Heteroskedastcty, the sngle crossng property and ordered response models Andreas C. Drchouts Agrculural Unversty of Athens Panagots Lazards Agrculural Unversty of Athens Rodolfo M. Nayga, Jr. Texas AMUnversty Abstract Heteroskedastcty n ordered response models has not garnered enough attenton n the lterature. Econometrc software packages do not handle ths problem satsfactorly ether. We provde formulas to calculate heteroskedastcty corrected margnal effects and dscrete changes usng an approach that deals wth sngle crossng property, a very restrctve assumpton of ordered response models. We wsh to thank Xmng Wu for helpful comments on an earler draft of ths paper. Ctaton: Drchouts, Andreas C., Panagots Lazards, and Rodolfo M. Nayga, Jr., (006) "Heteroskedastcty, the sngle crossng property and ordered response models." Economcs Bulletn, Vol. 3, No. 3 pp. -6 Submtted: November 0, 006. Accepted: November 4, 006. URL: http://economcsbulletn.vanderblt.edu/006/volume3/eb-06c0009a.pdf
. Introducton Ordered response models orgnated from the bometrcs lterature and ther appearance n the socal scences s attrbuted to McKelvey and Zavona (975). Snce then, many applcatons and extensons of these models have appeared n the economcs lterature. Most recent s Boes and Wnkelmann s (005) attempt to develop more flexble models that wll overcome the restrctons nherent n standard models (.e., the sngle crossng property - that s the sgns of the margnal effects can only change once when movng from the smallest to the largest category). Despte the growng popularty of ordered response models, the lterature provdes no clear way of accountng for heteroskedastcty n these models. Very few econometrc packages can also account for heteroskedastcty, and even for those that could, the software generally cannot dstngush cases where heteroskedastcty s created by the same covarates that are ncluded n the model. Consequently, ths ssue s often gnored n model estmaton. Usually f one manages to derve heteroskedastcty corrected margnal effects, these are presented by softwares n two tables: one for the varable n the model and one for the varable n the heteroskedastc term. Clearly, the more approprate way would be to account for the smultaneous varaton of the varable n the model and n the heteroskedastc term. Snce the lterature s lmted and vague n ths area, we show how to derve the approprate formulas to account for heteroskedastcty n ordered response models. Furthermore, we also show that accountng for heteroskedastcty provdes a more flexble analyss of margnal effects snce the restrctve sngle crossng property vanshes.. The standard ordered response model Assume there s a latent varable y rangng from to + and s mapped to an observed varable y. The y varable s thought as provdng ncomplete nformaton about the underlyng y accordng to the measurement equaton: y = m f τ m y < τ m for m= to J () The τ s are called thresholds and the extreme categores and J are defned by open-ended ntervals wth τ 0 = and τ J =+. The structural model s: y = b'x + ε () where b s a vector of structural coeffcents. The observaton mechansm results from a complete censorng of the latent dependent varable as follows: y = 0 f y τ 0, = f τ 0 < y τ, = f τ < y τ, (3) L = J f y τ J. Maxmum lkelhood estmaton (ML) can be used to estmate the regresson of y on x. To use ML one has frst to assume a specfc form of the error dstrbuton. The ordered probt model flows from the assumpton that ε s dstrbuted normally wth mean 0 and varance, whle the ordered logt model results from the assumpton that ε has a logstc dstrbuton wth mean 0 and varance π 3. Snce y s not observable ts nterpretaton s of no nterest. The man focus n ordered data s on the condtonal cell probabltes gven by: Pr y = m x = F τ F (4) ( ) ( ) ( ) m τm
where F represents ether the standard normal dstrbuton functon or the logstc dstrbuton. The parameter estmates from ordered response models, such as ordered probt and ordered logt, must be transformed to yeld estmates of the margnal changes, that s, to determne how a margnal change n one regressor changes the dstrbuton of all the outcome probabltes. Takng the partal dervatve of (4) wth respect to x k yelds the margnal effect, Pr ( y = m x) F( τm ) F( ) MEmk = = = xk xk xk (5) = bk f ( ) f ( τm ) where f ( β ) = df ( β) dβ. Interpretaton usng the margnal effects can be msleadng when an ndependent varable s a dummy varable. Hence, t s more approprate to calculate the dscrete change whch s the change n the predcted probablty for a change n x k from the start value 0 to the end value. Δ Pr ( y = m x) DSmk = = Pr ( y = m x, xk = ) Pr ( y = m x, xk = 0) (6) Δxk It s clear from (5) that margnal effects are frst postve (negatve) and then negatve (postve) dependng on the sgn of b k. In the case of dscrete changes, the sngle crossng property s not mathematcally clear but s ntutve, consderng that equaton (6) nvolves the dfference between two probabltes that follow the bell-shaped dstrbuton functons of the standard normal and logstc dstrbuton. 3. Ordered response models wth heteroskedastcty In the case where the form of the heteroskedastcty adds no new parameters.e. Var ε = w, then the procedure s very smple snce (4) wll be, ( ) τm τm Pr ( y m ) F F = x = (7) w w And consequently (5) and (6) wll yeld: Pr ( y = m x) b k τm MEmk = = f f (8) xk w w w τm τm DSmk = F F F F (9) w w w xk w = x k = 0 In the case where heteroskedastcty adds an addtonal parameter vector (multplcatve heteroskedastcty),.e. Var ( ε ) exp( ) = γ'z, thngs mght be trcker. If the z vector contans no common varables wth the x vector, then margnal effects and dscrete changes can be calculated by (8) and (9) snce w = exp( γ'z ). However, f one or more varables n the z vector are common to the x vector, then the calculaton of the margnal effects and dscrete changes have to account for the fact that the varables of nterest appear both n the nomnator and the denomnator of equatons (8) and (9). Formally, assume that we can break the z vector nto two vectors z and x where the x vector contans parameters common to x. We can then wrte Var ( ε ) ( ) = exp γ 'z + γ 'x and (4) wll take the form:
Pr y = m = F F ( x ) τm τm exp( + ) exp( + ) γ 'z γ 'x γ 'z γ 'x Takng the partal dervatve of (0) wth respect to x k wll yeld the margnal effect, Pr ( y = m x) MEmk = = xk τ exp( ) ( ) exp m bk γ 'z + γ 'x + b'x γ k ( γ 'z + γ 'x ) = f exp( ) γ 'z + γ 'x ( exp( γ 'z + γ 'x) ) τ b exp k ( ) ( τm ) γkexp m γ 'z + γ 'x + b'x ( γ 'z + γ 'x) f = exp( + ) γ 'z γ 'x ( exp( γ 'z + γ 'x) ) b k b'x = f f + exp( γ 'z + γ 'x ) exp ( ) exp( ) γ 'z + γ 'x γ 'z + γ 'x γ k ( ) ( ) b'x + b'x f ( ) f exp γ 'z + γ 'x exp( + ) γ 'z γ 'x exp γ 'z + γ 'x ( ) The formulas are much smpler n the case of dscrete changes, snce (6) wll then be: τm DSmk = F F exp( ) exp( ) γ 'z + γ 'x γ 'z + γ 'x x k = () τm F F exp( + ) exp( + ) γ 'z γ 'x γ 'z γ 'x x k = 0 Note that the dfference between () and (9) s that the numerator wll vary smultaneously wth the denomnator. Clearly, () and () no longer rely on a sngle crossng property. The sgn n () s ndetermnate. Whle () stll nvolves the dfference between cell probabltes, these are scaled up or down dependng on the heteroskedastc terms. In all, f x k s subset of x then one should use equatons (8) and (9), but f x k s subset of x then equatons () and () are approprate. (0) () 4. Emprcal llustraton To llustrate how the presence of heteroskedastcty changes margnal effects and dscrete changes, we use the data from Drchouts et al. (005). The purpose of the paper was to assess the effect of several varables on nutrtonal label use. Specfcally, ther paper nvestgated whch factors may have an effect on how often consumers tend to read the onpack nutrton nformaton of food products when grocery shoppng. Label use was measured on a four lkert scale rangng from never, not often, often and always. We defne only a parsmonous form of the model snce our purpose s to llustrate the estmaton process wth and wthout heteroskedastcty. Therefore, we wll only assume that label use s affected by household sze (Hsze), age (Age40, Age55, Age56), educaton (Educ ) and ncome (Inc, Inc 3 ), Label Use = f (Hsze, Age40, Age55, Age56, Educ, Inc, Inc 3 ) (3) In (3), only household sze s a contnuous varable. All the other varables are dummes, ndcatng varous age groups, educatonal levels and ncome levels. In Table, we present the results of the above estmaton usng an ordered probt model. The frst half of the table 3
presents the results where no heteroskedastcty s assumed. The second half of the table shows results where we assumed that household sze and ncome are responsble for heteroskedastcty. Table shows that gnorng the presence of heteroskedastcty can be msleadng n terms of magntude but also n terms of the drecton of the effect. Notce also that the sngle crossng property vanshes snce the sgn of some varables n Table changes twce when movng from the smallest to the largest outcome category provdng a more flexble way to analyze the data. 5. Concluson The need to account for heteroskedastcty n ordered response models s a problem, f present, snce t can lead to erroneous results. That s, t can ether overestmate or underestmate the true varance and, hence the standard errors may therefore be ether understated or overstated. Ths ssue becomes more relevant consderng that known econometrc software packages do not handle ths problem satsfactorly. We provde formulas to calculate heteroskedastcty corrected margnal effects and dscrete changes. We also show that our approach deals wth a very restrctve assumpton of the ordered response models (.e., sngle crossng property). 4
Table. Margnal effects and dscrete changes of demographc varables on frequency of readng nutrtonal labels Varables No heteroskedastcty Groupwse heteroskedastcty Not Not Never often Often Always Never often Often Always Hsze 0.04 0.004-0.005-0.03-0.005 0.03 0.07-0.04 Age40-0.066-0.0 0.00 0.067-0.079-0.047 0.045 0.08 Age55-0.0-0.04 0.033 0.8-0.49-0.096 0.076 0.69 Age56-0.035-0.0 0.00 0.036-0.05-0.03 0.08 0.054 Educ -0. -0.03 0.036 0.08-0.0-0.06 0.07 0. INC 0.034 0.009-0.0-0.03-0.003 0.07 0.05-0.093 INC 3 0.03 0.05-0.037-0.09 0.04 0.068-0.078-0.3 References Boes, S. and R. Wnkelmann (005) "Ordered response models", Workng paper no. 0507, Socoeconomc Insttute, Unversty of Zurch. Drchouts, A. C., P. Lazards and R. M. Nayga, Jr. (005) "Nutrton knowledge and consumer use of nutrtonal food labels" European Revew of Agrcultural Economcs 3,, 93-8. McKelvey, R., and W. Zavona (975) :A statstcal model for the analyss of ordered level dependent varables" Journal of Mathematcal Socology 4, 03-0. 5