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3. Consider a forecasting model with a trend where t is the t index for t...
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...
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 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.
1. A series of observations is assumed to follow the trend model: where t (n +...
1. A series of observations is assumed to follow the trend model: where t (n + 1)/2 -5 and et is white noise. Write down the least squares estimates c, b of c,b in terms of y, y2.n The next term in the series, yio, is predicted as Express this in the form W, w2y2+ and obtain the numerical values of w2, . 10 marks
4. (20 pts) Consider the following regression model, i = 1,2. ,...n, N (0, 2/i) where...
4. (20 pts) Consider the following regression model, i = 1,2. ,...n, N (0, 2/i) where , 1,2,...,n are independent, but c; ~ (a) Do you think if it is suitable for the (ordinary) least square regression technique to apply the data (x4, Y;)? Give a brief reasoning (b) Construct a transformed model so that you can use the ordinary least square method (c) Find the parameter estimates for the transformed model in (b) WIS (d) Find the weighted least...
Problem 3 Consider the non-constant variance linear model Y, =Be + B111,1 + B,1,2 + ......
Problem 3 Consider the non-constant variance linear model Y, =Be + B111,1 + B,1,2 + ... + Bp-111.-1+€, (1) with G N(0,0?), i=1,..., 7. Define the reciprocal of the variance of as the weight w, and let w 0 0 W OW... 0... 02. We can estimate the non-constant variance model by minimizing the objective function w (9-Bo-Buca -.- Bp-111p-1) Task: Derive the weighted least squares equation (3) Bu = (XWX)-{XWY
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...
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...
2. (a) Let us consider a full model of a balanced (all t treatments have equal...
2. (a) Let us consider a full model of a balanced (all t treatments have equal number of observations r) CRD design with t treatments and r replications of each treatment, hence having n rt observations. . Minimizing sum of square error Δ/u114%)-ΣΊ ΣΊ (Vij-μ-%)2 with respect to μ and Ti find the least square estimators of μ and Ti as μ and T. Hint: Take derivative of the objective function with respect to μ and Ti and equate then...
Exercise 4.11 Consider the regression model Y Po PX+u Suppose that you know Bo 1. Derive...
Exercise 4.11 Consider the regression model Y Po PX+u Suppose that you know Bo 1. Derive the formula for the least squares estimator of p The least squares objective function is OA. n (v2-bo-bx?) i-1 Ов. O B. n (M-bo-bX) /# 1 n Click to select your answer and then click Check Answer. Exercise 4.11 OA n Σ (--B,χ?) O B. E (Y-bo-b,X)2 j= 1 n Σ (Υ-Βo-bΧ) 3. j= 1 D. n Σ (Υ-0-b,) i- 1 Click to select...
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...
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 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.
1. A series of observations is assumed to follow the trend model: where t (n + 1)/2 -5 and et is white noise. Write down the least squares estimates c, b of c,b in terms of y, y2.n The next term in the series, yio, is predicted as Express this in the form W, w2y2+ and obtain the numerical values of w2, . 10 marks
4. (20 pts) Consider the following regression model, i = 1,2. ,...n, N (0, 2/i) where , 1,2,...,n are independent, but c; ~ (a) Do you think if it is suitable for the (ordinary) least square regression technique to apply the data (x4, Y;)? Give a brief reasoning (b) Construct a transformed model so that you can use the ordinary least square method (c) Find the parameter estimates for the transformed model in (b) WIS (d) Find the weighted least...
Problem 3 Consider the non-constant variance linear model Y, =Be + B111,1 + B,1,2 + ... + Bp-111.-1+€, (1) with G N(0,0?), i=1,..., 7. Define the reciprocal of the variance of as the weight w, and let w 0 0 W OW... 0... 02. We can estimate the non-constant variance model by minimizing the objective function w (9-Bo-Buca -.- Bp-111p-1) Task: Derive the weighted least squares equation (3) Bu = (XWX)-{XWY
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...
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...
2. (a) Let us consider a full model of a balanced (all t treatments have equal number of observations r) CRD design with t treatments and r replications of each treatment, hence having n rt observations. . Minimizing sum of square error Δ/u114%)-ΣΊ ΣΊ (Vij-μ-%)2 with respect to μ and Ti find the least square estimators of μ and Ti as μ and T. Hint: Take derivative of the objective function with respect to μ and Ti and equate then...
Exercise 4.11 Consider the regression model Y Po PX+u Suppose that you know Bo 1. Derive the formula for the least squares estimator of p The least squares objective function is OA. n (v2-bo-bx?) i-1 Ов. O B. n (M-bo-bX) /# 1 n Click to select your answer and then click Check Answer. Exercise 4.11 OA n Σ (--B,χ?) O B. E (Y-bo-b,X)2 j= 1 n Σ (Υ-Βo-bΧ) 3. j= 1 D. n Σ (Υ-0-b,) i- 1 Click to select...
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