THE LUCAS MODEL OF ASSET PRICING: EXPERIMENTS

MotivationExperimental Setup

PredictionsResults

Pretending To Analyze Historical DataConclusion

THE LUCAS MODEL OF ASSET

PRICING: EXPERIMENTS

Elena Asparouhova, Peter Bossaerts,Nilanjan Roy, William Zame

Special Topics in Finance, Melbourne May 2014

Asparouhova, e.a. Lucas Experiments

PredictionsResults

Motivation

Why experiments on the Lucas asset pricing model?

underlies most of theoretical macro-financegives clean predictions

cross-sectionalintertemporalmutually reinforcing

tests with historical data assume equilibriumfocus on parametric variations (preferences,consumption, dividends...) of “stochastic Eulerequations”weak empirical supportexperiments can inform us about where the the modelworks and where it potentially fails

PredictionsResults

Motivation

underlies most of theoretical macro-finance

gives clean predictionscross-sectionalintertemporalmutually reinforcing

PredictionsResults

Motivation

PredictionsResults

Motivation

tests with historical data assume equilibrium

focus on parametric variations (preferences,consumption, dividends...) of “stochastic Eulerequations”weak empirical supportexperiments can inform us about where the the modelworks and where it potentially fails

PredictionsResults

Motivation

tests with historical data assume equilibriumfocus on parametric variations (preferences,consumption, dividends...) of “stochastic Eulerequations”

weak empirical supportexperiments can inform us about where the the modelworks and where it potentially fails

PredictionsResults

Motivation

tests with historical data assume equilibriumfocus on parametric variations (preferences,consumption, dividends...) of “stochastic Eulerequations”weak empirical support

experiments can inform us about where the the modelworks and where it potentially fails

PredictionsResults

Motivation

PredictionsResults

(Stochastic Euler Equations)

{∂ui (ci (t+1))

∂c∂ui (ci (t))

[p(t + 1) + d(t + 1)

]|I (t)

}= p(t),

Take historical price and consumption data

Fit equations for a choice of utility and information

PredictionsResults

{∂ui (ci (t+1))

∂c∂ui (ci (t))

[p(t + 1) + d(t + 1)

]|I (t)

}= p(t),

PredictionsResults

{∂ui (ci (t+1))

∂c∂ui (ci (t))

[p(t + 1) + d(t + 1)

]|I (t)

}= p(t),

PredictionsResults

Some Objections

Why lab test test a model that is ‘obviously wrong’ in the field?

Why lab test a model that is ‘obviously right’ in the field?

Why is the model wrong in the field?

Models are idealizations; the laboratory is an opportunityto test them in an idealized environment.

PredictionsResults

Some Objections

PredictionsResults

Some Objections

PredictionsResults

Some Objections

PredictionsResults

Objections (c’d)

“The model relies on risk aversion. Nobody should be riskaverse in the lab, so the model cannot possibly be right inthe lab.”

People are risk averse in the lab. This is a fact.

Whether this is decreasing marginal utility, as in thetheory, remains to be seenBut decreasing marginal utility explains phenomena atthe market level

Important message from our work:individual 6↔ market

Contrast economic thinking/social choice thinking

PredictionsResults

Objections (c’d)

PredictionsResults

Objections (c’d)

Whether this is decreasing marginal utility, as in thetheory, remains to be seen

But decreasing marginal utility explains phenomena atthe market level

PredictionsResults

Objections (c’d)

PredictionsResults

Objections (c’d)

PredictionsResults

Objections (c’d)

PredictionsResults

How one SHOULD think about the experiments

(We think)

We will observe many features that are “defining” for theLucas model (Pareto efficiency, cross-sectional andintertemporal pricing)Yet at the same time we observe phenomena that areexactly like in the field:

Excess volatilityIndividuals hardly behave as predicted in the theory

... without having to invoke design elements that areclaimed to be the reason for these phenomena in the field

Institutions (intermediaries, governments,...), Stochastics(ambiguity, rare events,...), Constraints (incompletemarkets, collateral, indivisibilities, ...)

PredictionsResults

(We think)

We will observe many features that are “defining” for theLucas model (Pareto efficiency, cross-sectional andintertemporal pricing)

Yet at the same time we observe phenomena that areexactly like in the field:

PredictionsResults

(We think)

PredictionsResults

(We think)

PredictionsResults

(We think)

PredictionsResults

What we learn...

Relative prices are correct, and intertemporally pricesmove with fundamentals

... but fundamentals explain only 18% of price changes(excess volatility)

Still, substantial Pareto improvements to autarky

Subject price forecasts are “almost” fulfilled

PredictionsResults

What we learn...

PredictionsResults

What we learn...

PredictionsResults

What we learn...

PredictionsResults

What we learn...

PredictionsResults

Take away...

Messages:

For theorists: Investigate equilibria where agents makesmall forecast errors... they look very different fromLucas!For empiricists: Euler equations might be misguided(because they assume prices are functions offundamentals only)For policy: excess volatility does not stand in the way ofsignificant Pareto improvements

PredictionsResults

Take away...

Messages:

PredictionsResults

Take away...

Messages:

For theorists: Investigate equilibria where agents makesmall forecast errors... they look very different fromLucas!

For empiricists: Euler equations might be misguided(because they assume prices are functions offundamentals only)For policy: excess volatility does not stand in the way ofsignificant Pareto improvements

PredictionsResults

Take away...

Messages:

For theorists: Investigate equilibria where agents makesmall forecast errors... they look very different fromLucas!For empiricists: Euler equations might be misguided(because they assume prices are functions offundamentals only)

For policy: excess volatility does not stand in the way ofsignificant Pareto improvements

PredictionsResults

Take away...

Messages:

PredictionsResults

Experiments vs. Theory (vs. Reality)

Standard treatment of the Lucas Model starts with Paretoefficient allocations

(the equilibrium equations are really first-order conditionsof the representative agent that one can constructbecause of Pareto efficiency!)

what economic structure could potentially generatePareto efficiency... while allowing for heterogeneity acrossagents?

impossible to have a complete set of marketsmaybe use dynamic completeness and induce a Radnerequilibrium? (Duffie-Huang [1985])

PredictionsResults

impossible to have a complete set of markets

maybe use dynamic completeness and induce a Radnerequilibrium? (Duffie-Huang [1985])

PredictionsResults

The EconomyPrice FormationExperimental TimelineMore Design

Setting

stationary (in dividend levels!), infinite horizon

two long-lived assets

tree: pays $0 (bad state) $1 (good state) each periodprobability p = 0.5 (i.i.d.)

bond : pays $0.50 each period

PredictionsResults

Setting

PredictionsResults

Setting

PredictionsResults

two (types of) infinitely lived agents

endowmentstype I

10 trees, 0 bondsincome: 15 even periods, 0 odd periods

type II

0 trees, 10 bondsincome: 15 odd periods, 0 even periods

PredictionsResults

endowments

type I

type II

PredictionsResults

endowmentstype I

type II

PredictionsResults

endowmentstype I

type II

PredictionsResults

Why these parameters ???

stationary-in-levels (time-invariant) aggregates

promote trade (otherwise bubbles – Duffy and Crockett[2013])

(may restrict shortsales)

PredictionsResults

How Will Prices Be Formed? Trade Through

Continuous Electronic Open Book...

PredictionsResults

How Will Prices Be Formed? Trade Through

Continuous Electronic Open Book...

PredictionsResults

(Graphical Display Of Book Of Orders)

PredictionsResults

(Graphical Display Of Book Of Orders)

PredictionsResults

Experimental timeline

Period 1 Period 2 Period 3

Dividendsfrom initial allocationof “Trees” and “Bonds” Income

Tradeto a nal allocationof “Trees,” “Bonds,” andCASH (=consumption)

Consumption

Dividendsfrom carried over allocationof “Trees” and “Bonds” Income

Consumption

PredictionsResults

Experimental timeline

Consumption

PredictionsResults

Novel Design Solutions

Problems:

discounting in the lab

consumption (“cash”) smoothing in the lab

infinite horizon in the lab

stationarity: end of lab session...

Solutions:

random termination

pay subjects only cash of last period (intermediate payoffsare forfeited)

termination rule: at -10 minutes: reduce to 2-periodsending probabilities = 1/6, 5/6

(exploits separability, iid dividends)

PredictionsResults

Problems:

Solutions:

random termination

PredictionsResults

Problems:

Solutions:

random termination

PredictionsResults

Problems:

Solutions:

random termination

PredictionsResults

Problems:

Solutions:

random termination

PredictionsResults

Problems:

Solutions:

random termination

PredictionsResults

Problems:

Solutions:

random termination

PredictionsResults

Problems:

Solutions:

random termination

(exploits separability, iid dividends)Asparouhova, e.a. Lucas Experiments

PredictionsResults

Back To Experimental Timeline

Appendix: Time Line Plot To Complement In-

structions

Tradeto a !nal allocationof “Trees,” “Bonds,” andCASH

Possible Termination of Session*If termination--keep CASH*If continuation--lose CASH, carry over “Trees” and “Bonds”

References

Klaus Adam, Albert Marcet, and Juan Pablo Nicolini. Stock market volatility and

learning. Working paper, 2012.

Elena Asparouhova, Peter Bossaerts, and Charles Plott. Excess demand and equi-

libration in multi-security financial markets: The empirical evidence. Journal of

Financial Markets, 6:1–21, 2003.

Ravi Bansal and Amir Yaron. Risks for the long run: A potential reso-

lution of asset pricing puzzles. The Journal of Finance, 59(4):1481–1509,

2004. ISSN 1540-6261. doi: 10.1111/j.1540-6261.2004.00670.x. URL

http://dx.doi.org/10.1111/j.1540-6261.2004.00670.x.

Nicholas Barberis, Ming Huang, and Tano Santos. Prospect theory and asset prices.

The Quarterly Journal of Economics, 116(1):1–53, 2001. ISSN 00335533. URL

http://www.jstor.org/stable/2696442.

PredictionsResults

Back To Experimental TimelineAppendix: Time Line Plot To Complement In-

structions

References

Klaus Adam, Albert Marcet, and Juan Pablo Nicolini. Stock market volatility and

learning. Working paper, 2012.

Elena Asparouhova, Peter Bossaerts, and Charles Plott. Excess demand and equi-

libration in multi-security financial markets: The empirical evidence. Journal of

Financial Markets, 6:1–21, 2003.

Ravi Bansal and Amir Yaron. Risks for the long run: A potential reso-

lution of asset pricing puzzles. The Journal of Finance, 59(4):1481–1509,

2004. ISSN 1540-6261. doi: 10.1111/j.1540-6261.2004.00670.x. URL

http://dx.doi.org/10.1111/j.1540-6261.2004.00670.x.

Nicholas Barberis, Ming Huang, and Tano Santos. Prospect theory and asset prices.

The Quarterly Journal of Economics, 116(1):1–53, 2001. ISSN 00335533. URL

http://www.jstor.org/stable/2696442.

PredictionsResults

Equilibrium NotionPricesNumerical Example – Homogeneity

(Radner) Equilibrium

asset prices p and consumption (“cash”) plans c I , c II

such that

agents optimize subject to budget constraintsmarkets clear

Agents know structure of uncertainty, true probabilities,asset payoffs, own endowments, utility function, currentasset prices

... and future asset prices

Equilibrium assumes perfect/correct forecasts!

PredictionsResults

such that

PredictionsResults

such that

PredictionsResults

such that

PredictionsResults

such that

PredictionsResults

such that

PredictionsResults

Prices – Allowing For Heterogeneity

cross-sectionaltree is less expensive than bondtree expected rate of return higher than bond

higher consumption betacaution: this is an equilibrium prediction

intertemporalprice levels correlate perfectly and positively withfundamentals(dividends of tree)

cross-sectional and intertemporal predictions reinforceeach other(countercyclical equity premium, or cyclical discount ofTree price relative to Bond price)

PredictionsResults

cross-sectional and intertemporal predictions reinforceeach other

(countercyclical equity premium, or cyclical discount ofTree price relative to Bond price)

PredictionsResults

Allocations

dynamic completeness:

Pareto optimality

smoothing: agents fully insure income fluctuationsdiversification: consumption is positively (rank)correlated across states (high, low dividend on tree)(If homothetic utilities: consumption shares are constantacross states/periods)

(price risk is hedged)

PredictionsResults

Allocations

dynamic completeness:Pareto optimality

PredictionsResults

Allocations

smoothing: agents fully insure income fluctuations

diversification: consumption is positively (rank)correlated across states (high, low dividend on tree)(If homothetic utilities: consumption shares are constantacross states/periods)

PredictionsResults

Allocations

smoothing: agents fully insure income fluctuationsdiversification: consumption is positively (rank)correlated across states (high, low dividend on tree)

(If homothetic utilities: consumption shares are constantacross states/periods)

PredictionsResults

Allocations

PredictionsResults

Allocations

PredictionsResults

Homogeneous Log Utility, β = 5/6

Prices and returns – Tree cheaper; Both assets cheaper inLow state; Countercyclical equity premium andpro-cyclical discount

State Tree Bond Price EquityPrice Return Price Return Discount Premium

High (H) $2.50 3.4% $3.12 -0.5% $0.62 3.9%Low (L) $1.67 55% $2.09 49% $0.42 6%

PredictionsResults

Prices and returns – Tree cheaper; Both assets cheaper inLow state; Countercyclical equity premium andpro-cyclical discount

State Tree Bond Price EquityPrice Return Price Return Discount Premium

High (H) $2.50 3.4% $3.12 -0.5% $0.62 3.9%Low (L) $1.67 55% $2.09 49% $0.42 6%

PredictionsResults

Holdings and trading: Type I (receives income in Evenperiods and buys Trees to hedge price risk)

Period Tree Bond (Total)Odd 7.57 0.62 (8.19)Even 2.03 7.78 (9.81)(Trade in Odd) (+5.54) (-7.16) (-1.62)

PredictionsResults

Holdings and trading: Type I (receives income in Evenperiods and buys Trees to hedge price risk)

Period Tree Bond (Total)Odd 7.57 0.62 (8.19)Even 2.03 7.78 (9.81)(Trade in Odd) (+5.54) (-7.16) (-1.62)

PredictionsResults

Prices: Cross-SectionalPrices: TemporalConsumption Across TypesPrice HedgingIndividual Choices

Sessions/Replications

Session Place Replication Periods SubjectNumber (Total, Min, Max) Count

1 Caltech∗ 4 (14, 1, 7) 162 Caltech 2 (13, 4, 9) 123 UCLA∗ 3 (12, 3, 6) 304 UCLA∗ 2 (14, 6, 8) 245 Caltech∗ 2 (12, 2, 10) 206 Utah∗ 2 (15, 6, 9) 24

(Overall) 15 (80, 1, 10)

(Starred sessions ended with prematurely halted replication)

PredictionsResults

Sessions/Replications

Session Place Replication Periods SubjectNumber (Total, Min, Max) Count

1 Caltech∗ 4 (14, 1, 7) 162 Caltech 2 (13, 4, 9) 123 UCLA∗ 3 (12, 3, 6) 304 UCLA∗ 2 (14, 6, 8) 245 Caltech∗ 2 (12, 2, 10) 206 Utah∗ 2 (15, 6, 9) 24

(Overall) 15 (80, 1, 10)

(Starred sessions ended with prematurely halted replication)

PredictionsResults

Tree cheaper; Both assets cheaper in low state;

But discount counter-cyclical

Data Tree Bond DiscountPrice Price (Bond - Tree)

Mean 2.75 3.25 0.50St. Dev. 0.41 0.49 0.40

High (State) 2.91 3.34 0.43Low (State) 2.66 3.20 0.54

Difference 0.24 0.14 -0.11across states

PredictionsResults

Tree cheaper; Both assets cheaper in low state;

But discount counter-cyclical

Data Tree Bond DiscountPrice Price (Bond - Tree)

Mean 2.75 3.25 0.50St. Dev. 0.41 0.49 0.40

High (State) 2.91 3.34 0.43Low (State) 2.66 3.20 0.54

Difference 0.24 0.14 -0.11across states

PredictionsResults

Discount (of tree price) and price differential

across states are positively correlated

Correlation is between the average (per replication)difference between bond and tree price, and the average(per replication) difference of prices (of a security)between high and low states.

Tree Bond

Correlation 0.80 0.52(St. Err.) (0.40) (0.40)

PredictionsResults

Discount (of tree price) and price differential

across states are positively correlated

Correlation is between the average (per replication)difference between bond and tree price, and the average(per replication) difference of prices (of a security)between high and low states.

Tree Bond

Correlation 0.80 0.52(St. Err.) (0.40) (0.40)

PredictionsResults

Prices move with fundamentals – but noisily

10 20 30 40 50 60 700

Period

1 2 3 4 5 6

PredictionsResults

Prices move with fundamentals – but noisily

10 20 30 40 50 60 700

Period

1 2 3 4 5 6

PredictionsResults

Apparent trend is not significant once allowing for

influence of state (change)

Table 10: OLS regression of changes in period-average transaction prices. (⇤ = significant

at p = 0.05.)

Explanatory Tree Price Change Bond Price Change

Variables Estim. (95% Conf. Int.) Estim. (95% Conf. Int.)

Change in State Dummy

(None=0; High-to-Low=-1, 0.19⇤ (0.08, 0.29) 0.10 (-0.03, 0.23)

Low-to-High=+1)

R2 0.18 0.04

Autocor. (s.e.=0.13) 0.18 -0.19

Closer inspection of the properties of the error term did reveal substantial depen-

dence over time, despite our including dummies to mitigate time series e↵ects. Table 9

shows Durbin-Watson (DW) test statistics with value that correpond to p < 0.001.

Proper time series model specification analysis revealed that the best model involved

first di↵erencing price changes, e↵ectively confirming the stochastic drift evident in

Figure 2 and discussed before. All dummies could be deleted, and the highest R2 was

obtained when explaining (average) price changes as the result of a change in the state.

See Table 10.13 For the Tree, the e↵ect of a change in state from Low to High is a

significant $0.19 (p < 0.05). The e↵ect of a change in state on the Bond price remains

insignificant, however (p > 0.05). The autocorrelations of the error terms are now

acceptable (marginally above their standard errors).

At 18%, the explained variance of Tree price changes (R2) is high. In theory,

one should be able to explain 100% of price variability. But prices are excessively

volatile, as already discussed in connection with Figure 2. Overall, the regression in

first di↵erences shows that, consistent with the Lucas model, fundamental economic

forces are behind price changes, significantly so for the Tree. But at the same time,

prices are excessively volatile, with no distinct drift.

Figure 3 displays the evolution of price changes, after chronologically concatenating

all replications for all sessions. Like in the data underlying the regression in Table 10,

the plot only shows intra-replication price changes. The period state (=1 if Low; 2 if

High) is plotted on top.

13We deleted observations that straddled two replications. Hence, the results in Table 10 are solely based

on intra-replication price behavior. The regression does not include an intercept; average price changes are

insignificantly di↵erent from zero.