Question

Problem 3. Suppose we have a set of observations (x1,y1),...,(xn,yn) coming from the simple linear model: yi = ?0 + ?1xi + ?i, 1 ? i ? n. (1) Least square method finds the minimizer of Q(?0,?1) = ? (yi - ?0 - ?1xi)". Define Sxx = ? (xi - x?)", Syy = ? (yi - y?)", Sxy = ? (xi - x?)(yi - y?), which are the sample variances of x and y, and the sample covariance between x and y, respectively. (a) Show that the least square estimates can be written as ?1 = Sxy/Sxx, and ?0 = y? - ?1x?. Note that the expression of ?1 appears different from what you see in class. Explain why the two expressions are in fact equivalent. (b) Show that yi - y? = ?1(xi - x?) + (yi - ?i). (c) Show that Syy = ? ?1"(xi - x?)" + ? (yi - ?i)". (d) The sample correlation between y and x is ? = Sxy/?SxxSyy. Show that the coefficient of determination R" is equal to the square of the sample correlation ?".

          Problem 3. Suppose we have a set of observations (x1,y1),...,(xn,yn) coming from the simple linear model:
yi = ?0 + ?1xi + ?i, 1 ? i ? n. (1)
Least square method finds the minimizer of
Q(?0,?1) = ? (yi - ?0 - ?1xi)".
Define
Sxx = ? (xi - x?)", Syy = ? (yi - y?)", Sxy = ? (xi - x?)(yi - y?),
which are the sample variances of x and y, and the sample covariance between x and y, respectively.
(a) Show that the least square estimates can be written as ?1 = Sxy/Sxx, and ?0 = y? - ?1x?. Note that the expression of ?1 appears different from what you see in class. Explain why the two expressions are in fact equivalent.
(b) Show that yi - y? = ?1(xi - x?) + (yi - ?i).
(c) Show that Syy = ? ?1"(xi - x?)" + ? (yi - ?i)".
(d) The sample correlation between y and x is ? = Sxy/?SxxSyy. Show that the coefficient of determination R" is equal to the square of the sample correlation ?".

Problem 3. Suppose we have a set of observations (x1,y1),...,(xn,yn) coming from the simple linear model:
yi = ?0 + ?1xi + ?i, 1 ? i ? n. (1)
Least square method finds the minimizer of
Q(?0,?1) = ? (yi - ?0 - ?1xi)".
Define
Sxx = ? (xi - x?)", Syy = ? (yi - y?)", Sxy = ? (xi - x?)(yi - y?),
which are the sample variances of x and y, and the sample covariance between x and y, respectively.
(a) Show that the least square estimates can be written as ?1 = Sxy/Sxx, and ?0 = y? - ?1x?. Note that the expression of ?1 appears different from what you see in class. Explain why the two expressions are in fact equivalent.
(b) Show that yi - y? = ?1(xi - x?) + (yi - ?i).
(c) Show that Syy = ? ?1"(xi - x?)" + ? (yi - ?i)".
(d) The sample correlation between y and x is ? = Sxy/?SxxSyy. Show that the coefficient of determination R" is equal to the square of the sample correlation ?".

Added by Julia R.

Question

Please give Ace some feedback