Is Happiness Contagious? Separating Spillover Externalities from the Group-Level Social Context

Abstract

We investigate whether individuals feel happier when others around them are happier in broadly defined worker groups. This will be a formal test of spillovers in happiness. Answering this question requires a careful handling of the reflection problem, as it may not be possible to separate the endogenous spillover effects from contextual effects unless an appropriately designed identification strategy is employed. Implementing such a strategy and using the 2008 release of the British Housing Panel Survey, we show that the group-level happiness does not have a statistically significant endogenous effect on individual-level happiness in the Great Britain. We report, however, statistically significant contextual effects in various dimensions including age, education, employer status, and health. These results suggest that higher group-level happiness does not spill over to the individual level in neither negative nor positive sense, while the individual-level happiness is instead determined by social context (i.e., the group-level counterparts of certain observed covariates). We also test the relevance of the “Easterlin paradox” and find that our result regarding the effect of income on happiness—controlling for social interactions effects—is the group-level analogue of Easterlin’s original results.

Appendices

Appendix 1: GHQ Questionnaire

A series of questions are asked in the General Health Questionnaire (GHQ). These questions are:

Have you recently:

• 1. Been able to concentrate on whatever you are doing?

• 2. Lost much sleep over worry?

• 3. Felt that you are playing a useful part in things?

• 4. Felt capable of making decisions about things?

• 5. Felt constantly under strain?

• 6. Felt you couldn’t overcome your difficulties?

• 7. Been able to enjoy your normal day to day activities?

• 8. Been able to face up to your problems?

• 9. Been feeling unhappy and depressed?

• 10. Been losing confidence in yourself?

• 11. Been thinking of yourself as a worthless person?

• 12. Been feeling reasonably happy all things considered?

Answers are coded on a four-point scale from “Disagree strongly” (coded 1) to “Agree strongly” (coded 4)—questions 1, 3, 4, 7, 8, and 12 are coded in reverse order—and added together to provide a total GHQ score of mental distress ranging in total from 12 to 48, which we call the “general happiness” score [see, e.g., Taylor (2006) for the construction principles]. Low scores correspond to low levels of stress/depression (i.e., high feelings of well-being). This approach is known as a Likert scale. Although our main focus is on the last question, which we call the “overall happiness,” we also perform our analysis using the general happiness score for robustness purposes.

Appendix 2: Details of the Econometric Framework

The linear-in-means equation that we estimate is:

$$\begin{aligned} \omega _{i_g} = \beta _0 + \beta _1 \varvec{X}_{i_g} + \beta _2 \varvec{Y}_g + J m_g + \epsilon _{i_g}, \end{aligned}$$

(8.1)

where the variables are as they are described in Sect. 4.2. Taking the conditional mathematical expectations in both sides of the Eq. (8.1) yields

$$\begin{aligned} m_g = \beta _0 + \beta _1 \varvec{X}_{g} + \beta _2 \varvec{Y}_g + J m_g, \end{aligned}$$

(8.2)

where $\varvec{X}_{g} = {\mathbb {E}}[\varvec{X}_{i_g}|g]$. The distinction between $\varvec{X}_{g}$ and $\varvec{Y}_{g}$ is key to understanding the identification problem. Any variable in $\varvec{Y}_{g}$ has to describe something “meaningfully” related to that group; such as, the fraction of males, the mean education, the fraction of non-whites, etc. $\varvec{X}_{g}$ is the group-level mean of individual-level observed characteristics and it may or may not coincide with $\varvec{Y}_g$, i.e., not every variable in $\varvec{X}_{g}$ has to correspond to an element in $\varvec{Y}_{g}$.

In Eq. (8.2), $m_g$ appears on both sides. Solving for $m_g$ yields the expression that

$$\begin{aligned} m_g = \frac{\beta _0}{1 - J} + \frac{\beta _1}{1 - J} \varvec{X}_{g} + \frac{\beta _2}{1 - J} \varvec{Y}_g. \end{aligned}$$

(8.3)

Manski (1993) defines the reflection problem as follows. Let $dim(\varvec{a})$ denote the dimension of some generic vector $\varvec{a}$. The reflection problem states that if $dim(\varvec{X}_{g}) = dim(\varvec{Y}_g)$, then linearity masks the econometric identification of the (endogenous) social interactions parameter $J$ and the vector of contextual effect parameters $\beta _2$. To see this clearly, one can plug Eq. (8.3) into Eq. (8.1) to obtain the outcome equation

$$\begin{aligned} \omega _{i_g} = \frac{\beta _0}{1 - J} + \beta _1 \varvec{X}_{i_g} + \frac{J \beta _1}{1 - J} \varvec{X}_{g} + \frac{\beta _2}{1 - J} \varvec{Y}_g + \epsilon _{i_g}. \end{aligned}$$

(8.4)

When the reflection problem is in effect, i.e., when $dim(\varvec{X}_{g}) = dim(\varvec{Y}_g)$, $J$ and $\beta _2$ cannot be distinguished from each other econometrically, which implies that social interactions cannot be identified. For expositional purposes, we abuse the notation and set $\varvec{X}_{g} = \varvec{Y}_g$, which yields the equation

$$\begin{aligned} \omega _{i_g} = \frac{\beta _0}{1 - J} + \beta _1 \varvec{X}_{i_g} + \frac{J\beta _1 + \beta _2}{1 - J} \varvec{Y}_g + \epsilon _{i_g}. \end{aligned}$$

(8.5)

Obviously, it is impossible to separate $J$ from $\beta _2$. One solution is to propose an additional $X_g$ which is not in $\varvec{Y}_g$; that is, we need to have $dim(\varvec{X}_{g}) = dim(\varvec{Y}_g) + 1$. If such a $X_g$ exists, then all the parameters in our linear-in-means model are identified. In other words, one individual-level variable, the mean of which cannot be regarded as a group-level variable, is required for identification of social interactions. To demonstrate this, let $\tilde{X}_g$ be an element of the vector $\varvec{X}_{g}$ and let $\tilde{\beta }_1$ be the coefficient associated with $\tilde{X}_g$. Let $\tilde{X}_g \notin \varvec{Y}_g$ and $dim(\varvec{X}_{g}) = dim(\varvec{Y}_g) + 1$. In other words, we let $\tilde{X}_g$ define that additional variable in $\varvec{X}_{g}$, which does not correspond to a contextual variable. Then, Eq. (8.4) can be rewritten as

$$\begin{aligned} \omega _{i_g} = \frac{\beta _0}{1 - J} + \beta _1 \varvec{X}_{i_g} + \frac{J \tilde{\beta }_1}{1 - J} \tilde{X}_{g} + \frac{J\bar{\beta }_1 + \beta _2}{1 - J} \varvec{Y}_g + \epsilon _{i_g}, \end{aligned}$$

(8.6)

where $\bar{\beta }_1$ consists of the elements of $\beta _1$ except $\tilde{\beta }_1$. In other words, $\beta _1 = [\tilde{\beta }_1 \bar{\beta }_1]$. From Eq. (8.6), $\tilde{\beta }_1$ can be identified, which implies that $J$ and $\beta _2$ can separately be identified within this framework. Again, the key point is the existence of a variable $\tilde{X}_g$, which does not correspond to a contextual variable $\varvec{Y}_g$. Individual-level variables such as gender, education, age, marital status, and so on necessarily correspond to contextual effects when averaged out. One should find an individual-level variable $\tilde{X}_g$ such that it cannot be interpreted as a group-level characteristic. If such a variable exists, it serves as an instrument and secures identification of $J$ and $\beta _2$ separately.

Appendix 3: Details of the Instrumental Variable

In this part, we explain the main intuition behind our IV strategy with an example. Suppose that there are many individuals in the population and each individual interacts with only a group of them. Individuals self-report the following happiness levels (everything is observed—the scale of the numbers is arbitrary, but the ordering of the days is consistent with the estimates reported in Table 2):

$$\begin{aligned}&M: (7+\epsilon _{M}) \\&T: (6+\epsilon _{T}) \\&W: (6+\epsilon _{W}) \\&T: (6+\epsilon _{T}) \\&F: (5+\epsilon _{F}) \\&St: (5+\epsilon _{St}) \\&S: (7+\epsilon _{S}). \end{aligned}$$

A lower score corresponds to a higher happiness level, just as in the BHPS dataset. Each individual’s type is given by the vector $\varvec{\epsilon }=\bigl (\epsilon _{M}, \epsilon _{T}, \epsilon _{W}, \epsilon _{T}, \epsilon _{F}, \epsilon _{St}, \epsilon _{S}\bigr )$. We assume that $\varvec{\epsilon }$ is independent and identically distributed (iid) across individuals. Elements of $\varvec{\epsilon }$ have all zero means, but different variances. Thus, the joint distribution of $\varvec{\epsilon }$ has mean 0 and a covariance matrix $\varvec{\Sigma }$. Individuals are identical apart from their $\varvec{\epsilon }$’s. As a result, on average, we will have a 7, 6, 6, 6, 5, 5, 7 pattern from Monday to Sunday, just as the ordering of our estimates suggest. The overall mean is 6. However, this is the case at the population level. The group-level means would be different depending on the configuration of the $\varvec{\epsilon }$’s.

To resolve the reflection problem, we need one variable that is correlated with the individual-level happiness score, but does not correspond to a contextual effect. The mean of this variable should affect the individual’s happiness level only through the group-level happiness. Let’s say we put only “Saturday” as an instrument. The mean of this variable could in fact correspond to a contextual effect, because we would be picking those who are more likely to feel happier due to a possible self-selection into a Saturday interview.

We first make the observation that groups with a higher fraction of individuals interviewed on, say, a Wednesday or Friday or Sunday have higher average happiness levels. To capture this effect, we construct a dummy variable $D=1$ if the individual is interviewed on either of these days, and 0 otherwise. Clearly, the group-level mean of $D$ is not a meaningful contextual variable. But the incidental correlation between $\epsilon _W$, $\epsilon _F$, and $\epsilon _S$ at the group level makes the group-level mean of $D$ a strong determinant of social interactions. So, the main identifying assumption is that groups with a larger fraction of individuals interviewed on Wednesdays, Fridays, or Sundays have higher happiness levels on average. The empirical justification is presented in Table 7.

Table 7 $p$—values of the day dummies

To sum up, we try to obtain a “synthetic” day-of-the-week variable that can potentially resolve the reflection problem.