Модель VAR / VECM. Тест Granger’a на причинность

Построение VAR

Построим VAR для рядов lint и linf. Определим порядок VAR. Для этого проведем тесты для определения порядка VAR.

VAR Lag Order Selection Criteria
Endogenous variables: LINF LINT
Exogenous variables: C
Date: 05/22/07 Time: 23:02
Sample: 1960 2005
Included observations: 25
Lag	LogL	LR	FPE	AIC	SC	HQ
0	-19.18795	NA	0.018674	1.695036	1.792546	1.722081
1	12.30502	55.42762*	0.002075*	-0.504401	-0.211871*	-0.423266
2	15.03675	4.370773	0.002316	-0.402940	0.084610	-0.267714
3	15.36220	0.468651	0.003170	-0.108976	0.573594	0.080340
4	17.55006	2.800458	0.003802	0.035995	0.913586	0.279402
5	21.57721	4.510412	0.004034	0.033823	1.106434	0.331320
6	23.22051	1.577568	0.005364	0.222359	1.489990	0.573946
7	30.48696	5.813161	0.004786	-0.038957	1.423694	0.366720
8	37.72311	4.631131	0.004621	-0.297849	1.359823	0.161919
9	44.01760	3.021355	0.005449	-0.481408	1.371284	0.032450
10	63.56025	6.253650	0.002806	-1.724820*	0.322891	-1.156872*
* indicates lag order selected by the criterion
LR: sequential modified LR test statistic (each test at 5% level)
FPE: Final prediction error
AIC: Akaike information criterion
SC: Schwarz information criterion
HQ: Hannan-Quinn information criterion

По большинству критериев модель VAR была выбрана 1 порядка. Оценим VAR(1).

Vector Autoregression Estimates
Date: 05/22/07 Time: 23:03
Sample(adjusted): 1961 2005
Included observations: 43
Excluded observations: 2 after adjusting
Endpoints
Standard errors in () & t-statistics in [ ]
	LINF	LINT
LINF(-1)	0.443605	-0.018611
	(0.11821)	(0.04180)
	[ 3.75281]	[-0.44519]

LINT(-1)	0.731735	0.994270
	(0.32353)	(0.11442)
	[ 2.26174]	[ 8.69002]

C	-1.631033	0.127436
	(1.71872)	(0.60782)
	[-0.94898]	[ 0.20966]
R-squared	0.597028	0.771163
Adj. R-squared	0.576879	0.759721
Sum sq. resids	5.842189	0.730667
S.E. equation	0.382171	0.135154
F-statistic	29.63121	67.39849
Log likelihood	-18.09832	26.59809
Akaike AIC	0.981317	-1.097586
Schwarz SC	1.104192	-0.974711
Mean dependent	5.572390	6.486922
S.D. dependent	0.587524	0.275722
Determinant Residual Covariance		0.002047
Log Likelihood (d.f. adjusted)		11.08395
Akaike Information Criteria		-0.236463
Schwarz Criteria		0.009286

Проверим VAR на наличие автокорреляции.

VAR Residual Serial Correlation LM Tests
H0: no serial correlation at lag order h
Date: 05/22/07 Time: 23:05
Sample: 1960 2005
Included observations: 43
Lags	LM-Stat	Prob
1	8.267909	0.0822
2	6.792159	0.1473
3	3.685835	0.4502
4	2.502930	0.6441
5	1.362375	0.8507
6	4.458230	0.3475
7	2.900400	0.5746
8	9.727034	0.0453
9	1.774730	0.7771
10	10.12151	0.0384
11	2.766267	0.5977
12	8.780658	0.0668
Probs from chi-square with 4 df.

Видно, что автокорреляция есть на 8 и на 10 лаге. Включим их в VAR.

Vector Autoregression Estimates

Date: 05/22/07 Time: 23:05

Sample(adjusted): 1970 2005

Included observations: 32

Excluded observations: 4 after adjusting

Endpoints

Standard errors in () & t-statistics in [ ]

LINF

LINT

LINF(-1)

0.388838

-0.012141

(0.12531)

(0.04985)

[ 3.10292]

[-0.24353]

LINF(-8)

0.301423

0.077930

(0.12853)

(0.05113)

[ 2.34517]

[ 1.52411]

LINF(-10)

0.117291

0.027252

(0.13287)

(0.05286)

[ 0.88278]

[ 0.51558]

LINT(-1)

1.014297

0.947487

(0.36118)

(0.14369)

[ 2.80826]

[ 6.59412]

LINT(-8)

-0.838051

0.099577

(0.48738)

(0.19389)

[-1.71949]

[ 0.51357]

LINT(-10)

-1.212959

-0.141701

(0.50935)

(0.20263)

[-2.38139]

[-0.69931]

8.022121

0.071844

(3.96211)

(1.57621)

[ 2.02471]

[ 0.04558]

R-squared

0.757200

0.834208

Adj. R-squared

0.698928

0.794417

Sum sq. resids

3.228167

0.510897

S.E. equation

0.359342

0.142954

F-statistic

12.99425

20.96516

Log likelihood

-8.704890

20.79115

Akaike AIC

0.981556

-0.861947

Schwarz SC

1.302185

-0.541317

Mean dependent

5.617068

6.490179

S.D. dependent

0.654897

0.315285

Determinant Residual Covariance

0.002055

Log Likelihood (d.f. adjusted)

8.188916

Akaike Information Criteria

0.363193

Schwarz Criteria

1.004452

Проверим VAR на наличие автокорреляции после включения лагов.

VAR Residual Serial Correlation LM Tests
H0: no serial correlation at lag order h
Date: 05/22/07 Time: 23:06
Sample: 1960 2005
Included observations: 32
Lags	LM-Stat	Prob
1	1.595499	0.8096
2	6.255415	0.1809
3	3.783082	0.4362
4	2.057536	0.7252
5	4.745309	0.3144
6	3.878486	0.4227
7	1.475367	0.8310
8	3.431380	0.4884
9	4.671107	0.3227
10	3.476089	0.4815
11	5.639265	0.2278
12	12.63686	0.0132
Probs from chi-square with 4 df.

Видно, что автокрреляция появилась на 12 лаге. Добавим 12 лаг для устранения автокорреляции.

Vector Autoregression Estimates
Date: 05/22/07 Time: 23:06
Sample(adjusted): 1972 2005
Included observations: 29
Excluded observations: 5 after adjusting
Endpoints
Standard errors in () & t-statistics in [ ]
	LINF	LINT
LINF(-1)	0.437080	-0.015028
	(0.13539)	(0.04624)
	[ 3.22820]	[-0.32502]

LINF(-8)	0.190536	0.040348
	(0.14259)	(0.04869)
	[ 1.33625]	[ 0.82859]

LINF(-10)	0.103648	0.046911
	(0.13513)	(0.04615)
	[ 0.76701]	[ 1.01655]

LINF(-12)	0.217332	0.008047
	(0.14707)	(0.05023)
	[ 1.47770]	[ 0.16021]

LINT(-1)	0.645629	0.899226
	(0.43649)	(0.14906)
	[ 1.47913]	[ 6.03257]

LINT(-8)	-0.109756	0.224018
	(0.66464)	(0.22697)
	[-0.16514]	[ 0.98698]

LINT(-10)	-1.303565	-0.066658
	(0.52969)	(0.18089)
	[-2.46101]	[-0.36851]

LINT(-12)	-0.619811	-0.453889
	(0.54637)	(0.18658)
	[-1.13442]	[-2.43262]

C	9.522590	2.122757
	(5.42847)	(1.85383)
	[ 1.75419]	[ 1.14507]
R-squared	0.786416	0.896581
Adj. R-squared	0.700983	0.855213
Sum sq. resids	2.505775	0.292230
S.E. equation	0.353962	0.120878
F-statistic	9.205018	21.67342
Log likelihood	-5.643101	25.51470
Akaike AIC	1.009869	-1.138945
Schwarz SC	1.434202	-0.714611
Mean dependent	5.621873	6.489489
S.D. dependent	0.647304	0.317674
Determinant Residual Covariance		0.001451
Log Likelihood (d.f. adjusted)		12.46967
Akaike Information Criteria		0.381402
Schwarz Criteria		1.230069

Проверим на автокорреляцию VAR c добавленным 12 лагом.

VAR Residual Serial Correlation LM Tests
H0: no serial correlation at lag order h
Date: 05/22/07 Time: 23:06
Sample: 1960 2005
Included observations: 29
Lags	LM-Stat	Prob
1	0.847773	0.9319
2	4.563143	0.3351
3	6.705258	0.1523
4	5.711181	0.2218
5	4.662481	0.3237
6	2.018002	0.7324
7	5.821397	0.2129
8	6.785999	0.1476
9	3.124147	0.5373
10	5.461704	0.2431
11	1.649165	0.7999
12	8.726010	0.0683
Probs from chi-square with 4 df.

Автокорреляция ушла.

Рассмотрим корни характеристического уравнения для данной VAR. Если хотя бы один из них лежит на единичной окружности, то в модели может существовать коинтеграция.

Видно, что некоторые корни лежат вне единичной окружности и некоторые на единичной окружности. Это можно подтвердить табличным представлением единичных корней.

Roots of Characteristic Polynomial

Endogenous variables: LINF LINT

Exogenous variables: C

Lag specification: 1 1 8 8 10 10 12 12

Date: 05/22/07 Time: 23:07

Root

Modulus

1.034424 + 0.130164i

1.042582

1.034424 - 0.130164i

1.042582

0.775304 + 0.648611i