2. Let 3₁ and 3₂ denote the estimated regression coefficients for the linear model y = X₁3₁ + X₂³₂ + €₂where X₁ is an n x k₁ matrix, X₂ is an n x k₂ matrix, and ₁ and 3₂ are of dimensions k₁ and k₂,respectively.(a) Show that 3₁ = Ŝ₁ + (X₁X₁)−¹X₁ X232, where ₁ is a vector of coefficients from the regressionof y solely on X₁.(b) Show that B₁ is an unbiased estimator of 3₁.(c) Under what conditions ₁ is an unbiased estimator for $₁.Hint: Note that B₁ =B₁ - (X₁X₁)-¹X₁ X₂B₂ = (X₁X₁)-¹X₁(y - X₂8₂).

In this problem, we are examining a linear regression model with two sets of independent variables,…

Answered: 2. Let 3₁ and 3₂ denote the estimated…

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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Consider the model Y = ß0 + B₁X1 + ß₂X2 + B3X³ + B4X4 + €. If it is suggested to you that the two variables Z₁ = X₁ + X3 and Z2 = X₂ + X4 might be adequate to represent the data, what hypothesis, in the form C3 = 0, would you need to test? (Give the form of C.)
Find the required linear model using least-squares regression. The following table shows the number of operating federal credit unions in a certain country for several years. 2011 2012 2013 2014 4426 4255 4094 3919 Year Number of federal credit unions (a) The linear model for these data is y = (Round to the nearest tenth as needed.) (a) Find a linear model for these data with x = 11 corresponding to the year 2011. (b) Assuming the trend continues, estimate the number of federal credit unions in the year 2018. X + 2015 (b) The estimated number of credit unions for the year 2018 is (Round to the nearest integer as needed.) 3752
Consider the multiple regression model with three independent variables, under the classical linear model assumptions MLR.1 through MLR.6: y = Bo + Bixi + Bzx2 + Bztz + u. You would like to test the null hypothesis H,: B1 – 3B2 = 1. (i) Let Bj and B denote the OLS estimators of ß1 and B,. Find Var(ß – 3B,) in terms of the vari- ances of Bj and Bz and the covariance between them. What is the standard error of Bi – 3B? |(ii) Write the t statistic for testing Họ: B1 – 3B2 = 1. |(iii) Define 0, = Bi – 3B2 and ô, = B1 – 3B. Write a regression equation involving Bo. 01. B2- and Bz that allows you to directly obtain ô, and its standard error.
An economist estimates the following regression model:y = β0 + β1x1 + β2x2 + εThe estimates of the parameters b1 and b2 are not very large compared with their respective standard errors. But the size of the coefficient of determination indicates quite a strong relationship between the dependent variable and the pair of independent variables.Having obtained these results, the economist strongly suspects the presence of multicollinearity. Since his chief interest is in the influence of X1 on the dependent variable, he decides that he will avoid the problem of multicollinearity by regressing Y on X1 alone.Comment on this strategy.
Derive the least squares estimates of a and ß for the centred form of the simple linear regression model given by Yi = a + B(x; – I) + €; i= 1,2,..,n. Check that the estimates do give a minimum in the same way as we saw for the standard form of the simple linear regression model.
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.
2
In a study to predict temperature from air pressure in a piston-cylinder device, 19 measurements were made of temperature in °F (y) and air pressure in psi (x). Three models were fit: the linear model y = Bo + B,x + ɛ, the quadratic model y = Bo + Bix + B2x² + ɛ, and the cubic model y = Bo + B,x + B2x? + Bx3 + E. The residuals and fitted values for each model are presented in the following table. Plot the residuals versus the fitted values for each model. For each model, state whether the model is appropriate, and explain. %3D Quadratic Model Fit Linear Model Cubic Model Residual Fit Residual Residual Fit -56.2 125.6 11.2 58.2 3.3 66.1 -34.0 153.1 -7.4 126.5 -6.7 125.7 8.4 179.8 4.9 183.4 9.2 179.0 21.4 207.2 -3.6 232.2 0.9 227.8 28.6 234.7 -8.2 271.5 -5.9 269.2 46.9 260.9 8.1 299.7 7.3 300.5 47.2 288.1 15.7 319.6 12.1 323.2 8.5 314.4 -7.0 329.8 -11.7 334.6 -7.1 342.0 4.0 330.9 1.0 333.9 -47.1 139.3 -1.3 93.6 -4.1 96.4 -1.6 166.2 9.1 155.5 12.1 152.5 38.0 220.9 5.9 253.0 9.5 249.4…
3. Find the the equation of the least squares regression line for the data as the linear model f(x) = ao + a₁x in the manner discussed in the textbook using the formula A = (XTX)-¹XTY. What are your values for ao and a₁?
Compute the sum-of-squares error (SSE) by hand for the given set of data and linear model. (7, 7), (8, 8), (9, 10); SSE = y = x - 1

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