Consider the multiple regression model containing three independent variables, under Assumptions MLR.1 through MLR.4: y = Bo + Biti + Bx, + Bzł3 + u. You are interested in estimating the sum of the parameters on x, and x2; call this 01 = Bo + B1. Show that ô, = Bi + Bz is an unbiased estimator of 01. |(ii) Find Var(@1) in terms of Var(ß),Var(ß,), and Corr(ß1. B2).
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- The following fictitious table shows kryptonite price, in dollar per gram, t years after 2006. t= Years since 2006 0 1 2 3 4 5 6 7 8 9 10 K= Price 56 51 50 55 58 52 45 43 44 48 51 Make a quartic model of these data. Round the regression parameters to two decimal places.In running a regression of the retunrs of stock XYZ against the returns on the market, the Std for the returns of stock XYZ is 20% and that of the market returns is 15%. If the estimated beta is found to be 0.75 : What is the maximum possible value of beta given that the standar deivation of the returns of stock XYZ is 20% and those of the market is 15% ?Consider the following simple linear regression model: y = β0 + β1x + u. Using a sample of n observations on x and y, you estimate the model by OLS and obtain the estimates βˆ 0, βˆ 1, and the R-squared of the regression, R2 . Then you scale this sample by a factor of 100, obtain a new sample {xi/100; yi/100} for i = 1, . . . , n, re-estimate the model by OLS, and denote the new coefficient estimates by β˜ 0, β˜ 1, and the new R-squared of the regression by R˜2 . a) Give the expression of β˜ 1 in terms of βˆ 1, and justify your answer.
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- If a set of paired data gives the indication that the regression equation is of the form μY|x = α · βx, it is cus-tomary to estimate α and β by fitting the line log ˆy = log ˆα + x · log βˆ to the points {(xi, log yi);i = 1, 2, ... , n} by the methodof least squares. Use this technique to fit an exponentialcurve of the form ˆy = αˆ · βˆx to the following data on thegrowth of cactus grafts under controlled environmentalconditions: Weeks after Heightgrafting (inches)x y1 2.02 2.44 5.15 7.36 9.48 18.3Consider the following datasets: X1=2,8,4 X2= 0.4, 7.10, 3.2 Y= 2.6, 9.2, 5.3 Statistically regress Y on X1 and X2, i.e. find a regression equation in which output variable is Y and input variable is X1 and X2. Show first two iterations of Gradient Descent method to solve part a. Initialize slopes and intercept at 0 value.Q11. A fitted linear regression equation is ŷ = 6–2x. If x = 3 and the corresponding observedvalue of y = 2, the residual at this observation is:a. –5b. 2c. –3d. –2
- The following table gives the marks obtained by 10 students in POLI 344 (X) together with the marks obtained in the exam in POLI 308 (Y). POLI 344 (X)8 8 9 10 10 11 12 13 13 11 14 POLI 443 (Y) 7 11 8 7 12 11 10 12 14 17 15 (i) State the two equation lines of the regression line. (ii) If a student was absent from POLI 443 but scored 18 in POLI 344 (X) state the regression line, which would be suitable for estimating his/her possible mark in POLI 443 and work out a fair estimate for his /her possible mark.We have been assigned to determine how the total weeklyproduction cost for Widgetco depends on the number ofwidgets produced during the week. The following modelhas been proposed:Y b0 b1X b2X2 b3X3 where X number of widgets produced during the weekand Y total production cost for the week. For 15 weeksof data, we found that SSR 215,475 and SST 229,228.For this model, we obtain the following estimated regressionequation (t-statistics for each coefficient are in parentheses):yˆ 29.7 19.8X 0.39X2 0.005X3(0.78) (0.62) (1.25)a For a 0.10, test H0: bi 0 against Ha: bi 0(i 1, 2, 3).b Determine R2 for this model. How can the high R2value be reconciled with the answer to part (a)?Given are five observations collected in a regression study on two variables. xi 2 6 9 13 20 yi 7 18 9 26 23 Compute b0 and b1 (to 1 decimal).b1 b0 Complete the estimated regression equation (to 1 decimal).^y = + x Use the estimated regression equation to predict the value of y when x = 6 (to 1 decimal).^y =