Exploring Nonlinear Relationships in Econometrics
Discover the complexities of nonlinear relationships through polynomials, dummy variables, and interactions between continuous variables in econometrics. Delve into cost and product curves, average and marginal cost curves, and their implications in economic analysis. Understand the application of dummy variables in modeling wages, property prices, and more. Uncover the significance of intercept dummy variables and the effect of characteristics on pricing. Gain insights into estimating wages, property values, and economic trends using advanced econometric principles.
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7.1 Polynomials 7.2 Dummy Variables 7.3 Applying Dummy Variables 7.4 Interactions Between Continuous Variables Principles of Econometrics, 3rd Edition Slide 7-2
7.1.1 Cost and Product Curves Figure 7.1 (a) Total cost curve and (b) total product curve Slide 7-3 Principles of Econometrics, 3rd Edition
Figure 7.2 Average and marginal (a) cost curves and (b) product curves Slide 7-4 Principles of Econometrics, 3rd Edition
= + + + 2 AC Q Q e (7.1) 1 2 3 = + + + + 2 3 TC Q Q Q e (7.2) 1 2 3 4 ( dQ ) dE AC = + 2 Q (7.3) 2 3 ( ) dE TC dQ = + + 2 2 3 Q Q (7.4) 2 3 4 Principles of Econometrics, 3rd Edition Slide 7-5
= + + + 2 WAGE EDUC EXPER+ EXPER e (7.5) 1 2 3 4 E WAGE ( EXPER ) = + 2 EXPER (7.6) 3 4 Principles of Econometrics, 3rd Edition Slide 7-6
Principles of Econometrics, 3rd Edition Slide 7-7
E WAGE ( EXPER ) = .3409 2( .0051)18 + = .1576 18 = EXPER EXPER = .3409/ 2( .0051) = = 2 33.47 3 4 Principles of Econometrics, 3rd Edition Slide 7-8
= + + PRICE SQFT e (7.7) 1 2 1 0 if characteristic is present if characteristic is not present = D (7.8) 1 0 if property is in the desirable neighborhood if property is not in the desirable neighborhood = D Principles of Econometrics, 3rd Edition Slide 7-9
= + + + PRICE D SQFT e (7.9) 1 2 + + = + = = ( ) when when 1 0 SQFT D D 1 2 ( ) E PRICE (7.10) SQFT 1 2 Principles of Econometrics, 3rd Edition Slide 7-10
Figure 7.3 An intercept dummy variable Principles of Econometrics, 3rd Edition Slide 7-11
1 0 if property is not in the desirable neighborhood if property is in the desirable neighborhood = LD = + + + PRICE LD SQFT e 1 2 = + + + + PRICE D LD SQFT e 1 2 Principles of Econometrics, 3rd Edition Slide 7-12
= + + + ( ) PRICE SQFT SQFT D e (7.11) 1 2 + + = + = = ( ) when when 1 0 SQFT D D ( ) 1 2 = + + ( ) E PRICE SQFT SQFT D 1 2 SQFT 1 2 + = = when when 1 0 D D ( ) E PRICE SQFT 2 = 2 Principles of Econometrics, 3rd Edition Slide 7-13
Figure 7.4 (a) A slope dummy variable. (b) A slope and intercept dummy variable Principles of Econometrics, 3rd Edition Slide 7-14
= + + + + ( ) PRICE D SQFT SQFT D e (7.12) 1 2 + + + = + = = ( ) ( SQFT ) when when 1 0 SQFT D D 1 2 ( ) E PRICE 1 2 Principles of Econometrics, 3rd Edition Slide 7-15
Principles of Econometrics, 3rd Edition Slide 7-16
( ) = + + + SQFT UTOWN PRICE UTOWN SQFT 1 1 2 (7.13) + + + + AGE POOL FPLACE e 3 2 3 Principles of Econometrics, 3rd Edition Slide 7-17
Principles of Econometrics, 3rd Edition Slide 7-18
= + + + (24.5 27.453) (7.6122 1.2994) 4.3772 1.6492 POOL + + .1901 PRICE SQFT AGE FPLACE = + + 51.953+8.9116 .1901 4.3772 1.6492 SQFT AGE POOL FPLACE = 24.5 7.6122 + + + .1901 4.3772 1.6492 PRICE SQFT AGE POOL FPLACE Principles of Econometrics, 3rd Edition Slide 7-19
Based on these regression results, we estimate the location premium, for lots near the university, to be $27,453 the price per square foot to be $89.12 for houses near the university, and $76.12 for houses in other areas. that houses depreciate $190.10 per year that a pool increases the value of a home by $4377.20 that a fireplace increases the value of a home by $1649.20 Principles of Econometrics, 3rd Edition Slide 7-20
7.3.1 Interactions Between Qualitative Factors ( ) = + + + + + WAGE EDUC BLACK FEMALE BLACK FEMALE e (7.14) 1 2 1 2 + + + = + + + + + EDUC WHITE BLACK WHITE BLACK MALE MALE FEMALE FEMALE 1 2 ( ( ( ) ) EDUC EDUC 1 1 2 ( ) E WAGE + 1 2 2 ) EDUC 1 1 2 2 Principles of Econometrics, 3rd Edition Slide 7-21
Principles of Econometrics, 3rd Edition Slide 7-22
( )/ K SSE SSE SSE N J = F R U /( ) U 4.9122 1.1385 = + WAGE EDUC (se) (.9668) (.0716) ( )/ K (31093 29308)/3 29308/995 SSE SSE SSE N J = = = 20.20 F R U /( ) U Principles of Econometrics, 3rd Edition Slide 7-23
= + EDUC+ SOUTH + + + WAGE MIDWEST WEST e (7.15) 1 2 1 2 3 ( ( ( ) ) ) + + = + + + + + EDUC EDUC EDUC WEST MIDWEST SOUTH NORTHEAST 1 3 2 1 2 2 ( ) E WAGE 1 1 EDUC 2 1 2 Principles of Econometrics, 3rd Edition Slide 7-24
Principles of Econometrics, 3rd Edition Slide 7-25
= + + + + ( ) PRICE D SQFT SQFT D e 1 2 + = = 1 0 SQFT D D ( ) 1 2 SQFT = + E PRICE 1 2 ( ) = + + + + + WAGE EDUC BLACK FEMALE BLACK FEMALE e 1 2 1 2 Principles of Econometrics, 3rd Edition Slide 7-26
( ) = + + + + + WAGE EDUC BLACK FEMALE BLACK FEMALE 1 2 1 2 ( ) ( ) + EDUC SOUTH + + SOUTH BLACK SOUTH (7.16) 1 2 3 ( ) ( ) + + FEMALE SOUTH BLACK FEMALE SOUTH e 4 5 + = + + + + + + + EDUC BLACK FEMALE 1 2 1 2 = 0 SOUTH ( ) BLACK FEMALE ( ) E WAGE + + + = ( ( ) ( ) FEMALE ) ( ) 1 EDUC + + BLACK FEMALE SOUTH 1 1 2 2 1 3 ( ) ( ) BLACK 2 4 5 Principles of Econometrics, 3rd Edition Slide 7-27
Principles of Econometrics, 3rd Edition Slide 7-28
= = = = = : 0 H 0 1 2 3 4 5 ( )/ K (29307.7 29012.7)/5 29012.7/990 SSE SSE SSE N J = = = 2.0132 F R U /( ) U Principles of Econometrics, 3rd Edition Slide 7-29
Remark: The usual F-test of a joint hypothesis relies on the assumptions MR1-MR6 of the linear regression model. Of particular relevance for testing the equivalence of two regressions is assumption MR3, that the variance of the error term is the same for all observations. If we are considering possibly different slopes and intercepts for parts of the data, it might also be true that the error variances are different in the two parts of the data. In such a case the usual F-test is not valid. Principles of Econometrics, 3rd Edition Slide 7-30
7.3.4a Seasonal Dummies 7.3.4b Annual Dummies 7.3.4c Regime Effects 1 0 1962 1965,1970 1986 otherwise = ITC = + + + + INV ITC GNP GNP e 1 2 3 1 t t t t t Slide 7-31 Principles of Econometrics, 3rd Edition
Principles of Econometrics, 3rd Edition Slide 7-32
= + + + PIZZA AGE INCOME e (7.17) 1 2 3 = + + + PIZZA AGE INCOME e 1 2 3 E PIZZA INCOME = ( ) 3 = + 342.88 7.58 .0024 PIZZA AGE INCOME ( ) ( 3.27) (3.95) t Principles of Econometrics, 3rd Edition Slide 7-33
= + + + AGE INCOME + 4( ) (7.18) PIZZA AGE INCOME e 1 2 3 E PIZZA AGE = + ( ) INCOME 2 4 E PIZZA INCOME = + ( ) AGE 3 4 = + AGE INCOME 161.47 2.98 .009 .00016( ) PIZZA AGE INCOME ( ) ( .89) (2.47) ( 1.85) t Principles of Econometrics, 3rd Edition Slide 7-34
