Question

Consider the regression model $Y_i = eta X_i + u_i$ Where $u_i$ and $X_i$ satisfy the assumptions specified here. Let $ar{eta}$ denote an estimator of $eta$ that is constructed as $ar{eta} = frac{ar{Y}}{ar{X}}$, where $ar{Y}$ and $ar{X}$ are the sample means of $Y_i$ and $X_i$, respectively. Show that $ar{eta}$ is a linear function of $Y_1, Y_2, ..., Y_n$ $ar{eta} = frac{ar{Y}}{ar{X}} = frac{1}{n} frac{(Y_1 + Y_2 + ... + Y_n)}{ar{X}}$ Show that $ar{eta}$ is conditionally unbiased. 1. $E(Y_i | X_1, X_2, ..., X_n) = eta X_i$ 2. $E(ar{eta} | X_1, X_2, ..., X_n) = E left[ frac{1}{n} frac{(Y_1 + Y_2 + ... + Y_n)}{ar{X}} ight] | (X_1, X_2, ..., X_n) = frac{eta ar{X}}{ar{X}} = eta$

          Consider the regression model
$Y_i = eta X_i + u_i$
Where $u_i$ and $X_i$ satisfy the assumptions specified here. Let $ar{eta}$ denote an estimator of $eta$ that is constructed as $ar{eta} = frac{ar{Y}}{ar{X}}$, where $ar{Y}$ and $ar{X}$ are the sample means of $Y_i$ and $X_i$, respectively.
Show that $ar{eta}$ is a linear function of $Y_1, Y_2, ..., Y_n$
$ar{eta} = frac{ar{Y}}{ar{X}} = frac{1}{n} frac{(Y_1 + Y_2 + ... + Y_n)}{ar{X}}$
Show that $ar{eta}$ is conditionally unbiased.
1. $E(Y_i | X_1, X_2, ..., X_n) = eta X_i$
2. $E(ar{eta} | X_1, X_2, ..., X_n) = E left[ frac{1}{n} frac{(Y_1 + Y_2 + ... + Y_n)}{ar{X}} 
ight] | (X_1, X_2, ..., X_n) = frac{eta ar{X}}{ar{X}} = eta$

$Consider the regression model Yi = eta Xi + ui Where ui and Xi satisfy the assumptions specified here. Let areta denote an estimator of eta that is constructed as areta = fracarYarX, where arY and arX are the sample means of Yi and Xi, respectively. Show that areta is a linear function of Y1, Y2, ..., Yn areta = fracarYarX = frac1n frac(Y1 + Y2 + ... + Yn)arX Show that areta is conditionally unbiased. 1. E(Yi | X1, X2, ..., Xn) = eta Xi 2. E(areta | X1, X2, ..., Xn) = E left[ frac1n frac(Y1 + Y2 + ... + Yn)arX ight] | (X1, X2, ..., Xn) = fraceta arXarX = eta$

Added by Emily C.

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