GENERATING MACROECONOMIC
SCENARIOS USING STATISTICAL
COPULAS
Enrique M. Quilis
Macroeconomic Research Department
DISCLAIMER
Any views expressed herein are my own and
INDEX
What is a macroeconomic scenario?
How do we generate a macroeconomic
scenario?
Application: simulating 2014 GDP growth. Application: simulating 2014 GDP growth.
Calibration.
Monte Carlo simulation.
Stress testing.
DEFINITION
What is a macroeconomic scenario?
A quantitative statement about the future rate of
growth of key macroeconomic aggregates.
Ideally, it should contain measures of the
uncertainty around those rates.
The individual scenarios can be plugged into
DEFINITION
Mean Std. Dev.
Households consumption C 0.2 0.4
Government consumption G -2.2 0.7
Fixed capital formation I -0.8 1.0
2014 Rate of growth Variable
Fixed capital formation I -0.8 1.0
GENERATION
How do you think macro scenarios come to live?
There are several methodologies that can be used in
a complementary way.
Here we propose the use of Monte Carlo simulation of Here we propose the use of Monte Carlo simulation of
a multivariate model.
The multivariate model is a Gaussian copula model
GENERATION: Marginals
Marginal pdfs (after
normalization):
) 1 ( x ) 1 ( x x 100 X − − − × =
Z=[C G I X M]
GENERATION: Copula function
C, G, I, X and M are not
independent, they covariate
model: Gaussian copula, key
parameter: correlation matrix =
parameter: correlation matrix =
R.
Origin of the model: valuation of
GENERATION: Monte Carlo simulation
1. Setting model parameters (calibration). 2. Simulating the multivariate standardized
gaussian distribution. Correlation matrix: R.
3. Deriving the implied uniforms linked to the 3. Deriving the implied uniforms linked to the
cdf (grades).
4. Computing the marginals according to their
univariate specification.
5. Generating GDP results: Y=wZ, w=weights
APPLICATION: 2014 GDP growth
Simulation of the GDP growth in 2014.
The simulation is based on consistent
scenarios for C, G, I, X and M. GDP will
be derived bottom-up.
The model is calibrated using the panel of
private forecasters compiled by FUNCAS
private forecasters compiled by FUNCAS
(November, 2013).
Numerical results are provided via Monte
Carlo simulation of the copula model.
Stress tests allow us to check the
APPLICATION: Calibration, FUNCAS panel
4 6 8
APPLICATION: Calibration
Calibration of the Beta parameters, after
suitable normalization on the simplex [0,1].
Min Max p q Mean Variance
Original panel Beta parameters Implied moments Normalized panel
Min Max p q Mean Variance
C -0.60 0.90 1.73 1.45 0.54 0.06
G -3.30 -0.70 1.24 1.60 0.44 0.06
I -2.50 1.70 1.36 2.10 0.39 0.05
X 4.50 7.10 1.03 1.13 0.48 0.08
APPLICATION: Calibration
The correlation matrix is calibrated using the
historical database of the REMS model.
We have used rates of growth from 1981 to
2012 (annual time series).
C G I X M
C 1.00 0.57 0.91 0.11 0.86
G 0.57 1.00 0.44 -0.28 0.37
I 0.91 0.44 1.00 0.11 0.88
X 0.11 -0.28 0.11 1.00 0.26
APPLICATION: Calibration
The GDP growth is derived as a weighted
average of the individual rates of growth.
The contributions are set according to their
weight on 2013 nominal GDP.
2011 2012 2013 2011 2012 2013
C 0.59 0.59 0.59
G 0.21 0.20 0.20
I 0.21 0.20 0.18
X 0.31 0.33 0.34
APPLICATION: Monte Carlo simulation
We assume that the contribution of the
changes in stocks to GDP growth is zero.
We run a Monte Carlo simulation of the
copula model (n=5000 draws).
Each simulation defines a consistent scenario
for the growth of C, G, I, X, M and, by aggregation, of Y (GDP).
Each scenario can be plugged into a system
APPLICATION: Results
5 -4 -2 0 G -1 0 1 C0 2 4
4 6 8
-5 0 5
-4 -2 0
-1 0 1
APPLICATION: Results
400 600 800 1000 1200
Mean
Std.
Deviation 2.5 25 50 75 97.5 0 0.7
0.73 0.34 0.08 0.49 0.74 0.96 1.39 0.01 0.47 Probability of growth lower than: Percentiles
-0.50 0 0.5 1 1.5 2
200
APPLICATION: Results
Mean Std. Dev.
Households consumption C 0.2 0.4
Government consumption G -2.2 0.7
Fixed capital formation I -0.8 1.0
2014 Rate of growth Variable
Fixed capital formation I -0.8 1.0
APPLICATION: STRESS TESTS
Stress test 1: Increased correlation between
C and X and between X and M.
Stress test 2: X evolves according to an
extreme value (EV) distribution (instead of a beta). The EV distribution is calibrated to
beta). The EV distribution is calibrated to yield only growth values lower than 5%.
Stress test 3: Combining previous stressed
APPLICATION: STRESS TESTS
Stress test 1: Increased (doubled)
correlation between C and X and between X and M. X is now more coupled to C and M.
Variable
Variable C G I X M
Variable C G I X M
Households consumption C 1.00 0.57 0.91 0.22 0.86
Government consumption G 0.57 1.00 0.44 -0.28 0.37 Gross capital formation I 0.91 0.44 1.00 0.11 0.88 Exports of goods and services X 0.22 -0.28 0.11 1.00 0.52
APPLICATION: STRESS TESTS
Stress test 2: X evolves according to an
extreme value (EV) distribution (instead of a beta). The EV distribution is calibrated to
yield only growth values lower than 5%.
This simulation tries to quantify the exposure
This simulation tries to quantify the exposure
of GDP to X’s tail risk.
Variable Threshold Location Scale
X 5% 0.1300 0.0441
APPLICATION: STRESS TESTS
1 1.5 G D P G ro w thBase (a) X: Increased correlation (b) X: tail risk (a) + (b)
APPLICATION: STRESS TESTS
Mean
Std.
Deviation 2.5 25 50 75 97.5 0 0.7
Probability of growth lower than: Percentiles
Simulation
Base 0.73 0.34 0.08 0.49 0.74 0.96 1.39 0.01 0.47
(a) Increased correlation 0.73 0.30 0.14 0.52 0.72 0.93 1.35 0.01 0.47
(b) Tail risk 0.40 0.28 -0.09 0.19 0.40 0.61 0.94 0.08 0.83
(a) and (b) 0.40 0.27 -0.09 0.19 0.40 0.61 0.92 0.07 0.84
MACRO SCENARIOS: CONCLUSIONS
The generation of macroeconomic scenarios
by means of the Monte Carlo simulation of a copula model suitably calibrated is explicit, objective and flexible, all of them atractive features to communicate and share with the final users of those scenarios.
The methodology can also encompass a wide
variety of stress tests that quantify its robustness to alternative hypothesis.
The results can be easily plugged in other
REFERENCES
Li, D. X. (2000) “On Default Correlation:
A Copula Function Approach”,
Journal of
Fixed Income,
vol. 9, p. 43—54.
Martin, R. (2004)
Credit Portfolio
Martin, R. (2004)
Credit Portfolio
Modeling Handbook
, Credit Suisse First
Boston.
Meucci, A., (2005)
Risk and Asset
THANKS FOR YOUR ATTENTION!!
Enrique M. Quilis
Macroeconomic Research Department
Ministry of Economy and Competitiveness. Spain.
