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

Transcription

1 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.

2 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 / 40

3 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 / 40

4 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 / 40

5 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 / 40

6 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 / 40

7 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 / 40

8 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 / 40

9 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 / 40

10 Household Consumption Functions Example An example solution: 2.50 Consumption Functions: Black Before Retirement, Red After Consumption Cash on hand Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

11 , max = , max = , max = , max = , max = , max = , max = 835 Density Survey of Consumer Finance, Federal Reserve Board Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

12 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 / 40

13 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 / 40

14 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 / 40

15 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, Pool simulated data to match age groups of SCF data Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

16 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 / 40

17 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 / 40

18 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 / 40

19 Motivation: Why Consider N Types? , max = , max = , max = , max = , max = , max = , max = 835 Density Survey of Consumer Finance, Federal Reserve Board Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

20 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 / 40

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

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

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

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

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

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

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

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

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

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

31 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 / 40

32 Very Preliminary Results 1 Original, median-only 4, N type = 1: β = 1.007, ρ = Original, median+iqr, N type = 1: β = 1.01, ρ = With N type = 2 6: β lo , β hi 1.04; ρ with low fraction 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 / 40

33 Very Preliminary Results 1 Original, median-only 4, N type = 1: β = 1.007, ρ = Original, median+iqr, N type = 1: β = 1.01, ρ = With N type = 2 6: β lo , β hi 1.04; ρ with low fraction 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 / 40

34 Very Preliminary Results 1 Original, median-only 4, N type = 1: β = 1.007, ρ = Original, median+iqr, N type = 1: β = 1.01, ρ = With N type = 2 6: β lo , β hi 1.04; ρ with low fraction 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 / 40

35 K-Folds CV on N Types 12 Mean Score 10 Cross-validation score Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

36 Zoom In: K-Folds CV: 2-6 Types Mean Score Cross-validation score Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

37 N=2 Consumption Functions 5 =1.04 and =4.7 5 =0.25 and = Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

38 Ugly Tables: Full Estimation Results, N (1,2,3,4) N β ρ β , , 0.99, , 0.41, 0.81, 1.04 frac n.a. 0.46, , 0.35, , 0.19, 0.09, 0.54 β lo frac lo n.a Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

39 Ugly Tables: Full Estimation Results, N (5,6) N β 5 6 ρ β 0.00, 0.01, 0.52, 0.79, , 0.16, 0.24, 0.28, 0.45, 1.04 frac 0.09, 0.07, 0.15, 0.17, , 0.17, 0.09, 0.08, 0.1, 0.54 β lo frac lo Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

40 Estimation Results, N (1,2,3,4,5,6) N β ρ β {lo,hi} 0.25, , , , , 1.04 frac {lo,hi} 0.46, , , , , 0.54 Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

41 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 / 40

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

43 Appendix: Regular Consumption Functions 5 =1.007 and =4.4 5 =1.01 and = Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

44 Appendix: K-Folds CV on N Types 16 Mean Score Cross-validation score /- 1 sd Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

45 Appendix: K-Folds CV on N Types 1.6 Mean Score +/- Stdev Cross-validation score /- 1 sd Number of types Nathan M. Palmer (OFR) MSQ Rule-of-Thumb Estimation GMU CSS / 40

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