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

question 17

Essay

(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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Definitions:

Marginal Utility

The additional satisfaction or utility received by a consumer from consuming one more unit of a good or service.

Total Utility

The sum satisfaction or benefit a consumer receives from consuming a particular quantity of a good or service.

Homogeneous Products

Homogeneous products are products that are considered identical by consumers and are indistinguishable in terms of quality, appearance, and performance.

Marginal Utility

The additional satisfaction or utility a consumer receives from consuming one more unit of a good or service.