3. Denoting the residuals from the estimated regression equation by ü,, show that the sum of the erors is equal to zero: 0 Вх, + й, 4. Show that: û, where y, = Y,-Y and x, х, — х is equivalent to: Consider an estimated simple linear regression model: Y, = B, + BX,+ ü,. Show that the average predicted 5. value of the dependent variable, Y , is equal to the sample mean of the dependent variable, Y . That is, show: Σ-7 п Consider the estimated linear regression model: Y = Bo +B,X,û, 6. Write the expressions for the least squares estimators of the intercept and slope coefficients. Do not derive them. A. В. Suppose that the same dependent variable in (A) is modeled as a function of an alternative independent variable, denoted by Z. The estimated model is given by: Y, = B BZ, +uf, where the * distinguishes the estimates from those in (A). Write the expressions for the least squares estimators of the intercept and slope coefficients. С. Next, suppose the variable Z is a linear function of X. Specifically, let Z, = aX, where a is a constant. Considering your results from A and B, show how: DA and A are related in terms of the constant a and are related in terms of the constant a Scanned with CamScanner 3. Denoting the residuals from the estimated regression equation by ü,,show that the sum of the erors is equal to zero: 0 Во + вх, + й, 4. Show that: û, where y = Y,- Y and x, х, - х is equivalent to: Consider an estimated simple linear regression model: Y = B + BX+u. Show that the average predicted 5. value of the dependent variable, Y , is equal to the sample mean of the dependent variable, Y . That is, show: Σ-Υ п Consider the estimated linear regression model: Y = Bo + B,X, +û, 6. Write the expressions for the least squares estimators of the intercept and slope coefficients. Do not derive them. A. В. Suppose that the same dependent variable in (A) is modeled as a function of an alternative independent variable, denoted by Z. The estimated model is given by: Y, = B^ £Z, + uf, where the * distinguishes the estimates from those in (A). Write the expressions for the least squares estimators of the intercept and slope coefficients. С. Next, suppose the variable Z is a linear function of X. Specifically, let Z, = aX where a is a constant. Considering your results from A and B, show how: A and A i),and are related in terms of the constant a are related in terms of the constant a Scanned with CamScanner