The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
38,675 characters
Modeling Macroeconomic Variations after COVID-19
\bibliographystyle{harvard}
\thispagestyle{empty}
\setcounter{page}{0}
\let\Large\large
\title{{\large{\bf Modeling Macroeconomic Variations after COVID-19}}}
\author{Serena Ng \bigskip \thanks{Department of Economics, Columbia University and NBER. 420 W. 118 St., New York, NY 10025, email: serena.ng at columbia.edu \newline The author thanks Sai Ma for many helpful discussions. Financial Support from the National Science Foundation (SES 2018368) is gratefully acknowledged. } \\ Columbia University and NBER }
\date{\today}
\maketitle
\begin{abstract}
The coronavirus is a global event of historical proportions and just a few months changed the time series properties of the data in ways that make many pre-\textsc{covid}\ forecasting models inadequate. It also creates a new problem for estimation of economic factors and dynamic causal effects because the variations around the outbreak can be interpreted as outliers, as shifts to the distribution of existing shocks, or as addition of new shocks. I take the latter view and use \textsc{covid}\ indicators as controls to 'de-covid' the data prior to estimation. I find that economic uncertainty remains high at the end of 2020 even though real economic activity has recovered and \textsc{covid}\ uncertainty has receded. Dynamic responses of variables to shocks in a VAR similar in magnitude and shape to the ones identified before 2020 can be recovered by directly or indirectly modeling \textsc{covid}\ and treating it as exogenous. These responses to economic shocks are distinctly different from those to a \textsc{covid}\ shock which are much larger but shorter lived. Disentangling the two types of shocks can be important in macroeconomic modeling post-\textsc{covid}.
\end{abstract}
\noindent Keywords: COVID-19, Factor Analysis, VAR, Outliers.
\noindent JOE Classification: E0, E6
\thispagestyle{empty}
\setcounter{page}{0}
\baselineskip=18.0pt
\newpage
\section{Introduction}
The 2019 coronavirus (hereafter \textsc{covid}) is a once in a century global pandemic and remains very much active as of the first draft of this writing, one year after it surfaced in early 2020. It is a health disaster that has resulted in a loss of over half a million lives. It has also disrupted economic activities to such an extent that the three trillion dollar relief package passed by the Congress was not enough to offset the economic disruptions \textsc{covid}\ has caused. \citet{lmn3-PP:21} suggests that \textsc{covid}\ constitutes a 192$\sigma$ costly disaster shock.
In addition to social and economic disruptions, \textsc{covid}\ has also created challenges for the modeling of economic time series. If we plot a randomly chosen series used in business cycle analysis, we will likely see a spike around March 2020 so large as to drawf five decades of observations preceding it. Without any adjustment, the post-covid\ observations will dominate to yield uninterpretable estimates, messing up the pre-covid fit. A case in point is factor estimation which is used in a variety of economic analysis. Without any adjustment, the first factor known to load heavily on real activity variables would be 19 standard deviations away from the mean of zero in March 2020 when by way of comparison, the financial crisis in 2008 registered four standard deviations.
There is no simple solution as there were only two pandemics in the post World War II era, and no lockdown was enforced in 1957-58 or 1968-69. As the historical data provide little guide to help understand the economic implications of global health shocks, how to econometrically handle pandemics is very much an open question. We may treat these irregular data points as temporary, but standard outlier adjustments would still leave the real activity factor 13 standard deviations below mean. It could be argued that the extreme values are not void of economic content and should not be 'dummied out'. Though treating the extreme observations as resulting from shifts to the underlying distributions may seem appealing, there are not enough post-\textsc{covid}\ data to model the instabilities adequately. Introducing restrictions and information may help. \citet{fms:20} corrects post \textsc{covid}\ forecasts using information from the 2008 financial crisis. \citet{primiceri-tambalotti:20} assumes that \textsc{covid}\ is a one-period shock that propagates differently from a typical macroeconomic shock, but whose trajectory can be approximated by a polynomial. Dynamic responses are then obtained by calibrating the polynomial to represent, for example, the scenario that the pandemic will dissipate by the end of 2020. Others incorporate information via priors. \citet{lenza-primceri:20} specifies a pareto distributed prior to the variance of the shocks while \citet{hkops:20} estimates an additive regression tree and uses flexible priors to deal with the extreme values during the pandemic. Many of these studies were prepared at the early stage of the pandemic and it is unclear whether the conclusions would hold up in an extended sample that include the subsequent waves.
The approach considered in this note also uses additional information, but I take as starting point that \textsc{covid}\ is not an economic shock, but rather a large and persistent health event with pervasive economic consequences. Under this view, the variations in the post-\textsc{covid}\ economic data are large not because of changes in distribution of variables already in the economic model, but because the economic data are no longer driven by economic shocks alone. The presence of a new, non-economic shock has implications for factor estimation as the principal components are no longer linear combinations of economic variations alone, for diffusion index forecasting as new predictors might be relevant, and for estimation of the dynamic causal effects of economic shocks as \textsc{covid}\ now becomes a confounder.
To address these issues, I use \textsc{covid}\ indicators either as controls in regressions to 'de-covid' the data so that economic factors and shocks can be identified, or as additional predictors to account for the persistent nature of \textsc{covid}. In a nutshell, I use the \textsc{covid}\ indicators to either remove or incorporate additional information relevant for the task. The \textsc{covid}\ indicators also allow the trajectory of \textsc{covid}\ to be determined by the data. Three measures of \textsc{covid}\ indicators are considered: hospitalization ($\mathcal H$), positive cases ($\mathcal P$), and deaths $(\mathcal D$). They enter the de-covid regressions in four ways reflecting different assumptions about March and April of 2020. I study their implications in the context of updating the JLN measure of economic uncertainty developed in \citet{jurado/ludvigson/ng:13} since the exercise has a factor estimation step and a forecasting step. I also explore the impact of \textsc{covid}\ for VAR modeling. In my set up, the issue \textsc{covid}\ creates is that a $n$ variable VAR now has \textsc{covid}\ as the $n+1$-th shock.
The main findings can be summarized as follows. All four models find economic uncertainty in March/April of 2020 to be at a historical high but there is no corresponding decline in the level of real activity after controlling for \textsc{covid}. A decomposition finds that while \textsc{covid}\ uncertainty has subsided and real activity rebounded by the end of the year, economic uncertainty even after controlling for \textsc{covid}\ remains high. But while the uncertainty estimates are qualitative similar across methods, the impulse response functions are strongly affected by the de-\textsc{covid}\ method used. Dynamic responses to economic shocks similar to the ones identified pre-\textsc{covid}\ can be obtained if we directly or indirectly remove the \textsc{covid}\ variations prior to VAR estimation, essentially assuming that \textsc{covid}\ is exogenous. These responses to economic shocks are distinctly different from those to a \textsc{covid}\ shock, reinforcing the need to distinguish the two types of shocks in post-\textsc{covid}\ estimation. The data support the exogeneity assumption.
\section{ Estimation of Common Factors }
The JLN concept of uncertainty is based on the premise that uncertainty arises because of lack of predictability with respect to information available, and macroeconomic uncertainty occurs only when the lack of economic predictability is board based. Three ingredients are needed to make the uncertainty measure operational: factor estimation, forecasting, and volatility estimation.\footnote{In practice, $r_m=8$ economic factors $F_m$ are estimated from a large panel of $N_m$ macro economic variables $X_M$, and $r_f=4$ financial factors $F_f$ from $N_f$ financial time series $X_F$. In implementation, data for $X_M$ are taken from \textsc{fred-md} and transformed as documented in \citet{fred-md}, and $X_F$ are based on those used in \citet{ludvigson-ng-jfe}. The data are demeaned and standardized prior to factor estimation.} \citet{lmn4} makes mean and standard deviation adjustments prior to factor estimation and add \textsc{covid}\ variables as predictors. The procedure worked well for the August release of FRED-MD but the adjustments treat too much of the subsequent variations as predictable, making uncertainty counter-intuitively low. \citet{moran-etal:20} uses the same approach to update a Q2 measure of Canadian uncertainty. Quarterly data are less affected by spikes created by \textsc{covid}\ because the month-to-month variations tend to average out, so the adjustments may perhaps not be necessary. My focus in what follows is the modeling of monthly data.
Generically let $X$ be a panel of data with $N$ columns. Let $T_o=720$ be the size of the pre-\textsc{covid}\ sample running from 1960:3-2020:2. The $T_0\times N$ matrix of pre-covid data $X$ are assumed to have a factor structure
\[ X_{it}-\mu_i=\Lambda_i^\prime F_t+e^X_{it}\]
where $F_t=(F_{1t},\ldots F_{rt})^\prime $ is a vector of $r$ common economic factors and $e^X_{it}$ is an idiosyncratic error associated with variable $i$.
Under conditions in \citet{bai/ng:02} for example, the principal components of $X$ will consistently estimate $F$ and $\Lambda$ up to a rotation matrix. In practice, the data are transformed by taking log and first or second difference, and adjusted for outliers data prior to estimation. The latter amounts to treating as missing those observations whose deviations from median are ten times larger than the difference between values at the top and bottom 25 percentiles, and imputing the missing values using the EM algorithm. Prior to \textsc{covid}\, this procedure affects only a few observations during the financial crisis of 2008.
Let $T_1=730$ be the size of the full sample spanning 1960:3-2020:12, so observations $T_0+1$ to $T_1$ are post \textsc{covid}. Instead of modeling \textsc{covid}\ effects through shifts to $F_t$ or $e_{it}^X$, I allow for a virus factor $ V_t$ so that the factor representation of the extended sample is
\[ X_{it}-\mu_i=\Lambda_i^\prime F_t+ \Gamma_i V_t+e_{it}^X\]
\paragraph{Assumption A}
\begin{itemize}
\item[i.] $V_t=0$ when $t<=T_0$, and for $t> T_0+1$, $V_t$ is a persistent process with innovations $v_t\sim (0,\sigma^2_v)$.
\item[ii.] Let $ F_{t}=\Phi^F(L)u^F_{t}$ where $u^F_{kt}\sim (0,\sigma^2_{F_k})$ and $ e^X_{it}=\Phi^X_i(L)u^X_{it}$ where $u^X_{it}\sim (0,\sigma^2_{X_i})$. The shocks ($u^F_{t},u^X_t$, $v_t$) are serially and mutually uncorrelated.
\end{itemize}
Assumption (i) states that $V_t$ is non-zero only after March 2020 and allows it to be persistent.\footnote{We can also model the non-zero values for the 1968-69 pandemic, but its health and especially the economic impact were small relative to \textsc{covid}. See \citet{doshi-08} and article in 'Solving the Mystery of the 1957 and 1968 Flu Pandemics in Bloomberg Opinion, March 11, 2021.} Assumption A.ii assumes that the shocks ($u^F_t, u^X_{t}, v_t)$ are serially and mutually uncorrelated.
Now $ V_t$ and $F_t$ are both common factors in the sense that they affect a sufficiently large number of series indexed by $i$. Unlike the pre-\textsc{covid}\ data, the principal components of $X$ over the full sample will no longer be spanned by the economic factors $F_t$ alone. Furthermore, as noted earlier, the full sample location and scale of the data will be dominated by the last few observations, creating spurious revisions to estimates obtained before \textsc{covid}.\footnote{The financial factors are much less impacted by the \textsc{covid}\ observations. The first financial factor based on unadjusted data is -4.71, comparable to -4.73 in October 2008 and -6.15 in October 1987.}
I consider estimation of economic factors $F_t$ from 'de-\textsc{covid}' data
\[ x_{it}=\begin{cases} X_{i,t}-\mu_{it}^0 &\quad t\le T_0\\
X_{it}-\mu^1_{it} &\quad t>T_0
\end{cases}
\]
for suitably defined $\mu^0_{it}$ and $\mu^1_{it}$. For the pre-\textsc{covid}\ sample, one can let $\mu^0_{it}=\mu_i^0$ for all $t\le T_0$, and a consistent estimate of $\mu_i^0$ is the mean of series $i$ over the sample up to and including $T_0$. Estimation of $\mu_{it}^1$ is more delicate as \textsc{covid}\ is persistent and so its trajectory also needs to be specified. Even if $V_t$ was observed, there are fewer than a year's worth of monthly data to work with. More problematic is that $F$ and $V$ are both latent, both pervasive, and both persistent. Thus recovering both from the post-\textsc{covid}\ data would require additional information. I make use of \textsc{covid}\ indicators.\footnote{Data are downloaded from \url{https://covidtracking.com/data/download/national-history.csv}. I use the February 21, 2021 vintage. The last release of data was March 7, 2021.}
\begin{table}[ht]
\caption{Proxies for $ V_t$ and $v_t$}
\label{tbl:table2}
\begin{center}
\begin{tabular}{l|rrr|rrrrr}
& \multicolumn{3}{c}{$ V$} & \multicolumn{3}{c}{$v_t$} \\ \hline
& $\mathcal H$ & $\mathcal P$ & $\mathcal D$ & $\mathcal H$ & $\mathcal P$ & $\mathcal D$\\ \hline
2020-1 & 0 & 2 & 0 & -& - & -\\
2020-2 & 0 & 16 & 5 & -& - & -\\
2020-3 & 6700 & 196830 & 4326 & 8.809& 9.4175 & 6.762\\
2020-4 & 38399 & 876304 & 55315 & 1.745& 1.493 &2.548 \\
2020-5 & 73150 & 718191 & 41137 & 0.644& -0.199 & -0.296\\
2020-6 & 31513 & 831681 & 19475 & -0.842& 0.147 & -0.747\\
2020-7 & 63105 & 1900163 & 25249 & 0.694& 0.826 & 0.259\\
2020-8 & 61144 & 1457252 & 30244 & -0.015& -0.265 &0.180\\
2020-9 & 37446 & 1192663 & 23329 & -0.490& -0.200 & -0.259\\
2020-10 & 53485 & 1892016 & 23545 & 0.356& 0.461 & 0.009\\
2020-11 & 92675 & 4475990 & 37065 & 0.549& 0.861 & 0.453 \\
2020-12 & 126244 & 6323266 & 77112 & 0.309& 0.346 & 0.732\\
2021-01 & 120837 & 6112572 & 95387 & -0.043 & -0.033 &0.212\\
2021-02 & 61054 & 2374243 & 71058& -0.682 & -0.945 & -0.294\\
\hline
\end{tabular}
\end{center}
\hspace*{1.0in}Note: $\mathcal H$ is HospitalizedIncrease, $\mathcal P$ is PostiveIncrease and $\mathcal D$ is DeathIncrease.
\newline \hspace*{1.0in} Daily data are aggregated to monthly.
\newline \hspace*{1.0in} Source: \url{covidtracking.com/data/download/national-history.csv}.
\end{table}
Table \ref{tbl:table2} shows the daily data for $\mathcal H=$ 'hospitalizedIncrease, $\mathcal P$='positiveIncrease', and $\mathcal D$='deathIncrease' aggregated to monthly. The cumulative sum of $\mathcal P$ agrees with 11 million cases documented for the United States in February 2021, while the cumulative sum of $\mathcal D$ is 336802 in December 2020. Of the three proxies, $\mathcal D$ tends to lag $\mathcal H$ and $\mathcal P$, while $\mathcal P$ may overstate the situation because one could be tested positive and yet asymptomatic. From these \textsc{covid}\ indicators, I construct three versions of $v_t$:
\[ v_t=\log\bigg(\frac{V_t}{ V _{t-1}}\bigg).\]
Notably, the $V$ series trend up throughout 2020 but the $v$ series are less persistent.\footnote{The data for $\mathcal H$ vary across source but the $v$ values are quite similar. Since there were zero hospitalizations in February but the March value for $v_t$ is crucial, the calculation assumes $\mathcal H=1$ in February.} While the data show large increases in $V$ in the summer and the fall, it is the extraordinary jumps at the outbreak of the pandemic that is problematic for estimation.
Given my presumption that \textsc{covid}\ and economic shocks co-exist, the first task is to isolate the (predictable and unpredictable) \textsc{covid}\ variations. It is natural to identify $ v_{T_0+1}$ by assuming ($u^F_{t},u^X_t$)=(0,0) at $t=T_0+1$.\footnote{See, for example, \citet{primiceri-tambalotti:20} and \citet{chudik-etal:20}.} But \textsc{covid}\ was spreading rapidly in April at the same time when the lockdown was in place, so different interpretations to April are possible. I consider four specifications of the following:
\[X_{it}=d_o + \gamma_i D_t + \beta_{i0} v_t + \beta_{i1} v_{t-1}+ \beta_{iq} v_{t-q}+ x_{it}.\]
\begin{center}
\begin{tabular}{l|l|l|l|ll}
$t$ & Model1* & Model 2 & Model 3& Model 4 \\ \hline
$D_{t=T_0+1}$ & 1 & 1 & 1 & -\\
$v_{t=T_0+1}$ & 0 & $v_{T_0+1}$ & $v_{T_0+1}$ & $v_{T_0+1}$ \\
$\beta_{i0}$ & $0$ & $0$ & $\ne 0$ & $\ne 0$\\
\hline
\end{tabular}
(*) data adjusted for outlier
\end{center}
In all models, $\beta_{i1},\ldots,\beta_{iq}$ are unrestricted and differ in the treatment for March and April of 2020. Model 1 replaces the outliers by the pre-\textsc{covid}\ means. Since the interquartile range is now computed on the full sample, the outliers are concentrated in the two months in question.
Models 1, 2, and 3 pick up the jump at the outbreak using a one time dummy $D_t$, essentially modeling March as a pure \textsc{covid}\ shock. By setting both $v_{T_0+1}$ and $\beta_{i0}$ to zero, Model 1 allocates all variations in April to economic sources and is expected to generate large (economic) residuals for that month. Models 2, 3, and 4 control the extreme values using \textsc{covid}\ data. Model 2 allows $v_t$ to enter only with a lag while Model 3 allows for contemporaneous effects. Model 4 allows for economic shocks in March by simply letting current and past effects of $v_t$ be removed from $X_{it}$ as determined by the regression. To be clear of what the models imply, I list the first few post-pandemic values of the regressors (from March to July of 2020) for the $\mathcal P$ version below.
\begin{center}
\begin{tabular}{ccc}
\text{Model 1} & \text{Model 2} \\
$
\begin{pmatrix*}[r]
1.000& 1.000 & 0.000 & 0.000 \\
1.000& 0.000 & 0.000 & 0.000 \\
1.000& 0.000 & 1.493 & 0.000 \\
1.000& 0.000 & -0.199 & 1.493 \\
1.000& 0.000 & 0.147 & -0.199 \\
\end{pmatrix*} $
&
$\begin{pmatrix*}[r]
1.000& 1.000 & 0.000 & 0.000 \\
1.000& 0.000 & 9.418 & 0.000 \\
1.000& 0.000 & 1.493 & 9.418 \\
1.000& 0.000 & -0.199 & 1.493 \\
1.000& 0.000 & 0.147 & -0.199 \\
\end{pmatrix*}
$
\\
\text{Model 3} & \text{Model 4} \\
$
\begin{pmatrix*}[r]
1.000& 1.000 & 9.418 & 0.000 & 0.000 \\
1.000& 0.000 & 1.493 & 9.418 & 9.418 \\
1.000& 0.000 & -0.199 & 1.493 & 0.000 \\
1.000& 0.000 & 0.147 & -0.199 & 1.493 \\
1.000& 0.000 & 0.826 & 0.147 & -0.199 \\
\end{pmatrix*} $ &
$ \begin{pmatrix*}[r]
1.000& 9.418 & 0.000 & 0.000 \\
1.000& 1.493 & 9.418 & 0.000 \\
1.000& -0.199 & 1.493 & 9.418 \\
1.000& 0.147 & -0.199 & 1.493 \\
1.000& 0.826 & 0.147 & -0.199 \\
\end{pmatrix*}
$
\end{tabular}
\end{center}
\bigskip
Estimation of the model on post-\textsc{covid}\ data gives the fit, which is $\mu^1_{it}$.\footnote{An alternative is to estimate the model on the full sample with an additional dummy that equals one after $T_0$.} The mean adjustments for March 2020 are quite similar across methods. For example, based on Model 4, $\hat \mu_0$ \textsc{napmnoi} (new orders) changes from 55.440 in February to $\hat \mu_1=$ 42.269 in March, \textsc{napm} from 52.983 to 49.122, \textsc{cumfns} (capacity utilization) from -0.013 to -3.617, \textsc{umscent} (consumder sentiment) from -0.10 to -12.68, \textsc{unrate} (unemployment rate) from 0.01 to 0.969, and
\textsc{claims} (unemployment claims) from 0.0 to 2.515, while housing variables such as \textsc{permit} as well as \textsc{awhman} (average man hours) are much less affected. These results are representative of all four models. However, the April estimates for Model 1 are quite different from those for Models 2, 3, and 4. Model 1 gives (53.941, 0.005, -0.102) for \textsc{napm}, \textsc{payems}. and \textsc{claims}, while Model 4 yields much larger changes of (42.081, -0.146, and 0.651) respectively, with the implication that the adjusted data $x$ for Model 1 will have more extreme values than the other three models.
Figure \ref{fig:fig0} plots the 2020 adjustments for eight selected series. The impact and subsequent adjustments vary significantly across series and over time. Compared to the Model 1 adjustments in the top panel, the Model 4 adjustments in the bottom panel tend to be more concentrated around April 2020. The differences are most notable for \textsc{rpi}, \textsc{payems}, \textsc{busloans} and \textsc{claims}.\footnote{The Model 1 adjustments do not include the outlier adjustments.}. According to Model 4, many series are back to the Janurary/February levels shortly after April.
Once $x_{it}=X_{it}-\hat\mu^1_{it}$ is available, I proceed to estimate the factors. Though $x_{it}$ is mean zero in the two respective subsamples, they may not be mean zero when pooled. The stacked data are demeaned and standardized prior to factor estimation by the method of principal components. In the 2020-02 (pre-\textsc{covid}) vintage of FRED-MD which reports data up to 2019-12, $F_1$ explains 16\% of the variation in the data. All three post-\textsc{covid}\ estimates of $F_1$ continue to capture about 16\% of the variations in the extended sample and are nearly identical up till 2020-02, with pairwise correlations exceeding 0.99.
Turning to the other factors, $\hat F_2$ in the pre-\textsc{covid}\ data loads heavily on term spreads and explains 7\% of variations pre-\textsc{covid}, while $\hat F_3$ which loads heavily on prices and explains about 6.7\% of the variations. In the post-\textsc{covid}\ sample, $\hat F_2$ still loads heavily on term spreads while and $\hat F_3$ still loads heavily on prices. These two factor estimates, along with $\hat F_ 4, \hat F_7,$ and $\hat F_8$ are nearly perfectly correlated with the pre-\textsc{covid}\ estimates in the overlapping sample. However, the $\mathcal H$ and $\mathcal P$ correlations for $\hat F_5$ and $\hat F_6$ are less than 0.8, much lower than the $\mathcal D$ correlations which are over 0.95. Figure \ref{fig:F_model4} plots the first three macros factors and the first financial for Model 4. Interestingly, the February values of $\hat F_1, \hat F_2$ and $\hat F_3$ in the de-\textsc{covid}\ data are more different from August/September than March/April when the lockdown started.
Figure \ref{fig:F1_acrossmodels} summarizes the different estimates of $\hat F_1$ in 2020 across models.
All suggest a rebound in June presumably due to the stimulus package.
However, only Model 1 suggests a sharp drop in April with the $\mathcal D$ version being the largest (-3.59), and ends the year on the negative side. Models 2, 3, and 4 differ in the magnitude of the rebound in June, but all three estimates of $F_1$ are above the February level at the end of the year.
\section{Forecasting and Measuring Uncertainty Post-Covid}
The second step of the JLN exercise involves generating $h$-step ahead prediction errors. Following \citet{stock-watson-diforc}, we form diffusion index forecasts by augmenting the estimated factors to an autoregression:\footnote{For properties of a factor-augmented regression, see \citet{baing-ecta:06,stock-watson-diforc, stock-watson:16}}.
\[ y_{jt+h}=\phi^y_{jh}(L) y_{jt}+\gamma^F_{jh}(L) \hat F_t+\hat \gamma_{jh}^W(L) W_t + v^y_{jt+h}
\]
Prior to \textsc{covid}\, $y_{jt}$ is one of the 134 series in FRED-MD (ie. $X_{jt}$) after standardization,
\[W_t=(\hat F_{mt}^\prime, \hat F_{ft}^\prime, \hat F_{m,1,t}^2, \hat G_{m,t})^\prime.\]
where $\hat F_{mt}$ is a set of eight macro factors, $\hat F_{ft}$ is a set of four financial factors, $G_m$ is the first factor in $X_m^2$. A $t$ test with a threshold of 2.56 is used to screen predictors.
\textsc{covid}\ changes this exercise in two ways. First, as there is now a new source of variation in the data, predictability of $y_{jt+h}$ as measured by $X_{jt+h}$ must be distinguished from predictability as measured by $x_{jt+h}$. Second, the lingering effects of \textsc{covid}\ on economic activity are not entirely unpredicted after the initial outbreak. \citet{schorfheide-song:20} finds it better to make forecasts shortly after \textsc{covid}\ using the model estimated pre-\textsc{covid}, but this cannot be a sustained solution.\footnote{The authors consider a mixed (monthly-quarterly) frequency VAR of eleven variables and finds that for forecasts made in the end of January, April, and May of 2020, the model estimated using data up to the end of 2019 are reasonable than those based on recursive estimation that includes the post \textsc{covid}\ data}. I expand the predictor set to include \textsc{covid}\ indicators: Let
\begin{equation}
\label{eq2}
W^+_t=(\hat F_{mt}^\prime, \hat F_{ft}^\prime, \hat F_{m,1,t}^2, \hat v^ P_t, \hat v_t^D)^\prime.
\end{equation}
Since $y_{jt}$ has been transformed to be stationary, I use $ v_t$ instead of $ V_t$ which is not stationary. I enter two measures of $ v_t$ in $W_t$ because the number of deaths tend to lag the number of positive cases, and the two may contain different information. However, $G_m$ is no longer a potential predictor set. This factor would have to be estimated from the panel of $X_{jt}^2$ which necessitates additional adjustments. It is noteworthy that based on the $t$ test criterion, the variables being selected are quite similar before and after \textsc{covid}. The lags of $\hat v_t$ are selected with frequencies ranging from 0.3 to 0.54.
After an $h$ period ahead diffusion forecast for series $j$ is obtained from the factor-augmented regression, step three of the JLN exercise estimates stochastic volatility models for the one-period ahead predictor error $\hat \varepsilon^y_{j,t+1}$ for each series $j$ and $\hat \varepsilon^F_{k,t+1}$ for each factor $k$. This volatility estimation step is unaffected by \textsc{covid}\ though the volatility estimates will be higher after \textsc{covid}. The three-steps yield $N$ estimates of individual uncertainties which are then aggregated to form macro-economic uncertainty by equal-weighting. I will denote the macro-uncertainty measure based on predictability of $x$ by $ U(x)$, to distinguish it from the one based on predictability of $X$, which is denoted $U(X)$.
Figure \ref{fig:U1} plots the $\mathcal P$ version of uncertainty for Models 1 and 4. $U(X)$ is higher than $U(x)$ because $X$ retains the \textsc{covid}\ variations which contribute to unpredicted forecast errors. Model 1 suggests higher uncertainty in 2020 than Model 4 because of the treatment of the April variations. But regardless of model and version,
uncertainty is high in 2020 by historical standard. There are now four episodes of uncertainty that exceed 1.65 standard deviations: the 2007-09, the 1981-82 and 1973-74 recessions, and \textsc{covid}.
The difference between $U_{M}(X)$ and $U_M(x)$ can be thought of as 'covid uncertainty'. This is plotted in the bottom panel of Figure \ref{fig:U1} for Model 4. Though covid uncertainty' is high in March, it has subsided by the end of 2020. Also of note is that the \textsc{covid}\ uncertainty series for Models 2, 3, and 4 are very similar, suggesting that the March dummy incorporated in Models 2 and 3 is unnecessary, and the agnostic approach used in Model 4 suffices. The next section shows that Method 4 is also useful in VAR analysis.
\section{Implications for Estimation of Dynamic Causal Effects}
The data issues created by \textsc{covid}\ also apply to other forms of time series modeling. Prior to \textsc{covid}, the model for the $n\times 1$ vector of variables $Y_t$ is
\[ Y_t=\alpha+A(L) Y_{t-1}+B e_t\]
where $e_t$ is a vector of $n$ economic shocks. Now extend the data to include ten months of \textsc{covid}\ data. Under the view that \textsc{covid}\ is a health shock, there are now $n+1$ shocks in the $n$ variable VAR which cannot be expected to recover $n$ economic shocks.
To illustrate, consider a VAR(p) in unemplyoment rate (UR) and log industrial production (IP). The top panel of Figure \ref{fig:var} shows the dynamic responses to a UR shock using parameters estimate over the sample 1961:1-2019:12 with $p=6$ lags. Both responses have a hump shape shock, peak after eight months, and are persistent.
The second panel of Figure \ref{fig:var} shows that the post-\textsc{covid}\ responses are generally larger in magnitude and distinctively different in shape from the pre-covid responses in the top panel. In particular, they are no longer hump shaped. Presumably, results like this have prompted \citet{lenza-primceri:20} to scale up the variance of the shocks during the pandemic, and \citet{ccmm:21} to incorporate outliers in a VAR with stochastic volatility. Instead of assuming changes to the distributions of economic shocks, I assume that there is an additional shock.
Following the arguments above, the idea is to purge $v$ from each of the $n \cdot p$ variables in the VAR. Note that this is not the same as running a VAR on $n$ de-\textsc{covid}\ variables. However, by Frish-Waugh arguments, it is simple to augment the $n$ variable VAR in un-adjusted data $Y$ with $v$ and its lags as exogenous variables.
\[ Y_t=\alpha+ \gamma D_t+ \delta 1_{t>T_0} +A(L) Y_{t-1}+\beta(L) v_t+B e_t \quad \tag{VAR-E}\]
where $D_t$ and each row of $\beta(L)$ are defined as in Models 1 to 4 above. A mean shift dummy is needed since the de-\textsc{covid}\ regressions are now run on the full sample. The role of $v$ and its lags is to partial out the effects of \textsc{covid}\ from the dependent variable and the regressors so that coefficients can be used to construct dynamic responses to economic shocks, holding $v$ and its lags fixed. The third panel of Figure \ref{fig:var} shows the $\mathcal P$ version of the Model 4 impulse responses based on (VAR-E) along with the 95\% confidence interval. Notably, they are very similar both in magnitude and in shape to the pre-\textsc{covid}\ responses in the top panel. The results for $\mathcal D$ and $\mathcal H$ versions are similar.
Methods 2 and 3 also retain the hump shaped responses but not quite as close the pre-\textsc{covid}\ estimates as Model 4.
It is of interest to consider a three variable VAR that includes a \textsc{covid}\ variable and ordered first. While VAR-E assumes no feedback from UR or IP to $v$, a three variable VAR allows for such feedback. The dynamic responses should be similar if \textsc{covid}\ is indeed exogenous.\footnote{A six-variable VAR finds likewise that adding the \textsc{covid}\ indicators as exogenous variables recovers the pre-\textsc{covid}\ impulse responses. The variables are unemployment rate (unrate), log of employment (payems), log real consumption of durables (cdur), log real consumption of services (cs), industrial production (ip), and log of the consumption expenditure deflator excluding food and energy.} The bottom panel of Figure \ref{fig:var} verifies that the responses of the three variable VAR to a UR shock are similar to the ones plotted for VAR-E. However, these responses to a UR shock are distinctly different from the responses to a \textsc{covid}\ shock. As seen from Figure \ref{fig:varcovid}, a \textsc{covid}\ shock has much larger but shorter lived effects on economic activity. The unemployment rate returns to control after about one year. The effect on IP, while large, is statistically significant for only two months. A two variable VAR without controlling for \textsc{covid}\ would confound the responses to the two different shocks, as seen from the second panel of Figure \ref{fig:var}.
\begin{table}[ht]
\caption{Orthogonalized Shocks}
\label{tbl:shocks}
\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r}
\hline
& \multicolumn{6}{c|}{UNRATE shock } & \multicolumn{6}{c}{IP shock} \\ \hline
& M0 & M1 & M2 & M3 & M4 & VAR3 & M0 & M1 & M2 & M3 & M4 & VAR3\\
\hline
march & 0.81 & 0.00 & 0.00 & 0.00 & -0.22 & 0.07 & -4.30 & 0.00 & 0.00 & 0.00 & -0.16 & -0.07\\
\hline
april & 9.62 & 19.56 & 0.00 & -0.04 & 0.25 & 0.09 & -11.84 & 0.68 & -0.12 & -0.14 & -0.25 & -0.16\\
\hline
may & -1.28 & -2.43 & 0.34 & 0.09 & 0.14 & 0.04 & 3.22 & 1.40 & -0.04 & -0.20 & 0.01 & -0.10\\
\hline
june & -0.71 & -2.46 & -1.90 & -1.41 & -0.23 & -0.08 & 1.74 & 1.12 & 0.83 & 1.11 & 0.15 & 0.07\\
\hline
cor & 0.83 & 0.81 & 0.96 & 0.97 & 0.97 & 1.00 & 0.95 & 0.96 & 0.97 & 0.98 & 0.98 & 1.00\\
\hline
\end{tabular}
\end{table}
A look at the orthogonalized shocks highlights the identification problem that \textsc{covid}\ creates.
The column labeled M0 is based on post-\textsc{covid}\ estimation without controls, while M1, M2, M2, M4 correspond to the four methods above. These are all bivariate VARs and to be distinguished from VAR3, which is a three-variable model for ($v$, UR, IP). The rows list the values of the identified shocks from March through June as well as the correlation with the corresponding shock identified pre-\textsc{covid}\ data over the overlapping sample. Evidently, the M2, M3, M4 correlations are high and the shocks identified for UNRATE and IP in April are small. With no or inadequate adjustments, the M0 and M1 correlations are lower, and the shocks identified for UR and IP in April are large. The analysis lends support to treating \textsc{covid}\ as exogenous which is the maintained assumption of Method 4 used to purge \textsc{covid}\ effects from the data prior to factor estimation.
Our \textsc{covid}\ variables are zero before March 2020 which can be approximated by time series with values arbitrarily close to zero except for the few spikes in 2020. Data with such extreme values have heavy tails, and one might wonder if our least squares used to estimate the three variable VAR or the de-\textsc{covid}\ regressions are valid. Intuitively, the issue is that in a standard regression framework, the response variable should have heavy tails if one of the predictors has heavy tails. Yet, our economic data have fat but not heavy tails. For such problems, \citet{davis-ng:21} uses a new (heavy-light) framework to dampen the coefficient on the heavy tailed predictor so that it can meaningfully affect the thin-tailed response variable. In that setup, the coefficient on the heavy tailed regressor is consistent at rate $T^{1/\alpha}$ where $\alpha\in(0,2)$ is the tail index, implying that the estimated coefficient on the heavy-tailed regressors are super-consistent.
\section{Conclusion}
\textsc{covid}\ can be treated as an outlier, a health shock with macroeconomic consequences, or an economic shock. I use \textsc{covid}\ indicators to purge the data of their effects so that economic factors can be estimated. Adding \textsc{covid}\ indicators to a VAR as exogenous controls also makes it possible to recover impulse responses to economic shocks similar to the ones estimated pre-\textsc{covid}. This note draws attention to the need to control for \textsc{covid}\ variations in macroeconomic modeling.
\begin{center}
\begin{figure}[ht]
\caption{$\hat\mu^1_{it}$, $\mathcal P$ version: 2020:01-2020:12}
\label{fig:fig0}
\centering Model 1
\includegraphics[width=7.0in,height=3.50in]{model1A_meanadj1_positives.png}
\centering Model 4
\includegraphics[width=7.0in,height=3.550in]{model4A_meanadj1_positives.png}
\end{figure}
\end{center}
\vspace*{-.75in}
\clearpage
\begin{center}
\begin{figure}[ht]
\caption{Factor Estimates: 2019:01-2020:12: Model 4}
\label{fig:F_model4}
\includegraphics[width=7.0in,height=2.750in]{compare3_fhat0.png}
\end{figure}
\end{center}
\vspace*{-.5in}
\begin{center}
\begin{figure}[ht]
\caption{Estimates of $F_1$ in 2020:01-2020:12}
\label{fig:F1_acrossmodels}
\includegraphics[width=7.0in,height=3.0in]{fig_compare4_fhat2020.png}
\end{figure}
\end{center}
Note: $(H,P,D)$ uses hospitalization data, positive cases, and number of deaths as controls respectively.
\begin{figure}[ht]
\caption{One Period Ahead Uncertainty Estimates}
\label{fig:U1}
\includegraphics[width=7in,height=2.50in]{compareUV1_model1.png}
\includegraphics[width=7in,height=2.50in]{compareUV1_model4.png}
\includegraphics[width=7in,height=2.50in]{compare4UV.png}
Note: U(X) is based on predictability of unadjusted data $X$. U(x) is based on predictability of the de-covid data, $x$.
\end{figure}
\begin{center}
\begin{figure}[ht]
\caption{Dynamic Responses to Unemployment (UR) Shock}
\label{fig:var}
\centering Bivariate VAR, Pre-covid
\includegraphics[width=7.0in,height=1.75in]{precovid.png}
\centering Bivariate VAR, Post-covid, no adjustment
\includegraphics[width=7.0in,height=1.75in]{postcovid0.png}
\centering Bivariate VAR, Post-covid, Model 4
\includegraphics[width=7.0in,height=1.75in]{postcovid14.png}
\centering Three variable VAR
\includegraphics[width=7.0in,height=1.75in]{postcovid15.png}
\end{figure}
\end{center}
\begin{center}
\begin{figure}
\caption{Dynamic Responses to Covid Shock}
\label{fig:varcovid}
\centering Three variable VAR
\includegraphics[width=7.0in,height=3.50in]{postcovid16.png}
\end{figure}
\end{center}
\clearpage
\bibliography{../../../prog/sydney/ll,../../../prog1/weekly/notes/weekly,metrics,macro,metrics2,factor,consum}