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Economics 105: Statistics

Economics 105: Statistics

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Economics 105: Statistics

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  1. Economics 105: Statistics Go over GH 21 due Wednesday GH 22 due Friday

  2. Nonlinear Relationships • The relationship between the outcome and the explanatory variable may not be linear • Make the scatterplot to examine • Example: Quadratic model • Example: Log transformations • Log always means natural log (ln) in economics

  3. Quadratic Regression Model Model form: • where: β0 = Y intercept β1= regression coefficient for linear effect of X on Y β2= regression coefficient for quadratic effect of X on Y εi = random error in Y for observation i

  4. Linear vs. Nonlinear Fit Y Y X X X X residuals residuals Linear fit does not give random residuals Nonlinear fit gives random residuals 

  5. Quadratic Regression Model Quadratic models may be considered when the scatter diagram takes on one of the following shapes: Y Y Y Y X1 X1 X1 X1 β1 < 0 β1 > 0 β1 < 0 β1 > 0 β2 > 0 β2 > 0 β2 < 0 β2 < 0 β1 = the coefficient of the linear term β2 = the coefficient of the squared term

  6. Testing the Overall Quadratic Model • Estimate the quadratic model to obtain the regression equation: • Test for Overall Relationship H0: β1 = β2 = 0 (X does not have a significant effect on Y) H1: β1 and/or β2 ≠ 0 (X does have a significant effect on Y) • F-test statistic =

  7. Testing for Significance: Quadratic Effect • t-test H0: β2 = 0 H1: β2 0

  8. Example: Quadratic Model • Purity increases as filter time increases:

  9. Example: Quadratic Model (continued) • Simple regression results: Purity = -11.283 + 5.985 Time ^ t statistic, F statistic, and r2 are all high, but the residuals are not random:

  10. Example: Quadratic Model • Quadratic regression results: • Purity = 1.539 + 1.565 Time + 0.245 (Time)2 (continued) ^ The quadratic term is significant and improves the model: r2 is higher and SYX is lower, residuals are now random

  11. Coefficient of Determination for Multiple Regression • Reports the proportion of total variation in Y explained by all X variables taken together • Consider this model

  12. Multiple Coefficient of Determination (continued) 52.1% of the variation in pie sales is explained by the variation in price and advertising

  13. Adjusted R2 • R2 never decreases when a new X variable is added to the model • disadvantage when comparing models • What is the net effect of adding a new variable? • We lose a degree of freedom when a new X variable is added • Did the new X variable add enough explanatory power to offset the loss of one degree of freedom?

  14. Adjusted R2 (continued) • Penalizes excessive use of unimportant variables • Smaller than r2 and can increase, decrease, or stay same • Useful in comparing among models, but don’t rely too heavily on it – use theory and statistical signif

  15. Adjusted R2 (continued) 44.2% of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables

  16. Average Effect on Y of a change in X in Nonlinear Models • Consider a change in X1 of ΔX1 • X2 is held constant! • Average effect on Y is difference in pop reg models • Estimate of this pop difference is

  17. Example

  18. Example • What is the average effect of an increase in Age from 30 to 40 years? 40 to 50 years? • 2.03*(40-30) - .02*(1600 – 900) = 20.3 – 14 = 6.3 • 2.03*(50-40) - .02*(2500 – 1600) = 20.3 – 18 = 2.3 • Units?!

  19. http://xkcd.com/985/

  20. Example

  21. Example