(Requires Appendix material and Calculus)The log of the likelihood function (L)for the
simple regression model with i.i.d.normal errors is as follows (note that taking the
logarithm of the likelihood function simplifies maximization.It is a monotonic
transformation of the likelihood function, meaning that this transformation does not affect
the choice of maximum): $$L = - \frac { n } { 2 } \log ( 2 \pi ) - \frac { n } { 2 } \log \sigma ^ { 2 } - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i } \right) ^ { 2 }$$ X
Derive the maximum likelihood estimator for the slope and intercept.What general
properties do these estimators have? Explain intuitively why the OLS estimator is
identical to the maximum likelihood estimator here.

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The Answer of (Requires Appendix material and Calculus)The log of the likelihood function (L)for the
simple regression model with i.i.d.normal errors is as follows (note that taking the
logarithm of the likelihood function simplifies maximization.It is a monotonic
transformation of the likelihood function, meaning that this transformation does not affect
the choice of maximum): $$L = - \frac { n } { 2 } \log ( 2 \pi ) - \frac { n } { 2 } \log \sigma ^ { 2 } - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i } \right) ^ { 2 }$$ X
Derive the maximum likelihood estimator for the slope and intercept.What general
properties do these estimators have? Explain intuitively why the OLS estimator is
identical to the maximum likelihood estimator here.

(Requires Appendix Material and Calculus)The Log of the Likelihood Function