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 KeaneWolpinREStat1994.ox

Replicate Keane and Wolpin (REStat 1994) Dynamic Roy Model using KW Approximation.

The Model

Solution Method
Keane-Wolpin Approximation
Clock:
Finite horizon, normal aging.
T= 40.
age = 16+t
Action:
α=(a)
where a is which task/sector to perform/join
Label   Code
--------------
white    0
blue     1
school   2
home     3
Exogenous states
ϵ=(e0e1e2e3) and ϵdN(0,Σ)
Semi-Exogenous η=()
Endogenous states:
θ=(x0x1x2pt)
accumulate sector experience: xs=xs+I{a=s}, s3
enrolled previous period: p=I{a=2}
Occupation-Specific Human Capital Accumulation
h0=(1x0x1x2)α0
h1=(1x0x1x2)α1
Utility
U=R(θ)
where R() are the occupation-specific returns
R0=eh0(θ)+e0R1=eh1(θ)+e1R2=β0β1I{x212}R3=γ+e3

Approximation

ε is a four dimensional iid vector.
EVαθ>[θ'] = ∑θ'[ ∑e'0e'2e'2e'3 V(ε',θ') P(θ';α,θ) / ε.D]
    = Emax[θ'] 
K&W denote this Emax, as in the expected value of the maximum over actions at the state next period. Since the transition for θ is deterministic, the outer sum is over a single state next period and P() = 1. However, the inner summation involves ε.D values, each of which involves the maximization over 4 options of v(α,ε',θ'). If, say, each component takes on 5 values, then ε.D = 54 = 625 points to sum up per point in the endogenous space θ.
K&W define
maxE[θ'] = V(ê,θ') 
where ê is a 1×4 vector of 0s, the expected value of ε'. Thus, this is the max over actions at the expected iid shock next period.
Approximate EV by evaluating directly at a subset of points in θ, which includes evaluating max v(α,ε',θ').
Track the maximum V(ε',θ') and the value at ê, maxE[θ'] and the vector of choice values v(α,ê,θ').
At each t run a regression of V on a non-linear expression involving v() and maxE, resulting in coefficients. In particular:
Emax(ε',θ') - maxE(θ')  ≈ d0+ (maxE-v(A,θ') )d1 + sqrt(maxE-v(A,θ'))d2.
d0 is an intercept, and d1 and d2 are 4× 1 vectors of coefficients on differences between choice values v(A,θ') at the mean shock and the maximum value at the mean shock. (When schooling is ruled out there are only 3 choice values.) These coefficients are estimated from a regression at each age at the randomly chosen subset ΘKW ⊂ Θ.
Êmax (θ') = max{ maxE , maxE+ d0 + (maxE-v)d1 + sqrt(maxE-v)2 }, for θ' ∉ ΘKW
         = Emax(θ'), θ ∈ ΘKW.
Note that Êmax is defined to be at least maxE, and when the algorithm has computed Emax it is used for Êmax.

Replication

Header File:

Ox File:

Output niqlow/examples/output/KeaneWolpinREStat1994.txt
Author:
© 2011-2023 Christopher Ferrall

 DynamicRoy

Public fields
 accept static accepted offer
 alph static const α in paper
 attended static enrolled last period
 bet static const β vector
 gamm static const γ
 mxcnts static const
 offers static offer block
 sig static const lower triange Σ
 xper static occupation experience array
Public methods
 Replicate static
 Utility Utility vector equals the vector of feasible returns.
Enumerations
 Anonymous enum 1
 ApproxParams Approximation Parameters.
 Dimens State Space Dimensions.
 Sectors Labels for choices/sectors.

 DynamicRoy

Enumerations
Anonymous enum 1 BruteForce, Approximate, Nmethods
ApproxParams Approximation Parameters. TSampleStart, Nsimulate, MinSample, SamplePercentage
Dimens State Space Dimensions. A1, Noffers, Age0, School0, HSGrad, MaxSch, MaxExp, BIGMODEL
Sectors Labels for choices/sectors. white, blue, school, home, Msectors

 accept

static decl accept [public]
accepted offer

 alph

static const decl alph [public]
α in paper

 attended

static decl attended [public]
enrolled last period

 bet

static const decl bet [public]
β vector

 gamm

static const decl gamm [public]
γ

 mxcnts

static const decl mxcnts [public]

 offers

static decl offers [public]
offer block

 Replicate

static DynamicRoy :: Replicate ( )

 sig

static const decl sig [public]
lower triange Σ

 Utility

DynamicRoy :: Utility ( )
Utility vector equals the vector of feasible returns.

 xper

static decl xper [public]
occupation experience array