Interpretation of Coefficients y=b+bX1+b2Xx+…+bXk+ Prediction:y=b+b水X1+b2x2+…+bXk 1. Slope(b a Estimated y changes by b, for each 1 unit increase in Xi, holding other variables constant y*+Ay=b+bx1+…+b(x+1)+…+bk More generally +4y=b+bx1+…+bx+4x)+…+b×k Ay= b AyAx= b 2. Y-Intercept(bo) ■ Estimated value of y when x1=Ⅹ2=….=Ⅹ Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-16
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-16 yi = b0 + b1x1i + b2x2i + … + bkxki + ei Prediction: y* = b0 + b1x1 + b2x2 + … + bkxk 1. Slope (bj ) ◼ Estimated Y changes by bj for each 1 unit increase in Xj, , holding other variables constant y* + Dy= b0 + b1x1 + …+ bj (xj+1)+… + bkxk Dy= bj More generally, y* + Dy= b0 + b1x1 + …+ bj (xj+Dxj )+… + bkxk Dy= bjDxj Dy/Dx = b1 2. Y-Intercept (b0 ) ◼ Estimated value of Y when X1 = X2 = … = Xk = 0 Interpretation of Coefficients
Parameter Estimation Example You work in advertising for the You've collected the New york times. you want to find the effect of ad size(sq in ) following data: newspaper circulation(000)on the Resp Size Circ number of ad responses(00) 41324 8356 281746 10 X X2 Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-17
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-17 ◼You work in advertising for the New York Times. You want to find the effect of ad size (sq. in.) & newspaper circulation (000) on the number of ad responses (00). You’ve collected the following data: Resp Size Circ 1 1 2 4 8 8 1 3 1 3 5 7 2 6 4 4 10 6 Parameter Estimation Example y x1 x2
Parameter Estimation Computer Output Parameter es七 imates Parameter standard t for ho Variab1 e DE es七 imate Error Param=0 Prob>IT I 工 NTERCEP 0.06400.25990.246 0.8214 ADs工ZE 1(0.20490.05883.656 0.0399 CIRC 1(0.28050.06864.089 0.0264 Slope(b): Responses to Ad is expected to increase by 2049(20.49) for each 1 sq in increase in Ad size Holding circulation Constant Slope(b,):# Responses to Ad is expected to increase by 2805(28.05 for each 1 unit(1,000) increase in circulation Holding ad Size Constant Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-18
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-18 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Param=0 Prob>|T| INTERCEP 1 0.0640 0.2599 0.246 0.8214 ADSIZE 1 0.2049 0.0588 3.656 0.0399 CIRC 1 0.2805 0.0686 4.089 0.0264 Parameter Estimation Computer Output ◼ Slope (b1 ): # Responses to Ad is expected to increase by .2049 (20.49) for each 1 sq. in. increase in Ad Size Holding Circulation Constant ◼ Slope (b2 ): # Responses to Ad is expected to increase by .2805 (28.05) for each 1 unit (1,000) increase in circulation Holding Ad Size Constant
Interpreting the Standard Error of the Estimate Assumptions a Observed y values are normally distributed around each estimated value of y ■ Constant variance se measures the dispersion of the points around the regression line ■Ifs=0, equation is a“ perfect" estimator se may be used to compute confidence intervals of the estimated value Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-19
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-19 ◼ Assumptions: ◼ Observed Y values are normally distributed around each estimated value of Y* ◼ Constant variance ◼ se measures the dispersion of the points around the regression line ◼ If se = 0, equation is a “perfect” estimator ◼ se may be used to compute confidence intervals of the estimated value Interpreting the Standard Error of the Estimate
Test of Slope Coefficient (bj) 1. Tests if there is a linear relationship between X& y after other variables are controlled for 2. Involves population slope B 3. Hypotheses Ho: Bi =0(Xi should not appear in the linear relationship) ■H2:B,≠0 4. Theoretical basis is sampling distribution of slopes Ka-fu Wong C2007 ECON1003: Analysis of Economic Data Lesson 11-20
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson11-20 1. Tests if there is a linear relationship between Xj & Y after other variables are controlled for. 2. Involves population slope bj 3. Hypotheses ◼ H0 : bj= 0 (Xj should not appear in the linear relationship) ◼ H1 : bj 0 4. Theoretical basis is sampling distribution of slopes Test of Slope Coefficient (bj)