(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(2π)- $\frac { n } { 2 }$ log σ² - $\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i } \right) ^ { 2 }$ 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.

Question

(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(2π)-

\frac { n } { 2 }

log σ² -

\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i } \right) ^ { 2 }

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(2π)- $\frac { n } { 2 }$ log σ² - $\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i } \right) ^ { 2 }$ 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 n2\frac { n } { 2 }2n​

(Requires Appendix Material and Calculus)The Log of the Likelihood Function $\frac { n } { 2 }$