Dataset:Food ExpWkly Income52.25258.358.32343.181.79425119.9467.5125.8482.9100.46487.7121.51496.5100.08519.4127.75543.3104.94548.7107.48564.698.48588.3181.21591.3122.23607.3129.57611.292.84631117.92659.682.13664182.28704.2139.13704.898.14719.8123.94720126.31722.3146.47722.3115.98734.4207.23742.5119.8747.7151.33763.3169.51810.2108.03818.5168.9825.6227.11833.384.9483498.7918.1141.06918.1215.4929.6112.89951.7166.251014115.431141.3269.031154.6 Heteroskedasticity arises because of non-constant variance of the error terms. We said.proportional heteroskedasticity exists when the error variance takes the following structure:var (e,)=o² = 0²x₁1But as we know, that is only one of many forms of heteroskedasticity. To get rid of thatspecific form of heteroskedasticity using Generalized Least Squares, we employed a specificcorrection - we divided by the square root of our independent variable x. And the reason whythat specific correction worked, and yielded a variance of our GLS estimates that was sigma-squared, was because of the following math:- (+) == 1)e₁var(e) = var-var(e,) = -0²³x₁ = 0²x₂var(e) = 0²Whereaccording to our LS assumptions. In other words, dividing everything by thesquare root of x made this correction work to give us sigma squared at the end of the expression.But if we have a different form of heteroskedasticity (ie a difference variance structure), wehave to do a different correction to get rid of it.(i) var(e) = ²√√x₁(ii) var(e) ==o²x²1(a) what correction would you use if the form of heteroskedasticity you encountered wasassumed to be each of the following? Show mathematically (like the equation above does) whythe correction you are suggesting would work.1(b) using the household income/food expenditure data found the dataset below, use GLS toestimate our model employing the corrections (one model for each correction) you suggested inpart (a) above. Fully report your results. ¶(c) use the Goldfeldt-Quandt test to determine whether your corrections worked from theprevious question were successful. Be sure to carry out all parts of the hypothesis tests. Basedupon the results of your GQ tests, which of your two corrections do you think works better?¶

The linear regression model is valid when it satisfies few assumptions like Linearity, Normality,…

Answered: Heteroskedasticity arises because of…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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As what we learned, heteroskedasticity arises because of non-constant variance of the error terms. We said proportional heteroskedasticity exists when the error variance takes the following structure: var(et)=σt^2=σ^2xt But as we know, that is only one of many forms of heteroskedasticity. To get rid of that specific form of heteroskedasticity using Generalized Least Squares, we employed a specific correction – we divided by the square root of our independent variable x. And the reason why that specific correction worked, and yielded a variance of our GLS estimates that was sigma-squared, was because of the following math: var(et*)=var(et/(square root of xt))=1/xt * var(et)= 1/xt σ^2 xt= σ^2 Where var(et)=σ^2 according to our LS assumptions. In other words, dividing everything by the square root of x made this correction work to give us sigma squared at the end of the expression. But if we have a different form of heteroskedasticity (i.e. a difference variance structure), we have to do…
Suppose X and Y are independent random variables with the same mean µ but possibly different variances σx2 and σy2. Imagine that X and Y are the averages of independent samples of size n1 and n2, respectively, taken from populations with the same mean µ and the same variance σ2. Apply below to identify the α that minimizes and the minimum value. Be sure to simplify both the new estimator and the minimum variance. Interpret the result. The variance of αX+(1-α)Y, which also has mean µ, is minimized with α=σy2/(σX2+σy2).
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Let X and Y be random variables with variances Var(X) = 1 and Var(Y ) = 2. (Note that X and Y might not be independent.) What is the maximum possible value of Var(3X − 2Y + 4)?
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Food Exp	Wkly Income
52.25	258.3
58.32	343.1
81.79	425
119.9	467.5
125.8	482.9
100.46	487.7
121.51	496.5
100.08	519.4
127.75	543.3
104.94	548.7
107.48	564.6
98.48	588.3
181.21	591.3
122.23	607.3
129.57	611.2
92.84	631
117.92	659.6
82.13	664
182.28	704.2
139.13	704.8
98.14	719.8
123.94	720
126.31	722.3
146.47	722.3
115.98	734.4
207.23	742.5
119.8	747.7
151.33	763.3
169.51	810.2
108.03	818.5
168.9	825.6
227.11	833.3
84.94	834
98.7	918.1
141.06	918.1
215.4	929.6
112.89	951.7
166.25	1014
115.43	1141.3
269.03	1154.6