Life satisfaction and age: Dealing with underidentification in age-period-cohort models

Abstract

Recent literature typically finds a U shaped relationship between life satisfaction and age. Age profiles, however, are not identified without forcing arbitrary restrictions on the cohort and/or time profiles. In this paper we report what can be identified about the relationship between life satisfaction and age without applying such restrictions. Also, we identify the restrictions needed to conclude that life satisfaction is U shaped in age. For the case of Germany, we find that the relationship between life satisfaction and age is indeed U shaped, but only under the untestable condition that the linear time trend is negative and that the linear trend across birth cohorts is practically flat.

Introduction

An increasing body of literature documents for the USA and most European countries a U shaped relationship between life satisfaction and age [e.g., Clark, 2007, Blanchflower and Oswald, 2008 and Stone, Schwartz, Broderick, and Deaton (2010) among many others]. Life satisfaction is found to decrease to midlife and to increase subsequently towards retirement. This result is suggestive, because the elderly are generally less wealthy and less healthy than the middle aged.

Studying the relationship between life satisfaction and age is interesting in its own right, but it is of additional interest to economists. A baseline life cycle model predicts that consumption does not decrease after retirement. People correctly anticipate a drop in income upon retirement and would save enough to maintain a constant level of consumption. It is often observed however that consumption drops right after retirement, which suggests that people do not save enough [see e.g., Banks et al. (1998)]. A U shaped relationship between life satisfaction and age means that the drop in consumption is not accompanied by a drop in well-being. Instead, the evidence indicates that people become more satisfied when they retire. Maybe, the retirement – consumption puzzle is not much of a puzzle after all and people should have saved even less?

Many studies finding a U shaped relationship between life satisfaction and age use panel data and try to take into account age, period/time and cohort effects at the same time. These studies face the problem that it is in principle impossible to uniquely identify age, period and cohort effects from longitudinal data, without imposing arbitrary restrictions on the age, time and/or cohort profiles. An identification problem arises due to the equality year of birth + age = calender year. This identification problem has been a topic of debate in social science from the 1970s onwards [see for example Hall, 1971, Mason et al., 1973, Heckman and Robb, 1985, Deaton and Paxson, 1994, Attanasio, 1998 and McKenzie (2006). See also, e.g., Ferrer-i Carbonell and Frijters, 2004, Wunder et al., 2009 and Glenn (2009) for a discussion of the problem within the context of the relationship between life satisfaction and age]. Intuitively, it is impossible to observe the life satisfaction of a single person or cohort with the same age at two different points in time to isolate the time effect. Similarly, it is not possible to observe a single person or cohort with different ages at a single point in time to isolate the age effect.

It is important however to consider cohort (generation) and time effects when studying the relationship between satisfaction and age. In a cross section of households a U shape in age may be due to cohort effects that are not accounted for. In other words, the old in a cross section may be more satisfied than the middle aged, not because they are old, but because they belong to cohorts that have lived through rougher times (WWII for example) and have adjusted their point of reference. The availability of panel data allows to account for differences across cohorts by including cohort effects (or fixed effects, henceforth FE). Modeling cohort or fixed effects however does not solve the identification problem. The empirical strategy would immediately confound age and time effects, which was not the case in a cross section [see Ferrer-i Carbonell and Frijters (2004) for a discussion]. Time effects arise when a population as a whole becomes more or less satisfied. Economic growth or improvements in health care are likely to benefit all.

The simultaneous identification of age, period and cohort effects can be reached by imposing restrictions on either one of the three profiles. By doing so a second issue becomes important: how do imposing different assumptions matter for final inferences about the age profile? In this paper we show that imposing slightly different identifying assumptions has far reaching consequences on the substantive results. This point has been raised previously by e.g., Wunder et al. (2009), however, they subsequently place restrictions on the model themselves by modeling cohort effects with substantive variables, such as life expectancy of the birth cohort. Frijters and Beatton (2008) also demonstrate the sensitivity of the results to alternative identifying assumptions.

In this paper we take a different route. Instead of solving the identification problem, we report what can be identified, and hence what can be concluded from the data, without making assumptions. It has been shown that age, time and cohort profiles can be fully identified in deviation from a linear trend in age, time and year of birth respectively, whereas only two (out of three) of the respective linear trends (age, time, cohort) can be identified [see Heckman and Robb (1985) and McKenzie (2006) for example]. Consequently, without imposing arbitrary restrictions on the age, time and/or cohort profiles it is possible to test whether life satisfaction data is consistent with a U shape in age.

Using data from the German Socio Economic Panel (henceforth GSOEP) we find that life satisfaction is indeed consistent with a U shape over most of the life cycle. Like Wunder et al. (2009) and Gwozdz and Sousa-Poza (2010) we also observe an additional decrease in life satisfaction towards very old age. Whereas we find that the age profile is mainly U shaped around a linear trend in age, the linear trend itself cannot be uniquely inferred from the data. This has important implications. For example, we show that the data is also consistent with a generally negative relationship between life satisfaction in age.

The paper is organized as follows. Section 2 discusses the fact that age profiles are not identified without imposing assumptions on the cohort and time profiles. Furthermore, it shows how sensitive the age profiles are to making different kinds of assumptions. In Section 3 we discuss which elements of the age, cohort and time profiles can be identified without making assumptions. Section 4 discusses the GSOEP data and our empirical results. Finally, Section 5 concludes.