1. Consider the following regression modely = x3 + u.(1)Let 3 denote the Ordinary Least Squares (OLS) estimator of B. The so-called Gauss-Markov assumptions are:• MLR.1: The true model in the population is given by (1).• MLR.2: We have a random sample of n observations {(ri, Yi), i = 1, 2, .., n}following the population model in (1).....• MLR.3: No one explanatory variable can be written as a linear combination of theremaining explanatory variables that is, there is no perfectcollinearity.• MLR.4: In the population, the error u has an expected value of zero given any valuesof the explanatory variables, that is Elu|x] = 0.• MLR.5: In the population, the error u has the same variance given any values of theexplanatory variables, that is Var[u|x] = o? , an unknown finite, positive constant.In the following scenarios, state whether 3 is an unbiased and consistent estimator of 3,and provide a brief justification for your answer in each case - but no formal mathematicalderivations are required:(a)MLR.1 does not.(Word limit: 50 words)Assumptions MLR.2, MLR.3, MLR.4 and MLR.5 hold but Assumption

Given information: The information about the regression model is given.

Answered: Consider the following regression model…

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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b) Suppose you are given the following information for an OLS regression: Σχι X; Y = 830102,T = 22,x=416.5, y = 86.65, x = 3919654, RSS = 130.6 i) Calculate the ordinary least square regression line. Calculate SE (a) and SE (B) Perform the hypothesis test such that Ho: a = -60 versus H₁: α = -60 Perform the hypothesis test such that Ho: P = 0.4 versus H₁: B = 0.4 In both cases use a significance of 5%. ii) iii) iv)
Example 15.11) The following table shows the marks obtained in two tests by 10 students: Marks in Ist Test (X) 9. 8. 8 7 6. 10 4 7 Marks in 2nd Test (Y) 8 7 10 8 10 9. 8 6. (a) Find the least square regression line of Y on X. / (b) Test the hypothesis that marks in the two tests are linearly related.
2. Two dimensional random variable (X, Y) is given by Y / X 1 -2 0,25 0,5 0,25 Find the regression line. [Ans. y = -2x + 2]
5
1.
7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
6. Consider the regression model Yi = βXi + ui, where ui and Xi satisfy the least squares assumptions in Key Concept 4.3. Let β ̄ denote an estimator of β that is constructed as β ̄ = Y ̄/X ̄, where Y ̄ and X ̄ are the sample means of Yi and Xi, respectively. (a) Show that β ̄ is a linear function of Y1 , ..., Yn . (b) Show that β ̄ is conditionally unbiased.
Example 15.11) The following table shows the marks obtained in two tests by 10 students: Marks in Ist Test (X) 8 8. 10 4 7 Marks in 2nd Test (Y) 8 7 7. 10 5 8. 10 6. (a) Find the least square regression line of Y on X. / 7,
1. When estimating the parameters of a linear regression model, the OLS estimator is a scalar value and the OLS estimate is a random variable. 2. For a linear regression model including only an intercept, the OLS estimator of that intercept is equal to the sample mean of the independent variable. 3. If the error term in a linear regression model is normally distributed, then the distribution of the OLS estimator, conditional on explanatory variables, is also normal. 4. The value of the coefficient of determination, R2 of Model 1: In yi = a + Bx; + E; (M1) is different from the value of R2 of the Model 2: Yi = a+ Bx; + Ei. (M2) 5. Consider an earnings function In wage = 0.6 + 0.16 · educ estimated on a sample of n = 1000 individuals. The standard error of the estimated slope coefficient equals 0.11. This allows us to interpret the estimated slope coefficient as: ceteris paribus expected returns to schooling are 16%.
If beta1_hat = 0.4571 use the data below to find beta0_hat for the simple linear model using the method of Least Squares. Y 17 2 3 11 4 20 15 18 13 13.07 50.88 24.57 32.11
MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER An economist is studying the job market in Denver area neighborhoods. Let x represent the total number of jobs in a given neighborhood, and let y represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs). 13 34 52 28 50 25 y 2 9 A USE SALT Complete parts (a) through (e), given Ex = 202, Ey = 27, Ex = 7938, Ey = 159, Exy = 1071, and r 0.7844. %3D %3D %3D (a) Draw a scatter diagram displaying the data. 50 8 3 20 30 40 50 4 7 8. x (total number of jobs (in hundreds)) x (total number of jobs (in hundreds)) 50 8 40 30 2. 40 30 20 y (number of entry-level jobs (in hundreds)) of entry-level jobs (in hundreds)) 3. 4. y (number of entry-level jobs (in hundreds)) entry-level jobs (in hundreds)
A researcher wishes to investigate the association between two variables, denoted xi and yi. The following results have been obtained from the analysis of a sample of 30 observations: Sxi = 119.2, Syi = 129.2, Sxi yi = 480.9, xSi2 = 493.6 (xi −x)2 = 20.4, ei2 =444.2 Calculate the ordinary least squares (OLS) estimates of the coefficients of the linear regression, yi = β1 + β2xi +ui. What does your estimate of β2 in part (i) suggest about the association between xi and yi. Calculate the standard error of the regression, ˆ , and the standard error of the estimated slope coefficient, se(ˆ2) . Test the null hypothesis H0: 2=0 against the alternative hypothesis H1: 20, using a significance level of 0.05. On the basis of your hypothesis test in part (iv), is there any evidence of an association between xi and yi?

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