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Forecasting Systemic Risks
Gianni De Nicolò and Marcella Lucchetta
Forecasting Systemic Risks*
Gianni De Nicolò
International Monetary Fund and CESifo
Marcella Lucchetta
University of Venice
First draft, September 2014
This draft, January 2014
Abstract
Reliable early warning signals of real and financial vulnerabilities are essential for timely
implementation of macroeconomic and macroprudential policies. This paper presents an
early warning system for systemic risks as a set of multi-period forecasts of indicators of
systemic real and financial risks. Forecasts are obtained from: (a) factor-augmented VARs
with linear GARCH volatility, and (b) factor-augmented predictive Quantile Regressions.
We use monthly U.S. data for the period 1974:1-2013:12 to forecasts our systemic risk
indicators with each model in pseudo-real time. We find that forecasts obtained with factor
augmented VARs significantly underestimate systemic risks, while forecasts obtained with
factor-augmented predictive Quantile Regressions deliver reliable early warning signals for
systemic real and financial risks up to a one-year horizon.
JEL Classification Numbers: C5, E3, G2
Keywords: Systemic Risks, Density Forecasts, Factor models, Quantile Regressions.
Authors’ e-mail addresses: [email protected]; [email protected].
* We thank without implications Fabio Canova, Gianni Amisano, and participants at the IMF
seminar and the September 2014 conference on “Macroeconomic Stability, Banking Supervision and
Financial Regulation” at the European University Institute, for comments and suggestions. The views
expressed in this paper are those of the authors and do not necessarily represent the views of the
International Monetary Fund.
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I. INTRODUCTION
The 2007-2009 financial crisis has spurred renewed efforts in systemic risk modeling,
Bisias et al. (2012) provide an extensive survey of the models currently available to measure
and track indicators of systemic financial risk. Yet, most of these models focus exclusively
on vulnerabilities in the financial system or some of its components, with no direct
assessment of either their impact on real activity, or on how vulnerabilities in the real sector
may affect the financial sector. Most importantly, the out-of-sample forecasting power of
many of the proposed measures is seldom assessed, making it difficult to gauge their
usefulness as early warning signals. Reliable early warning signals—where reliability is
defined as the ability of a model to issue signals with relatively small out-of-sample forecast
errors—are essential for timely implementation of macroeconomic and macroprudential
policies.
Building on our previous work (De Nicolò and Lucchetta, 2011, 2012), this paper
develops an early warning system for systemic risks (EWS). The modeling procedure
underlying our proposed EWS has three key features. First, as advocated in Group of Ten
(2001), and implied by explicit modeling of tail risks as recently proposed by Acemoglu et al
(2015), we make an important distinction between systemic real risk and systemic financial
risk, based on the notion that the potential adverse welfare consequences of the real effects of
systemic risks arising either from the real sector, or from the financial sector, or from both,
are what ultimately concerns policymakers, Second, our EWS delivers multi-period forecasts
of Value-at-Risk (VaR) indicators of systemic real and financial risk. These forecasts are
obtained using two types of models: factor-augmented VARs, with the volatility of each
indicator following a linear GARCH process, termed FAVARs, and factor-augmented
3
predictive quantile regressions, termed QRs. The blending of factor models with time varying
volatility and quantile regressions is a novel feature of our modeling framework. Third, the
out-of sample forecasting power of each model is fully assessed by evaluating their multiperiod (weighted) density and quantile forecasts using a scoring rule, as it is common in the
density forecast literature.1 The aim is to identify the model, or model combination, that
delivers the best set of early warning signals of real and financial vulnerabilities.
Our EWS can be viewed as a tool complementary to forecasting with DSGE models
and data-driven models such as VARs. As detailed in Schorfheide (2010), there is still lack
of conclusive evidence of the superiority of the forecasting performance of DSGE models
relative to data-driven models (see e.g. Herbst and Schorfheide, 2010). Amisano and Geweke
(2013) illustrate the usefulness of combination of these models’ forecasts in improving the
accuracy of density predictions. Yet, forecasting applications of DSGE and data-driven
models do not typically focus on tail risks, as we do. Contributing to develop early warning
systems for systemic risks based on an explicit account of the interactions between financial
and real sectors is a key objective of this paper.
Systemic risks are measured following a risk management approach. In this paper we
consider forecasts of two indicators of real activity and two indicators of financial stress. For
each indicator, the relevant systemic risk metric is given by its VaR at given low probability
levels, as detailed momentarily. Specifically, systemic real risks are measured by the VaR of
growth in industrial production (IP) and employment (EM). Systemic financial risks are
measured by the VaR of the inverse of realized volatility of the equity returns of a value
1
For a survey of density forecast evaluation using scoring rules, see Corradi and Swanson (2006),
4
weighted portfolios of (systemically important) banks (denoted by BR), and one including a
large set of non-financial firms (denoted by CR). Atkenson et al (2013) show that for a firm,
the inverse of realized equity volatility is a proxy of a firm’s distance to insolvency (DI)
which can be derived from a large class of structural financial models. Our measures proxy
systemic financial risks as tail realizations of the DI of portfolios of firms, and are germane
to other theory-based indicators of systemic financial risk used in recent studies (see e.g.
Acharya et al., 2010 or Brownlee and Engle, 2010).
We implement our EWS using monthly U.S. data for the period 1974:1-2013:12. We
collect time series of about 50 variables representing the dynamics of price and quantities in
the real and financial sectors, use them to extract factors by principal components, and these
factors are used as predictors in all models. As is common in the density forecast literature,
estimation and forecasting is conducted using a moving window of data of 120 months to
account for time variation in parameters and possible structural brakes. For each variable
underlying our systemic risk measures (i.e., IP, EM, BR and CR), we compute multi-period
density and quantile forecasts via FAVAR and QR at a three month, six months and 12
month horizons. The accuracy of these forecasts is assessed using the Quantile Weighted
Probability Score (QWPS) introduced by Gneiting and Ranjan (2011), as well as in terms of
standard indicators of directional forecasting accuracy. This evaluation allows us to identify
the set of models that deliver the best early warning signals.
We begin by estimating recursive density forecasts in pseudo-real time associated
with four parametric models under a standard Gaussian assumption, as well as their Equally
Weighted Pool (EWP), as defined in Geweke and Amisano (2011), The first two models are
taken as benchmarks with which to compare the forecasting performance of the factor
5
models. The first one is a simple auto-regressive model with the conditional variance
following a linear GARCH(1,1) process, termed AR; the second one is a vector autoregression model that includes the four variables considered, with the conditional variance of
each variable following a linear GARCH(1,1) process, termed VAR. We focus on two
FAVAR models. In both, the conditional variance of each variable follows a linear
GARCH(1,1) process, but they differ according to the number of factors. The model termed
FAVAR(x) uses the optimal number of factors selected according to the selection criterion
proposed by Ahn and Horenstein (2013), while the model termed FAVAR5 has five factors,
treated as a benchmark regarding the number of factors, as in Stock and Watson (2102).
Our analysis of FAVARs models delivers three results. First, the two factor models
deliver density and quantile forecasts significantly more accurate than those of the AR and
VAR models for each variable and each forecast horizon. This result reinforces the well
known result of better predictive ability of factor models with many predictors, since it
extends it to their entire density forecast at multiple forecasting horizons. Second, the
FAVAR5 model yields more accurate density forecasts than the FAVAR(x) model: this
result suggests that for forecasting purposes, using a number of factors chosen according to a
criterion such as the one proposed by Ahn and Horenstein (2013) does not necessarily
improve the accuracy of density forecasts. Third, the density forecast of the EWP delivers
density forecasts for each variable and each forecast horizon significantly more accurate than
those obtained from each of the four models in the pool, consistently with the results of
Amisano and Geweke (2011,2012). These three results complement those obtained in the
current literature and are of independent interest.
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Yet, our main objective is to identify a set of models on which to construct a reliable
early warning system. In this regard, our first key result is that all the foregoing models
deliver VaR forecasts that significantly underestimate systemic (or tail) risks. Furthermore,
relative to the whole sample, their forecasting accuracy decreases substantially for the
subsample that includes the recent financial crisis: this indicates that the reliability of the
early warnings of these indicators falls when reliability is needed most. The failure of this
class of models to capture tail risks are most likely related to their inability to capture
asymmetric effects and time varying distributional shapes owing to their
underlying
Gaussian assumption. This is not good news for tail-risk-averse policymakers, since the
forecasts of this class of models (as well as their DSGE versions) are used extensively in
central banks and international organizations as inputs for stress testing purposes.
We then move to semi-parametric models, focusing on direct estimation of VaRs
through factor-augmented quantile regressions, where the quantile of each variable at every
forecast horizon is a linear function of the principal component factors extracted at the
forecasting date. As is well known, a potential advantage of these models is that they do not
require assumptions about the underlying distribution, and they can in principle capture any
type of asymmetry. Quantile regression approaches have been increasingly used in
economics (see Komunjer, 2013 for an excellent review), as well as in other economic areas
(see e.g. Chernuzikhov et al., 2011) and disciplines, such as climate sciences. .
Our second key result is good news for tail-risk-averse policymakers. We find that
quantile models deliver reliable early warning signals for systemic risks for forecast horizons
up to one year. Importantly, reliability is preserved both for the entire sample period and the
subsample which includes the recent financial crisis. Indeed, these models seem to capture
7
parsimoniously and timely those changes in the shape of the distribution of a variable that
may anticipate significant changes in its left tail.
The remainder of the paper is composed of four sections and an Appendix. Section II
defines our systemic risk measures. Section III describes the models, the estimation and
forecasting procedures, and forecast evaluation. Section IV describes the forecasting
procedure and the empirical results. Section IV concludes. The Appendix describes data and
their sources.
II. SYSTEMIC RISK MEASURES
We consider four monthly measures of systemic real and financial risks.2 The first
real measure is Industrial Production-at-Risk, defined as the VaR of the (log) change in the
industrial production index IP , denoted by VaRD ( IP) . The second real measure is
Employment-at-Risk, defined as the VaR of the (log) change in total employment EM ,
denoted by VaRD ( EM ) .
We measure tail financial stress in the banking and corporate sectors. Systemic
financial risk in the banking sector is gauged by Banking Sector-at-Risk, defined as the VaR
of a portfolio version of the Distance-to-Insolvency (DI) measure recently introduced by
Atkeson et al. (2013), They show that DI is a measure of the adequacy of a firm’s equity
cushion relative to its riskiness, based on Leland’s (1994) structural model of credit risk, and
is bounded above by the inverse of its instantaneous equity volatility. In our implementation,
we consider the returns of a value weighted portfolio of banks. This portfolio represents a
2
These measures can be viewed as an extension of the measures we introduced in De Nicolò and Lucchetta
(2012) at a quarterly frequency, such as GDP-at-Risk.
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large portion of the banking system, including all banks considered as “systemically
important”. In this case, a “portfolio” DI is a lower bound of the probability of insolvency of
the banking system, as profits and losses are evened out in the portfolio. We measure the
inverse of equity return volatility of this portfolio using daily data, which in turn are used to
compute a proxy measure of the inverse of realized volatility at a monthly frequency,
denoted by BR. Thus, the first measure of systemic financial risk is the VaR of this bank
portfolio, denoted by VaRD ( BR ) . Systemic financial risk in the corporate sector is
constructed similarly, using as a portfolio a large set of non-financial firms. Thus, the second
measure of systemic financial risk is Corporate Sector-at-Risk, given by the VaR of the
market portfolio of non-financial firms CR, denoted by VaRD (CR ) . All these systemic risk
measures are computed a 5% and 10% probability levels (i.e. D
0.05 and D
0.10 ).
Table 1 reports descriptive statistics of IP, EM, BR and CR and their correlation
matrix. The signs of the correlations are as expected, with indictors of financial risk
negatively and significantly related to indicators of real activity.
III. MODELS
A. FAVAR models
Denote with yt a variable we wish to forecast, and with ft a vector of factors of
dimension q . As in Stock and Watson (2002), and more recently in Pesaran et al (2011),
these factors are estimated by principal components extracted from N series X it ( i  N ). A
FAVAR model associated with yt is given by the following equations:
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ª ft º
«y »
¬ t¼
§ A( L) B ( L) · ª ft 1 º ªKt º
¨
¸« » « »
© a ( L) b( L) ¹ ¬ yt 1 ¼ «¬u yt »¼
u yt
V yt H yt
V yt
a bV yt 1 c | u yt 1 |
(1)
(2),
The FAVAR described by Equation (1) is unrestricted since we do not impose
B ( L)
0 . Stock and Watson (2005) found this restriction rejected in the FAVAR version of
their approximate dynamic factor model. In our application this restriction is rejected in most
specifications. Equation (2) describes the linear GARCH model, where H yt are i.i.d. 0-mean
random variables distributed with a Gaussian cdf, and V yt is the conditional standard
deviation.
We estimate FAVARs for each yt  {IPt , EM t , BRt , CRt } . Thus, as noted, the structure
of our forecasting set-up is similar to set-ups of individual forecasts of several
macroeconomic variables— with factors as predictors and each variable modeled as a
univariate process—first considered by Stock and Watson (2002), and subsequently by many
others, including more recently Stock and Watson (2012). However, our set-up differs from
previous set-ups in two important dimensions. First, in our models we specify a simple
dynamics for the volatility of the variables of interest, since we wish to assess strengths and
weaknesses of an otherwise standard factor model in capturing particular shapes of the
forecast distribution. Second, we use forward iterations of the FAVAR and the linear
GARCH process for each yt and its variance to obtain multi-period density forecasts. This is
in contrast to direct forecasts obtained as projections of the forecast variable at different
horizons, based on current information. This choice is motivated by the empirical results of
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Marcellino and Watson (2006) and Pesaran et al. (2011) on the potential superiority of
iterated forecasts over direct forecasts for first moments, and the results concerning the
superiority of iterated forecasts of Ghysels et al (2009) for second moments.
Let yˆt h and Vˆ t h denote the mean and volatility forecasts of y at horizon h t 1. The
VaR of yt h at probability level D  (0,1) is the quantile forecast given by:
VaRD ( yt h )
QD ( yt h )
yˆ t h Vˆ t h F 1 (D )
(3),
where F 1 (D ) is the inverse Gaussian cdf .
We estimate two FAVAR models for each variable. In both, the conditional variance
of each variable follows a linear GARCH(1,1) process, but the number of factors differs. The
model termed FAVAR(x) is a factor model with the optimal number of factors selected
according to the eigenvalue ratio tests introduced by Ahn and Horestein (2013), who show
that their proposed tests improve on the more commonly used tests based on Bai and Ng
(2002) criterion. The model termed FAVAR5 is a factor model with five factors as in Stock
and
Watson
(2012).
As
benchmarks,
we
also
obtain
forecasts
for
each
yt  ( IPt , EM t , BRt , CRt ) using a simple auto-regressive model where the conditional variance
follows a linear GARCH(1,1) process, termed AR,
and a VAR that includes all four
variables, with the conditional variance of each variable follows a linear GARCH(1,1)
process, called VAR.
As detailed below, the lags of these models are all determined
according to the SBIC criterion.
For each model, we construct density forecasts at three horizons, indexed by h , with
h 3 (three months ahead), h 6 (six months ahead), and h 12 (12 months ahead). Since
11
IP and EM, are expressed in percent changes and first differences respectively, the multiperiod mean forecasts at horizon h are given by yˆt h
h
¦ yˆ
. Following Ghysels et al
t i
i 1
(2009) and Andersen et al (2010), the multi-period volatility forecasts at horizon h are
proxied by the expected quadratic variation, given by Vˆ t h
h
¦ Vˆ
t i
where each term of this
i 1
sum is the forward iteration of the linear GARCH equation. The variables BR and CR are in
levels, so that the relevant forecasts are just the iterated forecast values h periods ahead.
\
B.
Quantile models
For each variable yt  ( IPt , EM t , BRt , CRt ) , we estimate quantile regressions of the
following form:
yt h (D )
a(D ) b(D ) ft H t h
(4)
In Equation (4), estimated factors ft are the predictors of yt h (D ) at each forecasting
horizon (h=3,6, and 12) for quantiles D
0.05 and D
0.10 . The VaR of yt h at probability
level D  (0,1) is the quantile forecast given by:
VaRD ( yt h )
aˆ (D ) bˆ(D ) f t
(5),
where the “hat” denotes the estimated parameters of the quantile regressions (4).
Note that in this case, forecasts are direct rather than iterated, as in the case of factor
models. We estimate five quantile models: the first one has the first PCA factor as predictor,
the second one the first and the second PCA factors as predictors, and so on, up to the
12
maximum number of factors identified by the Ahn and Horenstein (2013) criterion, which
turned out to be five.
C. Forecast evaluation
We compare the accuracy of density and quantile forecasts of different models using
a scoring rule, consistently with our focus on assessing the ability of different models to
deliver forecasts of systemic risk indicators useful as early warning signals.3 As a scoring
rule, we use the Quantile-Weighted Probability Score (QWPS) proposed by Gneiting and
Ranjan (2011), which allows us to evaluate the accuracy of density forecasts not only when
we assign equal weights to each quantile, but also with reference to particular regions of a
distribution, such as its tails.
Denote with f a density forecast, with y a realization of the forecast variable, with
F the cdf corresponding to the density f , and with F 1 (D ) the predicted quantile at level
D  (0,1) . The (continuous) quantile-weighted probability score QWPS ( f , y ) is defined by:
1
QWPS ( f , y )
³ QS ( F
1
(D ), y, D )w(D ) dD
(6),
0
where
QS ( F 1 (D ), y, D ) { 2( I { y d F 1 (D )} D )( F 1 (D ) y ) (7)
is the quantile score, I {.} is an indicator function, and w(D ) is a non negative weighting
function on the unit interval. This score has negative orientation, with lower values
indicating better performance.
3
On the use of appropriate scoring rules for statistical comparisons of density forecasts, see Gneiting et al,
(2007).
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Gneiting and Ranjan (2011) show that (6) is a proper weighted scoring rule.4 They
also show that the quantile prediction F 1 (D ) is optimal when the ex-post loss is
L ( x, y , D )
2(1 D ) | y x | in case of an over-prediction ( x t y ), and L( x, y , D )
2D | y x |
in case of a under-prediction ( x d y ).
When w(D ) 1 for all D  (0,1) , each quantile is assigned equal weight. We are also
interested in evaluating the performance of the density forecast over the tails, and in
particular over the left tail, since our systemic risk indicators are all formulated so that higher
risk is materializing in the left tail. As suggested by Gneiting and Ranjan (2011), we
construct weighted scores that place (symmetric) heavier weights on the left tail, setting
w(D ) (1 D ) 2 , as well as on the right tail, setting w(D ) D 2 . In the empirical
implementation, we compute a discretized version of the QWPS using a grid of 100
quantiles.
To evaluate quantile forecasts at 5% and 10% probability, we use two metrics. First,
the QWPS is collapsed on the particular quantile of interest. In other words, we use the
quantile score defined in (7). Second, we compare standard prediction coverage ratios
associated with each model and variable. These coverage ratios are the percentages of events
where the realized value of a variable is lower than the predicted VaR of that variable. Given
4
A density forecast f of Y is best if the conditional sampling density of Y is just f . Therefore, given a loss
function S ( f , y ) , where y is the future realized value, S ( f , y ) is a proper scoring rule if for all density
forecasts f and g ,
E f S ( f , y)
³ f ( y)S ( f , y)dy d ³ f ( y) S ( g , y)dy
E f S ( g , y) .
14
a chosen probability level (in our case, D
0.05 or D
0.10 ), the finding of a coverage
ratio higher then this probability level would indicate that the relevant forecast
underestimates systemic risk, since it does not capture all adverse realizations of risk at that
given probability level. In such a case early warning signals delivering coverage ratios
significantly higher than the targeted probability level would be unable to predict a fraction
of adverse outcomes. Conversely, the finding of a coverage ratio lower then this probability
level would indicate that the relevant forecast overestimates systemic risk, issuing a
percentage of signals that are “false alarms”.
Following Amisano and Giannini (2007) and Gneiting and Ranjan (2011),
comparisons of the predictive power of density and quantile forecasts are carried out
applying adapted Diebold and Mariano (1995) tests of equal performance of two different
density forecasts. 5
T w n , where w is the data window length, and n is the number of forecasts. At each date
t w, w 1,..., w n h , density forecasts fˆt h and gˆ t h obtained through two models are generated for
each actually observed yt h . The scoring rules associated with these two density forecasts are denoted by
S ( fˆ , y ) and S ( gˆ , y ) . The average scores are given by
5
Let
t h
S ( f , n)
t h
t h
( n h 1) 1
t h
w n h
¦
S ( fˆt h , yt h ) and S ( g , n)
(n h 1) 1
t w
Then, the test statistics is t ( n)
V (n) (n h 1) 1
¦
S ( gˆ t h , yt h ) respectively.
t w
n ( S ( f , n) S ( g , n)) / V ( n) , where
k 1
w n h | j |
j ( k 1)
t w
¦
w n h
¦
't ,k 't | j|,k , and ' t , k
S ( fˆt h , yt h ) S ( gˆ t h , yt h ) . Under
standard regularity conditions, the t (n) statistics is asymptotically standard normal under the null of equality of
differences in scores. Given the negative orientation of the above scoring rules, f is preferred to g if
t ( n) 0 , and vice versa.
15
IV. IMPLEMENTATION
A. Data
We use monthly time series of the U.S. economy taken from Datastream and national
sources for the period 1974:1-2013:12, as detailed in the Appendix.We selected number and
type of series from which to extract factors trying to attain a balance between maximizing
information content and minimizing noise, as suggested by Boivin and Ng (2006).
Factors were extracted from series classified in four groups: (1) quantity indicators
for the real sector, such as capacity utilization and survey expectations; (2) price indicators
for the real sector, such as consumer and producer price inflation and indicators of labor
costs; (3) quantity indicators of the financial sector, such as variables related to credit
conditions, 6 bank lending and monetary aggregates; and (4) asset price indicators, such as
equity valuations and funding costs. All series were transformed to ensure their stationarity.
B. Estimation and forecasting
Estimation and forecasting was conducted to simulate real time forecasting, To
account for time variation in parameters and possible structural breaks, we use a rollingwindow forecasting scheme for each of the models described earlier. The length of the
rolling window is set at 120 months. Parameters, factors, and optimal lags according to the
SBIC criterion are re-estimated for each estimation window.
6
In this group we include the Financial Condition Index (FCI) produced by the FED Chicago. FCIs have been
produced extensively since the onset of the 2007-2008 financial crisis (see e.g. Hatzius et al. , 2010), As
documented in Aramonte et al.,( 2013), the existing evidence on whether FCI are useful as coincident indicators
or as early warning signals is mixed
16
Specifically, let T0 denote the starting date of the sample, and W the window length.
The first forecasting date is T0 W , where we generate multi-period density and quantile
forecasts at dates T0 W 3 , T0 W 6 , and T0 W 12 using data in [T0 , T0 W ] , and
forward iterations for the FAVAR and the GARCH(1,1) processes. The next forecasting date
is
T0 W 1 , where multi-period density forecasts are obtained using data in
[T0 1, T0 W 1] . At each forecasting date factors are re-estimated, and the lags of the
FAVARs and quantile models for each variable are re-selected according to the SBIC
criterion. This process is repeated until the last available date for the 12 months-ahead
forecast. Thus, the first estimation period is 1974:1-1983:12. The number of forecast dates is
357 at a 3 months horizon, 354 at a 6 months horizon, and 351 at a 12 month horizon.
C. Results
We illustrate first the results for the AR, the VAR, the two FAVAR models, together
with the EWP of these models, and then we detail the results obtained with quantile
regressions.
FAVAR Models
Table 2 reports QWPSs and simple statistics of forecast directional accuracy for each
variable and forecast horizon. Four results stand out. First, the QWPS for the factor models
exhibit more accurate density forecasts than those obtained from the AR and VAR models.
This is true for all variables and forecasting horizons under equal quantile weights (uniform),
as well as for right and left tail quantile weights, indicating that the use of factors as
17
predictors improves density forecasts significantly. Second, for all variables and all
forecasting horizons, the difference between the forecasts of the factor models with an
optimal number of factors and the maximum of 5 does not yield significant differences in
QWPSs. In our case, choosing the number of factors according to some statistical criterion
aimed at reducing their number does not seem to matter for forecasting purposes. Third, the
EWP density forecast beats all other forecasts in terms of overall and tail QWPSs, for any
variable and almost all horizons, consistent with the evidence reported in Geweke and
Amisano (2012). Fourth, all models deliver density forecasts with significant directional
predictive ability, with the percent correct signs greater than 50 percent for all variables at all
horizons. Yet, while directional predictive accuracy is relatively high for the real variables IP
and EM, it is fairly low for the financial variables BR and CR.
Most importantly for our purposes, a result common to all FAVAR models is that
density forecasts at all horizons for IP and EM are better in predicting the right tail (positive
changes) rather than the left tail (negative changes): this is shown by the lower value of the
left tail QWPS relative to the right tail QWPS, as well as the lower fraction of predicted
negative changes over positive changes. This suggests the presence of asymmetries that are
not adequately captured by a specification based on the symmetric Gaussian assumption. By
contrast, left and right tails QWPSs and directional forecast statistics of the financial
variables BR and CR are approximately the same for positive and negative changes, but
nonetheless are fairly low.
The inability of these models in providing reliable early warning signals is starkly
illustrated in Table 3, which shows coverage ratios at 5% and 10% probability levels of the
EWP, which is the best model according to the left tail QWPS. Statistics are reported for all
18
variables and forecasting horizons, as well as for the whole sample and the sample starting
from the beginning of the financial crisis in 2007. With the exception of VaR forecasts for
IP at the three month horizon, all coverage ratios for each variable and forecast horizon
significantly exceed the target coverage of 5% and 10% , and they often do so by a large
magnitude. In addition, VaR forecasts become significantly worse during the last years of the
sample period that includes the recent financial crisis. Thus, these models do not provide
reliable early warning signals for systemic risks. The decline in reliability of these forecasts
during a period that includes a severe crisis further shows that the early warning signals
issued by these models become even less reliable when reliability is needed most.
Quantile models
An entirely different picture emerges from the forecasting results of the quantile
models. Table 4 shows coverage ratios at 5% and 10% in a format identical to that of Table
3 for the quantile model with five factors, denoted QR5, whose forecasting accuracy is best
among the five quantile models considered.
Overall, the improvement in the precision of VaR forecasts of QR5 relative to the
EWP model is quite dramatic. IP VaR forecasts indicate a coverage even lower than the
target coverage for horizons shorter than 12 months. For all other variables, coverage is still
higher than the target coverage, but differences are relatively small and in many instances not
statistically significant. Furthermore, tail forecasts during the sample period that includes the
recent financial crises do not differ significantly from those of earlier periods, indicating that
the early warning signals issued by QR5 are robust to severe adverse developments in the
economy. These results are further illustrated in Table 5, which shows quantile scores at 5%
19
and 10% for the EWP and the QR5 model for each variable and each forecast horizon.
Without exception, that is for all variables, forecast horizons and subsamples, quantile scores
are significantly better for the QR5 model at standard significance levels
The reliability of the forecasts of the systemic risk measures obtained with QR5 is
further illustrated by Figures 1-4, which depict actual and predicted values of the systemic
risk indicators for each variable and forecasting horizon. For all variables and the three and
six month horizon, systemic risk forecasts predict significant declines in each of the
indicators considered fairly accurately. At the 12 month horizon, the accuracy of the
forecasts declines, but the early warning signals issued by the model are still useful, since the
decline of a variable still occurs within the forecasting period. In other words, a prediction of
a decline of an indicator h periods ahead signaled by a decline in the VaR of this indicator is
a useful early warning signal as long as the decline in actual values occurs within the
forecast horizon.
We conclude that quantile models such as QR5 provide reliable early warning signals
for systemic risks at appropriate forecasting horizons. A striking feature of this result is that a
model of relative simplicity and ease of implementation beats more complicated models in
the dimension of tail risk prediction by a large margin. This likely occurs because the
conditional quantile model captures parsimoniously and in a timely fashion those changes in
those portions of the distribution of a variable that may anticipate significant changes in the
left tail associated with an increasing probability of systemic risks’ realizations.
20
V. CONCLUSIONS
Building on our previous efforts, in this paper we developed an early warning system
for systemic risks, testing the ability of several forecasting models to issue reliable early
warning signals. Our key positive result concerning the ability of factor-augmented
predictive quantile models to deliver such reliable early warning signals for different
measures of systemic real and financial risks is encouraging, and motivates several
potentially useful extensions of our EWS.
Fruitful directions to improve and an extend our modeling framework may include
setting up EWSs for different countries or sets of country in a region, using more
disaggregated data in modeling and factor extraction, and identifying the economic drivers of
shifts in the probability distribution of systemic risks: these worth-pursuing efforts are
already part of our research agenda.
21
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24
Table 1
Descriptive statistics and correlations
Variable
Obs
Mean
Std. Dev.
Min
Max
IP
EM
BR
CR
480
480
480
480
0.166
0.107
0.082
0.0831
0.738
0.275
0.045
0.0323
-4.299
-0.852
0.008
0.0142
2.068
1.502
0.316
0.2414
Correlations
IP
EM
BR
CR
Significant at 5% confidence level*
IP
EM
BR
CR
1
0.3817*
0.1370*
0.1717*
1
0.2340*
0.1452*
1
0.5919*
1
25
Table 2
Quantile Weighted Probability Scores (QWPS) and directional forecast accuracy
Panel A: IP
Model
AR
VAR
RMSE
1.24
1.28
QWPS (uniform)
QWPS (right tail)
QWPS (left tail)
32.65
9.66
10.55
33.99
9.71
11.25
32.54
9.47
10.53
% correct sign forecast
correct + signs/total + signs (%)
correct - signs/total - signs (%)
80.4
83.2
51.52
75.3
81.5
29.55
79.6
85.2
48.21
RMSE
2.36
2.34
QWPS (uniform)
QWPS (right tail)
QWPS (left tail)
61.1
17.8
20.3
61.9
17.2
21.1
62.2
17.8
20.8
% correct sign forecast
correct + signs/total + signs (%)
correct - signs/total - signs (%)
81.8
84.5
41.7
77.5
83.4
22.2
81.2
86.3
43.2
RMSE
4.42
4.21
QWPS (uniform)
QWPS (right tail)
QWPS (left tail)
112.3
32.5
38.2
112.5
31.9
38.4
118.1
34.0
40.0
% correct sign forecast
correct + signs/total + signs (%)
correct - signs/total - signs (%)
83.6
85.8
28.6
78.6
85.0
12.1
82.8
86.9
34.5
Panel B: EM
FAVAR(x) FAVAR(5)
EWP
AR
VAR
1.15
0.47
0.45
32.55
9.52
10.58
30.63
8.93
9.89
13.19
3.78
4.44
12.73
3.76
4.12
11.82
3.65
3.73
11.45
3.55
3.56
11.38
3.39
3.60
79.4
85.1
47.37
81.5
84.0
57.14
78.6
80.2
50.0
77.7
80.5
44.8
81.2
81.8
72.7
82.6
84.5
67.4
80.2
81.6
61.5
B. 6 month horizon
2.25
2.23
2.19
0.78
0.74
B. 6 month horizon
0.65
0.64
0.67
61.8
17.6
20.9
58.0
16.3
19.5
21.51
6.03
7.43
20.69
5.89
7.01
18.31
5.51
5.91
18.27
5.52
5.88
18.14
5.23
5.97
82.0
86.2
46.2
82.6
85.3
48.1
82.6
84.5
47.4
81.2
84.6
39.3
85.3
85.9
73.7
85.5
87.6
64.7
84.5
85.6
65.0
C. 12 month horizon
4.36
4.27
4.16
1.36
1.31
C. 12 month horizon
1.18
1.17
1.21
119.6
33.6
41.6
110.1
31.3
37.5
36.22
9.35
13.40
35.76
9.57
12.82
32.29
9.03
11.23
32.22
9.20
10.97
31.51
8.41
11.20
83.4
87.2
37.9
84.5
86.1
38.5
84.7
86.4
38.5
82.8
86.5
29.2
86.9
87.5
69.2
87.7
88.9
68.2
86.3
87.2
61.5
A. 3 month horizon
1.18
1.17
FAVAR(x)
FAVAR(5)
A. 3 month horizon
0.40
0.39
EWP
0.41
26
Table 2 (cont.)
Quantile Weighted Probability Scores (QWPS) and directional forecast accuracy
Panel A: BR
Model
AR
VAR
FAVAR(x)
RMSE
0.03
0.03
QWPS (uniform)
QWPS (right tail)
QWPS (left tail)
0.95
0.29
0.31
0.92
0.29
0.29
1.00
0.31
0.32
% correct sign forecast
correct + signs/total + signs (%)
correct - signs/total - signs (%)
62.7
61.7
65.1
63.5
62.2
67.0
62.7
61.5
66.0
RMSE
0.03
0.03
QWPS (uniform)
QWPS (right tail)
QWPS (left tail)
1.04
0.32
0.33
1.01
0.31
0.33
1.07
0.32
0.35
% correct sign forecast
correct + signs/total + signs (%)
correct - signs/total - signs (%)
61.1
59.4
65.4
61.9
59.9
67.3
61.4
59.5
66.3
RMSE
0.04
0.04
QWPS (uniform)
QWPS (right tail)
QWPS (left tail)
1.1
0.3
0.4
1.1
0.3
0.4
1.1
0.3
0.4
% correct sign forecast
correct + signs/total + signs (%)
correct - signs/total - signs (%)
64.6
61.7
72.0
64.6
61.6
72.4
64.9
61.8
72.6
Panel B: CR
FAVAR(5)
EWP
AR
VAR
0.03
0.03
0.03
0.93
0.28
0.30
0.91
0.29
0.27
0.90
0.29
0.26
0.97
0.33
0.28
0.97
0.33
0.28
0.99
0.33
0.29
0.91
0.30
0.26
62.7
61.5
66.0
62.7
61.5
66.0
67.0
68.1
65.6
67.3
69.3
64.9
65.1
67.5
62.5
64.9
66.2
63.2
66.8
68.1
65.1
B. 6 month horizon
0.03
0.03
0.03
0.03
0.04
B. 6 month horizon
0.04
0.04
0.04
1.01
0.31
0.32
1.00
0.31
0.30
0.99
0.33
0.29
1.06
0.36
0.30
1.02
0.33
0.30
1.02
0.34
0.29
0.98
0.32
0.28
62.2
60.1
67.6
61.7
59.6
67.0
65.4
64.2
67.1
64.6
64.0
65.3
63.5
63.2
64.0
64.6
63.5
66.0
64.6
63.6
65.9
C. 12 month horizon
0.04
0.04
0.04
0.04
0.04
C. 12 month horizon
0.04
0.04
0.04
1.1
0.3
0.4
1.1
0.3
0.3
1.06
0.35
0.31
1.12
0.38
0.31
1.06
0.35
0.30
1.07
0.36
0.30
1.04
0.34
0.29
64.9
61.7
73.1
64.6
61.6
72.4
70.2
71.8
68.3
70.2
72.1
68.0
69.2
71.5
66.5
69.4
70.8
67.7
70.0
71.5
68.1
A. 3 month horizon
0.03
0.03
FAVAR(x) FAVAR(5)
A. 3 month horizon
0.03
0.03
EWP
0.03
27
Table 3
Coverage ratios of the Equally Weighted Pool (EWP) forecasts
of the FAVARs, AR and VAR Models
Sample period
IP
EM
BR
CR
Coverage 5%
0.08
0.08
A. 3 month horizon
0.08
0.20
0.18
0.30
0.14
0.20
1983:1-2013:12
2007:1-2013:12
0.13
0.17
B. 6 month horizon
0.14
0.24
0.24
0.36
0.17
0.25
1983:1-2013:12
2007:1-2013:12
0.16
0.20
C. 12 month horizon
0.18
0.28
0.30
0.44
0.21
0.31
1983:1-2013:12
2007:1-2013:12
Coverage 10%
1983:1-2013:12
2007:1-2013:12
1983:1-2013:12
2007:1-2013:12
1983:1-2013:12
2007:1-2013:12
0.12
0.10
A. 3 month horizon
0.13
0.27
0.25
0.36
0.18
0.23
0.17
0.18
B. 6 month horizon
0.17
0.29
0.31
0.40
0.21
0.29
0.20
0.21
C. 12 month horizon
0.21
0.33
0.35
0.49
0.25
0.31
28
Table 4
Coverage ratios of forecasts
of the Quantile Model with 5 factors (QR5)
IP
EM
BR
CR
0.02
0.02
A. 3 month horizon
0.05
0.07
0.06
0.04
0.06
0.04
0.06
0.06
B. 6 month horizon
0.06
0.08
0.08
0.07
0.08
0.07
0.08
0.08
C. 12 month horizon
0.09
0.13
0.07
0.07
0.07
0.07
0.13
0.12
0.13
0.12
Sample period
Coverage 5%
1983:1-2013:12
2007:1-2013:12
1983:1-2013:12
2007:1-2013:12
1983:1-2013:12
2007:1-2013:12
Coverage 10%
1983:1-2013:12
2007:1-2013:12
0.08
0.06
A. 3 month horizon
0.09
0.12
1983:1-2013:12
2007:1-2013:12
0.11
0.10
B. 6 month horizon
0.12
0.17
0.15
0.17
0.15
0.17
0.12
0.13
C. 12 month horizon
0.15
0.18
0.15
0.14
0.15
0.14
1983:1-2013:12
2007:1-2013:12
29
Table 5
Quantile scores of the EWP and the QR5 models
Model:
Sample period
EWP
IP
EM
QR5
BR
CR
IP
Quantile Scores 5%
EM
BR
CR
Quantile Scores 5%
1983:1-2013:12
2007:1-2013:12
A. 3 month horizon
0.1595712 0.0550693 0.0043826 0.0030342
0.2858559 0.1058641 0.0037042 0.0032248
A. 3 month horizon
0.1033566 0.0347299 0.0020396 0.0025904
0.1657562
0.034521 0.0011194 0.0020188
1983:1-2013:12
2007:1-2013:12
B. 6 month horizon
0.362737 0.1025715 0.0054362 0.0030053
0.7408015 0.2306563 0.0051772 0.0026635
B. 6 month horizon
0.1897938
5.50E-03
0.002187 0.0023346
0.3410011 0.0806617 0.0014027 0.0022837
1983:1-2013:12
2007:1-2013:12
C. 12 month horizon
0.7932504 0.2339903 0.0060732 0.003404
1.833009 0.5533323 0.0071991 0.0035644
C. 12 month horizon
0.3415327 0.0992656 0.0020159 0.0023789
0.6202426 0.1452739 0.0017146 0.0023942
Quantile Scores 10%
Quantile Scores 10%
1983:1-2013:12
2007:1-2013:12
A. 3 month horizon
0.2325078 0.082028 0.0064937 0.0052402
0.3729137 0.1384008 0.0050739 0.0054733
A. 3 month horizon
0.1679747 0.0603777
0.003831 0.0046083
0.2562135
0.060133 0.0024217 0.003629
1983:1-2013:12
2007:1-2013:12
0.4822449
0.893473
B. 6 month horizon
0.1461685
7.83E-03 0.0053401
0.2793333 0.0074855 0.0049827
0.3478128
0.5954645
B. 6 month horizon
0.0923721 0.0041145 0.0042568
0.1309304 0.0029111 0.0037267
1983:1-2013:12
2007:1-2013:12
0.971565
2.053451
C. 12 month horizon
0.301738 0.0084436 0.0058963
0.6277482 0.0093591 0.0058352
0.6072175
1.061803
C. 12 month horizon
0.1754266 0.0040291 0.0044733
0.2749356 0.0034175 0.0043155
30
Figure 1
-10
-5
0
5
IP and IP-at-Risk forecast at 5% probability (model QR5)
3 month horizon
1985m1
1990m1
1995m1
2000m1
time
ip3
2005m1
2010m1
2015m1
2010m1
2015m1
2010m1
2015m1
ipvar3q55
-30
-20
-10
0
10
6 month horizon
1985m1
1990m1
1995m1
2000m1
time
ip6
2005m1
ipvar6q55
-40
-30
-20
-10
0
10
12 month horizon
1985m1
1990m1
1995m1
ip12
2000m1
time
2005m1
ipvar12q55
31
Figure 2
EM and Employment-at-Risk forecast at 5% probability (model QR5)
-2
-1
0
1
2
3 month horizon
1985m1
1990m1
1995m1
2000m1
time
em3
2005m1
2010m1
2015m1
2010m1
2015m1
2010m1
2015m1
emvar3q55
-8
-6
-4
-2
0
2
6 month horizon
1985m1
1990m1
1995m1
2000m1
time
em6
2005m1
emvar6q55
-10
-5
0
5
12 month horizon
1985m1
1990m1
1995m1
em12
2000m1
time
2005m1
emvar12q55
32
Figure 3
BR and Banking Sector-at Risk forecast at 5% probability (model QR5)
0
.05
.1
.15
.2
3 month horizon
1985m1
1990m1
1995m1
2000m1
time
br3
2005m1
2010m1
2015m1
2010m1
2015m1
2010m1
2015m1
brvar3q55
-.05
0
.05
.1
.15
.2
6 month horizon
1985m1
1990m1
1995m1
2000m1
time
br6
2005m1
brvar6q55
0
.05
.1
.15
.2
12 month horizon
1985m1
1990m1
1995m1
br12
2000m1
time
2005m1
brvar12q55
33
Figure 4
CR and Corporate Sector-at-Risk forecast at 5% probability (model QR5)
0
.0
5
.1
.1
5
.2
.2
5
3 month horizon
1985m1
1990m1
1995m1
2000m1
time
fr3
2005m1
2010m1
2015m1
2010m1
2015m1
2010m1
2015m1
frvar3q55
0
.05
.1
.15
.2
.25
6 month horizon
1985m1
1990m1
1995m1
2000m1
time
fr6
2005m1
frvar6q55
0
.05
.1
.15
.2
.25
12 month horizon
1985m1
1990m1
1995m1
fr12
2000m1
time
2005m1
frvar12q55
34
APPENDIX
List of Variables
The frequency of all series is monthly. The data range of all variables described below is
1974:1-2013:12.
Forecast series:
DS identifier
US INDUSTRIAL PRODUCTION - TOTAL INDEX
US UNEMPLOYMENT RATE SADJ
USIPTOT.G
USUNTOT
DI: inverse of equity volatility obtained from Datastream bank index
DI: inverse of equity volatility obtained from Datastream non-financial s index
USBANKTOT
USNFTOT
Series used to contract PC factors
Group 1
Code
US THE CONFERENCE BOARD LEADING ECONOMIC INDICATORS INDEX SADJ
USCYLEADQ
US EXISTING HOME SALES: SINGLE-FAMILY & CONDO (AR) VOLA
USEXHOMEO
US PERSONAL CONSUMPTION EXPENDITURES (AR) CURA
USPERCONB
US EMPLOYED - NONFARM INDUSTRIES TOTAL (PAYROLL SURVEY) VOLA
USEMPALLO
US ISM PURCHASING MANAGERS INDEX (MFG SURVEY) SADJ
USCNFBUSQ
US CHICAGO PURCHASING MANAGER BUSINESS BAROMETER (SA) SADJ
USPMCHBB
US PHILADELPHIA FED OUTLOOK SURVEY-DIFFUSION INDEX,MFG. SADJ
USFRBPIM
US NEW PRIVATE HOUSING UNITS AUTHORIZED BY BLDG.PERMIT (AR) VOLA
USHOUSATE
US MFG - RATE OF CAPACITY UTILISATION SADJ
USMBS076Q
US CONSUMER - CONFIDENCE INDICATOR SADJ
USMCS002Q
US EMPLOYMENT RATE, ALL PERSONS (AGES 15 AND OVER) SADJ
USMLRT12Q
US CLI NET NEW ORDERS - DURABLE GOODS CURA
USOL2066D
US TOTAL RETAIL TRADE (VOLUME) VOLA
USOSLI15G
US AVG WKLY HOURS - TOTAL PRIVATE NONFARM VOLA
USHKIP..O
Group 2
add oil and raw materials?
US CPI - ALL ITEMS LESS FOOD & ENERGY (CORE) SADJ
USCPCOREE
US TERMS OF TRADE REBASED TO 1975=100 NADJ
USTOTPRCF
US CPI ALL ITEMS SADJ
USOCP009E
US REAL EFFECTIVE EXCHANGE RATES - CPI BASED VOLN
USOCC011
US HOURLY EARN: MFG SADJ
USOLC007E
US TOTAL PPI ENERGY NADJ
USOPIEN1F
US TOTAL PPI CONSUMER GOODS NADJ
USOPICG1F
US TOTAL PPI DURABLE CONSUMER GOODS NADJ
USOPICD1F
US TOTAL PPI INVESTMENT GOODS NADJ
USOPIVG1F
US TOTAL PPI NON DURABLE CONSUMER GOODS NADJ
USOPICN1F
Group 3
35
US MONETARY BASE CURN
USM0....A
US MONEY SUPPLY M1 CURN
USM1....A
US MONEY SUPPLY M2 (BCI 106) CONA
USM2....D
US CONSUMER CREDIT OUTSTANDING CURA
USCRDCONB
US COMMERCIAL BANK ASSETS - LOANS & LEASES IN BANK CREDIT CURA
USBANKLPB
US COMMERCIAL BANK ASSETS - COMMERCIAL & INDUSTRIAL LOANS CURA
USBCACI.B
US BROAD MONEY (M3) SADJ
USOMA001G
Group 4
US NATIONAL ASSOCIATION OF HOME BUILDERS HOUSING MARKET INDEX
USNAHBMI
US TRADE-WEIGHTED VALUE OF US DOLLAR AGAINST MAJOR CURRENCIES
USXTW..NF
US FEDERAL FUNDS RATE (MONTHLY AVERAGE)
USFDFUND
US TREASURY BILL RATE - 3 MONTH (EP)
USGBILL3
US INTERBANK RATE - 3 MONTH (LONDON) (MONTH AVG)
USINTER3
US TREASURY YIELD ADJUSTED TO CONSTANT MATURITY - 20 YEAR
USGBOND.
US RATE 3-MONTH EURO-DOLLAR DEPOSITS NADJ
USOIR075R
US OVERNIGHT INTERBANK ('FEDERAL FUNDS') RATE NADJ
USMIR060R
US PRIME RATE NADJ
USOIR039
US YIELD 10-YEAR FED GVT SECS NADJ
USOIR080R
US YIELD >10-YEAR FED GVT SECS (COMPOSITE) NADJ
USOIR061R
FCI: Chicago Fed’s National Financial Conditions Index (NFCI)