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Consider the least-squares residuals ei-yi-yi, 1, 2, . . . , linear regression model. Find the...
012. (a) The ordinary least squares estimate of B in the classical linear regression model Yi...
012. (a) The ordinary least squares estimate of B in the classical linear regression model Yi = α + AXi + Ui ; i=1,2, , n and xi = Xi-K, X-n2Xī i- 1 Show that if Var(B-.--u , no other linear unbiased estimator of β n im1 can be constructed with a smaller variance. (All symbols have their usual meaning) 18
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the...
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the mean of the yi, and let â and ß be the MLES of a and B, respectively. Let yi = â-+ Bxi be the fitted values, and let e; = yi -yi be the residuals a) What is Cov(j, B) b) What is Cov(â, ß) c) Show that 1 ei = 0 d) Show that _1 x;e; = 0 e) Show that 1iei =...
1. Consider the simple linear regression model where Bo is known. a) Find the least squares...
1. Consider the simple linear regression model where Bo is known. a) Find the least squares estimator bi of B (b) Is this estimator unbiased? Prove your result. (c) Find an expression for Var(b1x1, ,xn) in terms of x1, ,xn and σ2.
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 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...
In the multiple linear regression model with estimation by ordinary least squares, why must we make...
In the multiple linear regression model with estimation by ordinary least squares, why must we make an analysis of the scatter plot indices 1, 2,. . . , n and with the residuals ei for observations that are somehow ordered (for example, in time)? And what is the purpose of analyzing the sample autocorrelation function?
Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the...
Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the least squares estimates ỘT = (Bo ß1). We observe a new value of the predictor: x] = (1 xo). Show that the expression for the 100(1 - a)% prediction interval reduces to the following: . (xo – x2 Ēo + @130 Etap 11+ntan (x; – 7)2
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....
Consider the simple linear regression model where Bo is known. (a) Find the least squares estimator...
Consider the simple linear regression model where Bo is known. (a) Find the least squares estimator bi of β1- (b) Is this estimator unbiased? Prove your result
1. Consider a regression model Yi = x;ß +ei, i = 1,...,n. You estimate this model...
1. Consider a regression model Yi = x;ß +ei, i = 1,...,n. You estimate this model using the OLS estimator. (a) Present and discuss assumptions for the OLS estimation.
1. For the general multivariate regression model, the least squares estimator is given by Show that...
1. For the general multivariate regression model, the least squares estimator is given by Show that for the slope estimator in the simple (bivariate) regression case, this is equivalent to ja! įs] 2. In the general multivariate regression model, the variance of the least squares estimator, Va( is σ2(XX)". Show that for the simple regression case, this is equivalent to a. Var(B- b. Var(B)o i, Σ (Xi-X) 2 C. What is the covariance between β° and β,?
012. (a) The ordinary least squares estimate of B in the classical linear regression model Yi = α + AXi + Ui ; i=1,2, , n and xi = Xi-K, X-n2Xī i- 1 Show that if Var(B-.--u , no other linear unbiased estimator of β n im1 can be constructed with a smaller variance. (All symbols have their usual meaning) 18
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the mean of the yi, and let â and ß be the MLES of a and B, respectively. Let yi = â-+ Bxi be the fitted values, and let e; = yi -yi be the residuals a) What is Cov(j, B) b) What is Cov(â, ß) c) Show that 1 ei = 0 d) Show that _1 x;e; = 0 e) Show that 1iei =...
1. Consider the simple linear regression model where Bo is known. a) Find the least squares estimator bi of B (b) Is this estimator unbiased? Prove your result. (c) Find an expression for Var(b1x1, ,xn) in terms of x1, ,xn and σ2.
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 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...
Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the least squares estimates ỘT = (Bo ß1). We observe a new value of the predictor: x] = (1 xo). Show that the expression for the 100(1 - a)% prediction interval reduces to the following: . (xo – x2 Ēo + @130 Etap 11+ntan (x; – 7)2
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....
Consider the simple linear regression model where Bo is known. (a) Find the least squares estimator bi of β1- (b) Is this estimator unbiased? Prove your result
1. Consider a regression model Yi = x;ß +ei, i = 1,...,n. You estimate this model using the OLS estimator. (a) Present and discuss assumptions for the OLS estimation.
1. For the general multivariate regression model, the least squares estimator is given by Show that for the slope estimator in the simple (bivariate) regression case, this is equivalent to ja! įs] 2. In the general multivariate regression model, the variance of the least squares estimator, Va( is σ2(XX)". Show that for the simple regression case, this is equivalent to a. Var(B- b. Var(B)o i, Σ (Xi-X) 2 C. What is the covariance between β° and β,?
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