(a) When considering the linear regression model y = E(e) 0, find the least squares estimates of Bo, Bi, and B2. B0B12 B2z2 € with
(b) Are the least squares estimates of B1 and B2 obtained in (a) unbiased when the true model includes an interaction term B12, that is E(y) = Bo +B121+B222+B122122? Explain
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3. (20 pts) Suppose that we have 4 observations for 3 variables y , x\, X2...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider a problem of regressing y on two (qualitative) variables r, 2. Data: 22 obs no. y (Income) 2 (Management Status) I (Gender) 1 None Female 2 None Male Yes Female Yes Male 4 To handle the qualitative variables r, 12, we define dummy variables 1, 22 as for 1, 22= Yes Male for 1, 219 22 -1. for 22= None for 1= Female -1,...
Using 3 variables and 13 observations, we are going to predict 'y. According to 0.05 significant...
Using 3 variables and 13 observations, we are going to predict 'y. According to 0.05 significant level, present stepwise regression and find the value of R square Variables SSR (regression sum of squares) 1500 SSE (error sum of squares) 1000 X X₂, X2 xz, xz X2, X₃ X2, X2, X3 SST (total sum of squares) 2500 2500 2500 2500 2500 2500 2500 2500 1400 1300 1300 2250 2400 2435 2450 1100 1200 250 100 65 50
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
Question 4 1 pts Suppose that we are given the following model, where x1 and 2...
Question 4 1 pts Suppose that we are given the following model, where x1 and 2 are quantitative: E(y) = Bo + B121 + B2X2 + B3 21 22 What is the slope for x1 when x2 - 0? OBB ОВ,
4. Testing for significance Aa Aa Consider a multiple regression model of the dependent variable y on independent variables x1, x2, X3, and x4: Using data with n = 60 observations for each of the var...
4. Testing for significance Aa Aa Consider a multiple regression model of the dependent variable y on independent variables x1, x2, X3, and x4: Using data with n = 60 observations for each of the variables, a student obtains the following estimated regression equation for the model given: 0.04 + 0.28X1 + 0.84X2-0.06x3 + 0.14x4 y She would like to conduct significance tests for a multiple regression relationship. She uses the F test to determine whether a significant relationship exists...
2.4 We have defined the simple linear regression model to be y =B1 + B2x+e. Suppose...
2.4 We have defined the simple linear regression model to be y =B1 + B2x+e. Suppose however that we knew, for a fact, that ßı = 0. (a) What does the linear regression model look like, algebraically, if ßı = 0? (b) What does the linear regression model look like, graphically, if ßı = 0? (c) If Bi=0 the least squares "sum of squares" function becomes S(R2) = Gyi - B2x;)?. Using the data, x 1 2 3 4 5...
3. Suppose we have a data set with five predictors, X1 = GPA, X2 = IQ,...
3. Suppose we have a data set with five predictors, X1 = GPA, X2 = IQ, X3 = Gender (1 for Female and 0 for Male), X4 = Interaction between GPA and IQ, and X5 = Interaction between GPA and Gender. The response is starting salary after graduation in thousands of dollars). Suppose we use least squares to fit the model, and get Bo = 50, 8 = 20, B2 = 0.07, B3 = 35, B4 = 0.01, B5 =...
Suppose of the first drug is need to compare the effects of two drugs each administered...
Suppose of the first drug is need to compare the effects of two drugs each administered to n subjects. The model for the effect 0.1 we Y=Bo+81X\i + €li, (1) while for the second drug it is Y2=Bo+B2X2; + €2i, (2) 1,...,n and X = X2 = 0. Assume that all observations are and in each case i independent and that for each i both E1 and e are normally distributed with mean 0 and variance a2 1. Obtain the...
Suppose we are given data on n observations (zi, Y4), i-1, . . . , n,...
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...
Suppose we have the following values for a dependent variable, Y, and three independent variables, X1,...
Suppose we have the following values for a dependent variable, Y, and three independent variables, X1, X2, and X3. The variable X3 is a dummy variable where 1 = male and 2 = female: X1 X2 X3 Y 0 40 1 30 0 50 0 10 2 20 0 40 2 50 1 50 4 90 0 60 4 60 0 70 4 70 1 80 4 40 1 90 6 40 0 70 6 50 1 90 8 80 ...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider a problem of regressing y on two (qualitative) variables r, 2. Data: 22 obs no. y (Income) 2 (Management Status) I (Gender) 1 None Female 2 None Male Yes Female Yes Male 4 To handle the qualitative variables r, 12, we define dummy variables 1, 22 as for 1, 22= Yes Male for 1, 219 22 -1. for 22= None for 1= Female -1,...
Using 3 variables and 13 observations, we are going to predict 'y. According to 0.05 significant level, present stepwise regression and find the value of R square Variables SSR (regression sum of squares) 1500 SSE (error sum of squares) 1000 X X₂, X2 xz, xz X2, X₃ X2, X2, X3 SST (total sum of squares) 2500 2500 2500 2500 2500 2500 2500 2500 1400 1300 1300 2250 2400 2435 2450 1100 1200 250 100 65 50
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
Question 4 1 pts Suppose that we are given the following model, where x1 and 2 are quantitative: E(y) = Bo + B121 + B2X2 + B3 21 22 What is the slope for x1 when x2 - 0? OBB ОВ,
4. Testing for significance Aa Aa Consider a multiple regression model of the dependent variable y on independent variables x1, x2, X3, and x4: Using data with n = 60 observations for each of the variables, a student obtains the following estimated regression equation for the model given: 0.04 + 0.28X1 + 0.84X2-0.06x3 + 0.14x4 y She would like to conduct significance tests for a multiple regression relationship. She uses the F test to determine whether a significant relationship exists...
2.4 We have defined the simple linear regression model to be y =B1 + B2x+e. Suppose however that we knew, for a fact, that ßı = 0. (a) What does the linear regression model look like, algebraically, if ßı = 0? (b) What does the linear regression model look like, graphically, if ßı = 0? (c) If Bi=0 the least squares "sum of squares" function becomes S(R2) = Gyi - B2x;)?. Using the data, x 1 2 3 4 5...
3. Suppose we have a data set with five predictors, X1 = GPA, X2 = IQ, X3 = Gender (1 for Female and 0 for Male), X4 = Interaction between GPA and IQ, and X5 = Interaction between GPA and Gender. The response is starting salary after graduation in thousands of dollars). Suppose we use least squares to fit the model, and get Bo = 50, 8 = 20, B2 = 0.07, B3 = 35, B4 = 0.01, B5 =...
Suppose of the first drug is need to compare the effects of two drugs each administered to n subjects. The model for the effect 0.1 we Y=Bo+81X\i + €li, (1) while for the second drug it is Y2=Bo+B2X2; + €2i, (2) 1,...,n and X = X2 = 0. Assume that all observations are and in each case i independent and that for each i both E1 and e are normally distributed with mean 0 and variance a2 1. Obtain the...
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...
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