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This study presents a general happiness equation using econometric models of panel data methods. The model tries to observe and estimate the relationship between income and happiness after controlling for other factors. With advanced methods, we also test for the presence of personality bias and whether it correlates with income. Finally, we provide some analysis of our estimation results and briefly discuss alternative approaches in the literature.
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Empirical research on human happiness have only recently in the last few decades received serious attention from both economists and non-economists. The lack of national-level representative survey data and the difficulty to apply econometric techniques were the stumbling blocks for further research in the past. With the establishments of national socio-economic panel surveys as well as technological advancements that gave birth to neat econometric software packages, the literature experienced a surge in the amount of research as well as the popularity drawn to these works. Things began to look brighter and brighter, and as a result came the birth of a new field called “happiness economics”.
What happiness economists typically try to do is to estimate what they call happiness equations. Using econometric techniques, they could test for a causal link between income and happiness. After controlling for other factors that can cause happiness (eg. education, marital status, disability, unemployment etc.), early work which used simple cross sectional methods suggest a positive and statistically significant correlation. To run Ordinary Least Squares (OLS) regressions on cross sectional data sounds decent, but is in actual fact highly inadequate. What if happiness is also caused by another factor that is unobservable in the data, such as personality? Could it be that one’s happiness strongly depends on who he is as a person? On face value, it seems plausible or at least interesting to suggest that people’s capacity to be happy vary from individual to individual. Perhaps some people are born extrovert and optimistic, and as a result tend to be happier than others even if they have less income than them. Then simple OLS will suffer from an omitted variable bias problem, which causes one or more of its classical assumptions to be violated and hence estimates to be biased.
To solve this problem of unobserved heterogeneity bias, we can use panel data and propose a fixed effects model. We can run a pooled OLS regression on panel data, but it would still be susceptible to the omitted variable bias problem. Firstly, we can think of the personality variable as a time-constant effect. By exploiting the nature of panel data, which follows the same individual over time, we can eliminate this unobserved time-constant effect by doing some transformation on the data. The simplest way is to perform first-differencing. Namely, we take observations on an individual for two time periods and we calculate the differences. Then we run an OLS regression on these transformed values. In effect, we have removed all unobserved time-constant variables not only limited to personality. Maybe an individual’s thumbprints or DNA may be correlated with happiness, we do not know for sure. But the elegance of first-differencing makes it sure that we remove all nuisance unobserved time-constant variables that disturb our primary goal. Through transforming the data in such a way that we are now dealing with relative rather than absolute values, we have also mitigated the problem of heterogeneous scaling in subjective responses. Every individual have their own perception on the “happiness” score. A score of 7 may be other’s score of 6, and so on. This would make interpersonal (cross-sectional) comparisons meaningless, and is part of the reason why in the past empirical work on this literature have been viewed with scepticism by many economists. By reasonably assuming that a person’s metric or perception is time-invariant, this issue is dealt with in a fixed effects model.
There are other advanced transformation techniques that uses data on multiple time periods. One technique performs a time-demeaning transformation on the data. Again, all unobserved time-constant variables will be eliminated. But for details presented later, OLS regression on these transformed values provides more efficient estimators than on the first-differenced values for our purposes. Estimators that result from this method are called fixed effects (FE) estimators. While the fixed effects model allows for arbitrary correlation between the explanatory variables and the unobserved time-constant effect, a random effects model explicitly assumes that there is no such correlation. Estimation on this model is typically done by transforming the data using a method of quasi-demeaning, and then a Generalised Least Squares (GLS) regression is run on the transformed values. The resulting estimators are called random effects (RE) estimators. How these techniques are performed as well as the intuition behind them is explained with technical detail in Section 3.
Why we may want to use a random effects model over a fixed effects model is because we may believe that personality has no effect on any of the independent variables, including income. If this is true, then using FE estimators will result in relatively inefficient estimates than RE estimators. But intuitively, personality is likely to be correlated with the ability to make money, and thus income. Studies have shown that happy people tend to earn more in general (eg. see Lyubomirsky et al. 2005). If this were true, simple pooled OLS methods will lead to inaccurate estimates where the effect of income on happiness will be overstated or biased upwards. The fixed effects model allows for this correlation, and is thus more widely accepted in the literature to fit the data better.
Lastly, can we test for this assumption? Is the unobserved time-constant variable correlated with any of the explanatory variables? Which model fits the data better? We can do what is called a Hausman test, which tests for statistically significant differences in the coefficients on the time-varying explanatory variables between fixed effects and random effects. The intuition and decision rule on which model to accept will be described in detail later. For comparison, we present the results for pooled OLS, FE and RE estimations together.
Although this approach is one of the most popular one in the literature when it comes to estimating happiness equations, there are other alternatives ways. Powdthavee (2009)’s work was quite similar to this study, but in addition he used a method of instrumental variables (IV) which involved using another variable to instrument for income. Happiness equations may suffer from the problem of simultaneity, whereby the causal link between happiness and income runs both ways. To address this, he used data on the proportion of household members whose payslip has been shown to the interviewer as the instrument for income. He reasoned that household income is bound to be measured more accurately with a higher proportion of household members showing their payslip. With this direct correlation, as well as reasonably assuming that this proportion has little correlation with happiness, it would allow for an estimation based on an exogenous income effect. Besides his work, other work (eg. Frijters et al. 2004, Gardner & Oswald 2007) has attempted to address the endogeneity effect more directly using different types of exogeneous income effects.
Another line of thinking interprets the happiness scores as ordinal rather than cardinal. Here, simple OLS estimation would be inadequate. One solution to this would be to use ordered latent response models. Winkelmann (2004) was one example of this in which he performed an ordered probit regression with multiple random effects on subjective well-being data in Germany. To date, there is no statistical software package that could implement a fixed effects ordered probit regression. An alternative to this would be to convert the happiness scoring scale into a (0,1) dummy, thereby roughly cutting the sample into half, and then estimate by conditional logit regression, as attempted by Winkelmann & Winkelmann (1998) and later Powdthavee (2009). However, their work combined with Ferrer-i-Carbonell & Frijters (2004) seems to suggest that it makes no difference qualitatively whether to assume cardinality or ordinality on the happiness scores.
There is no one perfect model that can address all the problems. We believe that the FE & RE approach, not only simple, is also elegant and easier to understand. Coefficient estimates can be interpreted easily and the approach also addresses the most important of problems in the estimation, especially that of unobserved heterogeneity bias. Although bias in happiness equations come from many different sources, it is our belief that this source is one of the major ones and is easily removed using simple techniques.
We use data from the British Household Panel Survey (BHPS), a widely used data source for empirical studies in the UK. The BHPS surveys a nationally representative sample of the UK population aged 16 and above. The survey interviews both individual respondents and households as a whole every year in waves since 1991. To date has been 18 waves in total. Survey questions are comprehensive and they include income, marital status, employment status, health, opinions on social attitudes and so on. The data set is also an unbalanced panel; there is entry into and exit from the panel. Data can be obtained through the UK Data Archive website.
Our dependent variable, happiness, uses data on the question of individual life satisfaction. From Wave 6 onwards, the survey included a question which asks respondents to rate how satisfied they are with their lives from a rating of 1 (very dissatisfied) to 7 (very satisfied). This question is strategically located at the end of the survey after respondents had been asked about their household and individual responses in order to avoid any framing effects of a particular event dominating responses to the LS question. For ease of representation, we now refer to happiness as life satisfaction (LS).
For income, we use data on the total household net income, deflated by consumer price index and equivalised using the Modified-OECD equivalence scale. The initial value is worked out through responses in the Household Finance section which includes question on sources and amount of incomes received in a year. Inflation would seriously distort our estimation and so is accounted for. Equivalisation involves dividing the total household net income by a value worked out according to an equivalence scale. For example, a household with two adults would have their total household income divided by 1.5. The more adults are there in the household, the higher this value would be. Children would add relatively less to the value than adults. This method would provide an equivalent household income variable, which would account for the fact that different household sizes enjoy different standards of living on the same level of income per household member. Due to economies of scale in consumption, a household with three adults would typically have needs more than triple than that of a single member household. Equivalisation would make comparisons between households a lot fairer or more accurate. Lastly, we use the log form.
We use data on the years 2002-2006 (Waves 12-16). There are in total [unconfirmed] respondents with [unconfirmed] observations that have nonmissing information on LS. Descriptive statistics are provided in the Appendix section.
We denote as our dependent variable. We have explanatory (binary and non-binary) variables which includes income, employment status, marital status and so on. There are respondents , where . A simple pooled cross-section model would look like
where the first subscript denotes the cross-sectional units, the second denotes the time period and the third denotes the explanatory variables.
As mentioned earlier, this simple model does not address the issue of unobserved heterogeneity bias. To see why, we can view the unobserved variables affecting the dependent variable, or the error, as consisting of two parts; a time-constant (the heterogeneity bias) and time-varying component.
Thus if we regress by simple pooled OLS, we obtain
Here one of the key assumptions for OLS estimation to be unbiased has been violated, since the error term is correlated with .
The above model is called a fixed effects model. The variable captures all unobserved, time-constant factors that affect . In our analysis, personality falls under this variable. is the idiosyncratic error that represents other unobserved factors that change over time and affect . The simplest method to eliminate is as follows. First, we write the equation for two years as
By subtracting the equation on the first period from the second, we obtain
where denotes the change from to . In effect, we have transformed the model in such a way that we are only dealing with relative rather than absolute values. This technique is called first-differencing. We can then proceed to estimate the equation at (4) via OLS. Essentially, the error term here is no longer correlated with , as the time-constant effect has been “differenced away” or minused out of the equation. However this is only the case if and only if the strict exogeneity assumption holds. This assumption requires that the idiosyncratic error at each time, is uncorrelated with the explanatory variables in every time period. If this holds, then OLS estimation will be unbiased.
A more popular transformation technique in the literature is the time-demeaning method. Again, we begin from equation (3), and using (2) we rewrite it as
Then we perform the following transformation. First, we average (5) over time, giving
where and so on. Next, we subtract (6) from (5) for every time period, giving
where is the time-demeaned value of LS, and so on. Essentially again, has disappeared from the equation. With these new, transformed values, we can then use standard OLS estimation. Conditions for unbiasedness remain the same as in the first-differencing method, including the strict exogeneity assumption. As mentioned earlier, the resulting estimators are called FE estimators.
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In our analysis, we decided to use FE over first-differencing. It is important to state why we do this. The reasoning is as follows. When , their estimation is fundamentally the same. When , both estimations are still unbiased (and in fact consistent), but they differ in terms of relative efficiency. The crucial point to note here is the degree of serial correlation between the idiosyncratic errors, . When there is no serial correlation, FE is more efficient than first-differencing. We have confidence that we have included sufficient controls for other factors in our happiness equation, so that whatever that is left in the error term should be minimal and serially uncorrelated. In addition, FE is safer in the sense that if the strict exogeneity assumption is somehow violated, the bias tends to zero at the rate whereas the bias in first-differencing does not depend on T. With multiple time periods, FE can exploit this fact and be better than first-differencing. Another reason why FE is more popular is that it is easier to implement in standard statistical software packages, and is even more so when we have an unbalanced panel. With multiple time periods, the first-differencing transformation requires more computation and is less elegant overall than FE.
As mentioned earlier, if is uncorrelated with each explanatory variable in every time period, the transformation in FE will lead to inefficient estimators. We can use a random effects model to address this. We begin from (5), writing it as
with an intercept explicitly included. This is so that, without loss of generality, we can make the assumption that has zero mean. The other fundamental assumption is that is uncorrelated with each explanatory variable at every time period, or
With (9), the equation at (8) is called a random effects model. If the assumption at (9) holds, even simple cross section OLS estimation will provide us with consistent results. With multiple time periods, pooled OLS can be even better and also still achieve consistency. However, because is in the composite error from (2), then the are serially correlated across time. The correlation between two time periods will be
where and . This correlation can be quite substantial, and thus causes standard errors in pooled OLS estimation to be incorrect.
To solve this problem, we can use the method of Generalized Least Squares (GLS). First, we transform the data in a way that eliminates serial correlation in the errors. We define a constant as
Then in a similar way to the FE transformation, we quasi-demean the data for each variable,
where is the quasi-demeaned value of LS, and so on. takes a value between zero and one. As mentioned earlier, estimations on these values produce RE estimators. This transformation basically subtracts a fraction of the time average. That fraction, from (11), depends on , and . We can see here that FE and pooled OLS are in fact a special cases of RE; in FE, and in pooled OLS, . In a way, measures how much of the unobserved effect is kept in the error term. Now that the errors are serially uncorrelated, we can proceed by feasible GLS estimation. This will give us consistent estimators with large N and fixed T, which is suitable for our data set.
To summarize, if we believe that personality is an unobserved heterogeneous factor affecting LS then pooled OLS will give us biased estimators. To address this issue, we can use a fixed effects or random effects model. In the former case, we prefer the FE transformation over first-differencing. The choice between FE and RE depends on whether this factor is also correlated with one of our explanatory variables. We think that personality may be correlated with income. If so, then we use the transformation in FE to completely remove it. If this factor is uncorrelated with all explanatory variables at all time periods, then we do a transformation in RE to partially remove it as a complete removal will lead to inefficient estimates. In this scenario, RE is still better or more efficient than pooled OLS because of the serial correlation problem.
An additional characteristic that RE has over FE is that RE allows for time-constant explanatory variables in the regression equation. Remember in FE that every variable is time-demeaned; so variables like gender (does not vary) as well as age (varies very little) will not provide us with useful information. In RE, these variables are only quasi-demeaned, so we can still include these variables in our estimation.
We produce results for estimation by pooled OLS, FE and RE. Besides our key explanatory income variable, other