$2$ . Given the likelihood function; $\newline$ $\text { Lf }=\frac{1}{\sigma^{n}(\sqrt{2} \bar{x})^{n}} \exp \left(-\frac{-1 / 2 \sum\left(Y_{i}-\beta_{1}-\beta_{2} x_{2 i}\right)^{2}}{\sigma^{2}}\right)$ $\newline$ (a) Derive the log likelihood function as a function of $\beta_{1}, \beta_{2}$ , and $\sigma^{2}$ . $\newline$ Scanned with CamScanner $\newline$ (b) Choose. $\beta_{1}, \beta_{2}$ , and $\sigma^{2}$ to maximize the log likefihood function. $\newline$ (c) Dexive the maximum liketihood enor variance. Is it the same as the ceast equare minimization error variance? If nit, state the condition under which the two error vasiances will be the same.

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2

. Given the likelihood function;

\newline

\text { Lf }=\frac{1}{\sigma^{n}(\sqrt{2} \bar{x})^{n}} \exp \left(-\frac{-1 / 2 \sum\left(Y_{i}-\beta_{1}-\beta_{2} x_{2 i}\right)^{2}}{\sigma^{2}}\right)

\newline

(a) Derive the log likelihood function as a function of

\beta_{1}, \beta_{2}

, and

\sigma^{2}

\newline

Scanned with CamScanner

\newline

(b) Choose.

\beta_{1}, \beta_{2}

, and

\sigma^{2}

to maximize the log likefihood function.

\newline

(c) Dexive the maximum liketihood enor variance. Is it the same as the ceast equare minimization error variance? If nit, state the condition under which the two error vasiances will be the same.

Take Natural Logarithm: To derive the log likelihood function, take the natural logarithm of the likelihood function ＂Lf＂. $\log(L_f) = \log\left(\frac{1}{\sigma^{n}(\sqrt{2\pi})^n}\right) - \frac{1}{2} \cdot \sum\left(\frac{(Y_i - \beta_1 - \beta_2 \cdot x_i)^2}{\sigma^2}\right)$
Simplify Log Likelihood: Simplify the log likelihood function by distributing the logarithm. $\newline$ $\log(L_f) = -n \cdot \log(\sigma) - n \cdot \log(\sqrt{2\pi}) - \frac{1}{2} \cdot \sum\left((Y_i - \beta_1 - \beta_2 \cdot x_i)^2\right)/(\sigma^2)$
Maximize Log Likelihood: To maximize the log likelihood function, take partial derivatives with respect to $\beta_1$ , $\beta_2$ , and $\sigma^2$ and set them to zero. $\newline$ $\frac{\partial\log(L_f)}{\partial\beta_1} = 0$ , $\frac{\partial\log(L_f)}{\partial\beta_2} = 0$ , $\frac{\partial\log(L_f)}{\partial\sigma^2} = 0$
Solve System of Equations: Solve the system of equations from the partial derivatives to find the values of $\beta_1$ , $\beta_2$ , and $\sigma^2$ that maximize the log likelihood function. $\newline$ This involves solving a system of equations which is not shown here.
Derive Error Variance: Derive the maximum likelihood error variance by finding the second derivative of the log likelihood function with respect to $\sigma^2$ and evaluating it at the maximum likelihood estimates. $\newline$ This step involves calculus which is not shown here.
Compare Variances: Compare the maximum likelihood error variance to the least squares minimization error variance. The comparison involves understanding the relationship between the two variances under certain conditions, which is not shown here.