1. Consider a linear regression model y = XB+ e with E(e) = 0. The bias of the ridge estimator of obtained by minimizing Q(B) = (y - XB)¹ (y - XB) +r(BTB), for some r > 0, is -(X¹X (XTX -1 +rI)-¹B ² (X²X+rI)-¹B −r(X¹X +rI)−¹ß r(X¹X+r1) ¹8
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- bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.The following table provides values of the function f(x,y). However, because of potential; errors in measurement, the functional values may be slightly inaccurately. Using the statistical package included with a graphical calculator or spreadsheet and critical thinking skills, find the function f(x,y)=a+bx+cy that best estimate the table where a, b and c are integers. Hint: Do a linear regression on each column with the value of y fixed and then use these four regression equations to determine the coefficient c. x y 0 1 2 3 0 4.02 7.04 9.98 13.00 1 6.01 9.06 11.98 14.96 2 7.99 10.95 14.02 17.09 3 9.99 13.01 16.01 19.02An engineer studied the relationship between the input and output of a production process. In(,X1) B, X2 1. He considered the non-Ilinear multiple regression model: Y = Bo + In order to estimate the parameters with a software package, the engineer needs to transform the above equation to a linear equation. Let U, V denote the transformed variables for X1 and Y What are U and V? (As functions of X1 and X2)
- Assume that we have the usual Simple Linear Regression model given by: Y₁ = Bo + Bixi +8 Show that in this case: Where Txy R² = rzy is the correlation coefficient between x and y.The relationship between yield of maize (a type of corn), date of planting, and planting density was investigated in an article. Let the variables be defined as follows. y = maize yield (percent) x1 = planting date (days after April 20) x2 = planting density (10,000 plants/ha) The following regression model with both quadratic terms where x3 = x12 and x4 = x22 provides a good description of the relationship between y and the independent variables. y = ? + ?1 x1 + ?2 x2 + ?3 x3 + ?4 x4 + e (a) If ? = 21.05, ?1 = 0.652, ?2 = 0.0025, ?3 = −0.0204, and ?4 = 0.5, what is the population regression function? y = (b) Use the regression function in part (a) to determine the mean yield (in percent) for a plot planted on May 8 with a density of 41,182 plants/ha. (Round your answer to two decimal places.) % (c) Would the mean yield be higher for a planting date of May 8 or May 22 (for the same density)? The mean yield would be higher for . (d) Is it…2. An engineer studied the relationship between the input and output of a production process. He considered the non-linear regression model: y = In (x + B1) – In ( Bo x). - In order to estimate the model parameters using a statistical software, the engineer needs to transform it to a linear equation. (a) What transformation is needed? U = the transformed X Both X and Y need to be transformed. Denote: V = the transformed Y What are U and V? (as functions of X and Y) (b) After the variables were transformed according to your answer in Part (a), the following parameter estimates were obtained: Intercept: 0.4 Coefficient of U: 1.8 What would be the estimates for Bo and B1 in the original non-linear model?
- The relationship between yield of maize (a type of corn), date of planting, and planting density was investigated in an article. Let the variables be defined as follows. y = maize yield (percent) x₁ planting date (days after April 20) x₂ = planting density (10,000 plants/ha) The following regression model with both quadratic terms where x3 = x₁² and x4 = x₂² provides a good description of the relationship between y and the independent variables. y=a + B₁X₁ + B₂X2 + ₂xy + B₁X₁ + e (a) If a = 21.05, ₁ = 0.654, B₂ = 0.0023, B3 = -0.0207, and ₁=0.3, what is the population regression function? (b) Use the regression function in part (a) to determine the mean yield (In percent) for a plot planted on May 4 with a density of 41,179 plants/ha. (Round your answer to two decimal places.) (c) Would the mean yield The mean yield would higher for a planting date of May 4 or May 22 (for the same density)? higher for --Select-- (d) Is it appropriate to interpret , = 0.654 as the average change in…The relationship between yield of maize, date of planting, and planting density was investigated in an article. Let the variables be defined as follows. y = percent maize yield x = planting date (days after April 20) z = planting density (plants/ha) The following regression model with both quadratic terms where x₁ = x, X₂ = Z, X3 = x² and x4 = 2² provides a good description of the relationship between y and the independent variables. y =a +B₁x₁ + B₂X₂ + B3X3+B₁x₁ + e (a) If a = 21.07, B₁ = 0.653, B₂ = 0.0022, B3 = -0.0207, and B4 = 0.00002, what is the population regression function? y = 509 X (b) Use the regression function in Part (a) to determine the mean yield for a plot planted on May 7 with a density of 41,182 plants/ha. (Give the exact answer.) (c) Would the mean yield be higher for a planting date of May 7 or May 23 (for the same density)? The mean yield would be higher for [May 7 You may need to use the appropriate table in Appendix A to answer this question.Find the quadratic regression curve through the following points (2, 5), (3, 5), (5, 3) y = 3.0000x? 0.3333x + 1.6667 y = 3.0000x2 + 1.6667x – 0.3333 O y = -0.3333x2 + 1.6667x + 3.0000 O y = 1.6667x2 – 0.3333x + 3.0000
- Consider the multiple regression model Y₁ = Bo + B₁x1₁j + B₂x2j+B3 x 3j+ €j under the usual assumptions labelled A1, A2, A3, A4, A5, A6. Briefly explain which type of graphs are performed in the analysis of residuals.Suppose that the sales of a company (Y) is regressed on advertising expenditure (x) and labor cost (z), and the estimated regression equation is Y = 5 + 0.5x + 0.7z + u (where u is the error term). Here, sales, advertising expenditure and labor cost are measured in million Tk. Standard error for the coefficient of x is 0.04, standard error for the coefficient of z is 0.01, and the sample size is 20. Can we conclude that advertising expenditure is a statistically significant variable?Construct an example of a regression model that satisfies the assumptionE(ui | Xi) = 0 but for which E(U | X ) ≠0n.
