**Problem 2.** Consider TSLS (Two-Stage Least Squares) estimation with a single included endogenous variable and a single instrument. The predicted value from the first-stage regression is $ \hat{X}_i = \hat{\pi}_0 + \hat{\pi}_1 Z_i $.(i) Use the definition of the sample variance and covariance to show that $ s_{X\hat{Y}} = \hat{\pi}_1 s_{ZY} $ and $ s^2_{X} = \hat{\pi}_1^2 s^2_{Z} $.(ii) Remember that $ \hat{\pi}_1 = s_{XZ} / s^2_{Z} $ and that the TSLS estimator $ \hat{\beta}^{TSLS} $ is the OLS (Ordinary Least Squares) estimator of the slope parameter in the regression of $ Y $ on $ \hat{X} $. Use these facts and the result of part (i) to prove that $ \hat{\beta}^{TSLS} $, obtained by the two-stage least squares procedure, can be written as\[\hat{\beta}^{TSLS} = \frac{s_{ZY}}{s_{ZX}}.\]

From the given information, The predicted value from the first-stage regression is X^i=π^0+π^1Zi. i)…

Answered: Problem 2. Consider TSLS estimation…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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State Per Capita Personal Income ($) Birth Rate Delaware 28,493 13.9 Maryland 28,674 13.0 District of Columbia 35,704 15.7 Virginia 26,109 13.2 18,724 11.3 West Virginia 14.2 23.168 North Carolina 13.6 20,508 South Carolina 15.8 23,822 Georgia 13.1 24,799 Florida (a) The Ts(in 3 significant figures) for the given data is equal to Blank 1. (b) The typre of relation (negative, positve, or no relation) that exist between per capita personal income and birt rate is Blank 2
4. The following regression is fitted using variables identified that could be related to tuition charges ($) of a university. TUITION = a+ B ACCEPT + y MSAT + 1 VSAT Where ACCEPT = the percentage of applicants that was accepted by the university, MSAT = Median Math SAT score for the freshman class and VSAT = Median English SAT score for the freshman class. The data was processed using MNITAB and the following is an extract of the output obtained: Predictor Coef StDev T P Constant -26780 6115 -4.38 0.000 ACCEPT 116.00 37.17 * 0.003 MSAT -4.21 14.12 VSAT 70.85 15.77 -0.30 4.49 0.767 ** S = 2685 R-Sq = 69.6% Analysis of Variance R-Sq (adj) = 67.7% Source DF SS MS F P Regression 3 808139371 Residual Error 49 353193051 269379790 7208021 37.37 0.000 Total 52 1161332421 a) Write out the regression equation. b) State the dependent and independent variable(s) c) Fill in the blanks identified by ** and ****. d) Is B significant, at the 10% level of significance? e) State one limitation of using…
If Linear correlation coefficient r = 0.794556 Coefficient of determination = 0.631319 Equation of regression line is y = 16.5523 + %3D 0.725938x then both variables are moving in the same direction. A True В False
Suppose that you run a regression of Y, on X, with 110 observations and obtain an estimate for the slope. Your estimate for the standard error of ₁ is 1. You are considering two different hypothesis tests: The first is a one-sided test: Ho: B1-0, Ha: 31>0, a = .05 The second is a two-sided test: Ho: 31-0, Ha: B1 0,a = .05 (a) What values of , would lead you to reject the null hypothesis in the one-sided test? (b) What values of , would lead you to reject the null hypothesis in the one-sided test? (c) What values of would lead you to reject the mill hypothesis in the one-sided test, but not the two-sided test? (d) What values of 3 would lead you to reject the null hypothesis in the two-sided test, but not the one-sided test?
QUESTION 3 Different foods have a different number of calories. The size in grams of an average serving and the weight of the contestant in lose weight program is being studied. The simple linear regression and correlation analysis is performed in a sample of 9 observations and give the results as below: SS = 8400 SSyy = 1067.68 a = 30.134 b = 0.2867 (a) Develop the equation of the simple linear regression line for this data. Then, give a brief interpretation of the values of the linear correlation coefficient, the intercept and the slope. (b) Test the hypothesis that the slope would be positive at the 10% significance level by showing your solution. (c) Construct an 80% confidence interval for B. Interpret your answer.
3) For a sample of 306 students in a basic business communications course, the sample regression line y = 58.813 + 0.2875x was obtained. Here y = final student score at the end of the course x = score on a diagnostic writing skills test given at the beginning of the course The coefficient of determination was 0.1158 and the estimated standard deviation of the estimator of the slope of population regression line was 0.04566. a) Define and interpret the estimates of the regression equation. b) Interpret the coefficient of determination. c) What will happen to the final student score if his diagnostic writing test score increases by 2? d) How can you inmprove this model?
29)
4. A simple linear regression is fit to a dataset. Unfortunately, the corresponding ANOVA table is not complete because some quantities in the table are missing due to unknown digital errors. Calculate the missing values, denoted by "?", in the ANOVA table based on the other available values. Is this regression model significant (α = 0.05)? Source Regression Residual Total df ? ? 21 SS ? ? ? MS = SS/df ? 1.13 F-Ratio 430.65 p-value ?
The table below lists measured amounts of redshift and the distances (billions of light-years) to randomly selected astronomical objects. Find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. There is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. For the prediction interval, use a 90% confidence level with a redshift of 0.0126. Redshift Distance 0.0238 0.31 a. Find the explained variation. 0.0543 0.74 (Round to six decimal places as needed.) b. Find the unexplained variation. (Round to six decimal places as needed.) c. Find the indicated prediction interval. 0.0722 1.02 billion light-years
Suppose that you have estimated coefficients for the regression model Y = B₁ + B₁X₁ + ß2X2 + ß3X3. Test the hypothesis that all three of the predictor variables are equal to 0, given the analysis of variance shown on the right. Use α = 0.05. Click here to view page 1 of a table of critical values of F. Click here to view page 2 of a table of critical values of F. Choose the correct null and alternartive hypotheses below. A. Ho: B₁ B₂ =B3 = 0 H₁: at least one ß; #0 C. Ho: B₁ B₂ = 3 = 0 H₁: B₁0, B₂0, B3 > 0 Find the critical value. The critical value is (Round to two decimal places as needed.) Source Regression Residual Error DF SS 3 9,654 23 2,400 B. Hō: at least one ß; ‡0 H₁: B₁ =B₂ =B3 = 0 D. H₁: B₁ = P₂ = ³3 = 0 H₁: B₁ B₂ B3 0 MS
Suppose that you perform a hypothesis test for the slope of the population regression line with the null hypothesis H0: β1 = 0 and the alternative hypothesis Ha: β1 ≠ 0. If you reject the null hypothesis, what can you say about the utility of the regression equation for making predictions?
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