Estimation of dynamic linear models in short panels with ... - Cemmap

ESTIMATION OF DYNAMIC LINEAR MODELS INSHORT PANELS WITH ORDINAL OBSERVATIONStephen PudneyTHE INSTITUTE FOR FISCAL STUDIESDEPARTMENT OF ECONOMICS, UCLcemmap working paper CWP05/05

Estimation of dynamic linear models in short panels withordinal observationStephen PudneyCentre for Microdata Methods and PracticeInstitute for Fiscal StudiesandInstitute for Social and Economic ResearchUniversity of EssexSeptember 2004, revised May 2005Abstract: We develop a simulated ML method for short-panel estimation of oneor more dynamic linear equations, where the dependent variables are only partiallyobserved through ordinal scales. We argue that this latent autoregression (LAR)model is often more appropriate than the usual state-dependence (SD) probit modelfor attitudinal and interval variables. We propose a score test for assisting in thetreatment of initial conditions and a new simulation approach to calculate the requiredpartial derivative matrices. An illustrative application to a model of households’perceptions of their financial well-being demonstrates the superior fit of the LARmodel.Keywords: Dynamic panel data models, ordinal variables, simulated maximumlikelihood, GHK simulator, BHPS.JEL classifications: C23, C25, C33, C35, D84Address for correspondence: Steve Pudney, ISER, University of Essex,Wivenhoe Park, Colchester CO4 3SQ. E-mail: spudney@essex.ac.uk.I am grateful to Sarah Brown and Karl Taylor for allowing me to use their BHPS dataset and to themand to Manuel Arellano, Olympia Bover, Stephen Jenkins, Cheti Nicoletti and participants at the ISERJESS seminar and CIDE conference, Venice 2005, for helpful comments. This work was supported bythe Economic and Social research Council through project “Fertility and poverty in developingcountries” (award no. RES000230462) and the UK Longitudinal Studies Centre.

1 IntroductionIn discrete data modelling there is an important distinction to be made betweeninherent and observational discreteness. Inherent discreteness refers to a case wherethe variables of interest are naturally discrete. For example, an individual is eitheremployed or not employed; she has a university degree or not; she is married or not.Observational discreteness arises when the variables of interest are naturallycontinuous, but the survey instrument used to observe them imposes discreteness via apre-specified ordinal scale of allowable responses. This applies to a wide range ofattitudinal questions, which ask respondents to record their perceptions or beliefs on aLikert scale. Examples of econometric analysis of attitudinal variables haveproliferated in recent years, with the development of the economic literature onhappiness and satisfaction (see Van Praag and Ferrer-i-Carbonell, 2004, for a recentsurvey). There has been important work on econometric methodology in this area,particularly the choice between random and fixed effects modelling in panels (Ferreri-Carbonelland Frijters, 2004), but there has so far been little discussion of thedynamics of perceptions or of the most appropriate type of dynamic model to use.Observational discreteness does not only arise with attitudinal data. It mayalso occur in survey questions about more ‘objective’ entities like income, whenrespondents are required to place themselves within one of a number of given incomeranges. The discreteness here is an essentially artificial consequence of questionnairedesign. For example, business surveys often ask about expectations of future sales orinvestment intentions. The respondent’s expected value for sales or investmentconditional on his information set is a continuous variable but the survey questionstypically ask for a response graded as “up”, “down” or “no change”.Most of the econometric literature dealing with discrete models forlongitudinal data assumes inherent discreteness. The pioneering work of Heckman(1978, 1981a,b) centred on binary response models of the form:yy*it= αyit−1it= 1( y*it+ β'x> 0)where 1(.) is the indicator function, x it is a vector of strictly exogenous covariates, u iis an unobserved individual effect uncorrelated with x it and ε it is a random residualuncorrelated across individuals and time. We refer to (1) as the state dependence (SD)model. It was developed primarily for applications in labour economics, whereit+ ui+ εit(1)- 1 -

discreteness is inherent in the problem and where past outcomes of y it represent statedependence. In these applications, the latent variable y * itis essentially an artificialconstruct and there is no reason whyy should appear in the model (1).*it −1However, attitudes, expectations and incomes are not inherently discrete andthe use of models like (1), although common in the applied literature, is questionable. 1If the discrete nature of y it is only an artificial construct imposed by the questionnairedesigner, then behaviour centres on the continuous variable y * it, rather than theobserved indicator y it . In these cases,*yit−1rather than y it-1 , should carry the dynamicfeedback if the dynamic equation is to be a description of behaviour.The paper has four main objectives. Firstly, (above and in section 2) we makethe case for using dynamics in y* it−1, rather than y it-1, in applications where thediscreteness is observational rather than inherent; and then explore the interpretationof the model and its dynamic implications. The second objective, which is the subjectof sections 3-4, is to consider identification and propose a practical method ofestimation. The third aim is to set up a procedure for dealing with the initialconditions problem, using new specification tests which are proposed in section 5.Fourthly, we propose a new simulation method of estimating the cross-derivativematrices required for these tests; this is described in the appendix. Section 6 of thepaper presents an illustrative application to a panel data model of individuals’financial expectations and section 7 concludes.2 The model2.1 The statistical structureWe work with a behavioural model specified in terms of the ‘natural’continuous variables as follows:y*it= α y−+ β'x + u + ε*it 1We refer to this as the Latent Autoregression (LAR) model. The vector x it is assumedstrictly exogenous and individuals are sampled independently from the underlyingpopulation. We make the standard assumption of Gaussian random effects so that theunobservables u i and ε it satisfy the following assumptions:itiit(2)- 2 -

imperfect observation: there is a lack of natural units for ‘happiness’ or ‘utility’ whichrenders the scale of β inherently ambiguous. In some cases, where there are naturalunits of measurement for y * , we can fix σ ε at a reasonable hypothetical value. Forexample, for an analysis of survey responses to a question about expected inflation wemight reasonably set σ ε at (say) half a percentage point to allow a rough but directinterpretation of the model in terms of the natural units. Note that α is identifiableindependently of σ ε . As a consequence, we can estimate unambiguously the speed ofadjustment. For example, following a shock, the proportion of disequilibrium which iseliminated within s periods is 1-α s and this is unaffected by normalisation.2.3 DynamicsThe SD and LAR processes (1) and (2) imply different patterns of dynamic behaviour.Consider the following artificial example:SD model:y*t= .8yt−10 + x + ε(8)tLAR model:y*t= 0.422y−+ 0.355 + 0. 770x+ ε*t 1t(9)where x = 0.5, ε t ~ N(0,1) and y 1 ( y* > 0). The parameters of the LAR process (9)t=thave been chosen to reproduce exactly three properties of the SD process (8): 2(i) Pr(y = 1) = 0.877;(ii) ∂Pr(y = 1 | x)/∂x = 0.246;(iii) Pr(y t ≠ y t-1 ) = 0.170.With the LAR parameters chosen in this way, the distributions of run lengths in states0 and 1 are identical for the two processes. However, the relationship betweensuccessive run lengths is not. This is reflected in the autocorrelation functions (Figure1). As we would expect, the LAR model has much higher autocorrelations than theSD model for*yt. For the observed y t , the ACF decays faster for the SD than the LARprocess, despite the fact that they have the same 1st-order autocorrelation byconstruction. Thus, an LAR model will display greater persistence than anobservationally similar SD model, in this quite subtle sense.2 Conditions (i) and (ii) are imposed analytically to determine β 0 and β 1 for given α; Monte Carlosimulation was then used to find the value of α to equalise the 1 st order autocorrelations.- 4 -

0.450.40.350.30.250.2SD model: ACF(y)SD model: ACF(y*)LAR model: ACF(y)LAR model: ACF(y*)0.150.10.0501 2 3 4 5 6 7LAG ORDERFigure 1ACFs for the SD and LAR modelsThe two models also differ in terms of the implied dynamic multiplier effectsof x on y. To illustrate this, consider again the binary case and focus on two importantfeatures: the impact on Pr(y it =1 | y it-1 , X i , u i ) of switching the conditioning event fromy it-1 = 0 to y it-1 = 1; and the impact of the history of {x it } on the probability of apositive response, without conditioning on y it-1 .For the former, the SD model is relatively simple:Pr( yit= 1| yit−1= 1, Xi, ui) − Pr( yit= 1| yit−1= 0, Xi, ui) =Φ( α + β'x + u ) − Φ( β'x + u )where Φ(.) is the cdf of the N(0,1) distribution. For the LAR model, we have instead:itiiti(10)Pr( yit= 1| yit−1= 1, X , u ) − Pr( yiiit= 1| yit−1= 0, Xi, u )i=Pr( yitPr( y= 1, yit−1it−1= 1| X , u )= 1| X , u )iiii−Pr( yit= 1| X , u ) − Pr( yi1−Pr( yiit−1it= 1, yit−1= 1| X , u )ii= 1| Xi, u )i=Pr( yit= 1, yit−1Pr( y= 1| X , u ) − Pr( yit−1ii= 1| X , u ) 1iiit= 1| X , u ) Pr( yi[ − Pr( y = 1| X , u )]it−1iiit−1i= 1| X , u )ii(11)Assume the process (2) is stable and long-established. Then:y*it=∞∑s=0∞s uisα β ' xit− s+ + ∑αεit−s(12)1−αs=0- 5 -

and therefore Pr(y it =1, y it-1 =1 | X i , u i ) = Φ * (µ it , µ it-1 ; α) and Pr(y it =1 | X i , u i ) = Φ(µ it ),where Φ * (.,.;α) is the bivariate standard normal cdf with correlation α and µ it is thescaled conditional mean (1-α 2 ) 1/2 [∑ s α s β′x it-s +u i /(1-α)]. Thus:Pr( yit= 1| yit−1= 1, X , u ) − Pr( yiiit= 1| yit−1Φ(µit, µit−1;α)− Φ(µit) Φ(µit= 0, Xi, ui) =Φ(µ ) 1it−1[ − Φ(µ )]it−1−1)(13)The important difference between (10) and (13) is that the former depends only on thecurrent vector x it , whereas the latter depends on the entire history of x it .Consider now the alternative summary measure, Pr(y it =1 | X i , u i ). The LARprocess gives a relatively simple form:Pr( y = 1| X , u ) = Φ(µ )(14)itiiitimplying that the lagged marginal response decays geometrically:X 2∂ Pr( y it= 1|i, ui)s= φ(µit) α 1−α β∂xit−s(15)where φ(.) is the standard normal pdf.For the state-dependence model, we can write:Pr( yit= 1| Xi, u ) = Pr( yi+ Pr( yitit= 1| y= 1| yRearrange and write this as a recursion:PitP itwhere:P it = Pr(y it =1 | X i , u i )δ it = Φ(α+β′x it +u i ) - Φ(β′x it +u i )ρ it = Φ(β′x it +u i ).Solving back to an arbitrary period 0:Pit−1it−1it= 0, X= 1, Xitii, u) Pr( y, u ) Pr( yiiit−1it−1= 0 | X= 1| Xii, u ), uii)(16)=−1δ + ρ(17)=t−1∏itP i 0j=01∏ j= −j= 0 it−jt−1it− j+ ∑s=0it−ss−1∏δ ρ δ(18)where we use the convention δ ≡ 1. On reasonable assumptions about the x-process, solving back indefinitely leads to the following representation:j=0it−jPit=∞∑s=0it−ss−1∏ρ δ(19)j=0it−j- 6 -

Thus:∂ Pr( yit= 1| X∂xit−s, u )ii==∂ρ∂x⎛⎜⎝s−1∏it−sj=0+it−sδ∞∑it−jk = s+1s−1∏j=0δ⎞⎟φ⎠⎛ ρ⎜⎝ δit−j( β'x + u )it−kit−s+it−sk −1∏j=0∞∑k = s+1δit−jρδit−si⎞⎟⎠it−kβk −1∏j=0δit−j∂δ∂x[ φ( α + β'x + u ) − φ( β'x + u )]β(20)The profile of ∂ Pr(y it =1 | X i , u i )/ ∂x it-s is thus considerably more complicated than thegeometric decay implied by the SD model (1).it−sit−sit−siit−si3 Estimation3.1 Initial conditionsIn the SD model, there are two alternative approaches for dealing with the randomeffects u. Heckman (1981b) specifies an approximation to the distribution of y i0 | X i ,u i , and then derives the distribution of y i1 ... y iT | y i0 , X i , u i using sequentialconditioning. The random effects are then integrated out by numerical quadrature.The alternative approach, used by Wooldridge (2000) is to specify instead thedistribution of u i | y i0 , X i . A semi-parametric variant due to Arellano and Carrasco(2003) involves the sequence of conditional means = E( u y ... y ,x ... x )λ ,iti|i0it i0which are estimated as nuisance parameters. The latter approach has many advantagesin models like (1) but is problematic in LAR models, where the lagged dependentvariable is not observable and cannot be conditioned on. Conditioning on itsobservable counterpart complicates matters enormously. For this reason, we use theHeckman treatment of initial conditions, together with an explicit hypothesis testingprocedure to control the bias induced by approximation error in the assumeddistribution of y i0 | X i , u i .Assume that we observe y and x over a period t = 0 … T. The LAR process (2)implies the following distributed lag representation: 3it3 In the case where the Γ r are not observable, we impose the normalisation σ ε = 1 and henceforthand σ u are re-interpreted accordingly.*yit, β- 7 -

y*itt= α y*i0+t−1∑s=0s 1−αsα β ' xα ε(21)t t−1it− s+ ui+ ∑1−αs=0it−sThis is a useful basis for estimation if either t is sufficiently large and α t decayssufficiently rapidly with t or if we can find a good empirical approximation forWrite this approximation to*i*yi0| X i , u i as:iii*yi0.y0= δ' w + γ u + η(22)* 0 0yi = r iff yi∈[Γr−, Γr), r 1...R(23)0 0 1=where w i is a vector constructed from X i ; δ and γ are parameters and, in the orderedprobit case, Γ 0rmay differ from Γ r . The random term η i satisfies the followingassumptions:η i ⊥ u i ⏐ X i (24)η i ⊥ ε it ⏐ X i for every t > 0 (25)u i ⏐ X i ∼ N(0, σ η 2 ) (26)Note that, even in the ordered probit case, η is not normalised to have unit variance.In principle, the vector w i may contain all distinct elements of {x i0 , X i }.However, in practice it may be found that w i = x i0 is adequate, or that limitedT0 ∑11summaries, such as w = { ,−ixiT xit} , work well. This is essentially an empiricalissue.With approximation (22)-(23), equation (21) becomes:y*itt−1t−1tsstδ ' wi+ ∑αβ'xit−s+ ctui+ α εit−sα ηi(27)s=0s=0= α ∑ +where c t = (1 - α t )/ (1 - α) + α t γ.The model now consists of equation (22) and a set of equations (27) for anycollection of periods t > 0. In practice, the initial conditions model (22) is only anapproximation and is a potential source of specification error. However, if |α | < 1 sothat α t → 0 as t → ∞, then the influence of the initial conditions declines as weconsider later periods. There is, therefore, a case for leaving a gap (of S periods)between the initial period 0 and the subsequent periods used to estimate the LARmodel. Consequently, we work with a system of (T-S+1) equations consisting of (22)and (27) for t = S+1…T. Data on {y i1 …y iS } are not used. The choice of S involves atrade-off between possible misspecification bias and efficiency, since increasing S- 8 -

educes both the influence of initial conditions and the amount of data used forestimation. Increasing S also reduces the scale of the computational problem. Thissystem is nonlinear in its parameters θ = {α, β, δ, γ, σ u , σ ε , σ η }, where σ ε = 1 in thecase of observable interval boundaries.3.2 IdentificationConsider the model with unobserved grading thresholds. Partition thecovariates into a common set of time-invariant variables ζ i and a sequence of timevaryingcovariates ξ it , so that x it = (ζ i , ξ it ). Assume a full specification of the initialcondition (9), so that w i = (ζ i , ξ i1 ... ξ iT ). Make the further assumption that the matrixplim(n -1 ∑w i w i ′) is positive definite. An ordered probit model for y i0 on w i willconsistently estimate the normed coefficient vector δ/v 0 , where v 2 0 = σ 2 η + γ 2 σ 2 u .Consider equation (27), for any period, t > 0. Rewrite it in standardised form:⎛ y⎜⎝ v*ittt⎞ ⎛αv⎟ =⎜⎠ ⎝ vt0t⎞ ⎛ (1 − α ) β⎜⎟ωi+⎠ ⎝ (1 − α)vtζ⎞⎟'ζ⎠i⎛ β+⎜⎝ vξt⎞⎟'ξ⎠it⎛αβ+⎜⎝ vt⎡+ ⎢ctu⎣iξ+⎞⎟'ξ⎠t 1∑ −s=0it−1α εt−1⎛αβ+ ... + ⎜⎝ vtsit−sξt ⎤+ α ηi⎥ / v⎦⎞⎟'ξ⎠ti1(28)where β ′ = (β ζ ′ , β ξ ′ ), v 2 t = c 2 t σ 2 u + (1-α 2t )/(1-α 2 ) + α 2t 2σ η and ω i is the variableδ′w i /v 0 which can be constructed from the coefficients of the initial conditions model(22). Rewrite (28) in simplified notation as:y*it/ v ω b ξ + υ(29)t= ati+t' ζi+ d0 t' ξit+ d1t' ξit−1+ ... + dt −1,t'i1Note that the covariates (ω i , ζ i , ξ i1 ... ξ it ) are (asymptotically) non-collinear. Thus,ordered probit estimation of (29) will generate consistent estimates of the scaledcoefficients (a t , b t , d 0t , ..., d t-1,t ). Identification then proceeds as follows. First, thevalue of α can be constructed as any element of any of the vectors of ratios d st /d s-1,t . Ifα is zero, the model becomes a static random effects ordered probit, so there is nonew identification issue; we consider the case α ≠ 0 henceforth. With α known, β canitbe inferred up to scale asg / g where g = [b t (1-α)/(1-α t ), d 0t ]. Thus, the keybehavioural parameters α and the direction of the vector β are essentially identifiablefrom only two waves of the panel.The ratio, R t , of a t to α t gives the value v 0 /v t , thus:- 9 -

then done sequentially, starting with S = 8, so that it initially involved only the y-observations from waves 0, 9 and 10. The skip rate S was then reduced sequentiallywhile the score test remained insignificant. We encountered no rejection at any stage,so our final specification uses all available waves of data. Consequently, the rectangleprobabilities involved in SML estimation are 11-dimensional.The analogous SD model is:*where d = ( y m )mit it=y*it4= ∑α mdmit−1+ β'xit+ ui+ εitm=1(60)1 . Estimated parameters for the SD model are given inAppendix Table A2.1. They were computed using 48-point Gauss-Hermitequadrature. Despite the fact that the SD model has 3 more parameters than the LARmodel, the latter achieves a substantially higher log-likelihood.In the model, economic circumstances are represented by the level ofhousehold income per capita, the proportion of household income earned by therespondent himself, and a dummy for owner-occupation, together with the estimatedvalue of the equity in the house. As expected, the level of household per capitaincome has a significant positive effect on perceptions of financial well-being. Themagnitude of the respondent’s personal contribution to household finances has asignificant positive influence on his reported perceptions. There is strong evidence tosupport the widely-held view that homeowners’ perceptions respond to rising housevalues. Human capital also appears to be an important element in perceived financialwell-being, since there is a strong positive influence of educational attainment. Recentchanges in circumstances are represented by the first differences in per capitahousehold income, the respondent’s own income and the estimated house value. Noneof these is statistically significant.However, these ‘objective’ financial factors are not sufficient to explain thedetermination and evolution of perceived financial well-being. Other explanatoryvariables are mostly are time-invariant. The small number of time-varying covariatesare included in the form of current levels and changes from the previous year.Ethnicity is represented by dummies for the Black and Asian groups and thereis evidence of a negative difference, which is statistically significant for the lattergroup. The effect of marital status is captured by dummies for beingmarried/cohabiting, divorced/separated or widowed. A further dummy identifies those- 16 -

Table 1 Estimates and tests (standard errors in parentheses)Dynamic model Initial conditionsCovariate βˆ Std. err.ˆδ /σˆη Std. err.α 0.331 0.012Black -0.224 0.290 0.153 0.602Asian -0.366 0.128 -0.030 0.235In relationship 0.081 0.043 -0.069 0.100Divorced/separated -0.161 0.079 -0.729 0.181Widowed 0.300 0.159 0.765 0.828Newly divorced/separated/widowed -0.178 0.114Employed 0.291 0.048 0.346 0.129Self-employed 0.234 0.060 0.250 0.155Unemployed -0.340 0.076 -0.653 0.171Newly unemployed -0.396 0.096Degree or other further education 0.222 0.055 0.448 0.111A-level 0.269 0.066 0.384 0.125O-level / GCSE / CSE / other qualification 0.130 0.062 0.373 0.110Household size 0.169 0.026 0.175 0.050Number of children in household -0.105 0.018 -0.137 0.039∆ number of children -0.026 0.029Homeowner 0.078 0.057 0.053 0.199Annual household income per head (£×10 -3 ) 0.199 0.013 0.474 0.053∆ household income per head (£×10 -3 ) 0.005 0.016∆ own income (£×10 -3 ) -0.018 0.019Own income as share of household income 0.356 0.061 0.498 0.639Value of house (£×10 -5 ) 0.211 0.062 0.245 0.268Proportionate change in value of house 0.000 0.005Age 18-30 -0.065 0.052 -0.293 0.119Age 31-40 -0.084 0.044 -0.101 0.129Age 41-50 -0.052 0.045 -0.009 0.130Age 51-65 0.117 0.052 -0.004 0.157Poor health 0.014 0.067 0.016 0.122Newly-developed ill-health 0.036 0.082Wave 1 dummy -0.270 0.130Wave 2 dummy -0.136 0.051Wave 3 dummy -0.075 0.051Wave 4 dummy -0.148 0.051Wave 5 dummy 0.041 0.051Wave 6 dummy 0.007 0.052Wave 7 dummy 0.018 0.051Wave 8 dummy -0.081 0.052Wave 9 dummy -0.111 0.052Γ 1 -1.542 0.168 -1.598 0.315Γ 2 -0.620 0.166 -0.577 0.309Γ 3 0.934 0.166 1.233 0.323Γ 4 2.463 0.167 2.870 0.3602σ u 0.375 0.027γ 1.005 0.0812σ η 2.666 0.193Score test for the S = 1 model χ 2 (59) 54.78 (P = 0.368)Log-likelihood -14,535.50- 18 -

7 ConclusionsWe have considered an alternative to the discrete state dependence (SD) model fordynamic modelling of ordinal variables from panel data. The alternative LAR modelinvolves ordinal observation of a latent autoregression, rather than lagged feedback ofthe previous period’s discrete outcome. It is argued that this specification is moreappropriate for a range of applications involving observational, rather than inherent,discreteness. Examples include interval regressions and models of expectations, andsatisfaction.We have developed a simulated maximum likelihood estimator and anassociated test procedure designed to assist in handling the initial conditions problem.As part of this procedure, a novel simulation algorithm has been implemented forcomputing a required numerical Hessian matrix.The method has been applied to a simple model of individual perceptions offinancial well-being, applied to UK household panel data. The LAR model provides arobust description of the evolution of financial perceptions over time, with asignificant role for lagged adjustment. The LAR model fits the data considerablybetter than the conventional SD model and has quite different equilibrium anddynamic properties. In particular, the SD model generally displays less persistencethan the LAR model, and when misused to model highly-persistent data, the estimatedvariance of the individual effect is biased upwards to compensate.ReferencesArellano, M. and Carrasco, R. (2003). Binary choice panel data models withpredetermined variables, Journal of Econometrics 115, 125-157.Arellano, M., Bover, O. and Labeaga, J. M. (1997). Autoregressive models withsample selectivity for panel data, unpublished working paper(http://rt001hfd.eresmas.net/obmajl97.pdf)Bover, O. and Arellano, M. (1997). Estimating dynamic limited dependent variablemodels from panel data, Investigaciones Economicas 21, 141-165.Chow, G. S. (1960). Tests for equality between sets of coefficients in two linearregressions, Econometrica 28, 591-605.Ferrer-i-Carbonell, A. and Frijters, P. (2004) How important is methodology for theestimates of the determinants of happiness? Economic Journal 114, 641-659.- 19 -

Hajivassiliou, V. and Ruud, P. (1994). Classical estimation methods for LDV modelsusing simulation, in Engle, R. F. and McFadden, D. L. (eds.) Handbook ofEconometrics 4, 2383-2441. Amsterdam: North-Holland.Heckman, J. J. (1978). Simple statistical models for discrete panel data developed andapplied to test the hypothesis of true state dependence against the hypothesis ofspurious state dependence, Annales de l’INSEE 30, 227-269.Heckman, J. J. (1981a). Statistical models for discrete panel data, in StructuralAnalysis of Discrete Data with Econometric Applications, Manski, C. F. andMcFadden, D. (eds.), 227-269. Cambridge MA: MIT Press.Heckman, J. J. (1981b). The incidental parameters problem and the problem of initialconditions in estimating a discrete time-discrete data stochastic process, inStructural Analysis of Discrete Data with Econometric Applications, Manski, C.F. and McFadden, D. (eds.), 179-195. Cambridge MA: MIT Press.Van Praag, B. and Ferrer-i-Carbonell, A. (2004). Happiness Quantified. A SatisfactionCalculus Approach. Oxford: Oxford University Press.Wooldridge, J.M. (2000), A framework for estimating dynamic, unobserved effectspanel data models with possible feedback to future explanatory variables,Economics Letters 68, 245-250.- 20 -

*Appendix 1: A simulation approximation to L~ψψOur aim is to estimate the probability limit of the partial Hessiann−1∂2L( ψˆ, ~ τ ) / ∂ψˆ∂ψˆ' . Let V be an asymptotically valid approximation to the*2covariance matrix of ψˆ : for example, we might use−1−1V = −n H(ˆ)θ . Since V reflectsthe variability of ψˆ and is O p (n -1 ), it provides a good metric for the approximation ofderivatives. Let K be the Choleski factor such that V = KK′. Note that K = O p (n -1/2 ).Now generate a sequence of independent pseudo-random N(0, I) vectors v 1 … v M andm 1mconstruct z n ( L ( ψˆ+ Kv , ~ τ ) − L *( ψˆ, ~ τ ))= − µ , where µ is a steplength*n22parameter. Note that z m , and any covariance (with respect to the distribution of v m ) ofz m with powers of v m , are O p (n -1/2 µ ). Expand z m in a Taylor series:z2m= µmm mdpvp+ ∑ 2∑ ∑ Dpqvpvq+ ∑p n p q>p 2npnµµDppm 2 m( v ) + ζ−1where d p and D pq are elements of d = K' ( n ∂L*( ψˆ, ~ τ ) / ψˆ2∂ )−12D = n K ( n ∂ L *( ψˆ, ~ τ ) / ∂ψˆ∂ψˆ') K'2are O p (1). We then have the following results:EEEp(A1)n and, and ζ is a remainder term. Note that d and Dm m µm m( z v ) d + E( v ζ )p=p p(A2)nm m mm m m( z v v ) = D + E( v v ζ ),p ≠ qpq2µnpqm m 2m m( z ( v ) ) 3D+ D + E( v ) ζ )ppµ 2⎡⎤⎢2ppqq ⎥ p2nq≠p= ∑⎣q⎦(A3)(A4)where the expectations are taken with respect to v m .If necessary, the process for generating the variates v m should be appropriatelymtruncated to ensure that L *(ψˆ+ µ Kv , ~nτ2) always exists and that terms of the formv3 *( v v ( ∂ L ∂ψ∂ψ∂ψ| +)E exist for any function ψ + (v) bounded byij/ p q r ψ = ψ ( v )( ψ ˆ , ψˆ+ µ Kv). In practice, this requires ensuring that simulation ofmˆ avoidsψ + µKvregions where the Γ r are non-ordered. A sufficiently small value for µ (for example- 21 -

0.001) will generally achieve this. Under these circumstances, the remainder terms in(A2)-(A4) are o p (1). Thus we can estimate the cross-partials consistently as follows:Dˆpqn=µ−1M m m mM ∑ m =z vpv2 1 q,p ≠ q(A5)To calculate the remaining second derivatives, we solve the simulated analogue of thesystem of equations (A4), to give:Dˆpp=nM2µNow transform D back to ψ-space:⎡MP−1m m 2∑z⎢vp− ⎜ ⎟∑m= 1 2 + P q=1⎣⎛⎝1⎞⎠vm 2q⎤⎥⎦(A6)Lˆ* −1−1ψψ≈ n K'D K(A7)One advantage of this method is that replications can be made sequentially and theprocedure stopped when the quantities (A5) and (A6) appear to have reachedconvergence. The simulated analogue of expression (A2) can be compared with aconventional numerical derivative to check on the negligibility assumption for theremainder terms. Antithetic variance reduction can be used to improve simulationprecision in (A5) and (A6).- 22 -

Appendix 2: SD model estimatesTable A2.1 Estimates for the SD model (standard errors in parentheses)Dynamic model Initial conditionsCovariate βˆ Std. err.ˆδ /σˆη Std. err.α 1 0.286 0.080α 2 0.631 0.071α 3 1.083 0.074α 4 1.524 0.076Black -0.254 0.327 0.186 0.603Asian -0.398 0.144 -0.034 0.247In relationship 0.088 0.045 -0.085 0.100Divorced/separated -0.173 0.083 -0.708 0.181Widowed 0.328 0.165 0.780 0.818Newly divorced/separated/widowed -0.160 0.113Employed 0.302 0.050 0.342 0.131Self-employed 0.239 0.062 0.270 0.156Unemployed -0.387 0.077 -0.634 0.172Newly unemployed -0.321 0.096Degree or other further education 0.240 0.059 0.454 0.111A-level 0.297 0.071 0.398 0.127O-level / GCSE / CSE / other qualification 0.144 0.066 0.369 0.111Household size 0.181 0.027 0.184 0.050Number of children in household -0.116 0.019 -0.131 0.039∆ number of children -0.016 0.029Homeowner 0.087 0.059 0.064 0.207Annual household income per head (£×10 -3 ) 0.215 0.012 0.484 0.056∆ household income per head (£×10 -3 ) -0.010 0.016∆ own income (£×10 -3 ) -0.010 0.019Own income as share of household income 0.365 0.063 0.574 0.639Value of house (£×10 -5 ) 0.215 0.065 0.215 0.290∆ value of house (£×10 -5 ) 0.000 0.005Age 18-30 -0.077 0.053 -0.283 0.118Age 31-40 -0.090 0.044 -0.092 0.128Age 41-50 -0.052 0.045 -0.007 0.129Age 51-65 0.124 0.053 -0.001 0.154Poor health 0.017 0.068 0.025 0.122Newly-developed ill-health 0.029 0.082Wave 1 dummy -0.175 0.053Wave 2 dummy -0.148 0.051Wave 3 dummy -0.084 0.051Wave 4 dummy -0.155 0.051Wave 5 dummy 0.031 0.051Wave 6 dummy 0.010 0.051Wave 7 dummy 0.020 0.050Wave 8 dummy -0.075 0.051Wave 9 dummy -0.109 0.050Γ 1 -0.954 0.132 -1.092 0.212Γ 2 -0.037 0.131 -0.367 0.213Γ 3 1.511 0.132 0.892 0.219Γ 4 3.028 0.132 2.035 0.2212σ u 0.684 0.022γ 0.945 0.071Log-likelihood -14,560.686- 23 -