workers and under-predicts for the rest. Most importantly it misses the strong downturn in expected wages for older wage-earners. The second projection fits much better. We can call this equation a quadratic projection since the function is quadratic in experience. Figure Linear and Quadratic Projections of log wage onto Experience CHAPTER 3. CONDITIONAL EXPECTATION AND PROJECTION 53 Invertibility and Identification The linear projection coefficient p E xx0 1 E xy exists and is unique as long as the k X k matrix Qxx E xx0 is invertible. The matrix Qxx is sometimes called the design matrix as in experimental settings the researcher is able to control Qxx by manipulating the distribution of the regressors x. Observe that for any non-zero a 2 Rk a0Qxxa E a0xx0a E a0x 2 0 so Qxx by construction is positive semi-definite. The assumption that it is positive definite means that this is a strict inequality E a0x 2 0. Equivalently there cannot exist a nonzero vector a such that a0x 0 identically. This occurs when redundant variables are included in x. Positive semi-definite matrices are invertible if and only if they are positive definite. When Qxx is invertible then p E xx0 1E xy exists and is uniquely defined. In other words in order for p to be uniquely defined we must exclude the degenerate situation of redundant varibles. Theorem shows that the linear projection coefficient p is identified uniquely determined under Assumptions . The key is invertibility of Qxx. Otherwise there is no unique solution to the equation QxxP Qxy. When Qxx is not invertible there are multiple solutions to all

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