Question: Answer all parts of this question Given

where \(\boldsymbol{y} \equiv \boldsymbol{X}\left[\begin{array}{c}b_{0} \\ b_{1}\end{array}\right]+\varepsilon\), we wish to choose \(b_{0}\) and \(b_{1}\) to minimise \(\varepsilon^{\prime} \varepsilon\).

1. Using sigma notation, express the minimand in terms of \(x_{t}, y_{t}, b_{0}\) and \(b_{1}\).
2. Partially differentiate the minimand with respect to each of the choice variables to derive its first order conditions.
3. Write the system of two first order conditions derived in part b, above, in matrix notation,
   \[M\left[\begin{array}{l} b_{0} \\ b_{1} \end{array}\right]=\boldsymbol{v}\]
   where \(M\) is a matrix and \(v\) a vector.
4. Solve system of equations 1 for \(\left(b_{0}, b_{1}\right)^{\prime}\). What must you assume to do so?
5. Calculate the following values in terms of \(T, \sum x_{t}, \sum x_{t}^{2}, \sum y_{t}\) and \(\sum x_{t} y_{t}:\) i. \(\boldsymbol{X}^{\prime} \boldsymbol{X}\)
   ii. \(\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1}\)
   iii. \(\boldsymbol{X}^{\prime} \boldsymbol{y}\)
   iv. \(\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{y}\)
6. Compare the result derived in part \(\mathrm{d}\) to that derived in subpart e iv.
7. Interpret the model you have been working with, and the calculations you have performed.

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