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

• 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

Nonlinear Relationships

Quadratic Regression Model

• where:β0 = Y interceptβ1 = regression coefficient for linear effect of X on Yβ2 = regression coefficient for quadratic effect of X on Yεi = random error in Y for observation i

Model form:

Linear fit does not give random residuals

Linear vs. Nonlinear Fit

Nonlinear fit gives random residuals

X

resi

dual

s

X

Y

X

resi

dual

s

Y

X

Quadratic Regression Model

Quadratic models may be considered when the scatter diagram takes on one of the following shapes:

X1

Y

X1X1

YYY

β1 < 0 β1 > 0 β1 < 0 β1 > 0

β1 = the coefficient of the linear term β2 = the coefficient of the squared term

X1

β2 > 0 β2 > 0 β2 < 0 β2 < 0

Testing the Overall Quadratic Model

• Test for Overall RelationshipH0: 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 =

• Estimate the quadratic model to obtain the regression equation:

Testing for Significance: Quadratic Effect

• t-testH0: β2 = 0

H1: β2 0

Example: Quadratic Model• Purity increases as filter time

increases:Purity

FilterTime

3 17 28 3

15 522 733 840 1054 1267 1370 1478 1585 1587 1699 17

Example: Quadratic Model (continued)

Regression Statistics

R Square 0.96888

Adjusted R Square 0.96628

Standard Error 6.15997

• Simple regression results:

Purity = -11.283 + 5.985 Time

  CoefficientsStandard

Error t Stat P-value

Intercept -11.28267 3.46805 -3.25332 0.00691

Time 5.98520 0.30966 19.32819 2.07E-10

FSignificance

F

373.5790 2.0778E-10

^

t statistic, F statistic, and r2 are all high, but the residuals are not random:

  CoefficientsStandard

Error t Stat P-value

Intercept 1.53870 2.24465 0.68550 0.50722

Time 1.56496 0.60179 2.60052 0.02467

Time-squared 0.24516 0.03258 7.52406 1.165E-5

Regression Statistics

R Square 0.99494Adjusted R Squar 0.99402Standard Error 2.59513

FSignificance

F

1080.733 2.368E-13

• Quadratic regression results:

Purity = 1.539 + 1.565 Time + 0.245 (Time)2^

Example: Quadratic Model(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

Regression StatisticsMultiple R 0.72213R Square 0.52148Adjusted R Squar 0.44172Standard Error 47.46341Observations 15

ANOVA   df SS MS F Significance FRegression 2 29460.027 14730.013 6.53861 0.01201Residual 12 27033.306 2252.776Total 14 56493.333      

  CoefficientsStandard

Error t Stat P-value Lower 95% Upper 95%Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888

52.1% of the variation in pie sales is explained by the variation in price and advertising

Multiple Coefficient of Determination(continued)

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

• 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

(continued)

Regression StatisticsMultiple R 0.72213R Square 0.52148

Adjusted R Squar 0.44172Standard Error 47.46341Observations 15

ANOVA   df SS MS F Significance FRegression 2 29460.027 14730.013 6.53861 0.01201Residual 12 27033.306 2252.776Total 14 56493.333      

  CoefficientsStandard

Error t Stat P-value Lower 95% Upper 95%Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888

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

(continued)Adjusted R2

• 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

Average Effect on Y of a change in X in Nonlinear Models

Example

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?!

http://xkcd.com/985/

Example

Example