Introduction to Cohort Analysis PRI SUMMER METHODS WORKSHOP June 16, 2008 Glenn Firebaugh

Cohort Analysis Objective -- To separate the effects of: Age (aging/maturation, life cycle status) Age (aging/maturation, life cycle status) Period (historical conditions that affect everyone) Period (historical conditions that affect everyone) Birth Cohort (each cohort experiences “a distinctive slice of history” - Ryder) Birth Cohort (each cohort experiences “a distinctive slice of history” - Ryder) - key notion of “imprinting” during impressionable years; imprinting may result in period-age interaction effects that create cohort differences which persist over time

Cohort Analysis Problem: Linear dependence Age (years since birth) = Period (current year) – Cohort (year of birth) Age (years since birth) = Period (current year) – Cohort (year of birth) So you cannot estimate the linear equation Y = α + β A Age + β P Period + β C Cohort + ε

Cohort Analysis There are various ways to think about the problem. One useful way -- as a problem of multicollinearity: r AC. P = -1.0 r AC. P = -1.0 r AP. C = +1.0 r AP. C = +1.0 r PC. A = +1.0 r PC. A = +1.0

Cohort Analysis r AC. P = -1: At a given point in time, everyone lies on the diagonal line for age by birth year:

Cohort Analysis What to do? Replace A with A*, then r A*C. P ≠ -1

Cohort Analysis In effect, we have replaced A with A*, a nonlinear function of A, where r A*C. P ≠ -1. The correlation r A*C. P still is close to 1.0  large standard errors, unless N is large The correlation r A*C. P still is close to 1.0  large standard errors, unless N is large What we are assuming is that, for the Y of interest, A* captures the age effect as well as does actual age A. What we are assuming is that, for the Y of interest, A* captures the age effect as well as does actual age A. Possible example: age and voting. Voting increases with age until some age threshold where it levels off due to declining health and mobility. In this approach, some set of parameters constrained to be equal. Possible example: age and voting. Voting increases with age until some age threshold where it levels off due to declining health and mobility. In this approach, some set of parameters constrained to be equal.

Cohort Analysis Strategy 1 – Transformed Variables Method: Identification by assuming equivalence of adjacent categories of A, C, or P to create A*, C*, or P*, respectively. Example: A (age in years)  A* (collapsed/recoded A)  Y A (age in years)  A* (collapsed/recoded A)  Y Because A has no direct effect on Y, net of A*, to get the age effect we can simply estimate effect of A* on Y (and A* is not linearly dependent with P and C).

Cohort Analysis Observations about Transformed Variables Method : Often C is the variable that is collapsed (e.g. “depression cohort,” “baby boomers,” etc.) Often C is the variable that is collapsed (e.g. “depression cohort,” “baby boomers,” etc.) Extreme case: collapse all the categories of A, P, or C. That’s what researchers do in effect when they omit A, P, or C (i.e., assume no effect for one of them). Extreme case: collapse all the categories of A, P, or C. That’s what researchers do in effect when they omit A, P, or C (i.e., assume no effect for one of them). Collapsing adjacent categories to create A*, P* and C* all goes back to “moving cases off the linear regression line” for r AC. P etc. Collapsing adjacent categories to create A*, P* and C* all goes back to “moving cases off the linear regression line” for r AC. P etc.

Cohort Analysis Age by cohort figure where cohort categories are collapsed ( r A*C. P ≠ -1 ):

Cohort Analysis Strategy 2 – Proxy Variables Method: Avoid linear dependence by substituting A**, P** or C** for A, P, or C, where ** measures capture what it is about age, period, or cohort that matters. Common example: Cohort size for C. Used in labor market studies where, e.g., wage is thought to depend on one’s age (hump-shaped), period, and the size of one’s birth cohort (C**). Common example: Cohort size for C. Used in labor market studies where, e.g., wage is thought to depend on one’s age (hump-shaped), period, and the size of one’s birth cohort (C**). Unlike A*-P*-C* measures, A**-P**-C** measures are not recoded functions of A, P and C Unlike A*-P*-C* measures, A**-P**-C** measures are not recoded functions of A, P and C

Same underlying assumption for both strategies Assumption: That A has no effect on Y net of A*, or that C has no effect on Y net of C*, or P has no effect on Y net of P*. Similarly for the proxy variables A**, C** and P**. The idea in both cases is that at least one * or ** variable must mediate all the effect. (Note parallels with Winship-Harding approach – for both * and ** methods.) For example, C*: Are we capturing all the cohort effect when we assume no effect within some range of birth years? Or, by collapsing birth years, are we simply identifying by adding measurement error? For example, C*: Are we capturing all the cohort effect when we assume no effect within some range of birth years? Or, by collapsing birth years, are we simply identifying by adding measurement error? C**: Are we capturing all the cohort effect when we use cohort size? C**: Are we capturing all the cohort effect when we use cohort size?

Natural experiments as a promising method (where possible) Are there instances “in nature” where, say, age and cohort effects are uncoupled? Consider voting & 19 th Amendment (“enduring effect of disenfranchisement”) : Best predictor of voting at time t is whether you voted at t-1, so learning to vote early matters. Best predictor of voting at time t is whether you voted at t-1, so learning to vote early matters. But women couldn’t vote in most states before 1920  different cohort experiences than men the same age But women couldn’t vote in most states before 1920  different cohort experiences than men the same age Sex is randomly assigned by family SES, etc. Sex is randomly assigned by family SES, etc. A “natural experiment” – compare sex differences in voting rates for men and women who came of age pre- and post-19 th Amendment (Firebaugh-Chen AJS 1995) A “natural experiment” – compare sex differences in voting rates for men and women who came of age pre- and post-19 th Amendment (Firebaugh-Chen AJS 1995)