Homework Answers
Problem 1: Consider the model Y = BO + Bi X+e, where e is a N(0,02)...
Problem 1: Consider the model Y = BO + Bi X+e, where e is a N(0,02) random variable independent of X. Let also Y = Bo + B1X. Show that E[(Y - EY)^3 = E[(Ỹ – EY)^3 + E[(Y – Y)1.
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...
1. Suppose that Yi = Bo + B1Xi + €¡ where ; is N(0,0.6), Bo =...
1. Suppose that Yi = Bo + B1Xi + €¡ where ; is N(0,0.6), Bo = 2 and 31 = 1. (a) What are the conditional mean and standard deviation of Yị given that Xi = 1? What is P(Yi < 3|X; = 1)? (b) A regression model is a model for the conditional distribution of Yị given Xị. However, if we also have a model for the marginal distribution of X; then we can find the marginal distribution of...
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...
3. Consider simple linear regression model yi = Bo + B12; + &; and B. parameter...
3. Consider simple linear regression model yi = Bo + B12; + &; and B. parameter estimate of the slope coefficient Bi: Find the expectation and variance of 31. Is parameter estimate B1 a) unbiased? b) linear on y? c) effective optimal in terms of variance)? What will be your answers if you know that there is no intercept coefficient in your model?
Consider the regression equation Y = Bo+B1Xi+u; where E[u;|Xi]=0 for all i = 1, ..., n....
Consider the regression equation Y = Bo+B1Xi+u; where E[u;|Xi]=0 for all i = 1, ..., n. Let B 1 be the OLS estimator for B 1. Which statement is the most irrelevant to the consistency of B1? Hint: see Lecture Note 2 (p.25-p.28) a. When n is large, the estimator B 1 is near the population parameter B1 O". Consistency of B1 is mathematically written as B1-B1 VB) is inversely proportional to the sample size n. Od. RMSE is close...
Yi = Bo + BiXi + Ui, where Xi = 1 if individual i was treated...
Yi = Bo + BiXi + Ui, where Xi = 1 if individual i was treated and 0 otherwise, Bo = E[Y/C), B1 = T, and Ui = YC - E[Y;-). (a) Calculate E[ui]. (b) Calculate E[u;X]. Hint: use law of iterated expectations. (c) Using your answers to (a) and (b), calculate Cov(Xị, Ui) (d) Using your answer to (c), show that endogeneity bias is equal to selection bias in this setting, i.e., Cov(Xi, ui iz 41) = E[Y;|X; =...
Let Y be a linear function of X, i.e. Y = bo + bịX where bo...
Let Y be a linear function of X, i.e. Y = bo + bịX where bo and bl are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the values of bo and bi that minimizes Q (b) Use(a) to show that the minimal value of Q is Co - Hint: You may use the fact that Q(bg, b) [(Y-Y*)2] = Var (Y-Y*) +E(Y-Y*)]2 where Y*-bg + bİX and bi,...
(2) Let Y be a linear function of X, i.e. Y- bo biX where bo and...
(2) Let Y be a linear function of X, i.e. Y- bo biX where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity we=E[rMb, that minimize«Q (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ-ar 2 Cov2(x,Y) m Hint: You may use the fact that Q(bo,Y-YVar (Y -Y)+E (Y -Y)where Y.-bg + bİX...
1. Fisher information i(): Let X ~ N(μ, 1), where μ is unknown. Calculate l(u) Let...
1. Fisher information i(): Let X ~ N(μ, 1), where μ is unknown. Calculate l(u) Let X ~ N(0,02), where σ2 is unknown. Calculate 1(σ2). Let X ~ Exp(λ), where λ is unknown. Calculate la).
Problem 1: Consider the model Y = BO + Bi X+e, where e is a N(0,02) random variable independent of X. Let also Y = Bo + B1X. Show that E[(Y - EY)^3 = E[(Ỹ – EY)^3 + E[(Y – Y)1.
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...
1. Suppose that Yi = Bo + B1Xi + €¡ where ; is N(0,0.6), Bo = 2 and 31 = 1. (a) What are the conditional mean and standard deviation of Yị given that Xi = 1? What is P(Yi < 3|X; = 1)? (b) A regression model is a model for the conditional distribution of Yị given Xị. However, if we also have a model for the marginal distribution of X; then we can find the marginal distribution of...
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...
3. Consider simple linear regression model yi = Bo + B12; + &; and B. parameter estimate of the slope coefficient Bi: Find the expectation and variance of 31. Is parameter estimate B1 a) unbiased? b) linear on y? c) effective optimal in terms of variance)? What will be your answers if you know that there is no intercept coefficient in your model?
Consider the regression equation Y = Bo+B1Xi+u; where E[u;|Xi]=0 for all i = 1, ..., n. Let B 1 be the OLS estimator for B 1. Which statement is the most irrelevant to the consistency of B1? Hint: see Lecture Note 2 (p.25-p.28) a. When n is large, the estimator B 1 is near the population parameter B1 O". Consistency of B1 is mathematically written as B1-B1 VB) is inversely proportional to the sample size n. Od. RMSE is close...
Yi = Bo + BiXi + Ui, where Xi = 1 if individual i was treated and 0 otherwise, Bo = E[Y/C), B1 = T, and Ui = YC - E[Y;-). (a) Calculate E[ui]. (b) Calculate E[u;X]. Hint: use law of iterated expectations. (c) Using your answers to (a) and (b), calculate Cov(Xị, Ui) (d) Using your answer to (c), show that endogeneity bias is equal to selection bias in this setting, i.e., Cov(Xi, ui iz 41) = E[Y;|X; =...
Let Y be a linear function of X, i.e. Y = bo + bịX where bo and bl are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the values of bo and bi that minimizes Q (b) Use(a) to show that the minimal value of Q is Co - Hint: You may use the fact that Q(bg, b) [(Y-Y*)2] = Var (Y-Y*) +E(Y-Y*)]2 where Y*-bg + bİX and bi,...
(2) Let Y be a linear function of X, i.e. Y- bo biX where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity we=E[rMb, that minimize«Q (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ-ar 2 Cov2(x,Y) m Hint: You may use the fact that Q(bo,Y-YVar (Y -Y)+E (Y -Y)where Y.-bg + bİX...
1. Fisher information i(): Let X ~ N(μ, 1), where μ is unknown. Calculate l(u) Let X ~ N(0,02), where σ2 is unknown. Calculate 1(σ2). Let X ~ Exp(λ), where λ is unknown. Calculate la).
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