esponse Y on days X following Suppose an automatic data collector collects obser
ID: 3311247 • Letter: E
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esponse Y on days X following Suppose an automatic data collector collects observations of r a simple linear regression model E(YX = x)-Ao +82 with variance function var(YX = z) = 2, here x = 1, 2, , 10. Rather than collecting a single observation on day 1, it collects r observations on day r assuming the all observations are independent. Therefore the raw data collected contains n = 55 = 1 + 2 + 3 + + 10 pairs (zi,Vi), which is available in "data-data Collector-01.csv". Alternatively, we could look into the daily average y and the raw data values (zi,Vi), i = 1, , 55 can be reduced to 10 data values (rHz), 2-1, , 10, which is available in "data-data Collector-02.cs (a) (2 points) Please use the datasets available to show that the WLS estimators 30 and given data (zHz) wight weights ur-x, x = 1, , 10, are identical to the ordinary least squares (OLS) estimators Ao and given data (zi, yi), i = 1, ,55 Alternatively, you can also use the formulas to prove mathematically. Hint: You need to argue that the numerator in the expression above for B is the same as the numerator in a corresponding expression for B1, and then the same argue that the corresponding denominators are (b) (2 points) If we were to use sums of r values yi collected on day r rather than the averages (that is our data is (,r) rather than (x,j), r1,2,, 10), how would you need to adjust the weights for the WLS estimators? Please fit the model on the sum data to show your selection of weights are correctExplanation / Answer
a) &b)when the weight is assumed to be x , then mathematically the model becomes minimizing
summation[(y-b0-b1x)^2/x)] where x is assumed to be the weight but when the observations are reduced to y(mean)on a particular day, then the number of observations get reduced to 10 and thus the weight would be modified as x for y(mean) on day-x.
Both of the above models when differentiated to find the optimal values of b0 and b1, would indicate that the results are similar with OLS as can be seen by differentiation.
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