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

A Simple Direct Estimate of Rule-of-Thumb Consumption Using the Method of Simulated Quantiles & Cross Validation Nathan M. Palmer The Office of Financial Research George Mason University Department of Computational Social Science November 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 stochastic general equilibrium problem Eg. you think that the complex structure of mortgage or repo markets mattered 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? Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 3 / 40

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),... Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 3 / 40

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 behavior Behavioral models are not nested Likelihood surface: very hard to compute, if it exists (semiparametric estimation) Easily available data on life cycle choices is not great Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 4 / 40

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 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 5 / 40

A Simple Implementation 0 th -order Example This presentation: a specific, simple implementation Specifically: Estimate a textbook structural, semi-parametric household life-cycle consumption-savings problem Then jointly re-estimate the model with additional parameter(s): number of agents with different beta Then 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 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 6 / 40

Outline 1 Motivation 2 Agent Problem and Solution 3 Estimation and Selection Method of Simulated Moments / Quantiles K-Fold Cross-Validation 4 Results, Summary, Next Steps Very Preliminary Results Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 7 / 40

Household Consumption Functions Example An example solution: 2.50 Consumption Functions: Black Before Retirement, Red After 2.25 2.00 1.75 Consumption 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Cash on hand Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 7 / 40

0.4 0.3 26 30, max = 129 31 35, max = 786 36 40, max = 12721 41 45, max = 843 46 50, max = 1583 51 55, max = 2062 56 60, max = 835 Density 0.2 0.1 0.0 20 0 20 40 60 80 100 Survey of Consumer Finance, Federal Reserve Board Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 8 / 40

Textbook Household Problem A household solves the T -horizon problem described by sequence of Bellmans: [ ] vt (m t ) = max u(c t ) + βb t E t Γ 1 ρ c t+1 v t t+1(m t+1 ) s.t. m t+1 = R t+1 (m t c t ) + ξ t+1 m 0 given where β, ρ are discount factor and risk aversion m t is total cash on hand: total assets + total income R t is risk-free return on assets ξ t are mean-1 temporary shocks to income Entire problem is normalized by permanent income process, not shown 2 2 See Carroll (2012a, b) for extensive discussion. Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 9 / 40

Estimation Method Denote behavioral parameters: φ {β,ρ} Denote structural parameters: ρ = {ρ t } T t=0 = {Γ t,ξ t,r t,b t, etc} T t=0 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 11 / 40

Estimation Method: Simulation Step Given ρ, choose φ, solve problem for optimal policy functions {c t }T t=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 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 12 / 40

Functions of Quantiles and Loss Function Construct functions of quantiles τ: ( ˆqτ,75 ˆq Empirical: ˆϕ τ = τ,25 ˆq τ,50 ) Theoretical: ϕ N φ,τ = ( q φ τ,75 qφ τ,25 q φ τ,50 ) Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 13 / 40

Functions of Quantiles Define vectors of functions of quantiles: ( ) ˆϕ = ˆϕ 1, ˆϕ 2,..., ˆϕ 7 ( ) ϕφ N = ϕφ,1,ϕ φ,2,...,ϕ φ,7 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 14 / 40

Loss Function Define the loss function: ϖ ρ (φ) = ( ) ( ) ˆϕ ϕφ N W ˆϕ ϕφ N where W is a positive definite matrix of weights Minimize the loss function, obtain φ Bootstrap for variance Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 15 / 40

Motivation: Why Consider N Types? 0.4 0.3 26 30, max = 129 31 35, max = 786 36 40, max = 12721 41 45, max = 843 46 50, max = 1583 51 55, max = 2062 56 60, max = 835 Density 0.2 0.1 0.0 20 0 20 40 60 80 100 Survey of Consumer Finance, Federal Reserve Board Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 16 / 40

Elementary Point of View Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 18 / 40 Quick Illustration Consider the following artificial data: 3 3 Discussion from Cosma Shalizi s Advanced Data Analysis from an

Polynomial Overfitting Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 19 / 40

R-squared Looks Great Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 20 / 40

Loss Function (SSE) Looks Great Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 21 / 40

However, Very Poor Fit Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 22 / 40

Data Selection, K = 5 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 23 / 40

Data Selection, K = 5 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 24 / 40

Data Selection, K = 5 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 25 / 40

Data Selection, K = 5 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 26 / 40

Very Preliminary Results 1 Original, median-only 4, N type = 1: β = 1.007, ρ = 4.4 2 Original, median+iqr, N type = 1: β = 1.01, ρ = 1.6 3 With N type = 2 6: β lo 0.25.45, β hi 1.04; ρ 4.7...with low fraction 0.46 4 Cross-validation: evidence for selecting N 2 types 4 Due to: p death, β τ. Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 28 / 40

K-Folds CV on N Types 12 Mean Score 10 Cross-validation score 8 6 4 2 1 2 3 4 5 6 Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 29 / 40

Zoom In: K-Folds CV: 2-6 Types 0.840 Mean Score 0.835 Cross-validation score 0.830 0.825 0.820 0.815 0.810 2 3 4 5 6 Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 30 / 40

N=2 Consumption Functions 5 =1.04 and =4.7 5 =0.25 and =4.7 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 31 / 40

Ugly Tables: Full Estimation Results, N (1,2,3,4) N β 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 β lo 1.01 0.25 0.69 0.34 frac lo n.a. 0.46 0.62 0.46 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 32 / 40

Ugly Tables: Full Estimation Results, N (5,6) N β 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 β lo 0.44 0.25 frac lo 0.47 0.46 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 33 / 40

Estimation Results, N (1,2,3,4,5,6) N β 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 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 34 / 40

Summary, Next Steps Alternative models of consumption-savings behavior have no closed form solution, extremely hard to calculate likelihood surface, and are not nested. None-the-less we would like to select between possible candidates. This project jointly estimates a basic structural life cycle consumption-savings problem multiple types and selects between number of types via k-fold CV. Fraction of low beta consumers is estimated at 0.46 Next steps: many Data update Robustness checks Selection with simple learning Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 35 / 40

Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 36 / 40

Appendix: Regular Consumption Functions 5 =1.007 and =4.4 5 =1.01 and =1.65 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 37 / 40

Appendix: K-Folds CV on N Types 16 Mean Score 14 12 Cross-validation score 10 8 6 4 2 0 +/- 1 sd 1 2 3 4 5 6 Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 38 / 40

Appendix: K-Folds CV on N Types 1.6 Mean Score +/- Stdev Cross-validation score 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 +/- 1 sd 2 3 4 5 6 Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 39 / 40

Appendix: K-Folds CV on N Types 0.02 Diff from N=4, Mean Score, +/- Stdev Diff from N=4 cross-validation score 0.00 0.02 0.04 0.06 +/- 1 sd 2 3 4 5 6 Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS 2017 40 / 40