Consider the following bivariate linear regression model...

Consider the following bivariate linear regression model that has no intercept: Y = ?1 * X + ?, and assume that we have collected a random sample of size N from the joint distribution of (X,Y), written {(Xi, Yi) : i = 1, ..., N}. (a) Write down the sum of squared distances (SSD) as a function of the data and the parameter to be estimated (see question 2!). (b) Show that the OLS estimator of ?1 solves the Normal equation ?_{i=1}^N ??_i * X_i = 0 where ??_i = Y_i ? ??1 * X_i is the fitted residual. Why is there only one Normal equation? (c) Using the result in (b), show that the OLS estimator of ?1 is ??1 = (?_{i=1}^N X_i * Y_i) / (?_{i=1}^N X_i^2). What is the main difference between this estimator and the estimator of ?1 for the bivariate linear regression model that includes the intercept ?0 (covered in class)? (d) Is it necessarily true that ?_{i=1}^N ??_i = 0? Explain!

Transcript

00:01 Hello students, today we will discuss about this question.

00:03 In this question, we need to consider the following bivariate linear regression model, that is y is equals to beta 1, x plus epsilon.

00:14 Now, and assume that we have collected a random sample of size capital n from the joint distribution of xy that return x i .y that is equals to where i that is equals to 1 to capital.

00:30 Now here in the part a we need to write down the sum of square distances that is ssd that is equals to question mark as a function of the data and the parameter to be estimated.

00:43 And then in the part b we need to show that ols estimator of beta 1 solves the normal equation that is ols estimator of beta 1 that solves the normal equation that is summation i is equals to 1 to n, ei hat xi that is equals to 0, where ei that is equals to yi minus beta 1 had multiplied by xi.

01:16 Now here in the part c using the result in part b, we need to show that ols estimator of beta 1, that is beta 1 hat that is equals to summation i is equal to 1 to 1 to n, xi, xi, y i divided by summation i is equals to 1 to n x i square and here in the part and here what is the main difference between the estimator and estimator of beta 1 so here we need to find the difference between estimator and estimator of beta 1 for the bivariate linear progression model and then in the part we need to find is it necessary true that whether it is true or not that i is equals to 1 to capital n epsilon i cap that is equal to zero whether it is true or not so here first of all for the part a here we can say that here that the sum of square distance that is also known as the sum of square due to the residual which is given by s s d that is equals to summation i is equals to 1 to capital n e i square so that is equals to summation i is equal to 1 to 1 to capital n y i that is equals to beta 1 cap y i minus beta 1 cap x i whole square so these will be the function of the data and parameter that to be estimated this will be the required answer for the part a now for the part b the least square estimator that are obtained by the differential calculate matter, so that is d divided by d beta, ssd, that is equals to 0.

03:14 So therefore, here we can write d divide by d beta, summation of i is equals to 1 to capital n, yi minus beta 1 cap xi whole square that is equals to 0.

03:26 So that implies that summation i is equals to 1 to n, ei cap xi that is equals to 0.

03:34 Since here summation i is equals to 1 to capital n, yi minus beta 1 cap xi that is equals to summation of ei cap.

03:46 Also only one normal equation because there is only one parameter is to be estimated here...

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