3. Consider the simple linear regression model: y = Bo+ Bi +, where the intercept term 30 is known. We assume that the errors have mean zero and unknown variance o². Further, the errors are assumed to be uncorrelated and independent of the predictor. We have n data points: (yi, xi) for i = 1, 2,..., n, and use the least-squares method to estimate 3₁ with the data. (a) Here, Bo is not a parameter to be estimated, as we know its value; for example, Bo is 5 or 20 or some other value. Consequently, the least-squares criterion is a function of 3₁. Write down the least-squares criterion, S(3₁). (b) Find a single least-squares normal equation. (c) Show that the solution to the normal equation is B₁ Ei=1 Yixi - Bo Σi-1Xi Σ i=1
3. Consider the simple linear regression model: y = Bo+ Bi +, where the intercept term 30 is known. We assume that the errors have mean zero and unknown variance o². Further, the errors are assumed to be uncorrelated and independent of the predictor. We have n data points: (yi, xi) for i = 1, 2,..., n, and use the least-squares method to estimate 3₁ with the data. (a) Here, Bo is not a parameter to be estimated, as we know its value; for example, Bo is 5 or 20 or some other value. Consequently, the least-squares criterion is a function of 3₁. Write down the least-squares criterion, S(3₁). (b) Find a single least-squares normal equation. (c) Show that the solution to the normal equation is B₁ Ei=1 Yixi - Bo Σi-1Xi Σ i=1
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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Problem 1P
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Transcribed Image Text:3. Consider the simple linear regression model:
y = Bo + Bi +,
where the intercept term Bo is known. We assume that the errors have mean zero and unknown variance o².
Further, the errors are assumed to be uncorrelated and independent of the predictor.
We have n data points: (yi, xi) for i = 1, 2, ..., n, and use the least-squares method to estimate 3₁ with the
data.
(a) Here, Bo is not a parameter to be estimated, as we know its value; for example, Bo is 5 or 20 or some
other value. Consequently, the least-squares criterion is a function of 3₁. Write down the least-squares
criterion, S(B₁).
(b) Find a single least-squares normal equation.
(c) Show that the solution to the normal equation is
3₁
=
Ei=1 Yixi - 30 i=1 Xi
Σ²=1x²
(d) Show that the expected value of 3₁ is E(Â₁) = ³₁.
02
Σ₁x²*
(e) Show that the variance of B₁ is Var (3₁)
=
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