firm dispersion and business cycles: estimating … · firm dispersion and business cycles:...

Firm Dispersion and Business Cycles:Estimating Aggregate Shocks using Panel Data

Jerome Williams Simon Mongey

NYU NYU

January 18, 2017

Introduction Model Estimation Results Appendix

Introduction

• Recent focus in business cycle literature on learning from micro-data.

• In particular, firm distribution becomes more “spread out” inrecessions.

• Std. dev. of sales growth across firms is countercyclical.

−.04

−.02

0

.02

.04

Devia

tion

from

tren

d

1990 1995 2000 2005 2010Std Dev of Sales Growth Aggregate Output

Aggregate Output is Real GDP, taken from BEA National Income and Product Accounts, logged and detrended using a one-sidedHodrick-Prescott filter. Std Dev of sales growth is the cross-sectional standard deviation of the percentage change in quarterly sales acrossfirms, using listed US firms in Compustat, with a cubic trend removed.

IQR Industry effects


Motivation and approach

Question: is increased dispersion a cause or an effect of cycle?

Our approach:

• Model of heterogeneous firms with aggregate shocks• Standard macro shocks

• productivity• discount factor• labor supply

• Uncertainty shock• direct shock to firm dispersion

• Estimate the model using• aggregate time series

• output, consumption, hours worked

• time series of firm dispersion• std. dev. sales growth over time, constructed from panel data

• Obtain estimates for• contribution of uncertainty shock to aggregates (cause)• contribution of macro shocks to dispersion (effect)


Results preview

• Q: Is firm dispersion a cause or effect of the business cycle?

• A: Neither!

• Estimation results show that• almost all fluctuation in aggregates is due to macro shocks• almost all fluctuation in dispersion is due to uncertainty shock

• Uncertainty shocks can generate recessions but don’t.


DSGE overview

Model

u(Ct) = Xβt βRtu(Ct+1)

−Xψt v(Nt) = u(Ct)

Yt = XZt Nνt K

1−νt

Yt = Ct + (Kt+1 − (1− δ)Kt)

+ Parameters

θ = (u, v , β, ν, δ)

⇒ Impulse responses

10 20 30 40

XZ

10 20 30 40

Xβ

10 20 30 40

Xψ

Yt

Ct

Nt

Impulse responses

+ Data

1990 1995 2000 2005 2010-0.05

0

0.05

⇒ LikelihoodP(data|θ)

+ Shock decomposition

1990 1995 2000 2005 2010-0.04

-0.02

0

0.02

0.04XZ

Xβ

Xψ


Model: environment

• Discrete time, infinite horizon.

• Household• supplies labor Ns

t at competitive wage Wt

• consumes consumption good Ct

• receives dividends Πt from firms.

• Firms Continuum of firms, each with state (kit , zit , ξit)• hire labor nit at competitive wage Wt ,• produce using labor nit and pre-installed capital stock kit

• decreasing returns to scale production function

yit = XZt zit

(nνitk

1−νit

)κ, κ < 1

• choose whether to adjust capital or not• pay ξit ∼ F

(ξ̄)

to adjust• if adjusting, choose next period’s capital stock ki,t+1

• pay out dividends πit


Firm problem (adjust or not)

• The firm’s state is (k , z , ξ; S).

• Let v(k , z , ξ; S) be the discounted present value of the firm.

• Each period, the firm chooses whether to adjust or not:

v(k , z , ξ; S

)= max

{−ξW (S) + v adj

(k , z ; S

), v stay (k, z ; S)

}• Gives threshold rule ξ∗ (k , z ; S).


Firm problem (continued)

Value of adjusting:

v adj(k, z ; S

)= max

d,k′≥0,nd + E [M (S,S′) v (k ′, z ′, ξ′; S′)]

subject to

d = XZ z(nνk1−ν)κ −W (S) n − (k ′ − (1− δ) k)

log z ′ = ρz log z + ε′z , ε′z ∼ N

(−1

2

(Xσσz)2

1 + ρz, (Xσσz)2

)S′ ∼ H

(S,S′

)

Value of not adjusting:

v stay same as v adj but k′−(1−δ)kk ∈ [−b, b]


Household problemLet W (S) be the household’s expected present discounted value:

W (S) = maxC ,N,A′

u (C − H)− Xψv(N) + βXβE [W (S′) |S]

C + Q (S)A′ = W (S)N + (Q (S) + Π (S))A

Optimality conditions:

• stochastic discount factor

M (S,S′) = βXβ (S)u′ (C (S′)− H(S′))

u′ (C (S)− H(S))

• labor supply

W (S) = Xψ (S)v ′ (N (S))

u′ (C (S)− H(S))


Aggregate state

• The aggregate state is the distribution of firms and the aggregateshocks:

S =(µ (k , z , ξ) ,XZ ,Xβ ,Xψ,Xσ

)where the aggregate shocks are:

• X Z , a shock to aggregate TFP,• Xβ , a shock to the household’s discount factor,• Xψ, a shock to the household disutility of labor,• Xσ, an uncertainty shock.

• Aggregate shocks follow AR(1) processes• in logs for (Z , β, ψ):

logX jt+1 = ρj logX j

t + σjεt , ε ∼ N(

0, (σj)2),

• in levels for σ:

Xσt+1 = ρσXσ

t + σσεt , ε ∼ N(

0, (σσ)2).


Equilibrium

An equilibrium of this model consists of

• firms’ value function v(k, z, ξ; S),

• firms’ policy functions k ′(k, z, ξ; S), n(k, z, ξ; S) and ξ∗(k, z; S),

• households’ policies C(S), N(S) and associated discount factor M(S,S’),

• a wage W (S),

• a law of motion for the distribution of firms Γ(µ, µ′; S),

such that

1. the firms’ policies are optimal,

2. the households’ policies are optimal,

3. the labor market clears,

N (S) =

∫n (k, z, ξ; S) dµ (k, z, ξ; S)

4. the law of motion of the distribution is consistent with the firm’s policies.


Solution method

Solve numerically using a computer. 4 steps:

1. Solve for “steady state”, i.e. eqbm without aggregate shocks.• A fixed-point problem in the wage W .

2. Construct discrete approximations of eqbm objects and conditions.

3. Take first-order Taylor approximation of discretized eqbm conditions.

• Expand around steady state from Step 1.

4. Solve resulting first-order difference equation using linear algebra.


Impulse response functions

10 20 30 40-2

0

2

4

6

8×10-3 TFP

10 20 30 40-2

0

2

4

6

8×10-3 Discount

10 20 30 40-2

0

2

4

6

8×10-3 Labor

OutputConsumptionLabor

10 20 30 40-2

0

2

4

6

8 ×10-4 TFP

10 20 30 40-2

-1.5

-1

-0.5

0 ×10-3 Discount

10 20 30 40-2

0

2

4

6

8 ×10-4 Labor

SD sales growth


Uncertainty shock

5 10 15 20 25 30 35 40Quarters

-6

-4

-2

0

2

4

Perc

ent d

evia

tion

from

SS

×10-4

OutputConsumptionLabor

5 10 15 20 25 30 35 40Quarters

-20

-15

-10

-5

0

5

Perc

ent d

evia

tion

from

SS

×10-4

InvestmentCapitalFraction of firms adjusting

-1 -0.5 0 0.5 1Log productivity

2

2.5

3

3.5

4

4.5

5

5.5

k′

Firm capital choiceSteady state1 period after uncertainty shock


Estimation

1. Fix some parameters externally:

θ1 = (β, σ, η, δ, ν)

2. Estimate steady-state parameters using simulated method ofmoments:

θ2 =(ξ̄, ρz , σz , κ

)• to match moments of long-run firm distribution (Compustat)

3. Estimate shock process parameters using Bayesian methods:

θ3 ={ρj , σj

}j∈{Z ,ψ,β,σ}

using as observable time series• Aggregate output, consumption and hours worked (NIPA)

• Cross-sectional time series of SD sales growth (Compustat)


Data: cross-sectional

• Compustat: quarterly accounting data for publicy listed firms.

• Sample selection

• US firms, 1989q1 to 2014q4• We drop from the sample

• quarters in which a firm is involved in M&A• financial firms and utilities• each firm’s first and last quarter

• Resulting sample: 360,591 firm-quarter observations

• Variable definitions

sales growthit =salesit − salesi,t−1

12 (salesit + salesi,t−1)

, investment rateit =ppeit − ppei,t−1

12

(ppeit + ppei,t−1

)

inaction ratet =1

Nt

Nt∑i=1

1{|investment rateit | > 0.01}


Data: time series• Aggregate time series

• Source: National Income and Product Accounts (BEA)• Variables

• Output Yt : Real Gross Domestic Product• Consumption Ct : Real Personal Consumption Expenditures• Labor Nt : Index of Aggregate Weekly Hours

• Logged and detrended using a one-sided HP-filter.

• Cross-sectional time series:• SD sales growth from Compustat.• Detrended using cubic time trend.

1990 1995 2000 2005 2010-0.06

-0.04

-0.02

0

0.02

0.04

Output Consumption Labor SD sales growth


Estimation (1)

(1) Fix some parameters externally:

u (C − H) = (C−H)1−σ

1−σ , v (N) = N1+η

1+η

Parameter ValueDiscount factor β 0.99Curvature of utility function σ 1Inverse Frisch elasticity η 2Depreciation rate δ 0.03Labor elasticity of output ν 0.65

(2) Estimate steady-state parameters using SMM:

Parameter Value Target Data ModelAdjustment costs ξ̄ 3.7 × 10−4 Investment inaction rate 0.263 0.270Decreasing RTS κ 0.744 Dividends / output 0.070 0.073Persistence idio. prod. ρz 0.311 SD sales growth 0.285 0.243Avg SD idio. prod. σ̄z 0.320 SD investment rates 0.154 0.181


Results: priors and posteriors

(3) Estimate shock process parameters using Bayesian methods:

Prior Posterior

Parameter name Type Mean SD Mode SD

Autocorrelation, TFP shock ρZ Beta 0.600 0.260 0.983 0.015

Autocorrelation, discount shock ρβ Beta 0.600 0.260 0.739 0.054

Autocorrelation, labor supply shock ρψ Beta 0.600 0.260 0.986 0.006Autocorrelation, uncertainty shock ρσ Beta 0.600 0.260 0.775 0.257

Standard deviation, TFP shock σZ Exp 0.100 0.100 0.012 0.001

Standard deviation, discount shock σβ Exp 0.100 0.100 0.002 0.001

Standard deviation, labor supply shock σψ Exp 0.100 0.100 0.012 0.001Standard deviation, uncertainty shock σσ Exp 0.100 0.100 0.002 0.005

Posterior mode and SD are computed by drawing a sample of 106 points from the posterior distribution using the Metropolis-Hastings

algorithm. We verify that Gelman’s√

R̂ statistic is less than 1.05 for each parameter separately.


Results: FEVDs

1 2 3 6 10 50

Forecast horizon

0

0.2

0.4

0.6

0.8

1S

ha

re o

f va

ria

nce

A. Output

TFP

Labor supply

Discount

Uncertainty

1 2 3 6 10 50

Forecast horizon

0

0.2

0.4

0.6

0.8

1

Sh

are

of

va

ria

nce

B. Consumption

1 2 3 6 10 50

Forecast horizon

0

0.2

0.4

0.6

0.8

1

Sh

are

of

va

ria

nce

C. Hours

1 2 3 6 10 50

Forecast horizon

0

0.2

0.4

0.6

0.8

1

Sh

are

of

va

ria

nce

A. SD sales growth

TFP

Labor supply

Discount

Uncertainty

1 2 3 6 10 50

Forecast horizon

0

0.2

0.4

0.6

0.8

1

Sh

are

of

va

ria

nce

B. SD investment rate (adj.)

1 2 3 6 10 50

Forecast horizon

0

0.2

0.4

0.6

0.8

1

Sh

are

of

va

ria

nce

C. Fraction of adjusters


Results: shock decomposition

1990 1995 2000 2005 2010-0.04

-0.02

0

0.02

0.04A. Output

1990 1995 2000 2005 2010-0.03

-0.02

-0.01

0

0.01

0.02B. Consumption

1990 1995 2000 2005 2010-0.1

-0.05

0

0.05C. Labor

TFPDiscountLaborUncertainty

1990 1995 2000 2005 2010-0.03

-0.02

-0.01

0

0.01

0.02

0.03D. SD sales growth


Uncertainty-driven business cycles

• Compute uncertainty shock series that matches output fluctuations.

• Counterfactual implications for SD sales growth:

1990 1995 2000 2005 2010-0.4

-0.2

0

0.2

0.4

0.6SD sales growth

Model Data


Conclusion

• Contribution• We build a macro model consistent with micro evidence on

dispersion.• We estimate the model on time series of aggregates and

cross-sectional dispersion.

• Findings• Macro shocks alone not sufficient to explain fluctuations in

dispersion.• Uncertainty shocks alone not sufficient to explain fluctuations in

aggregates.

• Future work• Bayesian estimation of full parameter vector.

• Computational speed is a barrier.

• More comprehensive data.• A better model?


Literature

1. Firm dispersion and uncertaintyBloom (2007), Bloom, Floetteto, Jaimovich, Saporta-Eksten and Terry (2014),Bachmann and Bayer (2014), Schaal (2012), Jurado, Ludgvigson and Ng (2015),Vavra (2014), Christiano, Motto, Rosagno (2014), Gilchrist, Sim and Zakrajsek(2014), Berger, Dew-Becker and Giglio (2016), Bachmann and Moscarini (2012)

2. Estimated macro modelsSargent (1989), Ingram, Kocherlakota and Savin (1994), Ireland (2004), Smetsand Wouters (2007), Christiano, Eichenbaum and Evans (2001), Justiniano,Primiceri and Tambalotti (2011), Jermann and Quadrini (2012)

3. Heterogeneous agent macro modelsHopenhayhn (1992), Khan and Thomas (2008), Khan, Senga and Thomas(2014), Clementi and Palazzo (2016), Winberry (2016), Gomes (2001), Gomesand Schmid (2010), Eisfeldt and Muir (2016)

4. Linearizing heterogeneous agent modelsReiter (2009), Reiter (2010), McKay and Reis (2010), Winberry (2016), Ahn,Kaplan, Moll, Winberry and Wolf (2016)


Sales growth interquartile range

−.05

0

.05

.1

.15

.2

1990 1995 2000 2005 2010Quarter

IQR sales growthMedian sales growth

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Sales growth dispersion by industry

.1

.2

.3

.4

.5

.6

1990 1995 2000 2005 2010 2015Year

Mining&ConstructionManufacturing (A)Manufacturing (B)Services

IQR sales growth

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