β 1 is the intercept for the female professor β 1 + δ is the intercept for the male professor

Transcription

1 Chaper 7 Usng Indcaor Varables Explanaory varables ha allow for qualave facors are of neres. Some applcaons nclude: me seres daa changes n governmen economc polcy (regme changes) seasonaly cross-secon daa. A labour force survey may record ndvdual characerscs such as: gender (male / female) educaonal achevemen (hgh school / unversy) maral saus (marred / sngle) Example A survey of professors a ABC Unversy has daa for: y annual salary for professor x years of eachng experence for professor A research queson s: Is here a salary dfferenal beween male professors and female professors? Ths dfferenal s known as he gender wage gap. Defne he dummy varable: f male D = for =,,..., N 0 f female Ths s he gender dummy varable. In general, a dummy varable s an arfcal varable ha assgns arbrary codes o dfferen groups. The use of 0- codes suggess ha he name bnary varable or ndcaor varable may be a more descrpve name han dummy varable. The erm dummy varable s wdely used n economercs work and so s he erm used here. Econ 6 - Chaper 7 Econ 6 - Chaper 7

2 The gender dummy varable can be ncluded n he salary deermnaon equaon o ge he lnear regresson equaon: y + δd + β x + e = β The equaon has a dfferenal nercep: The fgure below shows ha, wh δ > 0, he nercep dummy varable gves dfferen, bu parallel, regresson lnes for male professors and female professors. The vercal dsance beween he wo lnes s he amoun δ. β s he nercep for he female professor β + δ s he nercep for he male professor δ s he salary dfferenal beween male and female professors. The use of dummy varables as explanaory varables does no affec any of he sascal properes of he leas squares (OLS) esmaor. Esmaon and hypohess esng can proceed as before. annual salary male professor female professor years of eachng experence Econ 6 - Chaper 7 4 Econ 6 - Chaper 7

3 The parameers β, β and δ can be esmaed by he leas squares prncple. Is here a gender wage gap? To es he clam ha male professors earn more han female professors wh he dencal eachng experence consder: H 0 : δ = 0 agans H : δ > 0 Ths s a one-al es. The es sasc s he -sasc for a es of sgnfcance ha s repored as a sandard par of leas squares (OLS) esmaon oupu. If he coeffcen on he gender dummy varable s posve hen he p-value for hs one-al es s obaned by dvdng he p-value for a wo-al es by. For a sgnfcance level of 0.05, f he p-value s less han 0.05 hen he null hypohess s rejeced n favour of he alernave ha male professors have hgher salares han female professors. Anoher way o proceed s o defne he gender dummy varable as: F = 0 f f female male The dummy varable F can replace he dummy varable D n he salary deermnaon equaon. A gender wage gap would now be revealed by a negave coeffcen on he female dummy varable F. Alernave models, ha each gve numercally dencal resuls for he magnude of he gender wage gap, are: Regresson Equaon Gender Wage Gap y = β + δd + β x + e δ y = β + αf + β x + e α 0 y + = δd + δf + βx e δ δ Noe ha he fnal equaon conans no overall nercep. 5 Econ 6 - Chaper 7 6 Econ 6 - Chaper 7

4 A regresson equaon wh boh gender dummy varables and an overall equaon nercep s: y + δ D + δ F + β x + e = β Bu D + F = for all (he value of he varable aached o β ). Ths s a volaon of he assumpon ha explanaory varables canno be formed as lnear combnaons of oher explanaory varables. Therefore, s no possble o oban leas squares esmaes from hs model. Ths s called he dummy varable rap. The dummy varable rap s oo many dummy varables. Slope Dummy Varables A dfferen slope coeffcen for male and female professors can be nroduced wh he lnear regresson equaon: = β ( xd ) e y + δd + β x + γ + The varable ( x D) s called an neracon varable or a slope dummy varable. Ths se-up gves he salary funcons: E (y β + βx ) = ( β + δ) + ( β + γ) x for for female male professors professors The leas squares prncple can be used for esmaon and hypohess esng. 7 Econ 6 - Chaper 7 8 Econ 6 - Chaper 7

5 Ineracons beween Qualave Facors A regresson equaon can nclude several dummy varables as explanaory varables. For example, for he wage deermnaon equaon, a dummy varable for maral saus can be defned as: Marr = 0 f marred oherwse A wage (or salary) equaon ha ncludes a maral saus dummy varable s: y = β + α F + α Marr + β x + e The coeffcen α s he marrage premum. A lmaon of hs model s ha he marrage premum s assumed o be he same for men and women. A more general model, ha ncludes an neracon dummy varable F Marr, s: ( ) ( FMarr ) + βx e y = β + α F + α Marr + α + In hs se-up, he base group or reference group s sngle (unmarred) men. The base group s represened by he nercep β. The coeffcens on he dummy varables gve dfferences n wage relave o he base group. Wage funcons for each dummy varable combnaon can be saed: E (y β β ) = β β + β + α + α + α x + β + β + α x x + α + β x male, nomarred female, nomarred male, marred female, marred 9 Econ 6 - Chaper 7 0 Econ 6 - Chaper 7

6 The wage dfference beween sngle and marred men (he marrage premum for males) s: α The wage dfference beween sngle and marred women (he marrage premum for females) s: α + α The gender wage gap for sngle workers s: α The gender wage gap for marred workers s: ( α + ) α Followng model esmaon, hypohess esng exercses can consder he sascal sgnfcance of he dfferences beween he varous groups. Example Jeffrey Wooldrdge has colleced a cross-secon daa se wh N = 56 observaons. The varables nclude average hourly earnngs n dollars (y) and years poenal experence (x) as well as dummy varables for ndvdual characerscs. The frs 0 observaons of he daa se are: y x F Marr.0 0 female, no marred.4 female, marred male, no marred male, marred male, marred male, marred male, no marred female, no marred female, no marred male, marred The above shows he 0- codng for he dummy varables. Econ 6 - Chaper 7 Econ 6 - Chaper 7

7 Esmaon resuls are: ( FMarr ) x ŷ = F +.704Marr sasc (.4) (.) (6.00) ( 4.66) (0.97) p-value (< ) (0.) (< ) (< ) (0.) R = 0.8 Noe: As a sandard reporng syle he -sascs and p-values are for a wo-al es of he null hypohess of a zero coeffcen on he explanaory varable. The resuls show ha, for men, a marred worker wll earn abou $.70 per hour more han a sngle worker of smlar work experence. The -sasc value of 6.00 gves evdence ha hs marrage premum s sascally dfferen from zero. In conras, he marrage premum for female workers s esmaed as: ˆ + α = = 0. α ˆ Is hs sgnfcanly dfferen from zero? To answer hs queson he null hypohess s: H0 : α + α = 0. The es mehod was presened n he lecure noes for Chaper 5. For a calculaed -sasc he p-value for a wo-al es was repored as Ths hgh p-value suggess ha he null hypohess of no wage dfference beween marred and unmarred female workers s no rejeced. The esmaed coeffcen for he female dummy varable (F) s negave as expeced, bu s no sascally sgnfcan. Tha s, for he female dummy varable, he p-value of a wo-al es of a zero coeffcen s 0. whch exceeds he sgnfcance levels of boh 0.05 and 0.0. Therefore, for sngle workers, a wage gap beween males and females does no appear o exs. The gender wage gap for marred workers s esmaed as: αˆ + ˆ ) = ( ) =.4 ( α The nerpreaon of hs resul s ha, for marred workers, men earn abou $.4 dollars per hour more han her female colleagues of smlar work experence. For a es of H0 : ( α + α) = 0 compuaons gave a -sasc value of 8.96 accompaned by a p-value for a wo-al es less han o gve srong evdence o rejec he null hypohess and conclude ha, for marred workers, a gender wage gap s sascally sgnfcan. The hypohess ha neher maral saus nor gender affecs wages can also be esed. Ths s he jon hypohess es: H : α = 0, α = 0, and α 0 = H : a leas one s no zero The F-es procedure was descrbed n prevous lecure noes. 0 Econ 6 - Chaper 7 4 Econ 6 - Chaper 7

8 Example Qualave Facors wh Several Caegores A research area of neres s measurng he rewards for educaon. In he wage deermnaon equaon he level of educaon may be nroduced by ncludng an educaon varable ha gves he years of educaon for each ndvdual. Oher schemes can be used. Suppose a survey records he hghes level of educaon as one of: o less han hgh school o hgh school graduaon o a unversy degree o a posgraduae degree To work wh hs, defne a se of educaon dummy varables as: E0 = f less han hgh school; and E = f hgh school graduaon; and E = f a unversy degree; and E = f a posgraduae degree; and = 0 oherwse. = 0 oherwse. = 0 oherwse. = 0 oherwse. Snce each ndvdual n he survey s assgned o only one educaon group a propery s: For he wage deermnaon equaon nroduced earler, educaon dummy varables can be ncluded n he lnear regresson equaon: y = β + α γe F + γ + α E Marr + α + γ E + e ( FMarr ) + β The varable E0 s excluded o avod he dummy varable rap (oo many dummy varables). The base group or reference group s male, unmarred workers wh less han hgh school educaon. The choce of a base group s arbrary. In hs se-up, he parameer γ measures he wage dfferenal beween workers wh hgh school graduaon and hose who dd no fnsh hgh school (he base group), afer conrollng for gender, maral saus and experence (x). The parameer γ γ measures he wage dfferenal beween workers wh a unversy degree and hose wh hgh school graduaon only, afer conrollng for oher facors. Afer model esmaon a varey of hypohess esng exercses can be consdered. x + E0 + E + E + E = 5 Econ 6 - Chaper 7 6 Econ 6 - Chaper 7

9 A research queson s: Are here rewards o educaon? A jon hypohess es s: H : γ = 0, γ = 0, and γ 0 = H : a leas one s no zero The F-es procedure (descrbed n earler lecure noes) can be mplemened as follows. STEP The regresson equaon wh he full se of varables s he unresrced model. Esmaon gves he sum of squared resduals SSE U. 0 STEP The F-es sasc s: (SSER F = SSE U SSE ) J (N K) where J s he number of resrcons and K s he number of coeffcens n he unresrced model. In hs example, J = and K = 8. The p-value for he es, calculaed wh compuer sofware, s: p = P(FJ,N K) ( > U F) STEP The resrced model assumes he null hypohess s rue and can be wren as: ( FMarr ) + βx v y = β + α F + α Marr + α + Esmaon gves he sum of squared resduals SSE R. A sandard sgnfcance level s If he p-value s less han 0.05 hen here s evdence o rejec he null hypohess and conclude ha here are sgnfcan wage dfferences for hgher levels of educaon. 7 Econ 6 - Chaper 7 8 Econ 6 - Chaper 7

10 Conrollng for Tme A macroeconomc model may use quarerly me seres daa. Seasonal varaons can be mporan. Sascal agences have smoohng mehods ha adjus daa for seasonal effecs o gve quarerly seasonally adjused daa. For model esmaon, may be nformave o work wh daa ha s repored as quarerly seasonally unadjused. Seasonal varaons can be modelled by nroducng a se of seasonal dummy varables defned as: Q = f quarer ; and = 0 oherwse. Q = f quarer ; and = 0 oherwse. Q = f quarer ; and = 0 oherwse. A se-up of he daa se for he frs hree years s below. The frs four observaons are he daa for year one of he sample. The 0- codng for he quarerly seasonal dummy varables s shown. y x Q Q Q Q4 y x y x y x y4 x y5 x y6 x y7 x y8 x y9 x y0 x y x y x Q4 = f quarer 4; and = 0 oherwse. Wh quarerly daa, a model can be saed as: y = β + β Q + β Q + β Q4 + β x + e 4 5 for =,,..., T where he number of mes seres observaons s T, y s he dependen varable and x s an explanaory varable (he equaon may also nclude oher explanaory varables suggesed by he economc heory). For he regresson equaon, o avod he dummy varable rap, he Q varable s no ncluded. The dfferenal nerceps are: β quarer β + β quarer β + β quarer β + β 4 quarer 4 Hypohess esng can consder he sgnfcance of he seasonal effecs. 9 Econ 6 - Chaper 7 0 Econ 6 - Chaper 7

11 Tesng he Equvalence of Two Regressons Separae equaons may descrbe wo groups of daa. A smple model ha llusraes hs dea s: Group Group y + = α + αx e for =,..., N y + = β + βx e for = N+,..., N The sample s spl no wo groups. The frs group has N observaons and he second group has N observaons. The oal number of observaons s: N = N + N Example: Wh macroeconomc me seres daa he breakpon N may be suggesed by he nroducon of a new governmen economc polcy. When comparng he wo groups, a es of neres s: To develop he es, an assumpon s ha he equaon errors n he wo groups are ndependenly and normally dsrbued wh he same error varance σ. A es sasc s calculaed as follows. STEP STEP Combne (or pool) all observaons and oban leas squares esmaes of a common nercep and common slope coeffcen for he wo groups. Ths s called a pooled regresson. The sum of squared resduals from hs regresson s denoed by: SSE 0 Esmae separae regressons for each group and oban he sum of squared resduals: SSE for Group and SSE for Group H 0 : same coeffcens n he wo groups (srucural sably) H : dfferen coeffcens (srucural change) A es procedure s he Chow es (proposed by Gregory Chow n 960). Econ 6 - Chaper 7 Econ 6 - Chaper 7

12 STEP Calculae he Chow es sasc: F = ( SSE (SSE + SSE )) 0 (SSE / K + SSE )/(N K) The es sasc can be compared wh an F-dsrbuon wh (K, N K) degrees of freedom. The p-value of he es s calculaed as: p = P(FK,N K) ( > F) A p-value smaller han he chosen sgnfcance level gves evdence o rejec he null hypohess n favour of he alernave of dfferen coeffcens n he wo groups. The Chow es sasc does no say where he sources of dfferences are he nercep coeffcen, or he slope coeffcen, or boh. To address hs concern, an equvalen esng approach s o make use of dummy varables. Defne: D = 0 for observaons n Group for observaons n Group Pool he observaons and work wh he regresson equaon: y + = β + γd + βx + γ(xd ) e for =,,..., N Noe ha he equaon ncludes a slope dummy varable. Ths gves: E (y ( β + γ ) + ( β ) = β + βx + γ ) x for for Group Group Econ 6 - Chaper 7 4 Econ 6 - Chaper 7

13 An example of a daa se wh 0 observaons n group and 0 observaons n group s below. The frs 0 observaons are for group, followed by he 0 observaons for group. y D x (x D) y x x y x x y x x y4 x4 x4 y5 x5 x5 y6 x6 x6 y7 x7 x7 y8 x8 x8 y9 x9 x9 y0 x0 x0 y 0 x 0 y 0 x 0 y 0 x 0 y4 0 x4 0 y5 0 x5 0 y6 0 x6 0 y7 0 x7 0 y8 0 x8 0 y9 0 x9 0 y0 0 x0 0 slope dummy From he esmaon resuls for he general regresson, a varey of hypoheses can be esed. The es for srucural sably (he Chow es) s: H : γ = 0 and γ 0 = H : a leas one s no zero 0 Also, ndvdual ess for a dfferen nercep coeffcen ( γ 0) or a dfferen slope coeffcen ( γ 0) n he wo groups can be obaned from he -ess of sgnfcance repored as a sandard par of he compuer esmaon resuls. 5 Econ 6 - Chaper 7 6 Econ 6 - Chaper 7

14 Models wh a Log-Dependen Varable Example: Chaper 7 Texbook Compuer Exercse. Daa on he weekly sales of a major brand of canned una by a supermarke chan n a large mdwesern U.S. cy durng a md- 990s calendar year are conaned n a daa fle. The number of observaons s T = 5. A regresson equaon s: ln( y ) = β + β pr + β pr + β pr + β D + β D + e where y s un sales of Brand no. una n week, pr s prce n $ per can of Brand no. una n week, pr s prce n $ per can of Brand no. una n week, pr s prce n $ per can of Brand no. una n week, Leas squares esmaon resuls are presened n he able. Dependen Varable: log sales of Brand no. Varable Coeffcen Esmae Sandard Error p-value nercep < Prce of brand < Prce of brand Prce of brand Adversng dummy < Adversng dummy < R =0.84 Number of observaons 5 The repored p-values are for a wo-al es of sgnfcance where he null hypohess s a zero coeffcen on he explanaory varable. D s a dummy varable se o one f here s a sore dsplay (bu no newspaper ad) for Brand no. n week and zero oherwse, and D s a dummy varable se o one f here s boh a sore dsplay and newspaper ad for Brand no. n week and zero oherwse. 7 Econ 6 - Chaper 7 8 Econ 6 - Chaper 7

15 Dscuss and nerpre he esmaes of β, β and β 4. Before model esmaon, a commen can be made on he expeced sgns of he coeffcens on he prce varables. An ncrease n prce, holdng all else he same, should lead o a declne n sales. Ths suggess a negave coeffcen for β. Brands and can be vewed as subsues for Brand. An ncrease n he prce of subsues, holdng all else he same, may lead o a sales ncrease. Therefore, posve coeffcens for β and β 4 are expeced. The esmaon resuls show ha he esmaes for he hree prce coeffcens all have he expeced sgn. The p-values, gven n he fnal column of he able of esmaon resuls, reveal ha, a a 5% sgnfcance level, all esmaed prce coeffcens are sascally sgnfcan ha s, sgnfcanly dfferen from zero. Alhough, a a % sgnfcance level, he esmaed prce coeffcens for he subsue brands are no sascally sgnfcan. Now focus on he economc nerpreaon of he prce coeffcens. Wh a log-dependen varable, he coeffcens are nerpreed as proporonal changes n sales from a one-un change n prce. For example, a one dollar ncrease n he prce of Brand wll lead o a 00(.75) = 75% declne n sales (all oher varables held consan). An equvalen saemen s: a one cen change n prce wll promp a.75% drop n sales. An economs may be neresed n reporng an elascy defned as: ln(y) ln(y) = pr = β ln(pr ) pr pr For he sample, he average prce of Brand una was 78 cens per can. From he esmaon resuls, an esmae of he prce elascy evaluaed a he sample mean s: b (0.78) = (.746)(0.78) =.9 Ths says ha a one percen ncrease n he prce of Brand una wll gve a.9% reducon n sales. 9 Econ 6 - Chaper 7 0 Econ 6 - Chaper 7

16 The prce coeffcens on he subsue brands can also be nerpreed. A one dollar ncrease n he prce of Brand wll lead o a 5% ncrease n sales for Brand (all oher varables held consan). The average prce of Brand una was 8 cens per can. An esmae for he subsue prce elascy evaluaed a he sample mean s: b (0.8) = (.49)(0.8) = 0.9 Ths says ha a one percen ncrease n he prce of Brand una leads o a 0.9% ncrease n sales for Brand una. Are he sgns and relave magnudes of he esmaes of β 5 and β 6 conssen wh economc logc? In hs applcaon, he coeffcens on he dummy varables gve he mpac of dfferen promoonal mehods on sore sales. When workng wh a log-dependen varable care mus be aken o nerpre he coeffcens on dummy varables. Denoe S as sales wh no sore dsplay and S as sales wh a sore dsplay bu no newspaper ad. The model gves: ln( S) ln(s) = β 5 S Tha s, ln = β5 S S By akng an-logs: = exp( β ) S 5 Therefore, he proporonae change n sales generaed by he sore dsplay s: S S = exp( β S 5 ) The percenage change n sales generaed by he sore dsplay s: ( exp( ) ) 00 β 5 Econ 6 - Chaper 7 Econ 6 - Chaper 7

17 The esmaed coeffcen on he dummy varable for a sore dsplay (bu no newspaper ad) for Brand no. s b 5 = Ths value s used o oban an esmae of he percenage ncrease n sales generaed by he sore dsplay as: ( exp(b ) ) = 00( exp( 0.44 ) ) = % The esmaed coeffcen on he dummy varable for boh a sore dsplay and newspaper ad for Brand no. s b 6 =.4. Ths leads o an esmae of he percenage ncrease n sales resulng from boh a sore dsplay and newspaper ad as: ( exp(b ) ) = 00( exp(.4 ) ) = % The concluson s, a sore dsplay ncreases sales; a sore dsplay combned wh a newspaper adversemen ncreases sales by an even bgger amoun. To convnce he supermarke chan s execuves abou he mpac of a sore dsplay and adversng on sales some hypohess ess may be useful. To es he clam ha he combnaon of an adversemen and sore dsplay wll ncrease sales by more han can be acheved wh a sore dsplay only an approprae one-sded es s: H0: Ths can be saed as: β 6 β5 agans H : β 6 > β5 H0 6 5 : β β 0 agans : β β 0 H 6 5 > From he leas squares esmaon resuls, he -es sasc as: = = b 6 b vâr(b6) + vâr(b5) côv(b ( ) 5 5, b 6 ) = = 6.86 The p-value for hs rgh-al es s calculaed as: ( >6.86 ) p = P (46 ) < The p-value s less han any reasonable sgnfcance level (such as 0.05 or 0.0) and herefore he null hypohess s rejeced. There s srong evdence o suppor he clam ha newspaper adversng ncreases sales by more han can be acheved wh a sore dsplay alone. Econ 6 - Chaper 7 4 Econ 6 - Chaper 7

18 I s useful o consder some dagnosc ess ha may reveal general model msspecfcaon problems. Two ess o consder are: he Jarque-Bera es for normally dsrbued resduals, he Ramsey RESET es for model msspecfcaon. The model esmaon resuls repored a Jarque-Bera es sasc of 0.45 wh an accompanyng p-value of 0.8. Ths large p-value leads o he concluson ha he null hypohess of normal resduals s suppored by he daa. One verson of he RESET es augmens he model wh an exra varable creaed as he squared predcons of he dependen varable (n hs case, he log dependen varable). The dea s ha f he f of hs arfcal model s sgnfcanly mproved by he ncluson of an exra varable hen model msspecfcaon s revealed. For he una sales model, he p-value for a es of sgnfcance on he exra varable was 0.0. Ths s below a sgnfcance level of 0.05 o gve evdence ha he model may be nadequae. Ths suggess ha oher modellng sraeges, such as dfferen funconal forms, should be explored before arrvng a fnal conclusons abou he mpac of prce and adversng on sore sales. 5 Econ 6 - Chaper 7 6 Econ 6 - Chaper 7

19 Example More Work wh Funconal Forms Prevous lecure noes proposed he wage deermnaon equaon: where ( FMarr ) + βx e y = β + α F + α Marr + α + y s average hourly earnngs n dollars, x s experence n years, F = f female; = 0 f male (he gender dummy) and Marr = f marred; = 0 oherwse (he maral saus dummy) For he daa se wh N = 56 observaons he esmaed regresson equaon s: ( FMarr ) ln(x ) lnˆ(y ) = F Marr sasc (.4) (.6) (5.6) ( 4.6) (.) p-value (< ) (0.0) (< ) (< ) (0.00) The squared correlaon beween he an-log predced and observed values s: Rg = 0. 9 The wage-experence profle for male, marred workers s shown n he fgure. Ths assumes ha as experence ncreases wage wll ncrease a he same consan rae. In realy, may be expeced ha as experence ncreases, wage wll ncrease a a decreasng rae. How can hs behavour be modelled? wage One opon s o work wh log-ransformed daa. The wage deermnaon equaon can be saed: ( FMarr ) + βln(x) e ln( y ) = β + α F + α Marr + α + Noe: Dummy varables are no log-ransformed experence n years If 0 < β < hen, as experence ncreases, wage ncreases, bu a a decreasng rae. 7 Econ 6 - Chaper 7 8 Econ 6 - Chaper 7

20 From he esmaon resuls a gender wage gap can be suded. The nerpreaon of coeffcens on dummy varables when he model has a log-dependen varable was llusraed earler wh he canned una sales example. For he wage equaon he esmaed coeffcen on he female dummy varable reveals ha, for unmarred workers, he percenage dfference beween wages of female and male workers s roughly 4.%. A more accurae calculaon s o esmae he wage dfferenal beween female and male unmarred workers as: 00 (exp( 0.4) ) =.% For marred workers, he wage dfferenal for female workers relave o her male colleagues s calculaed as: Anoher modellng sraegy s o nclude a quadrac erm n he wage deermnaon equaon. The regresson equaon s: ( FMarr ) + βx + βx e y = β + α F + α Marr + α + Wh β > 0 and β < 0 he wage-experence funcon wll have an nvered-u shape. Esmaon resuls are: ( FMarr ) + 0.5x x ŷ = F +.0Marr () (9.57) (.4) (4.45) ( 4.65) (5.96) ( 5.90) R = 0. The wage-experence profle for male, marred workers s shown n he fgure. 00(exp( α ˆ + αˆ ) ) = 00(exp( ) ) = 9.7% wage experence n years 9 Econ 6 - Chaper 7 40 Econ 6 - Chaper 7

21 The nvered-u shape can be noed. A specal feaure s ha he wage-experence funcon peaks a a urnng pon. The urnng pon can be esmaed. The slope s: y = β x + β x The slope changes for dfferen levels of experence (x). The urnng pon, denoed by β * + β x = 0 Rearrangng gves: * x β β = * x, s found by solvng: Anoher funconal form ha may be favoured for he wage deermnaon equaon s o use a log-ransformed dependen varable and nclude lnear and quadrac erms for he experence varable. The regresson equaon s now: ( FMarr ) + βx + βx e ln( y ) = β + α F + α Marr + α + For he example daa se, esmaon resuls are: l nˆ(y ) = ( FMarr ) x x F + 0.5Marr ()(.) (.) (4.95) ( 4.) (6.4) ( 6.9) The squared correlaon beween he an-log predced and observed values s: Rg = 0. 4 Ths model gves a wage-experence profle for male, marred workers as llusraed n he fgure. Ths can be appled o he above esmaon resuls o ge he urnng pon: xˆ* = =.58 ( ) years wage experence n years 4 Econ 6 - Chaper 7 4 Econ 6 - Chaper 7

22 The urnng pon y = 0 x For hs model y ln(y) = y = y x x The dervave s zero when or β * + β x = 0 * x s he value a whch ( β + x) β For each funconal form, specal care mus be aken o assgn correc nerpreaon o he esmaed coeffcens. Tha s he nerpreaon depends crucally on he ypes of ransformed varables ha are nroduced n he regresson equaon. For he dfferen varaons of he wage deermnaon equaon presened here, each model gves dfferen numerc resuls for he measuremen of a gender wage gap among marred and unmarred workers. Afer model esmaon, dagnosc ess such as he Jarque-Bera es for normally dsrbued resduals and he RESET es may be useful o repor. These ess may gve evdence of general model msspecfcaon. * x β β = From he resuls wh a log-ransformed dependen varable an esmae of he urnng pon s years. 4 Econ 6 - Chaper 7 44 Econ 6 - Chaper 7