TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 16

CHAPTER 31 Residuals: Standardized, Predictive, “Studentized”. . Three Decisions about Plotting Residuals After running a regression it is always advisable to look at the residuals. Here one has to make three decisions. The first decision is whether to look at the ordinary residuals | CHAPTER 31 Residuals Standardized Predictive Studentized . Three Decisions about Plotting Residuals After running a regression it is always advisable to look at the residuals. Here one has to make three decisions. The first decision is whether to look at the ordinary residuals i yi - xT 3 xj is the ith row of X or the predictive residuals which are the residuals computed using the OLS estimate of gained from all the other data except the data point where the residual is taken. If one writes 3 i for the OLS estimate without the ith observation the defining equation for the ith predictive residual 795 796 31. RESIDUALS which we call i i is i i Vi - X W- The second decision is whether to standardize the residuals or not . whether to divide them by their estimated standard deviations or not. Since e My the variance of the ith ordinary residual is var j am a2 1 - hu and regarding the predictive residuals it will be shown below see that 22 a2 a2 var iW . . mii 1 hii Here hii X XTX -1Xi. Note that x is the ith row of X written as a column vector. hii is the ith diagonal element of the hat matrix H X XTX -1XT the projector on the column space of X. This projector is called hat matrix because y Hy . H puts the hat on y. . THREE DECISIONS ABOUT PLOTTING RESIDUALS 797 PROBLEM 362. 2 points Show that the ith diagonal element of the hat matrix H X X TX -1 X T is xi XTX 1Xi where x is the ith row of X written as a column vector. ANSWER. In terms of ei the n-vector with 1 on the ith place and 0 everywhere else Xi XTei and the ith diagonal element of the hat matrix is ef Hei ef Xi XTX - 1XTei x7 XTX -1Xi. PROBLEM 363. 2 points The variance of the ith disturbance is a2. Is the variance of the ith residual bigger than a2 smaller than a2 or equal to a2 Before doing the math first argue in words what you would expect it to be. What about the variance of the predictive residual Prove your answers mathematically. You are allowed to

• Kế toán - Kiểm toán
• thống kê học
• kinh tế lượng
• kinh tế học
• toán kinh tế
• giáo trình kinh tế
• Phương pháp thống kê
• Phương pháp chỉ số
• Tài liệu thống kê
• Tài liệu kiểm toán