Should We Condition on the Test for Pre-trends in
Difference-in-Difference Designs?
arXiv:1804.01208v1 [econ.EM] 4 Apr 2018

Jonathan Roth∗
April 5, 2018

Abstract
The common practice in difference-in-difference (DiD) designs is to check for parallel trends prior to treatment assignment, yet typical estimation and inference does
not account for the fact that this test has occurred. I analyze the properties of the
traditional DiD estimator conditional on having passed (i.e. not rejected) the test for
parallel pre-trends. When the DiD design is valid and the test for pre-trends confirms
it, the typical DiD estimator is unbiased, but traditional standard errors are overly
conservative. Additionally, there exists an alternative unbiased estimator that is more
efficient than the traditional DiD estimator under parallel trends. However, when in
population there is a non-zero pre-trend but we fail to reject the hypothesis of parallel
pre-trends, the DiD estimator is generally biased relative to the population DiD coefficient. Moreover, if the trend is monotone, then under reasonable assumptions the bias
from conditioning exacerbates the bias relative to the true treatment effect. I propose
new estimation and inference procedures that account for the test for parallel trends,
and compare their performance to that of the traditional estimator in a Monte Carlo
simulation.

1

Introduction

There is a standard playbook for difference-in-difference designs in applied economics. The
researcher first identifies a treatment group and a candidate control group, where the treatment group received some treatment at period t = 1 and the candidate control group did
∗

Department of Economics, Harvard University, jonathanroth@g.harvard.edu. I am grateful to Isaiah
Andrews, Kirill Borusyak, Edward Glaeser, Nathan Hendren, Ariella Kahn-Lang, Larry Katz, and Jann
Spiess for helpful comments. This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under Grant DGE1144152.

1

not receive the treatment but might plausibly have had parallel trends to the treatment
group in the absence of the treatment. The researcher then tests whether the treatment
and control group in fact had parallel trends prior to the treatment, and if the pre-trends
appear to be (roughly) parallel, the researcher accepts the research design and interprets the
difference-in-difference estimator as an estimate of the causal effect of treatment.
While testing for pre-trends is nearly universal, typical estimation and inference for the
treatment effects ignores the fact that this test has occurred. In this paper, I address the
following question: do the usual properties we expect of our difference-in-difference estimator
and its associated confidence interval hold conditional on having not rejected the test for
parallel pre-trends? In particular, I discuss whether the point estimate is unbiased and most
efficient among unbiased estimators, and whether the standard errors achieve their nominal
coverage rate. I also propose new estimation and inference procedures that recover some of
these properties in cases where they do not hold for the traditional difference-in-difference
procedure.
I consider the performance of the classical difference-in-difference estimator (and alternatives) conditional on having passed the test for pre-trends in two different cases: first, when
the parallel pre-trends assumption does in fact hold, and we have correctly confirmed it; and
second, when the parallel pre-trends assumption is false but we have failed to reject it in our
pre-test.
When the parallel pre-trends assumption is satisfied in population, the traditional differencein-difference estimator β̂post remains unbiased conditional on the pre-test. However, in this
case the traditional standard errors are overly conservative. Moreover, I show that there
exists an unbiased estimator β̃post that exploits the assumption of no pre-trends which is
more efficient than the traditional estimator β̂post .
When the parallel pre-trends assumption is false but we fail to reject it in our pretest, β̂post is generally biased relative to the population difference-in-difference coefficient,
which itself will differ from the true causal effect of the treatment if the pre-trend would
have continued into the post period. Moreover, if the pre-trend is monotone and would
have continued into the post-period, I show that the bias from conditioning exacerbates
the existing bias relative to the true treatment effect under certain reasonable conditions
on the covariance structure. To make this concrete, suppose that the true mean of y is
increasing by 1 unit per period relative to the control group in the pre-period, and this trend
would have continued absent treatment. Unconditionally, we would expect the differencein-difference estimator for the first period to be biased upwards by 1 relative to the true
treatment effect. Conditional on not having rejected the parallel pre-trends assumption, this
bias will be greater than 1. Thus, the usual “omitted variable bias” formula does not hold
2

once we condition on the pre-test. Unfortunately, the modified estimator β̃post discussed
above suffers from the same issue, and the bias can be even more severe.
I view these results as both “good news” and “bad news” for the applied researcher. In
some cases, we have strong a priori reason to believe that treatment assignment was random,
and the test for pre-trends serves only as a sanity check. In these cases, it seems reasonable
to use the modified estimator β̃post , which is more efficient when the parallel pre-trends
assumption holds but has undesirable properties when it does not. In other cases, however,
the researcher does not have such strong a priori reasons to believe in parallel trends – in
many cases, the pre-test itself is one of the main justifications for the research design. The
results here are problematic for this approach, because they show that this approach tends to
fail to reject bad research designs precisely in the cases where the estimated treatment effects
are farthest from the truth. This concern interacts with concerns about “file drawer effects”
or “publication bias,” because even if the probability of accepting a given bad design is small,
researchers may search over many bad designs until they find ones with apparently parallel
trends. Researchers should therefore be wary of relying on the absence of an observable
pre-trend as a justification in and of itself for the validity of a research design.
I also present two new estimators, along with corresponding confidence intervals, that adjust for the pre-test for parallel trends. The first procedure provides optimal median-unbiased
estimates (and valid confidence intervals) for the population difference-in-difference coefficient, i.e. the probability limit of OLS without conditioning. It thus removes the additional
bias imposed by conditioning on non-significant pre-trends when the parallel trends assumption fails, and hence allows us to make use of the usual intuition for how OLS difference-indifference estimates are affected by non-parallel trends. Note, however, that this estimator
fixes the bias from conditioning on the observed pre-trends but not the bias relative to the
true treatment effect from accepting a bad research design – if there is a pre-trend and it
would have continued into the post period, the population difference-in-difference coefficient
will differ from the true treatment effect.
The second estimator modifies the first to obtain median-unbiased estimates of the OLS
coefficient from a regression that allows the treatment group to be on a linear trend relative
to the control group.1 If in population there is a linear pre-trend but we fail to reject the
test for parallel trends, this estimator will remain unbiased for the true treatment effect
under the assumption that the linear trend would have continued into the post period. If
in population there is a pre-trend that is non-linear, the estimator will generally be biased
relative to the treatment effect, but may serve as a good approximation to the extent that
1

I show the estimator can also be modified to incorporate higher-order polynomial (e.g. quadratic) pretrends.

3

the extrapolation of the best linear fit to the pre-period serves as a close approximation to
the counterfactual trend.
Related Literature. This paper contributes to a large literature in econometrics and
statistics showing that, in a variety of contexts, problems can arise in both estimation and
inference if researchers do not account for a pre-testing or model selection step. Previous
research has investigated, for instance, the consequences of pre-testing for statistical significance to determine which covariates to include in a regression specification (Giles and
Giles (1993) provide a review), automated variable selection via the LASSO (Lee et al.,
2016), and selection between many models using goodness-of-fit criteria such as AIC or
BIC (see Leeb and Pötscher (2005) and references therein). I note that similar issues arise
in difference-in-differences, and show that under reasonable assumptions, when the parallel
trends assumption is violated the bias induced by the pre-test exacerbates the bias of the
difference-in-difference estimator relative to the true treatment effect.
I provide two types of corrections to account for the fact that researchers test for observable pre-trends in difference-in-difference designs. The first exploits the assumption of
parallel trends to obtain an unbiased treatment effects estimate β̃post that is more efficient
than the traditional estimator when the parallel trends assumption holds (both unconditionally, and conditional on observing a roughly parallel trend). The observation that the
traditional difference-in-difference estimator is inefficient under parallel pre-trends mirrors
a similar observation made by Borusyak and Jaravel (2016) in the context of event study
designs. I show, however, that this type of estimator can perform quite poorly in the case
where the parallel trends assumption does not hold but we fail to reject the hypothesis of
parallel trends in our pre-test. The second type of correction provides optimal medianunbiased estimates and valid confidence intervals for the population difference-in-difference
coefficient, or an adjustment thereof, even when the parallel pre-trends assumption is violated. These procedures build on results by Lee et al. (2016) and Andrews and Kasy (2017),
who respectively provide corrected inference procedures for the LASSO and for treatment
effects estimates under publication bias, as well as on earlier work by Pfanzagl (1994), who
developed optimal quantile-unbiased estimation results for exponential-family distributions.
By highlighting the need to account for the test for observably parallel pre-trends and
extending the difference-in-difference framework to correct for this test, this paper also contributes to work on the econometrics of difference-in-difference designs in particular. Previous work has illustrated the need to adjust standard errors for serial correlation and other
types of clustering in the data (see, e.g., Moulton (1990); Bertrand et al. (2004); Donald and
Lang (2007); Petersen (2009)). Other research has, for instance, extended the traditional
difference-in-difference framework to allow the treatment and control groups to have differ4

ent benefits from treatment (Athey and Imbens, 2006), or to relax the typical identification
assumption of parallel trends to allow for differences driven by observable characteristics
(Abadie, 2005).
Finally, this paper relates to the literature on selective publication of scientific results
(Rothstein et al. (2005) and Christensen and Miguel (2016) provide reviews). Interest in
this topic has been stoked recently by replication failures (Open Science Collaboration, 2015;
Camerer et al., 2016), as well as empirical evidence suggesting manipulation of p-values and
substantial publication bias (Brodeur et al., 2016; Andrews and Kasy, 2017). The existing
literature has primarily assumed (either implicitly or explicitly) that certain treatment effects
estimates are inherently more publishable than others – e.g. journals may prefer large or
statistically significant treatment effects. My results suggest, however, that even if journal
editors have no preferences over the estimated treatment effects per se, the distribution of
published treatment effects can still be affected if editors have preferences over the observed
pre-trends coefficients. This paper therefore has important implications for the design of
alternative publication processes to mitigate publication bias. For instance, Nyhan (2015)
proposes a results-blind peer review process, in which referees are provided information
on the research design but not the resulting treatment effects. The results here suggest,
however, that such a scheme may still create a distorted distribution of published estimates
if referees are provided evidence on the pre-trends coefficients as evidence in favor of the
research design.

2

Intuition

The intuition for why the distribution of estimated treatment effects is affected by the test
for pre-trends stems from the fact that estimation error in the reference period (t = 0) enters
both the estimated pre-trends coefficients and the estimated treatment coefficients. To see
this formally, let ∆Ȳt denote the difference in sample means between the treatment and
control groups in period t, and let ∆µt be the true difference in population means. The
difference-in-difference estimate for the treatment effect in period 1 is β̂post = ∆Ȳ1 − ∆Ȳ0 ,
whereas the coefficient for the kth pre-period is β̂−k = ∆Ȳ−k − ∆Ȳ0 – both of which contain
a ∆Ȳ0 term. Although not necessary for most of our results, for simplicity consider the case
where the estimation error ∆Yt − ∆µt is independent across t, as would occur if the data
come from repeated cross-sections. It it is then immediate that β̂post and β̂−k will both tend
to be above (below) their true expectation when ∆Ȳ0 − ∆µ0 is less (greater) than zero.
Conditioning on β̂−k being insignificant thus has implications for the distribution of
∆Ȳ0 − ∆µ0 , and hence for β̂post . These dynamics are captured in Figure 1, which shows the
5

simulated distribution of ∆Ȳ0 − ∆µ0 and β̂post − βpost by whether or not we accept the pretrends (meaning here that all of the β̂−k are insignificant, in this case for K = 4).2 In the top
left panel, we see that when the true model is parallel, ∆Ȳ0 − ∆µ0 has lower variance once we
condition on accepting the pre-trends, but continues to have mean 0. This is because under
parallel trends, β−k = 0, so β̂−k tends to be insignificant for all k when the measurement
error ∆Ȳ0 − ∆µ0 is small in magnitude – but the conditioning is symmetric around 0, so
there is no bias. As a result, β̂post − βpost is centered around zero but has lower variance
when we condition on accepting the pre-trends, as shown in the top right panel. By contrast,
the bottom left panel shows that when there is truly an upward trend, then conditional on
accepting the pre-trends, ∆Ȳ0 − ∆µ0 has mean below 0. This is because β−k < 0 for all k, so
β̂−k tends to be close to 0 when ∆Ȳ0 − ∆µ0 < 0. It follows that β̂post tends to be greater than
βpost when there is a positive underlying trend but we fail to reject the pre-test for parallel
trends, as shown in the bottom right panel.
Figure 1: Distribution of ∆Ȳ0 − ∆µ0 and β̂post − βpost by whether or not we accept the test
for parallel trends

2

Details on the simulation specification will be provided in Section 4.

6

3

Theoretical Results

3.1

Preliminary Set-Up

We assume that the researcher observes data on an outcome y for individuals in a treatment
group and a control group, where the treatment group receives the treatment of interest
beginning at period t = 1. We suppose that the researcher observes K + 1 periods prior to
the treatment. For ease of exposition, we will assume that the researcher only observes a
single period once the treatment occurs (i.e. t = 1), although the results extend easily to
estimating treatment effects in multiple subsequent periods. We suppose that the researcher
estimates the following canonical regression:

yit = αt +

X

βs × T reatmenti × 1[t = s] + βmain T reatmenti + it .

(1)

s6=0

Let β̂pre = (β̂−1 , . . . , β̂−K ) denote the vector of estimated coefficients for the pre-periods,
and β̂post = β̂1 denote the estimated coefficient for the post period. The researcher typically
evaluates the pre-trends by examining whether β̂pre is close to zero by some metric – e.g.
whether all of its elements are statistically insignificant. If β̂pre passes this test, the researcher
then evaluates β̂post as an estimate of the causal effect of the treatment on y. We are interested
in the distribution of β̂post conditional on β̂pre having passed the test for parallel pre-trends.
Note that under very mild regularity conditions, the Central Limit Theorem will imply
that (unconditionally):
√
N

β̂post
β̂pre

!
−

βpost
βpre

!!

0
0

→d N

!
,

Σ̃11 Σ̃12
Σ̃21 Σ̃22

!!
.

For the purposes of this note, I assume that this normal approximation holds exactly,
with known variance. That is, for fixed N ,
β̂post
β̂pre
with Σ =

Σ11 Σ12
Σ21 Σ22

!
∼N

βpost
βpre

!
,

Σ11 Σ12
Σ21 Σ22

!!

!
known.3

3

Note that Σ here represents Σ̃/N from the CLT above. That is, the square root of the diagonal of Σ
represents the standard errors.

7

3.2

Results when the parallel pre-trends assumption is correct and
we properly fail to reject it

I first examine the performance of β̂post conditional on having passed the test for pre-trends
when the pre-period trends are in fact parallel in population.
Assumption 1 (Parallel Pre-trends). The parallel pre-trends assumption states that βpre =
0.
We are interested in the distribution of β̂post conditional on the researcher having accepted
that the pre-period coefficients, β̂pre , are sufficiently close to zero. The standard used for
determining whether pre-trends look “okay” in applied work is not entirely clear – a commonly
used criterion is that all the pre-period coefficients be (individually) statistically insignificant,
although in other cases the researcher may follow other criteria (e.g. whether they can
reject the joint test that all the pre-period coefficients are equal to 0). We will denote by
p
BN S = {β̂pre : |β̂pre,j |/ Σjj ≤ cα for all j} the set of values for β̂pre such that no individual
coefficient is statistically significant at the α level (cα = 1.96 for α = 0.05), and we will
denote by B any arbitrary subset of the support of β̂pre . Some results will assume only that
we’ve conditioned on β̂pre ∈ B for some arbitrary B, whereas others will focus specifically
on the case where we’ve conditioned on β̂pre ∈ BN S .
Proposition 1. Suppose that the parallel pre-trends assumption holds. Let β̃post = β̂post −
Σ12 Σ−1
22 β̂pre . Then:
i
h
1. E β̂post |β̂pre ∈ BN S = βpost .
h
i
h
i
2. Var β̂post |β̂pre ∈ BN S ≤ Σ11 = Var β̂post .
h
i
3. E β̃post |β̂pre ∈ B = βpost for any B.
h

i

4. Var β̃post |β̂pre ∈ B = Σ11 −

Σ12 Σ−1
22 Σ21

h

i

≤ Var β̂post |β̂pre ∈ B for any B, and the
i
result holds with strict inequality if Var Σ12 Σ−1
β̂
|
β̂
∈
B
> 0.
pre
22 pre
h

Result 1 says that under the parallel pre-trends assumption, β̂post remains unbiased if
we condition on none of the pre-trend coefficients being significant. This result extends to
conditioning on β̂pre ∈ B for any B such that β̂pre has mean zero over the conditioning set.
Result 2 says that under the parallel pre-trends assumption, the variance of β̂post conditional
on not having significant pre-trend coefficients is smaller than the unconditional variance,

8

which suggests that traditional standard errors may be conservative.4 Result 3 says that under the parallel pre-trends assumption, the alternative estimator β̃post is unbiased regardless
of the conditioning set. Finally, Result 4 says that under the parallel pre-trends assumption,
β̃post is more efficient than β̂post . Note that Results 3 and 4 apply for arbitrary conditoning
sets, and thus, when the parallel pre-trends assumptions holds, β̃post is both unbiased and
more efficient than β̂post even when we do not condition on the observed pre-trends.

3.3

Results when the parallel pre-trends assumption is false but we
fail to reject it

I now consider the case where the parallel pre-trends assumption may not hold, i.e. when
βpre is potentially not equal to 0.
Proposition 2. For any βpost , with βpost potentially not equal to 0,
i

i
 h
h
β̂
|
β̂
∈
B
−
β
for any B.
E
1. E β̂post | β̂pre ∈ B = βpost + Σ12 Σ−1
pre
pre
pre
22
h
i
2. E β̃post | β̂pre ∈ B = βpost − Σ12 Σ−1
22 βpre for any B.
i
i
h
h
3. Var β̂post |β̂pre ∈ BN S ≤ Σ11 = Var β̂post .
Proposition 2 gives a general formula for the expectation of β̂post and β̃post , from which
we see that if the parallel pre-trends assumption does not hold, both estimators will tend to
be biased relative to the population difference-in-difference coefficient βpost , which itself will
differ from the true treatment effect if the pre-trend would have continued in the post period.
Nonetheless, if we condition on none of the pre-treatment coefficients being significant, the
true variance of β̂post is smaller than the unconditional covariance. Thus, confidence intervals
using the traditional standard errors may have coverage above or below the nominal rate
depending on the relative size of the bias of the coefficient relative to that of the standard
error – a fact which we confirm in our simulations below.
I now turn to understanding the direction of the bias in the case where the pre-trends are
monotone. To do this, I first introduce the following assumption on the covariance structure.
Assumption 2. Σ has a common term σ 2 on the diagonal and a common term ρ > 0 off
of the diagonal, with σ 2 > ρ.
4
Note that this does not account for possible errors in the estimation of standard errors owing to the
conditioning. The simulations below, which use estimates of Σ, suggest that this is likely second-order.

9

This is a natural assumption if the data used in the study come from repeated crosssections (e.g. the ACS or Census), and the outcome of interest has constant variance across
periods. To see why this is the case, again let ∆y¯t denote the difference in sample means between treatment and control in period t, and ∆µt the corresponding difference in population.
Note that for any t, β̂t = ∆ȳt − ∆ȳ0, and its
is βt = ∆µt − ∆µ0 .
 (unconditional) expectation

∆ ∆
∆
Let ∆
= Cov ∆
t = ∆ȳt − ∆µt . Then Cov β̂j , β̂k
j − 0 , k − 0 . In repeated crosssections, however,
the
error across periods shouldh bei independent, so that for

 measurement
 ∆
 
 ∆
+
Var
k ,
j 6= k, Cov β̂j , β̂k = Var 0 ≡ ρ > 0. Additionally, Var β̂k = Var ∆
0
 ∆
which is constant in k if Var k is the same across k. I note that for the result in Proposi 
tion 3, Assumption 2 can be relaxed so that Var ∆
k need only be equal for all k < 0, which
allows for heterogenous treatment effects. I leave it to future work to determine whether
the following result holds in a panel setting with serially correlated individual errors – I
conjecture that it should still hold so long as the serial correlation is not too large.
Proposition 3 (Sign of bias under upward pre-trend). Suppose that βt < 0 for all t < 0. If
Assumption 2 holds, then
i
h
1. E β̂post | β̂pre ∈ BN S > βpre .
h
i
2. E β̃post | β̂pre ∈ BN S > βpre .
The analogous result holds replacing ">" with "<" and vice versa.
Proposition 3 derives the direction of the bias from conditioning on not observing a
significant pre-trend in the case where in population there is an upward pre-trend, in the
sense that all of the population pre-period coefficients are less than zero. A leading example
of this is when the pre-trend is monotone, although the condition in Proposition 3 is weaker
than this. It follows from the proposition that when there is truly an upward pre-trend, β̂post
and β̃post are both biased upwards conditional on the pre-test, thus exacerbating the bias
relative to the true treatment effect, assuming the monotone trend would have continued
into the post period.

3.4

Median-unbiased estimation and corrected confidence intervals
under non-parallel pre-trends

In this section, I derive alternative estimators, along with associated confidence intervals,
that correct for the bias induced by conditioning on having not observed a significant pretrend when the parallel pre-trends assumption fails.
10

The first estimator is an optimal median-unbiased estimator for the population differencein-difference coefficient βpost , i.e. the population limit of OLS without conditioning. Although βpost is generally not equal to the treatment effect of interest when the parallel pretrends assumption fails, the extent to which βpost differs from the treatment effect is given
by the omitted variable bias formula, and so use of this estimator allows us to recover this
intuition from OLS even when we condition on having observed non-significant pre-trends.
I also show that we can modify the first estimator to obtain a median-unbiased estimate of
the population regression coefficient from a specification that allows the control group to have
a polynomial trend in the pre-period. Consider, for instance, the regression specification:

yit = αt + γpre × t × T reatmenti +

X

γs × T reatmenti × 1[t = s] + γmain T reatmenti + it .

s>0

(2)
If there is truly a linear pre-trend, then the population OLS coefficient γpost := γ1 will equal
the treatment effect in the first post-period assuming the pre-trend would have continued
absent treatment. If there is a non-linear pre-trend, then γpost may serve as a close approximation to the treatment effect of interest to the extent that the best-fit linear pre-trend
serves as a close approximation to the counterfactual trend. The second estimator I derive
provides median-unbiased estimates and valid confidence intervals for γpost , conditional on
having observed non-significant pre-trends. I also show that an analogous estimator can
be derived for modifications of specification (2) that allow for higher-order (e.g. quadratic)
polynomial pre-trends.
In order to derive these estimators, it is first useful to note that we can write γpost =
βpost − ηγ0 βpre for a suitably defined vector ηγ . Intuitively, in settings where each pre-period
is given equal weight in the regression – e.g. balanced panel or repeated cross-sections of
equal size – the difference between βpost in regression (1) and γpost in regression (2) will be
equal to the predicted value at t = 1 of the best-fit line through the pre-trends coefficients
β−K , . . . , β0 (where β0 = 0). That is, letting X be the matrix with 1s in the first column
and (0, −1, ..., −K) in the second column, and M−1 the selection matrix that selects all
but the first column, ηγ0 = (1, 1)0 (X 0 X)−1 X 0 M−1 . Lemma 6 in the Appendix formalizes
the relationship between βpost and γpost , and shows that if we instead wanted to include a
quadratic (or higher-order polynomial) trend in (2) in place of the linear trend, then we would
merely need to add an additional column to X with quadratic (or higher-order polynomial)
time terms.
Having noted that βpost and γpost are both linear combinations of β = (βpost , βpre ), we
11

can make use of results by Lee et al. (2016) and Andrews and Kasy (2017) to derive the
distribution of linear combinations of β̂ conditional on β̂pre falling in BN S . To do this,
we first note that we can define a matrix AN S and vector bN S such that β̂pre ∈ BN S iff
!
0
I
1×K
K×K
AN S β̂ ≤ bN S . In particular, it is easy to verify that this holds for AN S =
01×K −IK×K
!
p
cα × diag(Σ)
p
and bN S =
. We now derive the distribution for linear combinations of
cα × diag(Σ)
β̂ conditional on Aβ̂ ≤ b for arbitrary A and b, noting that this nests conditioning on
β̂pre ∈ BN S as a special case.
Proposition 4. Let β̂ = (β̂post , β̂pre ) and η 6= 0 be in RK+1 . Let B = {β|Aβ ≤ b} for some A
and b. Define c = Ση/(η 0 Ση) and Z = (I −cη 0 )β̂. Then the distribution of η 0 β̂ conditional on
β̂ ∈ B and Z is truncated normal with (untruncated) mean η 0 β and (untruncated) variance
η 0 Ση, with truncation points V − and V + given by

bj − (Az)j
{j:(Ac)j <0}
(Ac)j
bj − (Az)j
V + (z) = min
.
{j:(Ac)j >0}
(Ac)j

V − (z) =

max

(3)
(4)

Proposition 5. Let η 6= 0 be in RK+1 , and B = {β|Aβ ≤ b} for some A and b. Assume
[l,u]
that β̂ ∈ B with positive probability, and that Σ is full rank. Let Fµ,σ2 denote the CDF of the
normal distribution with mean µ and variance σ 2 truncated to [l, u]. Define b̂α (η 0 β̂, z) to be
[V − (z),V + (z)] 0
the value of x that solves Fx,η0 Ση
(η β̂) = α, for V − and V + as defined in Proposition
4. Then for any α ∈ (0, 1),


0

0



P b̂α (η β̂, Z) ≤ η β | β̂ ∈ B = α.
Further, if the parameter space for β is an open set, then b̂α is uniformly most concentrated in the class of level-α quantile-unbiased estimators, in the sense that for any other
level-α quantile unbiased estimator b̃α , and any loss function L(x, η 0 β) that attains its minimum at x = η 0 β and is increasing as x moves away from η 0 β,
h 

i
h 

i
E L b̂α (η 0 β, Z) , η 0 β | β̂ ∈ B ≤ E L b̃α (η 0 β, Z) , η 0 β | β̂ ∈ B .
Thus, conditional on β̂ ∈ B, b̂0.5 (η 0 β̂, Z) is an (optimal) median-unbiased estimate of η 0 β,
and the interval [b̂α/2 (η 0 β̂, Z), b̂1−α/2 (η 0 β̂, Z)] is a 1 − α confidence interval.
TN
Corollary 1 (Median Unbiased Estimator for βpost ). Let β̂post
:= b̂0.5 (η 0 β̂, Z) be the estimator

12

from Proposition 5 corresponding with A = AN S , b = bN S and η = e1 , the first basis vector.
TN
Then, conditional on β̂pre ∈ BN S , β̂post
is an optimal median unbiased estimator for βpost .
0
TN
Likewise, CIβ := [b̂α/2 (η β̂, Z), b̂1−α/2 (η 0 β̂, Z)] is a 1 − α confidence interval for βpost .
TN
:= b̂0.5 (η 0 β̂, Z) be the estimator
Corollary 2 (Median Unbiased Estimator for γpost ). Let γ̂post
from Proposition 5 corresponding with A = AN S , b = bN S and η = (1, ηγ ). Then, conditional
TN
on β̂pre ∈ BN S , γ̂post
is an optimal median unbiased estimator for γpost . Likewise, CIγT N :=
[b̂α/2 (η 0 β̂, Z), b̂1−α/2 (η 0 β̂, Z)] is a 1 − α confidence interval for γpost .

4
4.1

Simulations
Simulation DGP

I conduct simulations in which we observe a control group and a treatment group for K + 1
periods prior to treatment assignment and 1 period after treatment assignment. In the first
iid
specification, yit = it for  ∼ N (0, σ 2 ), regardless of treatment status, so the paralleltrends assumption holds and there is no treatment effect. In the second specification, yit =
iid
δ × t × T reatmenti + it for δ > 0, where again  ∼ N (0, σ 2 ), so there is a linear upward
pre-trend that continues into the post period. By construction, Assumption 2 holds.
For each of 1 million simulated datasets, I estimate regression (1). Below, I report the
distribution of β̂post and its confidence intervals conditional on the K pre-trends coefficients
being statistically insignificant for K = 0, . . . , 8 (with K = 0 denoting the unconditional
case). I do the same for the modified estimator β̃post , as well as the median unbiased estimators and adjusted confidence intervals.

4.2

Simulation results when the parallel trends assumption holds

The left panel of Table 1 shows the distribution of β̂post when the parallel trends assumption
is true. The first row (K = 0) shows that without conditioning on the pre-trends, the
average standard error matches the actual standard deviation of the estimator, and the
associated confidence intervals achieve nominal coverage. Looking at the other rows of
the table, however, we see that when we condition on insignificant pre-trends the average
standard error is below the actual standard deviation of the estimator, and the null rejection
probability (size) falls markedly below 0.05. This phenomenon becomes more acute as the
number of pre-periods we condition on grows. In all cases, however, β̂post is unbiased.
Turning to the right panel, we see that β̃post is also unbiased, has lower variability than
β̂post , and the estimated standard errors match the actual standard deviation of the estimator.
13

In the K = 1 case, these represent a 13% reduction in estimated standard errors, and an
11% reduction in the true variability of the estimator; in the K = 8 case, these numbers rise
to 26% and 15%.

K

Bias

0
1
2
3
4
5
6
7
8

-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
0.000
0.000

β̂post
Mean
Actual
S.E.
S.D.
0.127
0.127
0.127
0.123
0.127
0.120
0.127
0.118
0.127
0.116
0.127
0.115
0.127
0.113
0.127
0.112
0.127
0.111

Size

Bias

0.050
0.043
0.039
0.035
0.032
0.030
0.028
0.027
0.026

-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000

β̃post
Mean
Actual
S.E.
S.D.
0.110
0.110
0.103
0.103
0.100
0.100
0.098
0.098
0.097
0.097
0.096
0.096
0.095
0.095
0.094
0.094

Size
0.050
0.050
0.050
0.050
0.050
0.050
0.050
0.050

Table 1: Performance of β̂post and β̃post when the parallel trends assumption is true and we
condition on accepting pre-trends for K periods

4.3

Simulation results when there is actually an upward linear trend

Table 2 shows the performance of β̂post and β̃post when the true model has an upward pretrend. I set δ = 0.065, so that the slope of the pre-trend is roughly half of the unconditional
standard error for β̂post .
Looking at the first row, we see that unconditionally β̂post has a mean of βpost = 0.065, i.e.
precisely the slope of the pre-trend – this is the usual “omitted variable bias” intuition that
most people have in mind when evaluating the potential bias caused by non-parallel trends.
However, as we move down the table we see that this bias becomes substantially larger once
we condition on none of the pre-trends coefficients being statistically significant. Once we
condition on 4 insignificant pre-period coefficients, we see that the conditional mean is more
than twice that of the unconditional.
As before, however, the standard error of β̂post conditional on accepting the pre-trends
is less than the mean estimated standard error. As a result, for K small, when the bias is
not too large, the traditional confidence intervals cover βpost more than 95% of the time.
However, for larger K the bias dominates, and the CIs cover βpost less than 95% of the time.
In all specifications, the rejection probability at 0 exceeds 0.05 owing to the underlying trend,
but the over-rejection problem becomes more extreme as we condition on more insignificant
pre-period coefficients.
14

In the right panel, we see that while β̃post outperformed β̂post when the parallel trends
assumption was true, its performance is even worse under the linear pre-trend. The bias
relative to βpost is even larger than for β̂post , and the probability of rejecting βpost exceeds
0.05 in all cases.
β̂post
K
0
1
2
3
4
5
6
7
8

Accept
Pretrends
1.000
0.920
0.780
0.578
0.352
0.168
0.059
0.015
0.003

β̃post

Mean
β̂post

Mean
S.E.

Actual Reject
S.D.
βpost

Reject
0

Mean
β̃post

Mean
S.E.

Actual Reject
S.D.
βpost

Reject
0

0.065
0.073
0.088
0.109
0.136
0.167
0.201
0.238
0.278

0.127
0.127
0.127
0.127
0.127
0.127
0.127
0.127
0.127

0.127
0.122
0.118
0.114
0.110
0.107
0.105
0.102
0.100

0.081
0.081
0.090
0.112
0.154
0.224
0.327
0.454
0.608

0.097
0.130
0.162
0.195
0.227
0.260
0.292
0.327

0.110
0.103
0.100
0.098
0.097
0.096
0.095
0.094

0.110
0.103
0.100
0.098
0.097
0.095
0.094
0.093

0.144
0.242
0.367
0.510
0.652
0.774
0.870
0.940

0.050
0.043
0.039
0.042
0.057
0.089
0.143
0.230
0.367

0.060
0.096
0.164
0.262
0.388
0.528
0.670
0.799

Table 2: Performance of β̂post and β̃post when there is a linear upward pre-trend (slope =
0.065) but we do not reject parallel pre-trends for K periods

4.4

Performance of adjusted estimates and confidence intervals

Tables 3 and 4 show the performance of the median-unbiased estimates and truncated normal
confidence intervals developed in Section 3.4 relative to the traditional point estimates and
confidence intervals. The comparisons are shown for both the specification without any pretrends and with the upward linear trend. In Table 3, we see that regardless of how many
TN
pre-trends coefficients we’ve conditioned on, β̂post
has a median very close to the true βpost ,
TN
i.e. 0 in the case without pre-trends and 0.065 in the case with a linear trend. Likewise, γ̂post
has a median close to 0 (i.e. γ) in all cases. Table 4 shows that CIβT N and CIγT N achieve
(roughly) nominal rejection probabilities for βpost and 0 respectively.
However, the improved properties of the CIβT N and CIγT N come at a power cost in many
of the specifications, as evidenced by the median widths of the confidence intervals shown in
Table 4. In all specifications, the median widths for the traditional CIs are smaller than those
for the CIβT N intervals. The differences between the standard CIs and the CIβT N intervals are
fairly small when we condition on few periods, but increase as the number of conditioning
periods increases. By contrast, the gap between the median widths of the traditional CIs
and the CIγT N intervals is large for small K, but decreases as K grows. For K above 6,
15

Median Estimate
No Pre-trend
Linear Pre-trend
K
1
2
3
4
5
6
7
8

β̂post
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

TN
β̂post
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

TN
γ̂post
-0.000
0.000
0.000
-0.000
-0.000
0.000
0.000
0.000

β̂post
0.073
0.087
0.108
0.135
0.166
0.201
0.237
0.276

TN
β̂post
0.065
0.065
0.065
0.065
0.064
0.062
0.063
0.072

TN
γ̂post
-0.001
0.000
0.000
-0.000
-0.002
-0.002
0.002
0.007

Table 3: Traditional and Median-Unbiased Point Estimates for βpost
the CIγT N median widths are actually smaller than the traditional ones when the parallel
pre-trends assumption holds. Interestingly, the median widths for both the CIβT N and CIγT N
are much larger in the case with the linear pre-trend, so the simulations suggest that the
penalty for using the adjusted estimates is smaller when the parallel pre-trends assumption
actually holds.

5

Conclusion

This paper illustrates that many of the properties we expect of the traditional OLS differencein-difference estimator do not hold conditional on having passed the test for pre-trends that
is standard in empirical work. I describe two approaches to address this issue. The first
provides unbiased estimates and improved precision when the parallel trends assumption
is correct (both conditional on the pre-test and unconditionally), but can have undesirable
properties if the parallel trends assumption is false and we fail to reject it in our pre-test.
I recommend using this estimator only in cases where the researcher has strong ex ante
reasons to believe in the parallel trends assumption and the test of parallel trends serves
only as a sanity check. The second approach, which I recommend in most cases, provides
corrected point estimates and confidence intervals that condition on having not observed a
significant pre-trend, thereby recovering many of the desirable properties of OLS even when
the parallel trends assumption is false but we fail to reject it in our pre-test. I hope that
these corrections will be useful for the applied researcher. I caution, however, that these
corrections are not a panacea; a researcher who accepts a research design with a non-linear
underlying trend will generally not recover an unbiased estimate of the treatment effect
of interest even when applying these corrections. Therefore, in addition to applying these
16

No Pre-trend
Rejects βpost
K
1
2
3
4
5
6
7
8

CI T rad
0.043
0.039
0.035
0.032
0.030
0.028
0.027
0.026

CIβT N
0.050
0.050
0.050
0.050
0.050
0.050
0.050
0.050

Linear Pre-trend

Median Width
CI T rad
0.496
0.496
0.496
0.496
0.496
0.496
0.496
0.496

CIβT N
0.517
0.549
0.582
0.613
0.642
0.670
0.696
0.721

Rejects βpost
CI T rad
0.043
0.039
0.042
0.057
0.089
0.143
0.230
0.367

CIβT N
0.050
0.050
0.051
0.051
0.052
0.055
0.054
0.057

Median Width
CI T rad
0.496
0.496
0.496
0.496
0.496
0.496
0.496
0.496

CIβT N
0.521
0.580
0.678
0.820
1.018
1.262
1.459
1.974

(a) Comparison of Traditional CIs versus CIβT N

No Pre-trend
Rejects 0
K
1
2
3
4
5
6
7
8

CI T rad
0.043
0.039
0.035
0.032
0.030
0.028
0.027
0.026

CIγT N
0.050
0.050
0.050
0.050
0.050
0.050
0.050
0.050

Linear Pre-trend

Median Width
CI T rad
0.496
0.496
0.496
0.496
0.496
0.496
0.496
0.496

CIγT N
1.062
0.773
0.651
0.581
0.536
0.504
0.481
0.463

Rejects 0
CI T rad
0.081
0.090
0.112
0.154
0.224
0.327
0.454
0.608

CIγT N
0.050
0.050
0.050
0.050
0.050
0.051
0.049
0.048

Median Width
CI T rad
0.496
0.496
0.496
0.496
0.496
0.496
0.496
0.496

CIγT N
1.081
0.839
0.784
0.796
0.843
0.906
0.975
1.053

(b) Comparison of Traditional CIs versus CIγT N

Table 4: Comparison of Traditional and Truncated Normal Confidence Intervals
corrections, I also caution researchers not to exclusively rely on the test for pre-trends as
validation for their research design.

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A

Proofs

Lemma 1. Let β̃post = β̂post − Σ12 Σ−1
22 β̂pre . Then β̃post and β̂pre are independent.
Proof. Note that by assumption, β̂post and β̂pre are jointly normal. Since β̃post is a linear
combination of β̂post and β̂pre , it follows that β̂pre and β̃post are jointly normal. It thus
suffices to show that β̂pre and β̃post are uncorrelated. We have:


h
i

Cov β̂pre , β̃post = E β̂pre − βpre (β̂post − βpost ) − Σ12 Σ−1
(
β̂
−
β
)
pre
pre
22
= Σ12 − Σ12 Σ−1
22 Σ22
=0

We now prove Proposition 2 for the case of generic βpre , and then use these results to
prove Proposition 1 for the special case where βpre = 0.

19

Proof of Proposition 2
1. Note that by construction, β̂post = β̃post + Σ12 Σ−1
22 β̂pre . It follows that

h
i
h
i
h
i
E β̂post | β̂pre ∈ B = E β̃post | β̂pre ∈ B + Σ12 Σ−1
E
β̂
|
β̂
∈
B
pre
pre
22
h
i
h
i
= E β̃post + Σ12 Σ−1
22 E β̂pre | β̂pre ∈ B
h
i
h
i
−1
= E β̂post − Σ12 Σ−1
β̂
+
Σ
Σ
E
β̂
|
β̂
∈
B
12 22
pre
pre
22 pre
h
i
−1
−1
= βpost − Σ12 Σ22 βpre + Σ12 Σ22 E β̂pre | β̂pre ∈ B
 h
i

= βpost + Σ12 Σ−1
E
β̂
|
β̂
∈
B
−
β
pre
pre
pre
22
where the second line uses the independence of β̃post and β̂pre from Lemma 1, and the
third and fourth use the definition of β̃post , βpost , and βpre .
h
i
h
i
2. By Lemma 1, β̃post is independent of β̂pre , so E β̃post | β̂pre ∈ B = E β̃post for any B.
Further,
h
i
h
i
E β̃post = E β̂post − Σ12 Σ−1
β̂
= βpost − Σ12 Σ−1
pre
22
22 βpre .
3. Note that since β̂post = β̃post + Σ12 Σ−1
22 β̂pre , for any set B,
h

Var β̂post | β̂pre

i

h

i

h
i
−1
∈ B = Var β̃post | β̂pre ∈ B + Var Σ12 Σ22 β̂pre | β̂pre ∈ B


+ 2 Cov β̃post , Σ12 Σ−1
β̂
|
β̂
∈
B
pre
pre
22
i
i
h
h
β̂
|
β̂
∈
B
= Var β̃post + Var Σ12 Σ−1
pre
22 pre

(5)

h
i
where we use the independence of β̃post and β̂pre from Lemma 1 to obtain that Var β̃post | β̂pre ∈ B =
h
i
Var β̃post and that the covariance term equals 0. It follows that
h
i
h
i
h
i
h
i
−1
Var β̂post |β̂pre ∈ BN S − Var β̂post = Var Σ12 Σ−1
β̂
|
β̂
∈
B
β̂
−
Var
Σ
Σ
pre
pre
N
S
12
pre
22
22

h
i
h
i
0
= (Σ12 Σ−1
)
Var
β̂
|
β̂
∈
B
−
Var
β̂
(Σ12 Σ−1
pre
pre
NS
pre
22
22 )
h
i
h
i
which is (weakly) negative since Var β̂pre | β̂pre ∈ BN S − Var β̂pre is negative semidefinite by Lemma 3 below. 
20

Definition 1 (Symmetric Rectangular Truncation About 0). We say that B ⊂ RK is a
symmetric rectangular truncation around 0 if B = {y ∈ RK | − bj ≤ yj ≤ bj , for j =
1, . . . , K} for some non-negative constants b1 , . . . , bK .
Lemma 2. Suppose Y ∼ N (0, Σ) is a K-dimensional multivariate normal, and B is a
symmetric rectangular truncation around 0. Then E [Y | Y ∈ B] = 0.
Proof. Note that if Y ∼ N (0, Σ), then we also have −Y ∼ N (0, Σ). It follows that
E [Y | Y ∈ B] = E [−Y | (−Y ) ∈ B]. However, −bj ≤ yj ≤ bj iff −bj ≤ −yj ≤ bj , so
(−Y ) ∈ B iff Y ∈ B. It follows that

E [Y | Y ∈ B] = E [−Y | (−Y ) ∈ B]
= E [−Y | Y ∈ B]
= −E [Y | Y ∈ B]
which implies that E [Y | Y ∈ B] = 0.

Proof of Proposition 1
i
h
1. From Proposition 2, Part 1, it suffices to show that E β̂pre | β̂pre ∈ BN S − βpre equals
0. From
assumption, we have that βpre = 0, so this is equivalent
h the parallel pre-trends
i
to E β̂pre | β̂pre ∈ BN S = 0. Additionally, β̂pre is normally distributed with mean
βpre
h = 0. Since BNiS is a symmetric rectangular truncation around 0, it follows that
E β̂pre | β̂pre ∈ BN S = 0 by Lemma 2.
2. This is a restatement of Proposition 2, Part 3 (for the special case where βpre = 0).
h
i
h
i
3. By Lemma 1, β̃post is independent of β̂pre , so E β̃post | β̂pre ∈ B = E β̃post for any B.
And
h
i
h
i
E β̃post = E β̂post − Σ12 Σ−1
β̂
= βpost − Σ12 Σ−1
pre
22
22 βpre = βpost
since βpre = 0 under parallel pre-trends.
h
i
h
i
4. Note that by independence, Var β̃post | β̂pre ∈ B = Var β̃post . Further,

21

h

Var β̃post

i

i
h
−1
= Var β̂post − Σ12 Σ22 β̂pre
h
i
h
i




−1 0
−1
= Var β̂post + Σ12 Σ−1
Var
β̂
Σ
Σ
−
2Σ
Σ
Cov
β̂
,
β̂
pre
12 22
12 22
post
pre
22
−1
= Σ11 + Σ12 Σ−1
22 Σ21 − 2Σ12 Σ22 Σ21

= Σ11 − Σ12 Σ−1
22 Σ21
To prove the desired inequality, note that from (5),
h
i
h
i
h
i
Var β̂post | β̂pre ∈ B − Var β̃post = Var Σ12 Σ−1
β̂
|
β̂
∈
B
pre
pre
22
which is weakly positive. 
We now prove a series of Lemmas leading up to the proof of Proposition 3.
Lemma 3. Suppose Y is a k-dimensional multivariate normal, Y ∼ N (µ, Σ), and let
B = {y | aj ≤ yj ≤ bj , ∀j}, where aj < bj for all j. Letting Dµ denote the Jacobian operator
with respect to µ, we have
1. Dµ E [Y | Y ∈ B, µ] = Var [Y | Y ∈ B, µ] Σ−1 .
2. Var [Y | Y ∈ B] − Σ is negative semi-definite.
Proof.5
Define the function G : R2k → R by
Z

β1

Z

βk

···

G(α1 , . . . , αk , β1 , . . . , βk ) =
α1

φΣ (ỹ1 , ..., ỹk ) dy˜1 . . . dy˜k
αk

1

for φΣ (x) = det(2πΣ)− 2 exp(− 12 x0 Σ−1 x) the PDF of the N (0, Σ) distribution. Let αj (µ) =
aj − µj and βj (µ) = bj − µj , and define the function H : Rk → R by
H(µ) = G(α(µ), β(µ)).
R
We now argue that H is log-concave in µ. Note that we can write G(α, β) = y∈RK g1 (y, α, β)·
g2 (y, α, β) dy, for g1 (y, α, β) = φΣ (y) and g2 (y, α, β) = 1[α ≤ y ≤ β]. Observe that g1
and g2 are both log-concave in (y, α, β), and so their product is also log-concave. Moreover, the marginalization of a log-concave function with respect to one of its arguments is
5

I am grateful to Alecos Papadopolous, whose answer on StackOverflow to a related question inspired
this proof.

22

log-concave by Prekopa’s theorem (see, e.g. Theorem 3.3 in Saumard and Wellner (2014)),
so G(α, β) is log-concave. Finally, since α(µ) and β(µ) are linear in µ and G is log-concave
in (α, β), their composition, H, is log-concave in µ (see, e.g., equation (3.15) in Boyd and
Vandenberghe (2004)).
Now, applying Leibniz’s rule, we have that

∂ log H
=−
∂µ1

R

β2
α2

···

R βk
αk


Rβ
Rβ
φΣ (β1 , ỹ2 ..., ỹk )dỹ2 . . . ỹk − α22 · · · αkk φΣ (α1 , ỹ2 ..., ỹk )dỹ2 . . . ỹk
Rβ
R β1
· · · αkk φΣ (ỹ1 ..., ỹk )dỹ
α1

= f1 (a1 | Y ∈ B) − f1 (b1 | Y ∈ B)
where f1 (· | Y ∈ B) denotes the PDF of the marginal distribution of Y1 when Y is truncated
to B. By analogous argument for the other partial derivatives, we have that the 1 × k
gradient of log H is given by

Dµ log H = (∆f )0

(6)

where ∆f is the k × 1 vector with jthe element equal to fj (aj | Y ∈ B) − fj (bj | Y ∈ B).
Manjunath and Wilhelm (2012) show that E [Y | Y ∈ B, µ] = µ + Σ(∆f ). It thus follows
from (6) that

E [Y | Y ∈ B, µ] = µ + Σ(Dµ log H)0 .
Differentiating again with respect to µ, we have that the k×k Jacobian of E [Y | Y ∈ B, µ]
with respect to µ is given by

Dµ E [Y | Y ∈ B, µ] = I + ΣDµ (Dµ log H)0 .

(7)

Since H is log-concave, Dµ (Dµ log H)0 is the Hessian of a concave function, and thus is
negative semi-definite. Next, note that by definition,

R b1
a

···

E [Y | Y ∈ B, µ] = R1b1
a1

R bk

y φΣ (y − µ) dy
a
.
Rkbk
· · · ak φΣ (y − µ) dy

23

Thus, by the product rule,

R b1
Dµ E [Y | Y ∈ B, µ] =

R bk

y Dµ φΣ (y − µ) dy
+
R bk
φ
(y
−
µ)
dy
·
·
·
ak Σ
a1

Z
Z b 1
Z bk
y φΣ (y − µ) dy · Dµ
···
a1

···

ak

R b1

Z

−1

bk

φΣ (y − µ) dy

···

a1

ak

a1

b1

.

ak

(8)
Note that
Dµ φΣ (y − µ) = φΣ (y − µ) · (y − µ)0 Σ−1 .
The first term in (8) thus becomes

R b1
a1

Rb
· · · akk y(y − µ)0 φΣ (y − µ) dy −1
Σ =
R b1
R bk
·
·
·
φ
(y
−
µ)
dy
Σ
a1
ak

(E [Y Y 0 | Y ∈ B, µ] − E [Y | Y ∈ B, µ] µ0 ) Σ−1 .
Applying the chain-rule, the second term in (8) becomes

R b1
−

a1

R b1
R bk
y
φ
(y
−
µ)
dy
·
·
·
·
(y − µ)0 φΣ (y − µ) dy −1
Σ
ak
a1
ak
Σ =
hR
i2
R bk
b1
· · · ak φΣ (y − µ) dy
a1

−E [Y | Y ∈ B, µ] E [Y | Y ∈ B, µ]0 + E [Y | Y ∈ B, µ] µ0 Σ−1 .

···

R bk

Substituting back into (8), we have


Dµ E [Y | Y ∈ B, µ] = E [Y Y 0 | Y ∈ B, µ] − E [Y | Y ∈ B, µ] E [Y | Y ∈ B, µ]0 Σ−1
= Var [Y | Y ∈ B, µ] Σ−1

(9)

which establishes the first result. Additionally, combining (7) and (9), we have that

Var [Y | Y ∈ B, µ] Σ−1 = I + ΣDµ (Dµ log H)0

24

(10)

which implies that

Var [Y | Y ∈ B, µ] − Σ = Σ Dµ (Dµ log H)0 Σ.

(11)

Thus, for any vector x ∈ Rk ,

x0 (Var [Y | Y ∈ B, µ] − Σ) x = x0 (Σ Dµ (Dµ log H)0 Σ) x
= (Σx)0 (Dµ (Dµ log H)0 ) (Σx)
≤0
where the inequality follows from the fact that Dµ (Dµ log H)0 is negative semi-definite. Since
Var [Y | Y ∈ B, µ] − Σ is symmetric, it follows that it is negative semi-definite, as we desired
to show. 
Lemma 4. Suppose that Σ satisfies Assumption 2. Then for ι the vector of ones and some
0
c1 > 0, ι0 Σ−1 = c1 ι0 . Additionally, Σ12 Σ−1
22 = c2 ι , for a constant c2 > 0.
Proof. Note that we can write Σ = Λ + ριι0 , where Λ = (σ 2 − ρ)I. It follows from the
Sherman-Morrison formula that:

ρ2 Λ−1 ιι0 Λ−1
1 + ρ2 ι0 Λ−1 ι
ρ2 (σ 2 − ρ)−2 ιι0
.
= (σ 2 − ρ)−1 I −
1 + ρ2 (σ 2 − ρ)−1 ι0 ι

Σ−1 = Λ−1 −

Thus:
ι0 Σ−1 =


ρ2 (σ 2 − ρ)−2 ιι0
0
2
−1
ι (σ − ρ) I −
=
1 + ρ2 (σ 2 − ρ)−1 ι0 ι


ρ2 (σ 2 − ρ)−1 ι0 ι
2
−1
(σ − ρ)
1−
ι0 =
1 + ρ2 (σ 2 − ρ)−1 ι0 ι


1
2
−1
(σ − ρ)
ι0 .
1 + ρ2 (σ 2 − ρ)−1 ι0 ι
|
{z
}
:=c1

Since σ 2 − ρ > 0, all of the terms in c1 are positive, and thus c1 > 0, as needed. Additionally,
25

note that if Σ satisfies Assumption 2, then Σ22 also satisfies Assumption 2 and Σ12 = ρι0 . It
0
0
follows that Σ12 Σ−1
22 = ρc1 ι = c2 ι for c2 = ρc1 > 0.

Lemma 5. Suppose Y ∼ N (0, Σ) for Σ satisfying Assumption 2. Let B = {y ∈ RK | aj ≤
y ≤ bj for all j}, where −bj < aj < bj for all j. Then for ι the vector of ones, E [ι0 Y | Y ∈ B] =
E [Y1 + . . . + Yk | Y ∈ B] is elementwise greater than 0.
Proof. For any x ∈ RK such that xj ≤ bj for all j, define B X (x) = {y ∈ RK | xj ≤ y ≤
bj for all j}. Let b = (b1 , . . . , bK ). Note that B X (−b) is a symmetric rectangular truncation


around 0, so from Lemma 2, we have that E Y | Y ∈ B X (−b) = 0. Now, define



g(x) = E ι0 Y | Y ∈ B X (x) .
From the argument above, we have that g(−b) = 0, and we wish to show that g(a) > 0.
Note that by the mean-value theorem, for some t ∈ (0, 1),
g(a) = g(−b) + (a − (−b)) ∇g (ta + (1 − t)(−b))
= (a + b)∇g (ta + (1 − t)(−b))
=: (a + b)∇g(xt ).
By assumption, (a + b) is elementwise greater than 0. It thus suffices to show that all
∂g(xt )
> 0.
elements of ∇g (xt ) are positive. WLOG, we show that
∂xK
Using the definition of the conditional expectation and Leibniz’s rule, we have
∂g(xt )
=
∂xK

! Z
!−1 
Z b1
Z bK
Z bK
b1
∂ 
=
···
(y1 + . . . + yK ) φΣ (y) dy1 . . . dyK
···
φΣ (y) dy1 . . . dyK
∂xtK
xt1
xtK
xt1
xtK
!!
Z b1
Z bK
Z b1
Z bK−1
y−K
···
(y1 + . . . + yK ) φΣ (y) dy1 . . . dyK ×
···
φΣ
dy1 . . . dyK−1
xtK
xt1
xtK
xt1
xtK−1
!!
!
Z b1
Z bK−1
Z b1
Z bK
y−K
t
−
···
(y1 + . . . + yK−1 + xK ) φΣ
dy1 . . . dyK−1 ×
···
φΣ (y) dy1 . . . dyK
xtK
xt1
xtK−1
xt1
xtK
!−2
Z b1
Z bK
×
···
φΣ (y) dy1 . . . dyK
(12)
xt1

xtK

26

where φΣ (y) denotes the PDF of a multivariate normal with mean 0 and variance Σ. It
∂g(xt )
follows from (12) that
> 0 if and only if
∂xK
R b1
xt1

···

R bK

(y1 + . . . + yK ) φΣ (y) dy1 . . . dyK
>
R bK
·
·
·
φ
(y)
dy
.
.
.
dy
t
t
Σ
1
K
x
x

xt

R kb1
1

R b1
xt1

···

k

R bK−1
xtK−1

(y1 + · · · + yK−1 + xtK ) φΣ

R b1
xt1

···

R bK−1
xtK−1

φΣ

y−K
xtK

y−K
xtK

!!
dy1 . . . dyK−1

!!
dy1 . . . dyK−1

or equivalently,





E Y1 + . . . + YK | xtj ≤ Yj ≤ bj , ∀j > E Y1 + . . . + YK | xtj ≤ Yj ≤ bj , for j < K, YK = xtK .


It is clear that E YK | xtj ≤ Yj ≤ bj , ∀j > xtK , since xtK < bK and the Kth marginal density
of the rectangularly-truncated normal distribution is positive for all values in [xtK , bK ] (see
Cartinhour (1990)). It thus suffices to show that





E Y1 + . . . + YK−1 | xtj ≤ Yj ≤ bj , ∀j ≥ E Y1 + . . . + YK−1 | xtj ≤ Yj ≤ bj , for j < K, YK = xtK .
(13)
To see why (13) holds, let Ỹ−K = Y−K − Σ−K,K Σ−1
K,K YK , where a “−K” subscript denotes
all of the indices except for K. By an argument
to that in the Proof of Lemma
 analogous

1 for β̃post , one can easily verify that Ỹ−K ∼ N 0, Σ̃ for some Σ̃, and Ỹ−K is independent
of YK . Additionally, by an argument analogous to that in Proposition 1, Part 4, one can
show that Σ̃ = Σ−K,−K − Σ−K,K Σ−1
K,K ΣK,−K . We now argue that Σ̃ satisfies Assumption
2. Since Σ satisfies Assumption 2, so too does Σ−K,−K . Additionally, under Assumption 2,
−1
1
2
2
0
Σ−K,K = ρι and Σ−1
K,K = σ 2 , so Σ−K,K ΣK,K ΣK,−K equals ρ /σ times ιι , the matrix of ones.
The diagonal terms of Σ̃ are thus equal to σ̃ 2 = σ 2 − ρ2 /σ 2 , and the off-diagonal terms are
equal to ρ̃ = ρ − ρ2 /σ 2 , or equivalently ρ̃ = ρ(1 − ρ/σ 2 ). Since by Assumption 2, 0 < ρ < σ 2 ,
it is clear that σ̃ 2 > ρ̃. Additionally, 0 < ρ < σ 2 implies that 1 − ρ/σ 2 > 0, and hence ρ̃ > 0,
which completes the proof that Σ̃ satisfies Assumption 2.
Now, by construction, Y−K = Ỹ−K + Σ−K,K Σ−1
K,K YK . By Lemma 4, under Assumption 2,
−1
Σ−K,K ΣK,K = cι, for some c > 0, so Y−K = Ỹ−K + cιYK . From independence of Ỹ−K and
27

YK , it follows that



Y−K | YK = yk ∼ N c yk ι, Σ̃ .
h

i
Let h(µ) = E X|X ∈ B−K , X ∼ N µ, Σ̃ for B−K = {x̃ ∈ RK−1 |xtj ≤ x̃j ≤ bj , for j =


1, . . . , K − 1}. Then E ι0 Y−K | xtj ≤ Yj ≤ bj , for j < K, YK = yk = ι0 h(cyk ι). Hence,

∂ 0
E ι Y−K | xtj ≤ Yj ≤ bj , for j < K, YK = yk = ι0 (Dµ h|µ=cyk ι ) ι · c
∂yk
= ι0 Var [Y−K | Y−K ∈ B−K , YK = yk ] Σ̃−1 ιc
= ι0 Var [Y−K | Y−K ∈ B−K , YK = yk ] ιc1 c
≥0
where the second line follows from Lemma 3; the third line uses Lemma 4 to obtain that
Σ̃−1 ι = ιc1 for c1 > 0; and the inequality follows from the fact that Var [Y−K | Y−K ∈ B−K , YK = yk ]
is positive semi-definite and c1 and c are positive by construction. Thus, for all yk ∈ [xtk , bk ],



E Y1 + . . . + YK−1 | xtj ≤ Yj ≤ bj for j < K, YK = yk ≥


E Y1 + . . . + YK−1 | xtj ≤ Yj ≤ bj for j < K, YK = xtk .
By the law of iterated expectations, we have



E Y1 + . . . + YK−1 | xtj ≤ Yj ≤ bj , ∀j =
 


E E Y1 + . . . + YK−1 | xtj ≤ Yj ≤ bj for j < K, YK | xtj ≤ Yj ≤ bj , ∀j ≥
 


E E Y1 + . . . + YK−1 | xtj ≤ Yj ≤ bj for j < K, YK = xtK | xtj ≤ Yj ≤ bj , ∀j =


E Y1 + . . . + YK−1 | xtj ≤ Yj ≤ bj for j < K, YK = xtK
as we wished to show.

Proof of Proposition 3 From Proposition 2, Part 1, the first result is equivalent to
showing that
h
i
Σ12 Σ−1
E
β̂
−
β
|
β̂
∈
B
> 0.
pre
pre
pre
22
28

−1
0
By Lemma
h 4, Assumption 2 iimplies that Σ12 Σ22 = c1 ι for c1 > 0, so it suffices to show
that ι0 E β̂pre − βpre | β̂pre ∈ B is elementwise greater than zero. Note that by assumption
p
(β̂pre −βpre ) ∼ N (0, Σ22 ). Additionally, β̂pre ∈ BN S = {β̂pre : |β̂pre,j |/ Σjj ≤ cα for all j} iff
p
p
(β̂pre − βpre ) ∈ B̃N S = {β : aj ≤ βj ≤ bj } for aj = −cα Σjj − βpre,j and bj = cα Σjj − βpre,j .
Since βpre,j < 0 for all j, we have that −bj < aj < bj for all j. The first result then follows
immediately from Lemma 5.
From Proposition 2, Part 2, the second result is equivalent to Σ12 Σ−1
22 βpre < 0. But by
−1
0
assumption βpre < 0, and we’ve shown that Σ12 Σ22 = c1 ι > 0, from which the result follows.


Proof of Proposition 4
Proof. Follows immediately from Lemma 5.1 and Theorem 5.2 in Lee et al. (2016).
Proof of Proposition 5
Proof. The result follows immediately from Theorem 5 of Andrews and Kasy (2017), provided
we verify that the distribution of η 0 β̂ | Z, Aβ̂ ≤ b is continuous for almost every Z.
Note that by Proposition 4, η 0 β̂ | Z = z, Aβ̂ ≤ b is truncated normal with truncation
points V − (z) and V + (z), and hence, continuous if V − (z) < V + (z). Since conditional on
Aβ̂ ≤ b and Z = z, V − (z) ≤ η 0 β̂ ≤V + (z), we have V − (z)= V + (z) only if V − (z) = η 0 β̂.
It thus suffices to show that P η 0 β̂ = V − (Z) | Aβ̂ ≤ b = 0. Note though that



h 
i
P η 0 β̂ = V − (Z) = E P η 0 β̂ = V − (z) | Z = z
=0


where for any fixed value z, P η 0 β̂ = V − (z) | Z = z = 0 since η 0 β̂ and Z are independent
by construction (see Lee et al. (2016) for details) and the distribution of η 0 β̂ is continuous
since β̂ is normally distributed, Σ is full rank, and η 6= 0. It follows that



0 = P η 0 β̂ = V − (Z)

 


 

0
−
0
−
= P η β̂ = V (Z) | Aβ̂ ≤ b P Aβ̂ ≤ b + P η β̂ = V (Z) | Aβ̂ 6≤ b P Aβ̂ 6≤ b

 

≥ P η 0 β̂ = V − (Z) | Aβ̂ ≤ b P Aβ̂ ≤ b .

29





Since P Aβ̂ ≤ b > 0 by assumption, it follows that P η 0 β̂ = V − (Z) | Aβ̂ ≤ b = 0, as
needed.
Lemma 6. Consider the two regression specifications:

yit = αt +

X

βs × T reatmenti × 1[t = s] + βmain T reatmenti + it

(14)

s6=0

and

yit = αt +

X

γs × T reatmenti × 1[t = s] +

s>0

P
X

δp × T reatmenti × tp + it

(15)

p=0

where the latter allows for a P th order polynomial trend. Assume P ≤ K (so that the pretrend coefficients are identified). Then for any m > 0, the population OLS coefficient γm
0
from specification (15) can be written as βm − ηγ,m
βpre for βm and βpre from regression (14).
p
p
Additionally, let Xp = (0 , . . . , −K ) and let X be the (K + 1) × (P + 1) matrix with columns
X0 , . . . XP . Let M−1 be the selection matrix that selects all but the first column. Then

γm = βm − (m0 , . . . , mP )0 (X 0 X)−1 X 0 M−1 βpre

(16)

provided that the data-generating process puts equal weight on all periods in the sense that
a randomly-drawn observation (i, t) is equally likely to come from any period, and that
E [T reatmenti | t(i)] = E [T reatmenti ].
Proof. Let E∗ [Y |Z] denote the best linear predictor of Y given Z. Note that if E∗ [Y1 |Z] =
Zδ1 and E∗ [Y2 |Z] = Zδ2 , then E∗ [Y1 + Y2 |Z] = Z(δ1 + δ2 ).6 To prove the Lemma, we
will divide yit into three components. We will show that the regression of the first component on the RHS variables of (15) has coefficient 0 on 1[t = m] × T reatmenti ; the second
component has coefficient βm on 1[t = m] × T reatmenti ; and the third component has a
6

This is because
−1

E [X 0 Y ]

−1

E [X 0 (Y1 + Y2 )]

δ = E [X 0 X]
= E [X 0 X]

−1

−1

=δ1

=δ2

= E [X 0 X] E [X 0 Y1 ] + E [X 0 X] E [X 0 Y2 ] .
|
{z
} |
{z
}
See also FN 20 in Borusyak and Jaravel (2016).

30

coefficient on 1[t = m] × T reatmenti that is a linear combination of βpre , and is equal to
−(m0 , ..., mP )0 (X 0 X)−1 X 0 M−1 βpre if the DGP puts equal weight on all periods.
Note that from (14), we can write:

yit = αt + βmain T reatmenti + it +
|
{z
}
=:A
X
βs × T reatmenti × 1[t = s] +
s>0

|
X

{z

}

=:B

βs × T reatmenti × 1[t = s]

s<0

|

{z

}

=:C

where by construction, it is orthogonal to all of the RHS variables of (14). Note also that all
of the variables on the RHS of (15) can be written as linear combinations of the RHS variables
of (14), since all of the variables except for the polynomial terms are directly included in
P
(14), and T reatmenti × tp = s sp × T reatmenti × 1[s = t]. Hence, it is orthogonal to all
the RHS variables of (15) as well.
It is then clear that a regression of A on the RHS variables of (15) will load entirely on the
year effects and the main effect, and thus will have coefficient 0 on 1[t = m] × T reatmenti .
Likewise, a regression of B on the RHS variables of (15) will fit perfectly and have coefficient
βm on 1[t = m] × T reatmenti .
Now, consider the regression of C on the RHS variables of (15). Note that C is a linear
function of βpre , and therefore all of the regression coefficients from this regression will be a
linear combination of βpre , from which the first result follows. For the remainder of the proof,
we assume that the DGP puts equal weight on each period as defined in the statement of
the Lemma. By the Frisch-Waugh-Lovell theorem, we can first residualize the outcome and
all of the remaining covariates against the year dummies without changing the coefficients
on the other variables. The regression then becomes:

X

βs × (T reatmenti − T reatment) × 1[t = s] =

s<0

X

γ̃s × (T reatmenti − T reatment) × 1[t = s]+

s>0
P
X

δ̃p × (T reatmenti − T reatment) × tp + ˜it

p=0

where T reatment denotes the expected treatment probability. Additionally, note that the
31

LHS is always 0 for observations with t > 0, and that γ̃s gets positive weight only when
t = s > 0. It follows that δ̃ will be pinned down entirely by the observations with t ≤ 0, and
then we will have

γ̃s = −(s0 , ..., sP )0 (δ̃0 , ..., δ̃P )

(17)

so that the regression fits perfectly for observations with t > 0. To determine the values of
δ̃, we can therefore focus on the regression

P
X
(T reatmenti − T reatment) × tp × δ̃p + ˜it
βt (T reatmenti − T reatment) =

(18)

p=0

restricted to observations with t ≤ 0, where we define β0 = 0.
Consider first the problem of choosing δ̃ to minimize expected MSE for only the observations with T reatmenti = 1. Then the residuals are of the form: (1 − T reatment)2 (βt −
P
p 2
p δ̃p × t ) . Since by assumption we are equally likely to draw a treatment observation from any period, the expected mean squared
!
! error is minimized by setting δ̃ equal
0
0
,
to the coefficient from a regression of
on X, i.e. δ̃ = (X 0 X)−1 X 0
βpre
βpre
or equivalently, δ̃ = (X 0 X)−1 X 0 M−1 βpre . Likewise, consider minimizing expected MSE
for only the observations with T reatmenti = 0. Then the residuals are of the form:
P
(−T reatment)2 (βt − p δ̃p × tp )2 . Again, since a control observation is equally likely to
come from any period, expected MSE is minimized in this group for δ̃ = (X 0 X)−1 X 0 M−1 βpre .
Since this value of δ̃ minimizes expected MSE for both subgroups, it must minimize overall
MSE. Combining this expression for δ̃ with (17) gives the desired result.

32

