Econometrics. 5) Dummy variables

Transcription

1 30C00200 Econometrics 5) Dummy variables Timo Kuosmanen Professor, Ph.D.

2 Today s topics Qualitative factors as explanatory variables Binary qualitative factors Dummy variables and their interpretation Categorical variables Dummy variable trap Modeling ordinal scale data by dummy variables Slope dummies and their interpretation F-test of joint significance

3 Quantitative variables Thus far, our regressors (x) have been quantitative, interval or ratio-scale variables Examples Size of apartment in m 2 Year of construction Number of bedrooms

4 Qualitative factors In many cases, the dependent variable is influenced by qualitative factors Examples: Is there elevator in the building? Yes / no Is there sauna in the apartment? Yes / no Is there a balcony in the apartment? Yes / no

5 Dummy variables The qualitative factors can be coded numerically by using binary {0,1} numbers Example: Elevator = {1 if there is elevator; 0 if not} Sauna = {1 if there is sauna; 0 if not} Balcony = {1 if there is balcony; 0 if not} Such binary variables can be directly inserted to the regression model as explanatory variables (x), referred to as dummy variables.

6 Interpretation of dummy variables Example: Hedonic model with Elevator dummy: Price = β 1 + β 2 Size + + δ Elevator + ε i δ is the effect of elevator on price For apartments without elevator, the intercept is β 1 For apartments with elevator, the intercept is β 1 + δ

7 Example: hedonic model with elevator dummy SUMMARY OUTPUT Regression Statistics Multiple R 0,90764 R Square 0,82381 Adjusted R Square 0, Standard Error 80551,73 Observations 67 ANOVA df SS MS F Significance F Regression 4 1,88E+12 4,7E+11 72, ,12E-22 Residual 62 4,02E+11 6,49E+09 Total 66 2,28E+12 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept , ,42 2, , , size m2 7023,05 858,5364 8, ,93E , ,24 nr. Bedrooms , ,87-3, , ,2 Age -2553,42 558,2156-4, ,34E , ,57 Elevator 24616, ,74 1, , , ,46

8 Categorical variables Some qualitative factors have more than two possible values Example: Districts in Espoo Tapiola Espoonlahti Kivenlahti Leppävaara Olari Such qualitative factors with many categories can be coded numerically by using multiple dummy variables: D1 = {1 if district=tapiola; 0 otherwise} D2 = {1 if district=espoonlahti; 0 otherwise} D3 = {1 if district=kivenlahti; 0 otherwise}

9 SUMMARY OUTPUT Regression Statistics Multiple R 0, R Square 0, Adjusted R Square 0, Standard Error 53206,42 Observations 642 Example: hedonic model with a larger sample ANOVA df SS MS F Significance F Regression 14 5,67E+12 4,05E ,0872 1,7E-184 Residual 627 1,77E+12 2,83E+09 Total 641 7,45E+12 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 85388, ,18 4, ,8E , Size m2 4137, , , ,52E , ,29 Bedrooms ,7 5323,126-6,1794 1,16E ,4 Age -2584,75 151, ,096 4,51E , ,85 Elevator 6216, ,772 1, , , ,94 1st floor -9137, ,723-1, , ,7 2187,183 Top floor 9576, ,709 1, , , ,19 Energy class A-C 33899, ,676 3, , , ,58 Condition satisfactory -2680, ,14-0, , , ,98 Condition good 18120, ,51 1, , , ,51 Loc. Espoonlahti , ,91-1, , ,5 6219,35 Loc. Kivenlahti ,2 8488,473-2, , ,5-2072,91 Loc. Leppävaara -4761, ,203-0, , ,7 7915,191 Loc. Olari -714, ,031-0, , , ,94 Loc. Tapiola ,6 7110,665 17, ,58E ,2

10 Categorical variables When modeling qualitative factors with q different categories, we need to introduce q-1 dummy variables, and leave one level as a reference category, represented by the common intercept β 1 Other effects are expressed relative to the reference category D1 = {1 if district=tapiola; 0 otherwise} D2 = {1 if district=espoonlahti; 0 otherwise} D3 = {1 if district=kivenlahti; 0 otherwise} Price = β 1 + β 2 Size + + δ 1 D1 + δ 2 D2 + δ 3 D3 + ε i Intercept β 1 represent the regression equation for other districts of Espoo Parameter δ 1 captures the effect of Tapiola district

11 Dummy variable trap What if we included separate dummy for all categories? Example: D1 = {1 if district=tapiola; 0 otherwise} D2 = {1 if district=espoonlahti; 0 otherwise} D3 = {1 if district=kivenlahti; 0 otherwise} D6 = {1 if district= other ; 0 otherwise} Price = β 1 + β 2 Size + + δ 1 D1 + δ 2 D2 + δ 3 D3 + + δ 6 D6 + ε i But then, the effect of other districts is β 1 + δ 6 We could freely increase β 1 if we simultaneously decrease all δ parameters. Hence, the model is unidentified: the problem is referred to as the dummy variable trap.

12 Ordinal variables We can similarly use dummy variables for modeling ordinal variables. Example: condition of apartment 1) weak 2) satisfactory 3) good Variables measured on ordinal scale can be modeled using dummy variables, taking one of the levels (e.g. weak) as the reference category.

13 SUMMARY OUTPUT Regression Statistics Multiple R 0, R Square 0, Adjusted R Square 0, Standard Error 53206,42 Observations 642 Example: hedonic model with a larger sample ANOVA df SS MS F Significance F Regression 14 5,67E+12 4,05E ,0872 1,7E-184 Residual 627 1,77E+12 2,83E+09 Total 641 7,45E+12 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 85388, ,18 4, ,8E , Size m2 4137, , , ,52E , ,29 Bedrooms ,7 5323,126-6,1794 1,16E ,4 Age -2584,75 151, ,096 4,51E , ,85 Elevator 6216, ,772 1, , , ,94 1st floor -9137, ,723-1, , ,7 2187,183 Top floor 9576, ,709 1, , , ,19 Energy class A-C 33899, ,676 3, , , ,58 Condition satisfactory -2680, ,14-0, , , ,98 Condition good 18120, ,51 1, , , ,51 Loc. Espoonlahti , ,91-1, , ,5 6219,35 Loc. Kivenlahti ,2 8488,473-2, , ,5-2072,91 Loc. Leppävaara -4761, ,203-0, , ,7 7915,191 Loc. Olari -714, ,031-0, , , ,94 Loc. Tapiola ,6 7110,665 17, ,58E ,2

14 Quantitative dummy variables? Sometimes quantitative variables are modelled as dummy variables as well. Example: number of bedrooms in the hedonic model We implictly assumed that the change from 1 room apartment to 2 room apartment has the same effect on price as the change from 2 rooms to 3 rooms, etc. To capture the differences, we could use dummy variables: Assume the reference category is 3 bedrooms or more D1 = {1 if apartment has 1 bedroom; 0 otherwise} D2 = {1 if apartment has 2 bedrooms; 0 otherwise}

15 SUMMARY OUTPUT Regression Statistics Multiple R 0, R Square 0, Adjusted R Square 0, Standard Error 53524,35 Observations 642 ANOVA df SS MS F Significance F Regression 15 5,65E+12 3,77E ,5378 5E-182 Residual 626 1,79E+12 2,86E+09 Total 641 7,45E+12 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 23887, ,41 1, , , ,68 Size m2 3560, , , ,27E , ,866 1Bedroom 61666, ,29 5, ,18E , ,16 2Bedroom 18828, , , , ,58 Age -2643,66 153, ,257 7,07E ,5-2342,83 Elevator 7326, ,629 1, , , ,46 1st floor ,9 5827,307-2, , ,3-765,43 Top floor 7664, ,457 1, , , ,48 Energy class A-C 36051, ,075 3, , , ,35 Condition satisfactory -1955, ,55-0, , , ,75 Condition good 21884, ,75 1, , , ,63 Loc. Espoonlahti , ,76-1, , ,7 2839,068 Loc. Kivenlahti ,9 8549,151-2, , ,3-2548,37 Loc. Leppävaara -2237, ,609-0, , , ,86 Loc. Olari -352, ,148-0,0437 0, , ,36 Loc. Tapiola ,8 7170,01 17, ,46E ,

16 Slope dummies Thus far, we have assumed the dummy variables shift the regression line via the intercept, but do not affect the slopes. Dummy variables can also be used for modeling the effect on the slopes of quantitative variables (slope dummies). Example: the number of bedrooms can influence the marginal value of size Price = β 1 + β 2 Size + β 3 D1 Size + β 4 D2 Size + + ε i D1 = {1 if apartment has 1 bedroom; 0 otherwise} D2 = {1 if apartment has 2 bedrooms; 0 otherwise}

17 SUMMARY OUTPUT Regression Statistics Multiple R 0, R Square 0, Adjusted R Square 0, Standard Error 53914,47 Observations 642 ANOVA df SS MS F Significance F Regression 15 5,63E+12 3,75E ,0393 4,6E-180 Residual 626 1,82E+12 2,91E+09 Total 641 7,45E+12 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 47765, ,87 2, , , ,99 Size m2 3279, , , ,02E , ,91 Size X 1Bedroom 1295, ,891 4, ,53E , ,825 Size x 2Bedroom 143, ,5412 1, , , ,9625 Age -2626,49 153, ,0579 7,27E , ,12 Elevator 7783, ,02 1, , , ,83 1st floor ,3 5868,643-1,9615 0, ,27122 Top floor 7934, ,085 1, , , ,71 Energy class A-C 37274, ,31 3, , , ,18 Condition satisfactory -1648, ,8-0, , , ,82 Condition good 21748, ,1 1, , , ,34 Loc. Espoonlahti , ,61-1, , ,2 2368,139 Loc. Kivenlahti ,4 8612,746-2, , ,7-1879,98 Loc. Leppävaara -2373, ,595-0, , , ,3 Loc. Olari -116, ,084-0, , , ,15 Loc. Tapiola ,1 7201,74 17, ,17E , ,6

18 Dummies or separate regressions? Consider a model of single regressor x and a single dummy D: y = β 1 + β 2 x + δ D + γ D x + ε where δ is the coefficient of intercept dummy and γ is the coefficient of slope dummy. The model could be equivalently estimated in parts: split the sample in two subsamples (D=0, D=1), and run two separate regressions: For D=0: For D=1 y = β 1 + β 2 x + ε y = β 3 + β 4 x + ε [Note: β 3 = β 1 + δ; β 4 = β 2 + γ]

19 Dummies or separate regressions? Partitioning subsamples (D=0, D=1) to separate regression models is easy to implement, especially if there are many regressors. Pooling the sub-samples (D=0, D=1) in the same model enables us to test for significance of coefficients δ and γ by t-test. We can use the F-test for testing joint significance of the parameters δ and γ (see the next lecture)

20 Qualitative factors as dependent variables It is also possible to model binary {0,1} variables and ordinary variables as dependent variables. For example, regress variable y = elevator dummy on explanatory variables x (e.g., age, number of floors, etc.) Such models can be estimated using Probit and Logit approaches (based on maximum likelihood).

21 F-test of joint significance Statistical significance of the parameters β 1, β 2 can be tested by using t-test (one- and two-sided tests) p-value (two-sided tests only) confidence intervals (two-sided tests, 95% significance) In the multiple regression model, it is possible that some of the regressors are significant while others are not It is useful to test if the model as a whole is significant

22 SUMMARY OUTPUT Regression Statistics Multiple R 0, R Square 0, Adjusted R Square 0, Standard Error 53206,42 Observations 642 ANOVA df SS MS F Significance F Regression 14 5,67E+12 4,05E ,0872 1,7E-184 Residual 627 1,77E+12 2,83E+09 Total 641 7,45E+12 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 85388, ,18 4, ,8E , Size m2 4137, , , ,52E , ,29 Bedrooms ,7 5323,126-6,1794 1,16E ,4 Age -2584,75 151, ,096 4,51E , ,85 Elevator 6216, ,772 1, , , ,94 1st floor -9137, ,723-1, , ,7 2187,183 Top floor 9576, ,709 1, , , ,19 Energy class A-C 33899, ,676 3, , , ,58 Condition Satisfactory -2680, ,14-0, , , ,98 Condition Good 18120, ,51 1, , , ,51 Loc. Espoonlahti , ,91-1, , ,5 6219,35 Loc. Kivenlahti ,2 8488,473-2, , ,5-2072,91 Loc. Leppävaara -4761, ,203-0, , ,7 7915,191 Loc. Olari -714, ,031-0, , , ,94 22 Loc. Tapiola ,6 7110,665 17, ,58E ,2

23 F-test of joint significance Multiple regression model: y = β 1 + β 2 x 2 + β 3 x β K x K + ε H 0 : β 2 = β 3 = = β K =0 (true model: y = β 1 + ε) H 1 : β k 0 for some k = 2,,K Note: in the special case of a single regression model: y = β 1 + β 2 x 2 + ε H 0 : β 2 = 0 H 1 : β 2 0

24 F-test of joint significance Empirical fit of the estimated model measured by the coefficient of determination (R 2 ) When H 0 (y = β 1 + ε) is true, R 2 is equal to zero. Is R 2 of the estimated model significantly greater than zero?

25 Test statistic: F-test of joint significance ESS K R K F RSS n K R n K 2 / ( 1) / ( 1) 2 / ( ) (1 ) / ( ) If H0 holds, then the test statistic follows the F-distribution with (K-1) and (n-k) degrees of freedom: F[(K-1), (n-k)] In Excel: =FINV(prob.; df1; df2)

26 F-test of joint significance Plots of the density function of F distribution at different values of df

27 F-test of joint significance Critical values of the F distribution at different sample sizes n, single regressor case, 5% significance level: n critical F 20 4, , , , , , , ,843

28 F-test of joint significance In the single regression case, we could test for the hypothesis H 0 : β 2 = 0 using the F-test or the t-test. Which one is more powerful? In fact, the two tests are equivalent (See 2.12 for details) F 2 R / ( K 1) ( b ) (1 R ) / ( n K) s. e.( b ) t 2 Also the critical values of F-distribution are obtained as the squared values of the critical t

29 Next time Wed 7 March Topic: Model specification