ln(ˆ^ˆˆ

UNIVERSIDAD NACIONAL DE PIURA
FACULTAD DE ECONOMIA
SOLUCIÓN DE LA SEGUNDA PRÁCTICA DE ECONOMETRIA I
1º
El investigador especifica el siguiente modelo:
CT = a + b TA^c + d EMI + u
se le pide, estimar el modelo aplicando la serie de Taylor y mediante la transformación de Box y Cox (2
decimales). (6 puntos)
Vector gradiente:
∂f
= (1 TA^ c bTA^ c ln/(TA) EMI )
∂β
Aplicando la serie de Taylor:
 aˆ 
a
 ˆ
 
b
b
ˆ
ˆ
ˆ
ˆ
CT − aˆ − bTA^ cˆ − dEMI + 1 TA^ cˆ bTA^ cˆ ln(TA) EMI   ≅ 1 TA^ cˆ bTA^ cˆ ln(TA) EMI  
cˆ
c
 
 
 dˆ 
d 
 
 
(
)
(
)
multiplicando:
CT − aˆ − bˆTA^ cˆ − dˆEMI + aˆ + bˆTA^ cˆ + bˆcˆTA^ cˆ ln(TA) + dˆEMI ≅ a + bTA^ cˆ + cbˆTA^ cˆ ln(TA) + dEMI
simplificando:
CT + bˆcˆTA^ cˆ ln(TA) ≅ a + bTA^ cˆ + cbˆTA^ cˆ ln(TA) + dEMI
APROXIMACIÓN APLICANDO TAYLOR
I
T
R2
AJUSTADO
AKAIKE
SCHWARZ
1
2
108
108
0.963215
0.941409
24.70371
24.71266
24.80305
24.81200
Dependent Variable: _Y-15349.90935*1*_X^1*LOG(_X)
Method: Least Squares
Sample: 2001M01 2009M12
Included observations: 108
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
3509649.
_X^1
-564491.4
-15349.90935*_X^1*LOG(_X) -7.669287
_Z
39.18531
1569203.
285530.3
4.507449
1.017770
2.236580
-1.976993
-1.701470
38.50116
0.0274
0.0507
0.0918
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.964246
0.963215
54981.36
3.14E+11
-1330.000
0.261758
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
-544908.4
286668.5
24.70371
24.80305
934.9311
0.000000
2
TRANSFORMACION DE
BOX COX
LANDA
SR
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
3.23E+11
3.23E+11
3.24E+11
3.24E+11
3.24E+11
3.24E+11
3.25E+11
3.25E+11
3.25E+11
3.25E+11
3.26E+11
3.26E+11
3.26E+11
3.26E+11
3.27E+11
3.27E+11
3.27E+11
3.27E+11
3.28E+11
3.28E+11
3.28E+11
TRANSFORMACION DE
BOX COX
LANDA
SR
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
-1.1
-1.2
-1.3
-1.4
-1.5
-1.6
3.23E+11
3.23E+11
3.23E+11
3.23E+11
3.22E+11
3.22E+11
3.22E+11
3.22E+11
3.22E+11
3.21E+11
3.21E+11
3.21E+11
3.21E+11
3.21E+11
3.21E+11
3.21E+11
3.20E+11
3
-1.7
-1.8
-1.9
-2.0
3.20E+11
3.20E+11
3.20E+11
3.20E+11
TRANSFORMACION DE
BOX COX
2º
LANDA
SR
2.0
-2.1
-2.2
-2.3
-2.4
-2.5
-2.6
-2.7
-2.8
-2.9
-3.0
-3.1
-3.2
-3.3
-3.4
-3.5
-3.6
3.20E+11
3.20E+11
3.20E+11
3.20E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
3.19E+11
El investigador especifica el siguiente modelo:
CT = a E^(b SPREAD) + c EMI + u
se le pide, estimar el modelo por mínimos cuadrados no lineales y máxima verosimilitud. (4 puntos)
Dependent Variable: CT
Method: Least Squares
Sample: 2001M01 2009M12
Included observations: 108
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
EMI
130027.9
39.32318
13680.41
1.087786
9.504679
36.14974
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.924972
0.924264
62790.46
4.18E+11
-1345.372
0.168739
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
573724.7
228161.7
24.95134
25.00101
1306.804
0.000000
4
Dependent Variable: CT
Method: Least Squares
Sample: 2001M01 2009M12
Included observations: 104
Convergence achieved after 23 iterations
CT =C(1)*EXP(C(2)*SPREAD)+C(3)*EMI
C(1)
C(2)
C(3)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Coefficient
Std. Error
t-Statistic
Prob.
46229.92
0.001510
42.18189
11948.94
0.000283
0.918779
3.868955
5.338812
45.91080
0.0002
0.0000
0.0000
0.956722
0.955865
48783.92
2.40E+11
-1268.744
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
576072.3
232211.5
24.45661
24.53289
0.390209
System: MV
Estimation Method: Full Information Maximum Likelihood (Marquardt)
Sample: 2001M01 2009M12
Included observations: 104
Total system (balanced) observations 104
Convergence achieved after 16 iterations
C(1)
C(2)
C(3)
Coefficient
Std. Error
z-Statistic
Prob.
130027.8
0.000440
37.55821
23617.71
0.000248
1.227542
5.505522
1.772307
30.59626
0.0000
0.0763
0.0000
Log Likelihood
Determinant residual covariance
-1286.597
3.26E+09
Equation: CT =C(1)*EXP(C(2)*SPREAD)+C(3)*EMI
Observations: 104
R-squared
0.938993
Mean dependent var
Adjusted R-squared
0.937785
S.D. dependent var
S.E. of regression
57920.23
Sum squared resid
Durbin-Watson stat
0.193173
3º
Obtener los efectos de SPREAD y TA sobre el crédito total. (3 puntos)
APROXIMACION DE TAYLOR:
576072.3
232211.5
3.39E+11
5
EEMI = 39.18531
ETA = c(2)*c(3)*ta^(c(3)-1) = 8.71E-06.
MCNL:
EEMI = 42.18189
ESPREAD = c(1)*c(2)*exp(c(2)*spread) = 152.5652
MV:
EEMI = 70.87313
ESPREAD = c(1)*c(2)*exp(c(2)*spread) = 70.87313
4º
Comente y fundamente su respuesta. (7 puntos)
4.1.
La estimación por mínimos cuadrados no lineales y máxima verosimilitud son equivalentes.
4.2.
Todo modelo no lineal se estima por mínimos cuadrados después de transformarlo en lineal.