19. Given the following econometric model Y₁ = P + B₁X₁ + U₁ What is the conditional expectation of E(Y,IX,)? 20. Given we know the population = 2.1, ₁ = 3, and an individual in the survey has an X with the value of 4. What is the conditional expectation of Y for this individual, E(Y₁|X= 4)?
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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.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 following population linear regression model of individual food expenditure: Y = 50 + 0.5X + u, where Y is weekly food expenditure in dollars, X is the individual’s age, and 50+0.5X is the population regression line. Suppose we generate artificial data for 3 individuals using this model. This artificial sample, which consists of 3 observations, is shown in the following table: Answer the following questions. Show your working. (a) What are the values of V1 and V4? (b) Suppose we know that in this artificial sample, the sample covariance between X and Y is 150, and the sample variance of X is 100. Compute the OLS regression line of the regression of Y on X. (Hint: Assume these summary statistics and the OLS regression line continue to hold in parts (c)-(e).) (c) What are the values of V5 and V7?
- 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. –2Consider the following regression Yi = βXi + ui. Show that (a) the OLS estimator of β is βˆ = (Σ(XiYi))/(Σ(Xi^2)) (b) (Σ(uˆi)/)n = 1/n(Σ(Yi − Yˆi))=/0We 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)?
- .The worker has noticed that the more time he spends at work (x), the less money he is likely to make (y) in conducting transactions for his firm. Which of the regression equations MOST suggests such a possibility?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.The following estimated regression model was developed relating yearly income (y in $1000s) of 30 individuals with their age (x1) and their gender (x2) (0 if male and 1 if female).ŷ = 30 + 0.7x1 + 3x2Also provided are SST = 1200 and SSE = 384.The yearly income of a 24-year-old female individual is _____.
- 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 model:? = ?? + ?,known as the Classical Linear Regression Model (CLRM), where y is the dependent variable, X is the set of independent variables, ? is the vector of parameters to be estimated and ? is the error term. Present and discuss the R2 and the adjusted R2. Discuss pros and cons of each of the two statistics.The following estimated regression equation based on 10 observations was presented. ŷ = 29.1260 + 0.5306x1 + 0.4680x2 The values of SST and SSR are 6,728.125 and 6,215.375, respectively. (a) Find SSE. SSE = (b) Compute R2. (Round your answer to three decimal places.) R2 = (c) Compute Ra2. (Round your answer to three decimal places.) Ra2 = (d) Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.) The estimated regression equation provided a good fit as a small proportion of the variability in y has been explained by the estimated regression equation.The estimated regression equation did not provide a good fit as a small proportion of the variability in y has been explained by the estimated regression equation. The estimated regression equation provided a good fit as a large proportion of the variability in y has been explained by the estimated regression equation.