a simple direct estimate of rule-of-thumb consumption...

A Simple Direct Estimate of Rule-of-Thumb Consumption Using the Method of Simulated Quantiles & Cross Validation

Nathan M. PalmerThe Office of Financial Research

George Mason University Department of Computational Social ScienceNovember 3, 2017

Views expressed in this presentation are those of the speaker and not necessarily of the Office of Financial Research or other government organizations.

Motivation: Want to Build Agent-Based Macro Models

Assume you want to solve a “very complicated” dynamic stochasticgeneral equilibrium problem

Eg. you think that the complex structure of mortgage or repo marketsmattered for the crisis

Problem: rational expectations solutions are intractable – what to do?

Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 2 / 40

Motivation: Selecting Household Behavior

Agent-based literature: “rules of thumb.”

Which rules to use?


Motivation: Selecting Household Behavior

Agent-based literature: “rules of thumb.”

Which rules to use?

Growing literature: use optimization framework, approximate /learning solutions:

Howitt & Ozak (2014), Evans & McGough (2015), Lettau and Uhlig(1999), Allen and Carroll (2001) [and cottage literature], Arifovic(many), Gabaix QJE (2014), ...


Technical Difficulties in Selecting Household Behavior

Note: this only side-steps the question.

We’d still like to select between those. They are fairly different.

Difficulties:

No closed form solution for behaviorBehavioral models are not nestedLikelihood surface: “very hard” to compute, if it exists (semiparametricestimation)Easily available data on life cycle choices is not great


Possible Solution: Simulation-Based Estimation, Selection

Possible solution: semi-parametric, simulation-based estimation.Specifically:

“Method of Simulated Quantiles,” Dominicy & Veredas (2013)

...coupled with k-fold cross-validation

Highly general approach to both estimation and selection


A Simple Implementation 0th-order Example

This presentation: a specific, simple implementation

Specifically:

Estimate a “textbook” structural, semi-parametric household life-cycleconsumption-savings problemThen jointly re-estimate the model with additional parameter(s):number of agents with different betaThen use k-fold cross-validation to formally select between models

Roughly a micro version of Campbell and Mankiw (1989, 1990)*

Work is preliminary and very much in progress


Outline

1 Motivation

2 Agent Problem and Solution

3 Estimation and SelectionMethod of Simulated Moments / QuantilesK-Fold Cross-Validation

4 Results, Summary, Next StepsVery Preliminary Results


Household Consumption Functions Example

An example solution:

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Cash on hand

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Consu

mpti

on

Consumption Functions: Black Before Retirement, Red After


20 0 20 40 60 80 100

0.0

0.1

0.2

0.3

0.4

Den

sity

2630, max = 1293135, max = 7863640, max = 127214145, max = 8434650, max = 15835155, max = 20625660, max = 835

Survey of Consumer Finance, Federal Reserve Board


“Textbook” Household Problem

A household solves the T -horizon problem described by sequence ofBellmans:

v∗t (mt) = maxct

u(ct) + β BtEt

[Γ

1−ρ

t+1 v∗t+1(mt+1)]

s.t.

mt+1 = Rt+1(mt − ct) + ξt+1

m0 given

where

β , ρ are discount factor and risk aversion

mt is total “cash on hand:” total assets + total income

Rt is risk-free return on assets

ξt are mean-1 temporary shocks to income

Entire problem is normalized by permanent income process, notshown2

2See Carroll (2012a, b) for extensive discussion.Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 9 / 40

Outline

1 Motivation





Estimation Method

Denote behavioral parameters:

φ ≡ {β ,ρ}

Denote structural parameters:

ρ = {ρt}Tt=0

= {Γt ,ξt ,Rt ,Bt, etc}Tt=0


Estimation Method: Simulation Step

Given ρ, choose φ , solve problem for optimal policy functions {c∗t }Tt=1

Simulate large panel of artificial wealth data under φ

From SCF data, create weath distributions for 7 age groups:

21-30, 31-35, 31-40, 41-45, 41-50, 51-55, 51-60

Pool simulated data to match age groups of SCF data


Functions of Quantiles and Loss Function

Construct functions of quantiles ∀ τ:

Empirical: ϕ̂τ =

(q̂τ,75− q̂τ,25

q̂τ,50

)

Theoretical: ϕNφ ,τ =

(q

φ

τ,75−qφ

τ,25

qφ

τ,50

)


Functions of Quantiles

Define vectors of functions of quantiles:

ϕ̂ =(

ϕ̂>1 , ϕ̂>2 , ..., ϕ̂>7

)>

ϕNφ =

(ϕ>φ ,1,ϕ

>φ ,2, ...,ϕ

>φ ,7

)>


Loss Function

Define the loss function:

ϖρ (φ) =(

ϕ̂−ϕNφ

)W(

ϕ̂−ϕNφ

)where W is a positive definite matrix of weights

Minimize the loss function, obtain φ

Bootstrap for variance


Motivation: Why Consider N Types?

20 0 20 40 60 80 100

0.0

0.1

0.2

0.3

0.4

Den

sity

2630, max = 1293135, max = 7863640, max = 127214145, max = 8434650, max = 15835155, max = 20625660, max = 835

Survey of Consumer Finance, Federal Reserve Board


Outline

1 Motivation





Quick Illustration

Consider the following artificial data:3

3Discussion from Cosma Shalizi’s “Advanced Data Analysis from an Elementary Point of View”


Polynomial Overfitting


R-squared Looks Great


Loss Function (SSE) Looks Great


However, Very Poor Fit


Data Selection, K = 5


Outline

1 Motivation





Very Preliminary Results

1 Original, median-only4, Ntype = 1: β = 1.007, ρ = 4.4

2 Original, median+IQR, Ntype = 1: β = 1.01, ρ = 1.6

3 With Ntype = 2−6:

βlo ≈ 0.25− .45, βhi ≈ 1.04; ρ ≈ 4.7...with “low” fraction ≈ 0.46

4 Cross-validation: evidence for selecting N ≥ 2 types

4Due to: pdeath, βτ .Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 28 / 40

K-Folds CV on N Types

1 2 3 4 5 6Number of types

2

4

6

8

10

12

Cros

s-va

lidat

ion

scor

e

Mean Score


“Zoom In:” K-Folds CV: 2-6 Types

2 3 4 5 6Number of types

0.810

0.815

0.820

0.825

0.830

0.835

0.840

Cros

s-va

lidat

ion

scor

e

Mean Score


N=2 Consumption Functions

0 1 2 3 4 5

0

1

2

3

4

5

= 1.04 and = 4.7

0 1 2 3 4 5

0

1

2

3

4

5

= 0.25 and = 4.7


Ugly Tables: Full Estimation Results, N ∈ (1,2,3,4)

Nβ 1 2 3 4

ρ 1.65 4.65 4.94 4.24

β 1.01 0.25, 1.04 0.29, 0.99, 1.05 0.01, 0.41, 0.81, 1.04

frac n.a. 0.46, 0.54 0.26, 0.35, 0.38 0.17, 0.19, 0.09, 0.54

βlo 1.01 0.25 0.69 0.34

fraclo n.a. 0.46 0.62 0.46


Ugly Tables: Full Estimation Results, N ∈ (5,6)

Nβ 5 6

ρ 4.70 4.74

β 0.00, 0.01, 0.52, 0.79, 1.04 0.11, 0.16, 0.24, 0.28, 0.45, 1.04

frac 0.09, 0.07, 0.15, 0.17, 0.53 0.03, 0.17, 0.09, 0.08, 0.1, 0.54

βlo 0.44 0.25

fraclo 0.47 0.46


Estimation Results, N ∈ (1,2,3,4,5,6)

Nβ 2 3 4 5 6

ρ 4.65 4.94 4.24 4.70 4.74

β{lo,hi} 0.25, 1.04 0.69, 1.05 0.34, 1.04 0.44, 1.04 0.25, 1.04

frac{lo,hi} 0.46, 0.54 0.62, 0.38 0.46, 0.54 0.47, 0.53 0.46, 0.54


Summary, Next Steps

Alternative models of consumption-savings behavior have no closedform solution, extremely hard to calculate likelihood surface, and arenot nested. None-the-less we would like to select between possiblecandidates.

This project jointly estimates a basic structural life cycleconsumption-savings problem multiple types and selects betweennumber of types via k-fold CV.

Fraction of “low beta” consumers is estimated at ˜0.46

Next steps: many

Data updateRobustness checksSelection with simple learning


Appendix: “Regular” Consumption Functions

0 1 2 3 4 5

0

1

2

3

4

5

= 1.007 and = 4.4

0 1 2 3 4 5

0

1

2

3

4

5

= 1.01 and = 1.65


Appendix: K-Folds CV on N Types

1 2 3 4 5 6Number of types

0

2

4

6

8

10

12

14

16

Cros

s-va

lidat

ion

scor

e

Mean Score

+/- 1 sd




0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Cros

s-va

lidat

ion

scor

e

Mean Score +/- Stdev

+/- 1 sd




0.06

0.04

0.02

0.00

0.02

Diff

from

N=4

cro

ss-v

alid

atio

n sc

ore

Diff from N=4, Mean Score, +/- Stdev

+/- 1 sd


a simple direct estimate of rule-of-thumb consumption...

Documents