1
MINIATURE
Die wesentliche Beschreibung des ökonometrischen Modells
the basic description of the econometric model
le formalisme du modèle économétrique
descripción basica del modelo econométrico
la essenziale descrizione del modello econometrico
eine Minaturdarstellung eines Modells
a miniature of a model
une maquette d'un modèle
maqueta de un modelo, una versión simplificada del modelo
una rappresentazione a miniatura del modello, un modello semplificato
A problem related to aggregation is the construction of miniature models. For expository
reasons the really large models, say of 1000 or more equations, are condensed into smaller
models, orders of magnitude smaller, say to 10 to 15 equations. They are supposed to be easy
to explain and easy to comprehend.
If one denotes the endogenous variables by Y respectively y, and the exogenous variables by X
respectively x, then the miniaturizaton of model “S” to model “s” is a transformation
RN ×RM: S(Y, X) → Rn×Rm: s(y, x) , N>>n, M>>m.
The evident claim with respect to the miniature is the idea, that there are essential invariance
properties, i.e. substantial properties of the model are maintained. Only messy detail is deleted.
For almost all large models such explanatory miniature models are produced.
For the basic idea of miniature models see e.g.:
Gibbard A., Varian H.R., Economic Models, Journal of Philosophy 75, No. 11,
664-677.
Dominique Bureau, Didier Miqueu, Michel Norotte, La maquette désir, Économie
&Prevision, Paris, Numero 65, 1984/44-54
... “Les maquettes constituent alors un intermediaire entre deux termes: des modelès
empiriques, elles retiennent la pratique des simulation numeriques; leur nombre réduit
d’equations permet de mieux les fonder théoriquement et leur assure une grande souplesse
d’utilistion....”
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Illustration 1: A simple GNP-type model of Project Link
Klein L.R, Van Peeterssen A., Forecasting world trade within Project Link,
Chapter 13 of J. Ball, ed., The economic model of Project Link, North-Holland,
Amsterdam, 1973.
(1) private consumption
C t = a1 + a2Xt + a3C t-1 + e1,t
(2) accelerator; investment
It = b1 + b2Xt + e2,t
(3) exports
Et = g1 + g2WTt + g3PWt /PXt + g4Et-1 + e3,t
(4) imports
Mt = d1 + d2Xt + d3Xt /XCt + e4,t
(5) GDP identity
Xt ≡ C t + It + Gt + Et - Mt
(6) GNP - deflator
Pt = ε1 + ε2Xt /XCt + ε3PMt + e5,t
(7) Prices of exports
DPXt = ξ1 + ξ2DPt + ξ3D(PW/R(T)) + e6,t
(8) production capacity
XC = f1 + f2t + e7,t
There are 8 endogenous variables:
C Real domestic consumer expenditure, I Real investment, including inventory change, X Real
GNP, E Real exports of goods and services, P GNP deflator, M Real imports of goods and
services, XC Trend of capacity output, estimated as line through historical peaks of X, PX Export
price index;
and there are 5 exogenous variables:
PM Import price index, r $-exchange rate, G Real government expenditures on goods and
services, PW Deflator of world trade, dollar index, WT Volume of world trade in constant $.
Additional notation is given by
D the difference operator;
ai , b i , gi , di , εi , ξi , fi parameters to be estimated;
ej,t (j=1,2,...,7) random variables.
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Illustration 2: A Miniature of the NIESR Model of the UK
Henry S. G. B., Karakitsos E., Savage D., On the Derivation of the "Efficient"
Phillips Curve, Manchester School of Economics 50, 1982, 156-159.
(1)
Y
≡
C+I+G+X-Q-F
(2)
(3)
(4)
(5)
C
I
X
Q
=
=
=
=
C(Yd,BLP/P,P), ∂C/∂Y, ∂C/∂L>0, ∂C/∂P<0
I(DY.CU,BLP/P,R,p), ∂I/∂Y, ∂I/∂L, ∂I/∂p>0, ∂I/∂R<0
X(WT,PX.e/WP), ∂X/∂WT>0, ∂X/∂RL<0
Q(Y,WP/e.P), ∂Q/∂Y>0, ∂Q/∂RL<0
(6)
(7)
(8)
Yd
T
w
≡ Y-T
= tY
= W(Pe,U), ∂W/∂Pe>0, ∂W/∂U<0 or
(8)'
w
(9)
(10)
P
E
= WR(Pe,W/P(1 - T/W)), ∂WR/∂Pe>0, ∂WR/∂T<0
= P(w,WP/e), ∂P/∂w, ∂P/∂WP>0
= E(Y), ∂E/∂Y>0
(11)
(12)
(13)
U
PX
e
≡ L-E
= PX(P,WP/e), ∂PX/∂P>0, ∂PX/∂WP>0
= e(WP/P,CA/P,R - Rw), ∂e/∂RL, ∂e/∂CA, ∂e/∂R>0
(14)
(15)
(16)
(17)
CA
CR/P
SD/P
DB
≡
=
=
=
(18)
(19)
DBLG
≡ G - T - DB - DCR - DFO
D(BLP/P) = BL(R,Y,DV/P), ∂BL/∂R, ∂BL/∂V<0, ∂BL/∂Y>0
(20)
DTD
≡
(21)
(22)
(23)
DV
DM
R
≡ rB + Y - C - I - T ≡ DM + DB
= DCR + DSD + DTD
= R(r), ∂R/∂r>0
X.PX - Q.WP/e
CR(Y), ∂CR/∂Y>0
SD(R,Y), ∂SD/∂R<0, ∂SD/∂Y>0
B(R,r,DV), ∂B/∂R<0, ∂B/∂r, ∂B/∂V>0
DBLG - DSD + DBLP
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Glossary of variables:
DZ = Zt - Zt-1 (difference operator)
zt
Y
C
I
G
X
Q
Yd
P
P
BLP
BLG
T
CU
R
p
WT
PX
WP
e
CA
Rw
B
CR
FO
M
SD
TD
V
Pe
w
PR
U
L
E
r
F
RL
=
=
=
=
=
=
=
=
=
dZt/dt (z= P,w; a dot over a variable indicates its time derivative)
real national income
real consumption
real investment including stockbuilding
real government expenditure
real exports
real imports
real disposable income
price level
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
inflation rate
bank lending to the private sector
bank lending to the government
taxes
capacity utilization
market interest rate
profitability
index of world trade
export prices
world prices
exchange rate
current account
world interest rate
fixed price bonds
currency held by the private sector
overseas finance
Money supply (M3)
sight deposits
time deposits
private sector wealth
= expected inflation rate
=
=
=
=
=
=
=
=
wage inflation rate
productivity
unemployment
labor force
employment
interest rate on government debt
adjustment to factor cost
relative prices
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Beispiel / examples / exemples
Exemple
Example
Beispiel
Example
Example
Example
Example
Example
Example
Example
Exemple
Example
Example
Example
Example
Example
Exemple
1 (UN MODELE ECONOMETRIQUE DE L'ECONOMIE FRANCAISE,
Model France 69/1)
2 (Five miniature models of the ORANI model, Australia 75/1)
3 (Ein einfaches Modell im Project Link, simple GNP-type models)
4 (A miniature of the Danish model ADAM)
5 (A miniature model of the Cambridge Economic Policy Group)
6 (A miniature model of the U.S. economy)
7 (Miniatures of UK models: the LBS model and the NIESR model)
8 (Rappresentazione sintetica del modello,
a miniature of the Bologna model (Prometeia, model Italy 72/1))
9 (THE PACIFIC BASIN MODEL OF KINOSHITA ET AL)
10 (A simple neoclassical model of one-sector, one-occupation, one-region,
one-age-cohort, two-factor inputs, and three-final-demand components in
sixteen equations)
11 (Maquette représentative)
12 (Model Sweden 82/2,
The equations of the one-sector version of the model)
13 (A miniature of the VARUNA-model)
14 (A miniature of the Bank of Italy quarterly econometric model)
15 (A linear/loglinear model for the U.K.)
16 (A miniature model of the economy, simplification of large scale
macroeconometric models)
17 (La modelisation economique)