A fixed effects ordered choice model with flexible thresholds with an application to life-satisfaction

Transcription

1 Department of Economics A fixed effects ordered choice model with flexible thresholds with an application to life-satisfaction Patricia Cubí Mollá 1 City University London Firat Yaman City University London Department of Economics Discussion Paper Series No. 14/10 1 Corresponding author: Patricia Cubí-Mollá, Department of Economics, City University London. Patricia.Cubi-Molla.1@city.ac.uk

2 A fixed effects ordered choice model with flexible thresholds with an application to life-satisfaction Patricia Cubí-Mollá 1 Fırat Yaman 2 JEL classification: C23, C25, I31 Keywords: Ordered choice, fixed effects, subjective well-being, life-satisfaction Abstract In many contexts reported outcomes in a rating scale are modeled through the existence of a latent variable that separates the categories through thresholds. The literature has not been able to separate the effect of a variable on the latent variable from its effect on threshold parameters. We propose a model which incorporates (1) individual fixed effects on the latent variable, (2) individual fixed effects on the thresholds and (3) threshold shifts across time depending on observables. Importantly, the latent variable and the threshold specifications can include common variables. In order to illustrate the estimator, we apply it to a model of life satisfaction using the SOEP dataset. We demonstrate that important differences can arise depending on the choice of the model. Our model suggests that threshold shifts are statistically and quantitatively important. We find that changes in reported life-satisfaction arise as a combination of changes in both the latent variable and changes in the threshold parameters. 1 Department of Economics, City University London. Patricia.Cubi-Molla.1@city.ac.uk. 2 Department of Economics, City University London. Firat.Yaman.1@city.ac.uk.

3 1 Introduction Many variables and outcomes of interest in the social sciences are reported in rating scales. Some of these separate the categories along pre-specified thresholds, such as categorical grades in school (A to E, or 10 to 1, etc.), but many others do not. Surveyed individuals are often asked about their opinion on a certain statement or political issue with answer categories ranging from Strongly agree to Strongly disagree, their self-assessed health or well-being on a range from 0 to 10, or 1 to 5, how much they value certain things in life, such as family, friends, work, etc. with answers ranging from very much to not at all, or how they assess their proficiency in a certain skill or task, such as language fluency with answers ranging from very well to not at all. These rating scales are in wide use in disciplines as diverse as economics, psychology, and medicine. The ordinal (non-cardinal) nature of these variables has given rise to models of ordered choice such as the ordered probit and ordered logit models, and recent contributions have developed consistent and/or efficient (in the sense of using all sample observations with variation in the dependent variable) estimators for the ordered logit, which are all based on a dichotomization of the dependent variable and the application of Chamberlain s (1980) conditional logit model (Winkelmann and Winkelmann (1998), Das and van Soest (1999), Hamermesh (2001), Ferrer-i-Carbonell and Frijters (2004), Baetschmann et al. (2015)). All of these applications have assumed that the thresholds that divide one category from the other are fixed over time (but not necessarily across individuals). This is due to the fact that in ordered choice models the effect of a variable on the level of the latent variable cannot be separately identified from the variable s effect on the level of the thresholds. We suspect that researchers have been long aware of this, but the first explicit exposition of this problem goes back to Terza (1985). If thresholds do change systematically with observed variables, then the coefficient estimates in the aforementioned papers are not effects on the level of the latent variable, but rather on the level change of the latent variable relative to threshold locations (which might have changed themselves), and thus contain very little information, since even the sign cannot be interpreted in its effect on the latent variable. We propose a model which can identify effects of variables on the latent variable from effects on the 1

4 thresholds. To our knowledge this is the first paper which does this in the absence of objective measures of the latent variable (as done by Lindeboom and van Doorslaer (2004) for health) or of explicit anchoring vignettes (such as Bago d Uva et al. (2011)). We use the Amalgamated Conditional Logit Regression (ACLR) proposed by Mukherjee et al. (2008) 3 and extend it by including dependent variables which reflect the survey individuals answers to questions about their current outcome, but also their answers relating to the previous survey period s outcome. The latter is NOT the lagged dependent variable. Rather it is the individual s assessment today about her outcome last year. The key identifying assumption in this model is that survey respondents apply current thresholds to categorize both their current outcome (which depends on current values of the explanatory variables) AND their recalled past outcome (which depends on both current and past values of the explanatory variables). We illustrate our model by applying it to the outcome of life-satisfaction in the German Socio-Economic Panel. The application suggests that threshold shifts are statistically and quantitatively important. Whether a change in the reported outcome is due to a change in the underlying latent variable or due to a change in the threshold might at first seem like an arcane question, but the inference, implications, and possibly political consequences can be widely different between the two cases. Consider first a simple example: In the British higher education system a grade of at least 70 (out of 100) is considered a first class grade. Conceivably, a first class graduation grade might be considered a necessary condition for a popular and attractive employer to consider an applicant for a job interview. Now suppose the proportion of first class grades increases over time. Since the threshold is fixed, one would be inclined to infer that students seem to be getting better in some sense, but that is so only if we interpret the latent variable as the actual grade (a number between 0 and 100). However, the result might be due to the university becoming more generous in its marking, so that a particular student might receive 70 marks today, but would have received less for the same performance a few years earlier. If we interpret the latent variable as knowledge or quality of knowledge, then the increased proportion of first class grades could be a reflection of either more knowledgeable students, or of a laxation of marking standards. Clearly the distinction would be of interest to educational policy-making. 3 Baetschmann et al. (2015) call this estimator blow-up and cluster (BUC) and demonstrate its strong small sample properties in Monte-Carlo simulations. 2

5 Another increasingly important field in which the distinction between latent variable and threshold changes is crucial is the measurement and study of subjective well-being and happiness. Measures of wellbeing are increasingly suggested as substitute or at least complementary measures to GDP that should be targeted by governments (see for example HM Treasury Budget (2010), OECD (2011) or Dolan et al. (2011)). Since these measures reflect the entirety of the human experience, it is argued, they have the potential to be more complete and even accurate compared to GDP which includes only those goods which can be priced in the market (thus excluding things like the value of clean air, social and physical safety, biodiversity etc.). A non-market good x could in principle be priced by inferring from regression coefficients the amount of income that an individual would give up to compensate for a unit-increase in x to keep her latent variable constant. 4 But what exactly should the social welfare function be? If it is the sum of all individual reported levels of well-being (Y = N i=1 y i), we need not worry about the source of changing values of y, since both threshold and latent variable changes will be observationally equivalent. But if the social welfare function is over the latent variable (Y = N i=1 y i ), as it probably should be if we consider this to be the actual emotional state of an individual, then the distinction is important. Threshold shifts to the left (making it easier to report higher values of well-being) would increase Y, but leave Y unchanged. We imagine a policy-maker would like to know to what extent changes in Y are reflecting changes in Y. 2 Modeling We follow here the conventional choice of setting up the ordinal model as a latent variable model. That is the individual i at survey time t has a subjective evaluation of the question she is being asked (her health status, opinion, life-satisfaction, etc.). The question can only be answered by picking one out of an ordered list of answers and can refer either to a present outcome or a past outcome (reference time). The importance of the distinction between survey and reference time will become clear in the next subsections. We index the possible answer categories by k {1,...K}. The individual s evaluation at survey time t of her underlying latent variable at reference time s we denote y s it. This evaluation translates into the reported outcome ys it, such that y s it = k λ k 1 it < y s it λ k it (1) 4 It is not our intention to participate in a debate about the merits of using well-being instead or along with GDP. We only illustrate that threshold shifts can occur and will have different implications from latent variable shifts. 3

6 Thus, we assume that the individual applies survey time thresholds to any questions asked at the survey time, even if the question refers to a different reference time. To clarify matters and anticipating our application, consider a life-satisfaction example: Equation 1 says that a person asked about her current life-satisfaction will categorize her latent variable according to her current thresholds. When asked about her life-satisfaction one year ago she will categorize her remembered life-satisfaction according to her current thresholds, too, rather than recalling her past thresholds. 5 The latent variable for a question about a current outcome is specified as y t it = X it β 0 + X it β 1 + α i + ε it (2) where the distribution of α i is left unspecified and is allowed to be dependent with X i, and ε it is i.i.d. logistic (with location 0 and scale 1) across i and t. By including the differenced values of X we are anticipating here the later application to life-satisfaction. Prospect theory (Kahneman (2011)) advocates that changes are more important for perceived utility than levels. To what extent this finding would extend to a more general concept such as life-satisfaction and a time horizon of a year is unclear and we leave it to the empirical application to answer this question. 2.1 Threshold model If λ k it = λk i and we only want to model y t t, we have all the necessary building blocks to apply Chamberlain s conditional logit model for a pre-specified dichotomization of y, or to apply one of the estimators based on the conditional logit model which use all possible dichotomizations (minimum distance, ACLR). This specification of the threshold parameters is quite flexible, but it does impose that distances between any two thresholds are preserved over time and that the difference of the k th threshold between two individuals is constant over time: λ k it λ k jt = λ k i λ k j t It also implies that y t it > ys is yt it > y s is. In this paper we decompose the thresholds into an individual- and threshold-specific, time-invariant 5 The order of the questions answered probably matters. In our application the question on current life-satisfaction precedes the question on life-satisfaction in the previous year, setting current life-satisfaction as a benchmark for the past. We believe this order is in accordance with our assumption and with equation 1. 4

7 component, and a component which modifies the thresholds linearly in parameters: λ k it = τ k i + Z it γ (3) Thus, λ k it λ k jt = τ k i τ k j + (Z it Z jt )γ and y t it > ys is does not necessarily imply yt it > y s is and vice versa. Equations 1, 2 and 3 imply y t it = k τ k 1 i < X it β 0 + X it β 1 Z it γ + α i + ε it τ k i (4) Clearly, in this equation β 0 is not separately identified from γ for common variables in X and Z, or in other words, the estimate of β 0 will incur a level-bias on the order of γ. 2.2 The remembered outcome In most surveys the surveyed individual is asked to rate her current outcome. However, the surveyed individual might in addition be asked about the current outcome and about her outcome at some point in the past. Thus, y t it is the reported outcome for individual i surveyed at time t and where the evaluation refers to the time at which the individual is surveyed. In contrast y t 1 it is the reported outcome for individual i surveyed at time t but where the evaluation refers to the previous time at which the individual is surveyed: it is how the individual remembers her outcome. The economics literature provides very little guidance as to how a subjective outcome for which survey and reference time do not coincide should be modeled. The only paper we are aware of which models the perceived change in an outcome variable is Pudney (2011) who models answers to the question of current financial well-being (y t t ) and to the reported change in financial wellbeing (y t t yt t 1 ). He specifies the former as a function of current values of variables and their changes with respect to the previous year (x t, x t ), and the latter as depending on current and past values of variables and their change with respect to the previous year (x t,x t 1, x t ). We follow this approach and retain a rich specification of the equation describing a recalled outcome. We are looking for a model of the form y t 1 it = f (y t 1 i,t 1,X it,α i,ũ it θ) 5

8 where y t 1 i,t 1 is i s past outcome as perceived in the past, X it are current values of explanatory variables, α i the individual fixed effect, ũ it a white noise recall error and θ is a vector of parameters. The exact specification we estimate is y t 1 i,t = X it δ 0 + (X i,t 1 β 0 + X i,t 1 β 1 )δ 1 + α i δ 2 + ũ it (5) This equation incorporates a number of insights from psychological research. The first element in the sum, X it δ 0 allows for the possibility that an individual s recollection of the past is influenced by the present state, that is, that memories are reconstructive. O Brien et al. (2012) provide evidence that individuals who are negatively primed in a certain area of their life (reminded of currently not having a partner) tend to remember their past in that area (romantic life) more positively than individuals in a control group who were not primed 6. The following terms up to and including α i δ 2 reflect the non-random component of the true latent variable in the previous year, but allow for an amplification (δ 1 > 1), dampening(1 > δ 1 > 0), or reversal (δ 1 < 0) of the previous life-satisfaction when remembered today. In our model we will allow this effect to vary across different individual characteristics. Therefore, an individual may amplify some components of her previous life-satisfaction (as past romantic life) and dampen others. Finally, ũ it is a recollection error, with logistic density, location 0, and scale parameter σ. To appreciate the advantages of our model it is helpful to contrast it to the most simple specification. Assuming the individual to have an accurate recollection of her previous outcome, and having time-invariant thresholds, a difference between y t it and y t 1 it would be due to a change in the individual s latent variable. Our threshold specification also allows this to be due to changes in the threshold parameters provided that the thresholds of time t are applied to all questions at survey time t. A prominent example would be the effect of income or wealth on an outcome variable which records the individual s assessment about how rich she is: the barrier that divides the rich from the non-rich always seems to be above one s own wealth. Another possibility would be that the individual does not accurately recall her previous state. Our model has a rich structure to incorporate past outcomes, current state variables and random recall errors to allow for differences between experienced and recalled outcomes. Finally, defining ( ) as element by element multiplication the answer to a remembered outcome can be 6 See also Wilson et al. (2003) on the reconstruction of memories. 6

9 formalized as: y t 1 it = k τ k 1 i + Z it γ < X i,t δ 0 + X i,t 1 (β 0 δ 1 ) + X i,t 1 (β 1 δ 1 ) + α i δ 2 + ũ it τ k i + Z it γ or y t 1 it = k τk 1 i α i δ 2 δ 0 < X i,t σ σ Z it γ σ + X i,t 1 (β 0 δ 1 ) σ + X i,t 1 (β 1 δ 1 ) σ + u it τk i α iδ 2 σ y t 1 it = k τ k 1 δ 0 i < X i,t σ Z it γ σ + X i,t 1 (β 0 δ 1 ) σ + X i,t 1 (β 1 δ 1 ) σ + u it τ k i (6) where u now has location 0 and scale 1. Equations 6 and 4 identify β 0 and γ separately, even if X and Z share common variables. Equation 4 yields an estimate of β 1. Together with (β 0 δ 1 ) σ and (β 1 δ 1 ) σ from equation 6 we obtain β 0, and hence again from equation 4 a value for γ. We emphasize that this derives from the application of thresholds λ it rather than λ i,t 1 to ANY question asked at survey time t. We think that for most cases it is reasonable to assume that people apply their current criteria in answering a survey question, even if the question refers to an event in the past. Furthermore, identification also requires that only levels (Z) and not differences ( Z) enter the threshold specification (else an estimate for β 1 could not be obtained). 3 Estimation The estimation is a very straightforward extension of the estimator in Mukherjee et al. (2008), who call it amalgamated conditional logistic regression (ACLR) and Baetschmann et al. (2015) 7 who call it blow up and cluster (BUC) estimator. The idea of these estimators is the following: for a given cutoff value k, dichotomize the ordinal variable, e.g. ỹ it = 1(y it > k). Chamberlain s conditional logit model is derived from the likelihood of the sequence of ỹ i = (ỹ i1,...,ỹ it ) conditional on the number of ones in the sequence of answers being equal to T i t=1 ỹit. Denoting this by Pi k, the ACLR and BUC combine all the possible dichotomizations (with K categories, there are K-1 possible dichotomizations) in one log likelihood and 7 We are very thankful to the authors for bringing these models to our attention and for providing the Stata code for implementation. 7

10 maximize it over b max b LL = N K 1 i=1 k=1 lnp k i (X,b) (7) Under the assumptions on ε it, an individual s likelihood Pi k (X,b) is given by P k i (X,b) = T i t=1 exp(ỹ it X it β) di D i T i t=1 exp(d it X it β) where d it {0,1}, d i = (d i1 d iti ) and D i is the set of all distinct d i such that t d it = t ỹ it. Our extension to this model is the following. Suppose individual i has T pr i for present), and T pa i observations on y t 1 it and over all t, followed by y t 1 it observations on y t it (where the superscript pr stands (past). We stack first the reported outcomes y t it for individual i for individual i and over all t, these are the first T pr i + T pa i elements of the outcome vector. The vector is then appended by the next individual etc. until we reach the end of our sample. Since the u it are assumed to be independent of the ε it, statistically we can treat the observations for y t 1 it and y t it as distinct individuals even if in reality they are responses by the same individual. 8 The vector for the explanatory variables are (x it, z it, x it ) for responses referring to the present period, and (x it,x i,t 1, z it, x i,t 1 ) for responses referring to the past period. With these modifications to the data, the likelihood given in 7 is maximized 9. Standard errors are clustered by distinct ε it and u it. 4 Application: Life satisfaction over the life cycle We illustrate the working of our model by way of an application to life satisfaction. We use the Socio- Economic Panel (SOEP) from Germany. This annual panel contains the question How satisfied are you at present with your life as a whole? in all waves (the first panel wave is 1984). For the waves the SOEP then followed with the question How satisfied were you a year ago with your life? Both questions could be answered on a scale from 0 (totally satisfied) to 10 (totally unsatisfied), where we collapse the lowest three categories into a single one due to the small number of observations in them. We restrict the sample to all individuals between ages 25 and 64. In the framework of life-satisfaction, it is natural to ask what we mean by or how we interpret changes in the latent variable and changes in thresholds. We defer the 8 If this assumption is violated we are maximizing the partial likelihood function. The estimator is still consistent, but not efficient, as long as ε and u are independent of any of the explanatory variables. We are currently expanding this paper to efficient estimation. 9 Since present and past life-satisfaction errors are assumed to be independent, and equations 4 and 6 do not have any common reduced-form parameters, the two equations can also be maximized separately. 8

11 discussion of this interesting question to the discussion and conclusion section. Since equation 4 requires values on the difference of the explanatory variables, we cannot use y t it for Furthermore, we cannot use y t 1 it for the years 1984 and 1985 since this question requires the lag of the difference (thus, observations for x t 2 ). We thus have 3 answers to the question about current lifesatisfaction and 2 answers to the question about past life-satisfaction. In principle more waves could be used for the former. However, parameters need not stay stable over time, and we stop the sample two years before the fall of the Berlin wall and any structural break that might have accompanied it. We model life-satisfaction similarly to Baetschmann et al. (2015), and incorporate at least one variable belonging to the most important domains that have been identified to matter for life-satisfaction: income, employment status (employment, hours of work), family variables (marriage, partner, children), and health 10. The explanatory variables are described and summarized in table 1. Source: SOEP Table 1: Variables Variable name Description Range Mean Outcome variables LS Current life satisfaction LSPast Past life satisfaction Explanatory variables Health Self assessed health Chronic Has chronic health condition 0, Worries Finance Worries about financial situation Worries Job Worries about job situation Age2 Age squared / H 20less Works less than 20 hours 0, H Works between 21 and 34 hours 0, H Works between 35 and 45 hours 0, H 46more Works more than 45 hours 0, Unemployed Is unemployed (less than a year) 0, Long Unemployed Is unemployed (at least a year) 0, NILF Not in labor force 0, Partner Lives with partner 0, Children Has children 0, Income Ln (Equivalized household income) (-0.43,10.09) We do not claim to have a complete model of life-satisfaction which would be outside of the scope of this paper. However, we think that the chosen variables cover most of the determinants of life satisfaction that have been considered in the literature. To emphasize the main contribution of this paper, all of those variables are also included as threshold shifters. 10 See for example Dolan et al. (2008) for a comprehensive overview of the life-satisfaction literature. 9

12 Before showing regression results, table 2 demonstrates the discrepancy in the outcome variables y t 1 t 1 (life satisfaction referring to t-1 as reported in t-1) and y t 1 t (life satisfaction referring to t-1 as reported in t). A cell entry is the percentage of those who report y t 1 t, conditional on reporting y t 1 t 1 (so the rows sum to 100). The diagonal elements would be the consistent answers. Of those who reported category y t 1 t 1 = k, a plurality of observations record the same category for y t 1 t only for categories 5 (27%), 7 (28%), 8 (38%) and 10 (36%). All off-diagonal elements have positive entries (one rounded to 0), and people seem to revise their life-satisfaction both upwards and downwards with a slight tendency for downwards revision: there are 8,569 observations in the lower and 6,384 observations in the upper triangle of the table. The observations in the diagonal cells are 6,307. This phenomenon lends strong support to the hypothesis of flexible thresholds and/or the presence of a recall error. Table 2: Cross-tabulations, Life satisfaction in % LS in t-1 as repor- Life satisfaction in t-1 as reported in t: y t 1 t ted in t-1: y t 1 t Source: SOEP LS: Life satisfaction. A cell entry is the percentage of those who report y t 1 t yt 1 t 1. Rows sum to 1. conditional on reporting Our regression results are presented in Table 3. In general the coefficients have the expected signs, but very few of them are individually significant (we have not yet tested for the joint significance of β 0 and β 1 ). Note that the estimated level effects on the latent variable if threshold shifts were not identified would be equal to β 0 γ. For ease of interpretation we will classify variables into four categories based on their level effects on the latent variable (β 0 ) and their level effects on the threshold (γ). Variables are good if β 0 > 0, and bad otherwise. If a good variable has α 0 < 0 we call it reinforcing, and else adaptative. If a bad variable has α 0 < 0 it is adaptative and else reinforcing. 10

13 Table 3: Life satisfaction determinants (1) (2) (3) β 0 γ β 1 Health ( ) ( ) ( ) Chronic ( ) ( ) ( ) Worry Finance *** ( ) ( ) ( ) Worry Job *** ( ) ( ) ( ) Children ( ) ( ) ( ) Age ( ) ( ) ( ) Income ( ) ( ) ( ) H 20less ( ) ( ) ( ) H ( ) ( ) ( ) H 46more ( ) ( ) ( ) Unemployed ( ) ( ) ( ) Long Unemployed ( ) ( ) ( ) NILF ( ) ( ) ( ) Partner *** ( ) ( ) ( ) Source: Standard errors (in parentheses) are clustered by individual and reported outcome (past vs. present). Standard errors for columns (1) and (2) computed by Delta-Method. β 0 is the level effect on the latent variable, α 0 is the level effect on the threshold, β 1 is the effect of a change on the latent variable. *** p<0.01, ** p<0.05, * p<

14 5 Results 5.1 Health To understand the intuition of this terminology consider the first variable, self-assessed health. Individuals with better health experience a higher value of life-satisfaction, thus health is good. At the same time, good health shifts threshold parameters to the left, so that even if two individuals are equally satisfied (in terms of the latent variable), the healthier individual is likely to report higher life-satisfaction. In this sense health is a reinforcing good. Similarly, a chronic condition is an adaptative bad. It decreases life-satisfaction, but at the same time makes it easier for a person with a chronic condition to report higher life-satisfaction compared to a person with equal life-satisfaction and no chronic condition. The positive β 1 suggests that the latent variable effect is lower in the first year of the chronic condition (since the combined effect would be ), and that the condition is felt to further reduce life-satisfaction after the first year. 5.2 Worries Worrying about the financial situation reduces life satisfaction, but the effect on the threshold is much stronger, making financial worries a reinforcing bad. Worrying about the job has a very counter-intuitive positive sign, but also a high standard error (few individuals in our data do worry about their job-situation). However, the combined latent variable and threshold effect results in lower reported life-satisfaction for people who worry about their jobs. 5.3 Family The point-estimates of the partner coefficients are perhaps the most striking result of our estimates. With due caution because of the huge standard errors it seems that having a partner greatly increases life-satisfaction, but the adaptative effect is also very large. Possibly once a person has a partner, her attention quickly is focused on other areas in life which then provide the new defining targets for the thresholds. Children seem to leave the latent variable mostly unaffected, but pull down the thresholds. 12

15 5.4 Income and Employment Similarly to children, income has no important effect on life-satisfaction. The usual finding of a positive effect is driven almost entirely by income being a reinforcing good. Unemployment is as expected negative, and its effect reinforcing. Once a person has been unemployed for a long time, her life-satisfaction is not only back to, but in excess of the state of no unemployment, but the thresholds are shifted to the right by a huge amount. In terms of working hours, working more seems to increase the latent variable (the omitted category is working between 35 and 45 hours), but also increases the threshold values. 6 Discussion and conclusion It has been a long-standing insight that in ordered-choice models there is an observational equivalence between a variable s effect on the latent variable and on the threshold, and that only the combined effect is identified. However, in practice the difference between changes in the latent variable and the threshold can be important for inference and policy-making. We are proposing a model of ordered choice which accommodates the inclusion of 1) individual fixed effects in the latent variable, and 2) individual specific thresholds which are allowed to change through time. Crucially, our model can incorporate the same variables for the latent variable and the threshold and identify their separate effects. We apply our estimator to a simple model on life satisfaction and demonstrate that variables usually included in life-satisfaction models have statistically and quantitatively significant effects on the thresholds, which if omitted in the threshold specification are absorbed in the coefficient of the latent variable specification. Quantitatively important differences in the values of variables relating to healthy, family, employment and income, and worries arise between models with and without threshold shifts. But how can we interpret a difference between a change in the latent variable and a change in the threshold value in the framework of life-satisfaction? We can only offer tentative interpretations. Life-satisfaction is usually understood to be a quiet comprehensive or general evaluation of one s own life. This evaluation encompasses distinct aspects such as one s own and others present and past experiences, affective happiness, meaningfulness and purposefulness (Loewenstein (2009) and Dolan and Kahneman (2008)). Possibly each component bears on the latent variable and the threshold with different weights, for example affective happiness influencing the latent variable, and meaningfulness the thresholds. Thus, having children might not increase affective happiness (possibly decreasing it due to sleep deprivation 13

16 and increased levels of stress) but gives the parent a stronger sense of purpose and meaning. In some sense our paper raises more questions than it provides answers. Despite some experimental work in psychology, we do not really know how exactly people reconstruct their past experiences in general, and how they evaluate their past life-satisfaction in particular. While we have proposed a fairly general model in this paper, many other specifications in terms of functional form, time-reference, and relevant variables remain to be explored. References [1] Bago d Uva, T. B., Lindeboom, M., O Donnell, O., & Van Doorslaer, E. (2011). Slipping anchor? Testing the vignettes approach to identification and correction of reporting heterogeneity. Journal of Human Resources, 46(4), [2] Baetschmann, G., Staub, K.E. & Winkelmann, R. (2015). Consistent estimation of the fixed effects ordered logit model. Journal of the Royal Statistical Society: Series A (Statistics in Society), forthcoming. [3] Chamberlain, G. (1980). Analysis of covariance with qualitative data. Review of Economic Studies, vol. 47, [4] Das, M. & Van Soest, A. (1999). A panel data model for subjective information on household income growth. Journal of Economic Behavior & Organization, 40(4), [5] Dolan, P., Layard, R. & Metcalfe, R. (2011) Measuring subjective wellbeing for public policy: recommendations on measures. Centre for Economic Performance special papers, CEPSP23. Centre for Economic Performance, London School of Economics and Political Science, London, UK. [6] Dolan, P., Peasgood, T. & White, M. (2008). Do we really know what makes us happy? A review of the economic literature on the factors associated with subjective well-being. Journal of economic psychology, 29(1), [7] Dolan, P. & Kahneman, D. (2008). Interpretations of utility and their implications for the valuation of health. Economic Journal, 118(525),

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