Chapter 9: The Regression Model with Qualitative Information: Binary Variables (Dummies)

Transcription

1 Chapter 9: The Regression Model with Qualitative Information: Binary Variables (Dummies) Statistics and Introduction to Econometrics M. Angeles Carnero Departamento de Fundamentos del Análisis Económico Year M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

2 Introduction The variables we have considered so far have been quantitative variables (wages, income, labour experience, years of education, etc.) The gender or race of an individual, the region where he lives, the industrial sector where a firm belongs, are examples of qualitative factors that often appear in empirical models. In this Chapter, we analyse the regression model with qualitative variables. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

3 The qualitative factors often appear offering binary information: The individual is either male or female an individual participates or not in a professional training programme a firm offers a retirement plan to its workers or not, etc. These qualitative variables can be represented with binary variables taking values 0 or 1 and are denoted as dummy variables. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

4 A unique binary variable We start analysing the following example: consider the dummy variable 1 if the individual is a female female = 0 if the individual is a male We include this variable in a model for hourly wage as a function of education as seen in Chapter 6 wage = β 0 + δ 0 female + β 1 educ + u (1) If in this model we compute the mean of wages for males and females with the same years of education, we have Females! E(wage j female = 1, educ) = β 0 + δ 0 + β 1 educ Males! E(wage j female = 0, educ) = β 0 + β 1 educ (2) M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

5 Therefore, δ 0 = E(wage j female = 1, educ) E(wage j female = 0, educ) ) δ 0 captures the difference in the average wage between females and males with the same years of education In this model, the value of δ 0 determines if there is difference in wages between males and females: If δ 0 < 0, we have that for the same level of education, the average wage of females is smaller than the average wage for males. On the contrary, if δ 0 = 0, we have that for the same level of education, the average wage of females is the same as for males. Therefore, in this model, the hypothesis that there are not differences in wages between males and females is δ 0 = 0. If δ 0 < 0, we have graphically M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

6 M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

7 We could have also included the dummy variable for male 1 if the individual is a male male = 0 if the individual is a female where the dummy for being male has been included instead of the dummy for female. In this case we have: wage = α 0 + γ 0 male + α 1 educ + u (3) If we compute now the mean wage for males and females with the same years of education we have that: Therefore Females! E(wage j male = 0, educ) = α 0 + α 1 educ Males! E(wage j male = 1, educ) = α 0 + γ 0 + α 1 educ γ 0 = E(wage j male = 1, educ) E(wage j male = 0, educ) ) γ 0 captures the difference in the average wage between males and females with the same years of education (4) M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

8 Since both in (2) and in (4) the mean wage is expressed as a linear function of the years of education but the only difference is that we allow for difference in the constant term between males and females, both models (1) and (3) are equivalent. Comparing (2) and (4) we have that β 0 + δ 0 + β 1 educ = α 0 + α 1 educ β 0 + β 1 educ = α 0 + γ 0 + α 1 educ and therefore γ 0 = δ 0. 8 < ) : β 1 = α 1 β 0 + δ 0 = α 0 β 0 = α 0 + γ 0 This relationship between the parameters of the models (1) and (3) is also verified for the OLS estimators of both models as the following example illustrates. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

9 Example 1 Using the data from file WAGE1 in Wooldridge on a sample of n = 526 individuals, models (1) and (3) have been estimated and the following results have been obtained: [wage = (0.6725) [wage = (0.6523) (0.2790) female educ (5) (0.0504) male educ (0.2790) (0.0504) where wage is the hourly wage in dollars and educ are the years of education We can check that in fact bβ 1 = bα 1 = bβ 0 + bδ 0 = = = bα 0 bα 0 + bγ 0 = = = bβ 0 According to this model, the hourly wage of males is on average dollars higher than the wage of females with the same years of education M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

10 Example 1 (cont.) If we want to test, versus a two-sided alternative, if the average wage of females is equal to the average wage of males with the same years of education, we have to test in model (1) The test statistic is H 0 : δ 0 = 0 H 1 : δ 0 < 0 t = b δ 0 se(bδ 0 ) t 523 under H 0 In this sample t = 2.273/ = 8.16 and the p-value is basically equal to zero. Therefore, we can reject H 0 at any reasonable significance level and conclude that the average wage of females is lower than the average wage for males with the same years of education. Note that alternatively we could have used model (3) and test H 0 : γ 0 = 0 vs H 1 : γ 0 > 0. We should have draw the same conclusion since the p-value is the same in both tests. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

11 Lastly, we can also write the model including both gender dummies (male and female) without including the constant term, that is, we could specify the model wage = β 0 male + α 0 female + β 1 educ + u (6) in this model β 0 is the constant term for males and α 0 is the constant term for females. Model (6) is equivalent to models (1) and (3). However, model (6) has the disadvantage with respect to the other two models that if we want to test whether there are differences in wages against females, we must test H 0 : β 0 = α 0 H 1 : β 0 > α 0 which is more difficult to test than H 0 : δ 0 = 0 vs H 0 : δ 0 < 0 in model (1), or to test H 0 : γ 0 = 0 vs H 0 : γ 0 > 0 in model (3). M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

12 Comments: We cannot specify a model including both dummies for gender and a constant term. The reason is that, since for each observation in the sample we have that male + female = 1, if both gender dummies and the constant term are included there will be perfect multicollinearity in the model (the model will be misspecified) We can include a binary variable in the model with more explanatory variables and the interpretation of the coefficient for this dummy variable would be analogous. For example, in the model wage = β 0 + δ 0 female + β 1 educ + β 2 exper + u (7) δ 0 captures the difference in the average hourly wage between females and males with the same years of education and the same years of labour experience. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

13 Comments (cont.): The interpretation of the coefficients of the rest of the variables is the same as in the case in which the model did not include dummy variables. For example, in model (7), β 1 captures the increase in the average wage given an increase in one year of education, holding labour experience and gender fixed, that is, we capture the effect either for males or for females. If the dependent is in logs, the coefficient of the dummy variable multiplied by 100 captures percentage differences. We see with an example how the estimated coefficient of a dummy variable should be interpreted when the dependent variable is in logs. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

14 Example 2 Consider the model log(wage) = β 0 + δ 0 female + β 1 educ + u (8) Using the same sample as in example 1, this model has been estimated obtaining the following results: \ log(wage) = (0.094) (0.039) female educ (9) (0.007) In this model, the estimated coefficient of female has the following interpretation: is the average estimated difference between the log wage of females and the log wage of males with the same years of education. Therefore, since the percentage differences is approximately equal to the difference of logs multiplied by 100, this model estimates that females earn on average 36.1% less than males with the same years of education. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

15 Binary variables with multiple categories Assume that we want to analyse now whether there are differences between different regions. The region of residence is also a qualitative factor, but as opposed to gender that can only take 2 values, countries are generally divided into more than 2 regions. Assume that we have J regions in a country, we have to consider as a reference region one of them, for example, region 1 and define J 1 dummy variables as follows: R 2 = R J = 1 if the individual lives in region 2 0 if the individual does not live in region 2. 1 if the individual lives in region J 0 if the individual does not live in region J M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

16 Once these dummy variables have been defined, we can consider model In this model log(wage) = β 0 + δ 2 R δ J R J + β 1 educ + u 100δ 2 captures the percentage difference in wages between those individuals living in region 2 and those living in region 1 with the same years of education δ J captures the percentage difference in wages between those individuals living in region J and those living in region 1 with the same years of education The hypothesis that there are not differences in wages between regions is δ 2 =... = δ J = 0. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

17 We could have used as a reference category any of the other regions For example, if we use region 2 as the reference category we should define the dummy corresponding to region 1 R 1 = 1 if the individual lives in region 1 0 if the individual does not live in region 1 and consider a model including the following dummy variablesr 1, R 3,.., R J. When we use region 2 as the reference category, the coefficients of the rest of dummies reflect the percentage differences in wages between the different regions and region 2. The model defined excluding any of the regional dummies is equivalent and the relationship between the parameters of both models would be obtained analogously to the case of the gender dummies. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

18 Example 3 Consider the model log(wage) = β 0 + δ 1 northcen + δ 2 south + δ 3 west + β 1 educ + u where the reference category is the east region. Using the same sample as in example 1, this model has been estimated obtaining the following results: log(wage) \ = (0.107) (0.061) northcen 0.060south (0.057) (0.067) west educ (10) (0.0076) n = 526, R 2 = 0.193, SCR = This model estimates that the hourly wage for individuals with the same years of education is on average 7% less in the north region than in the east region, 6% less in the south region than in the east region and 4.6% larger in the west region than in the east. In order to test, for example, if there are differences in wages between the west region and the east region, we must test H 0 : δ 3 = 0 versus H 1 : δ 3 6= 0. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

19 Example 3 (cont.) The test statistic is t = b δ 3 se(bδ 3 ) t 521 under H 0 In this sample t = 0.046/0.067 = 0.68 and the p-value = Prob(jt 521 j > 0.68) = Therefore, we cannot reject H 0 at any reasonable significance level. There is no evidence to state that, controlling for the years of education, there are wage differences between regions east and west. Note that we can also compute the estimated difference in wages for individuals with the same years of education living in any two regions. For example, comparing regions west and east, we have E(log(wage) j west = 1, educ) = β 0 + δ 3 + β 1 educ E(log(wage) j northcen = 1, educ) = β 0 + δ 1 + β 1 educ Since bδ 3 bδ 1 = ( 0.07) = 0.116, we have that the hourly wage for individuals with the same years of education is on average 11.6% larger in region west than in the north region. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

20 Example 3 (cont.) In order to test if the differences in wages between the west region and the north region are statistically significant we must test H 0 : δ 3 δ 1 = 0 versus H 1 : δ 3 δ 1 6= 0. Another possibility to do this test is to use the north region as the reference category, that is log(wage) = γ 0 + γ 1 east + γ 2 south + γ 3 west + β 1 educ + u To test in this model if the differences in wages between the west region and the north region are statistically significant we have to test H 0 : γ 3 = 0 versus H 1 : γ 3 6= 0. The estimated model is log(wage) \ = east south (0.106) (0.061) (0.055) (0.066) (0.0076) In this sample t = and the p-value is Therefore, we can only reject H 0 for levels above 8%. There is evidence to state that, controlling for the years of education, there are differences in wages between the regions west and east, but this evidence is not very strong. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

21 Example 3 (cont.) If we want to test if there are differences in wages between regions, we must test the null hypothesis H 0 : δ 1 = δ 2 = δ 3 = 0 In order to perform this test we must estimate the restricted model log(wage) = β 0 + β 1 educ + u and compute the sum of squared residuals, SCR r = The test statistic is F = (SCR r SCR nr ) /3 SCR nr /( ) F 3,521 under H 0 and since SCR nr = and SCR r = , we have that F = ( ) / /521 = 1.44 The p-value of this test is 0.23 and therefore there is no difference against the null hypothesis. We conclude that, controlling for the years of education, there is not enough evidence to state that there is difference in wages between regions M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

22 The effect of the education certificates on wages Consider the model log(wage) = β 0 + β 1 educ + β 2 exper + u We know that, in this model, 100β 1 captures the return to equation, i.e. the percentage variation in wages due to an increase in the number of years of education, by holding xed labour experience and the rest of the factors affecting y. This model implies that a one-year increase implies a percentage change that is the same when we increase education from 1 to 2, from 3 to 4, etc.. regardless of whether this additional year implies the end of a particular certificate such as Bachiller or University Degree. Additionally, in many data sets there is no information about the years of education but we have information about the maximum certificate each individual in the sample achieves. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

23 In order to take into account this "certificate" effect on wages, we try to define a series of dummy variables that capture the higher degrees achieved by individuals: 8 < 1 if individual did not finish certificate educational lev educ0 = : educ1 = 8 < : 0 otherwise 1 if individual finished certificate educational level 1 but he did not finish level 2 0 otherwise educj =. 8 < : 1 if individual finished certificate educational level J 0 otherwise M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

24 Once these variables are defined we consider the following model: log(wage) = β 0 + δ 1 educ δ J educj + β 1 exper + u (11) where the excluded variable is not having finished the first education level δ 1 captures the percentage difference in wages between individuals with educational level 1 and those that have not finished level 1 with the same labour experience, δ J captures the percentage difference in wages between individuals with educational level J and those that have not finished level 1 with the same labour experience. In this model, the hypothesis on the absence of wage differences among educational levels is δ 1 = δ 2 =... = δ J = 0. We could also write the model including dummy educ0 and excluding any of the other variables. The obtained model would be equivalent to model (11) and the equivalences among the parameters could be obtained analogously to example 1. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

25 Example 4 Using a subsample for year 2001 of 4450 Spanish workers in data set "European Community Household Panel", the following model has been estimated where log(salary) = β 0 + δ 1 bach + δ 2 univ + β 1 age + β 2 agesq + u salary is monthly wages in euros bach takes value 1 if individual has finished bachillerato but he does not have a degree and univ takes value 1 if individual has a university degree, the omitted category is not having finished high-school. age is age and agesq = age 2. The data for this sample (SalaryECHP2001) is available in folder "datos" from campus virtual. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

26 Example 4 The results of this estimation are: log(salary) \ = bach (0.088) (0.021) age agesq (0.0046) ( ) n = 4450, R 2 = (0.018) univ According to these results, the monthly wage for individuals with the same age is on average 15.9% larger for those individuals who finished bachillerato than for those individuals that did not finish it and 43.5% larger for those individuals who have a university degree than for those that did not even finish bachillerato. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

27 Models with many qualitative factors We can also construct models in which we include many qualitative factors at the same time. For example, we can consider civil status for which we also define two categories: married and not married, and we can define the following dummy variable and the model married = 1 if the individual is married 0 if the individual is not married log(wage) = β 0 + δ 0 female + γ 0 married + β 1 educ + u (12) where wage is wage. Note that in this model the reference category for the gender is male and for the civil status is not being married. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

28 If we compute in this model the mean of the log of wage for males and females, married and not married, with the same years of education we have: E(log(wage) j female = 0, married = 0) = β 0 + β 1 educ E(log(wage) j female = 1, married = 0) = β 0 + δ 0 + β 1 educ E(log(wage) j female = 0, married = 1) = β 0 + γ 0 + β 1 educ E(log(wage) j female = 1, married = 1) = β 0 + δ 0 + γ 0 + β 1 educ where these expectations are also conditional on educ. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

29 and therefore 8 >< ) >: δ 0 = E(log(wage) j female = 1, married = 0) E(log(wage) j female = 0, married = 0) = E(log(wage) j female = 1, married = 1) E(log(wage) j female = 0, married = 1) γ 0 = E(log(wage) j female = 0, married = 1) E(log(wage) j female = 0, married = 0) = E(log(wage) j female = 1, married = 1) E(log(wage) j female = 1, married = 0) 100δ 0 captures the percentage difference in wages between females and males with the same years of education and civil status 100γ 0 captures the percentage difference in wage between married and not married with the same years of education and same gender M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

30 We could also write the model using the dummy variable for male instead of female and for the dummy of not being married instead of the dummy for married. These models are equivalent and we could obtain the relationship between the parameters of both models analogously as in Example 1. Example 5 With the same sample as in example 1, the following model for the hourly wage has been estimated log(wage) \ = female married (0.0386) (0.0394) (0.0069) educ These results show that the wage for women is on average 32.8% smaller than wage for males with the same years of education and the same civil status, while wage for married is 20.9% larger than wage for those individuals who are not married with the same years of education and same gender. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

31 Interactions with binary variables Interactions between binary variables The model we analysed in the previous section does not allow for the differences in wages between males and females with a certain number of years of education to be different depending on the civil status of the individual. Thus, in the previous example the percentage difference in wage between married women and married male is 100δ 0, and for not married is also 100δ 0. In order to allow that the difference in wages between males and females can be different for different civil status, and for the differences between married and not married to be different between males and females, we should add to the model a variable which is the product between the dummy variable female and the dummy variable married. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

32 We define the variable female married = female married (note that this variable is a dummy that takes value 1 for married women and 0 otherwise) and we consider the model: log(wage) = β 0 + δ 0 female + γ 0 married (13) +φ 0 female married + β 1 educ + u In this model the mean of the log wages for males and females, married and not married, with the same years of education is: E(log(wage) j female = 0, married = 0) = β 0 + β 1 educ E(log(wage) j female = 1, married = 0) = β 0 + δ 0 + β 1 educ E(log(wage) j female = 0, married = 1) = β 0 + γ 0 + β 1 educ E(log(wage) j female = 1, married = 1) = β 0 + δ 0 + γ 0 + φ 0 + β 1 educ where the expectations are also conditioned on educ. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

33 and therefore δ 0 = E(log(wage) j female = 1, married = 0) E(log(wage) j female = 0, married = 0) δ 0 + φ 0 = E(log(wage) j female = 1, married = 1) E(log(wage) j female = 0, married = 1) γ 0 = E(log(wage) j female = 0, married = 1) E(log(wage) j female = 0, married = 0) γ 0 + φ 0 = E(log(wage) j female = 1, married = 1) E(log(wage) j female = 1, married = 0) M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

34 When we introduce the interaction between the two dummy variables we allow for the difference in wages between males and females with the same years of education to be different between married and not married. In this model the percentage difference in wages between females and males with the same years of education is 100δ 0 for not married individuals and 100(δ 0 + φ 0 ) if married. Also, the difference in wages between married and not married with the same years of education can also be different between males and females. In this model the percentage difference in wages between married and not married with the same years of education is 100γ 0 for males 100(γ 0 + φ 0 ) for females. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

35 Example 5 (cont) With the same sample of example 1 the following results have been obtained: log(wage) \ = female According to this results: (0.0602) (0.0554) married female married educ (0.0771) (0.0067) The wage of not married females is on average 10.3% smaller than the wages of not married males with the same years of education. The wage of married females is on average 47.3% (100 ( )) smaller than the wage of married males with the same years of education. The wage of married males is on average 40% larger than for unmarried males with the same years of education. The wage of married females is on average 3% (100 ( )) larger than for unmarried females with the same years of education. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

36 Example 5 (cont) In order to test in this model the null hypothesis that the difference in wages between males and females are the same for married and unmarried individuals, we have to test: H 0 : φ 0 = 0 H 1 : φ 0 6= 0 The test statistic under the normality assumption of the errors is t = b φ 0 se(bφ 0 ) t 521 under H 0 The value of the statistic in the sample is t = 0.37/ = 4.8 and the p-value is basically zero. Therefore, we can reject H 0 at any reasonable significance level and conclude that the difference in wages between males and females with the same years of education are different between married and unmarried individuals. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

37 How to allow different slopes So far we have been considering different examples on how to use dummy variables to allow for the constant terms of different groups to differ in the context of the regression model. In all these examples the slopes were the same for all the groups and therefore, the partial effects of the explanatory variables were the same for all the groups. In practice, there are situation in which we can think that the slopes of the model can be different for different groups. For example, there can be differences between males and females in their returns to education. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

38 In order to take into account those possible differences in the returns to education between males and females we can consider the following model: log(wage) = β 0 + δ 0 female + β 1 educ + δ 1 female educ + u (14) where female educ = female educ. In this model we note that for males (female = 0), the constant term if β 0 and the slope is β 1. While for females (female = 1), the constant term if β 0 + δ 0 and the slope is β 1 + δ 1. Therefore, δ 0 captures the difference in the intercept between females and males and δ 1 captures the difference in the slope between females and males, that is, the difference in the returns to education between males and females. If δ 0 < 0 and δ 1 < 0, graphically we have M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

39 M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

40 In order to test whether there are differences in the returns to education between males and females, we must test the null hypothesis H 0 : δ 1 = 0 in model (14). Note that the null hypothesis δ 1 = 0 does not impose any restriction in δ 0, so that under this hypothesis the constant term of the wage equation can be different for males and females. Therefore, this hypothesis allows for wage differences between females and males but these wage differences must be constant (in percentage terms) for all the educational levels. We could also test whether there are differences in wages between males and females with the same years of education. To do so, we must test the null hypothesis H 0 : δ 0 = δ 1 = 0 in model (14). M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

41 Example 6 Using again the same sample as in example 1, the model (14) has been obtained with the following results: log(wage) \ = (0.119) 0.360female educ (0.185) (0.009) female educ (0.0145) n = 526, R 2 = 0.300, SCR = According to these results, the return to education is on average 7.7% for males and basically the same for females. The estimated difference in the returns to education between males and females is on average %. This difference is negligible from the economic point of view and additionally is not statistically significant since the t-ratio is very small, t = / = M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

42 Example 6 (cont.) The coefficient of female is very large from the economic point of view, since this implies large differences in wages of 36% between males and females with the same years of education. However, the associated t-ratio is relatively small, t = 0.36/0.185 = 1.946, if this is compared with the results we obtained when the interaction term is not included (see equation (9)). The p-value of the one-sided test is Prob(t 522 < 1.946) = so that we could reject at 5% (but not at 1%) the hypothesis that H 0 : δ 0 = 0 versus H 1 : δ 0 < 0. Note that however the estimated coefficient for female is quite similar to the one we obtained when the interaction is not included. The difference is that the standard error is 5 times larger now than in equation (9), which implies that the statistical evidence against the null hypothesis must be weaker now than when the interaction was not included. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

43 Example 6 (cont.) The reason why the standard error is so large is because there is multicollinearity since the correlation between female and female educ is very high. Note that the interpretation of the estimated coefficient of female (with the opposite sign) as the constant percentage difference between males and females with the same years of education is correct in this example because the coefficient female educ is basically equal to zero. In general, if the model includes an interaction term between the dummy variable and one of the explanatory variables in the model, the dummy variable does not capture the difference in the dependent variable between the two groups with the same values of the explanatory variables. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

44 Example 6 (cont.) We test now if there are differences in wages between males and females with the same level of education, that it is, we test the null hypothesis H 0 : δ 0 = δ 1 = 0. To do this test we should estimate the restricted model log(wage) = β 0 + β 1 educ + u and compute the sum of squares residuals, SCR r = The test statistic is F = (SCR r SCR nr ) /2 SCR nr /( ) F 2,522 under H 0 and since SCR nr = and SCR r = , we have that F = ( ) / /522 = The p-value of the test is 0 and therefore there is strong evidence against the null hypothesis. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

45 Testing for differences in the regression functions between different groups Now, we will see how to test whether two populations or groups follow the same regression function versus the alternative hypothesis that at least one of the slopes is different between groups. Example 7 Suppose we want to test whether the same model describes college grade point averages (GPA) of male and female athletes. cumgpa = β 0 + β 1 sat + β 2 hsperc + β 3 tothrs + u (15) where cumgpa is the college grade point average over a scale of four points, hsperc is high school rank percentile (so that if hsperc = 5 the individual is among the best 5% of graduated students), sat is the result of a school test and tothrs is total hours of college courses. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

46 Example 7 (cont.) If we are interested in testing whether there are differences between males and females in this model, we must add one of the gender dummies, for example female, and the interactions of this dummy with all the variables in the model in order to allow for the constant term and all the slope to differ between groups: cumgpa = β 0 + δ 0 female + β 1 sat + δ 1 female sat + β 2 hsperc (16) +δ 2 female hsperc + β 3 tothrs + δ 3 female tothrs + u Parameter δ 0 is the difference of the constant term between females and males, δ 1, is the difference between females and males in the effect of sat on cumgpa, and so on The null hypothesis that cumgpa follows the same model for males and females is H 0 : δ 0 = δ 1 = δ 2 = δ 3 = 0 M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

47 Example 7 (cont.) Using the data of the spring term in file GPA3 of Wooldridge, the following results have been obtained: cumgpa = 1.48 (0.21) (0.0014) (0.411) female sat female sat ( ) ( ) hsperc female hsperc (0.0032) female tothrs (0.0016) n = 366, R 2 = 0.406, SCR = ( ) tothrs Neither female nor any of its interactions are individually significant at 5%. In fact, at 10% only the interaction with sat is significant. However, we know that this does not mean that there is no evidence to reject the null hypothesis δ 0 = δ 1 = δ 2 = δ 3 = 0. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

48 Example 7 (cont.) To test this hypothesis we have to estimate the restricted model cumgpa = β 0 + β 1 sat + β 2 hsperc + β 3 tothrs + u and compute the sum of squared residuals, SCR r = The test statistic is F = (SCR r SCR nr ) /4 SCR nr /( ) F 4,358 under H 0 and since SCR nr = and SCR r = , we have that F = ( ) / /358 = 8.18 The p-value is basically equal to 0 and therefore there is strong evidence against the null hypothesis. The model for GPA is different for males and females despite the fact that neither female nor any of its interactions are individually significant at 5%. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

49 In a model like the one above with only three explanatory variables, it is quite easy to add all the interaction terms for the differences between groups. However, in other examples, with many explanatory variables, it is convenient to compute the F statistic in a different way. The test statistic F can be easily computed without the need of estimating the model with all the interaction terms. Consider a general model with k explanatory variables and a constant, and assume that we have two groups that are denoted with g = 1 and g = 2, and that we want to test if the constant term and all the slopes are the same for all the groups. We write the model as for g = 1 and g = 2. y = β g,0 + β g,1 x β g,k x k + u (17) The hypothesis that the parameters of the model are the same for the two groups is equivalent to k + 1 linear restrictions. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

50 The unrestricted model includes 2(k + 1) variables: the constant term, one of the group dummies, the k explanatory variables and the k interactions of the group dummy variable with the explanatory variables. It can be shown that the SSR of the unrestricted model can be computed as SSR nr = SSR 1 + SSR 2, where SSR 1 is the SSR of model (17) estimated with the observations of group 1 and SSR 2 is the SSR of model (17) estimated with the observations of group 2. Therefore the test statistic is F = (SCR r (SCR 1 + SCR 2 )) /(k + 1) (SCR 1 + SCR 2 ) /(n 2(k + 1)) This statistic is denoted as the Chow statistic. F k+1,n 2(k+1) under H 0 (18) M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

51 Example 7 (cont.) If in Example 7 we estimate model (17) using the data for 90 women in the sample, we have that SSR 1 = and using data for 276 males we have that SSR 2 = Therefore SSR 1 + SSR 2 = = , which coincides with the SSR of the unrestricted model which has been computed above. An important limitation of the Chow test is that the null hypothesis does not allow for any differences between groups. In many cases, it is more interesting to allow for differences in the constant term and test whether there are differences in the slopes. In this case, the restricted model includes a dummy for the group and the unrestricted model a dummy for the group and all the possible interactions. SSR nr can be computed, as in the previous test, separately estimating the model with two subsamples and computing the sum of the SSR. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

52 Example 7 (cont.) In example 7, in order to test if there are not differences in the slopes, allowing for differences in the constant term, we should test in model (16). The restricted model is H 0 : δ 1 = δ 2 = δ 3 = 0 cumgpa = β 0 + δ 0 female + β 1 sat + β 2 hsperc + β 3 tothrs + u (19) and the SSR of this model (estimated using the 366 observations) is SSR r = The test statistic is F = (SSR r SSR nr ) /3 SSR nr /( ) F 3,358 under H 0 and since SSR nr = and SSR r = , we have that F = ( ) / /358 = 1.53 The p-value of the test is and therefore there is not evidence against the null hypothesis. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53

53 Example 7 (cont.) Not rejecting the null hypothesis that there are no differences in the slopes suggests that the best model is the one where there are only differences in the constant term, that is, model (19). The results of the estimation of this model are: cumgpa = female sat + (0.180) (0.059) ( ) hsperc tothrs (0.0012) ( ) n = 366, R 2 = female is now quite significant and its estimated coefficient implies that, for given values of sat, hsperc and tothrs, the GPA of females is on average 0.31 points larger than the GPA for males, which means an important difference. M. Angeles Carnero (UA) Chapter 9: Dummies Year / 53