Appendix B. Validation of 2SLS specification
This appendix accompanies Jochem, Ladika, and Sautner (2016) and further develops the
paper’s 2SLS identification strategy. In particular, it addresses the potential concern that
the main results are biased because the baseline sample includes firms that did not accelerate
option vesting in any year. These are firms that chose not to accelerate in the year that FAS
123-R took effect (i.e., the year they received treatment). These firms may differ from those
that accelerated in either 2005 or 2006 because they received treatment in that year.
We examine whether our 2SLS estimates are affected by the inclusion of firms that did
not accelerate option vesting in any year. We first derive the 1st - and 2nd -stage estimators
for our baseline sample, and then compare these expressions to the estimators obtained from
a subsample of firms that accelerated option vesting at least once (the “accelerating firms
subsample”). The analysis shows that the baseline 2SLS estimates are unbiased when the
determinants of firms’ option acceleration decisions do not vary across fiscal year ends. Note
that this condition is encapsulated by the Exclusion Restriction, and thus holds as long as
our instrument affects executive turnover only by influencing firms’ decisions whether or not
to accelerate option vesting.
We also explain that early fiscal-year-end firms—i.e., those that waited one year to comply with FAS 123-R—may have experienced changes in the determinants of their acceleration
decision from 2005 to 2006. It is possible that some of these firms would have accelerated
in 2005 if they had been treated then, but did not accelerate when treated in 2006. In
other words, variation in FAS 123-R compliance dates affects not only the timing of option
acceleration, but also the decision whether or not to accelerate. We show that as a result,
2SLS estimates in the accelerating firms subsample can contain selection bias. As a result,
our main results are estimated on the baseline sample.
1
Model of acceleration decision
We begin the analysis by modeling firms’ option acceleration decisions, by including explicit
determinants of the acceleration decision in the 1st Stage of the 2SLS model from Section
II.C.1. This framework allows us to subsequently compare the 2SLS estimators for the
baseline sample and accelerating firms subsample.
We assume that boards decided whether or not to accelerate option vesting by comparing
its perceived costs and benefits to the firm’s market value. Company disclosures indicate
that boards perceived the primary benefit of option acceleration to be avoiding accounting charges on unvested options (Choudhary et al. (2009)). An important cost of option
acceleration is that the value of executives’ outside opportunities increases, leading to valuereducing turnover. For conciseness we do not include other benefits or costs in the model,
but our primary findings do not change if they are included. Our model of the 1st Stage is:
accelf,t = π1 FAS 123-R Takes Effect f,t + π2 xf,t − ρOutside Opp f,t + ηf,t
(1)
Outside Opp f,t represents the personal value of top executives’ outside opportunities, which
affects the cost of option acceleration. FAS 123-R Takes Effect f,t represents the benefits,
because option acceleration prevents earnings from decreasing only in the year of compliance
with the regulation. Note that in this model, all firms derive the same benefit from option
acceleration in the year of FAS 123-R compliance. Therefore, among treated firms the
variation in option acceleration is determined solely by differences in Outside Opp f,t . We
later consider the case where the benefits of option acceleration also vary across treated firms.
The residual in the revised model is ηf,t . We set E[η] = 0 because xf,t includes a constant
term. We also assume that E[ [FAS 123-R Takes Effect Outside Opp X]> η] = 0.
Our baseline sample contains three types of firms: those that did not accelerate option
vesting in any year, those that accelerated both in the year of FAS 123-R compliance and in
a non-compliance year, and those that accelerated only when they had to comply. Following
the treatment effects literature, we label these firms never-takers (NT ), always-takers (AT ),
and compliers (C ).1 By definition,
• Firm f = N T if E[accel|FAS 123-R Takes Effect = 1] ≤ 0 and E[accel|FAS 123-R Takes Effect =
0] ≤ 0. This condition is satisfied for firms with Outside Opp f,t ≥ (π1 + π2 xf,t )/ρ.2
• Firm f = AT if E[accel|FAS 123-R Takes Effect = 1] ≥ 0 and E[accel|FAS 123-R Takes Effect =
0] ≥ 0. This condition is satisfied for firms with Outside Opp f,t ≤ π2 xf,t /ρ.3
• Firm f = C if E[accel|FAS 123-R Takes Effect = 1] > 0 and E[accel|FAS 123-R Takes Effect =
0] < 0. This condition is satisfied for firms with π2 xf,t /ρ < Outside Opp f,t < (π1 + π2 xf,t )/ρ.4
In our model, never-takers are the sample firms with the highest cost of option acceleration—
they are most likely to experience costly executive turnover following a drop in unvested
1
We omit any defier firms that accelerated options only in a non-compliance year from the analysis.
Note that Outside Opp f,t ≥ (π1 + π2 xf,t )/ρ ⇒ E[π1 + π2 X − ρOutside Opp] ≤ π1 + π2 E[X] −
ρ × (π1 + π2 E[X])/ρ = 0 ⇒ E[accel|FAS 123-R Takes Effect = 1] ≤ 0. This immediately implies that
E[accel|FAS 123-R Takes Effect = 0] ≤ 0.
3
E[π2 X − ρOutside Opp] ≥ π2 E[X] − ρ × π2 E[X]/ρ ≥ 0 ⇒ E[accel|FAS 123-R Takes Effect = 0] ≥ 0. It
immediately follows that E[accel|FAS 123-R Takes Effect = 1] ≥ 0.
4
Outside Opp f,t < (π1 + π2 xf,t )/ρ ⇒ E[π1 + π2 X − Outside Opp] > π1 + π2 E[X] − ρ × (π1 + π2 E[X])/ρ =
0 ⇒ E[accel|FAS 123-R Takes Effect = 1] > 0. Additionally, Outside Opp f,t > π2 xf,t /ρ ⇒ E[π2 X −
ρOutside Opp] < π2 E[X] − ρ × π2 E[X]/ρ = 0 ⇒ E[accel|FAS 123-R Takes Effect = 0] < 0.
2
equity holdings. Similarly, always-takers have the lowest cost of option acceleration.
To simplify exposition, we allow Outside Oppf,t to take one of three values: ΩH = π1 +
π2 xf,t )/ρ, ΩL = π2 xf,t /ρ, or ΩM such that ΩL < ΩM < ΩH . We first consider the case where
these values do not vary over time, and then allow different values in 2005 and 2006.
We let N denote the total number of firms in the baseline sample, and nN T , nC , and nAT
denote the number of never-taker, complier, and always-taker firms, respectively. Therefore
fraction of nN T /N firms have Outside Opp N T = ΩH , nC /N have Outside Opp C = ΩM , and
nAT /N have Outside Opp AT = ΩL . We sample each firm twice, so there are 2N total observations. The population average E[Outside Opp] is Ω̄ = (nN T × ΩH + nAT × ΩL + nC × ΩM )/N .
We denote δN T as the fraction of never-takers with a late fiscal year end, and fraction
(1 − δN T ) as the fraction of never-taker firms with an early fiscal year end. Similarly, δAT
and δC are the fraction of always-taker and complier firms with a late fiscal year end.
We allow Outside Opp f,t to affect not only a firm’s acceleration decision, but also its
turnover rate. In our model, the reason that firms consider executives’ outside opportunities
when deciding whether to accelerate option vesting is precisely because these opportunities directly affect turnover. We additionally assume that Outside Opp f,t is unobservable
to the econometrician, because it is partly determined by executives’ unobservable utility
functions.5
This framework leads to the following modified version of our baseline 2SLS model:
accelf,t = π1 FAS 123-R Takes Effect f,t + π2 xf,t + vf,t
[ f,t + γ2 xf,t + uf,t+1
turnoverf,t+1 = γ1 accel
(1st Stage)
(2nd Stage)
where vf,t = ρ(Outside Opp f,t − Ω̄) + ηf,t and uf,t+1 = γ1 × ρ(Outside Opp f,t − Ω̄) + f,t+1 .
Note that E[v] =E[u] = 0; because xf,t contains a constant term, we can subtract Ω̄ from
each observation. However, the residuals do not necessarily equal 0 when conditioning on the
type of firm. Specifically, E[vN T ] = ρ(ΩH − Ω̄), E[vAT ] = ρ(ΩL − Ω̄), and E[vC ] = ρ(ΩM − Ω̄).
In other words, the variation across firms in the costs of option acceleration is reflected in the
regression residuals. (The conditional expectations of u are the same, except scaled by γ1 .)
2
Deriving 2SLS estimators
We now formally derive the key 2SLS estimators γ1 and π1 . We also derive the potential
bias of the regression estimates γ̂1 and π̂1 calculated over the baseline sample. We show that
5
Prior work explains only about half of the variation in option acceleration (Choudhary et al. (2009)),
so some of the costs or benefits of option acceleration are likely unobservable.
these coefficients are unbiased when the Exclusion Restriction holds. We then compare the
2SLS estimators obtained using the baseline sample and accelerating firms subsample.
2.1
2nd -stage estimator for baseline sample
The 2nd Stage of our 2SLS model can be re-expressed using matrix notation as:
y = WΓ + u
(3)


"
#
"
#
turnover1,2005


γ
accel
1
.
 is a 2N ×1 vector, Γ =
..
where y = 
is a k×1 vector, W =


γ2
X
turnoverN,2006
is a 2N × k matrix, and u is a 2N × 1 vector of residuals with individual elements uf,t .
Also, let Z > = [FAS X] be a k × 2N matrix that includes our instrument for accel. (For
conciseness, we abbreviate FAS 123-R Takes Effect to FAS.)
−1
A 2SLS regression of model (3) produces the estimates Γ̂ = E[Z > W ]
E[Z > y] =
−1
E[Z > u]. This formula implies that γ̂1 equals the causal effect of option
Γ + E[Z > W ]
acceleration on executive turnover, γ1 , plus a bias term that is proportional to E[FAS > u].6
This bias term can be simplified as follows:
E[FAS> u] = E[FAS> {γ1 × ρ(Outside Opp − Ω̄)}] + E[FAS> ]
= γ1 × ρ × E[FAS> (Outside Opp − Ω̄)]
where E[FAS > ] = 0 due to the Exclusion Restriction. Further,
1
×
E[FAS> (Outside Opp − Ω̄)] =
2N
=
!
FAS f × (Outside Opp f − Ω̄)
t=2005 f =1
1
× nN T × ΩH + nC × ΩM + nAT × ΩL − N × Ω̄
2N
1
= ×
2
=
2006 X
N
X
nN T × ΩH + nC × ΩM + nAT × ΩL
− Ω̄
N
1
× (Ω̄ − Ω̄) = 0
2
This derivation shows that γ̂1 is an unbiased estimator of the causal effect of option accel6
We assume throughout the analysis that the Relevance Condition holds, so that E[Z > W ] has full rank.
eration on executive turnover, even when the 2SLS regression does not explicitly control
for firms’ costs of option acceleration. In other words, our 2SLS specification will produce
unbiased estimates for the baseline sample.
Importantly, this result is obtained because Outside Opp f does not vary over time. We
sample each firm once in the year of FAS 123-R compliance and once in a non-compliance
year, so FAS 123-R Takes Effect f,t cannot be correlated with time-invariant firm characteristics. However, firms’ costs of option acceleration may vary across time if, for example, the
competitiveness of the managerial labor market changes from 2005 to 2006. Therefore, we
now allow Outside Opp f,t to vary across time. For never-taker firms, Outside Opp N T,t equals
either ΩH,2005 or ΩH,2006 ; it similarly varies by year for complier and always-taker firms. We
further denote the sample average in each year as either Ω̄2005 or Ω̄2006 .7
In this case, the bias term in γ̂1 is proportional to:
1
×
E[FAS (Outside Opp − Ω̄)] =
2N
>
=
2006 X
N
X
!
FAS f,t × (Outside Opp f,t − Ω̄t )
t=2005 f =1
n
1
× nN T × δN T × ΩH,2005 + nC × δC × ΩM,2005 + nAT × δAT × ΩL,2005
2N
− (nN T × δN T + nC × δC + nAT × δAT ) × Ω̄2005
+ nN T × (1 − δN T ) × ΩH,2006 + nC × (1 − δC ) × ΩM,2006 + nAT × (1 − δAT ) × ΩL,2006
o
− [nN T × (1 − δN T ) + nC × (1 − δC ) + nAT × (1 − δAT )] × Ω̄2006
This bias term may not reduce to 0. However, when δN T = δC = δAT = δ, it simplifies to:
h
1
× nN T × δ × ΩH,2005 + nC × δ × ΩM,2005 + nAT × δ × ΩL,2005 − N × δ × Ω̄2005
2N
i
+nN T × (1 − δ) × ΩH,2006 + nC × (1 − δ) × ΩM,2006 + nAT × (1 − δ) × ΩL,2006 − N × (1 − δ) × Ω̄2006
δ
nN T × ΩH,2005 + nC × ΩM,2005 + nAT × ΩL,2005
− Ω̄2005
= ×
2
N
(1 − δ)
nN T × ΩH,2006 + nC × ΩM,2006 + nAT × ΩL,2006
+
×
− Ω̄2006
2
N
δ
(1 − δ)
= × (Ω̄2005 − Ω̄2005 ) +
× (Ω̄2006 − Ω̄2006 )] = 0
2
2
This shows that γ̂ is unbiased when the distribution of never-takers, compliers, and alwaystakers does not vary across firms’ fiscal year ends. Note that if this condition does not hold,
7
Our regression model includes both a constant term and year fixed effects, which means that regression
residuals are de-meaned by the annual average of Outside Opp f,t across the entire sample.
the Exclusion Restriction would be violated—Outside Opp f,t would vary with FAS 123-R
Takes Effect f,t , but the Exclusion Restriction requires our instrument to be uncorrelated
with any unobservable determinant of executive turnover. Therefore, our 2SLS estimates for
the baseline sample are unbiased as long as the Exclusion Restriction holds.
Without observing all costs and benefits of option acceleration, we cannot determine
with certainty which firms are never-takers, compliers, or always-takers.8 However, these
different types of firms are indeed equally distributed by fiscal year ends, then the proportion of late fiscal-year-end firms that accelerated in 2005 should match the proportion of
early fiscal-year-end firms that accelerated in 2006. This is precisely what we find—16.4%
of late fiscal-year-end firms accelerated option vesting in 2005, compared to 14.4% of early
fiscal-year-end firms in 2006.
2.2
1st -stage estimator for baseline sample
The above results also apply to the 1st -stage estimator. In matrix notation, the 1st Stage is:
a = ZΠ + v


"
#
"
#
accel1,2005


π
FAS
1
.
 is a 2N × 1 vector, Π =
..
where a = 
is a k × 1 vector, Z =


π2
X
accelN,2006
is a 2N × k matrix, and v is a 2N × 1 vector of residuals with individual elements vf,t .
−1
A regression of the 1st Stage produces coefficient estimates Π̂ = E[Z > Z]
E[Z > a] =
−1
Π + E[Z > Z]
E[Z > v]. Similarly to the 2nd -stage estimator, this formula implies that
π̂1 equals the causal effect of FAS 123-R compliance on option acceleration, π1 , plus a bias
term that is proportional to E[FAS > v]. By assumption E[FAS > η] = 0, so the bias term is
proportional to E[FAS > {ρ(Outside Opp − Ω̄)}]. This is exactly the same as the bias term
for the 2nd -stage estimator (except for the scaling factor γ1 ). Therefore, the estimated effect
of FAS 123-R compliance on option acceleration is unbiased as long as either (i) firms’ costs
of accelerating option vesting are constant across time; or (ii) the proportion of never-takers,
always-takers, and compliers does not vary with firms’ fiscal year ends.
8
The set of firms that did not accelerate option vesting in either 2005 or 2006 contains all of the
never-taker firms. However, it could also contain some complier firms, if the determinants of option
acceleration change from 2005 to 2006.
2.3
Comparison to accelerating firms subsample
Next, we compare the 2SLS estimators obtained from the baseline sample and accelerating
firms subsample. Following the above results, we assume that the Exclusion Restriction
holds and that the estimators are unbiased. Our purpose in this part of the analysis is to
study the relative size of the estimators across the two samples.
In order to facilitate comparison, we derive explicit expressions for γ̂1 . This requires
omitting the firm controls xf,t , so that (E[Z > W ])−1 simplifies to (E[FAS > accel])−1 . The
general expression for Γ̂ then reduces to a single estimate:
γ̂1 = E[FAS > accel]
−1
E[FAS > y]
(4)
Note that (4) equals the ratio of coefficients from two reduced form regressions, which separately regress executive turnover and option acceleration on FAS 123-R Takes Effect.
When estimated over the baseline sample, γ̂1 equals:
−1
E[FAS > y] =
E[FAS > accel]
=
P
PN
1
2×(nN T +nAT +nC )
×
1
2N
×
1
2N
2006
t=2005
f =1 FAS f,t × turnoverf,t+1
P
PN
2006
FAS
×
accel
×
f,t
f,t
t=2005
f =1
P
2006
t=2005
1
2×(nN T +nAT +nC )
PN
f =1 FAS f,t × turnoverf,t+1
× (nC + 2 × nAT )
In the denominator of the last expression, nC firms accelerate option vesting in the year of
FAS 123-R compliance (δ × nC in 2005 and ((1 − δ) × nC in 2006), while nAT always-taker
firms accelerate option vesting twice, in both the compliance and non-compliance year.
The accelerating firms subsample contains only compliers and always-takers, or Ñ =
nC + nAT total firms and 2Ñ total observations. Expanding (4) for this subsample yields:
−1
E[FAS > accel]
E[FAS > y] =
1
2Ñ
×
P
1
2Ñ
2006
t=2005
f =1 FAS f,t × turnoverf,t+1
P
PÑ
2006
×
FAS
×
accel
f,t
f,t
t=2005
f =1
1
=
PÑ
2×(nAT +nC )
×
P
2006
t=2005
1
2×(nAT +nC )
PÑ
f =1 FAS f,t × turnoverf,t+1
× (nC + 2 × nAT )
The main difference between these 2SLS estimators is that FAS f,t × turnoverf,t+1 is
summed over all firms in the baseline sample, and over complier and always-taker firms
in the accelerating firms subsample. The estimators are exactly the same when FAS f,t ×
turnoverf,t+1 sums to 0 among never-taker firms. This occurs whenever the Exclusion Restriction is satisfied. This condition states that variation in fiscal year ends should only
impact turnover through its effect on option acceleration, which means that FAS 123-R
Takes Effect should not affect turnover among firms that did not accelerate option vesting.
The above expressions also show that the denominator of γ̂1 varies across the two samples. This denominator equals π̂1 , the 1st -stage coefficient of option acceleration on FAS
123-R compliance. The results show that π̂1 is larger in the accelerating firm subsample (the
denominator is scaled by 2Ñ < 2N ), because the impact of FAS 123-R is larger when firms
that did not accelerate option vesting are omitted. However, this does not affect γ̂1 because
the reduced-form coefficient of FAS 123-R compliance on turnover is also larger by the same
proportion (the numerator is also scaled by 2Ñ ). This is again because FAS 123-R’s impact
on executive turnover is concentrated among firms that accelerated option vesting.
3
Variation in benefits of option acceleration
We conclude our analysis by studying how variation in the perceived benefits of option acceleration can affect 2SLS estimates for the accelerating firms subsample. When benefits
vary across time, the subsample may exclude some complier firms that did not accelerate
option vesting, only because FAS 123-R took effect when their benefits of doing so were low.
We show that excluding these firms (essentially mischaracterizing them as never-takers) can
induce selection bias.9
3.1
Revised model of acceleration decision
The acceleration model (1) we have used so far assumes that the perceived benefit from accelerating option vesting is the same for all companies complying with FAS 123-R. However,
benefits may have varied because the size of accounting charges depended on the value of
outstanding unvested options, which differed across firms.
We add to our model a variable Impact f,t which represents the size of firms’ accounting charges on unvested options under FAS 123-R. Firms with larger amounts of unvested
options should have higher values of Impact f,t , and may be more likely to accelerate. Our
9
This result can also apply to changes in the cost of option acceleration—e.g., some complier firms may
have chosen not to accelerate because FAS 123-R took effect in a year when competition for managerial
talent was high. We analyze time-varying benefits to option acceleration, because it is clear that firms with
delayed FAS 123-R compliance dates could allow more options to vest under their normal schedule (thus
reducing the benefits of acceleration).
revised model is:10
accelf,t = π1 FAS 123-R Takes Effect f,t + π2 xf,t + σImpact f,t − ρOutside Opp f,t + ηf,t (5)
To simplify our derivations, we assume that Impact f,t takes one of two values in each
year: IH for a fraction φ of firms, and IL for a fraction (1 − φ) of firms, with IH > IL . Also,
the population average E[Impact] is I¯t = φ × IH,t + (1 − φ) × IL,t .
We allow Impact f,t to affect executive turnover directly, because the overall value of unvested stock options influences an executive’s retention incentives. We also assume that
Impact f,t is empirically unobservable. While data exists on the value of executives’ unvested
options, the sensitivity of net income or executive turnover to option value is not known.
Under these assumptions, the 2nd Stage of our 2SLS model becomes:
[ f,t + γ2 xf,t + ũf,t+1
turnoverf,t+1 = γ1 accel
(6)
¯ + f,t+1 .
where ũf,t+1 is γ1 × ρ(Outside Opp f,t − Ω̄) + γ1 × σ(Impact f,t − I)
3.2
2SLS estimators for accelerating firm subsample
The amount of outstanding unvested options changes over time, as previously granted options vest under their normal schedule and firms grant new unvested options. This means
that early fiscal-year-end firms had an alternative to option acceleration in 2006—they could
have allowed previously granted options to vest as scheduled, and substituted stock options
for other forms of pay when granting new annual compensation.11 Impact therefore may have
decreased from 2005 to 2006, in which case some early fiscal-year-end firms may have chosen
not to accelerate, even if they would have in 2005 if required to comply with FAS 123-R in
that year. These compliers would be excluded from the accelerating firm subsample.
To see how this can affect our results, we examine the case where complier firms with
Impact C,2006 = IL chose to not accelerate option vesting. Always-takers, on the other hand,
accelerate option vesting regardless of the value of Impact AT,2006 . We assume that δ and
φ are the same for compliers and always-takers, so that the distribution of the costs and
10
This model describes the benefits of option acceleration as a linear, additively separable function of two
variables, but conceptually the true function may be non-linear. One way to motivate a linear function is
that FAS 123-R Takes Effect f,t is a proxy variable for the benefits of acceleration, and Impact f,t represents
measurement error.
11
Prior work finds that after FAS 123-R took effect, firms adjusted compensation policies to minimize
accounting expenses under the new rules (e.g., Cadman et. al (2012)). Delayed FAS 123-R compliance also
may have allowed early fiscal-year-end firms to undertake other actions that minimized the impact of FAS
123-R, such as cutting expenses from 2005 to 2006.
benefits of option acceleration is the same across different types of firms. We also assume
that the value of Impact f,t did not affect firms’ acceleration decisions in 2005.12
Under these assumptions, the accelerating firms subsample contains Nsub = nAT + nC −
(1 − δ) × (1 − φ) × nC firms. Of the (1 − δ) compliers that are treated by FAS 123-R in
2006, (1 − φ) have Impact = IL . These firms do not accelerate option vesting in 2006 despite
being treated, and they also do not accelerate option vesting when untreated in 2005. The
number of complier firms that remains in the sample is ñC = nC − (1 − δ) × (1 − φ) × nC .
The general expression for the 2nd -stage estimator is the same as (3), except the bias
term is proportional to:
¯ + E[FAS> ]
E[FAS> u] = E[FAS> {γ1 × ρ(Outside Opp − Ω̄)}] + E[FAS> {γ1 × σ(Impact − I)}]
¯
= γ1 × σ × E[FAS> (Impact − I)]
Note that we continue to assume that FAS f,t is uncorrelated with Outside Opp f,t and f,t .
The bias term can be expressed as:

¯ =
E[FAS > ×(Impact−I)]
1
×
2Nsub
2006
sub
X NX

FAS f,t ×(Impactf,t −I¯t )
t=2005 f =1
n
1
× δ×ñC ×[φ×IH,2005 +(1−φ)×IL,2005 ]+δ×nAT ×[φ×IH,2005 +(1−φ)×IL,2005 ]−(δ×ñC +δ×nAT )×I¯2005
2Nsub
o
+(1−δ)×ñC ×IH,2006 +(1−δ)×nAT ×[φ×IH,2006 +(1−φ)×IL,2006 ]−[(1−δ)×ñC +(1−δ)×nAT )]×I¯2006
=
Because φ is distributed equally across firms in 2005, I¯2005 = φ × IH,2005 + (1 − φ) × IL,2005 .
The term reduces to:
=
o
1−δ n
× ñC ×IH,2006 +nAT ×[φ×IH,2006 +(1−φ)×IL,2006 ]−(ñC +nAT )×I¯2006
2Nsub
=
o
1−δ n
× ñC ×(IH,2006 −I¯2006 )+nAT ×[φ×IH,2006 +(1−φ)×IL,2006 −I¯2006 ]
2Nsub
(7)
Because compliers that remain in the sample in 2006 have higher values of Impact, the
expression for I¯2006 differs from I¯2005 :
I¯2006 =
12
h
i
1
× (nAT + δñC ) × (φ × IH,2006 + (1 − φ) × IL,2006 ) + (1 − δ) × ñC × IH,2006
Nsub
We make this assumption because firms that complied with FAS 123-R in 2005 likely did not have
time to take actions to mitigate option expensing (for many of these firms, the final compliance date was
set just before the regulation took effect). However, our results do not depend on this assumption.
Note that IH,2006 > I¯2006 > φ × IH,2006 + (1 − φ) × IL,2006 .
The 2nd -stage estimator is unbiased only when (7) equals 0. This requires:
ñC × (IH,2006 − I¯2006 ) + nAT × [φ × IH,2006 + (1 − φ) × IL,2006 − I¯2006 ] = 0
⇒ ñC × (IH,2006 − I¯2006 ) = −nAT × [φ × IH,2006 + (1 − φ) × IL,2006 − I¯2006 ]
In our sample ñC > nAT ; few firms accelerated option vesting in both 2005 and 2006,
suggesting that the number of always-takers is far smaller than the number of compliers. Therefore the 2nd -stage estimator is unbiased only when the absolute magnitude of
φ × IH,2006 + (1 − φ) × IL,2006 − I¯2006 is substantially larger than IH,2006 − I¯2006 . This condition
likely does not hold for many reasonable parameter values.
This result shows that the accelerating firm subsample excludes complier firms with low
values of Impact in 2006, and this can lead to selection bias in the estimate of γ̂1 . In particular, when Impact is also negatively correlated with executive turnover, then γ̂1 is biased downward toward 0.13 This is likely the case when Impact represents the value of unvested options,
because executives’ departure costs are higher when many of their options are unvested.
4
Summary
This appendix examines how our identification strategy is affected by including or omitting
firms that did not accelerate option vesting in either 2005 or 2006. It derives the 2SLS
estimators for our baseline sample of all firms, and also for a subsample of only firms that
accelerated option vesting.
The analysis produces two main results. First, we show that the inclusion or omission of
firms that did not accelerate option vesting does not bias our 2SLS estimates, as long as the
variables that determined firms’ acceleration decisions do not co-vary with fiscal year ends.
In other words, our estimates are unbiased as long as the Exclusion Restriction is satisfied.
Second, we show that omitting firms that did not accelerate option vesting in either
year can possibly induce selection bias. The determinants of option acceleration can change
from 2005 to 2006, and this could cause some early fiscal-year-end firms to avoid option
acceleration in 2006, even though they would have accelerated options in 2005 had they been
required to comply with FAS 123-R. Removing these firms can induce correlation between
¯
¯ > 0
The bias term is E[FAS> u] = γ1 × σ × E[FAS> (Impact − I)].
Note that E[FAS> (Impact − I)]
because compliers with above-average values of Impact remain in the subsample, and γ1 > 0 under our
main hypothesis that option acceleration positively affects turnover. Therefore the sign of the bias depends
on σ, which is negative when higher values of Impact reduce executive turnover.
13
unobservable determinants of firms’ acceleration decisions and our instrument, leading to a
violation of the Exclusion Restriction and biasing our estimates.
The analysis indicates that our results are most accurate when estimating on the baseline
sample of all firms, and estimates using the accelerating firm subsample may understate the
true effect of option acceleration on executive turnover.
References
Cadman, Brian, Jayanthi Sunder, and Tjomme Rusticus, 2013, Economic determinants of
stock option vesting periods, Review of Accounting Studies 18, 1159–1190.
Choudhary, Preeti, Shivaram Rajgopal, and Mohan Venkatachalam, 2009, Accelerated vesting of employee stock options in anticipation of FAS 123-R, Journal of Accounting Research
47, 105–146.
Jochem, Torsten, Tomislav Ladika, and Zacharias Sautner, 2016, The Retention Effects of
Unvested Equity: Evidence from Accelerated Option Vesting, Working paper, University of
Amsterdam.