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:

$\alpha = (a)$

where a is which task/sector to perform/join

Label   Code
--------------
white    0
blue     1
school   2
home     3

Exogenous states

$\epsilon = \pmatrix{ e_0& e_1 & e_2 & e_3}$ and $\epsilon \sim dN(0,\Sigma)$

Semi-Exogenous $\eta = ()$

Endogenous states:

$\theta = \pmatrix{x_0 & x_1& x_2& & p & t}$

accumulate sector experience: $x'_s = x_s + I_{\{a=s\}}$, $s \ne 3$

enrolled previous period: $p' = I_{\{a=2\}}$

Occupation-Specific Human Capital Accumulation

$h_0 = \pmatrix{ 1 & x_0 & x_1 & x_2 }\alpha_0$

$h_1 = \pmatrix{ 1 & x_0 & x_1 & x_2 }\alpha_1$

Utility

$U = R(\theta)$

where R() are the occupation-specific returns

$$\eqalign{ R_0 &= e^{ h_0(\theta) + e_0 }\cr R_1 &= e^{ h_1(\theta) + e_1 }\cr R_2 &= \beta_0 - \beta_1 I_{\{x_2\ge 12\}}\cr R_3 &= \gamma + e_3\cr}$$

Approximation

ε is a four dimensional iid vector.

EV_{αθ>[θ'] = ∑_θ'[ ∑_e'₀ ∑_e'₂ ∑_e'₂ ∑_e'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 Θ_KW ⊂ Θ.

Êmax (θ') = max{ maxE , maxE+ d₀ + (maxE-v)d₁ + sqrt(maxE-v)₂ }, for θ' ∉ Θ_KW
         = Emax(θ'), θ ∈ Θ_KW.

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]

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