Solution Found!
A sample of 100 observations produces the following sample data: y1 = 1, y2 = 2, y 1y1 =
Chapter 14, Problem 1(choose chapter or problem)
A sample of 100 observations produces the following sample data:
\(\begin{aligned}
\bar{y}_{1} & =1, \quad \bar{y}_{2}=2, \\
\mathbf{y}_{1}^{\prime} \mathbf{y}_{1} & =150, \\
\mathbf{y}_{2}^{\prime} \mathbf{y}_{2} & =550, \\
\mathbf{y}_{1}^{\prime} \mathbf{y}_{2} & =260 .
\end{aligned}\)
The underlying bivariate regression model is
\(\begin{array}{l}
y_{1}=\mu+\varepsilon_{1}, \\
y_{2}=\mu+\varepsilon_{2} .
\end{array}\)
a. Compute the OLS estimate of \(\mu,\) and estimate the sampling variance of this estimator.
b. Compute the FGLS estimate of \(\mu\) and the sampling variance of the estimator.
Questions & Answers
QUESTION:
A sample of 100 observations produces the following sample data:
\(\begin{aligned}
\bar{y}_{1} & =1, \quad \bar{y}_{2}=2, \\
\mathbf{y}_{1}^{\prime} \mathbf{y}_{1} & =150, \\
\mathbf{y}_{2}^{\prime} \mathbf{y}_{2} & =550, \\
\mathbf{y}_{1}^{\prime} \mathbf{y}_{2} & =260 .
\end{aligned}\)
The underlying bivariate regression model is
\(\begin{array}{l}
y_{1}=\mu+\varepsilon_{1}, \\
y_{2}=\mu+\varepsilon_{2} .
\end{array}\)
a. Compute the OLS estimate of \(\mu,\) and estimate the sampling variance of this estimator.
b. Compute the FGLS estimate of \(\mu\) and the sampling variance of the estimator.
ANSWER:Step 1 of 3
The model can be written as :
\(\left[ {\begin{array}{*{20}{c}}{{y_1}}\\{{y_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}i\\L\end{array}} \right]\mu + \left[ {\begin{array}{*{20}{c}}{{ \in _1}}\\{{ \in _2}}\end{array}} \right]\)
The ols estimator can be”
\({\rm{m = (i'i + i'i}}{{\rm{)}}^{ - 1}}{\rm{(i'}}{{\rm{y}}_{\rm{1}}}{\rm{ + i'}}{{\rm{y}}_{\rm{2}}}{\rm{) }}\)
\({\rm{ = (n }}\overline {{y_1}} + n\overline {{y_2}} )/\left( {n + n} \right)\)
\(= \left( {\overline {{y_1}} + \overline {{y_2}} } \right)/2 = 1.5\)