Hidden Information
Robert A. Miller
47-901 Advanced Economic Analysis
Miller (47-901 Advanced Economic Analysis)
January 2017
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January 2017
1 / 53
Motivation
Background
SOX is a legislative response taken in 2002 by the U.S. government
and applied to almost all public …rms.
SOX attempts to restore investors’con…dence mainly by improving
disclosure quality
corporate governance
Compensation is a crucial mechanism in corporate governance and
thus an important angle in any overall evaluation of SOX.
Miller (47-901 Advanced Economic Analysis)
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Motivation
Research Questions
Unanswered questions in literature:
Through which channels did SOX a¤ect CEO compensation?
This paper explicitly attributes SOX e¤ect to changes in di¤erent types
of agency cost and factors driven by changes in primitives.
Measures are derived from a dynamic principal-agent model of moral
hazard and hidden information.
To what extent did SOX a¤ect CEO compensation?
This paper quanti…es and compares the contribution of each channel
literature
Miller (47-901 Advanced Economic Analysis)
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Data
Data Sources
Main sources: ExecuComp, CRSP and Compustat.
Sample: S&P1500 …rms, 1993 to 2005.
12 …rm types (denoted as Znt for …rm n in period t):
three sectors (primary, consumer goods and services) based on GICS
code
2 2 categories in each sector de…ned by size (A: total assets) and
capital structure (C: debt/equity) with indicators L and S
…rm characteristics
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Data
Key Variables: Private States
Accounting return acc_retnt for …rm n in period t:
acc_retnt =
Assetsnt Debtnt + Dividendnt
Assetsn,t 1 Debtn,t 1
Proxy of private state snt 2 f1, 2g (CEOs observe and report):
snt
1 (bad) if acc_retnt < mean(acc_ret j Z )
2 (good) otherwise
Miller (47-901 Advanced Economic Analysis)
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Data
Key Variables: Compensation
Performance measure:
gross abnormal return xnt
e
xnt + wnt /Vn,t
distribution estimated using kernel density.
1,
with probability
Total compensation (w
ent ):
change in wealth (Antle and Smith 1985, 1986, Hall and Liebman
1998, Margiotta and Miller 2000, etc.).
Optimal compensation: a function of gross abnormal return
conditional on bond price, public and private states
N
wt (xn jZ , S, bt ) =
∑m =1 wemt I fZm =Z ,Sm =S gK ( xmthx xnt )K
N
∑m =1 I fZm =Z ,Sm =S gK ( xmthx xnt )K
bm bt
hb
bm bt
hb
bt (bond price) = the present value of an annuity of $1 Treasury Bill
paid for 30 years.
Miller (47-901 Advanced Economic Analysis)
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Testing for Structural Change
Compensation (Table 1)
Accounting matters: compensation in bad states is lower than that in good states.
Primary sector: CEO compensation signi…cantly increased after SOX.
Sox compressed compensation is all sectors
Sector
(A,C)
(S,S)
(S,L)
Primary
(L,S)
(L,L)
(S,S)
(S,L)
Consumer
Goods
(L,S)
(L,L)
(S,S)
(S,L)
Service
(L,S)
(L,L)
Miller (47-901 Advanced Economic Analysis)
Pre
521
(8244)
-29
(7356)
3429
(10015)
3368
(10847)
-1203
(15811)
-567
(11016)
1569
(23109)
3858
(21787)
454
(14951)
1814
(13211)
5351
(30923)
6666
(22752)
Bad
Post
1526
(12516)
2432
(5528)
6849
(14023)
6586
(13337)
-1594
(11796)
65
(8268)
2279
(26279)
5588
(25096)
558
(13127)
2600
(12876)
5502
(29914)
6964
(19732)
t/F
1.3
0.4
4.0
1.8
2.7
0.5
3.9
0.7
-0.4
1.8
0.5
1.8
0.3
0.8
0.9
0.8
0.1
1.3
0.6
1.1
0.1
1.1
0.3
1.3
Auctions and Contracts 6
Pre
4096
(15743)
3453
(9835)
6501
(14415)
6015
(12867)
7301
(27128)
4494
(16528)
12432
(38644)
12923
(31014)
6929
(25125)
5145
(19218)
18610
(46350)
14321
(29764)
Good
Post
7206
(14913)
6299
(11433)
9607
(16390)
10569
(17228)
7494
(22425)
4528
(13700)
13560
(33589)
16759
(35861)
7782
(23095)
6497
(23743)
17536
(37877)
16451
(33072)
t/F
2.7
1.1
2.3
0.7
2.0
0.8
4.1
0.6
0.1
1.5
0.0
1.5
0.3
1.3
1.4
0.7
0.7
1.2
0.5
0.7
-0.3
1.5
1.1
0.8
January 2017
7 / 53
Testing for Structural Change
Two Types of Change
A structural change in compensation occurs if
the distribution of abnormal returns changes
the relationship between abnormal returns and CEO compensation
changes
We need to test for equality, between the pre-and post-SOX eras, of
the probability density functions for gross abnormal returns
the shape of the compensation schedule
Miller (47-901 Advanced Economic Analysis)
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Testing for Structural Change
Change in the distribution of abnormal returns
Denote the set of 24 categorical variables (formed from 3 sectors, 2
…rm sizes, 2 capital structures, and 2 accounting states) by Z ,
Let fpre (xnt jznt ) denote the probability density function of abnormal
returns in the pre-Sox era conditional on znt 2 Z .
Also de…ne fpost (xnt jznt ) in a similar manner.
Under the null hypothesis of no change fpre (x jz ) = fpost (x jz ) for all
(x, z ) 2 R Z .
Li and Racine (2007, page 363) propose a one-sided test for the null,
in which the test statistic is asymptotically distributed standard
normal.
nonp results
Miller (47-901 Advanced Economic Analysis)
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Testing for Structural Change
Change in the shape of the contract:
Let wpre (xnt , znt ) denote CEO compensation as a function of
(xnt , znt ) in the pre-SOX era
and similarly de…ne wpost (xnt , znt ) in the post-SOX era.
A straightforward way of testing whether the two mappings are equal
is to include an indicator variable for the post-SOX regime in
nonparametric regressions of compensation on the gross abnormal
return xnt for each znt .
The one-sided test of the null hypothesis of equality is asymptotically
standard normal.
nonp results
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Testing for Structural Change
Nonparametric Tests Results (Table 2)
A: Test on PDF of Gross Abnormal Returns
Sector
(A,C)
(S,S)
(S,L)
(L,S)
(L,L)
Primary
Bad
Good
18.05
10.34
5.88
5.02
3.29
4.16
29.46
8.57
consumer
Bad
Good
12.51
12.39
1.26
2.27
3.74
2.03
9.03
8.68
B: Test on Contract Shape
Sector
Firm Type
(S,S)
(S,L)
(L,S)
(L,L)
Primary
Bad
Good
10.06
1.58
6.82
6.45
19.67
7.34
10.32
23.38
backnonp result
Service
Bad
Good
14.25
14.55
14.70
5.29
9.01
19.69
71.68
29.56
back1nonp result
Consumer
Bad
Good
2.89
1.09
3.30
1.71
5.51
3.52
3.69
6.74
Service
Bad
Good
1.54
1.47
4.08
6.85
5.66
8.74
7.37
10.65
Note: Statistics in both one-sided tests follow N(0,1).
The critical value of 5% con…dence level is 1.64.
Changes occur in both tests in almost all cases among the 3 sectors.
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Testing for Structural Change
Nonparametric density and compensation schedules
Miller (712-010 Structural Econometrics)
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12 / 53
Testing for Structural Change
Where we need a model
Was the reaction in compensation design a response to
the new distribution of abnormal returns?
CEO function changes?
We estimate a dynamic principal-agent model of CEO compensation
which is built on primitives that might change with the
implementation of SOX.
Miller (47-901 Advanced Economic Analysis)
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Model
Four Key Features
(1) Moral hazard is one contracting friction
- Hidden action is the primal force of using equity-based compensation (Margiotta
and Miller 2000, Gayle and Miller 2009)
(2) Hidden information is an additional contracting friction
- CEOs bene…t from private information of …rm perspective (Gayle and Miller 2015)
(3) Accounting is a channel for CEOs to reveal private information
(4) Long-term contract is decentralized to a series of short-term contracts
- US corporation governance has been quickly reacting to legal and economic
changes (Holmstrom and Kaplan 2003)
Miller (47-901 Advanced Economic Analysis)
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Model
Timeline
t
t+1
(1) Board
(2) CEO decides
(3) CEO observes
proposes a
whether to stay
…rm’s prospect s
an e¤ort level
…rm’s asset is
compensation
in the …rm or
from {good, bad}
from {work, shirk}
realized and
schedule
leave; picks real
and reports their
w(r, x)
consumption
private information
on state as r(s)
(4) CEO chooses
(5) Return on
CEO gets
paid at
w(r(s), x)
from {good, bad}
Miller (47-901 Advanced Economic Analysis)
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Model
CEO’s Indirect Utility in a one period model
In a one period model the CEO optimally chooses from {reject, shirk, work}
to maximize
E [U j action, report]
8
>
>
>
<
>
>
>
:
1
2
∑ s =1
ϕst
(
b
α1tt
2
∑s =1 ϕst α2t
1
1
R∞
x
bt
1
1
exp
R∞
x
exp
γt w r (s )t (x )
t
b t +1
γt w r (s )t (x )
t
b t +1
g st (x )fst (x )dx
)
fst (x )dx
if reject
if shirk
if work
the probability of privately observed state st 2f1,2 g
the coe¢ cient of absolute risk aversion
αjt : e¤ort cost parameter for e¤ort choice j 2f1,2 g
fst (x ): density of performance measure (x ) on the equilibrium path
g st (x ): likelihood ratio of the density of x o¤ the equilibrium path over
tech assumptions
that on the equilibrium path
b t : the price of a bond that pays a unit of consumption each period
from period t onwards, relative to the price of a unit of consumption in
period t .
ϕst :
γt :
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Measuring SOX E¤ects
Framework
τ 1t
: optimal compensation if there is neither hidden action nor hidden information.
yst (x )
: optimal compensation if there is hidden action only.
Miller (712-010 Structural Econometrics)
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optimal contract
November 2016
17 / 53
Optimal Contract
Shareholders solve two problems
Minimize expected compensation subject to
- participation constraint (in both states):
E [U j work, honest] outside option 0
- incentive compatibility constraint (in both states):
E [U j work, honest] E [U j shirk, honest] 0
- truth-telling constraint (in good state):
E [U j work, honest] E [U j work, under-report]
- sincerity constraint (in good state):
E [U j work, honest] E [U j shirk, under-report]
0
0
Maximize expected net bene…ts after compensation given CEO works
in both private states among all possible e¤ort choices of the CEO.
formal
optimal contract
back1constraints
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January 2017
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Optimal Contract
The Optimal Compensation Schedules
Note: The excess return is approximated by one-side truncated normal distribution .
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Identi…cation
Where we are
From the data:
we have known the joint distribution of performance measure and
compensation
From the model:
we have developed several measures relevant to evaluating SOX, which
can be quanti…ed from the data if parameter values are known
Before estimation, we need to answer: does the model have empirical
content?
model speci…cation test: the model can be rejected if there exist no
parameter values that satisfy the restrictions implied by the model.
criterion function: summarize model restrictions and penalize deviations
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January 2017
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Identi…cation
Two Steps to Construct the Criterion Function
Step 1: Shrink the parameter space
equality restrictions involving both observables and unobserved
parameters ) a platform to derive the mappings from the risk aversion
parameter γ to other primitives α1t , α2t , g1t (x ), g2t (x )
Step 2: Set identify γ
equality and inequality restrictions bound γ up to a set rather than a
point
construct a criterion function Q (γ) to summarize these restrictions
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January 2017
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Identi…cation
Step 1: Shrink the Parameter Space
Unknown primitives: γ, α1t , α2t , g1t (x ), g2t (x )
Equality restrictions:
First order conditions
Binding constraints: Participation, Incentive Compatibility
Complementary-slackness conditions: Truth-telling, Sincerity
Mappings: vst (x ,γ) exp ( γw st (x )/bt +1 ) and v st (γ) exp ( γw st /bt +1 )
α1t
α2t
g 1t (x )
g 2t (x )
=
=
=
=
α2t
"
v 2t (γ)
1
E 2 v2t (x , γ)
v 2t (γ)
1
E 2 [v2t (x , γ)]
1
1
#b t
1
1 bt
E [vst (x , γ)]
n
v 1t (γ) 1 v1t (x , γ)
1
+ η 3t h t
v 1t (γ)
v 2t (γ)
v 2t (γ)
1
1
v2t (x , γ)
E [vst (x , γ)]
oh
1
α1t
α2t
i
1
1 bt
η 4t g 2t (x )h t (x )
+ η 3t h t
1
E 2 [v2t (x , γ)
identi…cation of KT multipliers
Miller (47-901 Advanced Economic Analysis)
1
h t (x )
1]
identi…cation illustration
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optimal contract
January 2017
22 / 53
Identi…cation
Step 2: Set Identi…cation
Equality restrictions:
Ψlt (γ )=0
Likelihood ratio is non-negative with unit mass:
Ψ1t (γ )=E 1 [I fg 1t (x ,γ ) 0 g 1 ]=0
α1t is the same between two private states
Between truth-telling and sincerity, there is at least one binding
Complementary-slackness conditions for truth-telling and sincerity
Inequality restrictions:
Λkt (γ ) 0
Shareholders prefer working to shirking:
Λ2t (γ )=E [(Vx w )jworking ] E [(Vx w )jshirking ]
Kuhn Tucker multipliers are positive
Identi…ed set:
Γ
constraints
(
T
γ > 0 : Q (γ)
∑ ∑ Ψlt (γ)
t =1 l
2
T
+
∑ ∑ min [0, Λkt (γ)]
t =1 k
2
=0
)
identi…cation of KT multipliers
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Estimation
Con…dence Region of the Identi…ed Set
Determined by the critical value cδ associated with test size δ:
n
o
(N )
Γδ
γ > 0 : Q (N ) ( γ ) cδ
Intuition: an estimate of γ is penalized if it makes the sample analogue
of the criterion function deviate from its theoretical value (zero).
How: a consistent estimate of cδ can be determined numerically by
following subsampling procedures described in the paper, which follows
Chernozhukov Hong and Tamer (2007).
Miller (47-901 Advanced Economic Analysis)
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Results
The 95% Con…dence Regions of Risk-aversion
Period
Pre
Post
Years
1993-2001
2004-2005
Risk Aversion
(0.0695, 0.6158)
(0.0695, 0.6158)
Certainty Equivalent
(34722, 290206)
(34722, 290206)
The estimated risk aversion is comparable to previous studies in terms of certainty
equivalent which is the dollar amount that a CEO at that risk aversion level would
like to pay to avoid a gamble that has half chance of losing or winning $1 million.
We do not reject the null hypothesis that no signi…cant change in risk attitude
occurred after SOX.
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Results
Framework
τ1
τ2
τ3
τ4
Miller(712-010 Structural Econometrics)
ρ2 ρA3
(FedAeA2o0Nu1is6 . Baru2c6h/CNove5o3llege,
Results
Administrative Cost (measured in thousands of 2006 US$, Table 3)
Sector
Primary
Consumer
Service
(A,C)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
∆τ 1
(2076, 2194)
(3692, 3879)
(6254, 7054)
(1846, 2181)
(2203, 2438)
(587, 614)
(-3929, -2025)
(218, 581)
(-1819, -1295)
(-230, 158)
(-5403, -4827)
(-1742, -1191)
+
+
+
+
+
+
+
=
-
Note: A: Size. C: Debt/Equity. S: Small. L: Large.
τ 1 γ 1 bbtt +11 ln α2,pre .
framework
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Results
Aggregate Agency Costs (measured in thousands of 2006 US$, Table 4)
Sector
Primary
Consumer
Service
(A,C)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
∆τ 2
(20, 190)
(3, 30)
(76, 611)
(43, 379)
(-527, -59)
(21, 156)
(182, 1812)
(81, 459)
(-360, -41)
(45, 395)
(113, 355)
(53, 529)
+
+
+
+
+
+
+
+
+
+
Note: A: Size. C: Debt/Equity. S: Small. L: Large.
τ 2 ∑2s =1 ϕs ,pre Es ,pre [ws ,pre (x )] τ 1 .
framework
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Results
Welfare Costs of Moral Hazard (measured in thousands of 2006 US$, Table 5)
Sector
Primary
Consumer
Service
(A,C)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
∆τ 3
(228, 1532)
(-39, 165)
(96, 1774)
(265, 380)
(-4387, -600)
(-817, -202)
(-3111, -649)
(-3848, -332)
(-328, -150)
(-399, 268)
(-6438, 433)
(-2621, 479)
+
=
+
+
=
=
=
Note: A: Size. C: Debt/Equity. S: Small. L: Large.
τ 3 ∑2s =1 ϕs ,pre Es ,pre [ys ,pre (x )] τ 1 .
framework
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Results
Welfare Costs of Hidden Information (in thousands of 2006 US$, Table 6)
Sector
Primary
Consumer
Service
(A,C)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
∆τ 4
(-1342, -208)
(-135, 42)
(-1163, -20)
(-227, 87)
(540, 3860)
(223, 973)
(831, 4923)
(413, 4307)
(-32, 217)
(-218, 795)
(-320, 6788)
(-348, 3150)
=
=
+
+
+
+
=
=
=
=
Note: A: Size. C: Debt/Equity. S: Small. L: Large.
τ 4 ∑2s =1 ϕs ,pre Es ,pre [ws ,pre (x ) ys ,pre (x )].
framework
Miller (47-901 Advanced Economic Analysis)
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January 2017
30 / 53
Results
Shareholders’Losses from Shirking (%, Table 7)
Sector
Primary
Consumer
Service
(A,C)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
∆ρ1
(-2.69, -1.96)
(-6.92, -4.75)
(-2.82, -2.10)
(-1.96, -1.95)
(-9.16, -8.72)
(2.12, 12.21)
(-0.40, 1.54)
(-2.68, -2.11)
(-8.93, -6.34)
(-3.02, -1.03)
(-16.59, -15.37)
(-5.97, -5.07)
+
-
Note: A: Size. C: Debt/Equity. S: Small. L: Large.
ρ1 ∑2s =1 ϕs ,pre Es ,pre fx [1 gs ,pre (x )]g.
framework
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January 2017
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Results
CEOs’Bene…t from Shirking (in thousands of 2006 US$, Table 8)
Sector
Primary
Consumer
Service
(A,C)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
∆ρ2
(122, 221)
(-57, -24)
(1716, 2125)
(100, 380)
(-3213, -2091)
(287, 476)
(18, 792)
(-1078, -654)
(-780, -487)
(67, 446)
(-7697, -5721)
(-2041, -1985)
+
+
+
+
+
+
-
Note: A: Size. C: Debt/Equity. S: Small. L: Large.
ρ2 bt +1 [(bt 1) γ] 1 ln(α2,pre /α1,pre ).
framework
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January 2017
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Results
Change in Moral Hazard Cost due to Signal Quality (in thousands of 2006 US$, Table 9)
Sector
(A,C)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
(S,S)
(S,L)
(L,S)
(L,L)
Primary
Consumer
Service
ρ3
(216, 1261)
(-37, 198)
(-617, -151)
(59, 225)
(180, 1254)
(-1575, -272)
(-3202, -930)
(-224, 1181)
(-130, 398)
(-503, 212)
(1503, 24282)
(724, 4470)
+
=
+
+
=
=
=
+
+
Note: A: Size. C: Debt/Equity. S: Small. L: Large.
ρ3 τ 3 (αj ,post ,g s ,post ) τ 3 (αj ,post ,g s ,pre ).
τ 3 (αj ,post ,g s ,pre ) ∑2s =1 ϕs ,pre E s ,pre [ys (x ,αj ,post ,g s ,pre )] τ 1 (αj ,post ,g s ,pre ).
framework
optimal contract formula
Miller (47-901 Advanced Economic Analysis)
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January 2017
33 / 53
Conclusion
Rule out Competing Explanations
The structural changes in CEO compensation across 2002 could be
attributed to
(1) change in CEOs’risk attitude
(2) change in aggregate economy
We rule out (1) by showing no signi…cant change in the estimated
risk aversion.
contradicting concerns that CEOs would overreact to SOX provisions
and exercise undue caution in investment decisions thus destroying
shareholder value. (Coats and Srinivasan 2014).
We rule out (2) by anchoring the estimation of agency costs at the
same bond prices for both pre and post periods.
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Conclusion
Our Explanations
We attribute the structural changes in CEO compensation to the new
regulations passed in 2002.
The main impact of SOX was to increase the administrative burden
from compliance in small …rms, and in the primary sector more
generally.
complementing Coats and Srinivasan (2014) who document the direct
costs from control system expenditures incurred due to SOX’s new
requirements.
SOX also increased agency costs in large …rms, as well as in the
primary sector more generally. The most important reason for the
increased agency costs in the primary sector proved to be the higher
compensating di¤erential between shirking and working.
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Conclusion
Policy Implication
SOX e¤ect is …rm type dependent
Primary: both administrative and agency costs increased within all …rm
types
Consumer goods and services: for small …rms, declines in agency costs
roughly o¤set increases in administrative costs, whereas for large …rms,
it was the other way around.
Policy implication: there may be scope for relaxing some of the
regulations that SOX introduced, either by rolling back its harsher
provisions, or tailoring them to di¤erent …rm types.
Miller (47-901 Advanced Economic Analysis)
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January 2017
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- - - - Backup Slides - - - -
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Motivation
Literature on SOX E¤ect
…rm behavior: extensive evidence
switching earnings management methods (Cohen et al 2008), reducing
investment (Bargeron et al. 2010, Cohen et al. 2007, Kang et al.
2010), delisting (Engel et al. 2006, Leuz et al. 2007)
stock market reaction: mixed evidence
Zhang 2007, Jain and Rezaee 2006, Leuz 2007, Dey 2010, Livtak 2007,
Hochberg et al. 2009
compensation practice: crucial but underexplored
exceptions: Carter et al. 2009, Nekipelov 2010, Cohen et al. 2007
from literature
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Data
Firm characteristics
back…rm characteristics Table 1: Cross-section Summary of Firm Characteristics
(Asset in millions of 2006 US$, Standard deviations in parentheses)
Sector
Primary
Consumer
Service
(1)
Pre
(2)
Post
(3)
Pre
(4)
Post
(5)
Pre
(6)
Post
Total Asset
4597
(7677)
6477
(11117)
3516
(8103)
5168
(12439)
13499
(42502)
17568
(70163)
Debt/Equity
1.819
(1.450)
2.102
(2.575)
1.571
(1.646)
1.537
(2.119)
3.650
(5.263)
2.982
(4.323)
Accounting
Return
1.108
(0.228)
1.138
(0.217)
1.126
(0.282)
1.108
(0.253)
1.175
(0.335)
1.115
(0.261)
Abnormal
Return
-0.031
(0.319)
0.063
(0.277)
-0.029
(0.383)
0.014
(0.320)
0.022
(0.433)
0.044
(0.353)
6213
1620
5012
1394
7629
2656
Period
Observations
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Results
Risk Preference
The 95% Con…dence Regions of Risk-aversion and Corresponding Certainty Equivalent
(in $)
A: Full Sample
Period
Years
Pre
1993-2002
Post
2003-2005
Common
B: Restricted Sample
Period
Years
Pre
1993-2001
Post
2004-2005
Common
Risk Aversion
(0.0784, 0.2335)
(0.0616, 0.2335)
(0.0784, 0.2335)
Certainty Equivalent
(39160, 115704)
(30781, 115704)
(39160, 115704)
Risk Aversion
(0.0695, 0.6158)
(0.0695, 0.6158)
(0.0695, 0.6158)
Certainty Equivalent
(34722, 290206)
(34722, 290206)
(34722, 290206)
Note: The subsampling procedure was performed using 100 replications of subsamples
with 80 percent of full sample observations, each using 100 grid points on the searching
interval [0.0003,54.598]. The certainty equivalent corresponding to one particular value
of the risk aversion in the estimated con…dence region is the dollar amount that a CEO
at that risk aversion level would like to pay to avoid a gamble that has an equal
probability of losing or winning one million dollars.
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Model
Technology
backtech assumptions
For each private state st 2f1,2 g, denote by fst (x ) the probability density function for
return conditional on the agent working, and let fst (x )gst (x ) denote the probability
density function for return when the agent shirks. Then gst (x ), the ratio of the two
densities, is a likelihood ratio and is nonnegative for all x :
Est [gst (x )]
Z
gst (x ) fst (x ) dx = 1
We assume that the principal prefers the agent working to shirking
Est [xgst (x )]
Z
xfst (x ) gst (x ) dx <
Z
xfst (x ) dx
Est [x ]
We assume the likelihood of shirking declines to zero as abnormal returns increase
without bound:
lim [gst (x )] = 0
x !∞
We assume the weighted likelihood ratio of the second state occurring relative to
the …rst given any observed value of excess returns, x 2R converges to an upper
…nite limit as x increases, such that:
lim [ ϕ2t f2t (x )/ϕ1t f1t (x )]
x !∞
Miller (47-901 Advanced Economic Analysis)
lim [ht (x )] = sup [ht (x )]
x !∞
Auctions and Contracts 6
x 2R
h t < ∞.
January 2017
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Feasible Contracts
Participation and Incentive Compatibility
To induce an honest, diligent manager to participate, her expected
utility from employment must exceed the utility she would obtain
from outside option:
1/(b t 1 )
α2t
∑
2
s =1
Z ∞
x
ϕst exp ( γt wst (x )/bt +1 ) fst (x ) dx
1
Given her decision to stay with the …rm one more period, and to
truthfully reveal the state, the incentive compatibility constraint
induces the manager to prefer working diligently to shirking in each
state st 2 f1, 2g:
1/(b t 1 )
α2t
1/(b 1 )
α1t t
Miller (47-901 Advanced Economic Analysis)
Z ∞
x
Z ∞
x
exp ( γt wst (x )/bt +1 ) fst (x ) dx
exp ( γt wst (x )/bt +1 ) gst (x ) fst (x ) dx
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Feasible Contracts
Truth Telling and Sincerity
Incentives must be provided to persuade the manager not to
understate them:
1/(b t 1 )
α2t
Z ∞
x
1/(b t 1 )
α2t
Z ∞
x
exp ( γt w2t (x )/bt +1 ) f2t (x ) dx
exp ( γt w1t (x )/bt +1 ) f2t (x ) dx
An optimal contract also induces the manager not to understate and
shirk:
1/(b 1 )
α2t t
1/(b t
α1t
Z ∞
x
Z ∞
1)
x
exp ( γt w2t (x )/bt +1 ) f2t (x ) dx
exp ( γt w1t (x )/bt +1 ) g2t (x ) f2t (x ) dx
backformal
Miller (47-901 Advanced Economic Analysis)
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Optimal Contract
ρ3
If there is neither hidden information nor hidden action, the agent gets γ 1 bbtt +11 ln α2t
If there is only hidden action, the optimal compensation derived from a pure moral
hazard model is back1optimal contract
yst (x ) = γ
where
p
η st
1
b t +1
ln α2 + γ
bt 1
1
b t +1 ln [1 + η pst (α2t /α1t )1/(bt
1)
η pst g st (x )]
is the unique positive solution to:
Z ∞
x
g st (x ) (α2t /α1t )1/(bt 1 )
fs (x )dx = 0
1 + η pst (α2t /α1t )1/(bt 1 ) η pst g st (x )
Solving the two problems generates a group of equilibrium restrictions and the
optimal contacts expressed in FOCs for each state as
exp [γt w 1t (x )/b t +1 ]
=
1/(b t 1 )
α2t
h
+ η 1t (α2t /α1t )1/(bt
backoptimal contract
=
1/(b t
α2t
g 1t (x )
i
η 4t (α1 /α2 )1/(bt 1 ) g 2t (x ) h t (x )
h
i
1)
+ η 2t (α2t /α1t )1/(bt 1 ) g2t (x ) + η 3t + η 4t
η 3t h t (x )
exp [γt w 2t (x )/b t +1 ]
1)
back2optimal contract
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Identi…cation
Shadow prices of constraints
Shadow prices of constraints: η 0t (Participation), η 1t (Incentive Compatibility for
(Incentive Compatibility for st =2), η 3t (Truth-telling), η 4t (Sincerity)
De…ne
st =1 ), η 2t
vst (x ) exp ( γt w st (x )/b t +1 )
η 0t (γ)
=
η 1t (γ)
=
η 2t (γ)
=
v 2t (γ)
η 3t (γ)
=
E 2 [v2t (x , γ)]
η 4t (γ)
=
α2t (γ)
α1t (γ)
α2t (γ)
1/(1 b t )
1
E 1 [v1t (x ,γ)]
E [vst (x ,γ)]
h
n
v 1t (γ)
E 2 v2t (x , γ)
1
E [vst (x , γ)]
1
+ η 3t h t
1
E [vst (x , γ)] 1
n
E 1 [v1t (x , γ)h t (x )] E 2 [v2t (x , γ)]
o
η 4t
i
α1t (γ) 1/(1 b t )
α2t (γ)
E 1 [v1t (x , γ)g 2t (x , γ)h t (x )]
1
E [vst (x , γ)]
1
o
1
E 1 [v1t (x , γ)h t (x )]
backidenti…cation of KT multipliers
back1identi…cation of KT multipliers
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Identi…cation
Example
α2t can be identi…ed from binding participation constraint.
participation constraint:
1/(b 1 )
α2t t
∑
2
s =1
Z ∞
x
ϕst exp ( γt wst (x )/bt +1 ) fst (x ) dx
1/(b t 1 )
i.e. α2t
E [vst (x, γ)]
1
1
rearrange items when it is binding to get the mapping
α2t (γ) = E [vst (x, γ)]1
bt
backidenti…cation illustration
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Identi…cation
Criterion Function (detailed)
Equality restrictions:
Ψjt (γ)=0
Likelihood ratio is non-negative with unit mass: Ψ1t (γ)=E 1 [I fg1t (x ,γ) 0 g 1 ]
α1t is the same between two states st 2f1,2 g:
Ψ2t (γ)=E 1 [g 1t (x ,γ)v 1t (x ,γ)]/E 1 [v1t (x ,γ)] E 2 [g 2t (x ,γ)v2t (x ,γ)]/E 2 [v2t (x ,γ)]
Between truth-telling and sincerity, there is at least one binding:
1/(b 1 )
1/(b 1 )
E 2 [v2t (x ,γ)] α2t t
E 2 [v1t (x ,γ)] and
Ψ3t (γ) Ψ4t (γ)=0 , where Ψ3t (γ)=α2t t
1/(b t
Ψ4t (γ)=α2t
1)
1/(b t
E 2 [v2t (x ,γ)] α1t
1)
E 2 [v1t (x ,γ)g 2t (x ,γ)]
Complementary-slackness conditions for truth-telling and sincerity:
Ψ3t (γ) η 3t (γ)=0, Ψ4t (γ) η 4t (γ)=0,
Inequality restrictions:
Λkt (γ) 0
Kuhn Tucker multipliers are positive: Λkt (γ)= η kt (γ) for k =1,3,4
Shareholders prefer working to shirking:
Λ2t (γ)=E [(Vx w )jworking ] E [(Vx w )jshirking ]
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Measuring SOX E¤ects
Welfare Costs
Administrative cost: CEOs’pecuniary cost of working under perfect
monitoring.
bt +1
τ 1t
γ 1
ln α2 .
bt 1
Aggregate agency cost:
2
τ 3t
E [wst (x )]
τ 1t =
∑
Z ∞
s =1 x
ϕs (z ) wst (x )
γt 1 ln α2 fs (x )dx.
Cost to hidden action only:
τ 3t
Est [yst (x )]
τ 1t
Cost to hidden information only:
Miller (47-901 Advanced Economic Analysis)
τ 4t
Es [wst (x )
Auctions and Contracts 6
yst (x )]
January 2017
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Measuring SOX E¤ects
Factors A¤ecting Moral Hazard Cost
Losses of shareholders from CEO shirking
Est [x
ρ1t
xgst (x )]
Interest alignment: CEO’s pecuniary bene…t from shirking instead of
working under perfect monitoring.
ρ2t
γt
1
bt +1
ln
bt 1
α2
α1
.
Signal quality e¤ect:
ρ3t
τ 3 (αj ,post , g s ,post )
Miller (47-901 Advanced Economic Analysis)
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τ 3 (αj ,post , g s ,pre )
January 2017
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Identi…cation
A quick summary
Consolidating the restrictions directly applied to the model, we de…ne:
9
8
Λi (γ) 0 for i 2 f1, 2, 3g
>
>
>
>
<
=
η j (γ) 0 for j 2 f1, 3, 4g
Γ
γ>0:
>
>
Ψ1 (γ) = 0 and Ψk (γ) 0 for k 2 f3, 4g
>
>
;
:
Ψ3 ( γ ) Ψ4 ( γ ) = Ψ3 ( γ ) η 3 ( γ ) = Ψ4 ( γ ) η 4 ( γ ) = 0
where:
Λi (γ) refers to suboptimal choices by shareholders of shirking and
diligence in the two states (that is shirking in one or both).
η j (γ) are representations of the Kuhn Tucker multipliers (for incentive
compatibility in the …rst state, truth telling and sincerity).
Ψk (γ) refers to incentive compatibility and sincerity conditions, a
positivity condition on g1 (x, γ), and a condition that the taste for
shirking is independent of the state.
Miller (47-901 Advanced Economic Analysis)
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January 2017
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Estimation
T
e)
QH (γ
T
+
∑
t =1
∑
h
i2
γ
min
0,
η
+
(
)
∑
j
9
t =1 j =1
e )2 +
Ψ4t (γ
T
T
7
∑ ∑ [Ψ5t (γe ) Ψjt (γe )]2
t =1 j =6
∑ [Ψ6t (γe ) Ψ8t (γe )]2 +
t =1
Miller (47-901 Advanced Economic Analysis)
T
3
∑ ∑ min [0, Λkt (γe )]2 .
(1)
t =1 k =1
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Empirical Implementation
Approximating the Q function
The identi…ed set of risk parameters de…ned Γ has a simple empirical
analogue.
Suppose we have N cross sectional observations on (xn , wn ) on
identical …rms and their managers.
To estimate Q (γ) , we replace w with w (N ) max fw1 , . . . , wN g
and substitute sample moments for their population corresponding
expectations
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Empirical Implementation
Convergence of the approximation
Our tests are based on the fact that if γ 2 Γ then sampling error is
the only explanation for why Q (N ) (γ) might be negative.
Clearly Q (N ) (γ) converges at the rate of its slowest converging
component.
For simplicity suppose there exists some x < ∞ such that g (x ) = 0
for all x > x.
In words, there is a revenue threshold that shirking cannot achieve.
Thus compensation is ‡at at w for all pro…ts
p levels above x, and
N
(
)
w
converges to w at a faster rate than N.
N ) γ are sample moments, we
Since all the other components of Q (p
( )
N
(
)
conclude Q
(γ) converges at rate N.
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