1. Derive the critical values of 0 and 3₁ that minimize the residual sum of squares for the following sample regression model Y₁ = 0 + ₁ Xi + ei
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- Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?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.3. The research shows that both X₁ and X₂ are theoretically valid, meaning that they both belong in the true regression relationship. Yi Bo + B₁X1i + B₂X₂i + Ei The researcher decides to estimate two regressions for the sample: Y₁ = Bo + B₁X₁i + Ei Y₁ = Bo + B₂X2i + €₁ Would the estimated regression coefficients for ₁ and ₂ be the same as if the researchers had estimated the true regression relationship? How will the error term compare between models? How will the R-squared values for these two regressions compare to the R-squared of the true relationship? Explain and support your answers with relevant formulas.
- If the points (x1, y1), (x2, y2),..., (xn, yn) lie on a straight line, what can you say about the regression line associated with these points?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?Consider the two variable regression model: Y = Bo+P,Edu+ B2EXP21 + u, where Y denotes the average monthly income, Edu denotes the number of years of education, Exp denotes the number of years of experience, and u, denotes the error term. Suppose the researcher wants to test whether the effect of education on average monthly income and the effect of experience on the average monthly income of an individual are the same or not So, the test the researcher wants to conduct is Ho: B = 6, vs. H, f,+P2 The hypotheses can be tested by modifying the original regression equation to turn the restriction into a restriction on a single regression coefficient. Suppose the regression function is modified in the following way: Y, Po+Y,Edu, + B2W +u, where y, =P,-P2 and W, = Edu,, + Exp2i %D Since y, = B, - B2, the test the researcher wants to conduct will now be Ho: y=0 vs. H y0. Let y, and SE(y,), denote the estimated slope coefficient of y, and the standard error of 7, respectively. is .D IS If…
- 3. Derive the solution for the estimated intercept (₁) in case of a univariate linear regression.An 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)4. Write a Julia function to calculate CV and GCV in the case of smoothing by local linear regression.
- Consider the following linear regression model that relates income per capita in thousand dollars of a country i (GDP P Ci), with its percentage of the population in the agricultural sector (P Ai): Model : GDP P Ci = β0 + β1P Ai + ui (a) Explain in words how to interpret parameters β0 and β1. What sign do you think these parameters might have? Explain. (b) Draw the (population) regression line associated with this model assuming that parameters β0 and β1 have the sign you have indicated in answering question (2a). Explain the meaning of this regression line.For the regression model Yi = b0 + eI, derive the least squares estimator.1. Consider two least-squares regressions and y = Xíễ tế y = Xí$i+ XzB2 tê Let R2 and R2 be the R-squared from the two regressions. Show that R22 R2.
