Consider the simple linear regression Y: = Bo + B1a, +E, (a) Derive the weighted least squares estimators for 3, and B1. (b) Express the weighted least squares estimator for B1 in terms of the centered variables y, - Ju and z, – Fw, where gu, and fu are the weighted means.
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- 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?Which of the following are feasible equations of a least squares regression line for the annual population change of a small country from the year 2000 to the year 2015? Select all that apply. Select all that apply: yˆ=38,000+2500x yˆ=38,000−3500x yˆ=−38,000+2500x yˆ=38,000−1500x8)Suppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 11 members of the population, where for each member, the value of Y as well as the values of the three explanatory unknowns were observed. The data is entered into a computer using linear regression software and the output summary tells us that R-square is 0.86, the linear model coefficient of the first explanatory unknown is 7 with standard error estimate 2.5, the coefficient for the second explanatory unknown is 11 with standard error 2, and the coefficient for the third explanatory unknown is 15 with standard error 4. The regression intercept is reported as 28. The sum of squares in regression (SSR) is reported as 86000 and the sum of squared errors (SSE) is 14000. From this information, what is MSE/MST? .5000 NONE OF THE OTHERS .2000 .3000 .4000
- 9)Suppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 11 members of the population, where for each member, the value of Y as well as the values of the three explanatory unknowns were observed. The data is entered into a computer using linear regression software and the output summary tells us that R-square is 0.79, the linear model coefficient of the first explanatory unknown is 7 with standard error estimate 2.5, the coefficient for the second explanatory unknown is 11 with standard error 2, and the coefficient for the third explanatory unknown is 15 with standard error 4. The regression intercept is reported as 28. The sum of squares in regression (SSR) is reported as 79000 and the sum of squared errors (SSE) is 21000. From this information, what is the adjusted R-square? .8 .7 NONE OF THE OTHERS .6 .5A set of paired data has a least squares regressionline with equation yn = 0.50x + 2.0 and a correlationcoefficient of r = 0.80. Suppose we convert the datafor each variable to z-scores and then compute the newregression line. What will the equation be?A) zˆy = 0.50zx B) zˆy = 0.64zxC) zˆy = 0.80zx D) zˆy = 0.50zx + 20E) zˆy = 0.80zx + 2017) Suppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 41 members of the population, where for each member, the value of Y as well as the values of the three explanatory unknowns were observed. The data is entered into a computer using linear regression software and the output summary tells us that R-square is 0.9, the linear model coefficient of the first explanatory unknown is 7 with standard error estimate 2.5, the coefficient for the second explanatory unknown is 11 with standard error 2, and the coefficient for the third explanatory unknown is 15 with standard error 4. The regression intercept is reported as 28. The sum of squares in regression (SSR) is reported as 90000 and the sum of squared errors (SSE) is 10000. From this information, what is the number of degrees of freedom for the t-distribution used to compute critical values for hypothesis tests and confidence intervals for the individual…
- Consider the fitted values from a simple linear regression model with intercept: yˆ = 5 + 6x. Assume that the total number of observations is n = 302. In addition, the R-squared of the regression is R2 = 0.6 and Pn i=1(xi − x¯) 2 = 15, where ¯x is the sample mean of x. Under the classical Gauss-Markov assumptions, a) What is the standard error of the estimated slope coefficient?Consider the following population model for household consumption: cons = a + b1 * inc+ b2 * educ+ b3 * hhsize + u where cons is consumption, inc is income, educ is the education level of household head, hhsize is the size of a household. Suppose a researcher estimates the model and gets the predicted value, cons_hat, and then runs a regression of cons_hat on educ, inc, and hhsize. Which of the following choice is correct and please explain why. A) be certain that R^2 = 1 B) be certain that R^2 = 0 C) be certain that R^2 is less than 1 but greater than 0. D) not be certainA researcher collected data on the cholesterol level, CC, and the age, AA, of 24 people selected at random. Using the data, the researcher calculated the least-squares regression line to be Cˆ=182+2.2AC^=182+2.2A and the standard error of the slope to be 0.38. If the conditions for inference are met, which of the following is closest to the value of the test statistic to test the hypotheses H0:β=0H0:β=0 versus Ha:β≠0Ha:β≠0 ?
- In a study measuring the relationship between height in centimeters and annual income in dollars, it has been determined that for Group 1, r2 =0.15 and for Group 2, r2 =0.10 where r denotes the correlation between the two variables. Least-squares regression lines are fitted to the observations from each group. Which of the following statement is true: A. There could be a positive relationship between the two variables for Group 1 and a negative relationship between the two variables for Group 2 B. The sum of the residuals for Group 1 is greater than the sum of the residuals for Group 2. C. Measuring the height in inches would increase the value of r2 for both groups. D. None of the answer options is true Can you also explain the difference between r and r2, and why least square regressions are used?Find the least-squares regression line y^ =b0+b1xy^=b0+b1x through the points (−3,1),(1,8),(6,13),(7,18),(12,27). For what value of is y^ =0? x =?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.…
