(Requires Appendix material and Calculus)The logarithm of the likelihood function (L)for estimating the population mean and variance for an i.i.d. normal sample 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(2πσ²)- $\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \mu _ { Y } \right) ^ { 2 }$ Derive the maximum likelihood estimator for the mean and the variance. How do they differ, if at all, from the OLS estimator? Given that the OLS estimators are unbiased, what can you say about the maximum likelihood estimators here? Is the estimator for the variance consistent?

Question

(Requires Appendix material and Calculus)The logarithm of the likelihood function (L)for estimating the population mean and variance for an i.i.d. normal sample 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(2πσ²)-

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

Derive the maximum likelihood estimator for the mean and the variance. How do they differ, if at all, from the OLS estimator? Given that the OLS estimators are unbiased, what can you say about the maximum likelihood estimators here? Is the estimator for the variance consistent?

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The Answer of (Requires Appendix material and Calculus)The logarithm of the likelihood function (L)for estimating the population mean and variance for an i.i.d. normal sample 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(2πσ²)- $\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \mu _ { Y } \right) ^ { 2 }$ Derive the maximum likelihood estimator for the mean and the variance. How do they differ, if at all, from the OLS estimator? Given that the OLS estimators are unbiased, what can you say about the maximum likelihood estimators here? Is the estimator for the variance consistent?

(Requires Appendix Material and Calculus)The Logarithm of the Likelihood Function n2\frac { n } { 2 }2n​

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