Looking beyond linearity: A
nonparametric analysis of subjective
well-being and income
Dani¨
el Vullings, 2221551
Masters Thesis EORAS: Econometrics Supervisor: Dr. D. Ronchetti
Looking beyond linearity: A
nonparametric analysis of subjective
well-being and income
Dani¨
el Vullings, 2221551
Abstract
Contents
1 Introduction 2
2 Model 4
3 Estimation 5
3.1 Estimators . . . 5
3.2 Asymptotic distribution of local constant and local linear estimator . . . 8
3.2.1 Asymptotic distribution of density estimator . . . 10
3.3 Confidence band with heteroscedastic errors . . . 10
4 Variables and data 11 4.1 Variable selection . . . 11
4.2 Variable explanation . . . 12
5 Data analysis 14 5.1 Unconditional relation between income and SWB . . . 16
5.2 Conditional relation between income and SWB given health, age, and hours worked . . . 20
5.2.1 Health . . . 21
5.2.2 Age . . . 22
5.2.3 Hours worked . . . 24
1 Introduction
In recent years the literature that addresses subjective well-being (SWB) has exploded. Diener (2000) defines SWB as peoples cognitive and affective valuations of life. Both psy-chologists and economists argue that increases in SWB might be a more complete goal for governments to pursue than increases in standards of material welfare such as GDP. In Hicks, Tinker, and Allin (2013), for example, the authors looked at how to measure SWB in Great Britain as in British politics a similar idea is on the rise. From the 1990’s onward much research has been done around this topic. Diener (2012) is an overview of the psy-chological literature, where many factors such as income, health, and social economic status are presented as factors for SWB. Another overview, where more sophisticated econometric techniques are considered, is given by Dolan, Peasgood, and White (2008).
Currently there is no consensus and empirical proof about what makes people feel happy, and it is unclear how people react to certain circumstances. SWB is complex and, as with many social concepts, it is difficult to define a model that captures SWB. Whereas in eco-nomics, theory is often used to design a model that should describe the variable of interest well, for an abstract concept such as SWB this is nearly impossible as many factors are potentially important and many interactions could be crucial. The effect of better health could, for example, depend on the level of income, as wealthier people are better capable to buy supportive products and might therefore suffer less from their worse health. One can also imagine that a small decrease in health is considered to be just inconvenient, while a large decrease in health can have major consequences. A nonparametric procedure is very suitable to study complex social concepts where no clear structure for the model can be defined. This allows the researcher to estimate the effect of variables on SWB under much weaker conditions then when using parametric models. The results will therefore be much more robust, and essentially driven by real data.
Though much research has been done in this field, there is a lack of research that uses advanced econometric techniques. Fixed effects models are considered in some articles, such as in Ferrer-i Carbonell (2005) where the importance of income on SWB is estimated. However, more extensive research using these kinds of techniques is necessary. In all of the above mentioned literature the models used to explain SWB are just linear models. It is mentioned in Dolan et al. (2008) that whereas interaction effects were found to be important, little clear results have been found on what interaction effects to include in the models. As most of the literature makes use of linear models, occasionally with some interaction effects, the conclusions found make the crucial assumption that linear models are appropriate. This is a very strong assumption. Since no clear pattern between factors and SWB is typically detected, the assumption of linearity is over-restrictive. In addition, an increase or decrease in SWB may be steeper above a certain minimum wealth, as found in Stevenson and Wolfers (2013). All these findings, and the fact that within the literature results are often ambiguous furthermore raises suspicion that linear models are wrongly specified.
no structure, see for example Matzkin (2003) where a nonadditive model is estimated, to semi-parametric models that, for example, assume just the additivity of the regressors. This makes nonparametric techniques very flexible and useful for a wide variety of problems. This freedom along with the weak conditions do not come without its costs though. Correctly specified parametric models are more efficient. If more structure is added to a model, estima-tion will be more efficient. However, often there is no clear idea what the true structure of a certain model is. Then nonparametric estimation is an extremely useful tool to have in ones arsenal. As there is ever more data available and computers are ever faster, the drawbacks of nonparametric estimation become less of a problem. Due to the work of Henderson, Car-roll, and Li (2008), where a nonparametric fixed effects estimator is derived, and Mammen, Stove, and Tjostheim (2009), where an additive semi-parametric fixed effects estimator is derived, popular parametric models now have semi-parametric and nonparametric counter-parts, which makes the use of nonparametric techniques ever more appealing. Although the latter techniques are computationally very intensive and especially the nonparametric fixed effects estimator requires high computation times, using supercomputers, these estimation procedures are, or at least will become in the upcoming years, feasible estimation strategies. For a complex concept such as SWB the advantages of nonparametric estimation easily out-weigh the disadvantages, as the assumption that the SWB is in truth linear is dubious to say the least and linear models are therefore likely inconsistent. Therefore, we have little choice, but to use nonparametric techniques.
Despite the recent developments in both nonparametric estimation and the analysis of SWB, few have ever used nonparametric techniques to estimate a model for SWB.1 As there are now nonparametric counterparts to the more complex fixed effects models, there is no excuse not to explore the great potential that nonparametric techniques have for enriching the literature on SWB. A positive exception is given by Sacks, Stevenson, and Wolfers (2010) and Stevenson and Wolfers (2013). In Sacks et al. (2010), the authors used both standard linear models and Lowess models to investigate the relation between income and subjective well-being. However, the nature of the paper was to find a correlation between log income and subjective well-being and the authors have therefore not corrected for other variables. Stevenson and Wolfers (2013) yields similar results and the authors conclude that the nonparametric estimate is very similar to a linear model. In Binder and Coad (2011) the authors used these results and again tested whether the relation between log income and SWB is linear. Again the authors found that the relationship between SWB and log income is nearly linear, although they note that future research should look into this topic. A criticism on all three of these papers is that the authors give no detail about the procedure used to estimate the nonparametric model. By choosing very large bandwidths, even when inappropriate, standard nonparametric estimators will represent a linear model, independent of the data. Hence, more research is necessary to investigate the validity of the claimes.
In this paper, we will estimate a model for SWB where mainly the effect of income is considered. A panel that ranges from 1995 to 2014 and contains a wide variety of variables is used. In Van Praag, Frijters, and Ferrer-i Carbonell (2003) it is explained that SWB is
1As noted by Dolan et al. (2008), the problem of interaction effects that are difficult to estimate in
largely determined by several domains of satisfaction and the model will include variables representing domains as control variables. As this is one of the first papers that uses a nonparametric model to estimate more than a simple model that only contains income and SWB, a relatively simple cross sectional model is used, where the model is estimated for all cohorts separately. This allows us to demonstrate what the effects of different variables are on the level of SWB and to investigate whether the effects change over time. One can imagine that after the crisis, in which many individuals received an income shock, income has a different effect on SWB than in the nineties when the sky was the limit in terms of economic growth. Tang (2014) finds that for China SWB changed over time, and a similar development could be present in the Netherlands. Nonparametric techniques are suitable to extract the structure of the effect variables have on each other and capture whether structures change over time. Results obtained from our research could give crucial insights in the effect that income as well as health, age, and hours worked have on SWB. Even if parametric models are justified, this can only be checked in a concise way, when nonparametric methods are used. Otherwise the alternative to hypothesis tests could be inconsistent as well, thus yielding faulty tests, as described in Li and Racine (2007). This paper will therefore open the path to a whole new branch of research on SWB where the strong and implausible assumptions made so far in the literature are much weaker and more believable. When the benefits of this approach are illustrated, it becomes worthwhile to invest in more computationally intensive techniques such as nonparametric fixed effects models to get a more complete understanding of SWB.
The structure of this paper is as follows. In section 2 the model adopted to study the behaviour of the SWB is described. In section 3 the econometric technique used for estimation of the model is explained. Section 4 gives a description of the variables we consider as possible determinants of SWB and the data. The results are presented in section 5, and section 6 contains the conclusion.
2 Model
In this section we describe the model used to explain the relation between income and SWB. We assume a general nonparametric model with additive errors of the form
Yi = g(Xi) + i, i = 1, . . . , n,
where Y = (Y1, . . . , Yn)0 is a n × 1 vector of dependent variables, X = (X1, . . . , Xk), with
Xi = (Xi1, . . . , Xin)0 for i = 1, 2, . . . , k, is a n × k matrix with one continuous and k − 1
discrete explanatory variables, = (1, . . . , n) is a n × 1 vector of errors with zero expected
value and constant variance. g(·) is a smooth unknown function. Examples of g(·) could be a linear function g(xi) = (x1i, . . . , xki)β, where β is a k × 1 vector of unknown parameters,
or a quadratic function g(xi) = (x21i, . . . , x2ki)β. The variables are not serially autocorrelated.
interactions with income, the assumption is reasonable.2
In order to estimate the model consistently, a crucial assumption is that the error term is independent of the regressors. The main question is whether there are omitted variables in that influence the regressors as well. In section 4.1 the included variables are described and it is argued why we believe the assumption to be satisfied.
As opposed to the nonadditive model, the identification conditions for the nonparametric model with additive errors are weak. By imposing the conditions we do not seriously restrict the model and if the conditions would not hold, all parametric models would be invalid as well. Hence, in the literature on SWB these assumptions are made implicitly, along with much stronger conditions. The conditions are such that for nearly all concepts in society that one wishes to model, the assumptions can safely be made. In Li and Racine (2007), identification conditions for these models are not even considered. The interested reader can find the identification conditions in Matzkin (2012).
In the case of heteroscedastic errors the model is given by
Yi = g(Xi) + v(Xi)
1 2
i,
where X is independent of , using the same notation as before, and is assumed to be a standard normal random variable. Results obtained using the model with heteroscedastic errors can be used to investigate the robustness of confidence bands for the model that assumes homoscedastic errors.
3 Estimation
In this section we describe the techniques for estimation. The estimators for the model under the assumption of homoscedasticity are presented in section 3.1. Section 3.2 considers the asymptotic distribution of the estimators and the construction of confidence bands. The estimation of confidence bands with heteroscedastic errors is treated in section 3.3.
3.1 Estimators
Estimation of the model is done as described in Li and Racine (2007). Several strategies for nonparametric estimation can be deployed. We use kernel estimation. In the model, g(x) is the conditional mean of Y given X = x. Hence we have that E(Yi|Xi = x) = g(x). As
we have that E(Yi|Xi = x) =R yfy|x(y|x)dy =
R yfy,x(y,x)dy
f (x) , in order to estimate g(·) we can
either estimate the separate and joint distributions or directly the regression function. A consideration is whether to use a local constant estimator (Nadaraya-Watson estimator) or a local linear estimator. As explained in Hansen (2015) neither dominates the other, as it depends on the situation which should be chosen. The main advantage of the local linear estimator is the good boundary properties compared to the local constant approach. The latter performs poorly on the boundary, while local linear estimators do not have this problem. However, this comes at the cost of higher computational intensity, which could be problematic when large datasets with many variables are used. Due to the better properties at the boundary, the local linear estimator is gaining in popularity. In Yu and Jones (1997)
2A next step in this branch of research could be to relax this assumption and use techniques as described
the performance for the estimators for quantile functions are compared. The authors find that only at the boundaries there is much of a difference though and in the interior the performance is similar. We will focus on local constant estimators and perform a comparison with the local linear estimator. This will give an indication whether the choice of the estimator is important for the results.
In order to define the estimators, first some notation needs to be defined. We define Xc to be the n × 1 vector of the continuous variable in X, while Xd is the n × (k − 1) matrix of
discrete variables. Kernel estimation makes use of kernels and the kernel for the continuous variables is defined by k x c− Xc i h = √1 2πexp − xc−Xc i h 2 ,
where h is the bandwidth. Many different kernel functions exist and as the results are robust to different choices, we choose to work with a normal kernel. The idea of a kernel function is to return a value at a certain point, using weighted values of points in the neighbourhood of the point of interest. A normal kernel is very intuitive in this aspect, as the points closest to the point of interest get the highest weight.
For the discrete variables we have
L(Xid, xd) = Πk−1s=1l(Xids, xds), s = 1, . . . , k − 1, l(Xids, xds) = I(Xds
i =xds)
where IXds
i =xds denotes an indicator function that equals one if X
ds
i = xds and zero otherwise
and the s index denotes that we deal with the sth discrete variable. Given these kernels, we find that for a mixed data set the kernel is given by
Kh(Xi− x) = k (Xc i − xc) h L(Xid, xd).
Using the notation defined above, first the local constant estimator of g(·) is given. The estimators, ˆf (x) and ˆfy,x(y, x), of the density of x and the joint density of y and x respectively,
are given by ˆ f (x) = 1 nh n X i=1 Kh(Xi− x) ˆ fy,x(y, x) = 1 nh n X i=1 Kh(Xi− x) k y − Yi h0 .
Hence, the Nadaraya-Watson estimator is defined as
which is the minimizer of the local constant criterion
n
X
i=1
(Yi− a)2Kh(Xi− x),
where a = ˆg(x). An alternative estimator for g(·) is the local linear estimator, which can be found by minimizing
n
X
i=1
(Yi− a − (Xi− x)0b)2Kh(Xi− x)
over a and b, where a = ˆg(x) and b = ∂ ˆg(x)∂x is a k × 1 vector. Let δ(x) = (a, b0)0, then
ˆ δ(x) = [ n X i=1 Kh(Xi− x)(1, (Xi− x)0)0(1, (Xi− x)0)] × n X i=1 Kh(Xi− x)(1, (Xi− x)0)0Yi.
The local linear estimator gives both an estimate for the function g(·) and the derivative over the variables in X. The discontinuous support of the discrete variables in X requires a slightly altered estimator. In Li and Racine (2004) this situation is considered and the authors provide a simple alternative to the normal local linear estimator. The estimator, using the same notation as before, is given by
ˆ δ(x) = [X i=1 Kh(Xi− x)(1, (Xic− x c)0 )0(1, (Xic− xc)0 )] ×X i=1 Kh(Xi− x)(1, (Xic− x c)0 )0Yi,
where ˆδ(x) consists of ˆg(x) and the estimated derivatives of the continuous variables. In section 3.2, the asymptotic properties of both estimators are given.
the bandwidths, CV (h), that is the sum of the squared residuals, where the leave-one-out estimator, which is the Nadaraya Watson estimator solved without an observation, is used to estimate the function g(·). This criterion is
CV (h) = n−1 n X i=1 (Yi− ˆg−i(Xi))2M (Xi), where ˆg−i(Xi) = n P l6=i Kh(Xi−Xl)Yl n P l6=i Kh(Xi−Xl)
and M (Xi) is a weight function that prevents the function
from becoming undefined due to the denominator becoming zero in the leave-one-out esti-mator.
A clear downside of cross-validation is the high computation time and that it suffers from the curse of dimensionality. When for many variables bandwidths need to be chosen, using cross-validation can become problematic. However, as most variables for this research are discrete and frequency estimation will be used to estimate them, only the bandwidth for income needs to be computed, which can be done within reasonable time. Furthermore, the grid used to choose bandwidths will be rough in order to keep the computation time in check. Future research could use finer grids to obtain more suitable bandwidths. However, the rough grids are suitable for the purpose of the research, and are therefore chosen over finer grids to reduce computation time. 3
3.2 Asymptotic distribution of local constant and local linear estimator
In this section the asymptotic distributions of both the local constant (lc) estimator and the local linear (ll) estimator are given. First the case with a local constant estimator is treated, after which the slightly altered case for the local linear estimator is considered. The asymptotic properties are described to introduce the bootstrap method for the creation of the confidence bands of the estimated regression function. In Li and Racine (2007), Liaponov’s central limit theorem is used to derive the asymptotic distribution of ˆg(x). Assumptions necessary to derive the asymptotic distribution of ˆg(x) are that
· x is an interior point
· both g(x) and f (x) are three-times continuously differentiable · f (x) > 0.
Then as
n → ∞, h → 0, nh → ∞, nh7 → 0,
3Future research could, furthermore, look at using cross-validation to select smoothing parameters for
we have that √ nh[ˆg(x) − g(x) − B11(x)h2] d → N 0,κσ 2(x) f (x) ,
where κ =R k(v)2dv with k(v) a standard normal kernel and B
11(x) is given by
B11(x) = κ2[g11(x) + 2g1(x)f1(x)f (x)−1],
where g1(x) denotes the first derivative of g(x) over xc, g11(x) the second derivative of g(x)
over xc, f
1(x) is defined similarly, and κ2 =R v2k(v)dv, using similar notation as before. Note
that the first three assumptions that are necessary for this result to hold are very general and do not seriously restrict the model. By using cross-validation to select the bandwidths, the remaining assumptions hold as well.
The asymptotic variance of the estimator can easily be estimated using ˆf (x), the sample counterpart of f (x). However, this does not hold for the bias. Much research is dedicated to the topic of compensating for the bias. Hall and Horowitz (2013) developed a bootstrap method to construct confidence bands. The idea is to create a confidence band, that in at least 1 − ζ percent of the cases, covers the true value of g(x) with at least 1 − α percent chance. The band is least reliable at minima or maxima, as makes sense given the bias.
Following Hall and Horowitz (2013), we proceed by first estimating residuals by
ˆ
i = Yi− ˆg(Xi).
The estimate of the variance is then given by
ˆ σ2 = 1 n n X i=1 ˆ 2i.
Next, bootstrap samples are drawn in the following way
Yi∗ = ˆg(Xi) + ∗i, , i = 1, . . . , n,
where ∗i are drawn from ˆ1, . . . , ˆn with replacement. For each bootstrap sample an 1 − α0
percent confidence interval is estimated where the bias is ignored, and α0 is chosen in such
a way that ˆg(x) is covered by 1 − α percent of the bootstrap intervals. Next α1 is chosen as
the ζ quantile of the sequence of α0’s (sequence as for each x the α0 has to be found) and
the 1 − α percent confidence band is then given by
B(α) = ( (x, y) : x ∈ R, ˆg(x) − κˆσ(x) nh ˆf (x)z1−α12 ≤ y ≤ ˆg(x) + κˆσ(x) nh ˆf (x)z1−α12 ) , where z1−α1 2 is the 1− α1
2 critical point of the cumulative distribution function of the standard
The asymptotic distribution for the local linear estimator is similar to the asymptotic distribution of the local constant estimator. The local linear estimator is consistent under the conditions that
· {Xi, Yi} are i.i.d.
· g(x), f (x), and σ(x) = E(2
i|x) are twice differentiable
· K is a bounded second order kernel · as n → ∞, nh3 → ∞, and nh7 → 0.
The asymptotic distribution is then given by √ nhhˆg(x) − g(x) − κ2 2 g11(x)h 2i→ Nd 0,κσ 2(x) f (x) .
Again the bias is problematic. However, the bias can be compensated in a very similar way as with the local constant estimator. Using the same method as for the local constant estimator with straightforward adjustments, a confidence band can be constructed. The assumptions are again weak and suitable bandwidths are chosen using cross-validation.
3.2.1 Asymptotic distribution of density estimator
In order to get a clear picture of the data, estimating the density of specific variables is very useful. The density will be estimated using kernel functions, and the estimators itself are given in section 3. Following Li and Racine (2007), we have that under the conditions
· {Xi} are i.i.d.
· The PDF f (·) of x has three-times bounded continuous derivatives · x is an interior point of the support of X
· as n → ∞, h → 0, nh → ∞, and nh7 → 0,
the asymptotic distribution is given by √
nhhg(x) − g(x) −ˆ κ2
2f11(x)h
2i → N (0, κf (x)) ,d
with similar notation as before. The estimator is, similar to the lc and ll estimator, biased. As the density is only estimated to get a basic idea of the data, the bias will be ignored in the construction of confidence intervals. This serves the purpose of investigating the data and saves computation time.
3.3 Confidence band with heteroscedastic errors
linear estimator. This results in an estimate of σ that depends on x. The confidence band, B(α), is easily generalized by replacing ˆσ by ˆσ(x). In Yu and Jones (2004) it is mentioned that different bandwidths should be chosen for the different stages of estimation. Hence twice a cross-validation procedure is necessary to compute suitable bandwidths. This comes at the cost of high computation times.
4 Variables and data
In this section we describe the properties that appropriate regressors must have, in order to mitigate the sources of bias in our results. In particular we discuss in section 4.1 the choice of variables and describe the data used for for our analysis in section 4.2.
4.1 Variable selection
The assumption that the regressors in X are independent of the error terms, is often subject for debate. Especially in models where the variable of interest is likely to be par-tially caused by some individual specific characteristics that will have an influence on other variables as well, it is crucial to make a strong case why the independence assumption is satisfied.
As Dolan et al. (2008) provides a comprehensive overview of the SWB literature and as the main focus is more on checking the validity of the results found by using a parametric analysis, variables suggested by the authors to be important will be used here as well. Van Praag et al. (2003) provides a thorough analysis of several variables on SWB as well and found similar variables to be of interest. Variables included are given by gross total household income, age, hours worked per week, and health. The number of variables included in the model is limited as kernel estimation suffers from the curse of dimensionality. Furthermore, as we use frequency estimation for the discrete variables and this is only feasible with relatively many observations for each discrete variable, it is necessary to be careful with including too many variables in the model. One might wonder why gender is not included. The main reason is to reduce the number of discrete variables, and in literature the effect of gender is often found to be unimportant. By controlling for the variables found in literature to be most important, the assumption that the regressors are independent of the error term is believed to be satisfied. There are no obvious omitted variables and the model will at least greatly extent what has currently been done in literature on SWB using nonparametric estimation. Researchers often argue that individual specific effects will be correlated with variables such as SWB as well as explanatory variables such as health and income. A regular assump-tion is that the individual specific effects are time invariant. Panel data is a popular choice of data as this allows to easily correct for the individual effects by for example extracting from each variable its mean or by taking the model in first differences. The time invariant indi-vidual effects will then drop out. With nonparametric estimation this procedure is sadly not that easy to implement. As the function g(·) in the model is unknown, it is challenging to, for example, take the model in first differences and make sure the structure of the estimated function is the same for both the regular variables and the variables in first differences.
method is extremely computationally intensive though and for models with many variables and observations, computation time quickly becomes problematic. However, as computers become ever faster and the benefits of using a nonparametric fixed effects model are evident, future research should look at this option, using the results found in this paper to build on. An alternative approach that does not suffer from the curse of dimensionality is to esti-mate a semiparametric model where additivity is assumed. Although this method is much faster, it does have a clear downside that makes this approach less attractive. By imposing an additive structure on the variables, the possibility of interaction effects is neglected. In Mammen et al. (2009) an estimator is proposed where the authors allow both for time ef-fects and individual efef-fects. If it can be shown that there are no interaction efef-fects between the explanatory variables included in the model, this approach is justified and yields great potential.
Due to the disadvantages described above, we will not estimate a fixed effects nonpara-metric or semiparanonpara-metric model. The individual effects will be assumed to be independent of the regressors. This is a strong assumption that might not hold. However, as very little research has been done on SWB using nonparametric techniques, it is useful to start with a relative simple analysis that can give a general understanding of the relation between in-come and SWB. Future research can use this foundation to extend the results and use more advanced techniques, if the results obtained here indicate such strategies are necessary.
4.2 Variable explanation
In this paper use is made of data of the DNB Household Survey. The data set contains Dutch household data with a wide variety of variables from the years 1993 to 2014. The sample is constructed to be a good representation of the population. The variables of interest are reported happiness, gross total household income, reported health, hours worked, and age. The first two variables are the main interest, while the other variables are used as control variables. The questions the respondents were asked are given in appendix A. The respondents were asked to answer on a scale from one to six how happy they consider themselves, where a lower value means the respondent is happier. This variable represents the SWB and a lower value for this variable means a higher SWB. Few respondents have a value of more than three. However, as SWB is not an absolute scale, the variation in the answers between neither happy nor unhappy and very happy still allows us to estimate what the effect of the other variables is on SWB.4 Expected health is measured on a scale from one to five, where a lower value implies that the respondent considers himself healthier. The measure used for income, here gross total household income, is somewhat arbitrary. Many different kinds of measures could be used. Since some simple comparisons show the results of the paper are robust and that there is little difference between these measures of income, we will use gross total household income. Future research could check whether there are major differences for different measures of income. For convenience, we refer to gross total household income simply by income in the rest of the paper.
4Although the measure might not capture the cognitive and affective evaluation of life, and therefore
When observations are missing for some of the variables, the observation is deleted from the sample. This could introduce some bias, as it can be argued that certain groups of people are more likely to leave some questions open than others. We could employ imputation or replacement techniques for missing data. However, by deleting listwise we still retain many observations. Future research could be done on the robustness of our results to different imputation techniques.
Table 1 provides some descriptive statistics. The table gives the number of observations for each year as well as the mean and standard deviation of SWB and income, as these are the most relevant variables. The final column gives the mean age of the sample, as this gives some indication about whether the sample is representative for the population. Extreme outliers in the data were dropped. For example, in 1996 one person had a recorded total income of over six million Euro and the standard deviation was more than doubled compared to the next and previous year because of this single observation. As this could influence the results, this observation and similar outliers are dropped. Observations for which the total gross household income was zero, were dropped from the sample as well. In the Netherlands almost no one has an income of zero: even the poorest gets at least some social support from the government. When someone is stated to have zero income, there is a chance that something went wrong with the construction of the data set and these individuals are therefore deleted from the sample. Only individuals with an income of exactly zero were deleted from the sample. Any individual that had an income above zero is included. This implies that if someone would have an income of one, this individual would be included in the sample. Since some simple comparisons with different income thresholds show that the results of the paper are robust to different income thresholds, we will keep all observations with positive income. Table 1 shows that in general the level of SWB is quite stable over time. Income grows somewhat over time, which makes sense due to inflation. After 2002 the income is suddenly much smaller, which can be explained by the transition from the Dutch guilder to the Euro.
The mean age is much higher in 1995 than in other years. When we ignore 1995, the general picture is that individuals in the samples of 2002 and earlier are slightly younger than individuals in the samples from the years after 2002.
Some years are missing in table 1, as they are not included in the analysis. These years were dropped as they contained very few observations. The years included serve our goal of investigating the difference between reported happiness of individuals of different income groups and can be used to compare whether the differences changed over the years. The number of observations for the remaining years vary between different years as well. The first few years of the data set contain many observations as do the final 10 years.
In order to get some first insights in the relation between SWB and total gross household income, some scatter plots are provided in figures 1a to 1d. The figures do not give a clear picture about the relation between income and SWB. The people that report the lowest levels of happiness have in general low incomes. However, the people that report to be happiest does not consist of wealthier people than the group that reports to be somewhat less happy. The scatter plots appear to be very similar for the different years.
Table 1: Descriptive statistics
Year Observations SWB mean SWB std Income mean Income std Age mean
1995 1483 1.97 0.61 74350.1 49946.8 64.8 1996 1160 1.97 0.64 76607.6 54340.8 47.3 1997 801 1.98 0.66 74338.3 51884.6 47.1 1998 412 2.05 0.66 63705.1 64972.0 48.1 1999 430 2.00 0.65 66684.9 42120.0 48.3 2002 865 1.92 0.59 63350.9 41211.8 48.2 2005 1149 1.96 0.63 30983.6 25189.5 51.0 2006 1037 2.07 0.63 31350.7 18510.5 52.9 2007 1087 1.92 0.61 32931.6 18757.6 53.0 2008 1023 1.97 0.66 33843.9 18731.4 54.7 2009 1055 1.96 0.68 35194.0 19884.4 55.9 2010 1097 1.96 0.66 36477.4 26827.6 56.3 2011 1133 1.99 0.68 36088.2 22016.6 57.7 2012 1133 1.89 0.71 35575.6 23978.0 58.3 2013 1120 1.93 0.67 36234.4 22568.5 56.0 2014 1248 1.89 0.63 36103.2 23508.1 54.7
well. The bandwidth is chosen by hand, as the purpose of the density is only to give a broad picture of the distribution of the data. The estimated densities are given in figures 1e to 1h and the estimates are as expected. The estimated densities are similar for the different years, supporting the idea that the samples for different years are comparable. In figures 2a to 2d the estimated densities of income conditioned on SWB are given along with confidence intervals. The red lines give the group that reported a one for SWB, the blue lines the group that reported two and the green lines the group that reported three. While figure 2b shows little difference in the estimated density for groups with different levels of SWB, this does not hold for figures 2a, 2c, and 2d, where the estimated density for the groups with a SWB of three appears to have a higher weight on middle incomes and lower weight on high incomes. The confidence intervals of the groups with a SWB of one and a SWB of three in figures 2c and 2d barely overlap for parts of the interval. Even though the bias of the estimator is not corrected and we did not use cross-validation to compute the bandwidths, which makes the estimated confidence intervals somewhat unreliable, the figures do give some indications that there are differences in the distribution of income for groups with a different level of SWB.
5 Data analysis
Figure 1
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
1 2 3 4 5 6 HHIncome SWB
(a) Scatterplot SWB and income 2014
0e+00 1e+05 2e+05 3e+05 4e+05 5e+05 6e+05
1 2 3 4 5 6 HHIncome SWB
(b) Scatterplot SWB and income 1995
0e+00 1e+05 2e+05 3e+05 4e+05 5e+05 6e+05
1 2 3 4 5 6 HHIncome SWB
(c) Scatterplot SWB and income 2005 to 2014
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
1 2 3 4 5 6 HHIncome SWB
(d) Scatterplot SWB and income 1995 to 1999
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
5.0e−06 1.5e−05 HHIncome Prob (e) PDF income 2014 0 50000 100000 200000 0e+00 4e−06 8e−06 HHIncome Prob (f) PDF income 1995
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
Figure 2
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.0e+00
1.0e−05
2.0e−05
HHIncome
Prob
(a) PDF income 2014 conditional on SWB
0 50000 100000 200000 0.0e+00 6.0e−06 1.2e−05 HHIncome Prob (b) PDF income 1995 conditional on SWB
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.0e+00 1.0e−05 2.0e−05 HHIncome Prob (c) PDF income 2005 to 2014 conditional on SWB 0 50000 100000 200000 0.0e+00 6.0e−06 1.2e−05 HHIncome Prob (d) PDF income 1995 to 1999 conditional on SWB
5.1 Unconditional relation between income and SWB
weak relation between SWB and income for that year.
Again note that the axis of SWB must be read in the following way: a lower value implies a higher SWB. The figures show that per year there are quite some differences. Whereas the estimate for 1995 in figure 3a is very flat, the estimate for 1996 in figure 3b has a downward curve that increases as income increases. For all years it appears that the estimate either is very flat or increasingly curves downward as income increases. Exceptions are 2006 and 2007, figures 3e and 3f, where the curve starts flat for low income, then curves downward and becomes flat for the high incomes again. In general it appears that there is a positive relation between SWB and income. Although not all years show a positive relation, most do, and from figure 5a and 5b, it becomes clear that when the years are combined, there is a clear positive relation between SWB and income.
Figures 5a and 5b show that the relation between income and SWB has changed over the years. Given the best bandwidth for the models, the local constant estimator of SWB on income given in figure 5a for the years 1995 to 1999, is much flatter than the estimate for the years 2005 to 2014, shown in figure 5b. The 95 percent quantile for the data from 1995 to 1999 lies at 160560 guilders and at 71084 Euro for the years 2005 to 2014. Comparing these points in the graph, we notice that the level of SWB in figure 5b is much higher at the 95 percent quantile than in figure 5a at the 95 percent quantile. Even the confidence intervals barely overlap. It appears that whereas for the middle incomes the level of SWB has remained the same since the nineties, this does not hold for the lower and higher incomes. The lower incomes are slightly better off, while the higher incomes appear to be much better off. In relative terms the richest households have experienced the highest increase in SWB. As the confidence intervals barely overlap for these groups, the result is a strong indication that there has been a shift in society regarding the relation between income and SWB. Whereas in the nineties it appears that low and middle incomes were as happy or the latter group was a little happier, in the past ten years the lower incomes seem slightly happier than middle incomes. Furthermore, whereas in the nineties there appeared to be a large middle class that all had a similar level of SWB and only the richest one percent individuals had a substantially higher SWB, in recent years the richest five percent has a higher SWB than the individuals with low or middle incomes.
Most plots show two regions, where the first region contains the low and middle incomes and is nearly flat and the second region contains the high incomes and has a downward slope. This is in contrast with linear models, as linear models have the same slope over the entire support. Hence, though the results do not give an indication of causality, it does show that linear models are not suitable to model SWB. Especially in figure 5b the linear model lies often outside the confidence interval, underestimating the SWB of low and high incomes and overestimating the SWB of middle incomes, due to the change in the slope for high incomes. Linear models would therefore give a wrong idea about the relation between income and SWB.
Figure 3 0 50000 100000 200000 1.80 1.90 2.00 2.10 HHIncome SWB
(a) Lc estimator SWB on income 1995
0 50000 100000 200000 1.6 1.7 1.8 1.9 2.0 2.1 HHIncome SWB (b) Lc estimator SWB on income 1996 0 50000 100000 200000 1.5 1.7 1.9 2.1 HHIncome SWB (c) Lc estimator SWB on income 2002
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.75 1.85 1.95 2.05 HHIncome SWB (d) Lc estimator SWB on income 2005
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.7
1.9
2.1
HHIncome
SWB
(e) Lc estimator SWB on income 2006
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.6 1.8 2.0 HHIncome SWB (f) Lc estimator SWB on income 2007
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.7 1.8 1.9 2.0 2.1 HHIncome SWB
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
Figure 4
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.80
1.90
2.00
HHIncome
SWB
(a) Lc estimator SWB on income 2010
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.6 1.8 2.0 2.2 HHIncome SWB (b) Lc estimator SWB on income 2011
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.6 1.7 1.8 1.9 2.0 2.1 HHIncome SWB (c) Lc estimator SWB on income 2012
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.7 1.8 1.9 2.0 2.1 HHIncome SWB (d) Lc estimator SWB on income 2013
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.2 1.4 1.6 1.8 2.0 HHIncome SWB
(e) Lc estimator SWB on income 2014
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.2 1.4 1.6 1.8 2.0 HHIncome SWB
Figure 5 0 50000 100000 200000 1.7 1.8 1.9 2.0 2.1 HHIncome SWB
(a) Lc estimator SWB on income 1995 to 1999
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.7 1.8 1.9 2.0 HHIncome SWB (b) Lc estimator SWB on income 2005 to 1014
from 1995 there is some difference at the boundary, but it is minor and would not alter the conclusions much. Figure 6b shows that for the sample of 2014, there are barely any differences. Hence the lc estimator appears to be a suitable choice and using a ll estimator yields similar results.
5.2 Conditional relation between income and SWB given health, age, and hours worked
Figure 6
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.5 1.7 1.9 2.1 HHIncome SWB
(a) Lc and ll estimator SWB on income 1995
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.5 1.7 1.9 2.1 HHIncome SWB
(b) Lc and ll estimator SWB on income 2014
variables decreases the sample size, we only extend the model for the sample of 1995 to 1999 and the sample of 2005 to 2014. These samples have enough observations to condition on a number of variables, while this does not hold for the individual years. We control for health in section 5.2.1, for age in section 5.2.2, and for hours worked in section 5.2.3.
5.2.1 Health
In Van Praag et al. (2003) it is mentioned that, after financial satisfaction, health is the most important aspect for the level of SWB. The authors furthermore mention that higher educated people are in general healthier. As higher educated people often earn more as well, it is crucial to correct for health.
Figure 7 0 50000 100000 200000 0e+00 4e−06 8e−06 HHIncome Prob
(a) 1995-1999: PDF for groups with different reported health
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.0e+00
1.0e−05
2.0e−05
HHIncome
Prob
(b) 2005-2014: PDF for groups with different reported health
from 2005 to 2014, the picture is very different. From figures 8b and 8f, it becomes clear that the difference in SWB for individuals with different levels of income is very small for the groups with a reported health of one or three, and wealthier people have a slightly lower SWB. Figure 8d has a similar shape as figure 5b and shows clearly that people with higher income have a higher SWB. The difference between the groups that reported a health of one between the samples is striking. Although in general the SWB appears to be somewhat higher in recent years in general, the group that reported a health of one has a relatively much higher SWB in recent years than the same group in the nineties, whereas the SWB of the group with a reported health of three has remained stable. Figure 8e shows that in the nineties for individuals with an income between 180000 and 220000 guilders and a reported health of three, the level of SWB is lower than for the rest of the individuals. However, very few individuals fall in this range and no conclusions can therefore be drawn from this result. Interestingly, wealthy individuals report on average more often to have a health of one and in recent years the difference between individuals that reported a health of one compared to other health groups is much larger than it was in the nineties. This explains a part of the difference in the relation between income and health between the nineties and recent years.
5.2.2 Age
0 50000 100000 200000 1.2 1.4 1.6 1.8 2.0 HHIncome SWB
(a) 1995-1999: Lc estimator SWB on income, reported health is 1
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.4 1.6 1.8 2.0 HHIncome SWB
(b) 2005-2014: Lc estimator SWB on income, reported health is 1 0 50000 100000 200000 1.5 1.7 1.9 2.1 HHIncome SWB
(c) 1995-1999: Lc estimator SWB on income, reported health is 2
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.4 1.6 1.8 2.0 HHIncome SWB
(d) 2005-2014: Lc estimator SWB on income, reported health is 2 0 50000 100000 200000 1.0 2.0 3.0 4.0 HHIncome SWB
(e) 1995-1999: Lc estimator SWB on income, reported health is 3
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.6
2.0
2.4
HHIncome
SWB
will hold for the majority of the population, as we selected those traits that the majority of the individuals have. Future research, using larger datasets, could generalize the results to include the other groups.
In order to keep substantial samples, the sample was split in three groups. The first group consists of the individuals with an age from 30 to 50 years old, the second group consists of people between 51 and 65, and the third group consist of people above 65. Individuals younger than 30 were dropped, as there were too few to make a sensible analysis and as young people have in general a higher SWB than middle aged people according to Dolan et al. (2008), including them in the group of ages between 30 and 50 could influence the results undesirably.
Figures 9a to 9c show the local constant estimates for the different age groups. The first observation is that we do not find that middle aged people have a lower SWB than old people. Interestingly, all three figures show a different relation between income and SWB. Figure 9a indicates that for the age group between 30 and 50 the picture is different from the general sample, as SWB appears to be constant over income, except for those with very high incomes. For the group with ages 51 to 65, in figure 9b, the level of SWB appears to increase with income for incomes under 30000 Euro, it decreases between 30000 and 40000 Euro, and again increases for incomes higher than 40000 Euro. For low and middle incomes we find that the level of SWB is lower than for other age groups, while for high incomes the level of SWB is similar to the other groups. Hence the effect of income on SWB appears to be stronger for this group. Finally, for the oldest group, displayed in figure 9c, middle incomes appear to have the lowest level of SWB and for the highest incomes additional income decreases the level of SWB. However, at the boundaries, the lc estimator is imprecise and as less than one percent of the individuals had an income of over 80000 Euro in this sample, no strong conclusions can be drawn for the incomes higher than 80000 Euro. The effect of income on SWB between the incomes of 40000 and 80000 is smaller for the group over 65 than for the group between 51 and 65. The last result can be explained by the fact that in general people over the age of 65 have fewer responsibilities towards children and they are probably used to their income and know it will remain stable. They do not have to worry much for their future and therefore care possibly less about their income. However, this does not explain why the effect of income on SWB is very small for the group between 30 and 50 as well. Future research could look into the result found and come up with explanations why people between 51 and 65 appear to value income more than other groups.
Again we find that the relation between the variables is non-trivial and there appear to be numerous interaction effects. Similar to previous estimates, we find that the slope of the estimates differs for different income groups. This again illustrates the necessity of nonparametric techniques to find the relation between income and SWB.
5.2.3 Hours worked
though. Unemployed people have time to spare, however, they do not have a lot of leisure time by choice and probably would want less leisure time. On the contrary, someone that works many hours a week could attach more value to an additional hour of leisure time than people that barely work. Hence, we control for hours worked, as this variable could possibly bias the results when omitted. First, we estimate a model conditioning on health and hours worked and next we condition on health, hours worked and age. For health and hours worked we only select those individuals that work 40 hours a week and report a health of 2. Only individuals below the age of 65 are analysed, as too few over the age of 65 work 40 hours. The samples are somewhat small when we condition over health, hours worked and age, as the samples consist of a little under 500 observations. Conditioning on yet another group, again comes at the cost of not being able to make a statement about the whole population. Figures 9d and 9e represent the local constant estimate for SWB on income for all in-dividuals with a reported health of two and that work 40 hours a week. The estimates for both the samples from 1995 to 1999 and 2005 to 2014 are given. Again there appears to be a difference between the two periods. Whereas in the nineties, there is no effect of income, this is very different for the past ten years. In the past ten years, the group that worked 40 hours a week displays a similar relation between income and SWB to the population in general. People with higher incomes have a higher SWB. This results is more intuitive than the result for the years 1995 to 1999, where it appears that individuals are somewhat indifferent to their income. It could be that there has been some sort of a culture shift in the past decades that makes people value money more. However, the conditioned sample for 1995 to 1999 is somewhat small at a little over 400 observations, which makes drawing strong conclusions from the estimates impossible. For the sample from 2005 to 2014, figure 9e shows that for the individuals that actually work 40 hours, the effect of income on their SWB is bigger than for the population in general. Whereas for low and middle incomes the SWB is comparable to the general population, individuals with high incomes that work full-time show a higher SWB. Hence for the high income individuals the effect of additional income on SWB is stronger than for the general population. A possible explanation is that individ-uals that work full-time consider income as a status symbol, more so than other groups do. Furthermore, these individuals might work as much because they value income higher than other individuals do.
Finally, we conditioned on reported health is two, hours worked is 40 and we differentiated between two age groups. Namely individuals between 30 an 50 and individuals between 51 and 65. The results, obtained using lc estimators, can be found in figure 9f and 9g. The two estimates are somewhat similar to the estimate in figure 9e in that they imply that for low and middle incomes, additional income matters little for the level of SWB, whereas it does matter for high incomes. The figures again show that the estimate is constant at first and is downward sloping after some threshold.
6 Conclusion
Figure 9
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.5 1.7 1.9 2.1 HHIncome SWB
(a) Lc estimator SWB on income, reported health is two and age is between 30 and 50
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.5 1.7 1.9 2.1 HHIncome SWB
(b) Lc estimator SWB on income, reported health is two and age is between 51 and 65
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
1.5 1.7 1.9 2.1 HHIncome SWB
(c) Lc estimator SWB on income, reported health is two and age is over 65
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.8 1.2 1.6 2.0 HHIncome SWB
(d) 1995-1999: Lc estimator SWB on income, reported health is two and works 40 hours a week
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.8 1.2 1.6 2.0 HHIncome SWB
(e) 2005-2014: Lc estimator SWB on income, reported health is two and works 40 hours a week
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.8 1.2 1.6 2.0 HHIncome SWB
(f) Lc estimator SWB on income, reported health is two, hours worked is 40, and age is between 30 and 50
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
inconsistent estimates. As contradictory results were found in the literature about the effect of variables on SWB, there was a clear indication that linear models might not be suitable for the problem at hand. Dolan et al. (2008) noted that whereas it was found that interaction effects are relevant, it was unclear what interaction effects should be included. This makes defining linear or other parametric models extremely difficult. Nonparametric techniques are suitable alternatives to get more insight in the relation between income and SWB. Even though Sacks et al. (2010) used nonparametric techniques as well and found that there appeared to be a linear relation between income and SWB, the results did not include any control variables. Hence, no indication was given on how relevant interaction effects are.
Using the local constant estimator, we investigated the relation between SWB and income using several control variables. We find that the relation between income and SWB is not straightforward. Whereas we see that in general individuals with higher incomes are happier, this effect differs when we condition on other variables. This implies that there are many interaction effects that influence the effect of income on SWB. Our results somewhat contradict the results in Sacks et al. (2010). Whereas we do find a near linear relation for the sample from the nineties, this does not hold for the past decade. In the past decade, the slope changes after some threshold. Where for low and middle incomes, the slope is flat in most cases, after some threshold the slope is clearly negative. This implies that the relation between income and SWB is nonlinear. Hence, researchers should not take the results in Sacks et al. (2010) as evidence that linear models are suitable. Although the shift is partly caused by the increased SWB of the group with the best health, even when we control for health, the effect was nonlinear. Our sample consists of Dutch households, hence the obtained results might not carry over to other countries. However, the results do imply that the relation between income and SWB changes over time and there is little reason to assume this is only the case in the Netherlands. Furthermore, as conditioning the sample on basic characteristics shows very different results, there are interaction effects between income and the variables age, hours worked and health. When this is ignored in the estimation of a model of SWB on income, the results will be inconsistent. Many of the results obtained in recent years suffer from this problem and the conclusions from this research might therefore be invalid.
mentioned in Dolan et al. (2008)
Despite controlling for several important variables and making far fewer assumptions than most papers on the subject of SWB do, there are weaknesses in the methodology. First of all, even though we make far fewer assumptions than currently has been done in this field, we still make the assumption of additivity of the error term. As we showed that there are many interaction effects between income and other variables, it may be simplistic to assume all variables for which this holds were controlled for. There could very well be additional variables that we did not control for, which would make the additivity of the error term assumption invalid. This problem could be tackled using nonadditive models. However, due to extreme computational complexity this alternative is problematic. Another possible problem that we did not correct for is the presence of individual effects. As often argued in the literature, individual effects are likely not independent of income and by not controlling for this, the effect of income on SWB could be biased. As our sample was not suitable for estimating a fixed effects model, we urge future research to use fixed effects models to estimate the relation between income and SWB. Due to the work of Henderson et al. (2008), the techniques to estimate a nonparametric fixed effects model are available. Future research could furthermore look to extend the result to the general population. Due to the sample size, we focussed on showing that subgroups deviate from the general population. A next step is to consider the effect of income on SWB for many different groups.
We showed that nonparametric techniques should be used when performing research on SWB. Most results show that there is a clear threshold after which the slope of the estimate changes. This implies that estimates using linear models are inconsistent. Many results found and conclusions drawn in literature might therefore be invalid. Future research should continue the path we set out. The limitations of this paper can be addressed and when our model is extended to contain more control variables, the additivity assumption becomes plausible. Furthermore, when nonparametric techniques uncover the functional form of the relation between income and SWB, parametric models incorporating the found functional form can be constructed to efficiently estimate this relation. This will allow future research to uncover the causal effect of income on SWB. This paper is therefore a first step in showing the weaknesses of the current approaches applied in the field of SWB and in estimating a model of income on SWB without implausible assumptions.
Appendix A
The questions below are as they are asked to the respondents in the DNB household survey.
SWB: All in all to what extent do you consider yourself to be a happy person, where 1 represents very happy and 6 represents very unhappy.
Hours worked: How many hours per week [do / did] you on average in fact spend on your [last] (most important) job? For this question it doesnt make any difference whether overtime work [is / was] paid for or not.
Total income is calculated by
where all variables represent either income, government supplements for disabilities, retire-ment, child support and so on or profit from stocks and investments. For a more thorough explanation consult either the DNB household survey our contact us.
Reported health: In general, would you say your health is: one excellent to five poor.
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