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Economics 105: Statistics. Go over GH 21 due Wednesday GH 22 due Friday. 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
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Economics 105: Statistics Go over GH 21 due Wednesday GH 22 due Friday
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
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
Linear vs. Nonlinear Fit Y Y X X X X residuals residuals Linear fit does not give random residuals Nonlinear fit gives random residuals
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
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 =
Testing for Significance: Quadratic Effect • t-test H0: β2 = 0 H1: β2 0
Example: Quadratic Model • Purity increases as filter time increases:
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:
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
Coefficient of Determination for Multiple Regression • Reports the proportion of total variation in Y explained by all X variables taken together • Consider this model
Multiple Coefficient of Determination (continued) 52.1% of the variation in pie sales is explained by the variation in price and advertising
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?
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
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
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
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?!
