A Happiness Study Using Age-Period-Cohort Framework

Abstract

Age effects and birth cohort effects have not been differentiated in happiness studies. In this paper, age-period-cohort decomposition is applied to happiness data in the US. Since the relationship is linear, such as age = period − cohort, it is not possible to identify the three effects. This paper considers four identification models: the polynomial age-effect model, the proxy-variable model, the orthogonal period-effect model, and the principal component model. Happiness data are obtained from the General Social Survey for 1972–2008. Except for the polynomial age-effect model, three alternative models provide similar results. In particular, there is little difference between the decomposition results obtained by the orthogonal period-effect model and by the principal component model. The age effect shows downward movements for 18–55 and for 80–89, an upward movement for 56–69, and an almost flat movement for 70–79. The period effect shows cyclical movements slightly similar to unemployment rates fluctuations. The cohort effect shows a downward movement for the birth cohorts of 1894–1936, a dip for 1945–1958 (baby boomers), an upward movement for 1959–1969, and an almost flat movement for 1970–1987.

Appendix: The Identity Between the Parameter Estimates Obtained by the OPE Model and by the PC Model

Based on aims and scope of the journal, not a purely mathematical proof but a few hints concerning this identity are provided. The regression model (1) for I age groups, J survey periods, K birth cohorts, and N explanatory variables is re-specified as follows.

$$ Y = X\beta + e, $$

(3)

where Y is the T × 1 (T = I × J) vector of happiness data, X is the T × (M + N) (M = I + J + K − 3) matrix for age, period, and cohort effects dummies and explanatory variables, β is the parameter vector to be estimated, and e is the vector of the disturbance term. X can be partitioned as X = (DE), where D is T × (I + J + K − 3) design matrix corresponding to age, period, and cohort effects dummies, and E is T × N matrix composed of explanatory variables. The OLS estimates for the model (3) is obtained as

$$ \hat{\beta } = (X^{\prime}X)^{ - 1} X^{\prime}Y. $$

I newly define as (X′X)⁻¹ X′ = (P′Q′)′, where P is M × T matrix for age, period, and cohort effects, and Q is N × T matrix for explanatory variables. The parameter estimates for explanatory variables is calculated as QY. Q is obtained as follows.

$$ Q = \left( {E^{\prime}D(D^{\prime}D)^{ - 1} D^{\prime}E - E^{\prime}E} \right)^{ - 1} E^{\prime}D(D^{\prime}D)^{ - 1} D^{\prime} + \left( {E^{\prime}E - E^{\prime}D(D^{\prime}D)^{ - 1} D^{ - 1} E} \right)^{ - 1} E^{\prime} $$

(4)

The Eq. 4 shows that the matrix D(D′D)⁻¹ D′ is essential to the identity. Since the linear relationship between age, period, and cohort effects, the rank of the matrix D ^′ D is not M but M − 1, not ordinal inverse (D ^′ D)⁻¹ but generalized inverse matrix (D′D)⁻ is applied. Based on Rao and Mitra (1971), the following result is obtained.

$$ D^{\prime}D(D^{\prime}D)^{ - } D^{\prime}D = D^{\prime}D. $$

(5)

Furthermore, the generalized inverse for D′D is not unique. In the present paper, the OPE model and PC model provide different matrices of the generalized inverse for D′D, while both models make the rank of D′D(M − 1). The Eq. 5 shows that the matrix Q in (4) obtained by the OPE model is completely the same as that obtained by the PC model. Hence, the identity is obtained.