98 NV01 NV03 AZ04 CA08 CA31 CA36 NV04 99 NV02 N
6.1. Turning 8 matrices into 340 matrices: the theory of the relevant variable approach
We used our 4 models to compute 8 matrices: EA(,FL24, ), EA(,AZ07, ), EA(,NY14,v)
and EA(,WA09, ) for both assumptions A (Keynesian and Neoclassical).21 This was a large
but manageable computational task requiring 4480 annual solutions (4 models times 20 years times 14 shocks times 2 runs times 2 assumptions), repeated several times to incorporate refinements following analysis of preliminary results. How should we use these 8 matrices to develop Keynesian and Neoclassical matrices for all 170congressional districts?
One idea that can be quickly dismissed is that we should use the same Keynesian and Neoclassical matrices for each congressional district, some sort of average of matrices
21 E
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obtained from the USAGE-TERM simulations for the 4 models. However it is clear that the matrices should vary across congressional districts. For example, the effect on GDP of destruction of x per cent of the capital in a congressional district depends on the quantity of capital in that congressional district: the effect will be greater for districts that have a lot of capital than for districts that have only a small amount of capital. We would expect
destruction of x per cent of the capital in a congressional district with $150 billion worth of capital to reduce the nation’s GDP by about twice as much as the destruction of x per cent of the capital in a congressional district with $75 billion worth of capital. This leads us to the idea of relevant variables.
For each s, v and A, can we find an observable variable RV(s,d,v) for which there exists a coefficient, CA(s,v), independent of d, such that:
A A
E (s,d, v) C (s, v)*RV(s,d, v)
for all d? (6.1)We refer to RV as a relevant variable. The idea of the relevant variable is to capture data differences across regions that explain elasticity differences across regions. 22 If a relevant variable exists for s and v, and we know the value for a particular d of the elasticity, EA(s,d,v), then we can deduce the value of the coefficient CA(s,v). From there we can
compute EA(s,d,v) for all d.
To clarify, we consider the example of s equals capital destruction and v equals GDP. As we have already suggested it is reasonable to suppose that the elasticity of the nation’s GDP in year 1 with respect to capital destruction in any congressional district is proportional to the amount of capital in that district, that is, there exists a factor of proportionality, which we can denote by C, such that
A A
E (K-destruct,d,GDP) C (K-destruct,GDP)*RV(K-destruct,d,GDP)
for all d (6.2) where
EA(K-destruct,d, GDP) is the elasticity of GDP in year 1 with respect to capital
destruction in region d under assumption A;
RV(K-destruct,d,GDP) is the quantity of capital in region d, or more conveniently the share of the nation’s capital that is located in region d; and
CA(K-destruct, GDP) is the factor of proportionality under assumption A.
If we have evaluated EA for a particular d, say FL24, and we know the values of the RV’s,
then we can evaluate CA(K-destruct, GDP) as A A E (K-destruct, FL24, GDP) C (K-destruct, GDP) RV(K-destruct, FL24, GDP) (6.3)
allowing us to estimate EA(K-destruct,d, GDP) for all d via (6.2).
How can we find relevant variables and how can we know that they are legitimate, that is have the proportionality property described in (6.1)?
From our knowledge of the theory and data of USAGE-TERM we make guesses of relevant variables. For example, we have guessed that
VAL _ K(d) RV(K-destruct, d, GDP)
VAL _ K _ NAT
(6.4)
22 Notice that we assume that the same relevant variable will be adequate under either assumption A. This is not theoretically necessary but proved to be a non-damaging simplification.
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is a legitimate relevant variable for s equals capital destruction and v equals GDP where
VAL_K(d) is the value of capital in region d; and VAL_K_NAT is the value of capital in the nation.
To check the validity of the guesses for the relevant variable for any s,v pair we can calculate
A A FL24 FL24 guess E (s, FL24, v) C (s, v) RV (s, FL24, v) (6.5) A A AZ07 AZ07 guess E (s, AZ07, v) C (s, v) RV (s, AZ07, v) (6.6) NY14 NY14 A A guess E (s, NY14, v) C (s, v) RV (s, NY14, v) (6.7) WA09 WA09 A A guess E (s, WA09, v) C (s, v) RV (s, WA09, v) (6.8) where A FL24 E (s, FL24, v), A AZ07
E (s, AZ07, v), etc, are elasticities calculated from our 4 models, which we take as the true elasticities;
RVguess refers to our guess for the relevant variable, e.g. capital share; and
guess
RV
(s, FL24, v)
,RV
guess(s, AZ07, v)
, etc are the observed values of this variable forFL24, AZ07 etc.
We say that RVguess is a legitimate relevant variable for the s,v pair if there is little variation across the 4 values A
FL24 C (s, v), A AZ07 C (s, v), A NY14 C (s, v) and A WA09 C (s, v) for each A.
If for a given A the 4 values are not close, then we must think more deeply about the theory and data of the model to come up with a refined guess of the relevant variable. As well as meeting the immediate requirement of obtaining proportionality factors (C coefficients) that are consistent across our 4 models, the process of finding legitimate relevant variables is a valuable way of understanding key features of USAGE-TERM and of checking for
unrealistic specifications and errors. Once we were satisfied that the C coefficients were as uniform as possible across the 4 models, we averaged them and calculated the elasticities for the TRA groups according to:
TRA ave A A
E
(s,d, v) C (s, v)*RV(s,d, v)
(6.9) where TRA AE
(s,d, v)
is the value supplied to the TRA groups for the elasticity under assumption A (Keynesian or Neoclassical) of implication variable v with respect to shock s occurring in region d;ave A
C (s, v)
is the average value across the four models of the s,v-coefficients under assumption A; andRV(s,d,v) is the value for region d of the relevant variable for shock s and implication variable v.
The results from the four models for the coefficients, CA(s,v), and the definitions of the
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