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1 ISSN ISBN THE UNIVERSITY OF MELBOURNE DEPARTMENT OF ECONOMICS RESEARCH PAPER NUMBER 95 NOVEMBER 005 INTERACTIONS IN REGRESSIONS by Joe Hirschberg & Jenny Lye Deparmen of Economics The Universiy of Melbourne Melbourne Vicoria 3010 Ausralia.
2 Ineracions in Regressions J. Hirschberg and J. Lye 1 Deparmen of Economics Universiy of Melbourne November 005 Absrac Regression specificaions frequenly employ regressors ha are defined as he produc of wo oher regressors o form an ineracion. Unforunaely, hese models have a number of poenial difficulies when i comes o inerpreaion. In his paper, we discuss wo common aspecs of hese specificaions ha can lead o difficulies. The firs is he change in parameer esimaes and - saisics when differen definiions of dummy variables are used. The second involves an applicaion of Fieller s heorem o draw appropriae inferences from a regression wih ineracion variables. Key Words: Ineracion effecs; dummy variables; linear ransformaion, Fieller mehod JEL classificaion: C1; C51 1 Corresponding auhor, jnlye@unimelb.edu.au, Deparmen of Economics, Universiy of Melbourne, Melbourne, Vicoria 3010, Ausralia. The research carried ou in his paper was parially funded by a gran from he Faculy of Economics and Commerce a The Universiy of Melbourne. We wish o hank David Moreon for helpful commens. 1
3 1 Inroducion Regression models wih ineracion effecs are ofen proposed o allow for he marginal effec of one explanaory variable o depend on anoher. In his paper, we discuss wo aspecs of ineracions in regression specificaions. In Secion, we show how dummy variables are prone o linear affine ransformaions and discuss he implicaions for inerpreing an ineracion variable in he regression equaion when he ineracion variable involves a dummy variable. An example is included o highligh he impac on parameer esimaes and -saisics. In Secion 3, he inerpreaion of regressions wih ineracions is proposed by using an applicaion of Fieller s heorem. Two empirical examples are presened. Conclusions are presened in Secion. Variable Transformaions, Ineracions and Dummy Variables Griepenrog e al. (198) (GRS) show ha for models ha include an ineracive erm an affine linear ransformaion of one variable in he ineracive erm affecs he -saisic for he lower order of he oher variable. In his paper, we relae his resul o he case in which he ineracive erm involves a dummy variable. Following GRS, consider a linear model of he form: y = X β + ε (1) where y is an n 1 marix of observaions on he dependen variable, ε is an n 1 marix of disurbances, X is an n k marix of observaions on all of he explanaory variables, ha is, [ ] and = [ β β β β β β ] X 1 x w xw v v = 1 k coefficiens. akes he form: β is a k 1 vecor of k Suppose ha z, he h observaion, is formed by an affine linear ransformaion of x ha z = a + bx. () where a and b are consans. Consider an alernaive model o (1) defined in erms of Z as: y = Zγ + ε (3)
4 where Z is an n k marix of observaions on all of he explanaory variables, ha is, [ ] and = [ γ γ γ γ γ γ ] Z 1 z w zw v v = 1 k γ is a k 1 vecor of k coefficiens. The relaionship beween (1) and (3) is given by: y = XAγ = Xβ + ε () where A is he non-singular marix, A11 A1 Α1 A, in which A 11= 1 a b 0 0, a b A = 0, A = I. From () i follows ha 1, k k-,k- γˆ = -1ˆ A β, where ˆβ is he ordinary leas squares esimaor of β. We can define he - saisics for esing he null hypoheses H 0 : β i = 0, i = 1,, k in (1) as i βˆ i =, where c ij is he ijh elemen of he marix ( ) 1 σ c ii XX. The corresponding saisics for γˆ esing he null hypohesis H 0 : γ i = 0in (3) are * i i =, where d ii is he diagonal elemen of he σ d marix 1 1 ( ) ( 1 ) A XX A. I can be shown ha * i = sign( b) i when i = and and * i = i when i = 5 k. However, he -saisics for i = 1 and 3 are defined as ii * i β ˆ a ( ) ˆ i b β i + 1 a a ( b) ( b) = σ c c + c ii i( i+ 1) ( i+ 1)( i+ 1). In paricular, he -saisic for esing he hypohesis H : γ = 0, which is a es of he significance of he coefficien for variable w in (3), is no 0 3 equivalen o he -saisic for esing he hypohesis H 0: β 3 = 0, ha is, he es saisic for he same variable in (1). Thus, a ransformaion of one of he variables in he ineracive erm has implicaions for he -saisic relaing o he linear effec of he oher variable in he ineracive erm. This resul is of paricular ineres when he ineraced variable is a dummy variable since hese are ofen subjeced o linear affine ransformaions. In paricular, if x = 1 when observaions 3
5 represen he presence of a paricular characerisic, we could jus as easily define anoher variable z = 1 when he characerisic is no presen, ha is, z = 1 x or in his case, a = 1 and b = 1..1 An Applicaion of Models wih Ineracive Dummy Variables This example is based on daa from Bernd (1991, p. 193) in he form of observaions on 550 individuals from he May 1985 Curren Populaion Survey. The naural logarihm of average hourly earnings in dollars is he dependen variable, ED is years of schooling, EX is poenial years of experience, MR is a dummy variable ha akes he value 1 if he individual is married and F is a dummy variable ha akes he value 1 if he individual is female. The following wo models are esimaed in which being female and being male are used as indicaors of gender, respecively: ( ) Log( Wage ) =β +β F +β MR +β F MR +λ ED +λ EX +λ EX +ε (5) ( ) (( ) ) Log( Wage ) =γ +γ 1 F +γ MR +γ 1 F MR +λ ED +λ EX +λ EX +ε (6) The resuls are repored in Table 1. Table 1. A comparison of he resuls of he specificaions (5) and (6) Dependen Variable: LOG(WAGE) Sample: 1 53 Gender =F (5) Gender = (1 F) (6) Variable Coeff -sa Coeff -sa C Gender MR Gender MR ED EX EX^ R-squared S.E. of regression 0. Adjused R-squared 0.98 Sum squared resid A comparison of he coefficien esimaes and -saisics for he Gender variable and he ineracion beween Gender and MR in Table 1 indicaes ha hey are of he same magniude bu of opposie sign. However, in model (5), he coefficien esimae and he -saisic associaed wih MR indicae ha marriage has a posiive and significan impac whereas in (6), he esimaed parameer for he marriage dummy is negaive and insignifican. Thus, while he coefficien on he redefined dummy variable is expeced o change sign, he change in he effec of he unchanged variable (MR)
6 is no inuiive. From Secion.1, alhough he parameer on MR is expeced o change according o γ =β a 3 3 b β (in his case,.05 = ), he change in he -saisic is inversely relaed o he covariance beween ˆβ 3 and ˆβ. 3 The Parial Influence of Ineraced Regressors Consider he model: y = β + β x + β w + β xw + β v + β v + ε (7) k k- where y is he dependen variable, x, w, xw, v j are explanaory variables andε is he k j= 5 disurbance erm. When x and w are boh coninuous variables, he marginal effec of a change in x is: ( ) E yxw, x = β + β w (8) If x is a dummy variable and w is a coninuous variable, we define he difference in he regression equaion when x =1 and when x = 0 as: ΔE, Δx = β + β w (9) In he conex of (8) and (9), he difficuly of choosing an appropriae value of he oher regressor a which o evaluae hese compuaions arises. One approach is o se he value of his oher regressor o a paricular value, such as he mean, and hen make he compuaion. Anoher approach is o deermine he value of he oher regressor a which hese definiions become zero or change sign. The value of w ha resuls in x = 0 or E, * β * βˆ Δ x = 0 is w = β and is esimaed by wˆ = ˆ, ΔE, β where β ˆ i are he OLS esimaes of β, i =,. A confidence bound for i deermine if he impac of x on y is significan over he possible values of w. * ŵ can be consruced o Noe ha a similar analysis can be performed on E, w 5
7 3.1 Confidence Inervals for he Parial Influence Funcion For he regression model in (7), or E, x Δ is defined as β + β w. An esimae of ΔE, x his can be ploed wih a 100(1 α)% confidence inerval given by: where ( ˆ ˆ w) ( ˆ ˆ ˆ α w ) CI = β + β ± σ + σ + w σ (10) σ ˆ i is he esimaed variance of ˆ β i, i =,, ˆσ is he esimaed covariance beween ˆβ and * ˆβ. An esimae of he value of w ( ŵ ), where = 0 or E, x Δ = 0 is found by solving ΔE, x βˆ + β ˆ w = 0. Similarly, he bounds defining he 100(1 α )% confidence inerval on * ŵ are found by solving: ( ˆ ˆ w) ( ˆ ˆ ˆ α w ) β +β ± σ + σ + w σ = 0 (11) This is equivalen o solving he roos of he equaion: ( ˆ ˆ w) ( ˆ ˆ ˆ α w ) β +β - σ + σ + w σ = 0 (1) Equaion (1) is equivalen o he quadraic equaion, ( ax + bx + c = 0 ), where a, b and c are ( ˆ a= β ˆ α σ ), b ( ˆ ˆ ˆ α ) = β β σ and c= βˆ σ ˆ, α wih an α ( ) level of significance and T k degrees of freedom. α is he value from he -disribuion The confidence limis obained in his way are idenical o hose found applying a version of he Fieller mehod (Fieller 193, 195) o a raio of linear combinaions of regression parameers (Zerbe 1978). In order o have real roos, ha is, o have a finie inerval, we require a > 0, which implies rejecion of he null hypohesis ha β = 0 based on he -saisic (Buonaccorsi 1979). Besides he finie inerval, he resuling confidence inerval may be he complemen of a finie inerval, (b ac > 0, a < 0), or he whole real line, (b ac < 0, a < 0). These condiions are discussed in Scheffé (1970) and Zerbe (198). If x or E, Δ x is nonlinear and defined as g(w), hen is esimae, say gw, ˆ( ) can be ΔE, ploed along wih a 100(1 α)% confidence inerval given by: 6
8 where var ˆ { ˆ( )} α { ˆ } CI = gw ˆ( ) ± z var ˆ gw ( ) (13) gw is he esimaed variance of gw ˆ( ) and z α is he value from he sandardized normal disribuion wih an α ( ) level of significance. An esimae (or esimaes) of he value of w, where x or E, Δ x is found by solving gw= ˆ( ) 0; similarly, he bounds defining he ΔE, 100(1 α)% confidence inerval a his poin(s) are found by solving (13) equaed o zero. Because gw ˆ( ) is nonlinear, here may be more han one soluion and, hence, more han one se of confidence bounds. In many insances, i is possible o rule ou a number of he soluions by examining he range of he daa. 3. Example 1: The Demand for Economic Journals Sock and Wason (003, p. 7) analyse he relaionship beween he number of subscripions o a journal a US libraries (Y) and is library subscripions price by using daa from 000 on 180 economics journals. Price is measured in prices per ciaion, which represens an approximaion of dollars per idea. Oher explanaory variables include he age of he journal (Age) and he number of characers per year in he journal (Char). The regression equaion is: ( ) Log( Y ) = β + β Log( Price) + β Log( Age) + β Log( Price) Log( Age) +β Log( Char) + u (1) The regression resuls are repored in Table, which shows ha he coefficien on he ineracive erm is highly significan. The price elasiciy is: which is esimaed as: ( Age ) η =β 1+ β 3 Log( ) (15) η ˆ = Log( Age) (16) The raio ( 0.11 ) defines he value of Log(Age) a which η ˆ = 0. 7
9 Table : Regression Resuls, Demand For Economic Journals Dependen Variable: Log(Y) Sample: Variable Coefficien -Saisic 3 C Log(Price) Log(Age) Log(Price)*Log(Age) Log(Char) R-squared 0.633S.E. of regression Adjused R-squared 0.66 Sum squared resid Figure 1 plos he esimaed price elasiciy and is 95% corresponding confidence inerval. Since he acual values of Age in he daa have a minimum value of (Log(Age)=0.60) and a maximum value of 156 (Log(Age)=.193), we conclude from Figure 1 ha he price elasiciy is significanly differen from 0 for all values of Age in his daa se. Figure 1: Parial Influence Funcion, Demand For Economics Journals Example Elasiciy < Fieller Inerval >.5% Lower bound 97.5% Upper bound Prediced Log of Age However, i is common pracice o repor a summary price elasiciy such as he value of he mean of all he price elasiciies calculaed by using (16), which in his example equals Figure 1 plos a reference line a his value from which we can deermine he values of Age for which he elasiciy is significanly differen from he mean of all he price elasiciies. In his 3 The -saisics are based on Whie Heeroskedasic-Consisen Sandard errors 8
10 example, his is he case for young and old journals, ha is, hose for which Age is less han (Log(Age) =.) or greaer han 9. (Log(Age) = 3.9), respecively. 3.3 Example : California Tes Daa Scores In his example, daa from Sock and Wason (003, p. 30) is used on es scores from 0 California school disrics in The es score measure (TESTSCR) is he disric-wide average of reading and mah scores for fifh graders. Addiional variables are he disric-wide suden eacher raio (STR); he percenage of sudens qualifying for a subsidized lunch (Meal_pc); and he average annual per capia income in he school disric measured in housands of 1998 dollars (Avginc). Furher, HIEL is a dummy variable ha equals 1 if he percenage of sudens sill learning English in he disric is greaer han 10%. The regression equaion examines wheher he effec of he suden-eacher raio depends no only on he value of he suden-eacher raio bu also on he percenage of English learners. Ineracions beween HIEL and STR, STR and STR 3 are included o allow he regression funcions relaing es scores and STR o be differen for low and high percenages of English learners. The regression equaion is: ( ) ( ) ( ) TESTSCR = β + β STR+ β STR + β STR + β HIEL +β STR HIEL +β STR HIEL +β STR HIEL +β Meal _ Pc +β log( Avginc) +ε (17) Table 3 repors he regression resuls. Individually, β, β5, β6, β 7 are significan a he 5% level and a join F-es of H 0 : β =β5= β6= β 7=0 resuls in a p-value of less han is defined as: The difference in he es scores when HIEL akes he value 1 and when i akes he value 0 which is esimaed as: ( Tesscr x) Δ HIEL =β +β 5 STR +β 6 STR +β 7 STR (18) ( ) ( ) ( ) ΔE 3 Δ E ( Tesscr x) 3 Δ HIEL = STR STR 0.10 STR (19) Equaion (19) only has a real roo when STR =
11 Table 3: Regression Resuls, California Tes Daa Scores Dependen Variable: TESTSCR Sample: 1 0 Variable Coefficien -Saisic C STR STR^ STR^ HIEL STR*HIEL STR^*HIEL STR^3*HIEL Meal_pc LOG(Avginc) R-squared S.E. of regression 8.57 Adjused R-squared Sum squared resid Figure () plos (19) and he corresponding 95% confidence inerval. We find ha Δ E Tesscr x ( ) Δ is HIEL significanly differen from zero for values of STR ha are greaer han 16.7 and less han However, only he former range (values from 16.7) overlaps wih he acual range of STR values (1.0 o 5.8). Thus, we can conclude ha he effec of suden-eacher raios (STRs) in schools wih a high proporion of sudens learning English is only negaive for hose schools ha have STRs of over Figure : Parial Influence Funcion, California Tes Daa Scores Example Differences < Fieller > Inerval 97.5% Upper bound.5% Lower bound Prediced Difference Suden Teacher Raio The -saisics are based on Whie Heeroskedasic-Consisen Sandard errors. 10
12 Conclusions Ineracive erms are ofen included in regression equaions o deermine how he effec on he dependen variable of one independen variable depends on anoher independen variable. However, an affine linear ransformaion of one of he variables in he ineracive erm has he counerinuiive effec of changing he -saisic on he linear effec of he oher variable. We have shown he implicaions of his when ineracion erms are based on dummy variables, which are prone o affine linear ransformaions. In addiion, we have also shown ha inferences can be drawn from a regression wih ineracion variables by examining he confidence bounds of he parial influence funcion. In he conex of linear regression, his is equivalen o he applicaion of Fieller s Mehod for he consrucion of confidence bounds for raios of random variables. 11
13 References Bernd, E. R., 1991, The Pracice of Economerics: Classic and Conemporary, Addison Wesley, Massachuses, USA. Buonaccorsi, J. P., (1979), On Fieller s Theorem and he General Linear Model, The American Saisician, 33, 16. Fieller, E. C., (193), The Disribuion of he Index in a Normal Bivariae Populaion, Biomerika,, 8 0. Fieller, E. C., (195), Some Problems in Inerval Esimaion, Journal of he Royal Saisical Sociey. Series B, 16, Griepenrog, G. L., J. M. Ryan and D. Smih (198), Linear Transformaions of Polynomial Regression Models, The American Saisician, 36, Scheffé, H., (1970), Muliple Tesing versus Muliple Esimaion. Improper Confidence Ses. Esimaion of Direcions and Raios, The Annals of Mahemaical Saisics, 1, 1 9. Sock, J. H. and M. W. Wason, (003), Inroducion To Economerics, Addison Wesley, Boson, USA. Zerbe, G., (1978), On Fieller s Theorem and he General Linear Model, The American Saisician, 3, Zerbe, G., (198), On Mulivariae Confidence Regions and Simulaneous Confidence Limis for Raios, Communicaions in Saisics Theory and Mehods, 11,
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