Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables: One variable, denoted X, is regarded as the predictor, explanatory, or independent variable. The other variable, denoted Y, is regarded as the response, outcome, or dependent variable. Suppose that we are given n-i.i.d observations { (x;, y;)}"_, from the assumed simple linear regression model Y = BIX + Bo + . Answer the following questions on simple linear regression. 5-a. Denote 1 and Bo as the point estimators of B, and Bo, respectively, that are obtained through the least squares method. Show, step by step, that the two point estimators are unbiased. Derive the least squares estimator of of and determine whether it is unbiased or not. Show your work step by step. 5-b. Calculate _'_1(yi - Bix; - Bo) (Bix, + Bo). Determine whether the point (X, Y) is on the line Y = 1X + Bo. Explain your reasoning mathematically. 5-c. Using the maximum likelihood estimation (MLE) technique, derive a point estimator for the coefficient B1 and the intercept Bo, respectively. Determine whether the point estimators that you obtained via MLE are unbiased or not. Justify your conclusion mathematically. 5-d. Calculate the variance of the four estimators from Questions 5-a and 5-c, respectively. Show your work step by step. 5-e. Suppose that we are using the simple linear regression model Y = B1 X + Bo + 1 while the true model is Y = 1X1 + B2X2 + Bo + 82 where Bo, B1, and B2 are constants. We assume that the distributions of &, and e2 are both N(0,02), i.e., normal distribution with variance o?. We further assume that the two noise variables are uncorrelated. Find the least squares estimator of B, in this case and determine whether the point estimator that you obtain is biased or not. If it is biased, calculate the bias.1. We use the added variable technique to derive the variance ination factor (VIP). Consider a linear model of the form 91' =50+l31$1+l3213922+-"+}3p$a'p+zr, 5'3: 1:"'ana (1) where the errors are uncorrelated with mean zero and variance 02. Let X denote the n X p' predictor matrix and assume X is of full rank. We will derive the VIP for ip. The same derivation applies to any other coefcient simply by rearranging the columns of X. Let U denote the matrix containing the rst p' 1 columns of X and let z denote the the last column of X so that X = [U 2]. Then we can write the model in (1) as 50 x91 Y=[U z](,:J)+t-:=Ua+z6p+e with a: (2) x810. 1 Let 2 denote the vector of tted values from the least squares regression of z on the columns of U (Le. the regression of X.p on all the other variables), and let T : z 2 denote the residuals from that regression. Note that 'r' and 3 are not random, they are constant vectors obtained by linear transformations of z. (a) Show that the regression model in (2) can be rewritten in the form for some constant vector 6 of the same length as a. (Hint: z : i l 'r and 2? = U(UTU)_1UTz). (b) Show that UT? 2 0, a zero vector. (0) Obtain simplied expressions for the least squares estimators of 5 and 5?, showing, in particular, that 5,, : 'rTY/rT'r. (d) Based on Part (c) and the model assumptions, show that 0.2 ELK\"? _ is)? where :Eg-p is the LS tted value from regression X,D on the all the other predictor variables with an intercept. var(,p) : Other receipts for the couple were as follows: Dividends (all qualified dividends) $2.500 Interest income: Union Bank $ 220 State of Idaho-interest on tax refund 22 City of Boise school bonds 1.250 Interest from U.S. savings bonds 410 (not used for educational purposes) 2017 federal income tax refund received in 2018 2,007 2017 state income tax refund received in 2018 218 Idaho lottery winnings 1.100 Casino slot machine winnings 2,250 Gambling losses at casino 6.500 Other information that the Reyeses provided for the 2018 tax year. Mortgage interest on personal residence $11.081 Loan interest on fully equipped motor home 3.010 Doctor's fee for a face-lift for Mrs. Reyes 8.800 Dentist's fee for a new dental bridge for Mr. Reyes 3,500 Vitamins for the entire family 1 10 Real estate property taxes paid $ 5,025 DMV fees on motor home (tax portion) 1.044 DMV fees on family autos (tax portion) 436 Doctors' bills for grandmother 2,960 Nursing home for grandmother 10,200 Wheelchair for grandmother 1.030 Property taxes on boat 134 Interest on personal credit card 550 Interest on loan to buy public school district bonds 270 Cash contributions to church (all the contributions 6.100 were in cash and none more than $250 at any one time) Cash contribution to man at bottom of freeway off-ramp 25 Contribution of furniture to Goodwill-cost basis 4,000 Contribution of same furniture to listed above Goodwill-fair market value 410 Tax return preparation fee for 2017 taxes 625 Required Prepare a Form 1040 and appropriate schedules, Schedule A, and Schedule B for the comple- tion of the Reyes's tax return. They do not want to contribute to the presidential election cam- paign and do not want anyone to be a third-party designee. For any missing information, make reasonable assumptions. They had qualifying health coverage at all times during the year