(a) Please derive the least squares regression line, y = ax + b, for the points (x₁, y₁), i = 1,..., n. (Hint: find a and b such that minimize S = -₁(y₁ - axi — b)².)< - (b) Graph the three points (2, 2), (2, 1), (2.1, 1.5) and estimate the least squares regression line for these data by the formulas to find a better linear approximation for these points. (c) Repeat (b) if these points (-2, 2), (0, 1), (2, 1.5).
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- Consider the following regression model Yt = β0 + β1 Ut + β2 Vt + β3 Wt + β4Xt + ∈t , where U, V, W, X and Y are economic variables observed from t = 1, . . . , 75, β0 , . . . , β4 are the model parameters and ∈t is the random disturbance term satisfying the classical assumptions. Ordinary Least Squares (OLS) is used to estimate the parameters, producing the following estimated model: Yt = 1.115 + 0.790*Ut − 0.327*Vt + 0.763*Wt + 0.456*Xt (0.405) (0.178) (0.088) (0.274) (0.017) where standard errors are given in parentheses, the R-squared = 0.941, the Durbin-Watson statistic is DW = 1.907 and the residual sum of squares is RSS = 0.0757. In answering this question, use the 5% level of significance for any hypothesis tests that you are asked to perform, state clearly the null and al- ternative hypotheses that you are testing, the test statistics that you are using and interpret the decisions that you make.…(a) Derive the least squares estimates of α and β for the centred form of the simple linear regression model given by yi =α+β(xi −x ̄)+εi i=1,2,...,n. (b) Check that the estimates do give a minimum in the same way as we saw for the standard form of the simple linear regression model.A “Cobb–Douglas” production function relates production (Q) to factorsof production, capital (K), labor (L), and raw materials (M), and an errorterm u using the equation Q = λKβ1Lβ2Mβ3eu, where λ, β1, β2, and β3 areproduction parameters. Suppose that you have data on production and thefactors of production from a random sample of firms with the same Cobb–Douglas production function. How would you use regression analysis toestimate the production parameters?
- In the example of First-Order Model in Five Quantitative Independent Variables, the parameters, βi,i=1,…,5, are often called partial regression coefficients. What does "partial" mean here?Consider a k-variables linear regression model, i.e., Y = X 1β1 + X 2 β2 + ε, Where, X1 is (N . k1 ) , X 2 is (N . k2 ) and k = k1 + k2 . As you may recall, adding columns to the X matrix (including additional regressors in the model) gives a positive definite increase in R2. The adjusted R2 attempts to avoid this phenomenon of ever-increasing R2. Show that the additional k2 number of variables (regressors) in this model increases R2 if the calculated F-statistic in testing the joint statistical significance of coefficients of these additional regressors (β2 ) are larger than one.31 - Find the regression model. Regresyon modelini bulunuz. Y X 2 5 5 9 7 4 4 11A) y=-3,21+2,11xB) y=2,56+5,43xC) y=5,27+0,11x
- Find the least-squares regression line y^=b0+b1xy^=b0+b1x through the points (−1,1),(1,9),(4,13),(9,19),(11,27),(−1,1),(1,9),(4,13),(9,19),(11,27), and then use it to find point estimates y^y^ corresponding to x=2x=2 and x=7x=7. For x=2x=2, y^y^ = For x=7x=7, y^y^ =Two specimens of cold rolled steel sheet, which have differentcopper contents and annealing temperature are measured in hardness with the following results: First column = HardnessSecond column = Copper contentThird column = Annealing temperature a) Create a scatter plot to verify that it is reasonable to assume that the regression of Y on x is linear. b) Fit a straight line using the method of least squares. c) Fit an equation of the form (image 2), where x1 represents the copper content, x2 represents the annealing temperature, and y represents the hardness.Refer to the following nonlinear model which relates W to P, Q, and R: W = aPbQcRd The computer output form the regression analysis is: DEPENDENT VARIABLE: LNW R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 18 0.9023 43.12 0.0001 VARIABLE PARAMETER ESTIMATE STANDARD ERROR T-RATIO P-VALUE INTERCEPT 2.50 0.45 5.56 0.0001 LNP -5.10 1.75 -2.91 0.0113 LNQ 12.40 3.20 3.88 0.0017 LNR -6.00 1.50 -4.00 0.0010 Based on the information in the table, the nonlinear relation can be transformed into the following linear regression model:
- The table below shows the number of state-registered automatic weapons and the murder rate for several Northwestern states, where xx is thousands of automatic weapons and yy is murders per 100,000 residents. xx 11.5 8.5 6.7 3.5 2.9 2.7 2.7 0.9 yy 14.1 11 10 7.3 6.7 6.4 6.4 4.7 Use your calculator to determine the equation of the regression line and write it in the y=ax+by=ax+b form. Round to 2 decimal places. According to this model, how many murders per 100,000 residents can be expected in a state with 10.2 thousand automatic weapons? Round to 3 decimal places. According to this model, how many murders per 100,000 residents can be expected in a state with 5.8 thousand automatic weapons? Round to 3 decimal places.The number of murders and robberies per 100,000 population for a random selection of states are shown. Find the equation of the regression line y'=ax+b, and predict the number of robberies when x=4.5 murders murders,x: 2.4,2.7,5.6,2.6,2.1,3.3,6.6,5.7 robberies,y : 25.3,14.3,151.6,91.1,80,49,17.3,45.8The following table shows students’ test scores on the first two tests in an introductory physics class. Physics Test Scores First test, x 7373 6161 6969 4242 8888 8787 4242 4444 4040 4848 6868 6464 Second test, y 7474 6262 6262 4343 7878 6969 3838 4343 3030 4848 6666 5454 Step 1 of 2 : Find an equation of the least-squares regression line. Round your answer to three decimal places, if necessary. Step 2 of 2 : If a sudent scored a 75 on his first paper, make a prediction for his score on the second test. Assume the regression equation is appropriate for prediction. Round your answer to two decimal places, if necessary.