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The Covid-19 pandemic in Italy
Franco Peracchi∗
Georgetown University, EIEF, and University of Rome Tor Vergata
This version: March 29, 2020
1 Introduction
In this note I use the data from the website of Dipartimento della Protezione Civile (DPC), which
at the moment represent the main source of quantitative information on the Covid-19 pandemic in
Italy and are described in some more detail in Section 2. These data are updated daily, so I will also
update my note daily and post it shortly after the DPC data have been released.
What I try to do here is just to describe and summarize some basic aspects of the pandemic in
Italy with no attempt at structural modeling. Neverheless, I hope that the stylized facts presented
in Section 3 will help in sorting out possible interpretations of the process that is still unfolding.
However, in Section 4, I will venture into a simple forecasting exercise based on the extrapolation of
the observed trends.
2 Data
The DPC website contains several daily time-series at the national, regional and provincial level.
I focus on the following five series: the number of currently positive cases (“totale attualmente
positivi”), the number of new positive cases (“nuovi attualmente positivi”), the number of discharged
(“dimessi guariti”), the number of deaths (“deceduti”), and total number of confirmed cases (“totale
casi”). Notice that the number of currently positive cases is not equal to the number of currently
infected members of the population, but only to the number of those who have been tested positive.
The number of currently infected people is likely to be an order of magnitude larger (see e.g. Li et al.
2020). Further notice that the ratio of the currently positive to the currently infected cannot be taken
as approximately constant, as the criteria and the intensity of the testing process vary over time and
across regions. The new positive cases is the daily variation in the number of currently positive cases,
while the total number of confirmed cases is the sum of the number of currently positive cases, deaths
∗
E-mail address: fp211@georgetown.edu. I thank Luca Anderlini, Giuseppe De Luca, Raffaela Giordano, Luigi
Guiso, Caterina Peracchi, Marta Peracchi, and Daniele Terlizzese for comments and helpful discussions.
1
and discharged. These data are available, starting with February 24, 2020, both at the national and
the regional level, and are updated daily around 6:30 pm CET. Daily data on the number of total
confirmed cases by province are also available from the DPC website but are not used in this note.
The DPC data are very noisy and currently offer no breakdown of the cases by age and gender (for
a discussion of the value of this information in the present context see Beam Dow et al. 2020). The
noise reflects a number of factors that have been stressed by various commentators, such as delays in
recording and transmitting the information, uncertainty in the classification of cases, clerical errors,
etc. This is unsurprising, as people these days are battling for saving human lives, often in dramatic
conditions (see e.g. Nacoti et al. 2020), not for perfecting statistical information. However, because
the noise is large and pervasive, especially at the regional and provincial level, smoothing the data
becomes essential in order to detect underlying trends that would otherwise be obscured. Of course,
different smoothing methods may sometimes lead to different conclusions.
While I rely a lot on smoothing, though simple moving averages and polynomial trends, I do not
pre-process the original data except for merging the autonomous provinces of Trento and Bolzano
into a single region, “Trentino-Alto Adige”, resulting in a geographical disaggregation of the country
into 20 regions.
I produce the descriptive statistics in Section 3 and the forecasts in Section 4 using Stata/MP
version 16.1. The Stata code is available upon request.
3 Descriptive statistics
Figures 1 and 2 summarize the Covid-19 pandemic at the Italian level, and various versions of them
have appeared in the media.
Figure 1 shows the daily changes in the number of positive cases (blue profile), deaths (red profile),
discharged (green profile) and in the total number of confirmed cases (purple profile). In addition to
the original series (represented by the thinner profile), I also present a 5-day centered moving average
(represented by the thicker profile) as a visual aid in highlighting the trends. The time profile of the
changes in the number of positive cases is approximately hump-shaped, suggesting that the pandemic
may be slowing down.
Figure 2 shows the percentage daily changes in the number of positive cases, dead, discharged,
and total confirmed cases. It reveals an almost steady downward trend after the very high growth
rates of the number of deaths and positive cases in late February and early March.
Figures 3 and 4 are similar to Figures 1 and 2 but look separately at nine Italian regions selected
either for their size or for the particular intensity of the pandemic: Campania, Emilia-Romagna, Lazio,
Lombardia, Marche, Piemonte, Puglia, Sicilia, and Veneto (the figures for the other eleven regions are
available upon request). These two figures are meant to give a sense of the large differences between
2
Italian regions.
Figure 3 presents the daily changes in the number of positive cases, deaths, discharged and total
confirmed cases by region. Notice the very different vertical scales across regions. The regional dif-
ferences in the number of new positive cases are huge, with the Northern regions of Emilia-Romagna,
Lombardia, Piemonte and Veneto being the hardest hit. However, the qualitative differences in the
time profiles are much smaller.
Figure 4 presents the percentage daily changes by region over the last 20 days. Although the
regions with a lower incidence of the pandemic display much larger fluctuations, the similarity of the
process across regions is quite striking, with the observed peaks in the rate of increase of mortality
following by about 10–15 days the peaks in the rate of increase of new positive cases.
4 Forecasts
Figures 5 and 6 present the observed number of new positive cases at the national level (thin blue
profile) together with in-sample predictions and out-of-sample forecasts of the number of new positive
cases (thick orange profile) up to 50 days from the current date (March 29, 2020) using a common
model but different fitting criteria. The observed number of new positive cases is exactly the same
shown in Figure 1. To predict or forecast the number of new positive cases, I employ a polynomial
time trend model for the natural logarithm of the number of new positive cases at time (day) t,
denoted in what follows by Yt. Bonetti (2020) uses a similar strategy for analyzing the data on the
fraction of currently positive at the provincial level in Italy. This model provides a simple way of
allowing for the fact that Yt is a nonnegative number but can assume very large values.
Specifically, my statistical model for Yt is
ln Yt = β0 + β1t + β2t2
+ · · · + βktk
+ Ut, t = 0, 1, . . . , (1)
where β0, β1, β2, . . . , βk are unknown parameters, Ut is an unobservable random error with zero mean
and finite variance, and t = 0 corresponds to February 24, 2020. The statistical model (1) is just
a “black box” that I use for forecasting the future number of new positive cases under the strong
assumption of time invariance of the process underlying the Covis-19 pandemic in Italy. Given
estimates β0|T , β1|T , β2|T , . . . , βk|T of the model parameters based on all the data available up to the
current date T, my predictions of Yt at any date up to the current date T and my forecasts of Yt at
any future date t = T + 1, T + 2, . . . , are of the form
Yt|T = exp(β0|T + β1|T t + β2|T t2
+ · · · + βk|T tk
). (2)
The inverse exponential transformation in the forecast rule (2) automatically enforces the non-
negativity constraint on the predictions. After experimenting with various specifications, I found
3
that either a quadratic or a cubic trend model (k = 2 or 3) provide a reasonable compromise between
goodness of fit and parsimony. My choice is also supported by the results of a more formal model
selection method based on minimization of the Schwarz Information Criterion, also know as BIC. My
two fitting criteria are ordinary least squares (OLS), in which case Yt|T is interpreted as an estimate
of the mean of the probability distribution of new positive, and least absolute deviations (LAD), in
which case Yt|T is interpreted as an estimate of the median of the probability distribution of new
positives. Although the two estimates should be about the same if the probability distribution of
new positives is approximately symmetric and has moderate tails, I employ both because the LAD
estimates are much less sensitive than the OLS estimates to outliers, which represent a big problem
in these data.
Notice that my out-of-sample forecasts are based on all the data available up to the current
date T. As soon as new DPC data are released, I produce a new set of estimated parameters
β0|T+1, β1|T+1, β2|T+1, . . . , βk|T+1 and a new set of forecasts YT+2|T+1, YT+3|T+1, etc. Over time, with
the availability of more and more data, and possibly also with the revision of previously released data,
the uncertainty associated with the forecasts is likely to be reduced, especially if the pandemic (or
the data collection effort by the DPC) continues without major structural breaks.
Given the forecast rule (2), I can predict the day when the pandemic will end. This is just the
day t0 such that Yt0|T ≈ 0. Since Yt|T can only be zero in the limit, as t → ∞, I predict the end date
of the pandemic as the first day when the forecasted number of new positive cases is rounded to zero.
Notice that the predicted date t0 inherits the large statistical uncertainty associated with the entire
sequence of forecasts YT+1|T , YT+2|T , . . .
Figures 5 and Figure 6 present in-sample predictions and out-of-sample forecasts based on a
quadratic time trend model (selected on the basis of the BIC) and, respectively, the OLS and the
LAD criterion. The vertical black line marks the current date (March 29, 2020), while the vertical
orange line marks the predicted end date of the pandemic in Italy. Symmetric pointwise standard-
error bands of varying width are included to provide an indication of the large statistical uncertainty
associated with these forecasts. Since the standard errors increase rapidly with t, the shaded regions
tend to “curl-up” fast. For simplicity, I employ classical standard errors based on the homoskedasticity
assumption (see, for example, Stock and Watson 2019). The LAD standard errors are larger than the
OLS standard errors, so the forecasts based on the LAD estimates, though more robust to outliers,
are subject to greater uncertainty.
Figure 7 presents in-sample predictions and out-of-sample forecasts of the new positive cases up to
40 days from the current date by region, based on a cubic trend model, i.e. k = 3 in (1). My forecast
rule for the number Yrt of new positives in region r, where r is any of the 20 Italian regions, has the
same form as (2), namely Yrt|T = exp(β0r|T + β1r|T t + β2r|T t2 + β3r|T t3), where β0r|T , β1r|T , β2r|T and
4
β3r|T are parameters specific to region r estimated from the data available up to the current date
T. The figure excludes three regions (Marche, Molise, Sardegna) for which my low-order polynomial
trend models perform poorly, because of either the relatively small number of new positives or their
concentration at the beginning or the end of the sample period.
Figure 8 compares the aggregate OLS and LAD in-sample predictions and out-of-sample forecasts
with those obtained by aggregation of the regional LAD predictions and forecasts.
Finally, Table 1 presents the dates when the Covid-19 pandemic is predicted to end, both for Italy
as a whole and for each region for which the model can be estimated. The aggregate model (1) gives
different dates depending on whether it is estimated by OLS (more sensitive to outliers) or LAD (less
sensitive to outliers). The predicted end date for the country based on the regional disaggregation is
the latest end date predicted at the regional level.
References
Bonetti M. (2020). “Epilocal: a real-time tool for local epidemic monitoring.” Preprint. Available
at https://arxiv.org/abs/2003.07928.
Beam Dowd J., et al. (2020). “Demographic science aids in understanding the spread and fatality
rates of COVID-19.” Preprint. Available at https://www.medrxiv.org/content/10.1101/2020.
03.15.20036293v1
Li R., et al. (2020). “Substantial undocumented infection facilitates the rapid dissemination of novel
coronavirus (SARS-CoV2).” Science 10.1126/science.abb3221. Available at https://science.
sciencemag.org/content/early/2020/03/24/science.abb3221.abstract.
Nacoti M., et al. (2020). “At the epicenter of the Covid-19 pandemic and humanitarian crises in
Italy: Changing perspectives on preparation and mitigation.” NEJM Catalyst: Innovations in
Care Delivery, March 21, 2020. Available at https://catalyst.nejm.org/doi/full/10.1056/CAT.
20.0080.
Stock, J.H., and Watson, M.W. (2019). Introduction to Econometrics (4th ed.). Pearson: New York.
5
Table 1: Dates when the Covid-19 pandemic is predicted to end.
Region Forecast rule Expected date
Abruzzo LAD April 11
Basilicata LAD April 7
Calabria LAD April 17
Campania LAD April 20
Emilia-Romagna LAD April 28
Friuli-Venezia Giulia LAD April 10
Lazio LAD April 16
Liguria LAD April 7
Lombardia LAD April 22
Piemonte LAD April 15
Puglia LAD April 9
Sicilia LAD April 14
Toscana LAD May 5
Trentino-Alto Adige LAD April 6
Umbria LAD April 7
Valle d’Aosta LAD April 8
Veneto LAD April 14
Italy OLS aggregate May 16
Italy LAD aggregate May 9
Italy LAD disaggregate May 5
6
Figure 1: Daily changes in the number of cases: original series (thinner profile) and 5-day centered
moving average (thicker profile).
0
2000
4000
6000
8000
Dailychange
feb26 mar04 mar11 mar18 mar25 apr01
Day
New Dead Discharged Total
Figure 2: Percentage daily changes in the number of cases: original series (thinner profile) and 5-day
centered moving average (thicker profile).
0
20
40
60
Percentagedailychange
feb26 mar04 mar11 mar18 mar25 apr01
Day
Positive Dead Total
7
Figure 3: Daily changes in the number of cases by region: original series (thinner profile) and 5-day
centered moving average (thicker profile).
0
100
200
0
500
1000
0
100
200
0
2000
4000
0
100
200
300
0
200
400
600
0
50
100
150
0
100
200
0
200
400
600
feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01
feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01
feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01
Campania Emilia-Romagna Lazio
Lombardia Marche Piemonte
Puglia Sicilia Veneto
Positive Dead Discharged Total
Dailychange
Day
Figure 4: Percentage daily changes in the number of cases by region (last 20 days): original series
(thinner profile) and 5-day centered moving average (thicker profile).
0
50
100
150
200
0
10
20
30
40
0
20
40
60
0
10
20
30
40
0
10
20
30
40
0
20
40
60
80
0
50
100
0
20
40
60
0
10
20
30
40
mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01
mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01
mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01
Campania Emilia-Romagna Lazio
Lombardia Marche Piemonte
Puglia Sicilia Veneto
Positive Dead Total
Percentagedailychange
Day
8
Figure 5: Observed and predicted or forecasted number of new positive cases. OLS predictions and
forecasts . The vertical orange line marks the predicted end date of the pandemic in Italy.
0
1000
2000
3000
4000
5000
Newpositivecases
feb19 mar11 apr01 apr22 may13
Day
Observed Predicted
Figure 6: Observed and predicted or forecasted number of new positive cases. LAD predictions and
forecasts. The vertical orange line marks the predicted end date of the pandemic in Italy.
0
1000
2000
3000
4000
5000
Newpositivecases
feb19 mar11 apr01 apr22 may13
Day
Observed Predicted
9
Figure 7: Observed and predicted or forecasted number of new positive cases by region. LAD pre-
dictions and forecasts.
0
50
100
150
0
10
20
30
0
50
100
0
50
100
150
200
0
200
400
600
800
0
50
100
0
50
100
150
200
0
50
100
150
200
0
500
1000
1500
2000
0
200
400
600
0
50
100
150
0
50
100
150
0
100
200
300
0
50
100
150
200
0
50
100
0
20
40
60
80
0
200
400
600
mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01
mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01
mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01
mar01 apr01 may01 mar01 apr01 may01
Abruzzo Basilicata Calabria Campania Emilia-Romagna
Friuli-Venezia Giulia Lazio Liguria Lombardia Piemonte
Puglia Sicilia Toscana Trentino-Alto Adige Umbria
Valle d'Aosta Veneto
Observed Predicted
Newpositivecases
Day
Figure 8: Observed and predicted or forecasted number of new positive cases. Aggregate OLS and
LAD (OLS and LAD aggr) and aggregation of regional LAD prediction or forecasts (LAD disaggr).
0
1000
2000
3000
4000
5000
Newpositivecases
feb19 mar11 apr01 apr22 may13
Day
Observed OLS aggr LAD aggr LAD disaggr
10

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Italy Covid-19 pandemic trends and forecasts

  • 1. The Covid-19 pandemic in Italy Franco Peracchi∗ Georgetown University, EIEF, and University of Rome Tor Vergata This version: March 29, 2020 1 Introduction In this note I use the data from the website of Dipartimento della Protezione Civile (DPC), which at the moment represent the main source of quantitative information on the Covid-19 pandemic in Italy and are described in some more detail in Section 2. These data are updated daily, so I will also update my note daily and post it shortly after the DPC data have been released. What I try to do here is just to describe and summarize some basic aspects of the pandemic in Italy with no attempt at structural modeling. Neverheless, I hope that the stylized facts presented in Section 3 will help in sorting out possible interpretations of the process that is still unfolding. However, in Section 4, I will venture into a simple forecasting exercise based on the extrapolation of the observed trends. 2 Data The DPC website contains several daily time-series at the national, regional and provincial level. I focus on the following five series: the number of currently positive cases (“totale attualmente positivi”), the number of new positive cases (“nuovi attualmente positivi”), the number of discharged (“dimessi guariti”), the number of deaths (“deceduti”), and total number of confirmed cases (“totale casi”). Notice that the number of currently positive cases is not equal to the number of currently infected members of the population, but only to the number of those who have been tested positive. The number of currently infected people is likely to be an order of magnitude larger (see e.g. Li et al. 2020). Further notice that the ratio of the currently positive to the currently infected cannot be taken as approximately constant, as the criteria and the intensity of the testing process vary over time and across regions. The new positive cases is the daily variation in the number of currently positive cases, while the total number of confirmed cases is the sum of the number of currently positive cases, deaths ∗ E-mail address: fp211@georgetown.edu. I thank Luca Anderlini, Giuseppe De Luca, Raffaela Giordano, Luigi Guiso, Caterina Peracchi, Marta Peracchi, and Daniele Terlizzese for comments and helpful discussions. 1
  • 2. and discharged. These data are available, starting with February 24, 2020, both at the national and the regional level, and are updated daily around 6:30 pm CET. Daily data on the number of total confirmed cases by province are also available from the DPC website but are not used in this note. The DPC data are very noisy and currently offer no breakdown of the cases by age and gender (for a discussion of the value of this information in the present context see Beam Dow et al. 2020). The noise reflects a number of factors that have been stressed by various commentators, such as delays in recording and transmitting the information, uncertainty in the classification of cases, clerical errors, etc. This is unsurprising, as people these days are battling for saving human lives, often in dramatic conditions (see e.g. Nacoti et al. 2020), not for perfecting statistical information. However, because the noise is large and pervasive, especially at the regional and provincial level, smoothing the data becomes essential in order to detect underlying trends that would otherwise be obscured. Of course, different smoothing methods may sometimes lead to different conclusions. While I rely a lot on smoothing, though simple moving averages and polynomial trends, I do not pre-process the original data except for merging the autonomous provinces of Trento and Bolzano into a single region, “Trentino-Alto Adige”, resulting in a geographical disaggregation of the country into 20 regions. I produce the descriptive statistics in Section 3 and the forecasts in Section 4 using Stata/MP version 16.1. The Stata code is available upon request. 3 Descriptive statistics Figures 1 and 2 summarize the Covid-19 pandemic at the Italian level, and various versions of them have appeared in the media. Figure 1 shows the daily changes in the number of positive cases (blue profile), deaths (red profile), discharged (green profile) and in the total number of confirmed cases (purple profile). In addition to the original series (represented by the thinner profile), I also present a 5-day centered moving average (represented by the thicker profile) as a visual aid in highlighting the trends. The time profile of the changes in the number of positive cases is approximately hump-shaped, suggesting that the pandemic may be slowing down. Figure 2 shows the percentage daily changes in the number of positive cases, dead, discharged, and total confirmed cases. It reveals an almost steady downward trend after the very high growth rates of the number of deaths and positive cases in late February and early March. Figures 3 and 4 are similar to Figures 1 and 2 but look separately at nine Italian regions selected either for their size or for the particular intensity of the pandemic: Campania, Emilia-Romagna, Lazio, Lombardia, Marche, Piemonte, Puglia, Sicilia, and Veneto (the figures for the other eleven regions are available upon request). These two figures are meant to give a sense of the large differences between 2
  • 3. Italian regions. Figure 3 presents the daily changes in the number of positive cases, deaths, discharged and total confirmed cases by region. Notice the very different vertical scales across regions. The regional dif- ferences in the number of new positive cases are huge, with the Northern regions of Emilia-Romagna, Lombardia, Piemonte and Veneto being the hardest hit. However, the qualitative differences in the time profiles are much smaller. Figure 4 presents the percentage daily changes by region over the last 20 days. Although the regions with a lower incidence of the pandemic display much larger fluctuations, the similarity of the process across regions is quite striking, with the observed peaks in the rate of increase of mortality following by about 10–15 days the peaks in the rate of increase of new positive cases. 4 Forecasts Figures 5 and 6 present the observed number of new positive cases at the national level (thin blue profile) together with in-sample predictions and out-of-sample forecasts of the number of new positive cases (thick orange profile) up to 50 days from the current date (March 29, 2020) using a common model but different fitting criteria. The observed number of new positive cases is exactly the same shown in Figure 1. To predict or forecast the number of new positive cases, I employ a polynomial time trend model for the natural logarithm of the number of new positive cases at time (day) t, denoted in what follows by Yt. Bonetti (2020) uses a similar strategy for analyzing the data on the fraction of currently positive at the provincial level in Italy. This model provides a simple way of allowing for the fact that Yt is a nonnegative number but can assume very large values. Specifically, my statistical model for Yt is ln Yt = β0 + β1t + β2t2 + · · · + βktk + Ut, t = 0, 1, . . . , (1) where β0, β1, β2, . . . , βk are unknown parameters, Ut is an unobservable random error with zero mean and finite variance, and t = 0 corresponds to February 24, 2020. The statistical model (1) is just a “black box” that I use for forecasting the future number of new positive cases under the strong assumption of time invariance of the process underlying the Covis-19 pandemic in Italy. Given estimates β0|T , β1|T , β2|T , . . . , βk|T of the model parameters based on all the data available up to the current date T, my predictions of Yt at any date up to the current date T and my forecasts of Yt at any future date t = T + 1, T + 2, . . . , are of the form Yt|T = exp(β0|T + β1|T t + β2|T t2 + · · · + βk|T tk ). (2) The inverse exponential transformation in the forecast rule (2) automatically enforces the non- negativity constraint on the predictions. After experimenting with various specifications, I found 3
  • 4. that either a quadratic or a cubic trend model (k = 2 or 3) provide a reasonable compromise between goodness of fit and parsimony. My choice is also supported by the results of a more formal model selection method based on minimization of the Schwarz Information Criterion, also know as BIC. My two fitting criteria are ordinary least squares (OLS), in which case Yt|T is interpreted as an estimate of the mean of the probability distribution of new positive, and least absolute deviations (LAD), in which case Yt|T is interpreted as an estimate of the median of the probability distribution of new positives. Although the two estimates should be about the same if the probability distribution of new positives is approximately symmetric and has moderate tails, I employ both because the LAD estimates are much less sensitive than the OLS estimates to outliers, which represent a big problem in these data. Notice that my out-of-sample forecasts are based on all the data available up to the current date T. As soon as new DPC data are released, I produce a new set of estimated parameters β0|T+1, β1|T+1, β2|T+1, . . . , βk|T+1 and a new set of forecasts YT+2|T+1, YT+3|T+1, etc. Over time, with the availability of more and more data, and possibly also with the revision of previously released data, the uncertainty associated with the forecasts is likely to be reduced, especially if the pandemic (or the data collection effort by the DPC) continues without major structural breaks. Given the forecast rule (2), I can predict the day when the pandemic will end. This is just the day t0 such that Yt0|T ≈ 0. Since Yt|T can only be zero in the limit, as t → ∞, I predict the end date of the pandemic as the first day when the forecasted number of new positive cases is rounded to zero. Notice that the predicted date t0 inherits the large statistical uncertainty associated with the entire sequence of forecasts YT+1|T , YT+2|T , . . . Figures 5 and Figure 6 present in-sample predictions and out-of-sample forecasts based on a quadratic time trend model (selected on the basis of the BIC) and, respectively, the OLS and the LAD criterion. The vertical black line marks the current date (March 29, 2020), while the vertical orange line marks the predicted end date of the pandemic in Italy. Symmetric pointwise standard- error bands of varying width are included to provide an indication of the large statistical uncertainty associated with these forecasts. Since the standard errors increase rapidly with t, the shaded regions tend to “curl-up” fast. For simplicity, I employ classical standard errors based on the homoskedasticity assumption (see, for example, Stock and Watson 2019). The LAD standard errors are larger than the OLS standard errors, so the forecasts based on the LAD estimates, though more robust to outliers, are subject to greater uncertainty. Figure 7 presents in-sample predictions and out-of-sample forecasts of the new positive cases up to 40 days from the current date by region, based on a cubic trend model, i.e. k = 3 in (1). My forecast rule for the number Yrt of new positives in region r, where r is any of the 20 Italian regions, has the same form as (2), namely Yrt|T = exp(β0r|T + β1r|T t + β2r|T t2 + β3r|T t3), where β0r|T , β1r|T , β2r|T and 4
  • 5. β3r|T are parameters specific to region r estimated from the data available up to the current date T. The figure excludes three regions (Marche, Molise, Sardegna) for which my low-order polynomial trend models perform poorly, because of either the relatively small number of new positives or their concentration at the beginning or the end of the sample period. Figure 8 compares the aggregate OLS and LAD in-sample predictions and out-of-sample forecasts with those obtained by aggregation of the regional LAD predictions and forecasts. Finally, Table 1 presents the dates when the Covid-19 pandemic is predicted to end, both for Italy as a whole and for each region for which the model can be estimated. The aggregate model (1) gives different dates depending on whether it is estimated by OLS (more sensitive to outliers) or LAD (less sensitive to outliers). The predicted end date for the country based on the regional disaggregation is the latest end date predicted at the regional level. References Bonetti M. (2020). “Epilocal: a real-time tool for local epidemic monitoring.” Preprint. Available at https://arxiv.org/abs/2003.07928. Beam Dowd J., et al. (2020). “Demographic science aids in understanding the spread and fatality rates of COVID-19.” Preprint. Available at https://www.medrxiv.org/content/10.1101/2020. 03.15.20036293v1 Li R., et al. (2020). “Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2).” Science 10.1126/science.abb3221. Available at https://science. sciencemag.org/content/early/2020/03/24/science.abb3221.abstract. Nacoti M., et al. (2020). “At the epicenter of the Covid-19 pandemic and humanitarian crises in Italy: Changing perspectives on preparation and mitigation.” NEJM Catalyst: Innovations in Care Delivery, March 21, 2020. Available at https://catalyst.nejm.org/doi/full/10.1056/CAT. 20.0080. Stock, J.H., and Watson, M.W. (2019). Introduction to Econometrics (4th ed.). Pearson: New York. 5
  • 6. Table 1: Dates when the Covid-19 pandemic is predicted to end. Region Forecast rule Expected date Abruzzo LAD April 11 Basilicata LAD April 7 Calabria LAD April 17 Campania LAD April 20 Emilia-Romagna LAD April 28 Friuli-Venezia Giulia LAD April 10 Lazio LAD April 16 Liguria LAD April 7 Lombardia LAD April 22 Piemonte LAD April 15 Puglia LAD April 9 Sicilia LAD April 14 Toscana LAD May 5 Trentino-Alto Adige LAD April 6 Umbria LAD April 7 Valle d’Aosta LAD April 8 Veneto LAD April 14 Italy OLS aggregate May 16 Italy LAD aggregate May 9 Italy LAD disaggregate May 5 6
  • 7. Figure 1: Daily changes in the number of cases: original series (thinner profile) and 5-day centered moving average (thicker profile). 0 2000 4000 6000 8000 Dailychange feb26 mar04 mar11 mar18 mar25 apr01 Day New Dead Discharged Total Figure 2: Percentage daily changes in the number of cases: original series (thinner profile) and 5-day centered moving average (thicker profile). 0 20 40 60 Percentagedailychange feb26 mar04 mar11 mar18 mar25 apr01 Day Positive Dead Total 7
  • 8. Figure 3: Daily changes in the number of cases by region: original series (thinner profile) and 5-day centered moving average (thicker profile). 0 100 200 0 500 1000 0 100 200 0 2000 4000 0 100 200 300 0 200 400 600 0 50 100 150 0 100 200 0 200 400 600 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 feb19 mar04 mar18 apr01 Campania Emilia-Romagna Lazio Lombardia Marche Piemonte Puglia Sicilia Veneto Positive Dead Discharged Total Dailychange Day Figure 4: Percentage daily changes in the number of cases by region (last 20 days): original series (thinner profile) and 5-day centered moving average (thicker profile). 0 50 100 150 200 0 10 20 30 40 0 20 40 60 0 10 20 30 40 0 10 20 30 40 0 20 40 60 80 0 50 100 0 20 40 60 0 10 20 30 40 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 mar11 mar18 mar25 apr01 Campania Emilia-Romagna Lazio Lombardia Marche Piemonte Puglia Sicilia Veneto Positive Dead Total Percentagedailychange Day 8
  • 9. Figure 5: Observed and predicted or forecasted number of new positive cases. OLS predictions and forecasts . The vertical orange line marks the predicted end date of the pandemic in Italy. 0 1000 2000 3000 4000 5000 Newpositivecases feb19 mar11 apr01 apr22 may13 Day Observed Predicted Figure 6: Observed and predicted or forecasted number of new positive cases. LAD predictions and forecasts. The vertical orange line marks the predicted end date of the pandemic in Italy. 0 1000 2000 3000 4000 5000 Newpositivecases feb19 mar11 apr01 apr22 may13 Day Observed Predicted 9
  • 10. Figure 7: Observed and predicted or forecasted number of new positive cases by region. LAD pre- dictions and forecasts. 0 50 100 150 0 10 20 30 0 50 100 0 50 100 150 200 0 200 400 600 800 0 50 100 0 50 100 150 200 0 50 100 150 200 0 500 1000 1500 2000 0 200 400 600 0 50 100 150 0 50 100 150 0 100 200 300 0 50 100 150 200 0 50 100 0 20 40 60 80 0 200 400 600 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 mar01 apr01 may01 Abruzzo Basilicata Calabria Campania Emilia-Romagna Friuli-Venezia Giulia Lazio Liguria Lombardia Piemonte Puglia Sicilia Toscana Trentino-Alto Adige Umbria Valle d'Aosta Veneto Observed Predicted Newpositivecases Day Figure 8: Observed and predicted or forecasted number of new positive cases. Aggregate OLS and LAD (OLS and LAD aggr) and aggregation of regional LAD prediction or forecasts (LAD disaggr). 0 1000 2000 3000 4000 5000 Newpositivecases feb19 mar11 apr01 apr22 may13 Day Observed OLS aggr LAD aggr LAD disaggr 10