Figure 1: Model specification labels
Notes: Each node is one of the 16 model
specifications. B denotes a
specification without time-varying long-run growth, whereas T denotes a specification with
time-varying long-run growth; the suffixes SV, DH, and O are appended when stochastic
volatility, dynamic heterogeneity, and the outlier adjustment are
active, respectively. The 16 specifications form a 4-dimensional
hypercube, drawn here as two 3-dimensional cubes (left: constant
long-run growth; right: time-varying long-run growth) over the remaining
three features, with the outlier adjustment on the horizontal axis,
stochastic volatility on the vertical axis, and dynamic heterogeneity on
the diagonal axis within each cube. An edge connects two specifications
whenever they differ in exactly one feature: solid gray edges toggle
stochastic volatility, dynamic heterogeneity, or the outlier adjustment
within a cube, and dashed orange edges toggle long-run growth between
the two cubes.
A diagram of the 16 model specifications used in the paper, drawn as two cubes. The left cube contains the eight specifications without time-varying long-run growth (label prefix B), and the right cube contains the eight specifications with time-varying long-run growth (label prefix T). Within each cube, the outlier adjustment is on the horizontal axis, stochastic volatility is on the vertical axis, and dynamic heterogeneity is on the diagonal axis, so that each of the eight corners of a cube corresponds to one combination of those three features (labeled with suffixes O, SV, and DH as applicable). Solid gray lines connect specifications that differ in exactly one of these three within-cube features; dashed orange lines connect each specification in the left cube to its counterpart in the right cube, which differs only in whether long-run growth is time varying. Together the two cubes and the connecting lines represent all 16 combinations of the four binary features (long-run growth, stochastic volatility, dynamic heterogeneity, outlier adjustment) and every pair of specifications that differs in exactly one feature.
Figure 2: Balanced paired factorial contrasts by loss function and
period.
Notes: Points are percent effects, 100(exp (d̄j) − 1)
(or 100(exp (d̄jp) − 1)
for regime panels); horizontal bars are 90 percent bootstrap percentile
intervals from 5,000 target-quarter bootstrap replications. The vertical
dashed line marks zero effect. TV denotes time-varying.
A 2-by-4 grid of forest plots. The two rows correspond to the squared forecast error loss and the CRPS (density forecast) loss; the four columns correspond to the overall sample and the pre-pandemic, pandemic, and post-pandemic periods. Each panel plots, for the four model features (dynamic heterogeneity, stochastic volatility, time-varying long-run growth, and the outlier adjustment), a point estimate of the percent change in loss from including that feature and a horizontal bar showing the 90 percent bootstrap confidence interval; a vertical dashed line marks zero effect, and negative values indicate improved forecast performance. The key takeaway is that dynamic heterogeneity is the only feature whose 90 percent interval lies entirely below zero for both the squared-error and CRPS losses in the overall sample, lowering squared-error loss by about 17 percent and CRPS by about 8 percent; the other three features have intervals that include zero overall. By period, dynamic heterogeneity's gains concentrate before and after the pandemic; stochastic volatility's CRPS improvement is clearest before the pandemic; time-varying long-run growth shows a sizeable but imprecise squared-error improvement after the pandemic; and the outlier adjustment shows no clear benefit in any period for squared error, but raises (worsens) CRPS after the pandemic, with a 90 percent interval entirely above zero.
Figure 3: Balanced paired factorial contrasts for quantile scores, by
quantile level and period.
Notes: Points are percent changes in mean quantile
score relative to the matched feature-off specifications; horizontal
bars are 90 percent bootstrap percentile intervals from 5,000
target-quarter bootstrap replications. The vertical dashed line marks
zero effect. TV denotes time-varying.
A 4-by-4 grid of forest plots. The four rows correspond to the quantile levels 0.05, 0.10, 0.90, and 0.95 (5th, 10th, 90th, and 95th percentiles of the predictive distribution), and the four columns correspond to the overall sample and the pre-pandemic, pandemic, and post-pandemic periods. Each panel plots, for the same four model features as Figure 2, a point estimate of the percent change in mean quantile score from including that feature and a horizontal bar showing the 90 percent bootstrap confidence interval, with a vertical dashed line at zero and negative values indicating improved tail-forecast performance. The key takeaway is that dynamic heterogeneity is the only feature with a 90 percent interval entirely below zero in the overall sample, and only at the 90th percentile (cutting that quantile score by about 6 percent); by period, it improves lower-tail scores before and after the pandemic and produces its clearest upper-tail gains after the pandemic. Time-varying long-run growth raises (worsens) the 5th- and 10th-percentile scores overall, with intervals excluding zero, an effect driven mainly by the pre-pandemic period. The outlier adjustment improves the 90th- and 95th-percentile scores during the pandemic, though imprecisely, and turns detrimental for those same percentiles after the pandemic. Stochastic volatility shows scattered improvements at both tails, concentrated before the pandemic, without a clear overall pattern.
Figure A.1: Each of the 25 series used in the nowcasting exercise, in
the units in which it enters the model (Table 1), before
standardization, over the estimation sample through December
2024.
A 5-by-5 grid of line charts, one per variable in the data set (Table 1), each plotted in the units in which it enters the model (e.g., annualized quarter-over-quarter percent change for the three quarterly variables; month-over-month percent change, level change, or survey index for the monthly indicators), before standardization, over the estimation sample through December 2024. Series begin at their own availability date, so some panels (e.g., ADP nonfarm private payroll, JOLTS total job openings, real personal consumption expenditures, and the Empire State Manufacturing Survey) start later than 1990. The key takeaway is that nearly every series shows an extreme, isolated spike or trough around 2020Q2, consistent with the COVID-19 pandemic shock, standing well outside the range of fluctuations observed in the rest of the sample; this is the empirical motivation for the paper's stochastic-volatility and outlier-adjustment features.
Figure B.1: Real GDP growth forecasts and realizations by specification:
pre-pandemic period. Specification labels: \(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
A 4-by-4 grid of line-and-scatter charts, one panel per model specification (using the B/T, SV, DH, O labeling defined in Figure 1), covering target quarters from 2015Q1 through 2019Q4 plus one year of history. In each panel, a black line traces realized real GDP growth (annualized quarter-over-quarter percent), and colored dots show each specification's nowcasts at seven monthly horizons (from 5 months before quarter-end through 1 month after) relative to that quarter's realization. The key takeaway is that in this pre-pandemic period, forecasts from all 16 specifications track realized GDP growth fairly closely and converge toward the actual value as the horizon shortens, with no specification showing large or persistent misses; differences across specifications are visually modest.
Figure B.2: Real GDP growth forecasts and realizations by specification:
pandemic period. Specification labels: \(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
The same 4-by-4 grid of line-and-scatter charts as Figure B.1, but for target quarters from 2020Q1 through 2021Q4. Realized GDP growth swings sharply from about -30 percent in 2020Q2 to about +35 percent in 2020Q3 (annualized), an extreme pattern absent from other periods. The key takeaway is that specifications without the outlier adjustment (B, B-DH, B-SV, B-SV-DH, T, T-DH, T-SV, and T-SV-DH) produce several nowcasts that miss this swing by more than 100 percentage points; these extreme misses are plotted as triangles at the top or bottom edge of the panel with their numerical value annotated (the most extreme are about -193 and -228 percent). No such extreme misses appear in any of the eight specifications that include the outlier adjustment, illustrating that the adjustment contains the size of the largest individual forecast errors during the pandemic even though, as shown elsewhere in the paper, it does not improve average forecast accuracy in this period.
Figure B.3: Real GDP growth forecasts and realizations by specification:
post-pandemic period. Specification labels: \(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
The same 4-by-4 grid of line-and-scatter charts as Figures B.1-B.2, but for target quarters from 2022Q1 through 2024Q4 plus one year of history. Realized GDP growth fluctuates in a more moderate range (roughly -1 to 7 percent) than during the pandemic, including a temporary dip below zero in 2022Q2. The key takeaway is that nowcasts across all 16 specifications again track realized GDP growth reasonably well without extreme misses, though with somewhat more dispersion across horizons than in the pre-pandemic period, particularly around the 2022 dip.
Figure B.4: RMSE by horizon and specification: pre-pandemic period. Cell
entries are in percentage points of annualized quarter-on-quarter real
GDP growth. Colors are normalized within each horizon; a black outline
marks the lowest loss within that horizon. Specification labels: \(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
A heatmap with 7 rows (forecast horizons from 5 months before quarter-end through 1 month after) and 16 columns (model specifications), where each cell reports the root-mean-squared forecast error and is colored on a green-to-red scale normalized within its own row (green is the lowest, i.e., best, RMSE in that row; red is the highest). The key takeaway is that RMSEs in this period are broadly similar across specifications, ranging from about 1.06 to 1.55, with the two SV-only specifications (B-SV, B-SV-O) and the SV-plus-DH specifications generally near the low end and the base T and T-O specifications (without dynamic heterogeneity or stochastic volatility) generally near the high end within most horizons.
Figure B.5: RMSE by horizon and specification: pandemic period. Cell
entries are in percentage points of annualized quarter-on-quarter real
GDP growth. Colors are normalized within each horizon; a black outline
marks the lowest loss within that horizon. Specification labels: \(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
The same type of heatmap as Figure B.4 (7 horizons by 16 specifications, colored within each row), but for the pandemic period. The key takeaway is that RMSEs are far larger and much more dispersed across specifications than in the pre-pandemic period, ranging from about 4 to 98 depending on horizon and specification. At the longer horizons (5 and 4 months before quarter-end), specifications with stochastic volatility and/or the outlier adjustment tend to have much lower RMSE than those without either feature; for example, at the 4-month horizon, B-DH has an RMSE of about 98 while several SV- or outlier-adjusted specifications are near 17. This illustrates the pandemic-period value of these two features at longer horizons, even though, as reported elsewhere in the paper, their overall-sample benefit is less conclusive.
Figure B.6: RMSE by horizon and specification: post-pandemic period.
Cell entries are in percentage points of annualized quarter-on-quarter
real GDP growth. Colors are normalized within each horizon; a black
outline marks the lowest loss within that horizon. Specification labels:
\(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
The same type of heatmap as Figures B.4-B.5, but for the post-pandemic period. The key takeaway is that RMSEs return to a moderate range (about 1.2 to 2.9) similar in spirit to the pre-pandemic period, though somewhat higher overall; dynamic-heterogeneity specifications (B-DH, B-DH-O, T-DH, T-DH-O) tend to have the lowest RMSE within most horizons, while the SV specifications without dynamic heterogeneity (B-SV, T-SV) and their outlier-adjusted counterparts tend to have the highest.
Figure B.7: CRPS by horizon and specification: pre-pandemic period. Cell
entries are in percentage points of annualized quarter-on-quarter real
GDP growth. Colors are normalized within each horizon; a black outline
marks the lowest loss within that horizon. Specification labels: \(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
A heatmap with the same 7-horizon-by-16-specification layout as the RMSE heatmaps, but reporting average CRPS (a measure of density-forecast accuracy) instead of RMSE, again colored on a green-to-red scale normalized within each row. The key takeaway is that CRPS values are fairly similar across specifications within this period (roughly 0.66 to 0.95), with the four dynamic-heterogeneity specifications without stochastic volatility (B-DH, B-DH-O, T-DH, T-DH-O) and the dynamic-heterogeneity-plus-SV specifications consistently near the low (best) end within each horizon, and the base B and T specifications (without dynamic heterogeneity) consistently near the high (worst) end.
Figure B.8: CRPS by horizon and specification: pandemic period. Cell
entries are in percentage points of annualized quarter-on-quarter real
GDP growth. Colors are normalized within each horizon; a black outline
marks the lowest loss within that horizon. Specification labels: \(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
The same type of heatmap as Figure B.7, but for the pandemic period. The key takeaway is that CRPS values are both much larger and much more dispersed than pre-pandemic, ranging from about 2.5 to 35 depending on horizon and specification. At the longer horizons, specifications with stochastic volatility tend to have substantially lower (better) CRPS than specifications without it; for example, at the 4-month horizon, B-DH has a CRPS of about 35 while several SV specifications are near 10, though this pattern is less consistent than for RMSE and reverses at some shorter horizons (e.g., stochastic-volatility specifications have the worst CRPS at the 1-month-before horizon).
Figure B.9: CRPS by horizon and specification: post-pandemic period.
Cell entries are in percentage points of annualized quarter-on-quarter
real GDP growth. Colors are normalized within each horizon; a black
outline marks the lowest loss within that horizon. Specification labels:
\(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
The same type of heatmap as Figures B.7-B.8, but for the post-pandemic period. The key takeaway is that CRPS values return to a moderate range (about 0.84 to 1.67) and the dynamic-heterogeneity-plus-SV specifications (B-SV-DH, B-SV-DH-O, T-SV-DH, T-SV-DH-O) are consistently at or near the low (best) end within each horizon, while the base T and T-SV specifications without dynamic heterogeneity are consistently at or near the high (worst) end.
Figure B.10: Estimated economic activity factor and external growth
benchmarks by specification. The posterior mean and 68 percent credible
band of the factor are shown at calendar-quarter ends through 2024Q4.
The factor is measured in standardized model units on the left-hand
axis; PDFP and GDPplus are annualized quarter-over-quarter growth rates,
measured in percent on the right-hand axis. Both axes are common across
panels. PDFP denotes real final sales to private domestic purchasers;
GDPplus combines expenditure- and income-side estimates of real GDP
growth Aruoba et al., 2016.
Specification labels: \(B\) excludes
and \(T\) includes time-varying
long-run growth; \(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
A 4-by-4 grid of time-series charts, one per specification, each plotting the posterior mean and 68 percent credible band of that specification's estimated common factor ("economic activity," left axis, standardized units) from a single full-sample estimation, alongside two external annualized quarter-over-quarter growth benchmarks on the right axis: private domestic final purchases (PDFP) growth and the GDPplus indicator. All panels share a common calendar axis from the late 1980s through 2024 and a pronounced spike/trough around 2020. The key takeaway, discussed in the paper's text, is that specifications with dynamic heterogeneity (the DH suffix) track both external benchmarks far more closely than specifications without it, which instead exhibit a comparatively flat, low-amplitude factor that lags or misses several cyclical swings (most visibly around 2001 and 2008-09); this pattern holds regardless of whether a given specification also includes stochastic volatility or time-varying long-run growth.
Figure B.11: Quarterly long-run-growth component implied by \(g_t\) for \(T\text{-}SV\text{-}DH\text{-}O\): posterior
mean and 68 percent credible band from the same full-sample estimation
as Figure B.10, compared with PDFP and
GDPplus four-quarter growth. Only observations through December 2024 are
shown. Specification labels: \(B\)
excludes and \(T\) includes
time-varying long-run growth; \(SV\),
\(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
A single time-series chart plotting the posterior mean and 68 percent credible band of the estimated time-varying long-run growth component g, converted to a four-quarter growth rate, against the same PDFP and GDPplus four-quarter growth benchmarks used in Figure B.10, from the late 1980s through 2024. The key takeaway is that the estimated long-run growth component is much smoother than either benchmark, drifting gradually between roughly 1.5 and 3.5 percent and capturing slow-moving shifts, such as a modest decline in trend growth around the 2008-09 financial crisis and a partial recovery afterward, rather than tracking the sharp cyclical swings (including the 2020 pandemic spike) visible in the two benchmark series.
Figure B.12: Estimated idiosyncratic innovation standard deviation by
series for \(T\text{-}SV\text{-}DH\text{-}O\): posterior
mean and 68 percent credible band of \(\sqrt{\exp(h_{\eta_i,t})}\), rescaled to
the units reported in Table 1.
Calendar-quarter-end readings are shown through December 2024 and only
over each series’ own availability window, from the same full-sample
estimation as Figure B.10.
Specification labels: \(B\) excludes
and \(T\) includes time-varying
long-run growth; \(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
A 5-by-5 grid of time-series charts, one per variable in the data set, each plotting the posterior mean and 68 percent credible band of the baseline (stochastic-volatility) standard deviation of that series' idiosyncratic shock, in the series' own units, from the same full-sample estimation of the richest specification, over each series' own availability window through December 2024. The key takeaway is that nearly every series shows a pronounced, temporary rise in this baseline volatility around 2020, most dramatic for series such as all employees (total nonfarm), ADP nonfarm private payroll, and real personal consumption expenditures, with volatility generally receding toward pre-pandemic levels afterward, though not always fully.
Figure B.13: Change in the idiosyncratic innovation standard deviation
attributable to the multiplicative outlier adjustment, by series, for
\(T\text{-}SV\text{-}DH\text{-}O\):
posterior mean and 68 percent credible band of \(\sqrt{\exp(h_{\eta_i,t})\,o_{i,t}}-\sqrt{\exp(h_{\eta_i,t})}\)
in the units reported in Table 1.
Calendar-quarter-end readings through December 2024 are shown over each
series’ availability window, using the same full-sample estimation as
Figure B.10. Specification labels: \(B\) excludes and \(T\) includes time-varying long-run growth;
\(SV\), \(DH\), and \(O\) denote stochastic volatility, dynamic
heterogeneity, and outlier adjustment, respectively.
A 5-by-5 grid of time-series charts, one per variable, each plotting the posterior mean and 68 percent credible band of the additional standard deviation contributed by the multiplicative outlier scale (on top of the stochastic-volatility baseline shown in Figure B.12), in the series' own units, over each series' availability window through December 2024; a value of zero corresponds to no outlier adjustment being active. The key takeaway is that this excess standard deviation is close to zero for nearly the entire sample for most series, but spikes sharply and briefly around 2020 for many series (most visibly all employees, ADP payroll, and real personal consumption expenditures), showing that the outlier adjustment is activated almost exclusively during the pandemic shock rather than continuously throughout the sample.