Homework Answers
A simple linear regression model is given as follows Yi = Bo + B1Xi+ €i, for...
A simple linear regression model is given as follows Yi = Bo + B1Xi+ €i, for i = 1, ...,n, where are i.i.d. following N (0, o2) distribution. It is known that x4 n, and x = 0, otherwise. Denote by n2 = n - ni, Ji = 1 yi, and j2 = 1 1. for i = 1, ... ,n1 < n2 Lizn1+1 Yi. n1 Zi=1 1. Find the least squares estimators of Bo and 31, in terms of...
Suppose Yį = Bo + Bidi + Ei and εi ~ N(0,0%). Also suppose you have...
Suppose Yį = Bo + Bidi + Ei and εi ~ N(0,0%). Also suppose you have n observations of (yi, x;). Consider xi just a set of constants. (a) What is the distribution of Y;? (b) If you wanted to estimate Bo, B1, and oʻ, what likelihood function would you maximize?
Consider the zero intercept model given by Yi = B1Xi + ei (i=1,…,n) with the ei...
Consider the zero intercept model given by Yi = B1Xi + ei (i=1,…,n) with the ei normal, independent, with variance sigma^2. For this mode (i) find the sum of (Yi –Yi-hat). (ii) find the sum of (Yi – Yi-hat)Xi. (iii) find the estimator of the error variance, sigma^2. (iv) is the estimator of the error variance biased?
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...
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...
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; =...
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
Given a simple regression analysis, suppose that we have obtained the fitted regression model: ^yi=6+8xi and also...
Given a simple regression analysis, suppose that we have obtained the fitted regression model: ^yi=6+8xi and also the following statistics: SE=3.20, x̄=8,n=42, and ∑ (xi-x̄)2 =420 1) Find the 95% confidence interval for the point where x =18.
Consider the following regression model: Xi = Bo + Bixi + y; where yi is individual...
Consider the following regression model: Xi = Bo + Bixi + y; where yi is individual i's University GPA and xi is the individual's high school grades. a. What do you think is in ui? Do you think E[ulx) = 0? Suggest a variable that you think might affect University GPA that isn't included in the regression equation but should be. c. What sign of bias would you expect in an OLS regression of y on x? Briefly explain. d....
Find the sample variance of the dependent variable yi. What fraction of the sample variance in...
Find the sample variance of the dependent variable
yi.
What fraction of the sample variance in
yi can be explained by the linear regression
model? Find the coefficient of determination,
r2.
Find the adjusted r2
value.
Do we have enough evidence to conclude that there exists
a strong linear relationship between xi and
yi? Find the p-value for the
test.
Problem. Chris Pratt is almost ready to pop the question to the girl of his dreams. He collected the data...
A simple linear regression model is given as follows Yi = Bo + B1Xi+ €i, for i = 1, ...,n, where are i.i.d. following N (0, o2) distribution. It is known that x4 n, and x = 0, otherwise. Denote by n2 = n - ni, Ji = 1 yi, and j2 = 1 1. for i = 1, ... ,n1 < n2 Lizn1+1 Yi. n1 Zi=1 1. Find the least squares estimators of Bo and 31, in terms of...
Suppose Yį = Bo + Bidi + Ei and εi ~ N(0,0%). Also suppose you have n observations of (yi, x;). Consider xi just a set of constants. (a) What is the distribution of Y;? (b) If you wanted to estimate Bo, B1, and oʻ, what likelihood function would you maximize?
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...
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...
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; =...
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
Consider the following regression model: Xi = Bo + Bixi + y; where yi is individual i's University GPA and xi is the individual's high school grades. a. What do you think is in ui? Do you think E[ulx) = 0? Suggest a variable that you think might affect University GPA that isn't included in the regression equation but should be. c. What sign of bias would you expect in an OLS regression of y on x? Briefly explain. d....
Find the sample variance of the dependent variable
yi.
What fraction of the sample variance in
yi can be explained by the linear regression
model? Find the coefficient of determination,
r2.
Find the adjusted r2
value.
Do we have enough evidence to conclude that there exists
a strong linear relationship between xi and
yi? Find the p-value for the
test.
Problem. Chris Pratt is almost ready to pop the question to the girl of his dreams. He collected the data...
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