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1- The regression model we would like to study is: Y; = BX; + εi and...
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b)...
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vector, and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and in matrix form. aB (d) Solve : 0 for β, the MLE...
Consider the regression model where the εi are i.i.d. N(0,σ2) random variables, for i = 1,...
Consider the regression model
where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . .
. , n.
(a) (4 points) Show βˆ is normally distributed with mean β and
variance σ2 . 1 1SXX
Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
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,......
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...
I. In a regression analysis of banks, four types of banks were involved, namely, commercial banks,...
I. In a regression analysis of banks, four types of banks were involved, namely, commercial banks, mutual banks, community banks and savings banks. In this study, Y is the previous year's profit (in millions of dollars), X, is the size of the bank (in millions of dollars) and type of bank = X, X, X, which are indicator variables as coded below. Type of Bank X X X , Commercial 1 0 Mutual Community Savings 0 0 0 The first-order...
Consider the regression model: yi = β0 + β1Xi + εi for…. i = 1 Where...
Consider the regression model: yi = β0 + β1Xi + εi for…. i = 1 Where the dummy variable (0 = failure and 1 = success). Suppose that the data set contains n1 failure and n2 successes (and that n1+n2 = n) Obtain the X^T(X) matrix Obtain the X^T(Y) matrix Obtain the least square estimate b
D. We will use pinv function to find a linear regression model to map the measurement data to tru...
d. We will use pinv function to find a linear regression model to map the measurement data to truth data. Our model will be Y-AX. Y is the truth data matrix Y [ yı; y2i yai ya; ysi yg: y7 ]. A is the tmeasurement data matrix A=[ 1 mh; 1 m2. 1 mai 1 m4; 1 ms: 1 mt 6:1 m7 I . and X-[Xo: x1] is the coefficient of your regression model. Truth Data yn -1 0 Measurement...
Question 2: Suppose that we wish to fit a regression model for which the true regression...
Question 2: Suppose that we wish to fit a regression model for which the true regression line passes through the origin (0,0). The appropriate model is Y = Bx + €. Assume that we have n pairs of data (x1.yı) ... (Xn,yn). a) From first principle, derive the least square estimate of B. (write the loss function then take first derivative W.r.t coefficient etc) b) Assume that e is normally distributed what is the distribution of Y? Explain your answer...
mail/u/3/inbox?projector=1 For a multiple regression model Y = B. B.X.+ B.X.-B.X, BX, BX,+ € where is...
mail/u/3/inbox?projector=1 For a multiple regression model Y = B. B.X.+ B.X.-B.X, BX, BX,+ € where is the error term, to represent the relationship between Y and the four X-variables. We got the following results from the data: Source Sum of Squares degrees of freedom mean squares 110.92 Regression Residual Total 215.94 And also given: Variable B. values S(B) Degrees of freedom 0.02 0.056 -0.13 0.021 0.207 -0.05 0.21 0.067 0.001 0.067 Y-intercept is B. = 2.96 d. Find the regression...
2. Consider a simple linear regression i ion model for a response variable Y, a single...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
QUESTION 1 We consider the regression model Y= Bo+B1X u And we found for a sample...
QUESTION 1 We consider the regression model Y= Bo+B1X u And we found for a sample size of n 974 B1 -0.095 and S 0.02 Does X has a significant effect on Y at the 5 % level? True False
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vector, and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and in matrix form. aB (d) Solve : 0 for β, the MLE...
Consider the regression model
where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . .
. , n.
(a) (4 points) Show βˆ is normally distributed with mean β and
variance σ2 . 1 1SXX
Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
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...
I. In a regression analysis of banks, four types of banks were involved, namely, commercial banks, mutual banks, community banks and savings banks. In this study, Y is the previous year's profit (in millions of dollars), X, is the size of the bank (in millions of dollars) and type of bank = X, X, X, which are indicator variables as coded below. Type of Bank X X X , Commercial 1 0 Mutual Community Savings 0 0 0 The first-order...
d. We will use pinv function to find a linear regression model to map the measurement data to truth data. Our model will be Y-AX. Y is the truth data matrix Y [ yı; y2i yai ya; ysi yg: y7 ]. A is the tmeasurement data matrix A=[ 1 mh; 1 m2. 1 mai 1 m4; 1 ms: 1 mt 6:1 m7 I . and X-[Xo: x1] is the coefficient of your regression model. Truth Data yn -1 0 Measurement...
Question 2: Suppose that we wish to fit a regression model for which the true regression line passes through the origin (0,0). The appropriate model is Y = Bx + €. Assume that we have n pairs of data (x1.yı) ... (Xn,yn). a) From first principle, derive the least square estimate of B. (write the loss function then take first derivative W.r.t coefficient etc) b) Assume that e is normally distributed what is the distribution of Y? Explain your answer...
mail/u/3/inbox?projector=1 For a multiple regression model Y = B. B.X.+ B.X.-B.X, BX, BX,+ € where is the error term, to represent the relationship between Y and the four X-variables. We got the following results from the data: Source Sum of Squares degrees of freedom mean squares 110.92 Regression Residual Total 215.94 And also given: Variable B. values S(B) Degrees of freedom 0.02 0.056 -0.13 0.021 0.207 -0.05 0.21 0.067 0.001 0.067 Y-intercept is B. = 2.96 d. Find the regression...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
QUESTION 1 We consider the regression model Y= Bo+B1X u And we found for a sample size of n 974 B1 -0.095 and S 0.02 Does X has a significant effect on Y at the 5 % level? True False
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