Mobility over several periods for discrete variables: Theory and application to subjective wellbeing indicators in the United Kingdom

Transcription

1 Mobility over several periods for discrete variables: Theory and application to subjective wellbeing indicators in the United Kingdom Sanghamitra BANDYOPADHYAY Gaston YALONETZKY 13th June 2013 Abstract This paper discusses the meaning of intra-generational mobility when variables are discrete (ordinal or categorical). We propose concepts, and related desirable properties, of maximum and minimum mobility, along with mobility-inducing transformations. We also propose several functional forms for indices of individual mobility, and social mobility. It turns out that these indices measure mobility as diversity or instability in people s status (as measured by the ordinal variable) along the accounting period. We apply these indices to measure the extent of mobility as diversity in the responses to subjective wellbeing questions in the United Kingdom, using the British Household Panel Survey. Keywords: Intra-generational mobility, discrete variables, life satisfaction. JEL Classification: D30. Queen Mary, University of London; s.bandyopadhyay@qmul.ac.uk. University of Leeds; G.Yalonetzky@leeds.ac.uk. 1

2 1 INTRODUCTION 1 Introduction The study of fluctuations (and stability) in socioeconomic outcomes has long been of interest in the social sciences, often under the name of intra-generational economic mobility. Fields and Ok (1996) discuss several concepts of this form of mobility. Most of them apply to two-period analyses, in which several topics of interest include decomposition into structural and exchange components (e.g. Ruiz-Castillo, 2004, van Kerm, 2004), as well as the "pro-poor" nature of a growth experience (e.g. Deutsch and Silber, 2011). By contrast, mobility over several periods has been concerned mainly with the notion of mobility as equalizer of lifetime incomes (e.g. Shorrocks, 1978, Maasoumi and Zandvakili, 1986, Tsui, 2009, Fields, 2010). This literature deals with continuous variables. Unlike the two-period case, mobility with ordinal variables in a multi-period framework has not been explored yet. However some interesting questions are worth tackling in this context. For instance, how stable over time are people s expressions of life satisfaction (measured by ordinal indicators)? To what extent are these affected by life shocks, or conditioned by more stable socioeconomic and demographic characteristics? With datasets like the British Household Panel Survey answering these questions is now possible. This paper discusses the meaning of intra-generational mobility when variables are discrete (ordinal or categorical). We propose a concept of mobility as diversity in people s status (as measured by the discrete variable) over the accounting period. Accordingly we propose relative desirable properties, of maximum and minimum mobility, along with mobility-inducing transformations. We also propose several functional forms for indices of individual mobility, and social mobility. It turns out that the individual mobility indices are consistent with a mobility ordering that is identical to the Lorenz ordering. Our concept and measures of mobility for discrete variables is very similar to the notion of mobility as unpredictability put forward by Parker and Rougier (2001). The main difference between the two concepts (and related indices) is that our concept focuses on observing individuals over several periods, with the option of later aggregating their mobility measures into a social measure. By contrast, the concept of Parker and Rougier (2001) is based on two-period transition matrices and skips the individual evaluation stage. We apply these indices to measure the extent of mobility as diversity using responses to subjective wellbeing questions in the United Kingdom, using the British Household Panel Survey. The paper is organized as follows. In Section 2 we set the framework for conceptualising mobility measures for discrete variables, and present mobility measures for different functional forms. In Section 3 we illustrate the use of these measures using life satisfaction responses from the British Household Panel Survey. Section 4 concludes. 2

3 2 MOBILITY OVER SEVERAL PERIODS FOR ORDINAL VARIABLES 2 Mobility over several periods for ordinal variables 2.1 Preliminaries and notation Let x nt be the value of a discrete variable x for individual n in period t, such that x nt [1, S] N ++. The variable is observed for N individuals across T time periods. The unobserved individual probability that x nt = i in any time period is: p n (i) Pr[x nt = i]. In practice, the probability that x nt = i is estimated according to the following formula: p n (i) 1 T T I(x nt = i) (1) t=1 We also define the probability distribution of x for individual n, an S-dimensional vector, as: P n := [p n (1), p n (2),..., p n (S)]. An analogue definition applies to the estimated probability distribution, P n. 2.2 Concept and mobility orderings When we observe a discrete variable for each individual during several time periods we can ask: how stable is the individual s experience according to the variable? Does the individual always exhibit the same value? Or, on the other extreme, is the individual likely to exhibit any value at any given time period in such a way that, in each period, the value of the variable is unpredictable? In this context, we introduce the following two extreme situations of mobility understood as diversity (of outcomes) or instability: Definition 1. Minimum mobility: An individual exhibits minimum mobility if i [1, S] p n (i) = 1. According to definition 1, minimum mobility occurs when an individual always exhibits the same value of x. Definition 2. Maximum mobility: An individual exhibits maximum mobility if p n (j) = 1 S j [1, S]. According to definition 2, maximum mobility occurs when the individual probability distribution of x is uniform. Since we are considering mobility as instability or diversity, we can order probability distributions in terms of their degree of mobility, using the concepts of the inequality literature, in particular uniform majorization (Marshall, Olkin, and Arnold, 2010) and the traditional Pigou-Dalton transfer, but applied to probability distributions. We also define the following mobility orderings: the strong mobility ordering, such that P X P Y reads "distribution X is more mobile than distribution Y ". The weak mobility ordering such that P X P Y reads "distribution X is at least as mobile as distribution Y ". Finally the indifference mobility ordering such that P X P Y reads "distributions X and Y are equally mobile". Then we define a probability Pigou-Dalton transfer: 3

4 2 MOBILITY OVER SEVERAL PERIODS FOR ORDINAL VARIABLES Definition 3. Consider probabilities p n (i) and p n (j), such that p n (i) > p n (j). A probability Pigou-Dalton transfer (PPD) is a rank-preserving transfer of probability mass δ from the greater to the lower probability, that is: p n (i) δ p n (j) + δ We consider that a PPD should strictly increase mobility as diversity, i.e. if distribution Y is obtained from distribution X by a PPD then: P Y P X. More generally, we define a uniform majorization: Definition 4. Distribution Y is obtained from a uniform majorization of X if P Y where B is a bi-stochastic matrix. = BP X We consider that a uniform majorization should not decrease mobility as diversity, i.e. if distribution Y is obtained from distribution X by a uniform majorization then: P X P Y. Now note the following two details. Firstly, we say that a uniform majorization should not decrease mobility instead of asserting that it increases mobility strictly. The reason is that the domain of bi-stochastic matrices admits matrices whose elements are only either ones or zeroes, i.e. the identity matrix and its permutations. When multiplied by any of these matrices, the probabilities of X are not changed. At best they are reshuffled. In those circumstances we do not consider mobility as diversity to have increased. Secondly, any PPD can be implemented via a uniform majorization, choosing the appropriate bistochastic matrix. 2.3 Individual mobility indices We define an individual mobility index: M n : P n R S + [0, 1] R. The individual mobility index should fulfill the following desirable properties: Axiom 1. Maximum mobility: M n = 1 if and only if the individual exhibits maximum mobility. Axiom 2. Minimum mobility: M n = 0 if and only if the individual exhibits minimum mobility. Axiom 3. Sensitivity to PPD: If Y is obtained from X by PPD, then M n (Y ) > M n (X). Axiom 4. Sensitivity to uniform majorization: If Y is obtained from X by uniform majorization, then M n (Y ) M n (X). Proposition 1. The individual mobility index for multiple periods and discrete variables, M n fulfills the above four axioms if it is a Schur-concave function, and it is normalized so that M n = 1 only whenever p n (1) = p n (2) =... = p n (S) and M n = 0 only whenever i p n (i) = 1. Proof. Available upon request. Examples of M n include: M n = 1 1 S S i=1 [Sp n(i)] α 1 S α 1 α > 1 (2) 1 4

5 2 MOBILITY OVER SEVERAL PERIODS FOR ORDINAL VARIABLES M n = 1 M n = S S 1 2(S 1) S i=1 i=1 j=1 p n (i) (3) p n (i) p n (j) (4) Interestingly, when α = 2 in 2, then M n is a function of the Simpson index: M n = S S 1 p n (i)[1 p n (i)] (5) i=1 Note also that the indices in 2 and in 4 fulfill the above axioms. By contrast, the index in 3 fails to fulfill the axiom 2, since it can attain its minimum value of 0 in circumstances differing from minimum mobility. 2.4 Social mobility indices Just like in the poverty and wellbeing measurement literature, we can construct social mobility indices by aggregating individual mobility indices. The most natural proposal is the arithmetic average: M = 1 N M n (6) N The index in 6 fulfills the well-known properties of symmetry, population replication invariance, additive decomposability and subgroup consistency. It is also the case that: M = 0 M n = 0 n and M = 1 M n = 1 n. n=1 2.5 Normalization issues when the time period is small relative to the number of categories The above indices work well in theory. However in practice we need to compute the probabilities in 1 from the data. The situations of minimum mobility are easy to spot in the data as they only require one probability to be equal to one. By contrast, when the time period is short, the appearance of the probability distribution under a situation of maximum mobility depends on the relationship between S and T. Then, for any chosen individual mobility index, the maximum value varies accordingly. Therefore when the time period is short, mobility indices should be adjusted according to the following formula: A n = M n min M n max M n min M n (7) When the time period is relatively short, there are three cases of maximum mobility: Case 1. If T S: p n (i) = 1 T i [1, T ] and p n(i) = 0 i [T + 1, S]. Case 2. If T > S and T = λs where λ N ++ : p n (i) = 1 S i [1, S]. 5

6 3 EMPIRICAL ILLUSTRATION: LIFE SATISFACTION INDICATORS IN THE UNITED KINGDOM Case 3. If T > S and T = λs + R where λ, R N ++ and R < S: p n (i) = λ+1 T p n (i) = λ T i [R + 1, S]. i [1, R] and Consider, for example, the adjustment to the individual mobility index based on the Simpson index, as in 5: Case 1: A n = Case 2: A n = Case 3: A n = T T 1 S S 1 p n (i)[1 p n (i)] (8) i=1 p n (i)[1 p n (i)] (9) i=1 T 2 S i=1 p n(i)[1 p n (i)] R[T 2λ 1] + Sλ[T λ] Now consider the adjustment to the individual mobility index based on the Gini index, as in 4: Case 1: A n = S 1 T 1 1 2(T 1) Case 2: A n = 1 1 2(S 1) i=1 j=1 i=1 j=1 (10) p n (i) p n (j) (11) p n (i) p n (j) (12) Case 3: A n = T (S 1) T S S 2 i=1 j=1 p n(i) p n (j) T (S 1) R(S R) (13) 3 Empirical illustration: Life satisfaction indicators in the United Kingdom In this section we present estimates of the mobility indices proposed in Section 2, and illustrate their usefulness with an application to life satisfaction measures from the British Household Panel Survey (BHPS). Several international datasets provide longitudinal data on household characteristics and qualitative data. Among these, the BHPS features high-quality discrete data on self-reported levels of life satisfaction and happiness for all years covered. The BHPS follows the same representative sample of individuals over a period of 18 years from 1991 to Each annual interview round is called a wave: in our study we use the first 18 waves of data, and each wave is principally household-based, interviewing every adult member of sampled households. Each wave consists of over 5,500 households and over 10,000 individuals drawn from 250 areas of Great Britain. Samples of 1,500 households from Scotland and Wales (3,000 in total) were added to the main sample in 1999; later in 2001 a sample of 2,000 Northern Irish households was added. Our main variables of interest are life satisfaction and happiness (variables ghql and lfsato). The General happiness variable (ghql) features responses graded on a level of 1 6

7 3 EMPIRICAL ILLUSTRATION: LIFE SATISFACTION INDICATORS IN THE UNITED KINGDOM to 3, for the following categories: more than happy (3), same as happy (2) and less than happy (1). Life satisfaction (lfsato) is graded on a scale of 1 to 7, with decreasing levels of life satisfaction for increasing values of the variable. The following socio-economic indicators are considered as "determinants of life satisafaction", following their recurrent use in the "happiness" literature: Age, in years. Sex: male or female. Health: number of visits to GP since 1.9. year, in 5 categories. Marital status: married or not. Annual income: we use the log of annual income. Number of children in household. Educational attainment, measured on a decreasing scale, where 1 represents higher degree and 12 represents no academic qualifications. We have at least 3200 observations for each model that we estimated in Tables 1 to 4. The dataset used for each regression is balanced, i.e. there are no missing observations for each of the datasets. 1. The number of observations for each estimated model varies, as each regression has full availability of the socio-economic characteristics. We estimate the mobility indices based on the Gini and the Simpson functional forms, as described in equations 4 and 5. We call them the Gini Mobility Index and the Simpson Mobility Index. Figures 1 and 2 present the distribution of the mobility indices for both the happiness and life satisfaction variables. For the Gini mobility index we observe that the bulk of the distribution lies at the lower and middle section of the distribution, while for the Simpson we find that the distribution is heavier at the upper end. In other words, for a given set of categories of happiness, we find that the Simpson mobility index reveals greater mobility than the Gini. This is interesting to note, given that the happiness variable and the life satisfaction variable are categorised differently (the happiness variable has 3 states, while the life satisfaction variable ranges over 7 states). We now illustrate how one may use the estimated measures of mobility with an exercise popular in the subjective wellbeing literature and the mobility literature. We look at the socio-economic determinants of mobility when using life satisfaction measures as a proxy of one s wellbeing status. We use estimate the following relationship with ordinary least squares: LifeSatisfaction i = α + X i β + ɛ i where, LifeSatisfaction i is the measure of life satisfaction for individual i, X i is a vector of socio-economic variables described above, which function as determinants of life satisfaction, and ɛ i is assumed to be normally distributed, N(0, σ 2 ɛ). Tables 1 and 2 present the results using the Gini mobility measure, combined with the happiness variable (ghql) 1 Mobility indices were also estimated over different wave periods: 3 17, 1 12, 9 18, in order to observe different mobility patterns. The distribution of the estimated mobility indices was very similar to that observed in Figures 1 and 2, hence they are not reproduced for brevity and are available from the authors upon request. For the shorter wave periods, the number of observations for each wave was at least 4500, and for the shortest time period 9-18, we have over 6000 observations. 7

8 3 EMPIRICAL ILLUSTRATION: LIFE SATISFACTION INDICATORS IN THE UNITED KINGDOM Figure 1: Distribution of estimates of Gini mobility using happiness (ghql) and life satisfaction variables ( lfsato) variables 8

9 3 EMPIRICAL ILLUSTRATION: LIFE SATISFACTION INDICATORS IN THE UNITED KINGDOM Figure 2: Distribution of estimates of Simpson using happiness (ghql) and life satisfaction variables (lfsato) variables 9

10 3 EMPIRICAL ILLUSTRATION: LIFE SATISFACTION INDICATORS IN THE UNITED KINGDOM and the life satisfaction variable (lsfato ) respectively. Tables 3 and 4 present the results using the Simpson mobility measure, combined with the happiness variable (ghql) and the life satisfaction variable (lsfato ) respectively. Each column in the tables presents results of a regression where the individual mobility index was regressed upon the socio-economic variables in any one particular wave. The tables present regression results for waves 1, 4, 7, 10, 13, 16 are presented. The choice of these waves is ad-hoc, simply meant to represent all the years in the span of 1990 to 2008 equally. We have estimates using other waves as well; all results are stable across the waves and are available from the authors on request. Other strategies were also adopted and estimated, for example using the averages of the variables over the entire time period. For this, there were several difficulties encountered. First, the interpretation of several of the variables (such as education or marital status) were lost. Second, all waves do not have the same number of respondents on these questions and finally, the questionnaire changed and therefore the categorisation and coding of some of the variables changed as well (for example, marital status). Nevertheless, we estimated a model with averages of the socio-economic variables after having taken all of the above considerations into account, and the results are identical to those that we observe with the wave-specific setting. Estimates are available from the authors on request. Finally, in order to increase the number of observations per model, we also chose subsets of years, such as 3-12, 9-18, The estimates obtained are very similar to those we present below, and are again available from the authors on request. It is worth noting that the relationships between each socio-economic variable and the mobility index are remarkably stable across all four tables results. For all tables, sex is positively and significantly associated with the mobility indices, whether it is the Gini or the Simpson, and whether the mobility index is based on the happiness variable or the life satisfaction variable. Similarly health is also positively and significantly associated with the mobility index, as is number of children in household. Age is consistently negatively and (mostly) significantly associated with mobility. Interestingly we also found that lower education and higher income is associated with higher mobility in happiness, whereas higher education and lower income is associated with higher mobility in life satisfaction. These result are stable for the Gini and the Simpson indices and beg for further inquiry into the reasons behind them. In short, the respective relationships between the mobility index and its covariates are robust and stable for all models estimated, whether they are based on life satisfaction or happiness measures, and whether they resort to the Gini or the Simpson functional forms. 10

11 3 EMPIRICAL ILLUSTRATION: LIFE SATISFACTION INDICATORS IN THE UNITED KINGDOM WAVE sex 0.047*** 0.042*** 0.046*** 0.044*** 0.049*** 0.045*** (0.006) (0.006) (0.006) (0.006) (0.006) (0.006) health 0.023*** 0.021*** 0.024*** 0.028*** 0.028*** 0.027*** (0.003) (0.002) (0.003) (0.003) (0.003) (0.002) married * *** *** *** *** (0.007) (0.007) (0.007) (0.007) (0.007) (0.007) age *** *** *** *** *** *** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) education *** *** *** *** *** *** (0.001) (0.006) (0.001) (0.001) (0.001) (0.001) log income 0.006** ** 0.006* 0.011*** 0.007* (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) nkids (0.003) (0.003) (0.003) (0.003) (0.004) (0.004) constant 0.199*** 0.272*** 0.209*** 0.251*** 0.208*** 0.290*** (0.032) (0.036) (0.038) (0.038) (0.04) (0.043) N 3,229 3,245 3,251 3,253 3,234 3,229 R Notes: Heteroscedasticity robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 Table 1: Regressions for the Gini Mobility Index on socio-economic characteristics, with subjective wellbeing measured by general happiness WAVE sex 0.02*** 0.016*** 0.014*** 0.013*** 0.019*** 0.016*** (0.004) (0.005) (0.004) (0.004) (0.004) (0.004) health 0.014*** 0.014*** 0.017*** 0.018*** 0.018*** 0.019*** (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) married *** *** *** *** *** -0.06*** (0.005) (0.005) (0.005) (0.005) (0.005) (0.005) age * * *** *** *** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) education 0.004*** 0.004*** 0.003*** 0.003*** 0.004*** 0.003*** (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) log income ** *** *** ** (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) nkids 0.011*** 0.012*** 0.014*** 0.010*** 0.007*** 0.005* (0.002) (0.002) (0.002) (0.003) (0.003) (0.003) constant 0.138*** 0.180*** 0.202*** 0.227*** 0.188*** 0.260*** (0.022) (0.025) (0.026) (0.027) (0.028) (0.03) N R Notes Heteroscedasticity robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 Table 2: Regressions for the Gini Mobility Index on socio-economic characteristics, with subjective wellbeing measured by life satisfaction 11

12 3 EMPIRICAL ILLUSTRATION: LIFE SATISFACTION INDICATORS IN THE UNITED KINGDOM WAVE sex 0.075*** 0.068*** 0.073*** 0.07*** 0.078*** 0.071*** (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) health 0.035*** 0.033*** 0.036*** 0.043*** 0.043*** 0.041*** (0.004) (0.004) (0.004) (0.004) (0.004) (0.004) married * *** *** *** -0.07*** (0.011) (0.011) (0.01) (0.01) (0.01) (0.01) age *** *** *** *** *** *** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) education *** *** *** *** *** *** (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) log income 0.009** * *** 0.01* (0.005) (0.005) (0.005) (0.005) (0.005) (0.005) nkids (0.005) (0.005) (0.005) (0.005) (0.006) (0.006) constant 0.454*** 0.561*** 0.497*** 0.548*** 0.479*** 0.601*** (0.051) (0.057) (0.061) (0.06) (0.064) (0.069) N R Notes Heteroscedasticity robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 Table 3: Regressions for the Simpson Mobility Index on socio-economic characteristics, with subjective wellbeing measured by general happiness WAVE sex 0.037*** 0.029*** 0.026*** 0.025*** 0.034*** 0.031*** (0.008) (0.008) (0.008) (0.007) (0.007) (0.007) health 0.02*** 0.02*** 0.024*** 0.027*** 0.026*** 0.028*** (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) married *** *** *** *** -0.07*** *** (0.008) (0.008) (0.008) (0.008) (0.008) (0.008) age ** *** *** *** *** *** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) education 0.006*** 0.005*** 0.004*** 0.005*** 0.005*** 0.004*** (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) log income ** *** *** (0.003) (0.004) (0.004) (0.004) (0.004) (0.004) nkids 0.016*** 0.017*** 0.022*** 0.014*** 0.01** (0.004) (0.004) (0.004) (0.004) (0.004) (0.005) constant 0.51*** 0.6*** 0.64*** 0.662*** 0.59*** 0.691*** (0.038) (0.042) (0.045) (0.045) (0.047) (0.052) N R Notes Heteroscedasticity robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 Table 4: Regressions for the Simpson Mobility Index on socio-economic characteristics, with subjective wellbeing measured by life satisfaction 12

13 4 CONCLUSION 4 Conclusion In this paper we have conceptualised intra-generational mobility over several periods when variables are discrete, a topic not discussed in the mobility literature as yet. We proposed concepts, and related desirable properties, of maximum and minimum mobility, along with mobility-inducing transformations. A number of functional forms for indices of individual mobility and social mobility were also proposed. We showed that the proposed indices measure mobility as diversity or instability in people s status (as measured by the ordinal variable) for the period in question. Our illustration looked at the degree of mobility in life satisfaction measures in the UK. The indices were useful in uncovering interesting relationships between some key socio-economic characteristics and the degree of instability of life satisfaction responses in the British population. There are several avenues for future research. The obvious one is to conceptualize and propose measurements for directional mobility. Our current set-up is a measure of mobility as movement akin to the notion of mobility captured by indices like Shorrock s trace (in the two-period framework). Another avenue is the development of statistical inference tools for the proposed mobility measures. 13

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