Appendix B. Validation of 2SLS specification
This appendix accompanies Ladika and Sautner (2016) by further developing 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 corporate investment 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
2.3.1. 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 executives could more quickly unwind their equity incentives, possibly leading
to actions that temporarily boost firm value in the short term but also lead to long-term
reductions in value. 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 − ρMyopia Risk f,t + ηf,t
(1)
Myopia Risk f,t represents executives’ propensity to engage in short-termism. This in turn
depends on executives’ incentive horizons, and hence represents 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 Myopia Risk 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 Myopia Risk 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 Myopia Risk 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 Myopia Risk 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 < Myopia Risk f,t < (π1 + π2 xf,t )/ρ.4
1
We omit any defier firms that accelerated options only in a non-compliance year from the analysis.
Note that Myopia Risk f,t ≥ (π1 + π2 xf,t )/ρ ⇒ E[π1 + π2 X − ρMyopia Risk ] ≤ 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 − ρMyopia Risk ] ≥ 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
Myopia Risk f,t < (π1 + π2 xf,t )/ρ ⇒ E[π1 + π2 X − Myopia Risk ] > π1 + π2 E[X] − ρ × (π1 + π2 E[X])/ρ =
0 ⇒ E[accel|FAS 123-R Takes Effect = 1] > 0. Additionally, Myopia Risk f,t > π2 xf,t /ρ ⇒ E[π2 X −
ρMyopia Risk ] < π2 E[X] − ρ × π2 E[X]/ρ = 0 ⇒ E[accel|FAS 123-R Takes Effect = 0] < 0.
2
In our model, never-takers are the sample firms with the highest cost of option acceleration—
they are most likely to experience costly executive short-termism following a drop in unvested
equity holdings. Similarly, always-takers have the lowest cost of option acceleration.
To simplify exposition, we allow Myopia Riskf,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 Myopia Risk N T = ΩH , nC /N have Myopia Risk C = ΩM , and
nAT /N have Myopia Risk AT = ΩL . We sample each firm twice, so there are 2N total observations. The population average E[Myopia Risk ] 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 Myopia Risk f,t to affect not only a firm’s acceleration decision, but also its
investment rate directly. In our model, the reason that firms consider executives’ incentive
horizons when deciding whether to accelerate option vesting is precisely because the duration
of incentives can affect investment directly. We additionally assume that Myopia Risk 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
investmentf,t = γ1 accel
(1st Stage)
(2nd Stage)
where vf,t = ρ(Myopia Risk f,t − Ω̄) + ηf,t and uf,t = γ1 × ρ(Myopia Risk f,t − Ω̄) + f,t .
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 .)
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.
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
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)


"
#
"
#
investment1,2005


γ
accel
1
..
 is a 2N ×1 vector, Γ =
where y = 
is a k×1 vector, W =
.


γ2
X
investmentN,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
E[Z > y] =
A 2SLS regression of model (3) produces the estimates Γ̂ = E[Z > W ]
−1
Γ + E[Z > W ]
E[Z > u]. This formula implies that γ̂1 equals the causal effect of option
acceleration on corporate investment, 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(Myopia Risk − Ω̄)}] + E[FAS> ]
= γ1 × ρ × E[FAS> (Myopia Risk − Ω̄)]
6
We assume throughout the analysis that the Relevance Condition holds, so that E[Z > W ] has full rank.
where E[FAS > ] = 0 due to the Exclusion Restriction. Further,
E[FAS> (Myopia Risk − Ω̄)] =
=
1
×
2N
!
FAS f × (Myopia Risk 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 acceleration on investment, 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 Myopia Risk 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,
growth opportunities in the firm’s industry changed from 2005 to 2006. Therefore, we now
allow Myopia Risk f,t to vary across time. For never-taker firms, Myopia Risk 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> (Myopia Risk − Ω̄)] =
2N
=
2006 X
N
X
!
FAS f,t × (Myopia Risk 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
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 Myopia Risk f,t across the entire sample.
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
nN T × ΩH,2005 + nC × ΩM,2005 + nAT × ΩL,2005
δ
− Ω̄2005
= ×
2
N
(1 − δ)
nN T × ΩH,2006 + nC × ΩM,2006 + nAT × ΩL,2006
+
×
− Ω̄2006
2
N
(1 − δ)
δ
× (Ω̄2006 − Ω̄2006 )] = 0
= × (Ω̄2005 − Ω̄2005 ) +
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,
the Exclusion Restriction would be violated—Myopia Risk 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 investment. 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—16.9%
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


where a = 

8

"
#
"
#
accel1,2005

π
FAS
1
..
 is a 2N × 1 vector, Π =
is a k × 1 vector, Z =
.

π2
X
accelN,2006
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.
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(Myopia Risk − 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.
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 investment and option acceleration on FAS 123-R Takes Effect.
When estimated over the baseline sample, γ̂1 equals:
−1
E[FAS > accel]
E[FAS > y] =
=
1
2N
×
P
1
2N
2006
t=2005
PN
f =1 FAS f,t × investmentf,t
P
PN
2006
×
FAS
×
accel
f,t
f,t
t=2005
f =1
1
2×(nN T +nAT +nC )
×
P
2006
t=2005
1
2×(nN T +nAT +nC )
PN
f =1 FAS f,t × investmentf,
× (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:
E[FAS > accel]
−1
E[FAS > y] =
1
2Ñ
×
P
1
2Ñ
2006
t=2005
×
FAS f,t × investmentf,t
PÑ
2006
FAS
×
accel
f,t
f,t
t=2005
f =1
f =1
P
1
=
PÑ
2×(nAT +nC )
×
P
2006
t=2005
1
2×(nAT +nC )
PÑ
f =1
FAS f,t × investmentf,t
× (nC + 2 × nAT )
The main difference between these 2SLS estimators is that FAS f,t × investmentf,t 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 ×
investmentf,t 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 investment through its effect on option acceleration, which means that FAS 123-R
Takes Effect should not affect investment 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 investment is also larger by the
same proportion (the numerator is also scaled by 2Ñ ). This is again because FAS 123-R’s
impact on investment 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
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 investment opportunities were
low. 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).
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
revised model is:10
accelf,t = π1 FAS 123-R Takes Effect f,t + π2 xf,t + σImpact f,t − ρMyopia Risk 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 investment directly, because the overall value of unvested
stock options influences an executive’s incentive horizon. 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 short-termism to option value is not known. Under
these assumptions, the 2nd Stage of our 2SLS model becomes:
[ f,t + γ2 xf,t + ũf,t
investmentf,t = γ1 accel
(6)
¯ + f,t .
where ũf,t is γ1 × ρ(Myopia Risk 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
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
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
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(Myopia Risk − 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 Myopia Risk f,t and f,t .
The bias term can be expressed as:


2006
sub
X NX
1
¯ =
×
FAS f,t ×(Impactf,t −I¯t )
E[FAS > ×(Impact−I)]
2Nsub
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 .
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.
12
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.
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 =
h
i
1
× (nAT + δñC ) × (φ × IH,2006 + (1 − φ) × IL,2006 ) + (1 − δ) × ñC × IH,2006
Nsub
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 short-termism, then γ̂1 is
biased upward toward 0.13 This is likely the case when Impact represents the value of unvested options, because executives’ incentive horizons are longer 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
¯
¯ > 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 leads to investment cuts. Therefore the sign of the bias depends
on σ, which is negative when higher values of Impact reduce executive short-termism.
13
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
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.
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.
Ladika, Tomislav and Zacharias Sautner, 2016, Managerial Short-Termism and Investment:
Evidence from Accelerated Option Vesting, Working paper, University of Amsterdam.