Faculty of Economics and Business
Amsterdam School of Economics
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University of Amsterdam
The Income Effect on your Perception of Happiness
Thesis for MSc in EconometricsLiselotte Siteur
10662340Date - July 25, 2017 Semester 2 (2016 - 2017)
Supervisor: dr. J.C.M. van Ophem Second reader: mw. E. Aristodemou PhD
Statement of Originality
This document is written by Liselotte Siteur who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
University of Amsterdam Faculty of Economics and Business
The Income Effect on your Perception of Happiness
Liselotte Siteur
Date - July 25, 2017
Abstract
This thesis addresses the relation between income and happiness in an ordered response model with threshold. The focus will lie on the threshold location and the magnitude of the income effect. The estimated model is analysed with a Monte Carlo simulation to assess the quality and the finite sample properties. In addition to a grid search, approximation methods are used to overcome the discontinuity created by the threshold. The model specifications are implemented on data of the U.S General Social Survey using household income and self-reported happiness. The threshold is estimated at a value of $91000. Estimates incorporating multiple thresholds show that the effects of additional income on happiness decrease as income increases. Consequently, the threshold specification is questioned since a nonlinear decreasing effect of income on happiness cannot be captured in one single threshold.
Acknowledgements
For my parents and brother - For those who are most dear, for those who are always here. My solid gratitude and appreciation are as unmeasurable as the support and guidance I received. I need to apologise, as I took their time, made them unnecessary worried and did not show them the love and respect there should have been. Even when my plate was full, you trusted and supported me in pursuing new activities, although you advised against it. It is amazing that you could put up with me, as crazy and busy as I have been all those years. Therefore my never ending thankfulness will be here to support you now as I proceed in life as someone I believe I want to be.
To my partner in crime Robert, who has made my life as easy as could be where I made life difficult for myself. Your hard work, days and nights of studying, your dili-gence and competitive spirit gave me someone to look up to, hold on to, and grow to be. I will always remember that we as equals could rise above and solve the almost im-possible. I sincerely hope that one day our paths will cross again and we can excel together.
And finally to my supervisor, who guided me through the final steps of the last four years. The guidance but freedom I received gave me the opportunity to find my own way in the happiness research. Your encouraging words made me realise that enough can be enough and climbing only one mountain is also an achievement on its own.
Contents
Statement of Originality Abstract
Acknowledgements
1 Introduction 1
2 The Theory of Happiness 3
3 Ordered response model with threshold 8
4 Data overview 12
5 Results 15
5.1 Simulation . . . 15
5.2 Grid search procedures . . . 19
5.2.1 Grid Search over γ, assuming γ fixed . . . 19
5.2.2 Grid Search over γ, when γ estimated conditionally . . . 23
5.3 Discontinuity approximations . . . 25
5.3.1 Kernel Estimation . . . 26
5.3.2 Individual income coefficient . . . 29
5.4 Multiple thresholds . . . 32
6 Conclusion 37
References 39
Appendix A.1
A.I Data corrections . . . A.1 A.II Monte Carlo Specification and Results . . . A.3 A.III Results for Multiple thresholds . . . .A.13
1
Introduction
Most econometric research focusses on the relation between two or more well-measured variables, but what happens if you investigate a very subjective definition? Can you truly estimate the effect of a well-measured variable on an individual’s subjective opinion? This thesis focusses on the relation between income and subjective well-being, or simply: hap-piness.
The relation between income and happiness is a common question with no unique answer. Some argue that there is no relation at all, or provide arguments that money can only reduce suffering, or state that there must be some sort of positive effect. Oth-ers directly avoid the question and jump to the definition of happiness and wonder how they would define it. Whatever the exact relation between income and happiness is, any hypothesis, regardless of its statistical evidence, will have its opponents.
The complicated effect of income on happiness makes it an interesting topic as no straightforward relation is expected and the topic is always relevant as the dynamics will never stop changing. The main target of this thesis is to estimate the relation between income and subjective well-being by proposing a general model using only a few causal restrictions.
Perhaps the most common theory in the income-happiness field is that there exists a point of satiation. There must be a point at which additional income no longer contributes to happiness, either because your basic needs are fulfilled or additional wealth brings no more security or pleasure, or brings even worries instead. Such a threshold is estimated above which income is valued less or even has no or negative effect. The introduction of an unknown threshold with unknown magnitude serves as the instrument to correctly define the relation between income and happiness.
With this information, conclusions can be drawn towards the general effect of income on an individual’s self-reported happiness. This is summarised in two questions: “When does additional income have a diminishing effect on its contribution to happiness?” which naturally leads to “How does income affect an individual’s perception of happiness?”.
To find an answer to these questions a model will be proposed based on the paper of Hodge and Shankar (2016). It will form the foundation of this research and will be further improved to create more accurate estimates. We will use data from the General Social Survey to identify the relation between income and happiness. Happiness is measured on a three-point scale which results in an ordered response model specification. This model is extended with a threshold under the assumption that a kink can be found in the relation between income and subjective well-being.
The main issue in this model is the discontinuity created by the threshold. Multiple estimation methods will be used to overcome this. In addition to the grid search used by
Hodge and Shankar, the discontinuous point will be approximated to make a continuous evaluation possible. Most models will be evaluated with a simulation to show the finite sample behaviour and to assess the degree to which they are able to estimate the threshold correctly.
The analysis is extended by incorporating multiple thresholds. This implies that there can be multiple income groups at which the relation towards happiness is different. Additional thresholds may potentially identify even more relationships between happiness and income.
The research starts with a review of previous studies on happiness and its relation towards income in Section 2. In Section 3 the model as introduced by Hodge and Shankar is specified and a brief introduction of a basic estimation technique is mentioned. Section 4 provides an overview of the data from the General Social Survey and shows some descrip-tive statistics that indicate the first evidence of a posidescrip-tive relation between income and happiness. The results are reported in section 5 and contain various threshold estimations including a simulation to find the finite sample behaviour of the ordered response threshold model. The section concludes with a brief analysis of incorporating multiple thresholds in the model. Finally, conclusions are drawn in Section 6.
2
The Theory of Happiness
The question whether personal well-being is related to material pursuits has been raised for centuries. Financial security and the power to enjoy more than the basic needs is a luxury we all hope to find. It was only after World War II that surveys started to collect information on well-being in national population surveys. This set off a range of papers looking for the correct relation between income and happiness. Both empirical and data-driven investigations are discussed to gain a broad perspective on the possible arguments explaining this relation. This results in a theory on the most likely relation between income and subjective well-being.
Easterlin (1973) can be seen as a pioneer in the income-happiness research and brought together 30 surveys. He analysed the relation between income and happiness in both developed and less developed countries. With no exception, all surveys showed that the happiest had the highest average income. While Easterlin was very cautious in drawing too strong conclusions, he stated that in all societies individual wealth leads to higher individual happiness, but raising the income of all does not increase the happiness of all. The relation between income and happiness within a community is considered to be linear.
Easterlin argues that people are most concerned with their relative income. Material needs or wishes that can be fulfilled with income arise from the possessions of other community members. In addition, the standards at which an individual is raised may determine their perceptions. Economic and technological developments made that most children were able to exceed their parents’ living status. This may have increased the valuation of money, as there was clear evidence of what wealth could buy.
In addition, Easterlin states that also marital status, age and education are related to happiness. This results in the question how happiness can be defined, which can be seen as the most complicated part in happiness research. In some surveys, it was asked what people mean by happiness. Most common answers involved economics status, family relations and health, but some mention social or political arguments. The difference in interpretation relates to the feasibility of biased self-reported well-being which can be an issue in estimating the happiness effect. The findings of Easterlin are still leading in recent happiness papers and remain.
Kahneman and Deaton (2010) and Altman (2016) contribute to the discussion of the meaning of happiness by using the terms life satisfaction and emotional well-being. Life satisfaction refers to the thoughts about a person’s life and this information is mostly gained with a question like, “How satisfied are you with your life these days?”. Emotional well-being refers to the intensity and frequency of pleasant and unpleasant emotions. The Gallup organisation provides information on both outlines of happiness by asking many
questions about emotions and a detailed life satisfaction measure. Both Kahneman and Deaton and Altman use this survey in determining whether “money can buy happiness”. Altman (2016) puts the results of the Gallup survey in empirical perspective and provides insight in the income-happiness relation. He states that a reduction in income growth generates more loss in happiness than the standard linear income-happiness rela-tion proposed by Easterlin predicts. Altman proposes a threshold in the growth rate of income in relation to happiness, as an increase by a small percentage has less influence on the already wealthy. He is very sceptical about the general income-happiness relation, as un upward movement in living standards tends to demolish the effect of income growth that would be expected. In addition, he states that additional income can only increase well-being if goods and services are purchased that can contribute to happiness.
Kahneman and Deaton (2010) use the Gallup survey in an analytical way, as they regress the logarithm of the household income and multiple other explanatory variables on several measures of happiness. The introduction of the logarithm implies a concave relation between income and happiness such that we focus on a relative increase and not an absolute. From a plot of the happiness measures against the annual income, they conclude that above an income of $75000 there is no benefit in emotional well-being and a clear kink is found. However, when considering a different definition of happiness, the life satisfaction keeps increasing linearly. Therefore we need to carefully consider the definition of well-being when studying results in this happiness research, as income tends to effect no emotions after a certain threshold but people can still experience an increase in life satisfaction.
Kahneman and Deaton (2010) and Altman (2016) introduce a threshold in the rela-tion between income and happiness for individuals. This threshold funcrela-tions as a point at which additional income is less or not valued. The threshold of $75000 annual household income is probably analogous to a value at which a family can be financially secure and is resistant to small catastrophes. However, the location of this threshold is challenged as different papers found very different results and corresponding arguments.
Stevenson and Wolfers (2013) try to estimate the relation between countries’ Gross Domestic Product per Capita (GDP) and life satisfaction and incorporate a threshold. They analyse three different fixed threshold locations and define the optimal point at $15000. However, this threshold is not significant as no evidence is found of a clear kink in the population. They use data from different years and countries in which this inves-tigation focusses on the income-happiness relation between communities. The location of the threshold should be questioned as both developed and less-developed countries participated in this research and a limited amount of threshold locations is tested. The interpretation of the location of the income threshold at $15000 can be difficult as the wealth level differs greatly over the countries. However, in more developed countries this
amount is more related to a measure of poverty. This would result in the argument that money makes happy until you can supply the basic needs.
An extension of the analysis of Stevenson and Wolfers is made by Lien, Hu and Liu (2016) as they use analytical methods to find the threshold location. They focus on both data within one country and values for several countries. They use Hansen (2016) who proposes a grid search method for a threshold model with an observed dependent variable. Lien, Hu and Liu use the life satisfaction, a discreet measure, as a continuous variable and find a threshold around $7000 which is not significant nor stable over different survey waves.
Lien, Hu and Liu must conclude that no evidence for a threshold is found and there-fore no point of satiation of income is measured. The lack of evidence can be caused by the unjustified continuity assumption of the dependent variable, or the shortage of explanatory variables in this analysis. In addition, their grid search could have been too small as they leave out the top 10% of the observed income values and they regress on the log(GDP) which already incorporates a decrease in the happiness effect for higher incomes. This would suggest not to use the logarithm of income as this makes it more difficult to assess the threshold location.
Although the estimation methods in both the paper of Stevenson and Wolfers and Lien, Hu and Liu leave many suggestions open, they conclude with a threshold that is very low although not significant. As the analysis is made across countries, this threshold location would identify which countries are considered to be better-developed than others. They, however, do find a positive impact of income on happiness when analysing different countries, which is a contradiction of the Easterlin Paradox. This would suggest that in all situations and not only within communities the effect of income on subjective well-being is present.
So far we have focussed on the interpretation of happiness and the presence of a point of satiation. Before we continue to the threshold specification that seems most fit, attention must be paid to the implications of an ordered response variable. Because, in spite of the fact that all measures of happiness are discontinuous, most research neglects this issue. Afterwards, a paper will be discussed that addresses the problem of unobserved heterogeneity.
Boes and Winkelmann (2004) use a German Socio-Economic Panel on life satisfac-tion. They address the issue of an ordered response variable correctly whereas this is ignored in most other happiness-related research. They discuss whether the income effect on happiness is different per happiness level and indeed find evidence for such a relation. Boes and Winkelmann find that respondents in the highest happiness category do not have the highest average income. They focus on different estimation alternatives and reduce the distributional assumptions of the probability models. However, they do not incorporate
a threshold or point of satiation. Although they look for differences between happiness levels and do not focus on the relation of income to happiness, the difference in happi-ness groups and their significant results argues for the incorporation of a threshold. Their ordered response model is well analysed and much thought has been put in eliminating restrictive assumptions, which make their results form a good foundation for an argument to include a threshold.
Becchetti, Corrado and Rossetti (2011) use a British household panel on life satisfac-tion to correct for unobserved heterogeneity in the income-happiness spectrum. Especially in a subjective matter like your own perception of happiness, the unmeasurable feeling towards this topic can be removed with a fixed effect estimation. Almost all individuals in the population are only weakly sensitive to income changes as there is a high average income. They concluded that a point of satiation can be found, but do not specify an exact location. This conclusion results from a significantly large group of ‘frustrated achievers’, a large group of individuals earning more but reported to be less happy over the years.
Most frustrated achievers show characteristics like divorce or additional schooling over the years. For this group of individuals, the income-happiness relation tends to be negative which downgrades the population curve. An additional argument for the downturn of the happiness is the difference between realisation and expectation. A positive change in income and a lack of close friends to share the increased welfare may turn the positive relation between income and happiness negative. The combination of a weak relation between income and happiness, when income is already high, with arguments towards a downtrend for high incomes, results in a point of satiation.
The introduction of a threshold is one of the few conclusions most papers agree on. Although most state no more than a hypothesis, all accept the possibility that at some point people reach a level of satisfaction and additional income is either less valued or people are indifferent. The results about the location of the threshold are inconclusive and all use different methods and happiness measures. If a regression is employed, often the happiness response is interpreted as a continuous variable instead of an ordered response. Imported points that must be considered in estimation are the importance of community conditions to measure relative wealth and the corporation of the correct explanatory vari-ables, to reduce the omitted-variable bias caused by the different measures of the subjective well-being of all individuals.
The paper of Hodge and Shankar (2016) that is the foundation for this present re-search, tries to deal with multiple issues. By using an ordered response model and incorpo-rating a threshold around income, they try to correctly determine the correct relationship. They use life satisfaction as happiness measure in the estimation. They not only try to identify the threshold location but also look for evidence for the correct model specifica-tion. They follow the threshold estimation techniques of Hansen (2016) and adjust the
model by incorporating the ordered response. Hodge and Shankar also propose the testing procedure of Hansen to correctly prove the existence. A distinctive advantage of the model and method of Hansen is that the threshold location is unknown and the impact of this satiation point is therefore estimated conditionally.
Although Hodge and Shankar find evidence for a threshold, results are not very robust as threshold estimations range from $76000 to $93000 in household income. To assess the quality of their research, they also regress on self-reported health and find similar results with a threshold between $82000 and $91000. As self-reported health is concerned to be a measure of emotional well-being, these results are not in line with the findings of Kahneman and Deaton (2010). In their maximum likelihood procedure, they also find an increase in likelihood around $15000 which could host an argument about the absence of poverty.
The model of Hodge and Shankar is unique within the field of the income-happiness gradient research since they properly try to define a threshold location and prove a thresh-old exists. This research seems like a good opportunity to build on. Their estimation tech-niques are not very well developed and will be extended in this thesis to find a more robust result. With the same model and data, a proper identification of the threshold is the target. In addition, the model will be extended by including multiple thresholds to test even different patterns in the valuation of money. Since very different thresholds have been found in previous papers, perhaps a combination of both a low and high threshold could be a good solution. The exact model specification, in line with the propositions of Hodge and Shankar, will be explained in the next chapter.
3
Ordered response model with threshold
So far we have stated the hypothesis that income and happiness are positively related but after some income threshold, additional wealth is valued less. We consider the observed happiness measured as an ordered response variable with m different categories. Hodge and Shankar (2016) propose a threshold model based on the ordered response structure of the dependent variable. Their paper, together with the analysis of Hansen (2016), will function as the main sources for our model specification. Before the threshold model is explained, first the general framework of an ordered response model is discussed.
The starting point of the ordered response specification is a linear model based on a single latent variable yi∗:
yi∗= x0iβ + εi (1)
where happiness, yi will be explained by income and other explanatory variables
sum-marised by xi and where εi represents an error term. y∗i measures the relative happiness
and is converted in the ordered happiness ranking when yi∗ crosses boundaries µj. This
results in the following relation:
yi= j ⇔ y∗i ∈ [µj−1, µj] for j ∈ {1, ..., m} (2)
where µ0 = −∞ and µm = ∞ for m specified happiness categories. This model is estimated
without intercept because an identification problem would occur due to the shift of the location of µ. To estimate the maximum likelihood, the probability of each happiness level, given the explanatory parameters, needs to be calculated.
P[yi = j] = P[µj−1≤ yi∗ ≤ µj]
= P[µj−1≤ x0iβ + εi≤ µj]
= P[µj−1− x0iβ ≤ εi≤ µj − x0iβ]
= F (µj − x0iβ) − F (µj−1− x0iβ)
Here F (.) is equal to the cumulative distribution function of εi. Under the assumption of
an ordered probit model, this results in εi ∼ N (0, σ2). The variance σ2 is assumed to be
one, to overcome another identification problem. The scaling between β and µ results in an identification issue which can be resolved by a fixed variance of the error term. The log-likelihood is now constructed as the sum of probabilities that the observed happiness matches the real value for all individuals and is given below:
l(κ) = N X i=1 m X j=1 I{yi=j}· ln (P[yi= j]) = N X i=1 m X j=1 I{yi=j}· ln Φ(µj− x 0 iβ) − Φ(µj−1− x0iβ) (3)
where κ = {β, µ} and I{yi=j} an indicator function for the happiness of an individual. The
likelihood presented in (3) is continuous in all parameters and therefore easily estimated. Although the interpretation of the estimated coefficients is not very straightforward due to the latent dependent variable, the sign of β does state whether or not the corresponding coefficient increases yi∗, (Cameron & Trividi, 2005, Ch.15.9).
The linear equation (1) can be extended to a threshold model by allowing the β0s to vary, depending on a threshold parameter qi. In most literature qi is a function of xi as
will be in the general model in this thesis. The threshold introduces the assumption of two different observed groups that show a different preference. Including the threshold results in the structural equation of interest in (4):
y∗i = x0iβ1+ εi qi ≤ γ x0iβ2+ εi qi > γ (4)
The likelihood in (3) can be altered to reflect the threshold specification. The function is extended by considering two cases, the case when the the income is below the threshold, I{qi≤γ}, and when the income is above the threshold, I{qi>γ}. For both cases the correct
β0s need to be taken into account. The corresponding likelihood (5) is now a function of both κ and the threshold parameter γ.
l(κ, γ) = N X i=1 m X j=1 I{yi=j}I{qi≤γ}· ln Φ(µj− x 0 iβ1) − Φ(µj−1− x0iβ1) + I{yi=j}I{qi>γ}· ln Φ(µj− x 0 iβ2) − Φ(µj−1− x0iβ2) (5)
This model cannot be estimated with a simple optimisation principle as the threshold creates a discontinuity in γ. We will focus now on a grid search in a two-step procedure as this is the method proposed in Hodge and Shankar (2016). In the results in section 5, we will present two alternative methods that approximate (4) to overcome the discontinuity. Similar to a concentrated likelihood, in this two-step approach the parameter space is partitioned into a parameter set of interest θ, and remaining nuisance parameters η, as explained by Murphy and Van Der Vaart (2000). The coefficients η are then estimated first conditional on θ which results in ˆη(θ). The likelihood is now only dependent on θ, and the parameter space is reduced.
max
θ,η l(θ, η) ⇒ maxθ l(θ, ˆη(θ))
In the context of the threshold model, γ can be estimated either in the first step or in the second step. Hodge and Shankar initially propose to estimate γ in the first step, but even-tually estimate γ in the second step, as is done in most threshold research. Examples are the paper of Lie, Hu and Liu (2016) and the paper of Hansen (2016) which both estimate an unknown threshold. In all these studies a grid search over γ is used to overcome the
problem of discontinuity. The grid search implies that for a fixed set of γ’s, all other model parameters are estimated and the optimal γ results from the best estimated likelihood.
When using this approach, standard errors obtained from a hessian are not valid since a two-step approach is used. The standard error needs to be estimated with a boot-strap procedure to eliminate this problem. The bootboot-strap procedure does not eliminate all problems as the randomness of the first step parameters needs to be taken into ac-count. A bootstrap of the other model parameters, where γ is fixed in the second step, underestimates the variance as the randomness of the threshold is not taken into account. Therefore this bootstrap procedure is only valid for the standard errors of the estimated parameters in the second step. As the standard error of the income variables is the most interesting, the threshold γ must be estimated in the first step. This is in line with the theory proposed by Hodge and Shankar, but not with the actual estimation they employ. In addition to the change in estimation method Hodge and Shankar make, they change the notation of the structural equation. They state an equivalent model to (4) which represents the same relation between dependent and independent variables. Both models are widely used in the threshold analysis and authors often switch. The parameter λ is introduced such that λ = β1− β2 or β2 = β1− λ and that λ corresponds to the change
in the coefficient when crossing the threshold. The model has the following structural equation (6) and corresponding likelihood (7).
yi∗ = x0iβ1+ (xi· I{qi>γ}) 0λ + ε i (6) l(κ, γ) = N X i=1 m X j=1 I{yi=j}· ln Φ(µj− x0iβ1− (xi· I{qi>γ}) 0λ) − Φ(µj−1− x0iβ1− (xi· I{qi>γ}) 0 λ) (7)
The model will be estimated with both specifications throughout the analysis. Although the likelihood is equal for the same parameters, the optimum found is not the same, as we will discuss later in Section 5.
Until now, the assumption was made that the threshold specification is correct. The threshold specification is principally only valid when the difference between groups above and below the threshold is significant. To test the validity, a likelihood ratio test is used in which the threshold model results are compared with a restricted model assuming either λ = 0 or β1 = β2. This results in the following test statistic, as proposed by Hodge and
Shankar (2016):
LR = 2l(ˆθ) − l(ˆθr)
(8) in which l(ˆθ) results from the optimisation of either (5) or (7) and l(ˆθr) results from the
restricted model either implementing λ = 0 or β1 = β2 or removing the entire threshold
which k is equal to the amount of restriction or in other words the size of β1, (Cameron
& Trividi, 2005, Ch.7.3).
So far the estimated model is discussed with corresponding estimation techniques, which leads to the final question, how the estimated coefficients can be interpreted. The interpretation of the coefficients is not straightforward as the ordered response model has an unobserved variable y∗i and is non-linear. A measure that is often used and has an easy interpretation is the average partial effect (APE). The APE refers to the increase in the probability of being in a certain state, [µj−1, µj], when an increase in one of the
vari-ables occurs. It is expressed as the partial effect by using derivatives to the corresponding variables (Wooldridge, 2002). For any variable xk the partial effect would be equal to
δj(xik) = ∂P [yi = j|xi, qi, γ] ∂xik = − I{qi≤γ} φ(µj− x 0 iβ1) − φ(µj−1− x0iβ1) β1k − I{qi>γ} φ(µj− x0iβ2) − φ(µj−1− x0iβ2) β2k (9)
Throughout the estimation results, the APE will be interpreted as the additional proba-bility of being the happiest caused by the change in an explanatory variable. This results in the average partial effect below, which is specified separately for observations above and below the threshold.
δm(xik|qi ≤ γ) = − 1 PN i=1I{qi≤γ} N X i=1 I{qi≤γ} φ(µm− x 0 iβ1) − φ(µm−1− x0iβ1) β1k δm(xik|qi > γ) = − 1 PN i=1I{qi>γ} N X i=1 I{qi>γ} φ(µm− x 0 iβ2) − φ(µm−1− x0iβ2) β2k (10)
In the remainder of this thesis, the APE will be referred to by its coefficient rather than the corresponding variable. In the case of λ, the APE is not calculated, however the equivalent in which β2 is replaced by β1− λ is used to compare the APE between different notations
of the model.
With the tools to describe and estimate the model, we can start making the actual analysis. Two basic first estimation methods that can be implemented on the threshold model have been described. The finite sample behaviour of both model specifications with one of the estimation methods will be explored with a simulation first. Afterwards the model and methods can be implemented on the real data. A brief overview of the data used will follow to establish which exact relation needs to be estimated and tested.
4
Data overview
The General Social Survey (GSS) gathers data on American society to study its growing complexity. The personal-interview survey is constructed to monitor changes in social characteristics and current attitudes since 1972. It was conducted annually from 1972 to 1993 and is biennial since 1994, with available data until 2016.
This analysis focusses on life satisfaction as a scale of happiness which is measured with the question “Taken all together, how would you say things are these days, would you say that you are very happy, pretty happy or not too happy?”. The income measure is the family income earned in the previous year the survey was conducted. It is measured in 2010 equivalent dollars using the CPI-U-RS index from the Bureau of Labor Statistics, a consumer price index used to deflate the current population survey. In addition, the income measure is adjusted with the OECD-modified equivalence scale for the household size to correct for proportional needs of additional household members. This implies that every household head is assigned a value of 1 and 0.5 is added for each additional adult and 0.3 for each child (Hout, 2004). With an interpolation over the possible observed income intervals, the family income is converted to a continuous scale. All together this household income measure creates a good representation of available income adjusted for family size which is representative for all years of the survey to analyse the individuals’ impact of money on happiness. The income distribution can be found in Figure 1 and shows the distribution over all specific happiness levels and indeed shows that the density of ’very happy’ people is larger for the highest incomes. Similarly, the majority of the ’not too happy’ people has a relatively low income compared to the average population.
All additional explanatory variables used in this study, are equivalent to those used in Hodge and Shankar (2016) for the sake of reproduceability. This considers year and
regional dummies, age, gender, education, employment, religious attendance and marital status. An overview of the population distribution over all variables can be found in Table 1 together with a specification of all happiness levels.
The target is to find the influence of income on happiness, so the question is not how all other explanatory variables affect happiness, but how they relate to additional happiness from income. The objective is to find which variables relate to the income happiness measure and should be included to remove any false effect otherwise assigned to the effect of income.
The most straightforward variable is working status as this automatically results in more income. However, you not only gain happiness through income but also through social status, structure and achievement of goals which strongly influence happiness (Helliwell, Layard, & Sachs, 2017). The effect of the physical reward of employment, your income, is only measured correctly when introducing the working status variables to remove the additional effects.
An educational degree has similar effects, as is found in many studies that a higher degree will result in more income in the long run. A fact that could be considered is high student loans for higher degrees, which may decrease happiness in the early stages of life. The effect of a diploma on happiness is, however, more related to the qualifications in your surrounding than the actual achievement and the effect of education is valued lower than
Table 1: Data description with available data from 1972 to 2016
% In % Very % Pretty % Not too % In % Very % Pretty % Not too
Sample Happy Happy Happy Sample Happy Happy Happy
Income Working Status
- 15 000 15,7 24,2 54,6 21,2 Fulltime 51,7 33,7 57,7 8,6 15 000 - 30 000 20,6 28,3 57,3 14,4 Parttime 11,0 31,5 57,2 11,3 30 000 - 50 000 25,6 33,8 57,0 9,1 Temp. not 2,3 32,8 54,3 12,9 50 000 - 75 000 18,7 36,7 56,3 7,0 Unempl/laid off 3,4 17,6 53,8 28,6 75 000 + 19,5 43,1 51,8 5,2 Retired 11,1 37,8 50,5 11,7 Gender School 3,5 32,2 56,7 11,1
Female 53,2 34,2 55,0 10,8 Keeping house 15,4 36,5 51,5 12,0
Male 46,8 32,7 56,2 11,1 Other 1,7 22,5 50,8 26,7 Age Degree 18-24 12,3 28,6 59,3 12,1 Lt high school 20,1 31,3 51,9 16,7 25-44 42,4 32,4 57,4 10,2 High school 52,7 32,1 57,2 10,7 45-64 31,3 34,7 53,9 11,4 Junior college 5,6 33,6 57,3 9,1 65 - 14,0 38,5 50,5 11,0 Bachelor 14,6 38,2 55,1 6,7 Marital Graduate 7,0 40,8 53,0 6,2 Married 61,8 40,6 52,2 7,1 Race Widowed 5,8 23,9 56,9 19,3 White 82,8 35,1 55,4 9,5 Divorced/Separated 11,9 19,0 60,4 20,6 Black 12,0 24,2 56,0 19,9
Never married 20,6 23,2 62,4 14,4 Religious attendance
Region Never 16,8 26,4 58,2 15,5
Northeast 19,3 31,8 57,2 11,0 Once a year or less 21,5 28,5 59,6 11,9
Midwest 25,9 33,0 56,9 10,1 Several times 20,0 31,6 57,6 10,9
South 34,5 35,1 53,4 11,5 Often 14,0 35,4 54,8 9,8
that of income according to Helliwell, Layard and Sachs (2017).
Gender and age are taken into account to correct the change in valuation of money over the phases of life and lower wages for starters. In addition, males tend to have a different valuation towards money than women do. Marital status has a direct impact on happiness and income since a stable home life can be the basis of a happy life and a potential partner or family can adjust your income needs. The region is only taken into account to replicate the Hodge and Shankar (2016) paper but is of little interest as no state information is available and there is no interpretation for differences in such broad areas. As can be seen in Table 1 there is no clear variation in happiness between regions. The final variable that is considered is religious attendance and it has been found that increase in religious participation reduces income (Lipford & Tollison, 2003). A high income discourages religious participation while a lower income can encourage people to participate in religious activities. As Table 1 shows, people with a high rate of religious attendance are typically valuing their lives as happier than those who never or hardly ever attend.
In addition to dropping observations where information is missing, measures are taken to produce a correct national representative sample. The GSS itself provides a weight vari-able correcting for the sub-sampling of non-respondents, differential non-response across areas and the number of adults in a household. This last correction implies that multiple adults could serve as multiple observations of the same characteristics. As data of before and after 2004 is used this weight must be taken into account in the analysis, because the sub-sampling method is changed.
Hodge and Shankar (2016) follow Stevenson and Wolfers (2008b) and remove those Spanish languages interviews that would not have been held if English was the only op-tion. Stevenson and Wolfers also solve the problem of an oversample of the black ethnic population in 1982 and 1987 and the change in question order over several survey waves. They discuss the question order in an earlier paper (Stevenson & Wolfers, 2008a) and analyse that a difference can be found in waves in which the happiness question was part of a series of subjects among the importance of family, friends and leisure or in which the happiness question was never near any such mood influencing topics. Stevenson and Wolfers (2008b) propose a solution to this problem that is mostly caused by omitting the marital satisfaction question in multiple waves. They regress happiness on a dummy variable for those affected by the oversampling and question order and control for fixed effects for both married and unmarried people. This approach is used on the 2016 dataset and results of the change in weight can be found in the appendix in Table A.1. All weight taken into account, this results in a sample size of 50428.
5
Results
The ordered response threshold model is implemented to determine the income-happiness gradient. In the analysis to find the right identification for the relationship between income and happiness, the threshold can serve as a point of satiation above which income is not as highly valued. In addition to proving its existence, attention will be paid to the location of the threshold and possible additional ones.
Before an analysis can be made with the real data, first simulations are performed to assess the quality of estimation. This functions as the proof of being allowed to apply both notations of the model specification to real data. From these results in Section 5.1, conclusions can be drawn on the validity of the models which need to be considered in the further research.
Afterwards, a selection of various methods describe how to approximate the exact threshold and the income-happiness coefficients on the real data. Four different models are used with different approaches to determine the threshold and other coefficients of interest. First, a simple grid search will be used in a two-step approach as discussed in the model specification in Section 3. γ will be estimated in the second step in Section 5.2.1 and afterwards γ will be estimated in the first step, which is reported in Section 5.2.2. The next two models involve approximation techniques to solve the discontinuity problems. A kernel density is implemented in Section 5.3.1 and an approach is realised that adopts the income coefficient for each individual in Section 5.3.2. Finally in section 5.4, the last estimation method will be implemented to find the evidence of multiple thresholds. 5.1 Simulation
A Monte Carlo simulation is introduced by Hodge and Shankar (2016) as evidence for the finite sample behaviour of the ordered probability threshold model. They use a known data generating process on an ordered threshold probit model for different sample sizes and different magnitudes of the threshold effect. The magnitude, λ, equals the difference between β1 and β2 (β2 = β1 + λ) and measures the possibility of identification of two
independent groups. These effects are analysed within the model specified below. y∗i = β1xi+ θ1w1i+ θ2w2i+ εi xi ≤ γ β2xi+ θ1w1i+ θ2w2i+ εi xi > γ (11) A full specification of the data generating process can be found in Appendix II, which accounts for a description of the exact likelihood and gradients used for the analysis.
The model is approximated with a likelihood function conditioned on the estimated threshold. This implies that first an estimation of γ is made, after which the remaining parameters are estimated conditional on ˆγ. In the iteration process, this implies that in each step γ is estimated conditional on all other parameters. As the threshold parameter
γ creates a discontinuity in the model, it can not be estimated with any optimisation algorithm and a grid search is used. For the estimation of all other parameters, a trust-region algorithm is used based on the implemented gradients.
The result of the analysis with all different model specifications can be found in Table A.2. Some important results are shown in Table 2. The portraited bias is equal to the average percentage bias and is calculated as
Bias = 1 R R X r=1 ˆ θr− θ θ ! · 100 (12)
where R equals the 1000 Monte Carlo simulations, ˆθr the estimate of the parameter of
interest and θ the corresponding true value.
In Table A.2, the finite sample behaviour of the simulated model is clearly pictured. Especially the estimates for the smallest sample size strongly differ from the actual pa-rameters. This is mainly caused by the fact that certain groups, with either high or low yi ranking or xi above or below the threshold, are disproportionally represented in some
samples which causes difficulty in estimating all parameters properly. As the sample size increases, the probability of such an event occuring decreases rapidly and estimates become more accurate. For a sample size of 500, all coefficients are estimated within a boundary of a 10% bias of the original value. As the sample size increases to over a 1000 a maximum bias of 5% is observed. Therefore it can be concluded that for large samples the model is properly estimated and can be used on real samples.
When lowering the identification between the population, λ moving from −6 to −0.5, the parameter estimates are equally well estimated. This results in the assumption that under the notation and estimation of (11) the model performs well under all different magnitudes of the threshold effect. Together with the property that the model behaves well under a finite sample and has good estimates, it can be concluded that this ordered response model with threshold holds good estimates.
Table 2: Highlights of the Monte Carlo analysis
Effect/sample size Threshold Threshold coefficients Other Coefficients Auxiliary parameters λ = −6 γ = 0 β1= 4 β2= −2 θ1= 1 θ2= 3 µ1= −1 µ2= 2 µ3= 5 N=5000 Mean 0.0030 3.9990 -1.9958 1.0034 3.0099 -0.9916 2.0194 5.0338 Median 0.0100 3.9962 -1.9939 1.0030 3.0057 -0.9898 2.0179 5.0295 Bias -0.0245 -0.2091 0.3435 0.3293 -0.8436 0.9719 0.6768 Std 0.0563 0.1239 0.0724 0.0348 0.0754 0.0556 0.0683 0.1368 λ = −0.5 γ = 0 β1= 4 β2= 3.5 θ1= 1 θ2= 3 µ1= −1 µ2= 2 µ3= 5 N=5000 Mean 0.0117 3.9996 3.5232 1.0023 3.0080 -0.9815 2.0273 5.0344 Median 0.1100 3.9911 3.5189 1.0023 3.0059 -0.9828 2.0257 5.0338 Bias -0.0089 0.6636 0.2317 0.2658 -1.8518 1.3668 0.6889 Std 0.3710 0.1117 0.0857 0.0314 0.0671 0.0531 0.0619 0.1164
In addition to the estimation of (11) in which the structural equation is displayed with β1 and β2, an alternative notation is used considering β1 and λ. In this case the
structural equation changes to (13).
yi∗= β1xi+ λ(xi· I{xi>γ}) + θ1w1i+ θ2w2i+ εi (13)
The log-likelihood changes accordingly and is depicted in Appendix II. Important is to know that for the same parameter inputs and data the likelihoods corresponding to (11) and (13) are equal. The models only differ in their gradients which results in different movements in the optimisation iterations. The simulation results considering the model with alternative notation can be found in Table A.2 and a summary below in Table 3.
Table A.2 shows the behaviour of this model for increasing sample sizes. Consider-ing the identification specifications altogether, the results show that there is indeed an improvement for an increasing sample size. However, in all cases, it is not as good as the model where β2 was estimated. Especially when the identification in the model gets
lower, the estimations are not improving that much when the sample is increasing. A good measure is the median of γ. In many smaller samples, it hits the lower bound -2 of the grid search, which implies that estimation in these models is not reliable since no correct optimum was found. This raises concerns whether λ is identified when β1 ≈ β2.
Table 3 shows that even in a large sample the quality of estimation gets lower when there is only a small difference between groups. The estimate of λ is off by 63% which is related to the incorrect estimate of the threshold at -0.22. When the identification is high, λ = −6, the results are hardly biased and are in line with the data generating process, considering a decent sample size. However, when identification gets lower the model does not perform correctly and it must be concluded that in this case, this finite sample is not resulting in reliable estimates. Therefore it needs to be concluded that the model regarding (13) is only valid when identification is good.
Table 3: Monte Carlo Results using the alternative model notation with λ
Effect/sample size Threshold Threshold coefficients Other Coefficients Auxiliary parameters λ = −6 γ = 0 β1= 4 λ = −6 θ1= 1 θ2= 3 µ1= −1 µ2= 2 µ3= 5 N=5000 Mean 0.0032 3.9909 -5.9818 1.0028 3.0080 -0.9869 2.0221 5.0347 Median 0.0100 3.9879 -5.9783 1.0020 3.0045 -0.9858 2.0192 5.0283 Bias -0.2264 0.3025 0.2801 0.2670 -1.3130 1.1037 0.6931 Std 0.0627 0.1266 0.1862 0.0351 0.0761 0.0574 0.0697 0.1394 λ = −0.5 γ = 0 β1= 4 λ = −0.5 θ1= 1 θ2= 3 µ1= −1 µ2= 2 µ3= 5 N=5000 Mean -0.2215 3.7770 -0.1851 0.9994 2.9992 -0.9037 2.0909 5.1058 Median 0.2500 3.7869 -0.2610 0.9994 2.9969 -0.8945 2.0961 5.1083 Bias 0 -5.5742 -62.9781 -0.0602 -0.0277 -9.6313 4.5464 2.1161 Std 1.3077 0.2286 0.2822 0.0316 0.0684 0.0674 0.0711 0.1249
From the difference in estimation of the models (11) and (13) the question arises how this is caused. Since the derivative towards the coefficients of the income variable is the only difference, it is obvious to start looking for an argument here. In the gradient for β1
in the model estimating λ, all observations are summed, while in the model estimating β2,
when estimating β1, only those observations below the threshold are taken into account.
This results in a value of the gradient that is presumably higher, which would draw the optimisation in the direction of the optimum of β1 sooner than λ (Cameron & Trividi,
2005, Ch. 10). It then can be possible to reach a different maximum of the function, not directly assuming one solution is better than another. This argument is only valid if it is indeed found that the function can encounter both local and global maxima. Figure 2 shows that for one replication with a sample size of 5000 considering the model on a grid for γ, there are definitely more maxima to be found. The grim function directly proposes an argument for the grid search as discontinuity plays a great role, even on such a small interval.
There is a significant difference found in the estimation of model (11) and (13), with mostly inconsistent estimates when establishing λ instead of β2. Although the gradients
used for the optimisation differ, it is not entirely clear why other optimums are found. To find the correct optimum one should simply look at the highest found log-likelihood. Therefore it needs to be concluded that the models should not be mistaken for each other, although the interpretation of both is the same. This does not state that (13) is not a valid model to be estimated, but just that it works only for a large sample and large identification. In further research, it will be important to compare results of both models to find the correct optimum, as only the likelihoods can show the best optimum.
5.2 Grid search procedures
Using the established validity of the estimation technique for the ordered response model with threshold, we can now focus on finding the actual income threshold using real data. The indexed family income, Ii, and happiness valuation, Happyi, of over 50 000
partici-pants will be used to determine this relation. The following relation is specified:
Happyi∗ = β1Ii+ θ0zi+ εi if Ii ≤ γ β2Ii+ θ0zi+ εi if Ii > γ (14)
where γ is again the threshold variable and zi consists of dummies for all years, regions,
genders, marital statuses, religious attendance, working status, degrees and an age variable. First, we will focus on the replication of the results of Hodge and Shankar (2016) with the methods they used. We use a two-step approach where the model parameters are estimated in first instance for a fixed γ, after which the correct threshold location is found in the second step. Then, the model is slightly improved, something they claim in their model description but, in the end, not use. This time γ is estimated in the first step. Both methods are broadly used in the threshold literature and use a grid search over γ to solve the discontinuity the threshold creates. Both analyses will be done with the data up to and including 2010 and up to and including 2016, 2010 for the sake of comparison with Hodge and Shankar and 2016 to make the estimation as up to date as possible.
5.2.1 Grid Search over γ, assuming γ fixed
The method which Hodge and Shankar apply is a grid search over γ, considering a fixed threshold, from which they estimate all other model parameters. For a grid of {0, 1000, 2000, ..., 200 000} of γ, at each grid point, the model parameters are estimated with the corresponding likelihood. The overall maximum likelihood over the entire grid will now serve as the estimate for γ and all other parameters. The estimation relation is stated:
ˆ
γ = argγmax
κ l(κ|γ) (15)
where κ = {β1, β2, θ, µ}. The results of the grid search over γ can be found in Figure 3
and corresponding estimates of this model can be found in Table 4. The grid search was performed on the 2010 and 2016 dataset using all mentioned variables from section 4, as shown in Figure 3(b) and (d) respectively. In addition, the model was estimated excluding the marital status variables, these results are given in Figure 3(a) and (c). This variable is left out to compare results with Hodge and Shankar and test for the robustness of results. Note that in this analysis the income is measured in thousands.
Figure 3: Log-likelihood of Grid Search over γ
(a) 2010 excluding Marital Status (b) 2010
(c) 2016 excluding Marital Status (d) 2016
Table 4: Grid search results∗, corresponding to Figure 3
(a) (b) (c) (d)
Specification 2010 2016
Coeff. APE Coeff. APE Coeff. APE Coeff. APE
Income β1 0.0085 0.0029 0.0066 0.0022 0.0091 0.0031 0.0064 0.0022
Income β2 0.0045 0.0016 0.0037 0.0013 0.0052 0.0019 0.0037 0.0014
γ 95.000 95.000 75.000 91.000
Logl -44392.1889 -43509.7323 -51208.7300 -50219.0539
LR-Test 165.6314 80.5155 188.8496 88.6656
∗ Standard errors could not be obtained due to the high computational time and are not consistent as the randomness of the threshold is not taken into account.
All figures show two peaks, one around a threshold of 15 and the highest between 70 and 100 with the top mostly around 95. The roughness of the lines shows the great complexity caused by the discontinuous model and displays multiple local maxima close together, especially in (a) and (c) many maxima are found close together. However, the estimated threshold in (a) and (c) is rather different. In Figure 3(c) it is visible that although an optimum at γ = 75 is found, a value around 90 would hardly be less optimal. If this would have been the estimation result, only a noticeable difference would be found between the years and not with the specification of variables. This suggests that the estimation under this variable specification is not very robust to small changes in the data, as running the same analysis with merely including extra observations, results in very different estimates.
The estimates of (a) and (b) are comparable with the estimates of Hodge and Shankar (2016). With a slightly different sample size, they estimated a threshold of respectively γ = 76 and γ = 93. Their graphs also show multiple maxima around the optimum, to be precise in both specification (a) and (b) they find two. One close or at γ = 76 and the second close to γ = 93. The results show that they find the lower threshold to be slightly more optimal, while my research shows a small favour for the higher threshold value. Altogether this is another sign that the results might be not so robust and small variations can have a large impact.
Hodge and Shankar propose in their final conclusion that the threshold should lie at γ = 76 since this is the outcome for all analyses excluding marital status. In addition, some estimations regarding a different set of observations to test for robustness show no effect when marital status is added. However, including the marital status would theoretically be a logical step as it explains household income and it influences stability and happiness. In addition, the introduction of the marital status makes the grid search optimum much clearer when comparing Figure 3(a) with Figure 3(b). Where there were many local max-ima in specification (a), a clear global optimum is found when including the marital status variables in (b).
Table 4 shows the exact results at the optimal values of the threshold γ. The income coefficients β are estimated with the corresponding partial effects. The estimation shows that for an income below the threshold, an additional 1000 dollars would increase the probability of being very happy with 0.2-0.3%. When income is above the threshold, an equal raise would only increase the probability by around 0.15%.
The LR-test measures the evidence for the threshold, which is in all specifications significant on a 1% level. We conclude that it is indeed justified to estimate the model with a threshold. A final remark can be placed by the increase in likelihood when introducing the marital status. A likelihood-ratio test proves that including the marital status variables is preferred in this case.
A measure that is not reported or calculated, in contrast with the analysis of Hodge and Shankar, is the standard error. They use a bootstrap procedure that is not valid in this case. Since the randomness of γ is not taken into account they report standard errors conditional on the threshold value. This, in combination with a two-step approach with a very rough grid, results in invalid standard errors.
This grid search showed us that there is evidence for a threshold which will probably lie between 90 and 95. The marital status should be considered, or results must at least be compared with different specifications. One must be cautious of drawing too strong conclusions as no standard errors can be calculated and the results are not very robust to small changes, in both sample size and change in variables.
As discussed in the simulation analysis, the estimated relation in (14) can be rewritten by introducing λ = β1− β2. This results in the following equation:
Happyi∗= β1+ λ · I{Ii>γ} Ii+ θ 0
zi+ εi (16)
where I{Ii>γ}is an indicator function assessing either a high or low income. The grid search
is accomplished in the exact same way and estimation results can be found in Table 5. The average partial effect of the income variables is calculated according to the same formula (10) as mentioned in the model description, under the assumption that β1− λ = β2, to
make it comparable throughout all analyses.
The figures corresponding to these results are virtually identical to Figure 3 and are hence omitted. There are however small differences in the estimation. For example, in specification (c) a shift of the threshold is measured which can be found in Table 5. The optimum is now estimated at 91, as in (d) where the marital status is included. This again shows that the results are not robust, and although no optical difference appeared, the chosen approach can actually have a large impact.
Table 5: Grid search results∗, using the alternative λ notation
(a) (b) (c) (d)
Specification 2010 2016
Coeff. APE Coeff. APE Coeff. APE Coeff. APE
Income β1 0.0085 0.0029 0.0065 0.0022 0.0092 0.0031 0.0064 0.0022 λ 0.0040 0.0029 0.0040 0.0027 Income β2 0.0044 0.0016 0.0037 0.0013 0.0052 0.0019 0.0037 0.0014 γ 95 000 95 000 91 000 91 000 Logl -44392.3898 -43509.7152 -51208.3806 -50218.9050 LR-Test 165.2296 80.5496 189.5486 88.9632
∗ Standard errors could not be obtained due to the high computational time and are not consistent as the randomness of the threshold is not taken into account.
Comparing the likelihoods for specification (c) between Table 4 and 5 it is hard to assess which optimal point is best. Although the likelihood of the model considering λ is larger, the difference is marginal and no real conclusion can be drawn.
The difficulties that arose in the simulation for low identification and a relatively small sample size, have not occured using the real data. The estimation results between the models do differ, but there is no real estimation error in the process. In addition to the LR-test that proves the existence of a threshold, this demonstrates that the real data shows sufficient identification between groups with higher and lower income.
The analysis above showed that a threshold exists and suggested a threshold location. However, we found no satiation point, as additional income is always valued positively. A next step is now to create an estimation method that can generate valid standard errors and to further look at the robustness of results to get more certainty about the exact threshold location.
5.2.2 Grid Search over γ, when γ estimated conditionally
In the previous analysis, a grid search was performed over γ, where for this fixed γ the other model parameters were estimated. Now, this model is optimised by relocating the grid search inside the optimisation process of the other coefficients. This implies that conditional on all other parameters γ is estimated, whereafter all other parameters are estimated conditional on γ. This results in an iterative process stated below.
ˆ γ = argγmaxγl(γ|ˆκ) ˆ κ = argκmaxκl(κ|ˆγ) (17)
The procedure is equivalent to the two-step approach as mentioned in the model description in Section 3. The function l(κ, ˆγ(κ)) is optimised where ˆγ is a function of κ. This estimation technique is equal to the one used in the simulations. In most situations, this analysis is made to reduce the number of model parameters, however, in this case, only γ is conditioned and it is only used to speed up the computation process. The calculations of this model take less than 25% of the time used for the grid search over γ.
This estimation method is equal to that described by Hodge and Shankar but which they did not execute. In this study, a grid search is made over γ but now conditional on all other model parameters. As this threshold estimation is implemented in the estimation of κ, the resulting ˆκ takes into account the randomness of the threshold. This results in valid standard errors when performing a bootstrap procedure. Unfortunately, the computation process takes too long to obtain an appropriate number of bootstrap replications. This could be solved by taking a significantly smaller sample size, for example one thousand. However, since so many model parameters need to be estimated, including many year
dummies, a small sample size could cause biased results. Therefore no standard errors are calculated for this model.
All other model parameters are again estimated for both β1 and β2 and β1 and λ. In
both models the estimation is performed for the same year and variable specifications as before. Again, first an estimation is made of the β1/β2 model (14) in Table 6 after which
the results of the alternative model (16) are presented in Table 7.
The most important conclusion that can be drawn is the great similarity with the grid search approach. The estimation shows the same differences between the models with, in particular, the change in threshold location when switching between the models in specification (c). This model suffers again robustness issues as also multiple local maxima must exist close to the optimum.
The results of all other specifications, however, show that the estimation difference between the models is even smaller than in the grid search. This tells us that in essence the same maxima are found and there are no clear differences between the models. This estimation technique is equal to that in the simulation where different results were found.
Table 6: Estimation results∗, for γ estimated conditional on the model parameters
(a) (b) (c) (d)
Specification 2010 2016
Coeff. APE Coeff. APE Coeff. APE Coeff. APE
Income β1 0.0085 0.0029 0.0065 0.0022 0.0092 0.0031 0.0064 0.0022
Income β2 0.0045 0.0016 0.0037 0.0013 0.0052 0.0019 0.0037 0.0014
γ 95 000 95 000 75 000 91 000
Logl -44392.0871 -43509.5568 -51208.3723 -50218.8786
LR-Test 165.8335 80.8668 189.5653 89.0162
∗ Standard errors could not be obtained due to the high computational time.
Table 7: Estimation results∗, for γ estimated conditionally using the alternative notation
(a) (b) (c) (d)
Specification 2010 2016
Coeff. APE Coeff. APE Coeff. APE Coeff. APE
Income β1 0.0085 0.0029 0.0065 0.0022 0.0085 0.0029 0.0064 0.0022 λ 0.0040 0.0029 0.0039 0.0027 Income β2 0.0045 0.0016 0.0037 0.0013 0.0046 0.0017 0.0037 0.0014 γ 95 000 95 000 91 000 91 000 Logl -44392.0866 -43509.5567 -51209.0475 -50218.8786 LR-Test 165.8345 80.8669 188.2150 89.0162
This would suggest that this data set is large enough and the threshold is better identified, at least in the robust specifications.
As the results of the different estimation techniques are almost equal, no method is preferred over another. The grid search over γ has the advantage of a good visualisation of the results. On the other hand, the computational speed increases significantly when estimating γ conditional on all other model parameters and this approach can create valid bootstrap standard errors of the income variables, if time would permit such an analysis. A grid search on γ could be used to create valid standard errors for γ. However, this would take even more computational time which might be infeasible.
These models did give us insight in the existence of the threshold, the location and possible income effects. However, a large disadvantage of both models is that there is no possibility to find standard errors for all parameters. In addition, a very rough grid search is needed to solve the discontinuity caused by γ. Small model adjustments that would create a continuous likelihood function could be a large improvement and will be explored next.
5.3 Discontinuity approximations
The main issue in the optimization of a threshold is the discontinuity created by the threshold γ. We recall the estimated equation from (14):
Happyi∗ = β1Ii+ θ0zi+ εi if Ii ≤ γ β2Ii+ θ0zi+ εi if Ii > γ
Here the selection of the β’s corresponding with the level of income creates a clear kink in the model, under the assumption that the β’s are not identical. So far we have dealt with this difficulty by searching over a grid of γ. It has the great disadvantage of a very rough search, as computation of the model in each point takes up a lot of time. In addition, the standard errors are not valid as it can be seen as a two-step approach. The target is now to approximate the kink in the regression to make the model continuous. This leads to a simultaneous estimation of all parameters, valid standard errors and a precise threshold value.
First, we will focus on a kernel estimation where the indicator functions are approx-imated, second, a similar approach is used that creates an individual effect, based on the level of income and the threshold. This second model will lose all discontinuity while the kernel density estimation leaves some kinks. In both models, we use only the β notation, where before also the λ approach was considered. In addition, estimations are made with only the data up to and including 2016 and considering all explanatory variables including the marital status. This specification is chosen as this hosts the most recent data and all relevant variables.
5.3.1 Kernel Estimation
The first approach to solve the discontinuity is with the use of a kernel density estimator. The kernel will approximate the step that is created when moving across the threshold. The indicator functions I{Ii≤γ} and I{Ii>γ} are replaced by a kernel density estimator to
make this step continuous. The idea is that observations that are close to the threshold value do not obtain either β1 or β2 but a weighted average of both, corresponding to the
distance to γ. The value of I{Ii≤γ} will lie between 0 and 1 and consequently, the indicator
function I{Ii>γ}will get the counterpart value such that both sum up to 1. In this approach
the Epanechnikov kernel density is used which is usually represented as k(z) = 34(1 − z2) for |z| ≤ 1 (Cameron & Trividi, 2005, Ch.9.3) . We replace z by δ (Ii− γ) in which δ is the
inverse bandwidth. This results in the following approximation of the indicator functions in which K(z) is the cumulative distribution function.
I{Ii>γ} → K(z) = 0 if z < −1 3 4z − 1 4z 3+ 0.5 if |z| ≤ 1 1 if z > 1 and I{Ii≤γ} → 1 − K(z) (18)
As δ → ∞ this approximation indeed results in the indicator functions. The optimal value for δ needs to be chosen to create the best model performance. As δ is chosen too small, the bandwidth over which a weighted average is taken is unnecessarily small. As δ → ∞ the approximation gets too good and a continuous approximation measure will not find any results. The influence of δ is pictured in Figure 4 in which the relation between the threshold value measured in annual income and the value of β is shown.
The graph clearly shows that the approximation gets well very soon. A δ of 10 could even be too large in the approximation, as the model almost seems indifferentiable. The model is still not entirely continuous, as there is a kink when z = δ (Ii− γ) exceeds one in
absolute value. However, it is the case that a derivative in the threshold parameter exists and numerical optimisation can be used.
