1 YA+XB=U Schätzverfahren/estimation procedures/les méthodes

1
YA+XB=U
Schätzverfahren/estimation procedures/les méthodes des estimations/
método de estimación/procedura di stima
das lineare Modell (Linearität in den Koeffizienten (Parametern))
the linear model (linearity of coefficients (parameters))
le modèle du système des equations lineaires
(linearite des coefficients; un modèle à equations simultanées)
el modelo lineal (lineal en los coeficientes -parámetros-)
il modello lineare (linearita nei coefficenti (parametri))
the multi-variate model (univariate model)
YΓ + G(Y,X)∆ = U
Y∈RT,G, X∈RT,K , Γ∈RG,G, ∆∈RK,G , U∈RT,G
A special case is given by the linear model:
YΓ + X∆ = U
which here is abbreviated by “YA+XB=U”.
A further special case is given by the univariate model:
y = G(y,X)β + u,
respectively
y = Xβ + u, y∈RT, X∈RT,K , β∈RK , u∈RT
yt = Xt β + ut (t=1,2,...,T)
Illustration 1: Linearity in the coefficients: coefficient matrices Γ, ∆ (coefficent vector β);
data matrices Y, G (data vector y, data matrix G); U (u) as above), respectively
YΓ + X∆ = U (resp. for the univariate model: y = Xβ + u)
Illustration 2: Linearität in den Variablen: Y and X (bzw. y und X))
Ein kleines in den Parametern lineares (und auch in den Variablen lineares) Modell
(1) C t = a + b Yt + ut , t = 1,2,... T
(2) Yt = Ct + Zt
(Endogene Variable C, Y; exogene Variable Z; stochastische Variable u, E(ut ) = 0, var(ut )
= σ 2i >0, t =1,2,....T; Parameter a, b)
C1 Y1
1 Z1
1 Z2
C 2 Y2
Y:=
C3 Y3
CT YT
, X:=
1 Z3
1 ZT
u1 0
u2 0
, U:=
u3 0
uT 0
, Γ :=
1 -1
–a 0
, ∆:=
-b 1
0 –1
2
Illustration 3:
Ein kleines in den Parametern lineares und in den Variablen nichtlineares Modell
a model linear in parameters and nonlinear in variables
(1) y1,t = a xt
+ u1,t , t = 1,2,... T
(2) y2,t = b exp(y1,t ) + u2,t
(Endogenous variables y 1, y2; exogenous variable x; stochastic variable u1, u2;
E(ui,t ) = 0, var(ui,t ) = σ 2i >0, t =1,2,....T; parameters a, b)
y1,1 y2,1
y1,2 y 2,2
Y:=
y 1,3 y2,3
y1,T y2,T
, X:=
x1
exp(y1,1)
x2
exp(y1,2)
x3
exp(y1,3)
xT
exp(y1,T)
u 1,1 u 2,1
u1,2 u 2,2
, U:=
u 1,3 u 2,3
, Γ:=
10
–a 0
, ∆:=
01
0 –b
u 1,T u 2,T
Beispiele / examples / exemples
Beispiel
1 (Das lineare, in den Koeffizienten lineare, interdependente
ökonometrische Modell YΓ + X∆ + U = 0)
Beispiel 2 (Linearität in den Koeffizienten)
Beispiel 3 (Ein kleines lineares Makromodell)
Beispiel 4 (Zwei verbreitete Formulierungen des linearen/loglinearen Modells)
Example 5 (Two systems YΓ + X∆ = U, a linear model and a loglinear model)
Example 6 (A firm and/or sectoral model)
Beispiel 7 (Das Firmenmodell von Theil-Boot u.a.)
Example 8 (A submodel of exports and R&D)
Beispiel 9 (Ein Identifikationsproblem)
Example 10 (Identification of a two equations model)
Example 11 (A system linear in parameters YΓ + X∆ = U)
Example 12 (Demand for Refined Petroleum Products)
(HAAVELMO, NONLINEAR, SPECIFICATION, IDENTIFICATION, Y=XB+U, SELS)