# The Effects of the Pandemic on Market Power and Profitability111We thank
Dick Startz, Ted Frech, Javier Birchenall, Gonzalo Vazquez-Bare, Francis X.
Diebold, Pablo Kurlat, Hashem Pesaran and the applied economics reading group
and applied lunch seminar at UC Santa Barbara, and the LACEA-LAMES 2022
attendants for valuable comments in an earlier version of this manuscript.

Juan Andres Espinosa-Torres222Department of Economics, University of Southern
California. Address: 3620 S. Vermont Ave. Kaprielian Hall (KAP) 363 Los
Angeles, CA 90089-0253. Email<EMAIL_ADDRESS>
Jaime Ramirez-Cuellar333Office of the Chief Economist, Microsoft, Redmond, WA,
USA. Email<EMAIL_ADDRESS>
(
First version: May 27, 2022 )

###### Abstract

We explore firm-level markup and profit rates during the COVID-19 pandemic for
a panel of 3,611 publicly traded firms in Compustat and find increases for the
average firm. We offer conditions to give markups and profit rate forecasts a
causal interpretation of what would have happened had the pandemic not
happened. Our estimations suggest that had the pandemic not happened, markups
would have been 4% and 7% higher than observed in 2020 and 2021, respectively,
and profit rates would have been 2.1 and 6.4 percentage points lower. We
perform a battery of tests to assess the robustness of our approach. We
further show significant heterogeneity in the impact of the pandemic on firms
by key firm characteristics and industry. We find that firms with lower than
forecasted markups tend to have lower stock-exchange tenure and fewer
employees.
Keywords: COVID-19, markups, profitability, firm dynamics, market power.
JEL classification: D22, D43, E3, L1.

##  1 Introduction

The COVID-19 pandemic deeply disrupted both supply and demand in a myriad of
markets. In order to understand these disruptions, we analyze markup and
profit rates from 2020 to 2021. This analysis is relevant for two reasons:
First, markup rates summarize the interactions between supply and demand in a
market and can therefore provide insight into the effects of the pandemic on
markets. Second, it is uncertain whether the pandemic exacerbated previous
trends in markup or profit rates, or whether it created the conditions for an
increase in market power. The goal of this paper is not only to document
recent movements in markups and profitability during the pandemic but also to
determine whether pre-pandemic trends can predict these movements or whether
the pandemic itself triggered the conditions for an increase in market power.

This paper makes three contributions to the recent literature on the market
power and profitability of publicly-traded firms in the US during the COVID-19
pandemic. First, we document both the increasing markup and profit rates, and
more volatile profit rates. Specifically, after average markup rates decreased
from 1.61 to 1.49 between 2016 and 2019, they rebounded to 1.54 in 2021.
Additionally, the average profit rate reached its all-time high at 21.6% in
2021, at least since the start of historical records in 1955. We present our
findings using both yearly and quarterly firm-level data.

Second, to further understand the rising trends in market power and
profitability during the pandemic, we evaluate whether previous trends in
firm-level markup and profit rates predict these increases. To do so, we use
separate firm-level Bayesian structural time series models to compute firm-
level forecasts of markups and profit rates, using information up to 2019. We
then calculate the difference between the observed value and its forecast. Our
results indicate that, for the average firm, the observed markup rate was 4.3%
and 6.6% lower than its forecast in 2020 and 2021, respectively. In contrast,
the average firm had a profit rate of 21.6% in 2021, which is 6.4 percentage
points higher than its forecast value. These findings suggest that while
markup rates were lower than expected, profit rates were higher than expected.

Third, we analyze the heterogeneous effects of rising market power and
profitability during the pandemic. Firms that performed poorly in terms of
markups during the pandemic tend to have become recently publicly traded and
have fewer employees. As for profitability, there seems to be low effect
heterogeneity across firms with different levels of employment, tenure in the
stock exchange, or market share.

Alternatively, when analyzing effect heterogeneity in markups by industry,
information, real estate, and chemical manufacturing experienced lower markups
than expected, whereas warehousing and entertainment had higher markups than
expected. When analyzing heterogeneity in profit rates by industry,
transportation, entertainment, and hospitality tended to perform poorly,
whereas warehousing and real estate outperform. These findings suggest that
the impact of the pandemic on market power and profitability was uneven across
firms.

To support these contributions, we perform a battery of statistical tests to
assess the efficacy of the Bayesian structural time series model in producing
forecasts of firm-level markup and profit rates. Specifically, we perform
three exercises to evaluate the quality of our models. First, we statistically
test whether a critical assumption of our Bayesian structural time series
(BSTS) models holds for most firms. Second, we evaluate whether these firm-
specific models accurately reflect the observed firm’s markup and profit rates
before the pandemic. Third, we confirm that our forecasts have adequate
forecast quality for a sample before the pandemic.

In addition to evaluating the quality of the BSTS models, we offer robustness
checks for our results by considering two alternative modeling choices and
frameworks. First, we vary the modeling choices of the BSTS model by choosing
alternative hyperparameters of the priors in the Bayesian model. These
hyperparameters attribute either more or less variation to the latent
components of the structural model. Second, we alternatively implement
forecasts using the flexible approach of local linear projections (Jordà,
2005). Particularly, when comparing the forecast quality of BSTS and local
linear projections models, the later tend to have wide confidence intervals
which suggests that local linear projections do not have as much power for
this data. Despite these changes, our main results remain unchanged. These
robustness checks help to strengthen the reliability and robustness of our
findings.

Our results inform the recent literature on the rising markups of publicly-
traded firms in the US (De Loecker and Warzynski, 2012; Hall, 2018; De Loecker
et al., 2020; Autor et al., 2020; Berry et al., 2019). In particular, De
Loecker et al. (2020) documents the rising nature of markups from 1955 to
2016. We extend their analysis by providing evidence suggesting that previous
trends in markup and profit rates do not predict average market power behavior
during the pandemic.

Additionally, we uncover the average behavior in markups and profitability and
analyze its heterogeneity by key firm characteristics such as size, market
share, and tenure on the stock exchange. We further argue that our methods may
provide a causal accounting of the effects of the pandemic on markups and
profit rates. We support our identification strategy on a similar fact noted
in Jordà et al. (2022): pandemics are exogenous and unpredictable to the
economy and, even more so, to granular firm dynamics. A similar approach in
the context of country-level impact of the pandemic uses shocks to economic
growth expectations as identifying variation (Chudik et al., 2021). Our
methodology is related to the event study method commonly used in finance and
economics (MacKinlay, 1997). However, we do not limit our analysis to
estimating the impact on the stock price of a given sample of companies.

Furthermore, a growing literature documents the effects of the COVID-19
pandemic on macroeconomic aggregates, such as inflation (Cavallo, 2020; Ball
et al., 2022; Binder and Kamdar, 2022; Dietrich et al., 2022) and economic
output (Ludvigson et al., 2020; McKibbin and Fernando, 2021; Baqaee and Farhi,
2020), and the consequential public policy response (Guerrieri et al., 2022;
Chudik et al., 2021; Woodford, 2022; Bigio et al., 2020; Auerbach et al.,
2021). Brodeur et al. (2021) provides a comprehensive survey of the rapidly
growing body of literature on the economics of COVID-19. None of these papers
offers evidence of the disruption of the pandemic on competition dynamics.

This paper has six sections including this introduction. The second section
describes the data and outcomes of interest. The third section describes our
methodology including the Bayesian structural time series model. The fourth
section presents our main results. The fifth section introduces our robustness
checks. The sixth section concludes.

##  2 Data

We use yearly (and quarterly) information for 3,611 (3,192) publicly-traded
companies from Compustat, which provides a panel of financial statements since
1955 (1980) representing roughly 29% of US jobs (Davis et al., 2006). To make
our sample comparable to that in previous literature, we follow the same
sample selection steps as in De Loecker et al. (2020).444For a more detailed
description of the sample, please refer to Appendix A. Furthermore, to
evaluate whether our sample is comparable to the sample in previous
literature, we compute summary statistics for sales, COGS, capital stock,
SG&A, wage bill, and employment in both 1955–2021 and 1955–2016. Our numbers
are indeed quite similar to those in De Loecker et al. (2020). Differences
arise as Compustat updates and companies revise the corresponding financial
statements.555Table A1 in the Appendix reports summary statistics for these
variables for three samples: 1955–2016, 1955–2021, and the sample in De
Loecker et al. (2020). We use the same coding routine in Stata as those
authors, which is available in their published supplemental material. Although
the mean and median of the cost of goods sold differs between our sample and
the sample in De Loecker et al. (2020), we find a similar series of average
markup rates as those authors, and both series have a high positive
correlation (0.92).

###  2.1 Outcomes of interest

#### Markup rates

We compute markup rates using the production function approach (Hall, 1988; De
Loecker and Warzynski, 2012; De Loecker et al., 2020).666Alternatively, there
are two other competing methods to compute markup rates. First, the demand
approach by Berry et al. (1995), which requires information on the demand side
and, consequently, is more data-intensive and hard to carry out for an
extensive set of industries simultaneously. Second, the approach assumes
constant returns of scale, which implies that the markup rate can be estimated
from the ratio of sales to costs. This approach identifies the markup rate by
solving the firm’s cost minimization problem and two main assumptions: first,
the firm is a price taker in the input market. Second, there are two types of
factors, a variable factor, in which firms can flexibly decide how much to use
within a given period (typically a year), and a fixed factor, such as capital,
which remains constant within the same period. We compute markup rates,
$\mu_{ijt}$, at firm $i$, industry $j$, and period $t$ (either quarter or
year) as777We define an industry at the 2-digit level of the North American
Industry Classification System (NAICS).

$\displaystyle\mu_{ijt}=\theta_{jt}\frac{\text{Sales}_{ijt}}{\text{COGS}_{ijt}},$

where $\theta_{jt}$ is the industry-level output elasticity of variable
inputs, and $\text{COGS}_{ijt}$ are costs of goods sold.

#### Profit rates

Our measure of profitability follows standard accounting practice by
calculating the percentage of sales that remains after subtracting the costs
of goods sold and selling, general, and administrative expenses from total
sales.

$\displaystyle\text{Profit
rate}_{it}=\frac{\text{Sales}_{it}-\text{COGS}_{it}-\text{XSGA}_{it}}{\text{Sales}_{it}},$

where XSGA is selling, general and administrative expenses.

###  2.2 Entry and exit of stock exchange listings in Compustat

We further analyze the composition of firms in Compustat’s survey to evaluate
whether our sample exhibits similar patterns in the share of firms entering or
exiting before and during the pandemic. Figure 1 depicts the yearly entry and
exit rates and the total number of companies from 1955–2021.

Figure 1: Entry and Exit Rates Notes: The figure displays entry and exit rates
for the firms in Compustat. We classify a firm as entering at year $t$ if $t$
is the first year it appeared in the survey (the firm became publicly traded),
and we label a firm as exiting at time $t$ if it is the last period where we
observe the firm in the survey.

One potential issue with our sample of firms is that the COVID-19 pandemic may
have caused some firms to leave Compustat. However, our sample indicates that
only 7% of the firms in Compustat in 2019 left the survey in 2020, which is
only slightly above the 1965–2019 exit rate average of 5.7%. This exit rate
increased in the second year of the pandemic, with 17.6% of firms with data in
2020 leaving the survey in 2021.

Furthermore, firms that enter and exit Compustat’s survey each year are
similar to the firms that remain in the survey, at least in terms of markups
and profit rates. Figure 2 plots the average markup and profit rates for three
samples: first, a sample including all the firms in a given year; second,
firms entering the stock exchange; and third, firms exiting the stock
exchange. The markup rates for all three samples have been increasing since
1980, although entering and exiting firms’ markup rates tend to vary more
relative to the average markup rate. Likewise, entering and exiting firms tend
to have the same profit rates as the average firm. These findings suggest that
the firms that leave the survey are not significantly different from those
that remain.

Figure 2: Average Markup and Profit Rates for Entrant and Exiting firms.

(a) Average Markups

(b) Average Profit Rates

Notes: This figure plots the average markups and profit rates for three
samples: firms entering the stock exchange, firms exiting, and all firms in
the stock exchange.

###  2.3 Historical trends in markup and profit rates

We supplement this section with a historical analysis of markup and profit
rates. Figure 3 presents annual markup and profit rates for publicly-traded
firms in the US between 1955 and 2021. The average profit rate is at its
highest level in history, whereas the average markup rate is below its highest
recorded value in 2016.

The markup rate saw a significant decrease between 2016 and 2019, but this
trend reversed during the pandemic. In the first year of the pandemic, the
sales-weighted average markup rate increased from 1.49 to 1.52. In the second
year, it increased further to 1.54, which is slightly lower than the
historical maximum observed in 2016. As documented in De Loecker et al. (2020)
through 2016, the increase in markup rates was driven by the rising markups of
top firms, which continued from 2016–2021. For example, the third quartile
markup rate increased from 1.81 to 1.82 between 2016 and 2021, while the
markup rate for firms in the lower quartile decreased from 1.12 to 1.04.

Unlike markup rates, profit rates decreased in 2020 before recovering in 2021.
For instance, the sales-weighted profit rate increased from 17.6% to 21.6%
between 2016 and 2021, with a temporary decrease to 16.9% in 2020.
Interestingly, the most profitable firms were less affected than less
profitable firms: the third quartile firms by profit rate experienced profit
rates of 19.8% and 23.4% in 2020 and 2021, respectively, with the latter being
their highest profit rate since 195. In contrast, the lower quartile firms
reduced their profit rate from 3.1% to 1.2% between 2016 and 2020 before
rebounding to 3.7% in 2021.

In addition to the increase in markup and profit rates caused by the pandemic,
our findings indicate that their volatility was higher than usual during the
first two years of the pandemic. However, this increased volatility was still
within historical levels for most firms. Figure 4 shows how the volatility of
sales-weighted markup and profit rates changed from 1959 to 2021.888For a
given time series, The 5-year rolling coefficient of variation is calculated
as the absolute value of the ratio of the standard deviation and the average,
with the average and standard deviation being determined using data from the
previous five years. Consistent with the heterogeneous effects of the pandemic
on profit rates, the lower quartile experienced increasing volatility from
2000 to 2021, with a particularly pronounced increase during the pandemic.

Figure 3: Markup and Profit Rates

(a) Markup Rates

(b) Profit Rates

Notes: The figure shows the sales-weighted average, the first, second, and
third quartiles for the markup rates, based on De Loecker et al. (2020), and
profit rates for publicly traded firms collected in Compustat. Markup rates
are calculated as the multiplication of industry-level variable-input output
elasticities and the firm-level sales to cost of goods sold ratio. Profit
rates are total sales minus costs of goods sold and selling, general and
administrative expenses as a share of sales. Figure 4: Markup and Profit Rate
Volatility

(a) Markups

(b) Profit Rates

Notes: The figure shows the 5-year rolling coefficient of variation for the
sales-weighted average, the first, second, and third quartiles for the markup
rates and profit rates for publicly traded firms collected in Compustat.

##  3 Methodology

Let $Y_{i,t}$ be an observed outcome of interest, either the markup rate or
the profit rate, for a firm $i$ in period $t$. Let $Y_{i,t}(1)$ and
$Y_{i,t}(0)$ be the potential outcomes for a firm $i$ in time period $t$ when
a pandemic happens and a pandemic does not happen, respectively. We denote $T$
as the last period before the pandemic happens.

Our main parameter of interest is the effect of the pandemic in a firm-level
outcome, namely,

$\displaystyle\tau_{i,T+h}=Y_{i,T+h}(1)-Y_{i,T+h}(0),\quad h=1,2,\dots.$

We only observe the potential outcome when a pandemic happens after time
period $T$ for any firm, so $Y_{i,T+h}=Y_{i,T+h}(1)$ for $h=1,2,\dots$, but we
cannot observe the outcome had the pandemic not taken place, $Y_{i,T+h}(0)$.

To approximate the outcome had the pandemic not taken place, we propose to use
the forecast based on information up to period $T$ as a counterfactual for
each firm $i$, $Y_{i,T+h}(0)$. Specifically, let $I_{T}$ denote all
information available up to period $T$; we define our counterfactual as the
conditional expectation of $Y_{i,T+h}(0)$ given all observed information up to
period $T$, i.e., $\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}]$, which
potentially includes pre-treatment values of both outcomes and available
covariates.

Using the conditional expectation, we can compute an approximation to the
causal effect as the difference between the realized value of our outcome of
interest and the counterfactual

$\displaystyle\widetilde{\tau}_{i,T+h}=Y_{i,T+h}(1)-\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}].$

To operationalize our counterfactual, we pick a family of models to estimate
the conditional expectation of the outcome had the pandemic not taken place.
We explain this family of models in Section 3.1. In this sense, our firm-level
identification rests on the specific parametric family we use to estimate the
counterfactual. We denote our estimator as
$\widehat{Y}_{i,T+h}(0)\equiv\widehat{\operatorname{\mathbb{E}}}[Y_{i,T+h}(0)|I_{T}]$.
Our final estimator of the firm-level causal effect is

$\displaystyle\widehat{\tau}_{i,T+h}=Y_{i,T+h}-\widehat{Y}_{i,T+h}(0).$

The causal effect estimator, $\widehat{\tau}_{i,T+h}$, is a good approximation
to the true causal effect, $\tau_{i,T+h}$, under two conditions: first, the
difference between the actual potential outcome without the pandemic and its
counterfactual is near zero,
$Y_{i,T+h}(0)-\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}]\approx 0$, also
called conditional expectation error. Second, the difference between the
actual forecast and the counterfactual is approximately zero,
$\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}]-\widehat{Y}_{i,T+h}(0)\approx
0$, which we refer to as forecasting error. We make these conditions explicit
in the following decomposition

$\displaystyle\widehat{\tau}_{i,T+h}=\tau_{i,T+h}+(Y_{i,T+h}(0)-\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}])+(\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}]-\widehat{Y}_{i,T+h}(0)).$
(1)

In practice, it is difficult to assess these two conditions. We can, however,
aim to reduce the forecasting error as much as possible by selecting the best
model within a comprehensible family of models.

Furthermore, we can provide conditions to obtain a consistent estimator of the
average treatment effect across firms,
$\frac{1}{n}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\tau_{i,T+h}]$. In
particular, let $Y_{1},\dots,Y_{n}$ be a random sample, and define
$\operatorname{\mathbb{E}}[\tau_{T+h}]=\operatorname{\mathbb{E}}[\tau_{i,T+h}]$.
In Equation (1), summing across firms and dividing by the number of firms, we
obtain

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\widehat{\tau}_{i,T+h}=\frac{1}{n}\sum_{i=1}^{n}\left[\tau_{i,T+h}+(Y_{i,T+h}(0)-\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}])+(\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}]-\widehat{Y}_{i,T+h}(0))\right].$

By invoking the law of large numbers, the average conditional expectation
error should be close to zero as the number of firms in our sample increases,

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}(Y_{i,T+h}(0)-\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}])\xrightarrow[]{P}\operatorname{\mathbb{E}}[Y_{i,T+h}(0)]-\operatorname{\mathbb{E}}[\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}]]=0,$

where the equality follows from the law of iterated expectations, and
$\xrightarrow[]{P}$ means convergence in probability.

Finally, if our estimator of the conditional expectation is consistent in the
sense that

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}(\operatorname{\mathbb{E}}[Y_{i,T+h}(0)|I_{T}]-\widehat{Y}_{i,T+h}(0))\xrightarrow[]{P}0,$
(2)

we can conclude that

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\widehat{\tau}_{i,T+h}\xrightarrow[]{P}\frac{1}{n}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\tau_{i,T+h}]=\operatorname{\mathbb{E}}[\tau_{T+h}],$

where the equality follows from the random sample assumption.

###  3.1 Forecasting model

We use a Bayesian structural time series (BSTS) model forecast to create a
counterfactual at the firm level (Brodersen et al., 2015). These models are
highly flexible in the sense that they embed a broad class of time-series
models. Besides their flexibility, they are also modular: they encompass
different features of time-series data such as trend, seasonality, or holiday
effects.

Within the class of BSTS models, assume that the firm-level outcome of
interest, $Y_{i,t}$, has a stochastic trend representation (also known as a
local-level trend or random walk) as follows

$\displaystyle Y_{i,t}=\delta_{i,t}+\varepsilon_{i,t},$ (3)
$\displaystyle\delta_{i,t}=\delta_{i,t-1}+\eta_{\delta,i,t},$ (4)

where $\delta_{i,t}$ is a trend component at time $t$, and,
$\varepsilon_{i,t}$ and $\eta_{\delta,i,t}$ are mutually independent Gaussian
error components with variances $\sigma_{\varepsilon}^{2}$ and
$\sigma_{\mu}^{2}$, respectively. In the state-space representation
literature, Equation (3) is commonly known as the observation equation,
whereas Equation (4) is the state equation.

For the quarterly specification, the model also incorporates a seasonal
component. Specifically, let $\gamma_{i,t}$ be an additive quarterly component
such that we augment the model in Equations (3)-(4) with the following two
equations

$\displaystyle Y_{i,t}=\delta_{i,t}+\gamma_{i,t}+\varepsilon_{i,t},$ (5)
$\displaystyle\gamma_{i,t+1}=-\sum_{s=0}^{2}\gamma_{i,t-s}+\eta_{\gamma,i,t}.$
(6)

The seasonal specification in Equation (6) allows for the seasonal component
to change over time.999In contrast to a specification with seasonal indicator
variables, which restricts the seasonal effect to be fixed across years, the
specification in Equation 6 allows the seasonal component to change over time
(see Harvey, 1990, Section 2.3.4). To avoid perfect multicollinearity in the
estimation of the seasonal component, the seasonal component is restricted to
sum to zero across all seasons and time periods.

To avoid identification or endogeneity issues, we decide not to include any
contemporaneous variables on the right-hand side of Equations (3) or (5). In
principle, the pandemic may have affected all contemporaneous variables that
are relevant to markup and profit rates. The choice to exclude these variables
is supported in the bad control problem literature (Angrist and Pischke,
2008).

###  3.2 Estimation

To estimate the model in Equations (3)-(4) (Equations (3)-(6) in the quarterly
model), we implement Gibbs sampling. This method jointly estimates
unobservables, such as trend and seasonality components, as well as the
parameters of the model, such as the error term variances. To ensure that the
results are not sensitive to the initial values chosen and given the Markov-
chain nature of Gibbs sampling, we choose 10,000 Monte-Carlo iterations.

Table 1 shows the priors for the variances of the model and the initial values
of the state components of the model. As is common with linear models and
following Brodersen et al. (2015), we set an inverse-Gamma prior for any of
the variances $\sigma_{i}^{2}$ (either $\sigma_{\varepsilon,i}^{2}$,
$\sigma_{\delta,i}^{2}$, or $\sigma_{\gamma,i}^{2}$) as
$1/\sigma_{i}^{2}\sim\mathcal{G}(v/2,s/2)$ for firm $i$. Since the ratio of
the hyperparameters $s$ and $v$ is equal to the expected value of
$\sigma_{i}^{2}$, it is common to refer to $s$ as the sum of squares and $v$
as the degrees of freedom. A common choice for the hyperparameters is to set
$s/v=0.1s_{y_{i}}$, where $s_{y_{i}}$ is the sample standard deviation of the
outcome of interest $Y_{i}$ for firm $i$. As in Brodersen et al. (2015), we
have a prior belief that the error components are small so we set small values
for both the degrees of freedom, $v$, and the expected value of the variance,
$s/v$. We use weakly informative Gaussian priors for the initial values of the
state parameters.

Table 1: Priors on Bayesian Structural Model

[]  Parameter Prior density Hyperparameter 1 Hyperparameter 2
$\sigma_{\varepsilon,i}$ Inverse-Gamma 1 0.01 $\sigma_{\delta,i}$ Inverse-
Gamma 0.01 32 $\sigma_{\gamma,i}$ Inverse-Gamma 0.01 0.01 $\delta_{i,1}$
Normal $\frac{y_{0}-\overline{y}}{\hat{\sigma}_{y}}$ 1 $\gamma_{i,1}$ Normal 0
1

  * •

Notes: $\sigma_{\epsilon,i}$, $\sigma_{\delta,i}$, and $\sigma_{\gamma,i}$ are
the variances of the observation equation error (Equation (3)), trend, and
seasonal components, respectively, for a firm $i$. $\delta_{i,1}$ and
$\gamma_{i,1}$ are the initial values of the trend and seasonal state
components. When the prior density is inverse-gamma, hyperparameters 1 and 2
can be interpreted as the degrees of freedom and the sum of squared errors,
respectively (see Brodersen et al., 2015). When the prior density is normal,
hyperparameters 1 and 2 are the mean and standard deviation, respectively.

###  3.3 Assumption testing, model fit, and quality of forecasts

We perform a battery of statistical tests to evaluate whether the firm-level
markup and profit rates follow the data-generating process described in
Equations (3)-(4), which forms the base of our forecasts. Under this data-
generating process, the differentiated series of markup and profit rates,
$Y_{i,t}-Y_{i,t-1}$, should be normally distributed for each of the firms in
our sample. To test for the normality of the differentiated series, we perform
Jarque-Bera tests on these series. Indeed, we do not find evidence to reject
the hypothesis that a majority of these series have a normal distribution.
Specifically, for yearly (quarterly) markups, we find that 84.6% (64%) of the
cases among 3,604 (3,192) firms do not reject the null hypothesis of normality
at the 5% level after applying the Bonferroni correction for multiple testing.
Similarly, for yearly (quarterly) profit rates, we find that 82.2% (74%) of
our tests among 3,611 (3,192) firms do not reject the normality assumption
after controlling for the Bonferroni correction. We obtain similar results
after using other corrections for multiple testing, including the Holm,
Hochberg, and Hommel corrections. In summary, we do not find evidence to
reject the normality assumption embedded in our forecasting model.

We also evaluate whether the firm-specific models accurately reflect the
observed firm’s behavior before the pandemic. To this purpose, we calculate
the fraction of periods for each firm when the observed outcome, either markup
or profit rate, falls outside the model’s 95% Bayesian credible interval for
each period that the firm has data before 2020. Table 2 shows the average of
these fractions across firms for each outcome and frequency. On average, 3% of
the observations are not within the Bayesian credible interval for both markup
and profit rates. In addition, at least 75% of the firms have a model that
leaves no more than 6.3% of the periods outside of the 95% Bayesian credible
intervals (see Table 2, column 3). These results suggest that most of the
models correctly pin down the pre-pandemic trends.

Table 2:  Fraction of Periods that Observed Outcome Falls Outside of Bayesian
Credible Intervals

Outcome (frequency) | Average | Percentile 75th | Number of firms
---|---|---|---
Markups (quarterly) | 0.025 | 0.043 | 3,192
Markups (yearly) | 0.028 | 0.053 | 3,611
Profit rate (quarterly) | 0.029 | 0.045 | 3,192
Profit rate (yearly) | 0.035 | 0.063 | 3,611

  * •

Notes: The table introduces model fitness measures computed for markup rates
and profit rates during the pre-pandemic period. For each firm, we compute the
fraction of observations of the pre-pandemic period that do not belong to the
estimated 95% Bayesian credible intervals. The table displays the average and
the 75th percentile of these fractions across firms.

Furthermore, we find that our forecasts have adequate forecast quality. To
evaluate our model’s forecast quality, we compute forecasts for markup rates
and profit rates for 2018–2019 using firm-level models that are estimated with
observations up to 2017. We present some common forecast quality statistics in
Table 3. Two of these statistics confirm the forecast quality of our models.
First, as suggested by Equation (2), the objective is to minimize the mean
forecast error (ME) to be close to zero, which seems to be the case for the
markup rates. Second, for at least half of the firms, the median absolute
percentage error (MAPE) is 0.1 and 0.3 for markup and profit rates,
respectively. These metrics support the forecast quality of our Bayesian
models.

Table 3:  Forecast Quality Statistics of Bayesian Structural Time Series Model

Variable | Year | ME | RMSE | MAE | MAES | MAPE |  Median MAPE | $n$
---|---|---|---|---|---|---|---|---
Markup rate | 2018 | -0.008 | 0.827 | 0.373 | 0.215 | 0.204 |  0.107 | 2858
Markup rate | 2019 | -0.046 | 0.900 | 0.409 | 0.241 | 0.241 |  0.122 | 2872
Profit rate | 2018 | 19.923 | 147.174 | 28.151 | 3.402 | 2.951 |  0.335 | 2858
Profit rate | 2019 | 18.482 | 151.506 | 30.590 | 4.464 | 3.610 |  0.380 | 2872

  * •

Notes: The table introduces forecast quality statistics computed for 2018–2019
forecasts for markup and profit rates estimated using firm-level models with
samples up to 2017. The statistics computed are: mean error (ME), root mean
square error (RMSE), mean absolute error (MAE), standardized mean absolute
error (MAES), and the mean absolute percentage error (MAPE). Median MAPE is
the median MAPE across firms. $n$ is the number of firms.

##  4 Counterfactual analysis results

We introduce the results of the counterfactual effects of the pandemic in the
markups and profit rate in this section. We first summarize the findings for
the average firm. We then perform a heterogeneity analysis to elucidate which
firms fared better during the pandemic.

###  4.1 Aggregate data

Most firms had lower markup rates than what could have been observed without
the pandemic happening. To illustrate this point, we show in Figure 5(b) four
aggregates of both the observed and counterfactual markup rates, namely, its
sales-weighted average and quartiles in 2000–2021. For instance, for the
average firm, the observed markup rate was 4.3% and 6.6% lower than its
counterfactual value in 2020 and 2021, respectively. These effects are
significant as the observed markup rate falls outside the 95% credible
interval of the counterfactual. Furthermore, there is considerable
heterogeneity across firms in terms of quartiles of markup rates. For
instance, the firms in the third quartile of markup rates had the largest
negative effects in 2020 (2021) with a decrease of 5% (4.6%) relative to their
counterfactual value. In contrast, the median markup firm (quartile 2)
experienced less dramatic effects: a 3.3% (2.6%) lower markup rate with
respect to their counterfactual in 2020 (2021).101010Similar effects arise
when using quarterly data (see Figure 2(b) in Appendix C).

Figure 5: Comparison of Observed and Counterfactual Markup Rates using
Bayesian Structural Time Series Models

(a) Average

(b) Quartiles

Note: The figure shows four aggregate statistics of the yearly observed and
counterfactual markup rates. The statistics are the sales-weighted average
(shown in Panel (a)), the first quartile (shown in Panel (b) in red), the
second quartile (shown in Panel (b) in black), and the third quartile (shown
in Panel (b) in gray). The solid lines represent the observed values, while
the dotted lines represent the counterfactual values for 2020-2021, as
calculated using firm-level Bayesian structural time series models. The shaded
areas represent the 95%, 90%, 80%, and 50% (equally-tailed) credible interval
for each statistic, calculated based on 5,000 posterior simulations on Panel
(a). 95% (equally-tailed) credible intervals are displayed for the statistics
on Panel (b).  Figure 6: Comparison of Observed and Counterfactual Profit
Rates using Bayesian Structural Time Series Models

(a) Average

(b) Quartiles

Note: The figure shows four aggregate statistics of the yearly observed and
counterfactual profit rates. The statistics are the sales-weighted average
(shown in Panel (a)), the first quartile (shown in Panel (b) in red), the
second quartile (shown in Panel (b) in black), and the third quartile (shown
in Panel (b) in gray). The solid lines represent the observed values, while
the dotted lines represent the counterfactual values for 2020-2021, as
calculated using firm-level Bayesian structural time series models. The shaded
areas represent the 95%, 90%, 80%, and 50% (equally-tailed) credible interval
for each statistic, calculated based on 5,000 posterior simulations on Panel
(a). 95% (equally-tailed) credible intervals are displayed for the statistics
on Panel (b).

Unlike the observed negative effects of markups, we find that most firms had
higher profit rates than their counterfactuals during the pandemic. Figure
6(b) shows the (sales-weighted) average and quartiles of both the observed and
counterfactual profit rates across firms. When analyzing the counterfactual
effects of the pandemic on profit rates, most firms had a positive impact. For
instance, the average firm had a profit rate of 16.9% and 21.6% in 2020 and
2021, respectively, which are 2.1 and 6.4 percentage points higher than the
counterfactuals. These effects are significant as the observed markup rate
falls outside the 95% credible interval of the counterfactual. Additionally,
firms with higher markup rates experienced lower markup rates than what could
have been expected without the pandemic. For instance, the third-quartile firm
registered a 19.8% profit rate in 2020 which is 1.7 percentage points lower
than the profit rate these companies would experience had the pandemic not
happened. In contrast, firms in the first and second quartiles experienced no
effects on profit rates in 2020 while experiencing positive effects in the
second year of the pandemic.111111Similar effects arise when using quarterly
data (see Figure 3(b) in Appendix C).

Looking at both results for markups and profit rates jointly, it seems that
the uncertainty during the pandemic potentially promoted the decrease in
markups and the increase in profits(Altig et al., 2020). In the next section,
we explore the heterogeneity of these effects across firm characteristics as
well as the industries most affected.

###  4.2 Heterogeneity analysis

We investigate potential heterogeneity in the effects of the pandemic on
markup rates and profitability to elucidate whether firms with different
levels of key characteristics differ in their response to the pandemic. These
characteristics include pre-pandemic levels of cost of goods sold, sales,
employment, stock tenure, and market share. Specifically, to summarize the
effect heterogeneity in a convenient form, we run the following cross-
sectional regression

$\displaystyle\overline{\widehat{\tau}}_{i}$
$\displaystyle=\beta_{0}+\beta_{1}\log\overline{\text{COGS}}_{i}+\beta_{2}\log\overline{\text{Sales}}_{i}+\beta_{3}\log\overline{\text{Employment}}_{i}+\beta_{4}\overline{\text{Stock
tenure}}_{i}$ $\displaystyle\phantom{space}+\overline{\text{Market
share}}_{i}+\delta_{j}+\epsilon_{i},$

where $\overline{\widehat{\tau}}_{i}$ is the average effect in 2020–2021 for a
firm $i$ (on either markup or profit rates), the right-hand side variables are
firm-level averages for COGS, sales, employment, stock tenure, market share in
2015–2019, and $\delta_{j}$ is an industry-level fixed effect such that firm
$i$ belongs to industry $j=j(i)$. Tables 4 and 5 show the estimated regression
for the yearly specification for markups and profit rates, respectively. As
expected, those firms with higher tenure in the stock exchange have larger
positive effects on markup rates since they are more likely to be in a
consolidated industry or have survived a merger or an acquisition. Likewise,
firms with high sales or low costs have mechanically more positive effects on
profit rates. These regressions suggest that market share, number of
employees, or tenure in the stock market imply higher effect heterogeneity in
profit rates.

We find similar effect heterogeneity results when using the quarterly
specification for the markup and profit rates (see Tables A3 and A4,
respectively). Likewise, we find similar conclusions when we build scatter
plots of firm-level effects against pre-pandemic values of key firm
characteristics. In particular, we plot the effect in either markup or profit
rates against pre-pandemic values of markup rates, sales, stock exchange
tenure, profit rate, market share, and employment—the binscatter plots in
Appendix D illustrate this analysis.

We provide further evidence of the heterogeneity of the response to the
pandemic by tracking the fraction of firms with negative and positive effects
of the pandemic in each quarter in 2020–2021 (see Appendix C.1). In
particular, at the beginning of the pandemic, within those firms that are
significantly affected—by having an observed outcome outside the 95% credible
interval for the counterfactual outcome— most firms are negatively affected
both in terms of markups and profit rates. By the last quarter of 2021, the
share of firms with positive effects increased but is still lower than those
with negative effects. This suggests that there were more negatively affected
firms than positively affected firms in most quarters.

Table 4: Heterogeneity in the effects of the pandemic on markups
(Yearly Specification)

Dependent Variable: | Average markup rate pandemic effect in 2020–2021
---|---
| All | 2020 | 2021
Model: | (1) | (2) | (3) | (4) | (5) | (6)
_Variables_ |  |  |  |  |  |
COGS | 0.0856 | 0.0659 | 0.0490 | 0.0463 | 0.1300 | 0.0937
| (0.0851) | (0.0912) | (0.0893) | (0.0965) | (0.1497) | (0.1591)
Sales | -0.1534∗ | -0.0864 | -0.1393 | -0.0975 | -0.1689 | -0.0724
| (0.0905) | (0.0926) | (0.0946) | (0.0983) | (0.1594) | (0.1614)
Employment | 0.0519∗∗∗ | 0.0082 | 0.0951∗∗∗ | 0.0568∗∗∗ | -0.0016 | -0.0550∗
| (0.0156) | (0.0170) | (0.0183) | (0.0187) | (0.0260) | (0.0295)
Stock-exchange tenure | 0.0030∗∗∗ | 0.0031∗∗∗ | 0.0025∗∗∗ | 0.0018∗∗ | 0.0035∗∗∗ | 0.0043∗∗∗
| (0.0007) | (0.0007) | (0.0007) | (0.0008) | (0.0012) | (0.0013)
Market share | 0.8069∗∗ | -0.3262 | -0.0722 | -0.6241 | 1.818∗∗ | 0.0390
| (0.3312) | (0.4701) | (0.3854) | (0.5395) | (0.7961) | (0.7951)
_Fixed-effects_ |  |  |  |  |  |
2-digit NAICS industry |  | Yes |  | Yes |  | Yes
Mean | 1.700 |  | 1.674 |  | 1.732 |
Mean effect | -0.069 |  | -0.086 |  | -0.048 |
_Fit statistics_ |  |  |  |  |  |
Observations | 5,771 | 5,771 | 3,139 | 3,139 | 2,632 | 2,632
R2 | 0.00632 | 0.05509 | 0.01250 | 0.05801 | 0.00734 | 0.07110
Within R2 |  | 0.00303 |  | 0.00495 |  | 0.00561

  * •

Notes: The table presents cross-section firm-level regressions of the average
pandemic effect at the firm-level on the 2015–2019 average of the logarithm
cost of goods sold, logarithm of sales, logarithm of employment, years since
publicly-traded, and market shares. NAICS = North American Industrial
Classification System. Heteroskedasticity-robust standard-errors in
parentheses. Signif. Codes: ***: 0.01, **: 0.05, *: 0.1.

Table 5: Heterogeneity in the effects of the pandemic on profit rates
(Yearly Specification)

Dependent Variable: | Average profit rate pandemic effect in 2020–2021
---|---
| All | 2020 | 2021
Model: | (1) | (2) | (3) | (4) | (5) | (6)
_Variables_ |  |  |  |  |  |
COGS | -19.46∗∗∗ | -21.09∗∗∗ | -18.76∗∗∗ | -15.69∗∗ | -20.03∗ | -27.06∗∗
| (5.923) | (7.359) | (6.236) | (6.714) | (10.50) | (13.76)
Sales | 16.81∗∗ | 18.46∗∗ | 12.90 | 10.12 | 21.25 | 28.37∗
| (7.897) | (9.153) | (9.326) | (9.903) | (13.09) | (16.04)
Employment | -0.2620 | -0.2221 | 3.092 | 2.458 | -4.328 | -3.901
| (3.919) | (4.421) | (4.992) | (5.771) | (6.238) | (6.861)
Stock-exchange tenure | -0.0211 | -0.0288 | -0.0914 | -0.1333 | 0.0539 | 0.0761
| (0.0667) | (0.0796) | (0.0795) | (0.0978) | (0.1117) | (0.1303)
Market share | -5.302 | -29.90 | -10.50 | 15.34 | -1.726 | -84.34
| (26.36) | (72.81) | (37.57) | (84.41) | (37.10) | (127.3)
_Fixed-effects_ |  |  |  |  |  |
2-digit NAICS industry |  | Yes |  | Yes |  | Yes
Mean | 2.438 |  | 1.274 |  | 3.833 |
Mean effect | 7.145 |  | 5.443 |  | 9.183 |
_Fit statistics_ |  |  |  |  |  |
Observations | 5,771 | 5,771 | 3,139 | 3,139 | 2,632 | 2,632
R2 | 0.00534 | 0.01288 | 0.00867 | 0.02122 | 0.00411 | 0.01382
Within R2 |  | 0.00509 |  | 0.00664 |  | 0.00527

  * •

Notes: The table presents cross-section firm-level regressions of the average
pandemic effect at the firm-level on 2015–2019 average of the logarithm cost
of goods sold, the logarithm of sales, logarithm of employment, years since
publicly-traded, and market shares. NAICS = North American Industrial
Classification System. Heteroskedasticity-robust standard errors in
parentheses. Signif. Codes: ***: 0.01, **: 0.05, *: 0.1

###  4.3 Heterogeneity by industry

We supplement our heterogeneity results by analyzing the effects of industry.
Figure 7(b) shows the industry breakdown of the effects on markup and profit
rates. First, we observe that the transportation industry experienced negative
effects on profit rates in both 2020 and 2021. Following a negative demand
shock due to lockdowns, we also note a fall in this industry’s markups.
Interestingly, we observe similar behavior in the oil and gas industry; in
2020, lockdowns drove demand and prices down, which lead the industry to
report losses. However, the commodity price surge in 2021, partly in response
to a myriad of public policy responses, triggered increases in profit rates of
roughly 10%.

Figure 7: Effects on Markups and Profit Rates by Industry

(a) Markups

(b) Profit Rates

Note: The figure plots the sales-weighted average effects on markup and profit
rates by the 2-digit NAICS industry. The mean and percentiles 25 (P25) and 75
(P75) correspond to the aggregate effects for the sample of 3,611 firms in
Compustat (see also Figures 5(b) and 6(b)). Industries are organized in
ascending order according to their corresponding effect in 2020.

Similarly, hospitality displays adverse effects on profit rates in 2020, with
roughly no effect on markups. In contrast, information and technology,
warehousing, and real estate experienced profit rates above what could have
been predicted by previous trends. Likewise, professional and administrative
services, which could easily migrate to online environments, had profit rates
above their historical trends. Lastly, we see modest effects on the retail
industries for both markups and profit rates, which are known to have low
markups (Philippon, 2019).

###  4.4 Business cycle analysis

One may wonder that fluctuations in economic activity may affect either markup
or profit rates. To answer this question and under the assumption that the
pandemic does not affect these fluctuations, we further augment Equation (3)
with a cyclical component.

$\displaystyle Y_{it}=\delta_{it}+\alpha\ cycle_{t}+\varepsilon_{it},$ (7)

where $cycle_{t}$ is taken as the Hodrick-Prescott decomposition of the US
real gross domestic product in period $t$.

Figure 8: Histogram of Coefficient of the Cycle in Models (Yearly Frequency)

(a) Markup Rates

(b) Profit Rates

Note: The plots show the histograms for the average firm-specific coefficient
of the cycle using a yearly specification. The red dotted line shows the
average across all of these coefficients. Data were standardized before
estimating the model, so the coefficients are included in the $[-1,1]$
interval.

We find evidence for acyclical markups and profit rates on average across
firms. In particular, we estimate the firm-level coefficient of the GDP cycle
on both markup and profit rates, $\alpha$ in Equation (7). Figures 8(a) and
8(b) show histograms of these coefficients for separate models for markup and
profit rates. This distribution suggests that, on average, the cycle does not
seem to affect markups and profit rates. Furthermore, we compute the
percentage of significant cycle coefficients. In the case of markup rates, we
find 63% of the coefficients to be significant with a rather symmetrical share
of positive (30%) and negative (33%) coefficients. Similarly, in the case of
profit rates 62% of the coefficients are significant (34% positively
significant and 28% negative significantly).

These wide-ranging cycle coefficients may be connected to the literature on
the cyclicality of markups, which has identified potential reasons for both
pro- and counter-cyclical markups. In particular, the behavior of liquidity-
constrained firms that adjust their customer base to stay afloat during
recessions may explain counter-cyclical markups (Gilchrist et al., 2017), with
larger firms being able to further smooth these liquidity constraints out
(Hong, 2017), whereas Keynesian demand effects may explain pro-cyclical
markups (Nekarda and Ramey, 2020). It seems that these three conjectures may
have some weight in explaining the apparent average acylicality in the markup
and profit rates.

##  5 Robustness of findings to other modeling strategies and choices

In this section, we introduce two approaches to evaluate the robustness of our
main findings. In the first approach, we modify the hyperparameters associated
with the priors of the Bayesian structural time series (BSTS) models, which we
refer to as sensitivity analysis. In the second approach, we use the commonly-
used methodology of local linear projections (LLPs) to evaluate if our main
findings are robust to this fairly flexible strategy. Our sensitivity analysis
corroborates our main results in Section 4. The LLPs approach gives similar
point effects in markup rates compared to the Bayesian structural model, and a
smaller effect in profit rates; however, it seems that LLPs lack power for
this data as the confidence intervals seem to be too wide.

###  5.1 Robustness of findings to other modeling choices: a hyper-parameter
sensitivity analysis.

We validate the findings from the BSTS model by performing a sensitivity
analysis of our choice of hyperparameters. This analysis ensures that our
results are not driven by this choice. As noted in Section 3.2, the
hyperparameters govern the prior densities of the variances and initial values
of the trend and seasonality structural components. We perform two main
exercises: we either increase or decrease hyperparameters that govern the
scale of the priors densities by 25% compared to the baseline values in Table
1; by doing so, we allow for the increases in markup and profit rates to have
either more or less variability (Brodersen et al., 2015). As the location
hyperparameters associated with the initial values are calibrated with summary
statistics of the actual series, we do not change those location
hyperparameters. Table 6 shows the choice of hyperparameters for each of these
exercises.

Table 6: Sensitivity Analysis on Hyperparameters of Bayesian Structural Model

(a) Exercise 1: 25% Increase in Hyperparameters Parameter | Prior density | Hyperparameter 1 | Hyperparameter 2
---|---|---|---
$\sigma_{\varepsilon,i}$ | Inverse-Gamma | 1.25 | 0.0125
$\sigma_{\delta,i}$ | Inverse-Gamma | 0.0125 | 40
$\sigma_{\gamma,i}$ | Inverse-Gamma | 0.0125 | 0.0125
$\delta_{i,1}$ | Normal | $\frac{y_{0}-\overline{y}}{\hat{\sigma}_{y}}$ | 1.25
$\gamma_{i,1}$ | Normal | 0 | 1.25

(b) Exercise 2: 25% Decrease in Hyperparameters Parameter | Prior density | Hyperparameter 1 | Hyperparameter 2
---|---|---|---
$\sigma_{\varepsilon,i}$ | Inverse-Gamma | 0.75 | 0.0075
$\sigma_{\delta,i}$ | Inverse-Gamma | 0.0075 | 24
$\sigma_{\gamma,i}$ | Inverse-Gamma | 0.0075 | 0.0075
$\delta_{i,1}$ | Normal | $\frac{y_{0}-\overline{y}}{\hat{\sigma}_{y}}$ | 0.75
$\gamma_{i,1}$ | Normal | 0 | 0.75

  * •

Notes: $\sigma_{\epsilon,i}$, $\sigma_{\delta,i}$, and $\sigma_{\gamma,i}$ are
the variances of the observation equation error (Equation (3)), trend, and
seasonal components, respectively, for a firm $i$. $\delta_{i,1}$ and
$\gamma_{i,1}$ are the initial values of the trend and seasonal state
components. When the prior density is inverse-gamma, hyperparameters 1 and 2
can be interpreted as the degrees of freedom and the sum of squared errors,
respectively (Brodersen et al., 2015, see). When the prior density is normal,
hyperparameters 1 and 2 are the mean and standard deviation, respectively.

Figure 9(b) shows the change in the effects on markups and profit rates under
this sensitivity analysis. After either increasing or decreasing the
variability of our priors, our finding that the markup rate is negatively
affected for most firms does not change. Likewise, our findings for the profit
rates on the impact of the pandemic remain unchanged: on average, firms
performed better than the counterfactual without the pandemic. In general,
when we increase the variability of our hyperparameters, the effects are
larger for most firms both in terms of markup and profit rates.

Figure 9: Comparison of Observed and Counterfactual Markups and Profit Rates
Under Different Hyperparameter choices.

(a) Markup Rate

(b) Profit Rate

Note: The figure shows yearly sales-weighted average observed and
counterfactual markup and profit rates. The solid lines represent the observed
values, while the dotted lines represent the counterfactual values for
2020-2021, as calculated using firm-level local linear projection models. The
shaded areas represent the 95%, 90%, 80%, and 50% (equally-tailed) credible
interval for each statistic, calculated based on 5,000 posterior simulations.

###  5.2 Robustness of findings to other modeling strategies: local linear
projections.

We also validate the robustness of our findings by estimating forecasts using
the flexible approach of Local Linear Projections (LLPs) (Jordà, 2005; Montiel
Olea and Plagborg-Møller, 2021). This method creates forecasts using linear
regressions of the outcome of interest on its lags and possibly lags of other
exogenous variables, which offers a flexible linear approximation of the
conditional expectation of the outcome given exogenous variables. We estimate
LLPs with either firm-specific projection coefficients, which we called firm-
specific LLPs, or with projection coefficients constant across firms, which we
called panel LLPs.

Particularly, in the case of the firm-specific LLPs, for every forecast
horizon $h$ and firm $i$, we consider the linear projection of the outcome of
interest, $Y_{i,t+h}$, on a vector stacking a constant and outcome lags up to
period $t$, $\mathbf{X}_{it}=(1,Y_{i,t-1},Y_{i,t-2},\dots,Y_{i,t-p_{i}})$,
with $p_{i}$ a positive integer,

$\displaystyle Y_{i,t+h}=$ $\displaystyle\
\mathbf{X}_{it}^{\prime}\beta_{ih}+\xi_{ith},\quad h=\ 1,2,\ldots H;\
i=1,\dots,I,$ (8)

where $\beta_{ih}$ is a $p_{i}+1$ vector of projection coefficients,
$\xi_{ith}$ is the projection residual, and $H$ is the maximum forecast
horizon, i.e., either two years or eight quarters. Thus, Equation (8) provides
the firm-specific LLPs forecast $\widehat{Y}_{i,t+h}$ for either markup or
profit rates. To estimate these models, we select firm-specific lag lengths
$p_{i}$ according to the Akaike information criterion, with a lag of up to 3
years with annual data and up to 8 quarters with quarterly data. As with BSTS,
we also evaluate whether LLPs provide precise forecasts. Particularly, we find
that LLPs tend to produce similarly accurate forecasts (if not slightly more
accurate) compared to BSTS models according to most forecast quality
statistics we consider (see Tables 3 and 7).121212In Appendix B.1, we use the
same sample of firms and periods to calculate forecast quality statistics for
BSTS and LLPs – we do not see relevant changes to the comparison across
methods after using this sample.

Figure 10: Comparison of Observed and Counterfactual Profit Rates using Local
Linear Projections

(a) Markup Rate

(b) Profit Rate

Note: The figure shows yearly sales-weighted average observed and
counterfactual markup and profit rates. The solid lines represent the observed
values, while the dotted lines represent the counterfactual values for the
period of 2020-2021, as calculated using firm-level local linear projection
models. The shaded areas represent the 95%, 90%, 80%, and 50% (equally-tailed)
credible interval for each statistic, calculated based on 5,000 posterior
simulations.

Using LLPs, we replicate our analysis of aggregate markup and profit rates in
Section 4.1. In particular, Figure 10(b) shows the counterfactual markup and
profit rates estimated using firm-specific LLPs. These LLPs forecasts result
in similar point estimates for markup rates when compared to the BSTS, but
with higher point estimates for profit rates, which suggests that the firm-
specific LLPs predict a smaller effect of the pandemic on profit rates.

To test for significant effects using the firm-specific LLPs, we compute
aggregate confidence intervals using the bootstrap method. Specifically, we
create 10,000 resamples with replacement by sampling firms in each year of the
pandemic. We then create a 95% confidence interval by taking the 2.5th and
97.5th percentiles of the average response across the 10,000 resamples. These
confidence intervals appear to be wide, with a larger range than the BSTS’s
credible intervals, which only allows us to reject the null hypothesis of no
significant effect on the aggregate profit rate in 2021.

Table 7:  Forecast Quality Statistics of Local-Linear Projections

(a) Firm-Specific Local-Linear Projection Variable | Year | ME | RMSE | MAE | MAES | MAPE |  Median MAPE | $n$
---|---|---|---|---|---|---|---|---
Markup rate | 2018 | -0.019 | 0.697 | 0.264 | 0.152 | 0.150 |  0.058 | 2858
Markup rate | 2019 | -0.086 | 0.821 | 0.358 | 0.211 | 0.217 |  0.091 | 2751
Profit rate | 2018 | 23.490 | 488.429 | 30.526 | 3.689 | 2.294 |  0.203 | 2858
Profit rate | 2019 | 16.957 | 206.812 | 29.079 | 4.244 | 3.029 |  0.313 | 2762

(b) Panel Local-Linear Projection Variable | Year | ME | RMSE | MAE | MAES | MAPE |  Median MAPE | $n$
---|---|---|---|---|---|---|---|---
Markup rate | 2018 | 0.017 | 0.555 | 0.223 | 0.128 | 0.120 |  0.054 | 2808
Markup rate | 2019 | -0.027 | 0.743 | 0.309 | 0.182 | 0.180 |  0.081 | 2808
Profit rate | 2018 | 5.809 | 96.882 | 16.257 | 1.965 | 1.590 |  0.318 | 2843
Profit rate | 2019 | 2.536 | 75.113 | 18.855 | 2.752 | 2.305 |  0.391 | 2825

  * •

Notes: The table introduces common forecast quality measures computed for
markup rates and profit rates for 2018–2019 with firm-level models estimated
using information up to 2017. The measures computed are: mean error (ME), root
mean square error (RMSE), mean absolute error (MAE), standardized mean
absolute error (MAES), and the mean absolute percentage error (MAPE). Median
MAPE is the median of the MAPEs across firm-level MAPEs. $n$ is the number of
firms.

Additionally, we extend our LLPs framework to analyze whether potential
unobserved heterogeneity in the projection coefficient (Equation (8)) plays a
role in the forecast accuracy of LLPs.131313 When favoring either firm-
specific estimation (Equation (8)) instead of pooled estimation (Equation
(9)), Pesaran et al. (2022) argues that either (i) a high level of
heterogeneity in the projection coefficient, (ii) a large number of observed
periods for each firm, or (iii) models with outcome lags will favor firm-
specific estimation.

We refer to this extension as panel LLPs. To this purpose, we extend the
linear projection in Equation (8) to incorporate a constant projection
coefficient across firms and firm fixed effects as follows,

$\displaystyle Y_{i,t+h}=$ $\displaystyle\
\mathbf{X}_{it}^{\prime}\beta_{h}+u_{ih}+\zeta_{ith};\quad h=\ 1,2,\ldots,H;\
i=1,\dots,I;$ (9)

where $\mathbf{X}_{it}=(1,Y_{i,t-1},Y_{i,t-2},\dots,Y_{i,t-p})$ is a $p+1$
vector stacking a constant and lags of the outcome $Y_{it}$, $\beta_{h}$ is a
panel-level projection coefficient, $u_{ih}$ is a firm fixed effect, and
$\zeta_{ith}$ is a projection residual.141414In Equation (9), we do not
include time fixed effects as we cannot forecast time fixed effects for 2020
and 2021 which makes it infeasible to compute forecasts for the outcome of
interests. Similarly to the firm-specific LLPs specification, we choose the
lag length $p$ based on the Akaike information criterion.151515We also
estimate the following specifications not reported here: (i) Equation (9)
replacing firm-fixed effects with industry fixed effects, (ii) Equation (9)
without the firm fixed effect $u_{ih}$ (or pooled specification), and (iii)
Equation (9) with industry-specific projection coefficients $\beta_{h}$. We
find similar results in these specifications to our baseline specification in
Equation (9). Using this extended model, we compute individual forecasts for
each firm and bootstrapped standard errors as before, and plot these estimates
in Figure 11(b).161616It seems that panel LLPs are more accurate than firm-
specific LLPs (see Panel (a) and (b) in Table 7), which suggests that there is
no high unobserved heterogeneity in the projection coefficient, $\beta_{ih}$.
In Appendix B.1, we use the same sample of firms and periods to calculate
forecast quality statistics for BSTS and LLPs – we do not see relevant changes
to the comparison across methods after using this sample.

Figure 11: Comparison of Observed and Counterfactual Profit Rates using Panel
Local Linear Projections

(a) Markup Rate

(b) Profit Rate

Note: The figure shows yearly sales-weighted average observed and
counterfactual markup and profit rates. The solid lines represent the observed
values, while the dotted lines represent the counterfactual values for the
period of 2020-2021, as calculated using a panel data local linear projection
model. The shaded areas represent the 95%, 90%, 80%, and 50% (equally-tailed)
credible interval for each statistic, calculated based on 5,000 posterior
simulations.

The panel LLPs results are quite similar to the firm-level LLPs, i.e., there
are no effects in markups and profit rates but positive effects in profit
rates in 2021 (see Figures 10(b) and 11(b)). In summary, it seems that the
LLPs have limited power for this data, as their confidence intervals are
relatively wide and their forecast quality statistics indicate they are not as
precise as the BSTS.

##  6 Conclusion

We assess the impact of the COVID-19 pandemic on market power and
profitability. Specifically, we compare the realized series of markup and
profit rates to a forecast based on information up to the beginning of the
pandemic. Essentially, we argue that this forecast represents a counterfactual
of what would have happened if the pandemic had never occurred. Our
counterfactual reveals that the COVID pandemic adversely affected markup rates
for the average firm in 2020 and 2021. Likewise, we find that most firms had
record profit rates that are significant in the sense that pre-pandemic trends
cannot explain those trends.

We estimate counterfactuals using the arguably flexible framework of Bayesian
structural time series (BSTS), which can accommodate multiple sources of
variation, including local trends and seasonality. On the one hand, these
estimations suggest that markup rates were 4.3% and 6.6% lower than their
counterfactuals in 2020 and 2021, respectively. On the other hand, the
counterfactual suggests that had the pandemic not happened, the profit rate
would have been 15.2% in 2021 which is 6.4 percentage points lower than the
observed value in 2021. Furthermore, we find that the effects of the pandemic
are statistically significant.

Our findings are largely robust to changes in our modeling strategy.
Particularly, we evaluate the plausibility of our assumptions and the
robustness of our results by implementing three strategies. First,
specification and goodness-of-fit tests support the assumptions and the
quality of the model. Second, our results are largely resilient to changes to
modeling choices, such as different data frequencies (either quarterly or
yearly) or changing the hyperparameters of the BSTS priors. Third, similar
findings follow from counterfactuals constructed from other modeling
strategies, such as local linear projections, which is a commonly used and
flexible method to compute forecasts. In summary, specification and goodness-
of-fit tests validate our assumptions, and other modeling strategies indicate
that our results are robust.

Furthermore, this paper uncovers the heterogeneity of these effects in terms
of pre-pandemic baseline characteristics such as markup rates, firm size,
stock-exchange tenure, employment, profitability, and market shares. Our
findings show that companies with a longer history on the stock exchange tend
to have a larger impact on markup rates. Additionally, companies with high
sales or low costs tend to have a significant positive impact on profit rates.

We complement this heterogeneity analysis with a breakdown of the effects of
the pandemic by industry. Particularly, industries such as information, real
estate, and chemical manufacturing had lower markups than expected, while
warehousing and entertainment had higher markups. A similar analysis of profit
rates by industry shows that transportation, entertainment, and hospitality
tended to have lower profit rates than expected, whereas warehousing and real
estate had higher profit rates.

Our results have implications for the current academic and economic policy
debate on the impacts of the pandemic on businesses. For instance, this work
can help guide future research on the impact of fiscal and monetary COVID
relief policies. Likewise, understanding the changes, and the subsequent
policy response, in markup and profit rates during the pandemic can help
policymakers guide strategies for addressing the impacts of the pandemic on
markets. Additionally, future research will uncover whether firms may have
reacted to the increased uncertainty brought by the pandemic by reducing
markups in order to maintain or expand their customer base. Finally, the
pandemic-induced lockdowns may have created a reallocation within firms’ input
use, such as the rise of remote workers (Bloom et al., 2015; Bloom, 2020),
which could have decreased selling, general, and administrative expenses,
while increasing the value of the costs of goods sold.

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##  Appendix A Data and summary statistics

To calculate our measures of markup and profit rates, we remove all firms with
negative sales, cost of goods sold (COGS), and selling, general and
administrative expenses (SG&A). We also remove records below the first and
above the 99th yearly percentiles of the COGS and SG&A to sales ratios per
year. We removed records without either Compustat firm identification numbers
or industry codes. After the previous steps, we also remove values below the
first and above the 99th percentile of two cost-shares (the ratio of COGS to
the total of COGS and capital expenditure, and the ratio of COGS to the total
of COGS, capital expenditure, and SG&A). Before computing these ratios, we
deflate the respective series using the GDP deflator with 2010 as the base
year.

In addition to these restrictions on the sample, we further impose the
following conditions to estimate counterfactuals reliably. First, we require
firms with data for both the pre-pandemic and the pandemic, so we remove
18,022 (17,003) firms that exited the panel before or in 2019, of which 79%
left before 2010. Then, we eliminate 191 (431) firms that either entered the
panel during or after 2020 or had less than two years of available data before
this year. Finally, to compute counterfactuals for firms with at least 3 years
of observations, we drop an additional 802 (925) firms. After these filtering
steps, our sample comprises 3,611 (3,192) publicly-traded firms with yearly
(quarterly) data, no extreme values in costs ratios, information for at least
5 years before the pandemic and operating during it.

Table A1: Summary Statistics

Variable | Acronym | Mean | Median | N. obs
---|---|---|---|---
Panel A: Sample 1955–2021
Sales | Sale, PQ | 2,125,657 | 159,894 | 269,899
Cost of goods sold | COGS, V | 1,414,280 | 94,946 | 269,899
Capital stock | PPEGT,K | 1,655,290 | 62,386 | 269,899
SG&A | XSG&A,X | 380,988 | 32,308 | 269,899
Wage bill | CXLR,WL | 1,088,658 | 117,408 | 32,982
Employment | Emp,L | 8,910 | 900 | 241,083
Panel B: Sample 1955–2016
Sales | Sale, PQ | 1,953,210 | 149,483 | 248,378
Cost of goods sold | COGS, V | 1,306,434 | 90,116 | 248,378
Capital stock | PPEGT,K | 1,474,565 | 58,178 | 248,378
SG&A | XSG&A,X | 349,539 | 30,043 | 248,378
Wage bill | CXLR,WL | 1,106,980 | 131,148 | 28,321
Employment | Emp,L | 8,426 | 866 | 221,685
Panel C: Sample 1955-2016 from De Loecker et al. (2020)
Sales | Sale, PQ | 1,922,074 | 147,806 | 247,644
Cost of goods sold | COGS, V | 1,016,550 | 55,384 | 247,644
Capital Stock | PPEGT,K | 1,454,210 | 57,532 | 247,644
SG&A | XSG&A,X | 342,805 | 29,682 | 247,644
Wage bill | CXLR,WL | 1,093,406 | 130,486 | 28,116
Employment | Emp,L | 8,363 | 863 | 221,121

  * •

Notes: Thousands yearly US$ deflated using the GDP deflator with base year
2010. In column 2 of the table, we also report the Compustat acronym to keep
track of the specific variables with respect to that dataset.

##  Appendix B Counterfactual analysis using cost-of-goods weighting

In this section, we analyze the impact of our choice of weights on the results
for the aggregate markup and profit rates. In Figures 5(b) and 6(b) we use
sales weights, thus giving more importance to companies with higher sales.
Alternatively, one can use costs of good sold (COGS) as weights instead of
sales as presented in Figures 1(b). Therefore, we give more importance to
firms or industries with high cost structure, such as aviation, hospitality,
or oil and gas. The results for markups remain similar to those using sales-
weighting, i.e., the average firm has lower realized markups than its
counterfactual markups.

On the other hand, when weighting by COGS, the results for profit rate become
mixed: firms had negative effects in 2020 and exhibited a strong recovery in
2021. This is more likely related to the weighting: high cost firms seem to
have lost profits during 2020, whereas low cost firms increased their profits
(sales-weighting case).

Figure B.1: Comparison of Observed and Counterfactual Markup and Profit Rates,
weighting by Cost of Goods Sold (COGS)

(a) Markup Rate

(b) Profit Rate

Note: Note: The figure shows the cogs-weighted aggregate mean of the yearly
observed and counterfactual markups (Panel (a)) and profit rates (Panel (b)).
The solid lines represent the observed values, while the dotted lines
represent the counterfactual values for 2020-2021, as calculated using firm-
level Bayesian structural time series models. The shaded areas represent the
95%, 90%, 80%, and 50% (equally-tailed) credible interval for each statistic,
calculated based on 5,000 posterior simulations.

###  B.1 Forecast performance for BSTS and LLPs under a balanced sample

This subsection presents the measures on Table 3 and Table 7, but considering
a fully balanced panel. The results show that the presence of unbalanced
panels does not change the relative performance of each forecasting strategy.

Table A2:  Forecast Quality Statistics Across Models, Balanced Sample.

(a) Firm-Specific Bayesian Time Series Models Variable | Year | ME | RMSE | MAE | MAES | MAPE |  Median MAPE | $n$
---|---|---|---|---|---|---|---|---
Markup rate | 2018 | -0.015 | 0.790 | 0.361 | 0.210 | 0.200 |  0.105 | 2808
Markup rate | 2019 | -0.061 | 0.844 | 0.384 | 0.230 | 0.233 |  0.120 | 2693
Profit rate | 2018 | 19.632 | 146.502 | 27.645 | 3.224 | 2.957 |  0.333 | 2843
Profit rate | 2019 | 16.809 | 148.274 | 28.480 | 3.878 | 3.608 |  0.367 | 2718

(b) Firm-Specific Local-Linear Projection Variable | Year | ME | RMSE | MAE | MAES | MAPE |  Median MAPE | $n$
---|---|---|---|---|---|---|---|---
Markup rate | 2018 | -0.023 | 0.644 | 0.248 | 0.144 | 0.144 |  0.057 | 2808
Markup rate | 2019 | -0.086 | 0.767 | 0.337 | 0.203 | 0.206 |  0.089 | 2693
Profit rate | 2018 | 22.345 | 488.279 | 29.279 | 3.414 | 2.280 |  0.201 | 2843
Profit rate | 2019 | 16.425 | 206.869 | 27.893 | 3.798 | 3.032 |  0.309 | 2718

(c) Panel Local-Linear Projection Variable | Year | ME | RMSE | MAE | MAES | MAPE |  Median MAPE | $n$
---|---|---|---|---|---|---|---|---
Markup rate | 2018 | 0.017 | 0.555 | 0.223 | 0.128 | 0.120 |  0.054 | 2808
Markup rate | 2019 | -0.030 | 0.721 | 0.294 | 0.176 | 0.172 |  0.078 | 2693
Profit rate | 2018 | 5.809 | 96.882 | 16.257 | 1.896 | 1.590 |  0.318 | 2843
Profit rate | 2019 | 2.679 | 73.403 | 18.088 | 2.463 | 2.282 |  0.373 | 2718

Notes: The table introduces common forecast quality measures computed for
markup rates and profit rates for 2018–2019 with firm-level models estimated
using information up to 2017. The measures computed are: mean error (ME), root
mean square error (RMSE), mean absolute error (MAE), standardized mean
absolute error (MAES), and the mean absolute percentage error (MAPE). Median
MAPE is the median of the MAPEs across firm-level MAPEs. $n$ is the number of
firms and the sample is balanced to consider the same observations across
exercises.

##  Appendix C Quarterly Analysis

This section expands the aggregate counterfactual analysis in Figures 5(b) and
6(b), but using quarterly data instead of yearly data. Figures 2(b) and 3(b)
shows similar results for the markup rate: markups would have been higher had
the pandemic not happened. Particularly, for the average firm, the annual
average observed markup rate was 4.5% and 11.6% lower than its counterfactual
value in 2020 and 2021, respectively. Interestingly, in the case of profit
rates, the positive effects of the pandemic only seem to have started in the
third quarter of 2020. More precisely, the average firm had a profit rate of
16.9% and 21.6% in 2020 and 2021, respectively, which are 0.7 and 4.75
percentage points higher than the counterfactuals.

Figure C.2: Counterfactual Analysis of Average and Quartiles of Markup Rates

(a) Average

(b) Quartiles

Note: The figure shows four aggregate statistics of the quarterly observed and
counterfactual markup rates. The statistics are the sales-weighted average
(shown in Panel (a)), the first quartile (shown in Panel (b) in red), the
second quartile (shown in Panel (b) in black), and the third quartile (shown
in Panel (b) in gray). The solid lines represent the observed values, while
the dotted lines represent the counterfactual values for 2020-2021, as
calculated using firm-level Bayesian structural time series models. The shaded
areas represent the 95%, 90%, 80%, and 50% (equally-tailed) credible interval
for each statistic, calculated based on 5,000 posterior simulations on Panel
(a). 95% (equally-tailed) credible intervals are displayed for the statistics
on Panel (b).  Figure C.3: Counterfactual Analysis of Average and Quartiles of
Profit Rates

(a) Average

(b) Quartiles

Note: The figure shows four aggregate statistics of the quarterly observed and
counterfactual profit rates. The statistics are the sales-weighted average
(shown in Panel (a)), the first quartile (shown in Panel (b) in red), the
second quartile (shown in Panel (b) in black), and the third quartile (shown
in Panel (b) in gray). The solid lines represent the observed values, while
the dotted lines represent the counterfactual values for 2020-2021, as
calculated using firm-level Bayesian structural time series models. The shaded
areas represent the 95%, 90%, 80%, and 50% (equally-tailed) credible interval
for each statistic, calculated based on 5,000 posterior simulations on Panel
(a). 95% (equally-tailed) credible intervals are displayed for the statistics
on Panel (b).

###  C.1 Quarterly heterogeneity analysis

This section analyzes the heterogeneous effects of the pandemic using
quarterly data, which provides insights into the unfolding of the pandemic in
2020 and 2021.

####  C.1.1 Effect heterogeneity regressions

Finally, we validate the results in Section 4.2 by averaging the quarterly
effects into the annual frequency and repeating the accounting exercise, with
the results displayed on Table A3 and A4. The implied yearly effects backed by
the high-frequency analysis seem to be more reactive to cogs and sales, with
very similar results for employment, stock exchange tenure, and market shares.
On the other hand, the results for profit rates exhibit very similar results
for all of the variables. This suggests that the short-term effects on markups
may be more reactive to cogs and sales, in a way that when averaged out, they
still matter with respect to those calculated using the yearly data. This
could be driven by the immediate changes in very short-term demand movements
Gilchrist et al. (2017).

Table A3: Heterogeneous Effects of the pandemic on Markup Rates (Quarterly
Specification)

Dependent Variable: | Average markup rate pandemic effect in 2020–2021
---|---
| All | 2020 | 2021
Model: | (1) | (2) | (3) | (4) | (5) | (6)
_Variables_ |  |  |  |  |  |
COGS | -0.244∗∗∗ | -0.263∗∗∗ | -0.296∗∗∗ | -0.313∗∗∗ | -0.181∗∗ | -0.197∗∗
| (0.054) | (0.057) | (0.066) | (0.073) | (0.086) | (0.088)
Sales | 0.197∗∗∗ | 0.248∗∗∗ | 0.230∗∗∗ | 0.278∗∗∗ | 0.160∗ | 0.212∗∗
| (0.056) | (0.058) | (0.067) | (0.072) | (0.093) | (0.090)
Employment | 0.037∗∗∗ | 0.008 | 0.065∗∗∗ | 0.033∗∗ | 0.000 | -0.028
| (0.013) | (0.013) | (0.014) | (0.015) | (0.022) | (0.023)
Stock-exchange tenure | 0.002∗∗∗ | 0.002∗∗ | 0.003∗∗∗ | 0.002∗ | 0.001 | 0.001
| (0.001) | (0.001) | (0.001) | (0.001) | (0.002) | (0.002)
Market share | 0.837∗∗∗ | 0.002 | 0.235 | -0.012 | 1.481∗∗ | -0.047
| (0.301) | (0.308) | (0.253) | (0.287) | (0.680) | (0.566)
_Fixed-effects_ |  |  |  |  |  |
2-digit NAICS industry |  | Yes |  | Yes |  | Yes
Mean | 1.599 |  | 1.590 |  | 1.610 |
Mean effect | -0.040 |  | -0.045 |  | -0.033 |
_Fit statistics_ |  |  |  |  |  |
Observations | 5,017 | 5,017 | 2,752 | 2,752 | 2,265 | 2,265
R2 | 0.021 | 0.075 | 0.044 | 0.095 | 0.010 | 0.082
Within R2 |  | 0.018 |  | 0.034 |  | 0.009

  * •

Notes: The table presents cross-section firm-level regressions of the average
pandemic effect at the firm-level on the 2015–2019 average of the logarithm
cost of goods sold, logarithm of sales, logarithm of employment, years since
publicly-traded, and market shares. NAICS = North American Industrial
Classification System. Heteroskedasticity-robust standard errors in
parentheses. Signif. Codes: ***: 0.01, **: 0.05, *: 0.1

Table A4: Heterogeneous Effects of the pandemic on Profit Rates (Quarterly
Specification)

Dependent Variable: | Average profit rate pandemic effect in 2020–2021
---|---
| All | 2020 | 2021
Model: | (1) | (2) | (3) | (4) | (5) | (6)
_Variables_ |  |  |  |  |  |
COGS | -15.889∗∗∗ | -14.229∗∗∗ | -21.252∗∗∗ | -15.754∗∗∗ | -9.525 | -12.315
| (4.186) | (4.621) | (3.705) | (3.712) | (8.018) | (9.062)
Sales | 16.075∗∗∗ | 15.365∗∗∗ | 20.067∗∗∗ | 15.685∗∗∗ | 11.476 | 15.221∗
| (4.605) | (4.843) | (5.232) | (4.780) | (7.842) | (8.775)
Employment | 1.140 | -0.076 | 2.610 | 0.796 | -0.795 | -1.563
| (1.871) | (2.122) | (2.826) | (2.845) | (2.296) | (3.126)
Stock-exchange tenure | 0.149 | 0.005 | 0.140 | -0.081 | 0.143 | 0.086
| (0.093) | (0.086) | (0.115) | (0.107) | (0.153) | (0.140)
Market share | -71.887∗ | -20.439 | -100.702 | 9.398 | -43.641 | -59.176
| (39.196) | (44.425) | (70.899) | (67.204) | (27.539) | (38.484)
_Fixed-effects_ |  |  |  |  |  |
2-digit NAICS industry |  | Yes |  | Yes |  | Yes
Mean | 2.974 |  | -0.747 |  | 7.546 |
Mean effect | -1.070 |  | -4.573 |  | 3.235 |
_Fit statistics_ |  |  |  |  |  |
Observations | 5,017 | 5,017 | 2,752 | 2,752 | 2,265 | 2,265
R2 | 0.009 | 0.027 | 0.015 | 0.067 | 0.004 | 0.009
Within R2 |  | 0.005 |  | 0.006 |  | 0.006

  * •

Notes: The table presents cross-section firm-level regressions of the average
pandemic effect at the firm-level on 2015–2019 average of the logarithm cost
of goods sold, the logarithm of sales, logarithm of employment, years since
publicly-traded, and market shares. NAICS = North American Industrial
Classification System. Heteroskedasticity-robust standard errors in
parentheses. Signif. Codes: ***: 0.01, **: 0.05, *: 0.1

###  C.2 Tracking the fraction of firms with significant effects in the short-
term.

To elucidate whether these effects on markup rates are statistically
significant, we calculated the fraction of firms with significant effects, and
its decomposition into the fraction of firms with significant and positive
effects, and significant and negative effects. Figure 4(a) shows this
calculation for quarterly data in 2020–2021. We find that 19% of the firms had
a significant effect on their markups at the beginning of the pandemic, with
most of those firms having a negative effect and over 7% of the firms having a
positive effect. In most of 2020–2021, the number of firms with significant,
negative effects is higher than the number of firms with significant,
positive, effects. Counting the number of firms with at least one quarter with
a significant effect, we find that 1,122 out of 3,192 firms have a significant
effect.

Besides analyzing the statistical significance of the markup effects, we find
that only a small fraction of the effects on profit rates are significant.
Figure 4(b) shows this calculation for quarterly data in 2020–2021.
Particularly, we find that 19% of the firms had significant effects on profit
rates at the beginning of the pandemic, where most of the firms have a
negative effect and over 4% of the firms had a positive effect. By the first
quarter of 2021, the share of firms with positive effects increase but is
still lower than those with negative effects. When we count the number of
firms with at least one significant effect in one quarter, we find that 953
out of 3,152 firms have a significant effect.

Figure C.4: Fraction of Firms with Significant effects of the pandemic,
Bayesian Structural Time Series Model

(a) Markup Rates

(b) Profit Rates

Note: The figure plots the fraction of firms with significant (red solid),
positive and significant (blue dotted), and negative and significant (black
dotted) effects. A significant effect is defined when the corresponding 95%
posterior credible interval does not contain zero.

To avoid multiple hypothesis testing issues and given our sample has thousands
of firms corresponding to at least an equal number of hypothesis tests, we
implement the Benjamini-Hochberg algorithm to control for the false discovery
rate (FDR), which is the expected number of statistically significant effects
that are indeed null effects.171717We also implement four corrections that
control the family-wise error rate, i.e., the probability of detecting at
least one significant effect that is indeed null, including the Bonferroni,
Hochberg, Holm, and Hommel corrections, and find no significant effects.
However, such procedures are quite conservative, even more so when one has
thousands of hypotheses such as in our case (Storey and Tibshirani, 2003). In
addition to these procedures, we implement the $q$-value approach of Storey
and Tibshirani (2003) and found a similar number of tests that are significant
while controlling the FDR at 5%; however, the assumption on the uniform
distribution of $p$-values in the unit interval may not hold since the highest
$p$-value in our sample is just below 0.5. As this algorithm relies on the
computed $p$-values for each individual test to construct a joint test for
each firm for the entirety of the pandemic in our sample, we compute the firm-
level $p$-value for the 2020–2021 average effect on markup rates. When
controlling the FDR at 5% for the effects on markup rates, we find that there
are 1,240 out of 3,192 statistically significant effects. If we were to follow
the naive approach of counting the number of firms with statistically
significant effects at 5% using the $p$-values, we would consider 1,518
significant effects. Overall, the quarterly analysis using Figure 4(b) and the
FDR analysis of statistic significance are consistent with each other, with
the former producing a slightly higher fraction of firms with significant
effects, i.e., 35.2% versus 38.8%.

We repeat the previous multiple hypothesis correction to evaluate the
statistical significance of the effects on profit rates.181818Likewise, we
find similar effects when using the $q$-value correction of Storey and
Tibshirani (2003). We also implement four extremely conservative tests
correcting for the FDR (Bonferroni, Hochberg, Holm, and Hommel) and find no
significant effects. Consequently, when controlling the FDR at 5% for the
effects on profit rates, we find that there are 832 out of 3,152 statistically
significant effects. If we were to follow the naive approach of counting the
number of firms with statistically significant effects at 5% using the
$p$-values, we would consider 1,200 significant effects. Overall, the
quarterly analysis using Figure 4(b) and the FDR analysis of statistic
significance are consistent with each other, with the former producing a
slightly higher fraction of firms with significant effects, i.e., 29.9% versus
26.4%.

One concern when performing the Benjamini-Hochberg correction is that either
markup or profit rates may be highly correlated across firms. To alleviate
this concern, we perform both factor and correlation analyses as in Bermingham
and D’Agostino (2014). To have a sense of how correlated the series are across
firms, that study suggests estimating the correlation between each firm’s
markup or profit rate with the principal component calculated across firms. To
avoid losing a big fraction of our sample and given the pattern of entry and
exit of firms in the stock exchange, we estimate principal components in
5-year windows. We then compute the average $R^{2}$ for the firm-level
regressions of either the markup or profit rate on the corresponding first
principal component. We obtain an average $R^{2}$ of just 0.07 and 0.06 for
markup rates and profit rates, respectively. This result indicates that there
is not a sizable correlation across firms for either the markup or the profit
rates. Moreover, the average pairwise correlation in the markup and profit
rates across firms is very small and equal to 0.01 and 0.016, respectively.

In addition, we conduct the corresponding robustness checks by plotting this
decomposition for both the LLPs model and using the hyperparameters in the
sensitivity analysis presented in Section 5.

####  C.2.1 Fraction of firms with significant effects of the pandemic, LLPs
Model

Figure C.5: Fraction of Firms with Significant effects of the pandemic, LLPs
Model

(a) Markup Rates

(b) Profit Rates

Note: The figure plots the fraction of firms with significant (red solid),
positive and significant (blue dotted), and negative and significant (black
dotted) effects. A significant effect is defined when the corresponding 95%
posterior credible interval does not contain zero.

Using LLPs, we replicate our analysis for the aggregate markup and profit
rates in Section 4. Overall, both BSTS and LLPs forecasts find that there are
more negatively affected firms in the first two-quarters of the pandemic, with
LLPs reporting more significantly affected firms than BSTS. (see Figures 4(b)
and 5(b)). Additionally, both methods suggest that the share of firms
benefiting from the pandemic was initially low in 2020 and grew steadily up to
the end of 2021. LLPs seems to be more liberal as it tends to find more
significant effects on both markup and profit rates compared to BSTS.

The results tell the same story as those in Figure 4(b), but the LLPs model
shows more significant effects in both directions, with roughly 28%
significant effects in the first half of 2020. However, in Section 3.3 we
observe that the Bayesian model has better out-of-sample performance, so the
observed increase observed in the LLPs model might be driven by overestimated
effects due to forecasting errors.

####  C.2.2 Fraction of firms with significant effects of the pandemic,
Sensitivity Analysis

In this section, we repeat the analysis in Figure 4(b) and changing the
hyperparameters as detailed in Section 5. Interestingly, the results are quite
similar, with increased (decreased) variability producing higher (fewer)
significant effects, especially for those observing negative effects.
Nonetheless, these changes seem to be modest.

After changing the variability of the hyperparameters governing the priors of
the markups models, either by increasing or decreasing that variability, we
validate our finding that at the beginning of the pandemic, there was a higher
faction of negative firms that positively affected firms (see Panels (a) and
(b) of Figure C.6 and Panel (b) of Figure 4(b)).

Figure C.6: Sensitivity Analysis: Fraction of Firms with Significant effects
of the pandemic (Quarterly Specification)

(a) Markups, 125% Hyperparameters

(b) Markups, 75% Hyperparameters

(c) Profit Rates, 125% Hyperparameters

(d) Profit Rates, 75% Hyperparameters

Note: The figure plots the fraction of firms with significant (red solid),
positive and significant (blue dotted), and negative and significant (black
dotted) effects. A significant effect is defined when the corresponding 95%
posterior credible interval does not contain zero.

##  Appendix D Pandemic effect heterogeneity by quintiles of key firm
characteristics.

To scale the effect across firms with different baseline markup rates, we
divide the estimate of the causal impact for markup rates by the average value
of markup rates in 2015–2019 as follows

$\displaystyle\widehat{\tau}_{i,t+h}^{s}=\frac{Y_{i,t+h}-\widehat{Y}_{i,t+h}}{\overline{Y}_{i,2015\text{--}2019}},$

where $\overline{Y}_{i,2015\text{--}2019}$ is firm’s $i$ average markup rate
in 2015–2019. We acknowledge that choosing 2015–2019 as a scaling period is
somewhat arbitrary, but it has the advantage of at least taking a period that
is actually close to the pandemic years.191919Since the profit rate takes
negative values for some firms in some time periods, we do not scale those
rates to avoid interpretation issues.

Figures D.7 and D.8 report the results of this heterogeneity analysis for the
markup rates, both without and with scaling by the average markup rates in
2015–2019. Firms with low markups in 2015–2019 have experienced the strongest
reductions in their markup rates during the pandemic. In contrast, some of the
firms in the top 20% highest markups seem to have increased their markup rates
both in the first and second years of the pandemic (Panel 7(a)). Regarding
firm size heterogeneity, there are no big differences in the effects on
markups (Panel 7(b)). Firms with longer tenure as public firms seem to be
better at hedging the pandemic shock since they exhibit fewer negative effects
in the first year of the pandemic (Panel 7(c)). Furthermore, the top 20% firms
in the profit rate tend to have positive effects on markup rates in 2021,
whereas firms in the other quintiles tend to have negative effects on markups.
The effects are quite heterogeneous for those firms with negative profit rates
in 2015–2019. On the other hand, there are no noticeable differences in the
effects on markups across market shares (Panel 7(e)). Finally, firms in the
middle of the distribution of the number of employees tend to have lower
impacts on their markups (Panel 7(f)). We find similar patterns when using the
quarterly specification of the model (Figures D.10 and D.11).

In addition to showing the heterogeneous impacts of the pandemic on firms’
markup rates, we also explored heterogeneous effects on the profit rate.
Figure D.9 shows such a heterogeneity analysis. In terms of heterogeneity for
firms with different pre-pandemic markup rates, those firms with the highest
markups tend to have positive and higher profit rate effects due to the
pandemic (Panel 9(a)). Smaller firms tend to be less affected, yet the
differences across quintiles appear to be not significant (Panel 9(b)). Firms
that became public in the last half of the 2000s and in the early 1990s tend
to have more drastic negative impacts (Panel 9(c)). By profit rate, the bottom
20% of firms in the distribution of profit rate have highly heterogeneous
effects, whereas the most profitable firms in 2015–2019 tend to be better off
during the pandemic (Panel 9(d)). Furthermore, firms with larger market shares
tend to have larger negative effects on the profit rate (Panel 9(e)). Finally,
the very large employers and the very small employers seem to have the worse
profit-rate effects of the pandemic in 2020 (Panel 9(f)). When using the
quarterly specification of the model, we find similar patterns in the effects
on profit rates across the 2015–2019 average levels of profit rate, market
share, and employment (Figure D.12).

Figure D.7: Heterogeneity Analysis of Effect of Pandemic in Markups (Yearly
Specification)

(a) Markup Rate

(b) Sales

(c) Years in stock exchange

(d) Profit Rate

(e) Market share

(f) Employment

Note: $y$-axis: effect on markup rates as $Y_{it}-\widetilde{Y}_{it}$, where
$Y_{it}$ is firm $i$’s markup rate at time period $t$, and
$\widetilde{Y}_{i,t}$ is the firm’s counterfactual markup rate built upon a
Bayesian structural time series model with seasonal and local-level stochastic
trend. $x$-axis: firm’s $i$ average value of either (a) markup rates, (b)
sales (logarithm of constant 2010 thousand US dollars), (c) years since
publicly traded, (d) profit rates, (e) market shares, and (f) employment
(thousands of employees) in 2015–2019.  Figure D.8: Heterogeneity Analysis of
Effect of Pandemic in Scaled Markups (Yearly Specification)

(a) Markup Rate

(b) Sales

(c) Years in stock exchange

(d) Profit Rate

(e) Market Share

(f) Employment

Note: $y$-axis: effect on markup rates as
$(Y_{it}-\widetilde{Y}_{it})/\overline{Y}_{i,2015\text{--}2019}$, where
$Y_{it}$ is firm $i$’s markup rate at time period $t$, $\widetilde{Y}_{i,t}$
is the firm’s counterfactual markup rate built upon a Bayesian structural time
series model with the seasonal and local-level stochastic trend, and
$\overline{Y}_{i,2015\text{--}2019}$ is the firm’s average markup rate in
2015–2019. $x$-axis: firm’s $i$ average value of either (a) markup rates, (b)
sales (logarithm of constant 2010 thousand US dollars), (c) years since
publicly traded, (d) profit rates, (e) market shares, and (f) employment
(thousands of employees) in 2015–2019.  Figure D.9: Heterogeneity Analysis of
Effect of Pandemic on Profit Rate (Yearly Specification)

(a) Markup rate

(b) Sales

(c) Years in stock exchange

(d) Profit rate

(e) Market share

(f) Employment

Note: Each plot shows the 5-bin binscatter of the pandemic effect on profit
rates by the respective variable in each panel. $y$-axis: effect on profit
rates as $Y_{it}-\widetilde{Y}_{it}$, where $Y_{it}$ is firm $i$’s profit rate
at time period $t$, and $\widetilde{Y}_{i,t}$ is the firm’s counterfactual
profit rate built upon a Bayesian structural time series model with seasonal
and local-level stochastic trend. $x$-axis: firm’s $i$ average value of either
(a) markup rates, (b) sales (logarithm of constant 2010 thousand US dollars),
(c) years since publicly traded, (d) profit rates, (e) market shares, and (f)
employment (thousands of employees) in 2015–2019.  Figure D.10: Heterogeneity
Analysis of Effect of Pandemic in Markups (Quarterly Specification)

(a) Markup Rate

(b) Sales

(c) Years in stock exchange

(d) Profit Rate

(e) Market share

(f) Employment

Note: $y$-axis: effect on markup rates as $Y_{it}-\widetilde{Y}_{it}$, where
$Y_{it}$ is firm $i$’s markup rate at time period $t$, and
$\widetilde{Y}_{i,t}$ is the firm’s counterfactual markup rate built upon a
Bayesian structural time series model with seasonal and local-level stochastic
trend. $x$-axis: firm’s $i$ average value of either (a) markup rates, (b)
sales (logarithm of constant 2010 thousand US dollars), (c) years since
publicly traded, (d) profit rates, (e) market shares, and (f) employment
(thousands of employees) in 2015–2019. Figure D.11: Heterogeneity Analysis of
Effect of Pandemic in Scaled Markups (Quarterly Specification)

(a) Markup Rate

(b) Sales

(c) Years in stock exchange

(d) Profit Rate

(e) Market Share

(f) Employment

Note: $y$-axis: effect on markup rates as
$(Y_{it}-\widetilde{Y}_{it})/\overline{Y}_{i,2015\text{--}2019}$, where
$Y_{it}$ is firm $i$’s markup rate at time period $t$, $\widetilde{Y}_{i,t}$
is the firm’s counterfactual markup rate built upon a Bayesian structural time
series model with seasonal and local-level stochastic trend, and
$\overline{Y}_{i,2015\text{--}2019}$ is the firm’s average markup rate in
2015–2019. $x$-axis: firm’s $i$ average value of either (a) markup rates, (b)
sales (logarithm of constant 2010 thousand US dollars), (c) years since
publicly traded, (d) profit rates, (e) market shares, and (f) employment
(thousands of employees) in 2015–2019. Figure D.12: Heterogeneity Analysis of
Effect of Pandemic in Profit Rate (Quarterly Specification)

(a) Markup rate

(b) Sales

(c) Years in stock exchange

(d) Profit rate

(e) Market share

(f) Employment

Note: Each plot shows the 5-bin binscatter of the pandemic effect on profit
rates by the respective variable in each panel. $y$-axis: effect on profit
rates as $Y_{it}-\widetilde{Y}_{it}$, where $Y_{it}$ is firm $i$’s profit rate
at time period $t$, and $\widetilde{Y}_{i,t}$ is the firm’s counterfactual
profit rate built upon a Bayesian structural time series model with seasonal
and local-level stochastic trend. $x$-axis: firm’s $i$ average value of either
(a) markup rates, (b) sales (logarithm of constant 2010 thousand US dollars),
(c) years since publicly traded, (d) profit rates, (e) market shares, and (f)
employment (thousands of employees) in 2015–2019.