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244 ANALYSIS OF PANEL DATApermanent effects attached to the subjects. For binary outcomes, restrictions are neededfor identifiability. Thus, if a constant variance f of the e it is assumed, then it is necessarythat f ˆ 1 (or possibly some other preset value). Note, however, that time varyingvariances f t may be identifiable provided one of them (e.g. f 1 ) is set to a pre-specifiedvalue. Heckman extended this error structure to include coefficients on the a i . This isknown as the one factor modelZ it ˆ bX it l t a i k t e it (6:19)where, if the e it are taken as uncorrelated, the l t may be used to describe the correlationbetween time points t and s (see Example 6.6). For count outcomes, Dagne (1999)proposes a similar model with the loadings either over subjects or times. Thus,Y it Poi(m it ), and for loadings varying by subjectm it ˆ l i a i bX it e itOrdinal panel data Models for ordinal responses over time are important because inmany settings involving human subjects, classifications are on a graded scale, withprecise quantification not being possible. Examples include pre- and post-treatmentobservations on rankings of illness symptoms (e.g. no symptoms, mild, definite) orchanged illness states, as well as survey questions on changing views on controversialtopics. As discussed in Chapter 3, a continuous scale may often be envisaged to underliethe grading, with a series of thresholds t 1 , t 2 , : : t C1 defining which of C categories asubject lies in. Then with Z it taking one of the forms as above, for exampleZ it ˆ b 0 bX it e itwe have81 if Z it < t 1>< 2 if t 1 < Z it < t 2Y it ˆ . . .>: C 1 if t C2 < Z it < t C1C if Z it > t C1Alternative parameterisations are possible to ensure identifiability: either taking b 0 ˆ 0,or t 1 ˆ 0, or t C ˆ C ensure that the mean of the Z is identified (Long, 1997, p. 122). Thevariance of Z may be identified by taking e to be N(0, 1). This leads to the ordinal probitmodel, whereas taking e to be Student t(8) have variance 1 leads to the ordinal logitmodel.In panel setting one might consider shifts in the location of thresholds by making thecut-points time specific ± for example, if the analysis was intended to assess whetherthere had been a shift in attitudes. Random variations in intercepts between subjects orvariation in trends across time (random time slopes) may also be modelled in panelsettings because of the repetition of observations over subjects.6.3.1 Beta-binomial mixture for panel dataMany repeated observations of social choice processes are available as binary series attimes 1, 2, . . . T and aggregate to binomial series for total subject populations or subpopulationsof subjects. For example, the events migration, job change or divorceare binary though Y it ˆ 1 may sometimes include more than one event. As Davies