A Mechanism-Based Approach to the Identification of Age, Period, Cohort Models Christopher Winship and David J. Harding June 2008

Age, Period, Cohort Models Goal: Decompose Change in an Outcome Over Time into Components Associated with Age, Period and Cohort (e.g. various political attitudes, earnings, female labor force participation) Extensive Literature in Sociology in the 1970’s and 1980’s

Other Examples Immigration: Age, Age of Entry,Years in US. Mobility: Status of Son, Status of Father, Difference in Status Age, Age at Marriage, Marital Duration Age, Years of Education, Work Experience

Let the matrix X consist of a constant plus the variables: Age = Years Since Date of Birth Period = Current Date Cohort= Date of Birth Identity: Age = Period - Cohort

Estimate: Y= Xa + e = a 0 + Age a 1 + Period a 2 + Cohort a 3 + error Problem: (X’X) -1 fails to exist as Age = Period – Cohort Same issue exists If we set this up with dummy variables for APC

Traditional Solutions: I. Set the effect of Age, Period, or Cohort to zero II. Restrict a subset of parameters to be equal III. Use a proxy variable that is a nonlinear function of Age, Period, or Cohort

Problems: I. Lack of theoretical justification for restrictions 2. Lack of robustness if restrictions are mildly wrong 3. Failure to test identification restrictions 4. Lack of a well defined causal interpretation

Different Substantive Processes Age = Maturation, Life Cycle (e.g. work experience, martial status) Period = Effects of Time Period (e.g., Unemployment Rate, Party of President) Cohort = Effects of Birth Cohort (e.g. birth period – Depression, cohort size)

Alternative Approach Pearl’s Front Door Criteria Specification of Mechanisms by which Age, Period, and Cohort Affect Outcome

Pearl’s Identification Criteria l. Backdoor criteria – conditioning ll. Instrumental Variables lll. Front door criteria – mechanism focused

IV versus Front Door IV: Identification by extending model backwards from independent variables. Front door: Identification by extending model forwards from independent variables In both cases whether identification is achievable depends upon having a sufficiently theoretical rich model that is tenable.

Definitions Y = Outcome X= Constant, Age, Period, Cohort M= Matrix of m variables that potentially mediate the effects of Age, Period, and Cohort on the Outcome e, v error terms in equations below U = n x m matrix of errors

Basic Model ( 1) Y = Xa + e (2) M = XB + U (3) Y = Mc + v Substituting (2) into (3) (4) Y = XBc + Uc + v Goal: Estimate a = Bc

Theorem: In order to identify the “a” coefficients, it is only necessary that we fully specify all the mechanisms with one of Age, Period, and Cohort.

Key Point Models and their subcomponents will often be overidentified. When this is the case, one can test for both the fit of the overall model and of its subcomponents.

Empirical Example Changes in Political Alienation Kahn and Mason ASR 1987