1 EJC 18: Correlograma, FAC y FAP El siguiente cuadro representa

1
ECONOMETRIA 2 - ECON 3301 - SEMESTRE II - 08
Profesor: Ramón Rosales; rrosales@uniandes.edu.co
Profesor Taller: William Delgado; w-delgad@uniandes.edu.co
Profesor Taller: Juan Carlos Vasquez; jvasquez@uniandes.edu.co
Profesor Taller: Diego Marino; dmarino@uniandes.edu.co
Monitor: Alejandro Urrego; j-urrego@uniandes.edu.co
Monitor: Juan Sebastián Sánchez; jua-sanc@uniandes.edu.co
Monitor: Francisco Correa; fr-corre@uniandes.edu.co
Monitor: Carlos Morales; and-mora@uniandes.edu.co
EJC 18: Correlograma, FAC y FAP
El siguiente cuadro representa la serie de ventas de automóviles 4x4.
Y
848
1
Yt -Y prom
-189.41
(Yt -Y prom )2
2
Y t+1-Yprom
1x2
35877.77
115.59
-21893.59
3
Y t+2-Y prom
117.59
-22272.41
56.59
1x4
-10718.14
5
Yt+4-Y prom
47.59
1x5
-9013.41
115.59
13360.06
117.59
13591.23
47.59
5500.23
-77.41
-8947.99
1155
117.59
13826.40
56.59
6653.67
47.59
5595.40
-77.41
-9102.81
114.59
13473.64
1094
56.59
3201.94
47.59
2692.67
-77.41
-4380.54
114.59
6483.91
177.59
10048.81
47.59
6540.50
4
Yt+3-Y prom
1153
1085
56.59
1x3
2264.40
-77.41
-3683.81
114.59
5452.64
177.59
8450.54
-70.41
-3350.71
960
-77.41
5992.97
114.59
-8870.57
177.59
-13747.67
-70.41
5451.07
50.59
-3916.06
1152
114.59
13129.89
177.59
20348.79
-70.41
-8068.47
50.59
5796.40
46.59
5338.06
1215
177.59
31536.69
-70.41
-12504.57
50.59
8983.30
46.59
8272.96
-88.41
-15701.11
50.59
-3561.96
967
-70.41
4958.17
-3280.30
-88.41
6225.63
-146.41
10309.66
1088
50.59
2558.91
46.59
2356.57
-88.41
-4472.50
-146.41
-7406.47
-25.41
-1285.60
1084
46.59
2170.23
-88.41
-4118.84
-146.41
46.59
-6820.81
-25.41
-1183.94
93.59
4359.76
949
-88.41
7817.09
-146.41
12945.11
-25.41
2246.99
93.59
-8274.31
95.59
-8451.14
-51184.34
891
-146.41
21437.14
-25.41
3721.01
93.59
-13702.29
95.59
-13995.11
349.59
1012
-25.41
645.89
93.59
-2378.41
95.59
-2429.24
349.59
-8884.47
276.59
-7029.23
1131
93.59
8758.29
95.59
8945.46
349.59
32716.23
276.59
25884.47
217.59
20362.91
1133
95.59
9136.63
349.59
33415.40
276.59
26437.64
217.59
20798.09
-32.41
-3098.34
1387
349.59
122210.17
276.59
96690.41
217.59
76064.86
-32.41
-11331.57
-96.41
-33705.06
1314
276.59
76499.66
217.59
60181.10
-32.41
-8965.33
-96.41
-26666.81
159.59
1255
217.59
47343.54
-32.41
-7052.89
-96.41
-20978.37
159.59
34723.57
311.59
67796.60
1005
-32.41
1050.69
-96.41
3125.20
159.59
-5172.86
311.59
-10099.83
416.59
-13503.33
941
-96.41
9295.71
159.59
-15386.34
311.59
-30041.31
416.59
-40164.81
5.59
-538.54
1197
159.59
25467.60
311.59
49724.63
416.59
66481.13
5.59
891.40
163.59
26105.94
1349
311.59
97085.66
416.59
129802.16
5.59
1740.43
163.59
50970.97
6.59
2052.01
1454
416.59
173543.66
5.59
2326.93
163.59
68147.47
6.59
2743.51
258.59
107723.11
1043
5.59
31.20
163.59
913.74
6.59
36.79
258.59
1444.39
547.59
3058.66
1201
163.59
26760.29
6.59
1077.33
258.59
42300.93
547.59
89577.20
489.59
80089.23
1044
6.59
1296
258.59
66866.57
547.59
141597.84
489.59
126599.87
384.59
99448.37
296.59
76692.83
1585
547.59
299850.11
489.59
268090.14
384.59
210593.64
296.59
162406.10
-104.41
-57175.77
1527
489.59
239694.17
384.59
188287.67
296.59
145204.13
-104.41
-51119.74
347.59
170173.00
1422
384.59
147906.17
296.59
114062.63
-104.41
-40156.24
347.59
133676.50
417.59
160597.50
87963.09
-104.41
1334
296.59
43.37
258.59
1702.97
-30967.79
547.59
347.59
3606.24
103088.96
489.59
417.59
3224.27
123849.96
384.59
44139.13
752.59
2532.77
223206.17
933
-104.41
10902.34
347.59
-36292.91
417.59
-43601.91
752.59
-78580.70
378.59
-39529.76
1385
347.59
120815.83
417.59
145146.83
752.59
261588.04
378.59
131590.99
566.59
196937.10
1455
417.59
174377.83
752.59
314269.04
378.59
158091.99
566.59
236598.10
358.59
149740.27
1790
752.59
566385.26
378.59
284918.20
566.59
426404.31
358.59
269866.49
491.59
1416
378.59
143327.14
566.59
214501.26
358.59
135755.43
491.59
186107.33
429.59
162635.01
1604
566.59
321019.37
358.59
203169.54
491.59
278525.44
429.59
243397.13
-202.41
-114685.04
1396
358.59
128583.71
491.59
176275.61
429.59
154043.30
-202.41
-72582.87
-6.41
-2300.07
1529
491.59
241656.51
429.59
211178.20
-202.41
-99503.97
-6.41
-3153.17
114.59
56328.70
1467
369960.39
429.59
184543.89
-202.41
-86954.29
-6.41
-2755.49
114.59
49224.39
818.59
351652.73
835
-202.41
40971.54
-6.41
1298.34
114.59
-23193.79
818.59
-165693.44
616.59
-124805.76
1031
-6.41
41.14
114.59
-734.99
818.59
-5250.64
616.59
-3954.96
205.59
-1318.69
1152
114.59
13129.89
818.59
93798.23
616.59
70651.91
205.59
23557.19
92.59
10609.00
1856
818.59
670082.57
616.59
504728.26
205.59
168289.53
92.59
75789.34
-298.41
-244277.67
1654
616.59
380177.94
205.59
126761.21
92.59
57087.03
-298.41
-183997.99
-403.41
-248739.49
1243
205.59
42265.49
92.59
19034.30
-298.41
-61349.71
-403.41
-82936.21
-313.41
-64433.50
1130
92.59
8572.11
-298.41
-27628.90
-403.41
-37350.40
-313.41
-29017.69
-412.41
-38183.67
739
-298.41
89051.09
-403.41
120384.59
-313.41
93527.30
-412.41
123070.31
-194.41
58016.00
634
-403.41
162743.09
-313.41
126435.80
-412.41
166373.81
-194.41
78429.50
-286.41
115543.61
724
-313.41
98228.51
-412.41
129256.53
-194.41
60932.21
-286.41
89766.33
-387.41
121421.17
625
-412.41
170085.54
-194.41
80179.23
-286.41
118121.34
-387.41
159775.19
-353.41
145753.10
843
-194.41
37796.91
-286.41
55683.03
-387.41
75318.87
-353.41
68708.79
-461.41
751
-286.41
82033.14
-387.41
110960.99
-353.41
101222.90
-461.41
132155.64
-543.41
155641.61
650
-387.41
150089.83
-353.41
136917.74
-461.41
178758.49
-543.41
210526.46
-492.41
190768.33
684
-353.41
124901.66
-461.41
163070.40
-543.41
192050.37
-492.41
174026.24
-380.41
134443.84
576
-461.41
212903.14
-543.41
250739.11
-492.41
227206.99
-380.41
175528.59
-533.41
246124.97
494
-543.41
295299.09
-492.41
267584.96
-380.41
206722.56
-533.41
289864.94
-471.41
256173.26
545
-492.41
242471.83
-380.41
187321.43
-533.41
262660.81
-471.41
232131.13
-480.41
236562.86
657
-380.41
144715.03
-533.41
202918.41
-471.41
179332.73
-480.41
182756.46
-484.41
184278.11
504
-533.41
284530.80
-471.41
251459.11
-480.41
256259.84
-484.41
258393.50
-351.41
187449.40
566
-471.41
222231.43
-480.41
226474.16
-484.41
228359.81
-351.41
165661.71
-475.41
224117.09
557
-480.41
230797.89
-484.41
232719.54
-351.41
168824.44
-475.41
228395.81
-499.41
239925.76
553
-484.41
234657.20
-351.41
170230.10
-475.41
230297.47
-499.41
241923.41
-514.41
249189.63
686
-351.41
123492.00
-475.41
167067.37
-499.41
175501.31
-514.41
180772.53
-514.41
180772.53
562
-475.41
226018.74
-499.41
237428.69
-514.41
244559.90
-514.41
244559.90
-499.41
237428.69
538
-499.41
249414.63
-514.41
256905.84
-514.41
256905.84
-499.41
249414.63
0.00
0.00
523
-514.41
264622.06
-514.41
264622.06
-499.41
256905.84
0.00
0.00
0.00
0.00
523
-514.41
264622.06
-499.41
256905.84
0.00
0.00
0.00
0.00
0.00
0.00
538
-499.41
249414.63
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Σ
ΣY/n
0.0
8,805,257.0
189.4
7,090,566.8
73.8
5,864,622.7
-43.8
4,678,946.5
-100.3
4,484,064.2
0.0
125,789.4
2.7
101,293.8
1.1
83,780.3
-0.6
66,842.1
-1.4
64,058.1
89705.53
2
FAC: está representada por las autocorrelaciones simples:
70
∑ (Y
ρ t ,t +1 =
− Y )(Yt +1 − Y )
t
t =1
Cov(Yt , Yt +1 )
=
Var (Yt )
n
n
∑ (Y
t =1
t
− Y )2
n
70
∑ (Y
t
t =1
ρ1, 2 =
− Y )(Yt +1 − Y )
70
n
∑ (Y
t
t =1
− Y )2
7090567
70
=
=0.80527
8805257
70
70
FAP: representa la Autocorrelación parcial (FAP) ( φ ). Para la estimación de las
funciones de autocorrelación parcial, se utilizan las autocorrelaciones simples
estimadas anteriormente.
Método de Yule-Walker
φ11 = r1 = 0.80527
−1
−1
⎡φ 21 ⎤ ⎡ 1
⎢φ ⎥ = ⎢r
⎣ 22 ⎦ ⎣ 1
r1 ⎤ ⎡ r1 ⎤ ⎡ 1
0.805⎤ ⎡0.80500⎤ ⎡0.765⎤
= ⎢
=
⎥
⎢
⎥
1 ⎦ ⎣r2 ⎦ ⎣0.805
1 ⎥⎦ ⎢⎣0.66604⎥⎦ ⎢⎣0.050⎥⎦
⎡φ31 ⎤ ⎡ 1
⎢φ ⎥ = ⎢ r
⎢ 32 ⎥ ⎢ 1
⎢⎣φ33 ⎥⎦ ⎢⎣r2
r1
1
r1
r2 ⎤
r1 ⎥⎥
1 ⎥⎦
⎡φ 41 ⎤ ⎡ 1
⎢φ ⎥ ⎢ r
⎢ 42 ⎥ = ⎢ 1
⎢φ 43 ⎥ ⎢r2
⎢ ⎥ ⎢
⎣φ 44 ⎦ ⎣r3
r1
1
r2
r1
r1
r2
1
r1
−1
⎡ r1 ⎤
⎢r ⎥ =
⎢ 2⎥
⎢⎣ r3 ⎥⎦
r3 ⎤
r2 ⎥⎥
r1 ⎥
⎥
1⎦
−1
−1
0.80527 0.66604⎤ ⎡0.80527⎤ ⎡ 0.76761 ⎤
⎡ 1
⎢0.80527
1
0.80527⎥⎥ ⎢⎢0.66604⎥⎥ = ⎢⎢ 0.09018 ⎥⎥
⎢
⎢⎣0.66604 0.80527
1 ⎥⎦ ⎢⎣0.53138⎥⎦ ⎢⎣− 0.05249⎥⎦
0.80527 0.666 0.531⎤
⎡ r1 ⎤ ⎡ 1
⎢r ⎥ ⎢0.805
1
0.805 0.666⎥⎥
⎢ 2⎥=⎢
⎢r3 ⎥ ⎢0.666 0.805
1
0.805⎥
⎥
⎢ ⎥ ⎢
1 ⎦
⎣r4 ⎦ ⎣0.531 0.666 0.805
−1
⎡0.805⎤ ⎡ 0.7802 ⎤
⎢0.666⎥ ⎢ 0.0686 ⎥
⎢
⎥=⎢
⎥
⎢0.531⎥ ⎢− 0.2359⎥
⎢
⎥ ⎢
⎥
⎣0.509⎦ ⎣ 0.2390 ⎦
3
Método de Yule – Walter. Autocorrelación Parcial.
FAP
k
1.2
0
1
2
3
4
1
0.8
0.6
0.4
Phi
1.000
0.805
0.050
-0.052
0.239
0.2
0
-0.2
0
1
2
3
4
5
-0.4
Rezagos
Método de Durbin
K −1
φ KK =
rK − ∑ φ K −1, J .rK − J
J =1
K −1
1 − ∑ φ K −1, J .rJ
, donde φ KJ = φ K −1, J − φ KK φ K −1, K − J
J = 1,2,..., K − 1
J =1
Para φ11 =
r1 − φ0,1r0 r1 − 0
=
= r1 = 0.80527
1 − φ0,1r0 1 − 0
r2 − φ1,1r1 r2 − r12 0.666 − (0.8052) 2
φ 22 =
=
=
=0.04976
1 − φ1,1r1
1 − r12
1 − (0.8052) 2
φ33 =
r3 − (φ 2,1r1 + φ 2, 2 r1 )
0.53138 − [(0.765035) (0.666) + 0.05 (0.8053)]
=
= - 0.0524
1 − (φ 2,1r1 + φ 2, 2 r2 )
1 − [(0.765035) (0.8053) + 0.05 (0.666)]
4
Método de Durbin. Autocorrelación (FAC ó AC)
CORRELOGRAMA DE Y
Date: 10/29/08 Time: 19:07
5
Date: 10/29/08 Time: 19:10
Sample: 1 70
Included observations: 70
Autocorrelation
Partial Correlation
. |****** |
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. |**** |
. |**** |
. |*** |
. |*** |
. |****** |
.|. |
.|. |
. |** |
. |** |
.|. |
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.*| . |
1
2
3
4
5
6
7
8
9
10
AC
PAC
Q-Stat
Prob
0.805
0.666
0.531
0.509
0.562
0.561
0.546
0.488
0.435
0.358
0.805
0.050
-0.052
0.239
0.282
-0.041
0.038
0.022
-0.020
-0.165
47.365
80.244
101.48
121.29
145.79
170.59
194.47
213.83
229.49
240.23
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Método de regresión:
Yt = φ11Yt − 1
Yt = φ 21Yt − 1 + φ 22Yt − 2
Yt = φ 31Yt − 1 + φ 32Yt − 2 + φ 33Yt − 3
.
.
.
Yt = φk1Yt − 1 + φk 2Yt − 2 + φk 3Yt − 3 + ..... + φkkYt − k
Estimadas estas regresiones y observando los resultados de las regresiones con
cuatro rezagos, se ve que Yt − 2 y Yt − 3 , no son estadísticamente significativas; en
tanto que Yt − 4 si lo es.
Dependent Variable: Y
Method: Least Squares
Date: 10/29/08 Time: 19:14
Sample(adjusted): 2 70
Included observations: 69 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
Y(-1)
174.2162
0.828930
78.32220
0.071048
2.224352
11.66714
0.0295
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.670149
0.665226
207.7748
2892414.
-465.1074
2.053027
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
1040.159
359.1014
13.53934
13.60410
136.1222
0.000000
6
Dependent Variable: Y
Method: Least Squares
Date: 10/29/08 Time: 19:15
Sample(adjusted): 3 70
Included observations: 68 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
Y(-1)
Y(-2)
151.0134
0.788768
0.058173
83.02756
0.122261
0.123999
1.818834
6.451487
0.469143
0.0735
0.0000
0.6405
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.679607
0.669748
207.7478
2805346.
-457.8238
2.002197
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
1038.500
361.5047
13.55364
13.65156
68.93776
0.000000
Dependent Variable: Y
Method: Least Squares
Date: 10/29/08 Time: 19:16
Sample(adjusted): 4 70
Included observations: 67 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
Y(-1)
Y(-2)
Y(-3)
159.0598
0.786672
0.095341
-0.043155
88.53735
0.125898
0.158877
0.126263
1.796527
6.248508
0.600088
-0.341785
0.0772
0.0000
0.5506
0.7337
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.679940
0.664699
210.7438
2798016.
-451.4998
1.972182
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
1036.761
363.9465
13.59701
13.72863
44.61272
0.000000
Dependent Variable: Y
Method: Least Squares
Date: 10/29/08 Time: 19:18
Sample(adjusted): 5 70
Included observations: 66 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
Y(-1)
Y(-2)
Y(-3)
Y(-4)
95.63003
0.800154
0.074517
-0.259693
0.283000
90.90083
0.122881
0.156188
0.155324
0.123403
1.052026
6.511627
0.477099
-1.671939
2.293302
0.2969
0.0000
0.6350
0.0997
0.0253
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.705476
0.686163
205.4102
2573794.
-442.5008
2.182545
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
1035.894
366.6656
13.56063
13.72651
36.52853
0.000000