3 Please provide complete correct detailed solution Thank yo

Consider a sample of size N that can be divided into G \"buckets\", where each bucket has the property that individuals in bucket g have the same values for a vector of covariates xi. In other words, xi = xg if ¬ g (recall the example from lecture where each bucket is a complete specification of age, education, and gender, for example). In addition, there is an outcome variable y, that varies among people even in the same bucket. Suppose the model of interest is and consider two ways of estimating 0. Method 1 is to fit an OLS model, using the individual data. Method 2 is to first estimate the mean of the outcome variable for group g, yg, where and Ng is the number of observations in group g, and then to fit a weighted OLS model using the group level data: yg = x\'g + vg. Give the formula for the OLS estimator beta from method 1. (This should involve sums that run over i = 1,2, ...N.) Give the formula for a weighted OLS estimator 6 from method 2 using a weight wg > 0 for group g. (This should involve sums that run over g = 1,2,...G) Using your answers from (a) and (b), show that if you use weights wg = Ng/N you will get the same answer from method 2 as from method 1.

Solution

a) OLS stands for Ordinary Least Squares or Linear regression model of fitting a straight line like

Y = aX + b where Y represents Dependent variable and varies its values based on values of another Independent variable X ( greek symbols of alpha, beta, gamma, delta etc are not available so used this equivalent equation of model 1 mentioned in the question) The value of beta is that of value \'a\' in the Y = aX + b

In order to have minimum value of sum of the squares of the deviations between actual and forecasted values, the following normal equations are used to find the values of \'a\' and \'b\'.

Sigma Y = a * sigma X + N * b and Sigma X * Y = a * Sigma (X*X) + b * Sigma X

Formula copied from Wikipedia is as follows

The other formulas are also available on various websites such as wikipedia.org,   www.stst.ncsu.edu and www.amstat.org

It is not possible to write the complicated expressions with limitations on this space to print the symbols used in the question. Any advaced textbook on Statistics and or Economertics may be used for the remaining part of the question.