taylor rule at work: evidence from disaggregated data

OverviewLiterature and preliminary results

ModelEmpirical Results

Conclusion

Taylor Rule at Work:

Evidence from Disaggregated Data

Olesya V. Grishchenko1, Abraham Lioui2, and Marco Rossi3

1Federal Reserve Board2EDHEC Business School3Notre Dame University

The views expressed here do not necessarily reflect those of the FederalReserve Board and Federal Reserve System

New Economic School 20th Anniversary ConferenceMoscow, Russia

December 14, 2012

Grishchenko-Lioui-Rossi/NES/December 14, 2012 1 / 21



Conclusion

Summary

This paper examines the impact of the monetary policy on aggregateequity prices

We propose the model in which agents value both consumption and realmoney balances.

We depart from the representative agent paradigm

We estimate the model using Consumption Expenditures Survey (CEX)data set of the individual consumers/stockholders

We find the much lower RRA coefficient than in the model with the repagent paradigm

Overall, disaggregated data seem to provide more support for a modelthan aggregate data




Conclusion

Roadmap of the talk

1 Literature

2 Model

3 Empirical methodology

4 Preliminary results

5 Conclusion




Conclusion

Literature: I

Consumption CAPM (C-CAPM) (Lucas (1978), Breeden (1979)) providesa coherent theoretical framework to explain the cross-section of stockreturns

The empirical fit of C-CAPM has been poor: Breeden, Gibbons,Litzenberger (1989), Mankiw, Shapiro (1986), Mehra, Prescott (1985),Lettau and Ludvigson (2001), Yogo (2006), Cochrane (2008)...

There are two simultaneous extensions that we propose to improve the fitof the C-CAPM and to address the economic source of such animprovement




Conclusion

Extension I: Monetary factor

Monetary variables and stock returns: Hess and Lee (1999), Flannery and

Protopapadakis (2002), Bernanke and Kuttner (2005), Baele, Bekaert,

and Inghelbrecht (2010)...

Bernanke and Kuttner (1995): 25 b.p. cut in the Fed Fundsrate is associated with 1% increase in the stock indicesFlannery and Protopapadakis (2002): Inflation and monetaryaggregates are priced

Extensions of the Lucas CCAPM to a monetary economy: Marshall

(1992), Bakshi and Chen (1997), Lioui and Maio (2012)...

Additional monetary factor is pricedHowever, the performance of CCAPM is not improved relativeto a standard CCAPM, judged by the implied coefficient ofrelative risk aversion




Conclusion

Extension II: Heterogeneous agents

Growing literature in favor of modeling preferences of heterogeneous

agents in many aspects:

labor income: Heaton, Lucas (1996), Storesletten, Telmer,Yaron (2004)...consumption: Mace (1991), Constantinides (2002), Brav,Constantinides, Gezcy (2002), Balduzzi, Yao (2007),Grishchenko, Rossi (2012)...money demand: Andres, Lopez-Salido, Nelson (2009)




Conclusion

Contribution: Key model features

We provide a model with the money-in-the-utility (MITU) approach

Theory: Sidrauski (1967), Walsh (2003)Empirical support for the presence of real balances in theutility: Koenig (1990)

We depart from the representative agent paradigm

BOTH in consumption and real money holdingspreferences are not separable over both variables, so IMRS is afunction of both

The short rate can be easily related to the households’ RRA coefficient

preferences are not separable over both variables, so IMRS is afunction of both




Conclusion

No-arbitrage conditions

The economy is populated by i = 1, ..., I households, which are assignedto cohorts

Households trade in frictionless capital markets a set of securitiesj = 1, ..., J with gross returns Rj,t+1

cohorts are based on households’ demographic characteristics, namely,education, age, and income and classified according to some endogenousstatistical procedure following Grishchenko and Rossi (2012)

Consumption and real money growth rates for cohort k gk,t+1 andgmk,t+1 are defined as:

gk,t+1 =ck,t+1

ck,tand gmk,t+1 =

mk,t+1

mk,t

, (1)

where ck,t+1(mk,t+1) are cohort per capita consumption(real moneybalances levels)




Conclusion

No-arbitrage conditions, cont.

Assuming a representative agent within a cohort, Euler equation for acohort k must hold:

E [M(gk,t+1, gmk,t+1)Rj,t+1|Ft ] = 1, (2)

where M(gk,t+1, gmk,t+1) is an SDF

If averaged across cohorts, the following Euler equation must hold:

E

[

1

K

K∑

k=1

M (gk,t+1, gmk,t+1)Rj,t+1|Ft

]

= 1 (3)




Conclusion

Preferences

Utility function is a CES:

u (ct ,mt) =1

1− γ[εcρt + (1− ε)mρ

t ]1−γ

ρ (4)

Then a CES-based Euler equation is:

Et

[

β

(

εcρ

t+1 + (1− ε)mρ

t+1

εcρ

t + (1− ε)mρ

t

)

1−γ

ρ−1(

ct+1

ct

)

−γ

Rj,t+1

]

= 1 (5)

There is an additional factor in the Euler equation, real balances toconsumption ratio




Conclusion

Preferences, cont.

Two problems with Euler equation (5):

Measurement of real balances: M1, M2 in aggregate set up?Approximation of the SDF to see the importance ofconsumption growth factor

Thus we use a money demand equation, which relates the optimalconsumption to optimal money balances:

it

1 + it=

um (ct ,mt)

uc (ct ,mt)(6)

LHS: the opportunity cost of holding money

RHS: intratemporal MRS between real money and consumption




Conclusion

Preferences, cont.

In the CES set up Eq. (6) is:

it

1 + it=

(1− ε)mρ−1t

εcρ−1t

⇔mt

ct=

(

it

1 + it

) 1ρ−1

(

1− ε

ε

)

−

1ρ−1

(7)

Euler equation (5) becomes then:

Et

β

1 +(

it+1

1+it+1

)ρ

ρ−1 ( 1−ε

ε

)

−

1ρ−1

1 +(

it1+it

)ρ

ρ−1 ( 1−ε

ε

)

−

1ρ−1

1−γ

ρ−1

(

ct+1

ct

)

−γ

Rj,t+1

= 1 (8)

Eq. (8) is a basis for our empirical investigation.




Conclusion

A special case of the model

ρ = 0 yields Cobb-Douglas utility function:

u(ct ,mt) =1

1− γ

(

cǫ

t m1−ǫ

t

)1−γ

, (9)

This yields the following SDF:

SDFt,t+1 = β

(

ct+1

ct

)

−γ( it+1

1+it+1

it1+it

)−(1−ǫ)(1−γ)

. (10)

We test the model using this specification




Conclusion

Data

Consumption Expenditure Survey (CEX)

disaggregated consumption data based on households surveyed

interviews are conducted every month

households are interviewed for 4 quarters

and report consumption over the preceding quarter

we use quarterly consumption growth of nondurables and services atmonthly frequency

our sample period: Jan 1984 - Dec 2010

Market data

Asset returns: 6 size and BM Fama-French portfolios

Risk-free rate: return on a 30-day Treasury Bill portfolio

Federal Funds rate (for SDF)

Returns and consumption are expressed in real terms




Conclusion

Consumption growth measurement

Why not aggregate consumption?

Aggregate consumption series are too smooth, thus fit is poorand RRA is unrealistically high even with additional pricedfactor: Lioui, Maio (2012, JFQA)

Why not individual consumption?

available individual consumption series are short (12 obs perHH in CEX)reported consumption series are noisy: Brav, Constantinides,and Gezcy (2002, JPE), Balduzzi and Yao (2007, JME)




Conclusion

Consumption growth measurement, cont.

We choose something in between individual and aggregate consumption

We mitigate the measurement error by working with syntheticcohorts of households, more precisely, with clustersHouseholds are classified into cohorts (clusters) based oneducation, age, and incomeClassification is endogenous, we only decide on the number ofclusters, KGrishchenko and Rossi (2012, JBES) find that 9 clusters areoptimal to match the equity premium with the RRA coefficientroughly equal to 6

Money growth is substituted out by the function of the fed funds rate




Conclusion

CEX data: Summary statistics

Cluster Observations, N Sample Statistics

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Mean Standard Deviation

Min Med Max Min Mean Max Min Mean Max

3 85 284 796 0.9324 0.9958 1.0670 0.0014 0.0183 0.0545

9 1 104 530 0.8884 0.9961 1.1194 0.0189 0.0725 0.3195

15 1 62 316 0.8920 1.0045 1.1411 0.0268 0.1127 0.3705

21 1 45 288 0.8939 1.0073 1.1349 0.0353 0.1379 0.3766

27 1 33 261 0.9033 1.0074 1.1389 0.0504 0.1461 0.4188

30 1 29 259 0.9056 1.0076 1.1308 0.0524 0.1506 0.3993




Conclusion

Equity premium via CEX data

Clusters 3 9 18γ UEP p-val UEP p-val UEP p-val

0 1.52 0.00 1.52 0.00 1.52 0.00

2 1.55 0.00 1.49 0.01 1.48 0.01

4 1.59 0.00 1.32 0.04 1.28 0.04

61.65 0.00 0.39 0.40 -0.33 0.58

8 1.71 0.00 -4.56 0.78 -10.00 0.91




Conclusion

Preliminary model estimation results

Estimate unconditional Euler equations (9) with a monetary factor via atwo-step GMM

Parameters to be estimated: ǫ and γ

Assets in the estimation: 6 Fama-French portfolios

Parameter Coeff p-val LM estimates

γ 8.62 0.9994 88.36ǫ 1.24 0.9971 0.84J-stat 0.1333Prob(χ2(4) > J) 0.9979

γ is more reasonable compared with studies based on aggregate data

ǫ overloads on consumption

coefficients are not well identified in the unconditional set up

Model is not rejected overall




Conclusion

Conclusion

Introduce a model with disaggregated money and consumption growthfactors

Use of fed funds rate data to substitute out money growth

In our model, RRA is related to the short rate, and therefore, to themonetary policy and implicitly to the Taylor rule function

Use clusters of household consumption data to substitute out individualHH growth rates and mitigate a measurement error

Estimated risk aversion coefficient is much lower than in an otherwisesimilar model with aggregate consumption data




Conclusion

Work in progress

Calibrate the model in addition to GMM estimation

Estimate a full-fledged model with ρ 6= 0

Estimate a model with linearized SDF: further mitigate the effect of themeasurement error present in CEX data and avoid estimating highlynonlinear functions of consumption and money growths

Include explicitly Taylor Rule function as a part of Euler conditions


taylor rule at work: evidence from disaggregated data

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