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Problem 3 Consider the non-constant variance linear model Y, =Be + B111,1 + B,1,2 + ......
Q. 1 Consider the multiple linear regression model Y = x3 + €, where e indep...
Q. 1 Consider the multiple linear regression model Y = x3 + €, where e indep MV N(0,0²V) and V +In is a diagonal matrix. a) Derive the weighted least squares estimator for B, i.e., Owls. b) Show Bwis is an unbiased estimator for B. c) Derive the variances of w ls and the OLS estimator of 8. Is the OLS estimator of still the BLUE? In one sentence, explain why or why not.
4. +pt) When dealing with the problem of non-constant variance, the log transformation means using a....
4. +pt) When dealing with the problem of non-constant variance, the log transformation means using a. 1/X as the independent variable instead of X b. LogX as the independent variable instead of X c. Logy as the dependent variable instead of Y d. 1/Y as the dependent variable instead of Y 5. (4pt) A variable such as Z, whose value is Z = X1X2 is added to a general linear model in order to account for potential effects of two...
Problem 7. Consider the simple linear regression model Y1 = Bo + BiX; +€; for i=1,2,...,n...
Problem 7. Consider the simple linear regression model Y1 = Bo + BiX; +€; for i=1,2,...,n where the errors Eį are uncorrelated, have mean zero and common variance Varſei] = 02. Suppose that the Xį are in centimeters and we want to write the model in inches. If one centimeter = c inch with c known, we can write the above model as Yį = y +71 Zitki where Zi is Xi converted to inches. Can you obtain the least-squared...
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose...
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose that we would like to estimate the mean response at x = x*, that is we want to estimate lyx=* = Bo + B1 x*. The least squares estimator for /uyx* is = bo bi x*, where bo, b1 are the least squares estimators for Bo, Bi. ayx= (a) Show that the least squares estimator for...
6. This problem considers the simple linear regression model, that is, a model with a single...
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
P9:Consider this non-linear electrical circuit with R-64 4 and constant V (a) Derive the mathemat...
P9:Consider this non-linear electrical circuit with R-64 4 and constant V (a) Derive the mathematical model for charge q() in b) Find 1(on)
P9:Consider this non-linear electrical circuit with R-64 4 and constant V (a) Derive the mathematical model for charge q() in b) Find 1(on)
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear...
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear model y= +E, where u = (1, ..., and e is a vector of zero mean, uncorrelated errors with variance o'. This is a linear model in which the responses have a constant but unknown mean . We will call this model the location model. (a) If we write the location model in the usual form of the linear model y = X 8+...
Exercise 5 Consider a linear model with n = 2m in which Yi = Bo +...
Exercise 5 Consider a linear model with n = 2m in which Yi = Bo + Bici + Eigi = 1,..., m, and Yi = Bo + B2X1 + Ei, i = m + 1, ...,n. Here €1,..., En are i.i.d. from N(0,0), B = (Bo, B1, B2)' and o2 are unknown parameters, X1, ..., Xn are known constants with X1 + ... + Xm = Xm+1 + ... + Xn = 0. 1. Write the model in vector form...
Problem 3: Assume that 'nature' behaves according to the following linear additive model: Y = Bo...
Problem 3: Assume that 'nature' behaves according to the following linear additive model: Y = Bo + B1X +€, where ε is a Gaussian random variable N (0,02). Using this model, nature generates the following training dataset: D = {(Li, yi)}}–1 = {(–2, 47/2),(-1, -3), (0,0), (1,3), (2,7/2)}. Please, answer the questions below without the help of any computer software: a. Compute the estimates of Bo and @1 for a linear estimator û = Bo + 1X using the data...
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...
Q. 1 Consider the multiple linear regression model Y = x3 + €, where e indep MV N(0,0²V) and V +In is a diagonal matrix. a) Derive the weighted least squares estimator for B, i.e., Owls. b) Show Bwis is an unbiased estimator for B. c) Derive the variances of w ls and the OLS estimator of 8. Is the OLS estimator of still the BLUE? In one sentence, explain why or why not.
4. +pt) When dealing with the problem of non-constant variance, the log transformation means using a. 1/X as the independent variable instead of X b. LogX as the independent variable instead of X c. Logy as the dependent variable instead of Y d. 1/Y as the dependent variable instead of Y 5. (4pt) A variable such as Z, whose value is Z = X1X2 is added to a general linear model in order to account for potential effects of two...
Problem 7. Consider the simple linear regression model Y1 = Bo + BiX; +€; for i=1,2,...,n where the errors Eį are uncorrelated, have mean zero and common variance Varſei] = 02. Suppose that the Xį are in centimeters and we want to write the model in inches. If one centimeter = c inch with c known, we can write the above model as Yį = y +71 Zitki where Zi is Xi converted to inches. Can you obtain the least-squared...
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose that we would like to estimate the mean response at x = x*, that is we want to estimate lyx=* = Bo + B1 x*. The least squares estimator for /uyx* is = bo bi x*, where bo, b1 are the least squares estimators for Bo, Bi. ayx= (a) Show that the least squares estimator for...
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
P9:Consider this non-linear electrical circuit with R-64 4 and constant V (a) Derive the mathematical model for charge q() in b) Find 1(on)
P9:Consider this non-linear electrical circuit with R-64 4 and constant V (a) Derive the mathematical model for charge q() in b) Find 1(on)
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear model y= +E, where u = (1, ..., and e is a vector of zero mean, uncorrelated errors with variance o'. This is a linear model in which the responses have a constant but unknown mean . We will call this model the location model. (a) If we write the location model in the usual form of the linear model y = X 8+...
Exercise 5 Consider a linear model with n = 2m in which Yi = Bo + Bici + Eigi = 1,..., m, and Yi = Bo + B2X1 + Ei, i = m + 1, ...,n. Here €1,..., En are i.i.d. from N(0,0), B = (Bo, B1, B2)' and o2 are unknown parameters, X1, ..., Xn are known constants with X1 + ... + Xm = Xm+1 + ... + Xn = 0. 1. Write the model in vector form...
Problem 3: Assume that 'nature' behaves according to the following linear additive model: Y = Bo + B1X +€, where ε is a Gaussian random variable N (0,02). Using this model, nature generates the following training dataset: D = {(Li, yi)}}–1 = {(–2, 47/2),(-1, -3), (0,0), (1,3), (2,7/2)}. Please, answer the questions below without the help of any computer software: a. Compute the estimates of Bo and @1 for a linear estimator û = Bo + 1X using the data...
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...
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