The Expected Marginal Rate of Substitution in the United States
Transcript of The Expected Marginal Rate of Substitution in the United States
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The Expected Marginal Rate of
Substitution in the
United States and Canada
Andrew K. Rose
All materials (data sets, output, papers, slides) at:
http://faculty.haas.berkeley.edu/arose
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Two Objectives:
1. Derive new methodology to estimate and compare
the expected marginal rate of substitution (EMRS)
2. Illustrate technique empirically, and assess
integration of assets across markets
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The Paper in a Nutshell
1. Idiosyncratic Shocks are expected to earn expected
intertemporal marginal rate of substitution (EMRS)
2. There are LOTS of idiosyncratic shocks
o Noise is good, since it can be exploited
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Definition of Asset Integration
• Assets are integrated if satisfy asset-pricing condition:
)( 11j
tttj
t xmEp ++= (1)
• Completely standard general framework
• Note that mt+1 is the same for all j
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Paper Focus: Et(mt+1)
• Conditional Mean of Marginal Rate of Substitution/Stochastic
Discount Factor/Pricing Kernel/risk-free rate/zero-beta return
ties together all intertemporal decisions
• Subject of much research (Hansen-Jagannathan, etc.)
• Prices all assets (and intertemporal decisions!)
• Unobservable, even ex post (but estimable)
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Key:
• Should be identical for all assets in an integrated market
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Motivation: Who Cares about Integration and EMRS
• MRS is “DNA” of intertemporal economics
• Appears in RBC, new-Keynesian, and in between
• Whenever agents maximize an intertemporal utility function,
MRS is used
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Standard Macroeconomics
o Appears in IS curves that link interest rates and inflation
)*)(
*)(()1(
1
1
1
+
+=+ ttc
ttct
t qcuqcuE
iρ
Bonds/IS Curve
o Links prices with future firm revenues
)**)(
*)(( 11
1+
+
+= tttc
ttctt x
qcuqcuEp ρ Stocks/Investment
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In both Equations
1
11 *)(
*)(
+
++ =
ttc
ttct qcu
qcum ρ
• Is bond pricing integrated with stocks/investment-pricing?
• What arguments belong in IS curve?
• If stock and bond pricing are not integrated, different MRS
with possibly different arguments.
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International Finance
)*()1( 1
*1 ++=
+ tttt
t smEi
s Foreign-currency Bond, or
))1(*(1*
1*1
t
tttt s
ismE += +
+ Can rewrite as:
t
ttttt
t
tttt s
isEmEs
ismCOV )1(())1(*(1*
1*1
*1*
1+
++
= ++
++
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If domestic- and foreign-currency pricing is integrated,
)1(1)( 1
*1
ttttt i
mEmE+
== ++ then
)()1()1())1(*(1 1
**1*
1t
tt
t
t
t
tttt s
sEii
sismCOV ++
+ ++
++
=
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With lack of integration,however,
1*
1)( ++ ≠ tttt mEmE then
)()1()1())1(*(1 1
1
*1
**1*
1t
tt
tt
tt
t
t
t
tttt s
sEmEmE
ii
sismCOV +
+
+++ +
++
+=
1
*1
+
+=tt
ttt mE
mEθ is stochastic without integration
• Interpretation: domestic-currency bonds have higher liquidity
return than foreign-currency denominated bonds.
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• Rejection of UIP due ONLY to risk premium correlations?
o Or is 1≠tθ a factor also?
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Summary: Why Should we Care about EMRS?
o Links interest rates to inflation
o Links prices with future firm revenues
o Links leisure today with leisure tomorrow
o Links domestic and foreign asset prices (UIP deviations)…
• MRS of serious intrinsic interest
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Empirical Strategy
• Stocks have lots of noise and big cross-sections
Definition of Covariance/Expectation Decomposition:
).()(),()( 111111j
ttttj
tttj
tttj
t xEmExmCOVxmEp ++++++ +== (2)
Rearrange and substitute actual for expected x (WLOG):
,)](/1[),()](/1[ 111111j
tj
tttj
tttttj
t pmExmCOVmEx ++++++ ++−= ε
jt
jttt
jtt
jt xmCOVpx 1111 )),(( ++++ +−= εδ
(3)
where )(/1 1+= ttt mEδ and )( 11j
ttj
tt xEx ++ −≡ε
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3 Assumptions Traditionally Made for Estimation:
1) Rational Expectations: j
t 1+ε is assumed to be white noise,
uncorrelated with information available at time t,
2) Factor Model:
)1,1( jtxtmtCOV ++ =
it
ij
ij fββ Σ+0
, for the relevant sample,
3) Risk-Free Rate: Use Treasury-bill return for Et(mt+1)
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Two Approaches
An Asset Pricing/Factor Model is:
jti
it
jijtt
jt fpx 1
,1 )( ++ ++= ∑ εβδ
(4)
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Traditional Finance Asset Pricers: Use all 3 assumptions
• Normalize (4) by dividing by jtp
jti i
tfji
tij
tpj
tx 1),()1(/1 ++∑=+−+ εβ
• Delivers “good” estimates of factor loadings (β)
• Oriented towards estimating risk premia
• But no/poor estimates of Et(mt+1)
o It’s simply equated to T-bill! (alternatives
implausible/imprecise)
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New Approach
• Normalize (4) by dividing by jtp~ , defined as j
tp with
idiosyncratic part set to zero.
• Delivers estimates of EMRS, but no factor loadings at all!
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• Normalizing by jtp~ delivers:
jt
jt
jttt
jt
jtt
jt
jt pxmCOVpppx 1111 )]~/,()~/[(~/ ++++ +−= εδ
• First part (inside brackets) is an idiosyncratic function.
• Second part (covariance) a function of aggregate phenomena.
o Can therefore be ignored (as part of residual) without
affecting consistency of )(/1 1+= ttt mEδ
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Can estimate parameters of interest without covariance model!
• Adding Covariance (factor) model would improve efficiency of
estimating }{t
δ
o Potential Cost is inconsistency (mis-specified covariance
model)
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Notes
• Focus is on exploiting (not ignoring) idiosyncratic risk
o Idiosyncratic risk carries no risk premium
o Test involves estimating and comparing costs of carrying
purely idiosyncratic risk
• Don’t model covariances with factor model
o Instead substitute model of aggregate returns plus
orthogonality condition
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Strengths of the Methodology
1.Based on general intertemporal model
2.Do not model/parameterize MRS (with e.g., utility
function/consumption data); it varies arbitrarily
3.Requires only accessible, reliable data on prices, returns
4.Can be used at all frequencies
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5.Can be used for all types of assets
6.No special software required
7.Focus is on intrinsically interesting object, namely expectation
of marginal rate of substitution (EMRS)
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Empirical Implementation
• Cannot observe jtp~ ; must use observable empirical counterpart,
denoted jtp̂ .
• Use OLS to estimate J (= # assets) time-series regressions:
jtvtptpjbjaj
tpjtp +
−+=
−)1/ln(*)1/ln(
where tp is market-wide average price
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• Can then compute:
))1/ln(ˆˆexp(*1ˆ−
+−
≡ tptpjbjajtpj
tp
• No special attachment to this model; just need some model
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Estimation
• Equation to be estimated is linear:
jtuj
tpjtpt
jtpj
tx 1)ˆ/(ˆ/1 ++=+
δ
• May have non-trivial measurement error (hence inconsistency),
also generated regressor (hence incorrect standard errors)
• IV (using }{ tp as set of IVs) solves both problems; GMM too
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Data Sets
• Decade of monthly data (1994M1-2003M12)
• Year of daily data (2003)
o Could use different frequencies too
• American data from CRSP; Canadian (in $) from DataStream
o End-of-period prices and returns (with dividends)
o Use only firms with full span of data (selection bias?)
• Could use bonds/other assets …
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Monthly Data Set: 120 observations
• All 389 firms from S&P 500 traded on NYSE
• (Some firms from NASDAQ in S&P 500)
• 152 Firms from S&P/TSE index
Daily Data Set: 247 Business Days (both markets open)
• 440 firms from S&P 500 traded on NYSE
• 223 Firms from S&P/TSE index
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Portfolio Groupings
• Follow Finance tradition and group into sets of 20 portfolios
• Portfolios formed arbitrarily (alphabetical by ticker)
o Can use other grouping techniques (size/beta/…)
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Results
• Start with 400 firms from S&P 500 in Figure 1 (118 monthly
observations; lose observations because of lead/lag)
• First estimate EMRS with only 10 portfolios
o Plot mean, +/-2 standard error confidence interval
• 3 different estimation methods (OLS, GMM, IV)
o similar results
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What Does EMRS, }ˆ{δ , Look Like?
• Reasonable Mean (slightly over unity)
• Tight confidence intervals (estimation precision)
• Lots of time series volatility!
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0.5
1
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
Instrumental Variables
0.5
1
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
IV: Differences from last 10 Portfolios
0.5
1
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
Generalized Method of Moments
0.5
1
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
GMM: Differences from last 10 Portfolios
0.5
1
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
Ordinary Least Squares0
.51
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
OLS: Differences from last 10 Portfolios
Expected MRS, first 10 portfolios; S&P 500 1994M2-2003M11Deltas, with +/- 2 S.E. Confidence Interval
Figure 1: Estimated Expected MRS, Portfolios of 20 S&P500 firms, 1994M2-2003M11: Different Estimators
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Internal Integration
• Inside S&P 500, estimates of }ˆ{δ from different sets of (groups
of 10) portfolios similar
• Can test for joint equivalence with F-test
o Bootstrap because of non-normality (leptokurtosis)
o Cannot reject equality within S&P 500 portfolios, any
reasonable significance level
That is, do not reject integration
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Comparison with T-bill
• Similar means
• T-bills are much less volatile than EMRS
• Easily reject equality of EMRS and T-bill-equivalent
o F-test over 50!
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Other Markets
• 20 portfolios from NYSE (19 stocks) and TSE (7)
• Again, reasonable means, tight precision, much volatility
• Different estimators => similar results
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.81
1.2
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
NYSE
.81
1.2
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
TSE
.81
1.2
1994m1 1996m1 1998m1 2000m1 2002m1 2004m1
MRS implied by 3-month US T-bill
Expected MRS from 20 portfolios, IV, 1994M2-2003M11Deltas, with +/- 2 S.E. Confidence Interval
Figure 2: Estimates of Expected Marginal Rate of Substitution, 1994M2-2003M11: Different Markets
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Integration: Comparing EMRS Across Different Markets
• Estimate EMRS from portfolios of 19 stocks (20 from NYSE,
8 from TSE)
• Estimates of EMRS are positively correlated across markets
o Correlation of NYSE and TSE = .73
o But mean absolute error = .02; many > .1
o Can easily reject integration across markets
o F-tests > 8, strong rejection
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.8.9
11.
1
.8 .9 1 1.1
TSE against NYSE
.8.9
11.
1
.8 .9 1 1.1
US T-bill against NYSE
.8.9
11.
1
.8 .9 1 1.1
US T-bill against TSE
Monthly Data, 2004-2003
.96
.98
11.
021.
04
.96 .98 1 1.02 1.04
TSE against NYSE
.96
.98
11.
021.
04
.96 .98 1 1.02 1.04
US T-bill against NYSE
.96
.98
11.
021.
04
.96 .98 1 1.02 1.04
US T-bill against TSE
Daily Data, 2003
Expected MRS from 20 portfolios, IV
Figure 3: Scatter-plots of Estimated Expected MRS across Markets
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Daily Results
• Similar to monthly results
• Reasonable EMRS, precisely estimated, great volatility
• Internal integration, but easily reject integration across markets
• Strongly reject equality with T-bills (too smooth!); F-test > 150
• EMRS positively correlated across markets
o Still, easily reject integration across markets
o F-tests integration of NYSE w/ TSE > 17
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1.04
1.9
6
0 50 100 150 200 250
NYSE
1.04
1.9
6
0 50 100 150 200 250
TSE
1.04
1.9
6
0 50 100 150 200 250
MRS implied by 3-month US T-bill
Expected MRS from 20 portfolios, IV, Daily Data 2003Deltas, with +/- 2 S.E. Confidence Interval
Figure 4: Daily Estimates of Expected Marginal Rate of Substitution, 2003
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Future Agenda
• Adding Covariance Model?
• Different portfolio structure?
• Different model to estimate idiosyncratic risk?
o Different normalization?
• Forward-looking test for arbitrage profits from diverging
EMRS’s across markets, lack of equality with t-bill
• Explain reasons for lack of integration
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