introduction to cohort analysis pri summer methods workshop june 16, 2008 glenn firebaugh

Introduction to Cohort Introduction to Cohort AnalysisAnalysis

PRI SUMMER METHODS WORKSHOPPRI SUMMER METHODS WORKSHOP

June 16, 2008June 16, 2008Glenn Firebaugh Glenn Firebaugh

Cohort AnalysisCohort Analysis

ObjectiveObjective -- To separate the effects of: -- To separate the effects of: Age (aging/maturation, life cycle status)Age (aging/maturation, life cycle status) Period (historical conditions that affect Period (historical conditions that affect

everyone)everyone) Birth Cohort (each cohort experiences “a Birth Cohort (each cohort experiences “a

distinctive slice of history” - Ryder)distinctive slice of history” - Ryder)- key notion of “imprinting” during impressionable - key notion of “imprinting” during impressionable years; imprinting may result in period-age interaction years; imprinting may result in period-age interaction effects that create cohort differences which persist effects that create cohort differences which persist over timeover time


ProblemProblem: Linear dependence: Linear dependence

Age (years since birth) = Period (current Age (years since birth) = Period (current year) – Cohort (year of birth)year) – Cohort (year of birth)

So you cannot estimate the linear So you cannot estimate the linear equation equation

Y = Y = αα + + ββAAAge + Age + ββPPPeriod + Period + ββCCCohort + Cohort + εε


There are various ways to think about the There are various ways to think about the problem. problem.

One useful way -- as a problem of One useful way -- as a problem of multicollinearity:multicollinearity:

rrACAC..PP= -1.0= -1.0 rrAPAP..CC= +1.0= +1.0 rrPCPC..AA= +1.0= +1.0


rrACAC..PP= -1: At a given point in time, = -1: At a given point in time, everyone lies on the diagonal line for everyone lies on the diagonal line for age by birth year:age by birth year:


What to do? Replace A with A*, then rWhat to do? Replace A with A*, then rA*CA*C..PP ≠≠ - -11


In effect, we have replaced In effect, we have replaced AA with with A*, A*, a a nonlinear function of nonlinear function of AA, where r, where rA*CA*C..PP ≠≠ -1. -1.

The correlation rThe correlation rA*CA*C..PP still is still is closeclose to 1.0 to 1.0 large large

standard errors, unless standard errors, unless NN is large is large What we are assuming is that, for the What we are assuming is that, for the Y Y of interest, of interest,

A*A* captures the age effect as well as does actual captures the age effect as well as does actual age age A.A.

Possible example: age and voting. Voting increases Possible example: age and voting. Voting increases with age until some age threshold where it levels with age until some age threshold where it levels off due to declining health and mobility. In this off due to declining health and mobility. In this approach, some set of parameters constrained to approach, some set of parameters constrained to be equal.be equal.


Strategy 1 – Transformed Variables MethodStrategy 1 – Transformed Variables Method: : Identification by assuming equivalence of Identification by assuming equivalence of adjacent categories of adjacent categories of A, C, A, C, or or P P to create to create A*, C*, A*, C*, or or P*P*, respectively. Example:, respectively. Example:

AA (age in years)(age in years) A* (collapsed/recoded A) A* (collapsed/recoded A) Y Y

Because Because AA has no direct effect on has no direct effect on YY, net of , net of A*, A*, to get the to get the age effect we can simply estimate effect of age effect we can simply estimate effect of A*A* on on Y (Y (and and A*A* is not linearly dependent with is not linearly dependent with P P andand C) C)..


Observations about Observations about Transformed Variables Transformed Variables MethodMethod::

Often Often CC is the variable that is collapsed (e.g. is the variable that is collapsed (e.g. “depression cohort,” “baby boomers,” etc.)“depression cohort,” “baby boomers,” etc.)

Extreme case: collapse Extreme case: collapse allall the categories of the categories of A, P, A, P, or or C. C. That’s what researchers do in effect when they omit That’s what researchers do in effect when they omit A, PA, P, or , or CC (i.e., assume (i.e., assume no effectno effect for one of them). for one of them).

Collapsing adjacent categories to create Collapsing adjacent categories to create A*, P*A*, P* and and C*C* all goes back to “moving cases off the linear all goes back to “moving cases off the linear regression line” for rregression line” for rACAC..PP etc. etc.


Age by cohort figure where cohort Age by cohort figure where cohort categories are collapsed (categories are collapsed (rrA*CA*C..PP ≠≠ -1 -1):):


Strategy 2 – Proxy Variables MethodStrategy 2 – Proxy Variables Method: Avoid : Avoid linear dependence by substituting linear dependence by substituting A**, P**A**, P** or or C**C** for for A, PA, P, or , or CC, where ** measures capture , where ** measures capture what it is about age, period, or cohort that what it is about age, period, or cohort that matters.matters.

Common example: Cohort size for Common example: Cohort size for CC. Used in labor . Used in labor market studies where, e.g., wage is thought to depend market studies where, e.g., wage is thought to depend on one’s age (hump-shaped), period, and the on one’s age (hump-shaped), period, and the size of size of one’s birth cohortone’s birth cohort ( (C**C**).).

Unlike Unlike A*-P*-CA*-P*-C* measures, * measures, A**-P**-CA**-P**-C** measures are ** measures are notnot recoded functions of recoded functions of A, P A, P andand C C

Same underlying assumption for both Same underlying assumption for both strategiesstrategies

AssumptionAssumption: That : That AA has no effect on has no effect on YY net of net of A*,A*, or that or that CC has no effect on has no effect on YY net of net of C*,C*, or or PP has no effect on has no effect on YY net of net of P*.P*. Similarly for the proxy variables Similarly for the proxy variables A**, C** A**, C** andand P**. P**. The idea in both cases is that at least one * The idea in both cases is that at least one * or ** variable must mediate all the effect. (Note or ** variable must mediate all the effect. (Note parallels with Winship-Harding approach – for parallels with Winship-Harding approach – for bothboth * * and ** methods.)and ** methods.)

For example, For example, C*:C*: Are we capturing Are we capturing allall the cohort the cohort effect when we assume no effect within some range effect when we assume no effect within some range of birth years? Or, by collapsing birth years, are we of birth years? Or, by collapsing birth years, are we simply identifying by adding measurement error?simply identifying by adding measurement error?

C**C**: Are we capturing : Are we capturing allall the cohort effect when we the cohort effect when we use cohort size? use cohort size?

Natural experiments as a Natural experiments as a promising method (where promising method (where

possible)possible)Are there instances “in nature” where, say, age and Are there instances “in nature” where, say, age and

cohort effects are uncoupled? Consider voting & 19cohort effects are uncoupled? Consider voting & 19thth Amendment (“enduring effect of Amendment (“enduring effect of disenfranchisement”)disenfranchisement”)::

Best predictor of voting at time Best predictor of voting at time tt is whether you voted is whether you voted at at t-1, t-1, so learning to vote early matters.so learning to vote early matters.

But women couldn’t vote in most states before 1920 But women couldn’t vote in most states before 1920 different cohort experiences than men different cohort experiences than men the same the same ageage

Sex is randomly assigned by family SES, etc.Sex is randomly assigned by family SES, etc. A “natural experiment” – compare sex differences in A “natural experiment” – compare sex differences in

voting rates for men and women who came of age pre- voting rates for men and women who came of age pre- and post-19and post-19thth Amendment (Firebaugh-Chen AJS 1995) Amendment (Firebaugh-Chen AJS 1995)

introduction to cohort analysis pri summer methods workshop june 16, 2008 glenn firebaugh

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