The question is at end of Merged document.jpg. Ordinary Least Square (OLS) derivation• Here is one way to fit the regression line• Model: yi = Bo + Biri + Ei• We want to minimize the square error, f(Bo, B1) = (yi – Bo – Biei)?• A qudratic function with 2 variables Bn and B1. Minimum achieved when the derivatives are zero.af= 221 T:(Bo + B1t; – yi) = 0• (E )B1 +(E2i)Bo = ti or ( r)B1+nXßo = E TiYiaf= 2E(Bo + B1T; – Yi) =0• (E Ti)B1 +nBo =EYi or nXB1 + nBo = nỸ• Thus%3D%3D• Notice that we use B, to denoted estimated parameters B;• Let y; = Bo + B12; be the predicted value of y;• MLE derivation• Likelihood function: The probability that the data your observations arise from a specific probability distribution defined by a specific setof parameters.More succinctly, it is the likelihood of the data (Y) given the specific predictor variables (X) and a mapping fuction (f()), including theparameters that describe the distribution of the data.Now that last part of the description of the likelihood is the important part. This is why we have the assumption that e; is normally distributed.Remember that the probability distribution function for a normal distribution is1f(x|4, 0) =ov2nNow if we assume that e; are normally distributed, X is non-random, Bo and B1 are parameters (fixed numbers), it follows that Y is normallydistributed (what is its mean and standard deviation?). Thus we can assume that the likelihood is the product (I) of all PDFS for Ys, which arerandom variables from a normal distribution.11 OV27i-1In plain English, this says that the likelihood is the aggregated probability of observing a particular value of y, given the parameters we want toestimate. In this case we want to maximize this function, such that the data has the highest probability of arising from a model with a specificset of values for Bo, B1, and o.In practice it is easier to take the log of this function, called the log likelihood function (logL), which makes the problem boil down to moresimple algebra.logL(ßo, ß1,0) = log I[p(34|27; Bo, B1, 0)i=1log p(y:i; Bo, B1,0)i=11log(27) – n log(o) –-(yi – (Bo + Bz;))²202i=1It is clear that for any o, the object function for Bs is the same as the OLS objective function.OLS estimators = MLE estimators!- Estimation of the variance• Can we estimate the variance o? of e:? One of the measurement is a response variable yi, i = 1,2,..., n.• We have 1 explanatory variables 2;, i = 1,2, ...,n• We assume a linear relation between them%3DYi = Bo + Biai +• E; are Normal disturbance terms, e.g., due to measurement error• e is the only source of the randomness that we care about. Since we are interested in p(Y X), we can assume X has an arbitrarydistribution or non-random.• Bo and Bi are key parameters to be estimated.• each disturbance e; has mean 0, and the same variance o?E(Y) = B1r + AoN(8,23 + Ba, o")N(3 aa + do, o)N(81n + 30, o)• E; are independent from each otherS0, i=jcov(ei, es) ={, itjlo², itjShow that the MLE estimate for o? is -E-(yi - Bo - Bixi).

Question

The question is at end of Merged document.jpg. Ordinary Least Square (OLS) derivation• Here is one way to fit the regression line• Model: yi = Bo + Biri + Ei• We want to minimize the square error, f(Bo, B1) = (yi – Bo – Biei)?• A qudratic function with 2 variables Bn and B1. Minimum achieved when the derivatives are zero.af= 221 T:(Bo + B1t; – yi) = 0• (E )B1 +(E2i)Bo = ti or ( r)B1+nXßo = E TiYiaf= 2E(Bo + B1T; – Yi) =0• (E Ti)B1 +nBo =EYi or nXB1 + nBo = nỸ• Thus%3D%3D• Notice that we use B, to denoted estimated parameters B;• Let y; = Bo + B12; be the predicted value of y;• MLE derivation• Likelihood function: The probability that the data your observations arise from a specific probability distribution defined by a specific setof parameters.More succinctly, it is the likelihood of the data (Y) given the specific predictor variables (X) and a mapping fuction (f()), including theparameters that describe the distribution of the data.Now that last part of the description of the likelihood is the important part. This is why we have the assumption that e; is normally distributed.Remember that the probability distribution function for a normal distribution is1f(x|4, 0) =ov2nNow if we assume that e; are normally distributed, X is non-random, Bo and B1 are parameters (fixed numbers), it follows that Y is normallydistributed (what is its mean and standard deviation?). Thus we can assume that the likelihood is the product (I) of all PDFS for Ys, which arerandom variables from a normal distribution.11 OV27i-1In plain English, this says that the likelihood is the aggregated probability of observing a particular value of y, given the parameters we want toestimate. In this case we want to maximize this function, such that the data has the highest probability of arising from a model with a specificset of values for Bo, B1, and o.In practice it is easier to take the log of this function, called the log likelihood function (logL), which makes the problem boil down to moresimple algebra.logL(ßo, ß1,0) = log I[p(34|27; Bo, B1, 0)i=1log p(y:i; Bo, B1,0)i=11log(27) – n log(o) –-(yi – (Bo + Bz;))²202i=1It is clear that for any o, the object function for Bs is the same as the OLS objective function.OLS estimators = MLE estimators!- Estimation of the variance• Can we estimate the variance o? of e:? One of the measurement is a response variable yi, i = 1,2,..., n.• We have 1 explanatory variables 2;, i = 1,2, ...,n• We assume a linear relation between them%3DYi = Bo + Biai +• E; are Normal disturbance terms, e.g., due to measurement error• e is the only source of the randomness that we care about. Since we are interested in p(Y X), we can assume X has an arbitrarydistribution or non-random.• Bo and Bi are key parameters to be estimated.• each disturbance e; has mean 0, and the same variance o?E(Y) = B1r + AoN(8,23 + Ba, o&#34;)N(3 aa + do, o)N(81n + 30, o)• E; are independent from each otherS0, i=jcov(ei, es) ={, itjlo², itjShow that the MLE estimate for o? is -E-(yi - Bo - Bixi).

Accepted Answer

Let, 
       yi=β0+β1xi+ui be the population regression line where yi are normally and independently…

Show that the MLE estimate for o? is -E(vi – Bo - Bixi). Li%31