By Peter Kennedy
This is often the appropriate (and crucial) complement for all econometrics classes--from a rigorous first undergraduate direction, to a primary master's, to a PhD course.
* Explains what's going in textbooks choked with proofs and formulation
* bargains instinct, skepticism, insights, humor, and functional suggestion (dos and don’ts)
* comprises new chapters that hide instrumental variables and computational concerns
* comprises more information on GMM, nonparametrics, and an creation to wavelets
Read or Download A Guide to Econometrics (4th Edition) PDF
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Extra info for A Guide to Econometrics (4th Edition)
6 Exponential Model Let PÂ be an exponential distribution on the positive semiaxis with the parameter Â. 5. Â/. 7 Poisson Model Let PÂ be the Poisson distribution with the parameter Â. 6. Â/. jY1 j > y/ D e y= ; y 0; for some given > 0. Equivalently one can say that the absolute value of Y1 is exponential with parameter under P0 . Now define PÂ by shifting P0 by the value Â. jY1 Âj > y/ D e It is obvious that E0 Y1 D 0 and EÂ Y1 D Â. d. 7. Â/. d. with a density symmetric around zero and finite second moment 2 D E"21 .
1 (Cramér–Rao Inequality). d. PÂ /. d. Y ; Â/ almost surely. Â/ Âj2 Âj2 : Proof. Consider first the case of an unbiased estimate ÂQ with EÂ ÂQ Á Â. t. Y ; Â/. Y ; Â/ almost surely. This implies the first assertion. Now we consider the general case. The proof is similar. 10) continues to hold. Â/ and the second assertion follows. The last statement is the usual decomposition of the quadratic risk into the squared bias and the variance of the estimate. 2 Exponential Families and R-Efficiency An interesting question is how good (precise) the Cramér–Rao lower bound is.
2 Substitution Principle: Method of Moments 19 Proof. We use vector notation. e > h/e: Then by definition ˛Q D n 1=2 e > …1 Y and Q 2 D n 1 kY …1 Y k2 . In …1 /"k2 where In is the identity operator in Rn and In …1 is the projector on the hyperplane in Rn orthogonal to the vector e. 1. Check the details of the proof. Hint: reduce to the case of diagonal V . 2. ˛Q ˛/. Q 2 / . Show that ˛Q and Q 2 are independent. In …1 /"k2 . In …1 /" are independent if …1 is a projector and " is a Gaussian vector.