Do monetary subjective well-being evaluations vary across space? Comparing continuous and discrete spatial heterogeneity

ABSTRACT

Using subjective well-being estimations, this study analyzes whether compensating variations vary across space using a cross-sectional data set from Chile. To achieve this goal, it describes and compares two econometric ways of modelling unobserved spatial heterogeneity. Both approaches allow compensating variations to vary across spatial units by assuming some distribution a priori. One method assumes that the spatial heterogeneity can be represented by a discrete distribution (a group of regions that share the same coefficient) and the other that the preferences can be represented by a continuous distribution (each region has a different coefficient). The results show that focusing just on the average estimates of compensating variations, as the applied studies have done so far, masks useful local variation. More empirical studies are needed to assess the advantages and disadvantages of both econometric approaches and how their results compare across a wide range of conditions and samples.

KEYWORDS:

• life satisfaction
• subjective well-being
• compensating variation
• continuous and discrete spatial heterogeneity

JEL:

• C25
• I31
• R13

ACKNOWLEDGEMENT

The author is obliged to five anonymous referees; the editor for helpful comments; and Victor Iturra for enlightening conversations.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author.

ORCID

Mauricio Sarrias http://orcid.org/0000-0001-5932-4817

Notes

1 For a more comprehensive review of how these measures have been used in the field in economic, see Di Tella and MacCulloch (2006).

2 Some studies examine the spatial aspect of well-being in the economic literature (Berry & Okulicz-Kozaryn, 2011; Brereton, Clinch, & Ferreira, 2008; Moro, Brereton, Ferreira, & Clinch, 2008; Rehdanz & Maddison, 2008). Others have investigated the spatial distribution of the willingness to pay for amenities using choice experiments (Brouwer et al., 2010; Campbell, Hutchinson, & Scarpa, 2009).

3 Throughout, ‘location unit’, ‘region’ or ‘geographical area’ are used interchangeably.

4 Communes are the smallest geographical areas in Chile for which individual data are available.

5 Although the main focus of this paper is to highlight the spatial heterogeneity of the CVs, it is also important to underline that its results are not free of potential endogeneity problems. The fourth section discusses the potential biases that can be expected in terms of CVs. Thus, this study makes strong assumptions about the exogeneity of the variables, but offers valuable insights when analyzing the spatial heterogeneity of CVs.

6 In order to obtain anticipated SWB, they ask whether the respondent anticipates more life satisfaction for a particular hypothetical situation.

7 Some other studies have also found a strong and positive correlation between stated SWB measures and emotional expressions (e.g., Deiner et al., 1999; Ferrer-i Carbonell & Frijters, 2004, passim).

8 As further explained by Adler (2012), since the CV (or the marginal rate of substitution) is invariant to any monotonic transformation of the subjective well-being functions, no cardinal utility function is required. Furthermore, the direct effects or marginal utilities, captured by the coefficients, only shift the level of satisfaction up- or downwards, without changing the marginal rates of substitution between the life dimensions.

9 Unlike the geographically weighted regression (Brunsdon, Fotheringham, & Charlton, 1998), which forces a certain degree of correlation through geographical proximity, the present approach does not impose any restriction (beyond the distribution) and allows the coefficients to vary freely across space. In this sense, the present approach is close in spirit to the random coefficient model (Swamy, 1971), but it differs in that it uses data at the individual level in order to identify the parameters at the regional level.

10 For example, see Hensher and Greene (2003) for a discussion about some distributions and their implications.

11 This paper also ran some basic specifications using the original ordered categorical variable. However, no systematic differences were found between the binary and ordered probit model. These results are available from the author upon request.

12 All the estimations conducted in this study were carried out using R software (R Core Team, 2016). In particular, the Rchoice package (Sarrias, 2016) was used to estimate the models using simulated maximum likelihood (ML). The ML algorithm for the discrete case was also coded in the same software.

13 In order to check this assumption empirically, a model was also estimated by assuming that individuals’ income varies across space following a normal distribution. However, the standard deviation for this variable turned out to be non-significant. Given this result, it seems that assuming any kind of parametric distribution, at least for this sample, for the income coefficient is not appropriate.

14 One of the variables with more missing values is satisfaction with life. This is because the question is answered only for the household head. Table A1 of Supplementary data shows the differences in the mean between individuals who answer this questions and those who do not. In general, the results are as expected. For example, the proportion of inactive individuals is higher in the sample that do not answer the SWB question (no households head), while older individuals are more likely to answer the question. No significant differences were observed in the variables about neighbourhood perception.

15 The estimations were also carried out using other dichotomizations; however, no significant differences were observed. The results are available from the author upon request.

16 This is equivalent to US$235.00.

17 It is possible that self-perception of neighbourhood characteristics may suffer from measurement error, which could bias the estimates. However, the empirical literature has shown that subjective variables related to neighbourhood characteristics might in some cases perform better than the objective ones (Day, 2007). For example, Van Praag and Baarsma (2005) found that crucial information regarding noise pollution is not objective but the subjective perception of it. The argument is that when measuring the effect of location-specific amenities on subjective well-being, it is important that they are linked to the individual at the level at which s/he experience them.

18 The standard errors are computed using the Delta method.

19 Let W* = W(Z; log(Y)), where Y is household income pc. By total differentiation, setting W* = 0 and holding all the other variables constant yields:$\begin{array}{cc}\frac{\text{d}\log \left(Y\right)}{\text{d}Z}& =\frac{\mathrm{\partial }W/\mathrm{\partial }Z}{\mathrm{\partial }W/\mathrm{\partial }\log \left(Y\right)}.\end{array}$

Note that $\text{d}\log \left(Y\right)/\text{d}Z=\left(\text{d}Y/Y\right)/\text{d}Z$. The ratio then measures the ‘relative change’ in $Y$ needed to bring the individual to his original level of well-being given a marginal change in $Z$, other things being equal.

20 Other distributions have also been tried in the specification, such as the triangular, uniform and Johnson S_b distribution, and other different combinations. However, models showed a comparative lower fit than the models presented in Table 2.

21 Good performance of SML requires a very large number of draws. However, the maximization of SML can be very time consuming when estimating large and complex models. Researchers have gained speed with no degradation in simulation performance through the use of smaller number of Halton draws (Bhat, 2001; Train, 2000). Bhat’s (2001) Monte Carlo analysis found that the precision of the estimated parameters was less when using 100 Halton draws than 1000 pseudo-random numbers in the context of mixed logit. The present study found that anything beyond 100 Halton draws did not lead to significant changes in the estimated parameters.

22 The proportion of communes with a positive coefficient for some attribute is given by $\mathrm{\Phi }(\mathrm{at}\beta /{\hat{\sigma }}_{\beta })$, where $\mathrm{\Phi }(\cdot )$ is the cumulative distribution function of the standard normal distribution.

23 This paper also estimated versions of the continuous model where the mean of the random parameters varies according to functions of the geographical coordinates. However, due to the complexity of the estimation procedure, the models in some cases did not converge or flat regions of the simulated likelihood function producing a singular Hessian were encountered.

24 Specifically, the conditional probability of commune $c$ of belonging to class $q$ is computed as:${\hat{\pi }}_{cq}=\frac{{\hat{w}}_{cq}{\prod }_{i}f\left(W_{ci}\right|{\mathit{x}}_{ci},{\hat{\beta }}_{cq},{\hat{\theta }}_q)}{{\sum }_{q=1}^Q{\hat{w}}_{cq}{\prod }_{i}f\left(W_{ci}\right|{\mathit{x}}_{ci},{\hat{\beta }}_{cq},{\hat{\theta }}_q)}.$

25 To put in context the compensatory variation for disability for the second class, it can be considered that an average Chilean family would require an increase of household income pc of approximately US$900. This amount is realistic if one considers valuations made for similar long-term health problems. For example, Groot and Maassen van den Brink (2006) estimate that the monetary value for cardiovascular diseases is approximately £49,564 (US$ 65,000) for men and £17,503 (US$ 23,000) for women.

26 In cases where the research question is more direct, one could focus on particular communes if there are reasons to do so, or describe the distribution of specific quantiles of the location-specific CVs, as shown in Daziano and Achtnicht (2014).

27 The conditional mean of CVs for the rest of variables are shown in Figure 1.

Additional information

Funding

The author thanks the FONDECYT – Fondo Nacional de Desarrollo Científico y Tecnológico [grant number 11160104] for full financial support.