5 Multiple regression analysis with qualitative information

Transcription

1 5 Multple regresson analyss wth qualtatve nformaton Ezequel Urel Unversty of Valenca Verson: Introducton of qualtatve nformaton n econometrc models A sngle dummy ndependent varable 5.3 Multple categores for an attrbute Several attrbutes Interactons nvolvng dummy varables Interactons between two dummy varables Interactons between a dummy varable and a quanttatve varable Testng structural changes Usng dummy varables Usng separate regressons: The Chow test 15 Exercses Introducng qualtatve nformaton n econometrc models. Up untl now, the varables that we have used n explanng the endogenous varable have a quanttatve nature. However, there are other varables of a qualtatve nature that can be mportant when explanng the behavor of the endogenous varable, such as sex, race, relgon, natonalty, geographcal regon etc. For example, holdng all other factors constant, female workers are found to earn less than ther male counterparts. Ths pattern may result from gender dscrmnaton, but whatever the reason, qualtatve varables such as gender seem to nfluence the regressand and clearly should be ncluded n many cases among the explanatory varables, or the regressors. Qualtatve factors often (although not always) come n the form of bnary nformaton,.e. a person s male or female, s ether marred or not, etc. When qualtatve factors come n the form of dchotomous nformaton, the relevant nformaton can be captured by defnng a bnary varable or a zero-one varable. In econometrcs, bnary varables used as regressors are commonly called dummy varables. In defnng a dummy varable, we must decde whch event s assgned the value one and whch s assgned the value zero. In the case of gender, we can defne 1 f the person s a female female f the person s a male But of course we can also defne 1 f the person s a male male f the person s a female Nevertheless, t s mportant to remark that both varables, male and female, contan the same nformaton. Usng zero-one varables for capturng qualtatve nformaton s an arbtrary decson, but wth ths electon the parameters have a natural nterpretaton. 5. A sngle dummy ndependent varable Let us see how we ncorporate dchotomous nformaton nto regresson models. Consder the smple model of hourly wage determnaton as a functon of the years of educaton (educ): 1

2 wage educ u (5-1) 1 To measure gender wage dscrmnaton, we ntroduce a dummy varable for gender as an ndependent varable n the model defned above, wage female educ u (5-) 1 1 The attrbute gender has two categores: male and female. The female category has been ncluded n the model, whle the male category, whch was omtted, s the reference category. Model 1 s shown n Fgure 5.1, takng <. The nterpretaton of s the followng: s the dfference n hourly wage between females and males, gven the same amount of educaton (and the same error dsturbance u). Thus, the coeffcent determnes whether there s dscrmnaton aganst women or not. If < then, for the same level of other factors (educaton, n ths case), women earn less than men on average. Assumng that the dsturbance mean s zero, f we take expectaton for both categores we obtan: E( wage female 1, educ) educ wage female 1 1 E( wage female, educ) educ wage male 1 (5-3) As can be seen n (5-3), the ntercept s for males, and + for females. Graphcally, as can be seen n Fgure 5.1, there s a shft of the ntercept, but the lnes for men and women are parallel. wage β 1 β FIGURE 5.1. Same slope, dfferent ntercept. In (5-) we have ncluded a dummy varable for female but not for male, because f we had ncluded both dummes ths would have been redundant. In fact, all we need s two ntercepts, one for females and another one for males. As we have seen, f we ntroduce the female dummy varable, we have an ntercept for each gender. Introducng two dummy varables would cause perfect multcollnearty gven that female+male=1, whch means that male s an exact lnear functon of female and of the ntercept. Includng dummy varables for both genders plus the ntercept s the smplest example of the so-called dummy varable trap, as we shall show later on. If we use male nstead of female, the wage equaton would be the followng: wage male educ u (5-4) 1 1 educ

3 Nothng has changed wth the new equaton, except the nterpretaton of and : s the ntercept for women, whch s now the reference category, and + s the ntercept for men. Ths mples the followng relatonshp between the coeffcents: = + and + = = In any applcaton, t does not matter how we choose the reference category, snce ths only affects the nterpretaton of the coeffcents assocated to the dummy varables, but t s mportant to keep track of whch category s the reference category. Choosng a reference category s usually a matter of convenence. It would also be possble to drop the ntercept and to nclude a dummy varable for each category. The equaton would then be wage male female educ u (5-5) 1 1 where the ntercept s 1 for men and 1 for women. Hypothess testng s performed as usual. In model (5-), the null hypothess of no dfference between men and women s H : 1, whle the alternatve hypothess that there s dscrmnaton aganst women s H1: 1. Therefore, n ths case, we must apply a one sded (left) t test. A common specfcaton n appled work has the dependent varable as the logarthm transformaton ln(y) n models of ths type. For example: ln( wage) female educ u (5-6) 1 1 Let us see the nterpretaton of the coeffcent of the dummy varable n a log model. In model (5-6), takng u=, the wage for a female and for a male s as follows: ln( wagef ) 11 educ (5-7) That s to say ln( wagem ) 1 educ (5-8) Gven the same amount of educaton, f we subtract (5-7) from (5-8), we have ln( wage ) ln( wage ) (5-9) F Takng antlogs n (5-9) and subtractng 1 from both sdes of (5-9), we get wagef 1 1e 1 (5-1) wage M wagef wage wage M e M 1 M 1 1 (5-11) Accordng to (5-11), the proportonal change between the female wage and the 1 male wage, for the same amount of educaton, s equal to e 1. Therefore, the exact 1 percentage change n hourly wage between men and women s 1( e 1). As an approxmaton to ths change, 1 1 can be used. However, f the magntude of the percentage s hgh, then ths approxmaton s not so accurate. EXAMPLE 5.1 Is there wage dscrmnaton aganst women n Span? Usng data from the wage structure survey of Span for (fle wagesp), model (5-6) has been estmated and the followng results were obtaned: 3

4 ln( wage ) = female +.548educ (.6) (.) (.5) RSS=393 R =.43 n= where wage s hourly wage n euros, female s a dummy varable that takes the value 1 f t s a woman, and educ are the years of educaton. (The numbers n parentheses are standard errors of the estmators.) To answer the queston posed above, we need to test H : 1 aganst H1: 1. Gven that the t statstc s equal to -14.7, we reject the null hypothess for =.1. That s to say, there s a negatve dscrmnaton n Span aganst women n the year. In fact, the percentage dfference n.37 hourly wage between men and women s 1 ( e 1) 35.9%, gven the same years of educaton. EXAMPLE 5. Analyss of the relaton between market captalzaton and book value: the role of bex35 A researcher wants to study the relatonshp between market captalzaton and book value n shares quoted on the contnuous market of the Madrd stock exchange. In ths market some stocks quoted are ncluded n the bex35, a selectve ndex. The researcher also wants to know whether the stocks ncluded n the bex35 have, on average, a hgher captalzaton.. Wth ths purpose n mnd, the researcher formulates the followng model: ln( marketcap) 11bex35 ln( bookvalue) u (5-1) where - marktval s the captalzaton value of a company, whch s calculated by multplyng the prce of the stock by the number of stocks ssued. - bookval s the book value of a company, also referred to as the net worth of the company. The book value s calculated as the dfference between a company's assets and ts labltes. - bex35 s a dummy varable that takes the value 1 f the corporaton s ncluded n the selectve Ibex 35. Usng the 9 stocks quoted on 15 th November 11 whch supply nformaton on book value (fle bolmad11), the followng results were obtaned: ln( marketcap ) = bex ln( bookvalue) (.43) (.179) (.37) RSS=35.67 R =.893 n=9 The marketcap/bookvalue elastcty s equal to.69; that s to say, f the book value ncreases by 1%, then the market captalzaton of the quoted stocks wll ncrease by.675%. To test whether the stocks ncluded n bex35 have on average a hgher captalzaton mples testng H : 1 aganst H1: 1. Gven that the t statstc s (.69/.179)=3.85, we reject the null hypothess for the usual levels of sgnfcance. On the other hand, we see that the stocks ncluded n bex35 are quoted 99.4% hgher than the stocks not ncluded. The percentage s obtaned as follows:.69 1 ( e 1) 99.4%. In the case of, we can test H : aganst H1:. Gven that the t statstc s (.675/.37)=18, we reject the null hypothess for the usual levels of sgnfcance. EXAMPLE 5.3 Do people lvng n urban areas spend more on fsh than people lvng n rural areas? To see whether people lvng n urban areas spend more on fsh than people lvng n rural areas, the followng model s proposed: ln( fsh) 11urban ln( nc) u (5-13) where fsh s expendture on fsh, urban s a dummy varable whch takes the value 1 f the person lves n an urban area and nc s dsposable ncome. Usng a sample of sze 4 (fle demand), model (5-13) was estmated: ln( fsh ) = urban ln( nc ) (.511) (.55) (.7) RSS=1.131 R =.94 n=4 Accordng to these results, people lvng n urban areas spend 14% more on fsh than people lvng n rural areas. If we test H : 1 aganst H 1 : 1, we fnd that the t statstc s.1.1 (.14/.55)=.55. Gven that t t =.44, we reject the null hypothess n favor of the alternatve

5 for the usual levels of sgnfcance. That s to say, there s emprcal evdence that people lvng n urban areas spend more on fsh than people lvng n rural areas. 5.3 Multple categores for an attrbute In the prevous secton we have seen an attrbute (gender) that has two categores (male and female). Now we are gong to consder attrbutes wth more than two categores. In partcular, we wll examne an attrbute wth three categores To measure the mpact of frm sze on wage, we can use a dummy varable. Let us suppose that frms are classfed n three groups accordng to ther sze: small (up to 49 workers), medum (from 5 to 199 workers) and large (more than 199 workers). Wth ths nformaton, we can construct three dummy varables: 1 up to 49 workers small n other case 1 from 5 to 199 workers medum n other case 1 more than 199 workers large n other case If we want to explan hourly wages by ntroducng the frm sze n the model, we must omt one of the categores. In the followng model, the omtted category s small frms: wage medum large educ u (5-14) 1 1 The nterpretaton of the j coeffcents s the followng: 1 ( ) s the dfference n hourly wage between medum (large) frms and small frms, gven the same amount of educaton (and the same error term u). Let us see what happens f we also nclude the category small n (5-14). We would have the model: wage small medum large educ u (5-15) 1 1 Now, let us consder that we have a sample of sx observatons: the observatons 1 and correspond to small frms; 3 and 4 to medum ones; and 5 and 6 to large ones. In ths case the matrx of regressors X would have the followng confguraton: 1 1 educ1 1 1 educ 1 1 educ3 X 1 1 educ4 1 1 educ5 1 1 educ6 As can be seen n matrx X, column 1 of ths matrx s equal to the sum of columns, 3 and 4. Therefore, there s perfect multcollnearty due to the so-called dummy varable trap. Generalzng, f an attrbute has g categores, we need to nclude only g1 dummy varables n the model along wth the ntercept. The ntercept for the reference category s the overall ntercept n the model, and the dummy varable 5

6 coeffcent for a partcular group represents the estmated dfference n ntercepts between that category and the reference category. If we nclude g dummy varables along wth an ntercept, we wll fall nto the dummy varable trap. An alternatve s to nclude g dummy varables and to exclude an overall ntercept. In the case we are examnng, the model would be the followng: wage small medum large educ u (5-16) 1 Ths soluton s not advsable for two reasons. Wth ths confguraton of the model t s more dffcult to test dfferences wth respect to a reference category. Second, ths soluton only works n the case of a model wth only one unque attrbute. EXAMPLE 5.4 Does frm sze nfluence wage determnaton? Usng the sample of example 5.1 (fle wagesp), model (5-14), takng log for wage, was estmated: ln( wage ) = medum +.16large +.48educ (.7) (.5) (.4) (.5) RSS=46 R =.18 n= To answer the queston above, we wll not perform an ndvdual test on 1 or. Instead we must jontly test whether the sze of frms has a sgnfcant nfluence on wage. That s to say, we must test whether medum and large frms together have a sgnfcant nfluence on the determnaton of wage. In ths case, the null and the alternatve hypothess, takng (5-14) as the unrestrcted model, wll be the followng: H : 1 H1: H s not true The restrcted model n ths case s the followng: ln( wage) 1 educ u (5-17) The estmaton of ths model s the followng: ln( wage ) = educ (.6) (.6) RSS=433 R =.166 n= Therefore, the F statstc s RSSR RSSUR / q / F 66.4 RSSUR / ( n k) 46 / ( 4) So, accordng to the value of the F statstc, we can conclude that the sze of the frm has a sgnfcant nfluence on wage determnaton for the usual levels of sgnfcance. Example 5.5 In the case of Lyda E. Pnkham, are the tme dummy varables ntroduced sgnfcant ndvdually or jontly? In example 3.4, we consdered the case of Lyda E. Pnkham n whch sales of a herbal extract from ths company (expressed n thousands of dollars) were explaned n terms of advertsng expendtures n thousands of dollars (advexp) and last year's sales (sales t-1 ). However, n addton to these two varables, the author ncluded three tme dummy varables: d1, d and d3. These dummy varables encompass the varous stuatons whch took place n the company. Thus, d1 takes 1 n the perod and n the remanng perods, d takes 1 n the perod and n other perods, and fnally, d3 takes 1 n the perod and n the remanng perods. Thus, the reference category s the perod The fnal formulaton of the model was therefore the followng: sales t = + advexp t + sales t-1 + d1 t + d t + d3 t +u t (5-18) The results obtaned n the regresson, usng fle pnkham, were the followng: sales = advexp +.673sales d d -.5d3 t t t-1 t t t (96.3) (.136) (.814) (89) (67) (67) R =.99 n=53 6

7 To test whether the dummy varables ndvdually have a sgnfcant effect on sales, the null and alternatve hypotheses are: ìï H q í 1,,3 ï ïî H1 q ¹ The correspondng t statstcs are the followng: tˆ -1.5 tˆ 3. tˆ 3. q1 q q As can be seen, the regressor d1 s not sgnfcant for any of the usual levels of sgnfcance, whereas on the contrary the regressors d and d3 are sgnfcant for any of the usual levels. The nterpretaton of the coeffcent of the regressor d, for example, s as follows: holdng fxed the advertsng spendng and gven the prevous year's sales, sales for one year of the perod are $.684 hgher than for a year of the perod To test jontly the effect of the tme dummy varables, the null and alternatve hypotheses are ìï H q1 q q3 í ï ïî H1 H s not true and the correspondng test statstc s ( RUR RR )/ q ( ) / 3 F (1 R ) / ( nk) (1.99) / (53 6) UR For any of the usual sgnfcance levels the null hypothess s rejected. Therefore, the tme dummy varables have a sgnfcant effect on sales 5.4 Several attrbutes Now we wll consder the possblty of takng nto account two attrbutes to explan the determnaton of wage: gender and length of workday (part-tme and fulltme). Let partme be a dummy varable that takes value 1 when the type of contract s part-tme and f t s full-tme. In the followng model, we ntroduce two dummy varables: female and partme: wage female partme educ u (5-19) In ths model, 1 s the dfference n hourly wage between those who work parttme, gven gender and the same amount of educaton (and also the same dsturbance term u). Each of these two attrbutes has a reference category, whch s the omtted category. In ths case, male s the reference category for gender and full-tme for type of contract. If we take expectatons for the four categores nvolved, we obtan: E wage female, partme, educ educ Ewage female, fulltme, educ Ewage male, partme, educ Ewage male, fulltme, educ1 wage female, partme educ wage female, fulltme 1 1 educ wage male, partme 1 1 wage male, fulltme educ (5-) The overall ntercept n the equaton reflects the effect of both reference categores, male and full-tme, and so full-tme male s the reference category. From (5-), you can see the ntercept for each combnaton of categores. 7

8 EXAMPLE 5.6 The nfluence of gender and length of the workday on wage determnaton Model (5-19), takng log for wage, was estmated by usng data from the wage structure survey of Span for 6 (fle wage6sp): ln( wage ) = female-.87 partme +.531educ (.6) (.1) (.7) (.3) RSS=365 R =.35 n= Accordng to the values of the coeffcents and correspondng standard errors, t s clear that each one of the two dummy varables, female and partme, are statstcally sgnfcant for the usual levels of sgnfcance. EXAMPLE 5.7 Tryng to explan the absence from work n the company Buenosares Buenosares s a frm devoted to the manufacturng of fans, havng had relatvely acceptable results n recent years. The managers consder that these would have been better f absenteesm n the company were not so hgh. In order to analyze the factors determnng absenteesm, the followng model s proposed: absent 11bluecoll 1male age 3tenure 4wage u (5-1) where bluecoll s a dummy ndcatng that the person s a manual worker (the reference category s whte collar) and tenure s a contnuous varable reflectng the years worked n the company. Usng a sample of sze 48 (fle absent), the followng equaton has been estmated: absent = bluecoll +.49 male -.37 age -.151tenure-.44 wage (1.64) (.669) (.71) (.47) (.65) (.7) RSS= R =.76 n=48 Next, we wll look at whether bluecoll s sgnfcant. Testng H : 1 aganst H1: 1, the t statstc s (.968/.669)=1.45. As t.1/ 4 =1.68, we fal to reject the null hypothess for =.1. And so there s no emprcal evdence to state that absenteesm amongst blue collar workers s dfferent from whte collar workers. But f we test H : 1 aganst H1: 1, as t.1 4 =1.3 for =.1, then we cannot reject that absenteesm amongst blue collar workers s greater than amongst whte collar workers. On the contrary, n the case of the male dummy, testng H : 1 aganst H1: 1, gven that the t statstc s (.49/.71)=.88 and t.1/ 4 =.7, we reject that absenteesm s equal n men and women for the usual levels of sgnfcance. EXAMPLE 5.8 Sze of frm and gender n determnng wage In order to know whether the sze of the frm and gender jontly are two relevant factors n determnng wage, the followng model s formulated: ln( wage) 1 1female 1medum large educ u (5-) In ths case, we must perform a jont test where the null and the alternatve hypotheses are H : 1 1 H1: H s not true In ths case, the restrcted model s model (5-17) whch was estmated n example 5.4 (fle wagesp). The estmaton of the unrestrcted model s the followng: ln( wage ) = female +.38medum +.168large +.499educ The F statstc s (.6) (.1) (.3) (.3) (.4) RSS=361 R =.35 n= RSS RSS q R UR / /3 F 133 RSS / ( n k) 361/ ( 5) UR Therefore, accordng to the value of F, we can conclude that the sze of the frm and gender jontly have a sgnfcant nfluence n wage determnaton. 8

9 5.5 Interactons nvolvng dummy varables Interactons between two dummy varables To allow for the possblty of an nteracton between gender and length of the workday on wage determnaton, we can add an nteracton term between female and partme n model (5-19), wth the model to estmate beng the followng: wage female partme female partme educ u (5-3) Ths allows workng tme to depend on gender and vce versa. EXAMPLE 5.9 Is the nteracton between females and part-tme work sgnfcant? Model (5-3), takng log for wage, was estmated by usng data from the wage structure survey of Span for 6 (fle wage6sp): ln( wage ) = female-.198 partme+.167 female partme +.538educ (.6) (.) (.47) (.58) (.4) RSS=363 R =.38 n= To answer the queston posed, we have to test H : 1 aganst H : 1. Gven that the t statstc s (.167/.58)=.89 and takng nto account that t.1/ 6 =.66, we reject the null hypothess n favor of the alternatve hypothess. Therefore, there s emprcal evdence that the nteracton between females and part-tme work s statstcally sgnfcant. EXAMPLE 5.1 Do small frms dscrmnate aganst women more or less than larger frms? To answer ths queston, we formulate the followng model: ln( wage) 11female 1medum large (5-4) 1femalemedum femalelarge educ u Usng the sample of example 5.1 (fle wagesp), model (5-4) was estmated: ln( wage ) = female +.361medum +.179large (.7) (.34) (.8) (.7) female medum-.43 female large educ (.5) (.51) (.4) RSS=359 R =.38 n= If n (5-4) the parameters 1 and are equal to, ths wll mply that n the equaton for wage determnaton, there wll be non nteracton between gender and frm sze. Thus to answer the above queston, we take (5-4) as the unrestrcted model. The null and the alternatve hypothess wll be the followng: H : 1 H1: H s not true In ths case, the restrcted model s therefore model (5-) estmated n example 5.7. The F statstc takes the value RSSR RSSUR / q / F 5.55 RSSUR /( n k) 359/(7).1.1 For =.1, we fnd that F,1993 F,6 = As F>5.61, we reject H n favor of H 1. As H has been rejected for =.1, t wll also be rejected for levels of 5% and 1%. Therefore, the usual levels of sgnfcance, the nteracton between gender and frm sze s relevant for wage determnaton Interactons between a dummy varable and a quanttatve varable So far, n the examples for wage determnaton a dummy varable has been used to shft the ntercept or to study ts nteracton wth another dummy varable, whle keepng the slope of educ constant. However, one can also use dummy varables to shft the slopes by lettng them nteract wth any contnuous explanatory varables. For 9

10 example, n the followng model the female dummy varable nteracts wth the contnuous varable educ: wage educ femaleeduc u (5-5) 1 1 As can be seen n fgure 5., the ntercept s the same for men and women n ths model, but the slope s greater n men than n women because 1 s negatve. In model (5-5), the returns to an extra year of educaton depend upon the gender of the ndvdual. In fact, wage 1 for women (5-6) educ for men wage educ FIGURE 5.. Dfferent slope, same ntercept. EXAMPLE 5.11 Is the return to educaton for males greater than for females? Usng the sample of example 5.1 (fle wagesp), model (5-5) was estmated by takng log for wage: ln( wage ) = educ -.74educ female (.5) (.6) (.1) RSS=4 R =.9 n= In ths case, we need to test H : 1 aganst H1: 1. Gven that the t statstc s (-.74/.1) =-1.81, we reject the null hypothess n favor of the alternatve hypothess for any level of sgnfcance. That s to say, there s emprcal evdence that the return for an addtonal year of educaton s greater for men than for women. 5.6 Testng structural changes So far we have tested hypotheses n whch one parameter, or a subset of parameters of the model, s dfferent for two groups (women and men, for example). But sometmes we wsh to test the null hypothess that two groups have the same populaton regresson functon, aganst the alternatve that t s not the same. In other words, we want to test whether the same equaton s vald for the two groups. There are two procedures for ths: usng dummy varables and runnng separate regressons through the Chow test. 1

11 5.6.1 Usng dummy varables In ths procedure, testng for dfferences across groups conssts n performng a jont sgnfcance test of the dummy varable, whch dstngushes between the two groups and ts nteractons wth all other ndependent varables. We therefore estmate the model wth (unrestrcted model) and wthout (restrcted model) the dummy varable and all the nteractons. From the estmaton of both equatons we form the F statstc, ether through the RSS or from the R. In the followng model for the determnaton of wages, the ntercept and the slope are dfferent for males and females: wage female educ femaleeduc u (5-7) 1 1 The populaton regresson functon correspondng to ths model s represented n fgure 5.3. As can be seen, f female=1, we obtan wage ( ) ( ) educ u (5-8) 1 1 For women the ntercept s 1 1, and the slope. For female=, we obtan equaton (5-1). In ths case, for men the ntercept s 1, and the slope. Therefore, 1 measures the dfference n ntercepts between men and women and, measures the dfference n the return to educaton between males and females. Fgure 5.3 shows a lower ntercept and a lower slope for women than for men. Ths means that women earn less than men at all levels of educaton, and the gap ncreases as educ gets larger; that s to say, an addtonal year of educaton shows a lower return for women than for men. Estmatng (5-7) s equvalent to estmatng two wage equatons separately, one for men and another for women. The only dfference s that (5-7) mposes the same varance across the two groups, whereas separate regressons do not. Ths set-up s deal, as we wll see later on, for testng the equalty of slopes, equalty of ntercepts, and equalty of both ntercepts and slopes across groups. wage FIGURE 5.3. Dfferent slope, dfferent ntercept. EXAMPLE 5.1 Is the wage equaton vald for both men and women? If parameters 1 and are equal to n model (5-7), ths wll mply that the equaton for wage determnaton s the same for men and women. In order to answer the queston posed, we take (5-7), as educ 11

12 the unrestrcted model but express wage n logs. The null and the alternatve hypothess wll be the followng: H : 1 H1: H s not true Therefore, the restrcted model s model (5-17). Usng the same sample as n example 5.1 (fle wagesp), we have obtaned the followng estmaton of models (5-7) and (5-17): ln( wage ) = female +.539educ -.7educ female (.3) (.546) (.3) (.54) RSS=393 R =.43 n= ln( wage ) = educ (.6) (.6) RSS=433 R =.166 n= The F statstc takes the value RSSR RSSUR / q / F 1 RSSUR / ( n k) 393 / ( 4) It s clear that for any level of sgnfcance, the equatons for men and women are dfferent. When we tested n example 5.1 whether there was dscrmnaton n Span aganst women ( H : 1 aganst H1: 1 ), t was assumed that the slope of educ (model (5-6)) s the same for men and women. Now t s also possble to use model (5-7) to test the same null hypothess, but assumng a dfferent slope. Gven that the t statstc s (-.3319/.546)=-6.6, we reject the null hypothess by usng ths more general model than the one n example 5.1. In example 5.11 t was tested whether the coeffcent n model (5-5), takng log for wage, was, assumng that the ntercept s the same for males and females. Now, f we take (5-7), takng log for wage, as the unrestrcted model, we can test the same null hypothess, but assumng that the ntercept s dfferent for males and females. Gven that the t statstc s (.7/.54)=.49, we cannot reject the null hypothess whch states that there s no nteracton between gender and educaton. EXAMPLE 5.13 Would urban consumers have the same pattern of behavor as rural consumers regardng expendture on fsh? To answer ths queston, we formulate the followng model whch s taken as the unrestrcted model: ln( fsh) 11urban ln( nc) ln( nc) urban u (5-9) The null and the alternatve hypothess wll be the followng: H : 1 H1: H s not true The restrcted model correspondng to ths H s ln( fsh) 1ln( nc) u (5-3) Usng the sample of example 5.3 (fle demand), models (5-9) and (5-3) were estmated: ln( fsh ) = urban ln( nc ) -.75ln( nc) urban (.67) (1.95) (.87) (.15) RSS=1.13 R =.94 n=4 ln( fsh ) = ln( nc ) (.54) (.75) RSS=1.35 R =.887 n=4 The F statstc takes the value RSSR RSSUR / q / F 3.4 RSSUR /( n k) 1.13/(44) If we look up n the F table for df n the numerator and 35 df n the denomnator for =.1, we fnd F,36 F, As F>.46 we reject H. However, as F,36 F,35 3.7, we fal to 1

13 reject H n favour of H 1 for =.5 and, therefore, for =.1. Concluson: there s no strong evdence that famles lvng n rural areas have a dfferent pattern of fsh consumpton than famles lvng n rural areas. Example 5.14 Has the productve structure of Spansh regons changed? The queston to be answered s specfcally the followng: Dd the productve structure of Spansh regons change between 1995 and 8? The problem posed s a problem of structural stablty. To specfy the model to be taken as a reference n the test, let us defne the dummy y8, whch takes the value 1 f the year s 8 and f the year s The reference model s a Cobb-Douglas model, whch ntroduces addtonal parameters to collect the structural changes that may have occurred. Its expresson s: ln( q) 1 1ln( k) 1ln( l) y8 y8ln( k) y8ln( l) u (5-31) It s easly seen, accordng to the defnton of the dummy y8, that the elastctes producton/captal are dfferent n the perods 1995 and 8. Specfcally, they take the followng values: ln( Q) ln( Q) Q K(1995) 1 Q K(8) 1+ ln( K) ln( K) In the case that s equal to, then the elastcty of producton/captal s the same n both perods. Smlarly, the producton/labor elastctes for the two perods are gven by ln( L) ln( L) Q K(1995) 1 Q K(8) 1+ ln( K) ln( K) The ntercept n the Cobb-Douglas s a parameter that measures effcency. In model (5-31), the possblty that the effcency parameter (PEF) s dfferent n the two perods s consdered. Thus PEF(1995) 1 PEF(8) 1+ If the parameters 1, 1 and 1 are zero n model (5-31), the producton functon s the same n both perods. Therefore, n testng structural stablty of the producton functon, the null and alternatve hypotheses are: H (5-3) H H s not true 1 Under the null hypothess, the restrctons gven n (5-3) lead to the followng restrcted model: ln( q) 1 1ln( k) 1ln( l) u (5-33) The fle prodsp contans nformaton for each of the Spansh regons n 1995 and 8 on gross value added n mllons of euros (gdp), occupaton n thousands of jobs (labor), and productve captal n mllons of euros (captot). You can also fnd the dummy y8 n that fle. The results of the unrestrcted regresson model (5-31) are shown below. It s evdent that we cannot reject the null hypothess that each of the coeffcents 1, 1 and 1, taken ndvdually, are, snce none of the t statstcs reaches.1 n absolute value. ln( gva ) ln( captot) +.391ln( labor) (.916) (.185) (.185) y y8 ln( captot) -.94 y8 ln( labor) (.3) (.419) (.418) R = n=34 The results of the restrcted model (5-33) are the followng: ln( gva) ln( captot) +.311ln( labor) (.) (.36) (.4) R =.9939 n=34 As can be seen, the R of the two models are vrtually dentcal because they dffer only from the ffth decmal. It s not surprsng, therefore, that the F statstc for testng the null hypothess (5-3) takes a value close to : ( RUR RR )/ q ( ) / 3 F.38 (1 R ) / ( nk) ( ) / (34 6) UR 13

14 Thus, the alternatve hypothess that there s structural change n the productve economy of the Spansh regons between 1995 and 8 s rejected for any sgnfcance level Usng separate regressons: The Chow test Ths test was ntroduced by the econometrcan Chow (196). He consdered the problem of testng the equalty of two sets of regresson coeffcents. In the Chow test, the restrcted model s the same as n the case of usng dummy varables to dstngush between groups. The unrestrcted model, nstead of dstngushng the behavour of the two groups by usng dummy varables, conssts smply of separate regressons. Thus, n the wage determnaton example, the unrestrcted model conssts of two equatons: female : wage 11 1educ u (5-34) male : wage educ u 1 If we estmate both equatons by OLS, we can show that the RSS of the unrestrcted model, RSS UR, s equal to the sum of the RSS obtaned from the estmates for women, RSS 1, and for men, RSS. That s to say, RSS UR =RSS 1 +RSS The null hypothess states that the parameters of the two equatons n (5-34) are equal. Therefore 11 1 H : 1 H :No H 1 By applyng the null hypothess to model (5-34), you get model (5-17), whch s the restrcted model. The estmaton of ths model for the whole sample s usually called the pooled (P) regresson. Thus, we wll consder that the RSS R and RSS P are equvalent expressons. Therefore, the F statstc wll be the followng: RSSP RSS1RSS/ k F (5-35) RSS RSS / n k 1 It s mportant to remark that, under the null hypothess, the error varances for the groups must be equal. Note that we have k restrctons: the slope coeffcents (nteractons) plus the ntercept. Note also that n the unrestrcted model we estmate two dfferent ntercepts and two dfferent slope coeffcents, and so the df of the model are nk. One mportant lmtaton of the Chow test s that under the null hypothess there are no dfferences at all between the groups. In most cases, t s more nterestng to allow partal dfferences between both groups as we have done usng dummy varables. The Chow test can be generalzed to more than two groups n a natural way. From a practcal pont of vew, to run separate regressons for each group to perform the test s probably easer than usng dummy varables. In the case of three groups, the F statstc n the Chow test wll be the followng: 14

15 F RSSP ( RSS1RSS RSS3) /k ( RSS RSS RSS )/( n 3 k) 1 3 (5-36) Note that, as a general rule, the number of the df of the numerator s equal to the (number of groups-1)k, whle the number of the df of the denomnator s equal to n mnus (number of groups)k. EXAMPLE 5.15 Another way to approach the queston of wage determnaton by gender Usng the same sample as n example 5.1 (fle wagesp), we have obtaned the estmaton of the equatons n (5-34), takng log for wage, for men and women, whch taken together gves the estmaton of the unrestrcted model: Female equaton ln( wage ) = educ Male equaton (.4) (.41) RSS=14 R =.36 n=617 ln( wage ) = educ 1 (.31) (.3) RSS=89 R =.175 n=1383 The restrcted model, estmated n example 5.4, has the same confguraton as the equatons n (5-34) but n ths case refers to the whole sample. Therefore, t s the pooled regresson correspondng to the restrcted model. The F statstc takes the value RSSP ( RSSF RSSM) / k 433 (14 89) / F 1 RSSF RSSM) / ( n k) (14 89) / ( ) The F statstc must be, and s, the same as n example 5.1. The conclusons are therefore the same. EXAMPLE 5.16 Is the model of wage determnaton the same for dfferent frm szes? In other examples the ntercept, or the slope on educaton, was dfferent for three dfferent frm szes (small, medum and large). Now we shall consder a completely dfferent equaton for each frm sze. Therefore, the unrestrcted model wll be composed by three equatons: samall :ln( wage) female 1edu u medum :ln( wage) 1 1 female edu u (5-37) large :ln( wage) female 3edu u The null and the alternatve hypothess wll be the followng: H : H :No H Gven ths null hypothess, the restrcted model s model (5-). The estmatons of the three equatons of (5-37), by usng fle wagesp, are the followng: small medum large ln( wage ) = female +.396educ (.34) (.31) (.38) RSS=11 R =.16 n=81 ln( wage ) = female +.548educ (.51) (.39) (.46) RSS =13 R =.3 n=59 ln( wage ) = female +.554educ (.46) (.39) (.44) RSS =114 R =.73 n=69 The pooled regresson has been estmated n example 5.1. The F statstc takes the value 15

16 RSSP ( RSSS RSSM RSSL) /k F ( RSSS RSSM RSSL)/( n 3 k) 393 ( ) / ( ) / ( 33) For any level of sgnfcance, we reject that the equatons for wage determnaton are the same for dfferent frm szes. EXAMPLE 5.17 Is the Pnkham model vald for the four perods? In example 5.5, we ntroduced tme dummy varables and we tested whether the ntercept was dfferent for each perod. Now, we are gong to test whether the whole model s vald for the four perods consdered. Therefore, the unrestrcted model wll be composed by four equatons: salest 11 1advexpt 31 salest 1 ut salest 1 advexpt 3 salest 1 ut (5-38) salest 13 3advexpt 33 salest 1 ut salest 14 4advexpt 34 sales t1 ut The null and the alternatve hypothess wll be the followng: H : H :No H 1 Gven ths null hypothess, the restrcted model s the followng model: salest 1 advexpt 3salest1 ut (5-39) The estmatons of the four equatons of (5-38) are the followng: sales = advexp sales SSR = 3617 n = 7 t (63) (1.5) (.45) sales = advexp sales SSR = 465 n = 11 t (19) (.557) (.3) sales = advexp sales SSR = 1614 n = 15 t (11) (.115) (.87) sales = advexp +.34 sales SSR = n = t (134) (.41) (.111) The pooled regresson, estmated n example 3.4, s the followng: sales = advexp sales SSR = 5715 n = 53 t (95.7) (.156) (.915) The F statstc takes the value SSRP ( SSR1SSR SSR3 SSR4) /3k F ( SSR1 SSR SSR3 SSR4)/( n 4 k) 5715 ( ) / ( ) / (53 43) For any level of sgnfcance, we reject that the model (5-39) s the same for the four perods consdered. Exercses Exercse 5.1 Answer the followng questons for a model wth explanatory dummy varables: a) What s the nterpretaton of the dummy coeffcents? b) Why are not ncluded n the model so many dummy varables as categores there are? t-1 t-1 t-1 t-1 t-1 16

17 Exercse 5. Usng a sample of 56 famles, the followng estmatons of demand for rental are obtaned: qˆ p.96 y (.11) (.17) (.6) R =.371 n=56 qˆ p.9 y.341d y (.13) (.3) (.31) (.1) R =.38 where q s the log of expendture on rental housng of the th famly, p s the logarthm of rent per m n the lvng area of the th famly, y s the log of household dsposable ncome of the th famly and d s a dummy varable that takes value one f the famly lves n an urban area and zero n a rural area. (The numbers n parentheses are standard errors of the estmators.) a) Test the hypothess that the elastcty of expendture on rental housng wth respect to ncome s 1, n the frst ftted model. b) Test whether the nteracton between the dummy varable and ncome s sgnfcant. Is there a sgnfcant dfference n the housng expendture elastcty between urban and rural areas? Justfy your answer. Exercse 5.3 In a lnear regresson model wth dummy varables, answer the followng questons: a) The meanng and nterpretaton of the coeffcents of dummy varables n models wth endogenous varable n logs. b) Express how a model s affected when a dummy varable s ntroduced n a multplcatve way wth respect to a quanttatve varable. Exercse 5.4 In the context of a multple lnear regresson model, a) What s a dummy varable? Gve an example of an econometrc model wth dummy varables. Interpret the coeffcents. b) When s there perfect multcollnearty n a model wth dummy varables? Exercse 5.5 The followng estmaton s obtaned usng data for workers of a company: wage = tenure + college + 1 male where wage s the wage n euros per month, tenure s the number of years n the company, college s a dummy varable that takes value 1 f the worker s graduated from college and otherwse and male s a dummy varable whch takes value 1 f the worker s male and otherwse. a) What s the predcted wage for a male worker wth sx years of tenure and college educaton? b) Assumng that all workng women have college educaton and none of the male workers do, wrte a hypothetcal matrx of regressors (X) for sx observatons. In ths case, would you have any problem n the estmaton of ths model? Explan t. c) Formulate a new model that allows to establsh whether there are wage dfferentals between workers wth prmary, secondary and college educaton. 17

18 Exercse 5.6 Consder the followng lnear regresson model: y x d d u (1) 1 1 where y s the monthly salary of a teacher, x s the number of years of teachng experence y d 1 y d are two dummy varables takng the followng values: d1 1 f the teacher s male d otherwse 1 f the teacher s whte otherwse a) What s the reference category n the model? b) Interpret 1 and. What s the expected salary for each of the possble categores? c) To mprove the explanatory power of the model, the followng alternatve specfcaton was consdered: y x 1d1 d 3( d1d) u () d) What s the meanng of the term( d 1 d )? Interpret 3. e) What s the expected salary for each of the possble categores n model ()? Exercse 5.7 Usng a sample of 36 observatons, the followng results are obtaned: yˆ x 4.56 x.34 x t t1 t t3 (.1) (.34) (3.35) (.7) n yˆ y uˆ t t1 t1 n (The numbers n parentheses are standard errors of the estmators.) a) Test the ndvdual sgnfcance of the coeffcent assocated wth x. b) Calculate the coeffcent of determnaton, R, and explan ts meanng. c) Test the jont sgnfcance of the model. d) Two addtonal regressons, wth the same specfcaton, were made for the two categores A and B ncluded n the sample (n 1 =1 y n =15). In these estmates the followng RSS were obtaned: 11.9 y.17, respectvely. Test f the behavor of the endogenous varable s the same n the two categores. Exercse 5.8 To explan the tme devoted to sport (sport), the followng model was formulated sport b1+ d1female + j1smoker + bage + u (1) where sport s the mnutes spent on sports a day, on average; female and smoker are dummy varables takng the value 1 f the person s a woman or smoker of at least fve cgarettes per day, respectvely. a) Interpret the meanng of 1, j 1 and. b) What s the expected tme spent on sports actvtes for all possble categores? c) To mprove the explanatory power of the model, the followng alternatve specfcaton was consdered: depor b1+ d1mujer + j1fumador + g1mujer fumador () + d mujer edad + j fumador edad + b edad + u t 18

19 In model (), what s the meanng of 1? What s the meanng of and j? d) What are the possble margnal effects of sport wth respect to age n the model ()? Descrbe them. Exercse 5.9 Usng nformaton for Spansh regons n 1995 and, several producton functons were estmated. For the whole of the two perods, the followng results were obtaned: ln( q) = ln( k) +.75ln( l) f +.11f ln( k) -.5 f ln( l) (1) R R RSS n ln( q) = ln( k) +.6( l) () R.9567 R.955 RSS 1.7 Moreover, the followng models were estmated separately for each of the years: 1995 ln( q) = ln( k) +.75l (3) R R RSS =.65 ln( q) = ln( k) +.7l (4) R R RSS =.3331 where q s output, k s captal, l s labor and f s a dummy varable that takes the value 1 for 1995 data and for. a) Test whether there s structural change between 1995 and. b) Compare the results of estmatons (3) and (4) wth estmaton (1). c) Test the overall sgnfcance of model (1). Exercse 5.1 Wth a sample of 3 servce sector frms, the followng cost functon was estmated: cost = qty RSS = n = 3 (.5) where qty s the quantty produced. The 3 frms are dstrbuted n three bg areas (1 n each one). The followng results were obtaned: Area 1: cost = qty sˆ =.457 (.38) Area : cost = qty sˆ = (.96) Area 3: cost = qty sˆ = 4.55 (.1) a) Calculate an unbased estmaton of n the cost functon for the sample of 3 frms. b) Is the same cost functon vald for the three areas? Exercse 5.11 To study spendng on magaznes (mag), the followng models have been formulated: ln( mag) ln( nc) age male u (1) ln( mag) ln( nc) age male prm sec u ()

20 where nc s dsposable ncome, age s age n years, male s a dummy varable that takes the value 1 f he s male, prm and sec are dummy varables that take the value 1 when the ndvdual has reached, at most, prmary and secondary level respectvely. Wth a sample of 1 observatons, the followng results have been obtaned ln( mag) = ln( nc ) +.31age -.17 male (.14) (4) (.1) (.) RSS= R =.986 ln( mag ) = ln( nc ) +.3 age +.3male -.5 prm +.18sec (.) (.7) (.) (.3) (.4) (.5) RSS=.36 R =.9981 a) Is educaton a relevant factor to explan spendng on magaznes? What s the reference category for educaton? b) In the frst model, s spendng on magaznes hgher for men than for women? Justfy your answer. c) Interpret the coeffcent on the male varable n the second model. Is spendng on magaznes hgher for men than for women? Compare wth the result obtaned n secton a). Exercse 5.1 Let frut be the expendture on frut expressed n euros over a year carred out by a household and let r 1, r, r 3, and r 4 be dchotomous varables whch reflect the four regons of a country. a) If you regress frut only on r 1, r, r 3, and r 4 wthout an ntercept, what s the nterpretaton of the coeffcents? b) If you regress frut only on r 1, r, r 3, and r 4 wth an ntercept, what would happen? Why? c) If you regress frut only on r, r 3, and r 4 wthout an ntercept, what s the nterpretaton of the coeffcents? d) If you regress frut only on r 1 - r, r, r 4 -r 3, and r 4 wthout an ntercept, what s the nterpretaton of the coeffcents? Exercse 5.13 Consder the followng model wage 11female educ u Now, we are gong to consder three possbltes of defnng the female dummy varable. 1) 1 for female female ) for male for female female 3) 1 for male for female female for male a) Interpret the dummy varable coeffcent for each defnton. b) Is one dummy varable defnton preferable to another? Justfy the answer. Exercse 5.14 In the followng regresson model: wage 11female u where female s a dummy varable, takng value 1 for female and value for a male. Prove that applyng the OLS formulas for smple regresson you obtan that ˆ 1 wage M ˆ wage wage 1 F M

21 where F ndcates female and M male. In order to obtan a soluton, consder that n the sample there are n 1 females and n males: the total sample s n= n 1 +n. Exercse 5.15 The data of ths exercse were obtaned from a controlled marketng experment n stores n Pars on coffee expendture, as reported n A. C. Bemmaor and D. Mouchoux, Measurng the Short-Term Effect of In-Store Promoton and Retal Advertsng on Brand Sales: A Factoral Experment, Journal of Marketng Research, 8 (1991), 14. In ths experment, the followng model has been formulated to explan the quantty sold of coffee per week: ln( coffqty) 11advert ln( coffprc) advert ln( coffprc) u where coffprc takes three values: 1, for the usual prce,.95 and.85; advert s a dummy varable that takes value 1 f there s advertsng n ths week and f there s not. The experment lasted for 18 weeks. The orgnal model and three other models were estmated, usng fle coffee: ln( coffqty) = advert ln( coffprc ) advert ln( coffprc ) (.4) (.99) (.45) (.883) 1) R =.9468 n= 18 ln( coffqty) = advert ln( coffprc) ) (.4) (.57) (.393) R =.941 n = 18 ln( coffqty) = ln( coffprc) advert ln( coffprc) 3) (.4) (.513) (.58) R =.914 n = 18 ln( coffqty ) = ln( coffprc) 4) (.7) (.674) R =.7863 n = 18 a) In model (), what s the nterpretaton of the coeffcent on advert? b) In model (3), what s the nterpretaton of the coeffcent on advert ln(coffprc? c) In model (), does the coeffcent on advert have a sgnfcant postve effect at 5% and at 1%? d) Is model (4) vald for weeks wth advertsng and for weeks wthout advertsng? e) In model (1), s the ntercept the same for weeks wth advertsng and for weeks wthout advertsng? f) In model (3), s the coffee demand/prce elastcty dfferent for weeks wth advertsng and for weeks wthout advertsng? g) In model (4), s the coffee demand/prce elastcty smaller than -4? Exercse 5.16 (Contnuaton of exercse 4.39). Usng fle tmuse3, the followng models have been estmated: houswork educ age.337 padwork (3) (1.497) (.38) (.3) (1) R.14 n 1 houswork educ 1.775age.1568 padwork 3.11 female (.9) (1.356) (.79) (.1) (.16) () R.98 n 1 1

22 houswork educ 1.333age.871 padwork 3.75 female (35.18) (.35) (.5) (.3) (8.15).165educ female.119 age female.65 padwork female (.546) (.11) (.9) R.36 n1 a) In model (1), s there a statstcally sgnfcant tradeoff between tme devoted to pad work and tme devoted to housework? b) All other factors beng equal and takng as a reference model (), s there evdence that women devote more tme to housework than men? c) Compare the R of models (1) and (). What s your concluson? d) In model (3), what s the margnal effect of tme devoted to housework wth respect to tme devoted to pad work? e) Is nteracton between padwork and gender sgnfcant? f) Are the nteractons between gender and the quanttatve varables of the model jontly sgnfcant? Exercse 5.17 Usng data from Bolsa de Madrd (Madrd Stock Exchange) on November 19, 11 (fle bolmad11), the followng models have been estmated: ln( marktval ) = bex ln( bookval ) (1) where (.43) (.179) (.369) RSS=35.69 R =.8931 n=9 ln( marktval ) = bex ln( bookval ) (.75) (.778) (.43) +.31bex35 ln( bookval ) (.88) RSS=35.6 R =.8933 n=9 ln( marktval ) = bex ln( bookval ) (.31) (.785) (.45) +.369bex35 ln( bookval )-.6613servces consump (.89) (.36) (.1) For fnance= energy ndustry -. 7tt (.63) (.7) (.34) RSS=3.781 R =.978 n=9 ln( marktval ) = ln( bookval ) (4) (.34) (.35) RSS=41.65 R =.8753 n=9 ln( markval ) = ln( bookval ) (5) (.56) (.7) RSS=.741 R =.9415 n=13 - marktval s the captalzaton value of a company. - bookval s the book value of a company. - bex35 s a dummy varable that takes the value 1 f the corporaton s ncluded n the selectve Ibex servces, consumpton, energy, ndustry and tc (nformaton technology and communcaton) are dummy varables. Each of them takes the value 1 f the corporaton s classfed n ths sector n Bolsa de Madrd. The category of reference s fnance. (3) () (3)

23 a) In model (1), what s nterpretaton of the coeffcent on bex35? b) In model (1), s the marktval/bookval elastcty equal to 1? c) In model (), s the elastcty marktval/bookval the same for all corporatons ncluded n the sample? d) Is model (4) vald both for corporatons ncluded n bex 35 and for corporatons excluded? e) In model (3), what s nterpretaton of the coeffcent on consump? f) Is the coeffcent on consump sgnfcatvely negatve? g) Is the ntroducton of dummy varables for dfferent sectors statstcally justfable? h) Is the marktval/bookval elastcty for the fnancal sector equal to 1? Exercse 5.18 (Contnuaton óf exercse 4.37). Usng fle rdspan, the equatons whch appear n the attached table have been estmated. The followng varables appear n the table: - rdntens s expendture on research and development (R&D) as a percentage of sales, - sales are measured n mllons of euros, - exponsal s exports as a percentage of sales; - medtech and hghtech are two dummy varables whch reflects f the frm belongs to a medum or a hgh technology sector. The reference category corresponds to the frms wth low technology, - workers s the number of workers. 3