Economic Network Model: Forecast of Cascading Shocks
Team 8: Robert Feigenberg, Jungsun Kim, Carolina Lee, Daniela Savoia
Model Determination
Objectives
The United States economy can be modeled as
an interconnected network made up of various
sectors. The connections between sectors
determine the network structure and the
influence levels of each sector on the rest of the
economy. The sector relationships are
determined by the use of one sector’s output as
another sector’s input.
Network Diagram of the U.S. Economy
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Our approach for forecasting the shock effects in
an economy is to determine the network
structure and connections between sectors. A
mathematical model is used to measure the
random noise in the economy and differentiate it
from a larger shock in the economy. Using this
model, the effects of an individual shock are
forecasted and simulated over time.
The model’s accuracy is verified with data on past
shocks in the US economy. The model visually
displays the network structure in a diagram and a
table, highlighting the central and most
influential sectors in the economy. The forecasted
effects of a shock are simulated over time and
demonstrated graphically. The final product
design is an application that allows users to input
the origin and magnitude of a shock and analyze
its forecasted effects throughout the economy.
ADVISORS
Dr. Alejandro Ribeiro
Dr. Ali Jadbabaie
DEMO TIMES
Thursday, April 21, 2011
9:00-9:30 AM | 2:30-4:00 PM
Special Thanks To
Dr. Ken Laker
Dr. Peter Scott
Dr. Raymond Watrous
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Figure 1. U.S. Economy at All Levels of Connectivity
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Figure 2. U.S. Economy with Input-Output Proportion > 10%
• Figure 1 shows that all 61 sectors of the U.S. economy are highly interconnected. Due to such intricate relationship among the
sectors, economic shock in one sector propagates through other sectors.
• Figure 2 highlights sectors with higher degree of connectivity. The cascading effect is greater when the shocks occur in moreconnected sectors.
Mathematical Representation of the Model
V = Value Added Matrix [61 x 61]
•Average μ was calculated using a sample mean of the observed ε
• The sample covariance matrix Σ was calculated using the following equation where Σ is the sample mean of Σ(t)
Σ(t) = X(t+1) / V*A*X(t)
• There is a correlation between the terms in ε making a linear eigenvalue decomposition approach necessary
• The eigenvalues were computed and summed to determine the principal values of the system
• As 10 eigenvalues represent 99% of the system’s randomness, the U.S. economy has 10 degrees of freedom
Σ = UDUT
where U = Matrix with eigenvectors as columns [61 x 10]
D = Diagonal matrix with eigenvalues [10 x 10]
• It is possible to model ε By using the decomposition of Σ,
ε = μ + Σ1/2 δ = μ + UD1/2 δ
where δ ~ N(0,1)
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When a shock, or a large change in expected
output, occurs in a given sector, the effect of the
disturbance is experienced throughout the entire
economy. It is a useful tool to forecast the
magnitude and spread of the effects of a shock
over time.
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ε ~ N(μ, Σ)
Model Validation
To validate our model, we further observed the principle components of our empirical
covariance matrix Σ
• An out-of-sample Average μ was calculated using a sample mean of the observed ε
• Principal component analysis was applied to generate a matrix out-of-sample data
that should approach a normal:
(Σ-1/2 )T*(ε(t) – μ)≈ N(0,1)
• The result: 7 of our 10 principal components passed the following Chi-Squared test of
our data versus a standard normal:
H0: The sampled data comes from our specified distribution
and X2= ∑ (Ni-npi)2/npi
• Figure 5 is a normal probability plot, plotting our out-of-sample data vs. a standard
normal random variable. This visually confirms that the majority of the principal
component data follows a normal distribution:
Normal Probability Plot
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Probability
ABSTRACT
• Design a model to forecast change in output in the next period given knowledge of the historical level of output
• Forecast the effects of large initial disturbances
• Implement and validate the model based on historical data
• Create an interactive user-friendly Graphical User Interface (GUI) where a user can input specific details about the shock of interest
and observe the forecasted effect across sectors
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Data
Simulation Results of Potential Economic Shocks
• Historical data is available from the Bureau of Economic
Analysis (BEA) from 1987 to 2009
• Data is measured annually
• Forecasted output after 2009 is derived from the model
ε(t) = Noise vector at time t [61 x 1]
A = Input-Output Matrix [61 x 61]
• Matrix that represents proportion of intermediate good of one
sector used as inputs as the other sectors
• Row sum is 1 for all sectors
• Calculated from Bureau of Economic Analysis (BEA) data from
1998 to 2008
• Example Input-Output Matrix for a simple economy:
• 70% of Sector 1’s output at time t
serves as an input to Sector 1 at time t+1
• 10% of Sector 2’s output at time t
Figure 3. Sample Inputserves as an input to Sector 2 at time t+1
• Modeled as a Colored (Non-White) Gaussian Distribution using
historical data
• ε ~ N(μ,Σ) where μ and Σ were calculated using observed data
• μ is the average vector [61 x 1]
• Σ is the covariance matrix [61 x 61]
• Probability distribution
function (pdf) of ε the farms
sector for 2010 without any
external shocks:
Figure 7: Real Estate Sector Output Over Time with
Upper/Lower Forecast Bounds and Expected GDP
Figure 8: Construction Sector Output Over Time with
Upper/Lower Forecast Bounds and Expected GDP
SCENARIO #1: Housing Bubble Bursts
SCENARIO #2: Spike in Construction Output
 Negative shock in real estate sector: -18%
 Most affected sectors through comparison of
expected GDP without shock: Real Estate, Retail
Trade, Wholesale Trade
 Positive shock in construction sector: +6%
 Most affected sectors through comparison of
expected GDP without shock: Real Estate, Utilities
Retail Trade
Output Matrix
System Diagram
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Figure 6: Normal Probability Plot
Figure 4. Probability distribution function
for farms sector in 2010
UNIVERSITY OF PENNSYLVANIA
SCHOOL AND ENGINEERING AND APPLIED SCIENCE
Department of Electrical and Systems Engineering
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X(t) = Output vector at time t [61 x 1]
• Stationary diagonal matrix that represents the proportion of
output from 61 sectors that are used as final goods
• Each entry was computed by subtracting intermediate goods
from the total output per sector
V(t) = X(t+1)/(A*X(t))
• V is the average of V(t) for the years 1987 to 2009
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X(t)
Delay
X(t+1)
ε(t)
Figure 5: Block Diagram
“Richter” Scale of Shock Magnitude
(In percent change of output)
6%: “British Petroleum Oil Spill”
18%: “Technology Bubble Burst”
30%: “Subprime Mortgage Crisis”
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