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May 8, 2020

Economic Impact of COVID-19
Forecasting the COVID-19 Epidemic for the U.S.
By Paul Ho, Thomas A. Lubik, and Christian Matthes
Introduction
A key weapon in fighting the COVID-19 epidemic is understanding how the contagion has
spread through the U.S. population and how its
spread is likely to evolve in the future. Based on
such knowledge, public health measures can
be devised, whether they are social distancing
recommendations or more stringent lockdown
procedures. Understanding of the disease’s path
can be gained using theoretical or statistical
modelling techniques that allow researchers
to forecast its future course, which can then be
used as a basis for decisions about further public
health measures.
The coronavirus behind the COVID-19 pandemic
is a novel contagion that is highly infectious,
has a long incubation period, and can transmit
asymptomatically, that is, without an infected
person showing any signs of infection or disease.
At the same time, this also means that data on
infections and even deaths caused by the disease are difficult to collect, resulting in time lags
between infections, possible fatalities, and data
availability. In addition, the coronavirus is novel
enough that previous experiences, such as the
SARS pandemic of 2003, may not be immediately
applicable.
A particularly vexing feature of many attempts
to project the course of the pandemic in the
U.S. and across the world is that projections
May 2020 – Richmond Fed

have changed frequently, often in significant
ways. This is true of forecasting models that rely
on strong theoretical relationships, such as the
Imperial College model that informed the U.K.
government’s early response to the crisis, but
also of the statistical model developed by the
Institute for Health Metrics and Evaluation at the
University of Washington that was referenced in
the U.S. government’s response.
This aspect of forecasting the course of the pandemic is problematic insofar as frequent revisions
may cast doubt on the validity of the model.
Macroeconomic forecasters are familiar with
this challenge since the economy is buffeted by
shocks, the data are subject to measurement errors, and the underlying behavior of the variables
may change over the forecast horizon because
of policy interventions. All of these aspects are
present in the current situation when attempting
to forecast the path of the pandemic.
However, there is the danger that policymakers
and the public lose trust in the researchers’ and
forecasters’ ability to capture and describe the
disease. In such a forecasting environment, the
source of uncertainty needs to be carefully communicated and taken into account during the
decision-making process. Moreover, forecasters
should adapt to the changing nature of the data
and where forecasts went wrong.

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In this article, we describe a statistical model that we
use to estimate and forecast the path of infections
and deaths caused by COVID-19 in the U.S. We focus
on documenting the uncertainty surrounding the
estimates and projections, as our approach is not immune to the issues raised above. However, we argue
that understanding the source of uncertainty is an
important step in making public health decisions.
The Epidemic Forecasting Model and Data
We have developed a statistical model for estimating
and forecasting the number of infections and deaths
over the course of the pandemic. (Documentation
of the model and the sources can be found here).
Our model is almost entirely data-driven, in that it
tries to match the underlying time series properties
of the data at hand in a flexible manner while at the
same time relying on guidance from epidemiological
insights about how an epidemic runs its course.
The time path of the number of infections during an
epidemic follows a typical pattern. When a pathogen
enters a population that is susceptible to infection,
the number of infected cases is initially low. However,
the growth rate of new infections is high and tends
to rise sharply at an exponential rate because each
infected person creates a chain of new infections.
At some point, however, the pathogen runs out of
susceptible hosts, either because they are already
infected, are immune, or they are simply not physically present due to health policies such as social
distancing. At this inflection point, the growth rate of
infections falls until it eventually declines to zero.
In our empirical model, we attempt to replicate these
broad patterns of an epidemic. We do so by specifying a flexible functional form that describes the path
of infections over time as depending on the current
and lagged levels of the number of infections. The
model is loosely parameterized, whereby the parameters are estimated to provide best fit of the model
specification to the available data. In contrast to
theoretical epidemiological models, our specification
has more leeway to go where the data tell it to and is
not constrained by precise theoretical relationships
that may be specified incorrectly.

Identification of the model parameters is based
on the growth rate and changes in the growth
rate of infections. Early in an epidemic, the data
typically show exponential growth, rapid and
increasing, whereas after some time, as the stock
of susceptible hosts starts getting smaller, the
rise in the growth rate decelerates until it reaches
a peak. Afterward, the growth rate of new infections declines. These three distinct phases of an
epidemic can be associated with distinct parameters in our model, which are thus identified from
the data flow.
This is also where a problematic aspect of any epidemiological model lies. At first, data are sparse,
but the underlying course of the infection is such
that it should be easy to forecast. Put differently,
the epidemic develops a very strong trend with
exponential growth. Simply extrapolating from
this growth trend would produce good forecasts
for a while – until the spread starts slowing down
and gravitates toward an inflection point. While
epidemiological models based on the course of
previous epidemics confirm that there will be an
inflection point, estimates from the sparse initial
data are highly uncertain. Moreover, theoretical and statistical epidemiological models are
sensitive to small variations in parameters. It is
in this sense that model estimates and forecasts
should be interpreted with much caution at the
beginning of the pandemic, and uncertainty at
this stage should explicitly be taken into account
when making public health policy decisions.
In addition to modeling infections, we also
consider the mortality rate. Fundamentally, the
number of deaths is a function of the number of
infections. Not all infections are fatal, and an observed death is the outcome of a process that can
vary over time. We thus assume that the number
of deaths on any given day is proportional to the
average number of observed infections over a
time period. This captures the idea that there is
a minimum number of days that pass after an
initial infection can result in a fatality.

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A key aspect of our modeling approach is that we
explicitly capture the uncertainty of the model estimates and, perhaps more importantly, the uncertainty inherent in the forecast. The precision of a forecast,
or how tightly possible alternative forecast paths
are concentrated around the most plausible path,
is generally affected by two factors: first, the uncertainty of the model estimates in terms of overall fit
and parameter estimates since no statistical model
fits precisely; and second, by the extent to which
the model may be subject to further disturbances or
imprecision in data collection in the future. We take
both aspects into account to give a sense of how
uncertain forecasts in a pandemic truly are, especially
when the data flow is sparse at the beginning.
We fit our models to observed data on daily new
cases of infections and deaths. The estimated models are then used to forecast the future paths of the
respective variables, whereby we take into account
all potential sources of uncertainty. We collect data

from a variety of publicly available sources. The
estimates are performed on these data up to and
including May 3, 2020.
Estimates and Forecasts of the
Number of Infections
Figure 1 shows the cumulative number of cases,
i.e., infections, in the U.S. and the daily count of
new cases as a percentage of the population.
The grey line in the graph represents the actual
number of measured new infections, while the
orange lines are drawn from the estimated model. We show the best-fitting line and a 95 percent
confidence region around these estimates. In
other words, the estimates represent our assessment of the trend in number of infections as seen
through the lens of the empirical model. They
differ from the actual numbers because the latter
are subject to various errors, such as simple data
entry mistakes, different reporting guidelines
and dates across the 50 states, and other idiosyncratic variations in how the disease progresses.

Figure 1: Cumulative Cases and New Cases in the U.S.

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We estimate that the peak in the number of new
infections was reached by mid-April, around April
12. After this date, the number of new infections
has been falling slowly but steadily. In terms of the
cumulative case numbers, this suggests that the U.S.
is already past the inflection point and that measures
to suppress the spread of the pandemic have been
working to some degree. However, since mid-April
the incoming data on new cases have become
increasingly volatile. This appears largely driven by
the fact that infections have spread beyond a few
clusters with very high case numbers, specifically
New York City, to a wider swath of states. At the same
time, the volatility does not seem to affect the median estimated path as it shows a general downward
trend from the estimated peak.1
Given our last data point on May 3, we project the
time path of new infections and cumulative cases
forward until the start of August. We show the median forecast in Figure 1. The uncertainty region prior
to May 3 captures the estimation uncertainty of the
fitted model, while the uncertainty region after May
3 includes uncertainty from disturbances in the data.
We note that uncertainty about new case numbers
widens immediately, which reflects both the uncertainty about the dynamics of the pandemic and
the uncertainty inherent in the data process. More

specifically, wide uncertainty bands and volatile data
suggest that one should consider the broader trend
rather than extrapolate too much from a few recent
data points. Our forecasted range of new infections
includes the estimated peak, which indicates that
the U.S. is not out of the woods yet and that it may, in
fact, have reached a plateau.
As the pandemic runs its course, the degree of
uncertainty declines, however, and the incidence of
new cases becomes more precisely estimated as the
infection rate moves toward zero. The cumulative
case numbers in Figure 1 are projected to grow over
the next several months, albeit at a declining rate.
By the start of August, we project 0.71 percent of the
U.S. population will be infected, with a range of 0.66
percent to 0.78 percent.
In Figure 2, we take a closer look at how the passage of time and the availability of more data have
affected our projections. We estimate our model for
data that were available, respectively, 14 and 28 days
ago, before the current estimation date of May 3. The
projections as of April 5 are shown in green, those as
of April 19 in red, and the current estimate is in blue.
We only show the respective 95 percent confidence
regions.

Figure 2: Cumulative Cases and New Cases in the U.S.

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Overall, the estimates for the last two samples are
contained in the uncertainty region of the April 5
sample. As more information became available,
estimates of the underlying pattern in the infection
data became more precise and the model developed
a better sense of where the peak of new infections,
thus the inflection point of the pandemic, were.
Consequently, the projections became more precise.
The same pattern can be seen for the April 19 and
the May 3 sample. The latter is somewhat smaller,
but it is also shifted upward for both cumulative and
new cases. That is, the data flow over these 14 days
led to improved precision in the forecast, but also
in a revision of the projected path of the epidemic.
We can tie this pattern to the fact that observed new
infections appear to have plateaued over the last few
days.

Mortality Forecasts for the U.S.
Figure 3 shows our estimates of the mortality model
described above and our projections for cumulative
deaths through the end of July. These projections
depend on our models for both the number of cases
and the mortality rate, allowing for estimation uncertainty and disturbances in both models. Our median
projection of total fatalities by the start of August is
159,000, with a range of 140,000 to 181,000. We also
estimate that the number of daily deaths peaked
around April 20 at 2,300, but there is considerably
more uncertainty when compared with the infection
model.

Figure 3: Cumulative Deaths and New Deaths in the U.S.

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The peak of the mortality data comes with a delay of
about one week after new infections have peaked.
Given what we know so far about the course of
COVID-19, this lag appears short since the time from
infection to death appears to be about four to five
weeks. However, we are measuring as new cases
those who have been tested, and this group is dominated by those who have already developed more
serious complications. Figure 3 also shows the increased volatility of the recent mortality data, which
affects the precision of the projections. Specifically,
we cannot rule out that the peak of daily deaths has
been reached since the uncertainty region for several days out includes values that are considerably
higher.

In Figure 4, we perform the same exercise as before
where we estimate the mortality model for samples
up to 14 and 28 days ago. Forecast uncertainty based
on the April 5 data is very wide. The forecast left
open the possibility that cumulative deaths would
reach fewer than 50,000 by the start of August. At
the time of the estimates, the sample was simply too
short to result in tight inference. Moving the sample
ahead to include data up to April 19 changes the
outlook notably. In terms of cumulative deaths, the
error bands are now contained within the April 5
region, while moving to the current sample tightens
uncertainty further. The graph with the uncertainty
region for new deaths suggests, however, that the reduction in uncertainty is coming from bounding the
forecast distribution from below. That is, the model
now puts more weight on a higher number of fatalities than could have been expected on April 5.

Figure 4: Cumulative Deaths and New Deaths in the U.S.

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Conclusion
Using a statistical model of the COVID-19 pandemic
that attempts to capture the underlying patterns
and evolution of infections and deaths, we project
that by the start of August there will be 2.3 million observed cases of COVID-19 infections, which
translates to 0.71 percent of the U.S. population. At
the same time, we forecast 159,000 fatalities for the
same time period. Neither new infections nor daily
deaths are likely to have returned to zero by then.
The uncertainty surrounding these estimates is still
considerable, with deaths ranging between 140,000
to 181,000. As more data become available, the estimates of the underlying pattern of the epidemic will
become more precise and the uncertainty surrounding these forecasts will decline.

This article may be photocopied or reprinted in its
entirety. Please credit the authors, source, and the
Federal Reserve Bank of Richmond and include the
italicized statement below.
Views expressed in this article are those of the authors
and not necessarily those of the Federal Reserve Bank
of Richmond or the Federal Reserve System.

Our forecasts are implicitly predicated on the assumption that the public health policies that have
been put in place will not change over the course of
the forecast horizon. In that sense, our forecasts provide an assessment of whether and to what extent
these policies are successful. However, it is unlikely
that they will continue, which will then affect the
time path of the pandemic. The value of these forecasts thereby lies in highlighting the range of possible outcomes in a no-change scenario, which can
serve as a benchmark to evaluate alternative public
health measures against.
Paul Ho is an economist and Thomas Lubik is a
senior advisor in the Research Department of the
Federal Reserve Bank of Richmond. Christian Matthes is an associate professor in the Department of
Economics at Indiana University.
Endnotes
1

W
 e can contrast this estimate with the one reported in our
Regional Matters post “Forecasting the COVID-19 Pandemic in
the Fifth District” based on data up to April 20. We estimated
the peak to be several days earlier and the decline in new
infections much steeper. Since then, the new data seemed to
cluster around a plateau that by itself would have pushed out
the peak estimate further. However, our initial model specification was not well-suited to handle a data pattern that included
such plateauing. We therefore modified the model slightly by
including an additional parameter designed to capture this
pattern, which improved fit.

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