2. Suppose we are given data on n observations (zi,Y), i = 1, , n, and...
2. Suppose we are given data on n observations (zi, y), î i, . . . , n, and we have a linear model, so that E (Y,) = Ao +Ari. Let A = SXY/Sxx and A,-F-Ax be the least-square estimates given in lecture. (a) Show that E(SXY)-ASxx and E(y)-Ao +AT. (b) Use (a) to show that E (A)-A and E(A)-A- In other words, these are unbiased estimators (c) The fitted values Yī = β0+812 i are used as estimates...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we have a linear model, = SXY/SXX and A,-ㄚ-Ax be the least-square estimates so that E(X) = β0 +ATp Let given in lecture. (a) Show that E(5xx)-A5xx and E(Y)-Ao +A2. (b) Use (a) to show that E(A)-A and E(A)-A. În other words, these are unbiased estimators (c) The fitted values Yi = ArtAz; are used as estimates of E(K), and the residuals ei = Y-...
Suppose we are given data on n observations (zi, Y4), i-1, . . . , n, and we have a linear model so that E(X)-β0+B1zi. Let A-SXY/SXX and A-Y-Aī be the least-square estimates given in lecture. (a) Show that E(Sxy)-ASxx and E(Y)-A +AF (b) Use (a) to show that E(BB and E(B)In other words, these are unbiased estimators (c) The fitted values Yi-A+Azi are used as estimates of %), and the residuals e,-x-Y; are used as surrogates for the unobservable...
2 2. Suppose we are given data on n observations (i, Y),, and we have a linear model, so that E(X)-A, + ßiri-Let呙-SXY /SXX and β') = F-β,2 be the least-square estimates given in lecture (a) Show that E(SXY)-ASXX and E (T)-A] + β,7. (b) Use (a) to show that E(角)-βι and E(A) = 3). In other words, these are unbiased estimators. (c) The fitted values Yt = Atari are used as estimates of E(A), and the residuals e.-Yi for...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
4. We have n statistical units. For unit i, we have (xi; yi), for i-1,2,... ,n. We used the least squares line to obtain the estimated regression line у = bo +biz. (a) Show that the centroid (x, y) is a point on the least squares line, where x = (1/n) and у = (1/n) Σ¡ı yi. (Hint: E ) i-1 valuate the line at x = x. (b) In the suggested exercises, we showed that e,-0 and e-0, where...
Considering multiple linear regression models, we compute the regression of Y, an n x 1 vector, on an n x (p+1) full rank matrix X. As usual, H = X(XT X)-1 XT is the hat matrix with elements hij at the ith row and jth column. The residual is e; = yi - Ýi. (a) (7 points) Let Y be an n x 1 vector with 1 as its first element and Os elsewhere. Show that the elements of the...
Considering multiple linear regression models, we compute the regression of Y, an n x 1 vector, on an n x (p+1) full rank matrix X. As usual, H = X(XT X)-1 XT is the hat matrix with elements hij at the ith row and jth column. The residual is e; = yi - Ýi. (a) (7 points) Let Y be an n x 1 vector with 1 as its first element and Os elsewhere. Show that the elements of the...
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
linear statistics modeling and regression 2) Suppose you have multiple regression set up Ynxi XnxpBpxi Sxl and f ~ N(0nx1, σ21.). P Po X(X,X)-X, be the projection matrix on the column space of X. a) Show residual vector, e = (1,-P)Y. Here e is the vector of residuals ei S. b) Show that the variance of e, is 1 - Pi, where P is the i, j th entry of the matrix P c) Show that the sample covariance of...