Homework Answers
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...
7. In a simple regression model, suppose all of the assumptions of the classical linear regression...
7. In a simple regression model, suppose all of the assumptions of the classical linear regression morel apply, except that rather than assume E (ui | Xi) = 0, you assume that E (Ui / X;) = ali and E (xi) = 0 where a > 0 is a constant. (a) What is the conditional expectation of the OLS slope coefficient, i.e. E (B1 | 21, ..., XN)? (b) In this case, is ß1 an unbiased estimator of B1 or...
1. If a true model of simple linear regression reads: yi −y ̄ = β0 +β1(xi...
1. If a true model of simple linear regression reads: yi −y ̄ = β0 +β1(xi −x ̄)+εi for i = 1, 2, · · · , n, showβ0 =0andβˆ0 =0. (1pt) (hint: use the formula of estimator βˆ0 = y ̄ − βˆ1x ̄.)
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....
In a simple linear regression, the following information is given: x−x− = −39; y− y− =...
In a simple linear regression, the following information is given: x−x− = −39; y− y− = 40; Σ(xi−x− )(yi− y−)= −840;Σ(xi−x− )(yi− y−)= −840; Σ(xi− x−)2= 718Σ(xi− x−)2= 718 a. Calculate b1. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.) b1 b. Calculate b0. (Round intermediate calculations to 4 decimal places and final answer to 2 decimal places.) b0 c-1. What is the sample regression equation? (Negative value should be...
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...
Log-lin model in simple regression: yi=B0.B1Xi.eui take natural log of both sides lnyi=lnB0 + (lnB1)xi +...
Log-lin model in simple regression:
yi=B0.B1Xi.eui
take natural log of both sides
lnyi=lnB0 + (lnB1)xi +
ui . Take lnyi = wi,
lnB0= B*0 and
lnB1=B*1 so the former equation
equals to wi= B*0 +
B*1 + ui. It is said that, when
you increase x by 1 unit, you expect B*1 %
change in y.
B*1
=(dyi/yi) / dxi =
= (((yi-yi-1)/yi) /
(xi- xi-1))*100 . question is about the
numerator, (yi-yi-1)/yi)*100:
1)It is said that the numerator reflects...
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?
In a simple linear regression model, the intercept of the regression line measures
QUESTION 1In a simple linear regression model, the intercept of the regression line measuresa.the change in Y per unit change in X.b.the change in X per unit change in Y.c.the expected change in Y per unit change in X.d.the expected change in X per unit change in Y.e.the value of Y when X equals 0.f.the value of X when Y equals 0.g.the average value of Y when X equals 0.h.the average value of X when Y equals 0.QUESTION 2In a...
4. In the simple linear regression model yi = Bo + B, 21 +, a. Bcannot...
4. In the simple linear regression model yi = Bo + B, 21 +, a. Bcannot be estimated without first assuming (EU) = 0 b. B, could represent the average marginal association between 2 and y or the average effect of x on y c. we can directly observe e d. the B, estimate is unbiased only if E(€) = 0 e. None of the above
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...
7. In a simple regression model, suppose all of the assumptions of the classical linear regression morel apply, except that rather than assume E (ui | Xi) = 0, you assume that E (Ui / X;) = ali and E (xi) = 0 where a > 0 is a constant. (a) What is the conditional expectation of the OLS slope coefficient, i.e. E (B1 | 21, ..., XN)? (b) In this case, is ß1 an unbiased estimator of B1 or...
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....
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...
Log-lin model in simple regression:
yi=B0.B1Xi.eui
take natural log of both sides
lnyi=lnB0 + (lnB1)xi +
ui . Take lnyi = wi,
lnB0= B*0 and
lnB1=B*1 so the former equation
equals to wi= B*0 +
B*1 + ui. It is said that, when
you increase x by 1 unit, you expect B*1 %
change in y.
B*1
=(dyi/yi) / dxi =
= (((yi-yi-1)/yi) /
(xi- xi-1))*100 . question is about the
numerator, (yi-yi-1)/yi)*100:
1)It is said that the numerator reflects...
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?
4. In the simple linear regression model yi = Bo + B, 21 +, a. Bcannot be estimated without first assuming (EU) = 0 b. B, could represent the average marginal association between 2 and y or the average effect of x on y c. we can directly observe e d. the B, estimate is unbiased only if E(€) = 0 e. None of the above
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