Sparsity in Econometrics and Finance -1mm · Fan (2012), Fastrich (2013) ... Utility-based...
Transcript of Sparsity in Econometrics and Finance -1mm · Fan (2012), Fastrich (2013) ... Utility-based...
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Sparsity in Econometrics and Finance
David Puelz
Thesis defenseApril 27, 2018
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Outline
Motivating problem
Utility-based selection
ApplicationsPortfolio selectionSeemingly unrelated regression modelsMonotonic function estimation
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Motivating problem: Charles’ dilemma
He’d like to invest some money in the market.
He’s heard passive funds are the way to go.
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Motivating problem: Charles’ dilemma
He’d like to invest some money in the market.
He’s heard passive funds are the way to go.
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But ... ∃ thousands of passive funds
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The context for this talk
This problem (and many others like it!) can be studied usingvariable selection techniques from statistics to induce sparsity.
What’s typically done? (broadly speaking)
• Bayesian: Shrinkage prior design.• Frequentist: Penalized likelihood methods.
Common theme? Sparsity and inference go hand in hand.
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The context for this talk
This problem (and many others like it!) can be studied usingvariable selection techniques from statistics to induce sparsity.
What’s typically done? (broadly speaking)
• Bayesian: Shrinkage prior design.• Frequentist: Penalized likelihood methods.
Common theme? Sparsity and inference go hand in hand.
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Separating priors from utilities
Our view: Subset selection is a decision problem. We need asuitable loss function, not a more clever prior.
This leads us to think of selection in a “post-inferenceworld” by comparing models (or in this case, portfolios)based on utility.∗
*sparsity and statistical uncertainty play a key role in thispost-inference exercise.
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Separating priors from utilities
Our view: Subset selection is a decision problem. We need asuitable loss function, not a more clever prior.
This leads us to think of selection in a “post-inferenceworld” by comparing models (or in this case, portfolios)based on utility.∗
*sparsity and statistical uncertainty play a key role in thispost-inference exercise.
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Utility-based selection: Primitives
Let wt be a portfolio decision, λt be a complexity parameter, Θt bea vector of model parameters, and Rt be future data.
1. Loss function L(wt, Rt) – measures utility.
2. Complexity function Φ(λt,wt) – measures sparsity.
3. Statistical model Π(Θt) – characterizes uncertainty.
4. Regret tolerance κ – characterizes degree of comfortfrom deviating from a “target decision” (in terms ofposterior probability).
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Utility-based selection: Procedure
• Optimize E[L(wt, Rt) + Φ(λt,wt)], where theexpectation is over p(Rt,Θt | R).
• Calculate regret versus a target w∗t for decisions
indexed by λt.
→ ρ(wλt ,w∗t , Rt) = L(wλt , Rt)− L(w∗
t , Rt)
• Select w∗λt as the decision satisfying the tolerance.
→ πλt = P[ρ(wλt ,w∗t , Rt) < 0] (satisfaction probability)
→ Select wλ∗t s.t. πλ∗
t > κ
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What is innovative here?
Portfolio selection literature typically focuses on one of the following:
• Modeling inputs Θt = (µt,Σt): Jobson (1980), Ledoit and Wolf(2007), Garlappi (2007), DeMiguel (2009) ...
• Optimizing in a clever way: Jagananathan (2002), Brodie (2009),Fan (2012), Fastrich (2013) ...
Utility-based selection incorporates both modeling andoptimization through analysis of ρ(wλt ,w∗
t , Rt).
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What is innovative here?
Portfolio selection literature typically focuses on one of the following:
• Modeling inputs Θt = (µt,Σt): Jobson (1980), Ledoit and Wolf(2007), Garlappi (2007), DeMiguel (2009) ...
• Optimizing in a clever way: Jagananathan (2002), Brodie (2009),Fan (2012), Fastrich (2013) ...
Utility-based selection incorporates both modeling andoptimization through analysis of ρ(wλt ,w∗
t , Rt).
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Example I: Long-only ETF investing
• Let Rt be a vector of future ETF returns.• Let wt be the portfolio weight vector (decision) at time t.• We use the log cumulative growth rate for our utility.
Primitives:
1. Loss: − log(1+
∑Nk=1 wk
t Rkt
)2. Complexity: Number of funds in portfolio (think ∥wt∥0)3. Model: DLM for Rt parameterized by (µt,Σt | Dt−1)
Data: Monthly returns on 25 ETFs from 1992-2016.Target: Fully invested (dense) portfolio.
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Example I: Long-only ETF investing
• Let Rt be a vector of future ETF returns.• Let wt be the portfolio weight vector (decision) at time t.• We use the log cumulative growth rate for our utility.
Primitives:
1. Loss: − log(1+
∑Nk=1 wk
t Rkt
)2. Complexity: Number of funds in portfolio (think ∥wt∥0)3. Model: DLM for Rt parameterized by (µt,Σt | Dt−1)
Data: Monthly returns on 25 ETFs from 1992-2016.Target: Fully invested (dense) portfolio.
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Step 1: Constructing portfolio decisions
• Portfolio decisions have ≤ 5 funds.
• ≥ 25% in SPY
Decisions are found by minimizing expected loss for each time t.Results in a choice of 12,950 decisions to choose among!!
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Step 1: The expected loss
L(wt) = EΘtERt|Θt
[− log(1+ΣN
k=1wkt Rk
t ) + Φ(λt,wt)]
≈ EΘtERt|Θt
[−ΣN
k=1wkt Rk
t +12Σ
Nk=1Σ
Nj=1wk
t wjtRk
t Rjt +Φ(λt,wt)
]= −wT
t µt +12wT
t ΣNCt wt +Φ(λt,wt).
The past returns Rt enter into our utility consideration by definingthe posterior predictive distribution.
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Step 2: Compute and examine ρ for optimal decisions
λt−decisions ordered by increasing satisfaction probability − March 2002
Reg
ret (
diffe
renc
e in
loss
)
−0.
010
−0.
005
0.00
00.
005
0.01
00.
015
0.40
0.45
0.50
0.55
prob
abili
ty
E[Regret]πλt
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Step 3: Select decisions based on satisfaction threshold κ
Dates SPY EZU EWU EWY EWG EWJ OEF IVV IVE EFA IWP IWR IWF IWN IWM IYW IYR RSP
2003 25 - 58 - - - - - - - - - - 8.3 - - - 8.32004 25 - 43 - - 20 - 6.2 - - - - - - - - - 6.22005 25 - 25 - 6.2 13 - - - - - - - - - - 30 -2006 62 - - - 6.2 19 - - - - - - 6.3 - 6.2 - - -2007 75 - - 25 - - - - - - - - - - - - - -2008 44 - - - 8.3 21 - - - 26 - - - - - - - -2009 30 - - 6.2 - 41 - - - 17 6.3 - - - - - - -2010 75 - - 8.3 - - - - - - 8.3 - - - - 8.3 - -2011 58 - 25 - - - - - - - 8.3 - - - - 8.3 - -2012 29 8.3 - - - 54 - - - - - - - - - 8.3 - -2013 34 - - - - 49 - - - - 8.3 - - - - 8.3 - -2014 25 - - - - 37 26 - - 6.2 - 6.2 - - - - - -2015 45 - - - - 39 - - 8.3 - 8.3 - - - - - - -2016 35 - - - - 40 - 17 - - 8.3 - - - - - - -
Selected decisions for κ = 45% threshold.
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What happens when κ is varied?−
0.00
8−
0.00
40.
000
0.00
20.
004
Exp
ecte
d R
egre
t (di
ffere
nce
in lo
ss)
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
κ= 45%κ= 50%κ= 55%
Higher satisfaction threshold =⇒ lower expected regret! 14
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Comparing portfolios to their targets out of sample
out-of-sample statisticsSharperatio s.d. mean
returnsparse 0.40 14.98 6.02dense 0.45 14.41 6.47
Ex ante equivalence appears to carry over ex post.
There appear to be little ex post benefits of diversification.
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What about other models / variable selection tasks?
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Example II: Seemingly unrelated regressions
Y = βX + ϵ, ϵ ∼ N(0,Ψ)
• Y is q length response vector• X is p length covariate vector• β is q × p coefficient matrix• Ψ is non-diagonal matrix
finance: asset pricing, operations management: supply/demandstructural equations, marketing: consumer preferences, economics:capital structure, firm composition, macroeconomic indicators.
We are interested in the structure of β!
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Example II: Seemingly unrelated regressions
Y = βX + ϵ, ϵ ∼ N(0,Ψ)
• Y is q length response vector• X is p length covariate vector• β is q × p coefficient matrix• Ψ is non-diagonal matrix
finance: asset pricing, operations management: supply/demandstructural equations, marketing: consumer preferences, economics:capital structure, firm composition, macroeconomic indicators.
We are interested in the structure of β!17
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Meat science
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Factor selection for asset pricing
The Factor Zoo (Cochrane, 2011) – many possible factors ...
• Market• Size• Value• Momentum• Short and long term reversal• Betting against β• Direct profitability
• Dividend initiation• Carry trade• Liquidity• Quality minus junk• Investment• Leverage• ...
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Example II: Factor selection for asset pricing
Let the return on test assets be R, and the return on factors be F.R = γF + ϵ, ϵ ∼ N(0,Ψ)
Primitives:
1. Loss: L(γ, R, F) = − log p(R|F)2. Complexity: Φ(λ, γ) = λ ∥γ∥1.3. Model: R|F with normal errors and conjugate g-priors and F
via gaussian linear latent factor model.4. Regret tolerance: Let’s consider several κ’s.
Data: R: 25 Fama-French portfolios, F: 10 factors from finance literatureTargets: The λ = 0 model, i.e.: the fully dense graph
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Example II: Factor selection for asset pricing
Let the return on test assets be R, and the return on factors be F.R = γF + ϵ, ϵ ∼ N(0,Ψ)
Primitives:
1. Loss: L(γ, R, F) = − log p(R|F)2. Complexity: Φ(λ, γ) = λ ∥γ∥1.3. Model: R|F with normal errors and conjugate g-priors and F
via gaussian linear latent factor model.4. Regret tolerance: Let’s consider several κ’s.
Data: R: 25 Fama-French portfolios, F: 10 factors from finance literatureTargets: The λ = 0 model, i.e.: the fully dense graph
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ρ distributions for different sparse graphs
models ordered by decreasing λ
Reg
ret
0.0
0.1
0.2
0.3
0.4
0.5
0.6
prob
abili
ty
E[ρλ]πλselected model
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Factor selection graph, κ = 12.5%
R: 25 Fama-French portfolios, F: 10 factors from finance literature
Size1BM1
Size1BM2 Size1
BM3
Size1BM4
Size1BM5
Size2BM1
Size2BM2
Size2BM3
Size2BM4
Size2BM5
Size3BM1
Size3BM2
Size3BM3
Size3BM4
Size3BM5
Size4BM1
Size4BM2
Size4BM3
Size4BM4
Size4BM5
Size5BM1
Size5BM2
Size5BM3
Size5BM4
Size5BM5
Mkt.RF
SMB
HML
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Selected graphs under different satisfaction tolerances κ
κ = 2 %
SMALL.LoBM
ME1.BM2
ME1.BM3
ME1.BM4SMALL.HiBM
ME2.BM1
ME2.BM2
ME2.BM3
ME2.BM4
ME2.BM5
ME3.BM1
ME3.BM2
ME3.BM3
ME3.BM4
ME3.BM5
ME4.BM1
ME4.BM2
ME4.BM3
ME4.BM4
ME4.BM5
BIG.LoBM
ME5.BM2
ME5.BM3
ME5.BM4
BIG.HiBM
Mkt.RF
SMB
κ = 4 %
SMALL.LoBM
ME1.BM2
ME1.BM3
ME1.BM4
SMALL.HiBM
ME2.BM1
ME2.BM2
ME2.BM3
ME2.BM4
ME2.BM5
ME3.BM1
ME3.BM2
ME3.BM3
ME3.BM4
ME3.BM5
ME4.BM1
ME4.BM2
ME4.BM3
ME4.BM4
ME4.BM5
BIG.LoBM
ME5.BM2
ME5.BM3
ME5.BM4
BIG.HiBM
Mkt.RF
SMB
κ = 12.5 %
SMALL.LoBM
ME1.BM2ME1.BM3
ME1.BM4
SMALL.HiBM
ME2.BM1
ME2.BM2 ME2.BM3
ME2.BM4
ME2.BM5ME3.BM1
ME3.BM2
ME3.BM3
ME3.BM4
ME3.BM5
ME4.BM1
ME4.BM2
ME4.BM3
ME4.BM4ME4.BM5
BIG.LoBM
ME5.BM2
ME5.BM3
ME5.BM4
BIG.HiBM
Mkt.RF
SMB
HML
κ = 32.5 %
SMALL.LoBM
ME1.BM2
ME1.BM3ME1.BM4
SMALL.HiBMME2.BM1
ME2.BM2
ME2.BM3ME2.BM4
ME2.BM5ME3.BM1
ME3.BM2
ME3.BM3
ME3.BM4ME3.BM5
ME4.BM1
ME4.BM2
ME4.BM3
ME4.BM4ME4.BM5
BIG.LoBM
ME5.BM2ME5.BM3
ME5.BM4
BIG.HiBM
Mkt.RF
SMB
HML
RMW
CMA
QMJ
κ = 47.5 %
SMALL.LoBM
ME1.BM2
ME1.BM3ME1.BM4
SMALL.HiBM
ME2.BM1ME2.BM2 ME2.BM3
ME2.BM4
ME2.BM5
ME3.BM1
ME3.BM2
ME3.BM3
ME3.BM4
ME3.BM5
ME4.BM1
ME4.BM2
ME4.BM3
ME4.BM4
ME4.BM5
BIG.LoBM
ME5.BM2
ME5.BM3ME5.BM4
BIG.HiBM
Mkt.RFSMBHML
RMW
CMALTR
BAB
QMJ
κ = 49.75 %
SMALL.LoBM
ME1.BM2
ME1.BM3
ME1.BM4
SMALL.HiBM
ME2.BM1
ME2.BM2
ME2.BM3
ME2.BM4
ME2.BM5ME3.BM1 ME3.BM2
ME3.BM3
ME3.BM4
ME3.BM5
ME4.BM1
ME4.BM2
ME4.BM3 ME4.BM4
ME4.BM5
BIG.LoBM
ME5.BM2ME5.BM3
ME5.BM4
BIG.HiBM
Mkt.RFSMB
HML
RMW
CMA
LTR
STR
BAB
QMJ
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Example III: Monotonic function estimation
Goal: Describe expected returns with firm characteristics oraccounting measures (size, book-to-market, momentum, ...).
E[Rit | Xit−1] = f (Xit−1)
Rit: excess return of firm i at time tXit−1: vector of characteristics of firm i at time t
We would like to learn f !
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Example III: Monotonic function estimation
Goal: Describe expected returns with firm characteristics oraccounting measures (size, book-to-market, momentum, ...).
E[Rit | Xit−1] = f (Xit−1)
Rit: excess return of firm i at time tXit−1: vector of characteristics of firm i at time t
We would like to learn f !
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Portfolio sorts are one way to understand f ...
Jegadeesh and Titman (2001)
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Challenges and a solution
• Xit−1 is multidimensional.• Even if we had only 12 characteristics and sorted into quintiles
along each dimension, that requires constructing512 = 244140625 portfolios!
We propose modeling the CEF as an additive quadratic splinemodel (with monotonicity constraints and time variation):
E[Rit | Xit−1] = αt +K∑
k=1gkt(xki,t−1)
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Why monotonicity?
Finance theory often tells us that expected returns increase ordecrease in each characteristic. Ex: past high-performing firmshave higher returns than past weak-performing firms, on average.
Using this information is statistically advantageous!
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Why monotonicity?
Finance data is noisy – any bias aids in more precise estimation.
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Estimated functions at January 1978
0.0 0.2 0.4 0.6 0.8 1.0
−0.
010
0.00
00.
005
0.01
00.
015
0.02
0
no monotonicity
momentum
Exp
ecte
d R
etur
n
0.0 0.2 0.4 0.6 0.8 1.0
−0.
010
0.00
00.
005
0.01
00.
015
0.02
0
monotonicity
momentum
Exp
ecte
d R
etur
n
monotonicity is enforced by linear constraints on spline coefficients29
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How does the function vary over time?
0.0 0.2 0.4 0.6 0.8 1.0
−0.
010
0.00
00.
005
0.01
00.
015
0.02
0
Jan 1978
momentum
Exp
ecte
d R
etur
n
0.0 0.2 0.4 0.6 0.8 1.0
−0.
010
0.00
00.
005
0.01
00.
015
0.02
0
Jan 2014
momentum
Exp
ecte
d R
etur
n
dynamics are modeled by likelihood discounting, McCarthy and Jenson (2016)30
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A model with 36 characteristics - January 1978
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A model with 36 characteristics - January 1978
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Future work
Where to go from here?
• New utility specifications: value-at-risk and simulation based.Analyzing other properties of the regret distribution.
• New models: multinomial regression and classification models,nonlinear and nonparametric models.
• New application areas: corporate finance, marketing,macroeconomics.
Existing papers:Regret-based selection for sparse dynamic portfolios.submitted (2017). Thesis ch. 2.
Variable selection in SUR models with random predictors.Bayesian Analysis (2017). Thesis ch. 3.
Monotonic effects of characteristics on returns.working paper (2018). Thesis ch. 3.5. 33
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Concluding thoughts, and thanks!
• Passive investing, SUR model selection, and monotonicfunction estimation approached using new feature selectiontechnique.
• Utility functions can enforce inferential preferences thatare not prior beliefs.
• Statistical uncertainty should be used as a guide to avoidoverfitting.
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Extra slides
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Treatment effect estimation
Suppose we are trying to estimate the treatment effect of dietarykale on cholesterol level. But ... we only have observational data.
Yi = β0 + αZi + ϵi
• Yi is cholesterol level• Zi is amount of kale eaten.
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Problem: Gym rats tend to eat more kale!
In other words, exercise is predictive of cholesterol and kale intake!This leads to omitted variable bias.
Yi = β0 + αZi + ϵi
Because cov(Zi, ϵi) = 0 we can write:
Yi = β0 + αZi + wZi + ϵi
with cov(Zi, ϵi) = 0, we mis-estimate α as α+ w!
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Solution: “Adjust” for weekly exercise
By controlling for weekly exercise Xi in the regression
Yi = β0 + αZi + βXi + ϵi
we can “clear out” the confounding.
Conditional on Xi, cov(Zi, ϵi) = 0 and we’re all set!
But what if Xi is a big vector, and we don’t know whichcovariates to control for? (Enter sparsity).
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Regularized treatment effect estimation
Consider the model with no intercept and many covariates Xi:
Yi = αZi + XTi β + ϵi
We can induce sparsity with a ridge prior on β and leaving α
unpenalized. This injects bias into treatment effect estimate:
bias(αridge) = −(ZTZ)−1ZTX(
XTX + λIp − XTXZ)−1
λβ = 0
where (ZTZ)−1ZTX is a p-length vector of coefficients from punivariate regressions of each Xj on Z and XZ = Z(ZTZ)−1ZTXare the predicted values from these regressions.
This nonzero bias is referred to regularization-inducedconfounding (RIC). 39
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A different approach eliminates RIC
Consider the model where a likelihood is included for Z:
Selection equation: Zi = XTi γ + ϵi
Response equation: Yi = αZi + XTi β + νi
• Extract propensity from selection equation: Z ≈ Xγ
• Augment covariates with propensity Xnew =(Z Z X
)• Ridge estimate with Z and Z unpenalized mitigates RIC
Regularization and confounding in linear regression for treatment effect estimation.
Bayesian Analysis (2017).40
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A different approach eliminates RIC
The bias of the treatment effect becomes:
bias(αridge) = −(ZTZ)−1ZTX1(
XTX + λIp − XTXZ)−1
λβ ≈ 0
where Z =(Z Z
)and ·1 corresponds to the top row of the
matrix ·. (ZTZ)−1ZTX1 are the coefficients on Z in the twovariable regressions of each Xi on
(Z Z
).
Controlling for the propensity of the treatment wipes outregularization-induced confounding (RIC) in the treatment
effect estimate.
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Next steps
Selection equation: Zi = XTi γ + ϵi
Response equation: Yi = αZi + XTi β + νi
• Develop fast empirical Bayes approach to regularize twoequation system.
• Account for clustered observations using block boostrapping.• Many application in social science, including
micro/macroeconomics and corporate finance.• RIC still exists even in nonlinear, statistical learning based
models! Why? Because they especially need to be regularized.Extend this approach to random forests.
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A dynamic regression model giving moments (µt,Σt)
Rit = (βi
t)TRF
t + ϵit, ϵi
t ∼ N(0, 1/ϕit), β
it = βi
t−1 + wit, wi
t ∼ Tnit−1
(0,Wit),
βi0 | D0 ∼ Tni
0(mi
0,Ci0), ϕi
0 | D0 ∼ Ga(ni0/2, di
0/2),βi
t | Dt−1 ∼ Tnit−1
(mit−1,Ri
t), Rit = Ci
t−1/δβ ,
ϕit | Dt−1 ∼ Ga(δϵni
t−1/2, δϵdit−1/2),
RFt = µF
t + νt, νt ∼ N(0,ΣFt ), µF
t = µFt−1 +Ωt Ωt ∼ N(0,Wt,Σ
Ft ),
(µF0 ,Σ
F0 | D0) ∼ NW−1
n0 (m0,C0, S0),
(µFt ,Σ
Ft | Dt−1) ∼ NW−1
δFnt−1(mt−1,Rt, St−1), Rt = Ct−1/δc︸ ︷︷ ︸
µt = βTt µ
Ft
Σt = βtΣFtβ
Tt +Ψt
→ Moments are used in the expected loss minimization→ Predictive distribution is used to compute ρ
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Formulating as a convex penalized optimization
Define Σ = LLT.
L(w) = −wTµ+12wTΣw + λ ∥w∥1
=12∥∥LTw − L−1µ
∥∥22 + λ ∥w∥1 .
Now, we can solve the optimization using existing algorithms, suchas lars of Efron et. al. (2004).
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Example: Gross exposure complexity function
• Let Rt be a vector of N future asset returns.• Let wt be the portfolio weight vector (decision) at time t.• We use the log cumulative growth rate for our utility.
Primitives:
1. Loss: − log(1+
∑Nk=1 wk
t Rkt
)2. Complexity: λt ∥wt∥1
3. Model: DLM for Rt parameterized by (µt,Σt)
4. Regret tolerance: Let’s consider several κ’s.
Assume the target is fully invested (dense) portfolio.Data: Returns on 25 ETFs from 1992-2016.
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Example: Gross exposure complexity function
• Let Rt be a vector of N future asset returns.• Let wt be the portfolio weight vector (decision) at time t.• We use the log cumulative growth rate for our utility.
Primitives:
1. Loss: − log(1+
∑Nk=1 wk
t Rkt
)2. Complexity: λt ∥wt∥1
3. Model: DLM for Rt parameterized by (µt,Σt)
4. Regret tolerance: Let’s consider several κ’s.
Assume the target is fully invested (dense) portfolio.
Data: Returns on 25 ETFs from 1992-2016.
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Example: Gross exposure complexity function
• Let Rt be a vector of N future asset returns.• Let wt be the portfolio weight vector (decision) at time t.• We use the log cumulative growth rate for our utility.
Primitives:
1. Loss: − log(1+
∑Nk=1 wk
t Rkt
)2. Complexity: λt ∥wt∥1
3. Model: DLM for Rt parameterized by (µt,Σt)
4. Regret tolerance: Let’s consider several κ’s.
Assume the target is fully invested (dense) portfolio.Data: Returns on 25 ETFs from 1992-2016.
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Optimal decisions lined up for a snapshot in time
After optimizing expected loss for 500 λt’s, we compute regretρ(wλt ,w∗
t , Rt) (left axis) and πλt (right axis).
λt−decisions ordered by increasing satisfaction probability − March 2002
Reg
ret (
diffe
renc
e in
loss
)
0.00
00.
005
0.01
00.
015
0.40
0.42
0.44
0.46
0.48
0.50
prob
abili
ty
E[Regret]πλt
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Regret-based selection: Illustration
dλ : sparse decisions, d∗ : target decision.
πλ = P[ρ(dλ, d∗, Y) < 0]: probability of not regretting λ-decision.
Loss
Den
sity
sparse decisionstarget
0.0 0.1 0.2 0.3 0.4 0.5
decision 1
decision 2
Regret (difference in loss)
πdecision 2
−0.05 0.00 0.05 0.10 0.15 0.20 0.25
decision 1
decision 2
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Ex ante SRtarget − SRdecision evolution−
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Diff
eren
ce in
Sha
rpe
ratio
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
dense portfolio as targetSPY as target
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UBS for Monotonic function estimation
The regression model is:
Rit = αt +K∑
k=1fkt(xki,t−1) + ϵit, ϵit ∼ N(0, σ2)
Insight – with quadratic splines for all fkt, this can be written as apredictive regression:
Rt ∼ N(Xt−1Bt, σ
2t Int
)where
Xt−1 =[1nt Xt−1
], Bt =
[αt βt
]Xt−1 is matrix of size nt ×K(m+ 2), βt is vector of size K(m+ 2).Therefore, each firm is given a row in Xt−1, and each m + 2 blockof βt corresponds to the coefficients on the spline basis for aparticular characteristic, k. 49
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UBS for Monotonic function estimation
We can now proceed as Hahn and Carvalho (2015). The lossfunction is the negative log density of the regression plus a penaltyfunction Φ with parameter λt. Also, let the “sparsified action” forthe coefficient matrix At.
Lt(Rt,At,Θt) =12(Rt − Xt−1At)
T(Rt − Xt−1At) + Φ(λt,At).
After integrating over p(Rt,Θt), we obtain:
Lλt(At) =∥∥Xt−1At − Xt−1Bt
∥∥22 +Φ(λt,At)
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Modeling Time-dynamics: McCarthy and Jensen (2016)
• Power-weighted likelihoods let information decay over time• To estimate parameters at time τ , let δt = 0.99τ−t, such that
δ1 ≤ δ2 ≤ ... ≤ δτ = 1, the likelihood at time τ ∈ 1, ...,T is
p(R1, ...,Rτ |Θτ ) =τ∏
t=1p(Rt|Θτ )
δt .
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Model Summary
Rt|· ∼ N(αt1nt +
K∑k=1
fkt(xk,t−1), σ2t In
)δt
fkt(xk,t−1) = Xk,t−1βkt = Xk,t−1L−1Lβkt = Wktγkt
αt ∼ N(0, 10−2)
σ2t ∼ U(0, 103)
(γjkt|Ijkt = 1, σ2t ) ∼ N+(0, ckσ
2t )
(γjkt|Ijkt = 0) = 0Ijkt ∼ Bn(pjk = 0.2).
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Data
Freyberger, Neuhierl, and Weber (2017)’s dataset:
• CRSP monthly stock returns for most US traded firms• 36 characteristics from Compustat and CRSP, including size,
momentum, leverage, etc.• July 1962 - June 2014
Presence and direction of monotonicity is determined by importantpaper in the literature
53