SUNUMZaman Serilerinde Durağanlık
Analiz Kapsamı
Zaman Serilerinin Düzey Halleri
Zaman Serilerinin Korelogram Analizi
Logaritmik Dönüşümü Yapılmış Zaman Serileri
Logaritmik Dönüşümü Yapılmış Zaman Serilerinin Birinci Derecen Farkları
Fark Serileriyle Kurulan Doğrusal Modelin En Küçük Kareler Yöntemiyle Regresyon Analizi
Breusch Godfrey Otokorelayon Testi
Değişen Varyans Analizi İçin White Testi
Yapısal Kırılma Testleri (Cusum / Cusum of Squares)
Jarque-Bera Normal Dağılım Testi
Engle ve Yoo (1987) Eştümleşme Analizi
Statik (Ex-post) Öngörü
Dinamik (Ex-ante) Öngörü
Zaman Serilerinde Durağanlık
𝐸 𝜀𝑡 = 0
𝑉 𝜀𝑡 = 𝜎2
𝑉 𝜀𝑡 , 𝜀𝑠 = 0
𝜀𝑡~ 𝑁𝐷
𝐾𝑜𝑣𝑎𝑟𝑦𝑎𝑛𝑠 𝑑𝑢𝑟𝑎ğ𝑎𝑛𝑙𝚤𝑘
𝐺üç𝑙ü (𝐺𝑎𝑢𝑠𝑠𝑖𝑎𝑛) 𝑑𝑢𝑟𝑎ğ𝑎𝑛𝑙𝚤𝑘
Analiz Kapsamı
Değişkenler
: Tüketici Fiyat Endeksi (TÜFE) (2003=100): Tüketici Güven Endeksi: 2 Yıl Vadeli Devlet Tahvili: M2 Para Arzı
Gözlem Sayısı : 122Frekans : AylıkZaman Aralığı : Aralık 2005 – Ocak 2016Kullanılan Yöntem : En Küçük Kareler Yöntemi (EKK)
Model 1: Fark Serileriyle Kurulan Doğrusal Modelin En Küçük Kareler
Yöntemiyle Regresyon Analizi
Dependent Variable: DLTUFE
Method: Least Squares
Date: 03/03/16 Time: 22:14
Sample (adjusted): 2006M05 2016M01
Included observations: 117 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
DLM2 -0.112835 0.046330 -2.435437 0.0165
DLGUVEN -0.045788 0.017268 -2.651685 0.0092
DLTAHVIL 0.001577 0.000622 2.534316 0.0126
C 0.009981 0.001074 9.296794 0.0000
DLTUFE(-4) -0.284613 0.086239 -3.300287 0.0013
R-squared 0.199790 Mean dependent var 0.006664
Adjusted R-squared 0.171211 S.D. dependent var 0.007988
S.E. of regression 0.007272 Akaike info criterion -6.967812
Sum squared resid 0.005923 Schwarz criterion -6.849770
Log likelihood 412.6170 Hannan-Quinn criter. -6.919888
F-statistic 6.990815 Durbin-Watson stat 1.806012
Prob(F-statistic) 0.000046
∆𝑙𝑡𝑢𝑓𝑒𝑡 = 0,009981 − 0,112835∆𝑙𝑚2𝑡 − 0,045788∆𝑙𝑔𝑢𝑣𝑒𝑛𝑡 + 0,001577∆𝑙𝑡𝑎ℎ𝑣𝑖𝑙𝑡 − 0,284613∆𝑙𝑡𝑢𝑓𝑒𝑡−4
)0.001074 0.046330 0.017268 0.000622 (0.086239
)9,29 −2,43 −2,65 2,53 (−3,30
Model 1: Breusch Godfrey Otokorelayon Testi
Breusch-Godfrey Serial Correlation LM Test:
F-statistic 2.105190 Prob. F(2,110) 0.1267
Obs*R-squared 4.313220 Prob. Chi-Square(2) 0.1157
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Date: 03/16/16 Time: 23:41
Sample: 2006M05 2016M01
Included observations: 117
Presample missing value lagged residuals set to zero.
Variable Coefficient Std. Error t-Statistic Prob.
DLM2 0.017220 0.046711 0.368661 0.7131
DLGUVEN -0.002503 0.017240 -0.145166 0.8848
DLTAHVIL -0.000264 0.000642 -0.411482 0.6815
C -6.97E-05 0.001072 -0.064986 0.9483
DLTUFE(-4) -0.025378 0.087639 -0.289571 0.7727
RESID(-1) 0.099728 0.098900 1.008366 0.3155
RESID(-2) -0.185257 0.098423 -1.882259 0.0624
R-squared 0.036865 Mean dependent var -2.97E-20
Adjusted R-squared -0.015670 S.D. dependent var 0.007145
S.E. of regression 0.007201 Akaike info criterion -6.971186
Sum squared resid 0.005704 Schwarz criterion -6.805927
Log likelihood 414.8144 Hannan-Quinn criter. -6.904093
F-statistic 0.701730 Durbin-Watson stat 1.957250
Prob(F-statistic) 0.648789
𝐻0: 𝑂𝑡𝑜𝑘𝑜𝑟𝑒𝑙𝑎𝑠𝑦𝑜𝑛 𝑦𝑜𝑘𝑡𝑢𝑟𝐻1: 𝑂𝑡𝑜𝑘𝑜𝑟𝑒𝑙𝑎𝑠𝑦𝑜𝑛 𝑣𝑎𝑟𝑑𝚤𝑟
𝑇𝑒𝑠𝑡 𝑠𝑜𝑛𝑢𝑐𝑢 𝐻0 ℎ𝑖𝑝𝑜𝑡𝑒𝑧𝑖 𝑘𝑎𝑏𝑢𝑙 𝑒𝑑𝑖𝑙𝑚𝑖ş𝑡𝑖𝑟.𝑀𝑜𝑑𝑒𝑙 1 𝑑𝑒 𝑜𝑡𝑜𝑘𝑜𝑟𝑒𝑙𝑎𝑠𝑦𝑜𝑛 𝑏𝑢𝑙𝑢𝑛𝑚𝑎𝑚𝑎𝑘𝑡𝑎𝑑𝚤𝑟
Model 1: Değişen Varyans Analizi İçin White Testi*
𝐻0: 𝑀𝑜𝑑𝑒𝑙 1 𝑎𝑟𝑡𝚤𝑘𝑙𝑎𝑟𝚤𝑛𝑑𝑎 𝑧𝑎𝑚𝑎𝑛𝑎 𝑏𝑎ğ𝑙𝚤 𝑑𝑎ğ𝚤𝑙𝚤𝑚 𝑑𝑒ğ𝑖ş𝑚𝑒𝑚𝑒𝑘𝑡𝑒𝑑𝑖𝑟𝐻1:𝑀𝑜𝑑𝑒𝑙 1 𝑎𝑟𝑡𝚤𝑘𝑙𝑎𝑟𝚤𝑛𝑑𝑎 𝑧𝑎𝑚𝑎𝑛𝑎 𝑏𝑎ğ𝑙𝚤 𝑑𝑎ğ𝚤𝑙𝚤𝑚 𝑑𝑒ğ𝑖ş𝑚𝑒𝑘𝑡𝑒𝑑𝑖𝑟
𝑇𝑒𝑠𝑡 𝑠𝑜𝑛𝑢𝑐𝑢 𝐻0 ℎ𝑖𝑝𝑜𝑡𝑒𝑧𝑖 𝑘𝑎𝑏𝑢𝑙 𝑒𝑑𝑖𝑙𝑚𝑖ş𝑡𝑖𝑟.𝑀𝑜𝑑𝑒𝑙 1 𝑑𝑒 𝑑𝑒ğ𝑖ş𝑒𝑛 𝑣𝑎𝑟𝑦𝑎𝑛𝑠 𝑏𝑢𝑙𝑢𝑛𝑚𝑎𝑚𝑎𝑘𝑡𝑎𝑑𝚤𝑟
Heteroskedasticity Test: White
F-statistic 1.228216 Prob. F(14,102) 0.2669
Obs*R-squared 16.87837 Prob. Chi-Square(14) 0.2627
Scaled explained SS 12.54079 Prob. Chi-Square(14) 0.5630
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 03/16/16 Time: 23:42
Sample: 2006M05 2016M01
Included observations: 117
Variable Coefficient Std. Error t-Statistic Prob.
C 5.30E-05 1.17E-05 4.525277 0.0000
DLM2^2 -0.004903 0.017520 -0.279825 0.7802
DLM2*DLGUVEN 0.011040 0.013927 0.792703 0.4298
DLM2*DLTAHVIL 0.000298 0.000401 0.742982 0.4592
DLM2*DLTUFE(-4) 0.052604 0.060424 0.870583 0.3860
DLM2 2.57E-05 0.000803 0.032030 0.9745
DLGUVEN^2 0.000320 0.002087 0.153414 0.8784
DLGUVEN*DLTAHVIL -0.000125 0.000143 -0.875187 0.3835
DLGUVEN*DLTUFE(-4) 0.035278 0.025120 1.404354 0.1633
DLGUVEN -0.000138 0.000283 -0.488810 0.6260
DLTAHVIL^2 1.33E-06 2.82E-06 0.471393 0.6384
DLTAHVIL*DLTUFE(-4) -0.000669 0.000946 -0.707281 0.4810
DLTAHVIL 5.40E-06 1.12E-05 0.482438 0.6305
DLTUFE(-4)^2 -0.052176 0.069912 -0.746301 0.4572
DLTUFE(-4) -0.000591 0.001488 -0.397209 0.6920
R-squared 0.144260 Mean dependent var 5.06E-05
Adjusted R-squared 0.026805 S.D. dependent var 6.47E-05
S.E. of regression 6.39E-05 Akaike info criterion -16.36037
Sum squared resid 4.16E-07 Schwarz criterion -16.00624
Log likelihood 972.0815 Hannan-Quinn criter. -16.21660
F-statistic 1.228216 Durbin-Watson stat 1.635594
Prob(F-statistic) 0.266914
∗ White testi, normal dağ𝚤l𝚤m koşullu aramamaktad𝚤r.
Model 1: Yapısal Kırılma Testleri (Cusum / Cusum of Squares)
-40
-30
-20
-10
0
10
20
30
40
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
CUSUM 5% Significance
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
CUSUM of Squares 5% Significance
Model 1: Jarque-Bera Normal Dağılım Testi
0
2
4
6
8
10
12
-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015
Series: Residuals
Sample 2006M05 2016M01
Observations 117
Mean -2.97e-20
Median -0.000943
Maximum 0.016451
Minimum -0.018257
Std. Dev. 0.007145
Skewness 0.003546
Kurtosis 2.621661
Jarque-Bera 0.698055
Probability 0.705374
𝐻0: 𝑀𝑜𝑑𝑒𝑙 1′𝑒 𝑎𝑖𝑡 𝑎𝑟𝑡𝚤𝑘𝑙𝑎𝑟 𝑛𝑜𝑟𝑚𝑎𝑙 𝑑𝑎ğ𝚤𝑙𝚤𝑟.𝐻1:𝑀𝑜𝑑𝑒𝑙 1′𝑒 𝑎𝑖𝑡 𝑎𝑟𝑡𝚤𝑘𝑙𝑎𝑟 𝑛𝑜𝑟𝑚𝑎𝑙 𝑑𝑎ğ𝚤𝑙𝑚𝑎𝑚𝑎𝑘𝑡𝑎𝑑𝚤𝑟.
𝑇𝑒𝑠𝑡 𝑠𝑜𝑛𝑢𝑐𝑢 𝐻0 ℎ𝑖𝑝𝑜𝑡𝑒𝑧𝑖 𝑘𝑎𝑏𝑢𝑙 𝑒𝑑𝑖𝑙𝑚𝑖ş𝑡𝑖𝑟. 𝑀𝑜𝑑𝑒𝑙 1 𝑑𝑒 𝑎𝑟𝑡𝚤𝑘𝑙𝑎𝑟 𝑛𝑜𝑟𝑚𝑎𝑙 𝑑𝑎ğ𝚤𝑙𝑚𝑎𝑘𝑡𝑎𝑑𝚤𝑟.(𝐺𝑎𝑢𝑠𝑠𝑖𝑎𝑛 𝐷𝑢𝑟𝑎ğ𝑎𝑛𝑙𝚤𝑘 𝑠𝑎ğ𝑙𝑎𝑛𝑚𝚤ş𝑡𝚤𝑟)
Model 2: Engle ve Yoo (1987) Eştümleşme Analizi
𝐻0: 𝜌 = 0, 𝑒ş𝑡ü𝑚𝑙𝑒ş𝑚𝑒 𝑦𝑜𝑘𝑡𝑢𝑟. 𝑢𝑡 ~ 𝐼 1𝐻1: 𝜌 < 0, 𝑒ş𝑡ü𝑚𝑙𝑒ş𝑚𝑒 𝑣𝑎𝑟𝑑𝚤𝑟. 𝑢𝑡 ~ 𝐼(0)
∆𝑙𝑡𝑢𝑓𝑒𝑡 = 𝛼1 + 𝛼2∆𝑙𝑚2𝑡 + 𝛼3∆𝑙𝑔𝑢𝑣𝑒𝑛𝑡 + 𝛼4∆𝑙𝑡𝑎ℎ𝑣𝑖𝑙𝑡 + 𝛼5∆𝑙𝑡𝑢𝑓𝑒𝑡−4 + 𝜀𝑡
∆ 𝜀𝑡 = 𝜌 𝜀𝑡−1 + 𝑖=1
𝑘
𝜓𝑖∆ 𝜀𝑡−𝑖 + 𝑢𝑡 )𝑢𝑡~𝑖𝑖𝑑(0, 𝜎2
)𝜀𝑡~𝑛𝑖𝑖𝑑(0, 𝜎2
Dependent Variable: DKAL1
Method: Least Squares
Date: 03/03/16 Time: 22:35
Sample (adjusted): 2006M11 2016M01
Included observations: 111 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
KAL1(-1) -0.905691 0.093870 -9.648303 0.0000
DKAL1(-5) -0.143060 0.067824 -2.109270 0.0372
R-squared 0.466189 Mean dependent var 1.98E-05
Adjusted R-squared 0.461292 S.D. dependent var 0.009301
S.E. of regression 0.006827 Akaike info criterion -7.118082
Sum squared resid 0.005080 Schwarz criterion -7.069261
Log likelihood 397.0535 Hannan-Quinn criter. -7.098277
Durbin-Watson stat 1.887883
𝜏ℎ𝑠 = 𝜌
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑡 ℎ𝑎𝑡𝑎 𝜌= −9,64 < 𝜏𝑘𝑑 = 4,22
𝑇𝑒𝑠𝑡 𝑠𝑜𝑛𝑢𝑐𝑢 𝐻0 ℎ𝑖𝑝𝑜𝑡𝑒𝑧𝑖 𝑟𝑒𝑑𝑑𝑒𝑑𝑖𝑙𝑚𝑖ş𝑡𝑖𝑟.𝐷𝑒𝑛𝑘𝑙𝑒𝑚 𝑒ş𝑡ü𝑚𝑙𝑒ş𝑚𝑒 𝑟𝑒𝑔𝑟𝑒𝑦𝑜𝑛 𝑑𝑒𝑛𝑘𝑙𝑒𝑚𝑖 𝑜𝑙𝑎𝑟𝑎𝑘 𝑎𝑑𝑙𝑎𝑛𝑑𝚤𝑟𝚤𝑙𝑚𝑎𝑘𝑡𝑎𝑑𝚤𝑟.𝛼2, 𝛼3, 𝛼4, 𝛼5 𝑢𝑧𝑢𝑛 𝑑ö𝑛𝑒𝑚 𝑑𝑒𝑛𝑔𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑟𝑒𝑙𝑒𝑟𝑖𝑑𝑖𝑟.
Statik (Ex-post) Öngörü
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
LTUFE LTUFES
𝑳𝒕𝒖𝒇𝒆 𝑔𝑒𝑟ç𝑒𝑘 𝑑𝑒ğ𝑒𝑟𝑙𝑒𝑟𝑖, 𝒍𝒕𝒖𝒇𝒆𝒔 𝑖𝑠𝑒 𝑡𝑎ℎ𝑚𝑖𝑛𝑑𝑒ğ𝑒𝑟𝑙𝑒𝑟𝑖𝑛𝑖 𝑔ö𝑠𝑡𝑒𝑟𝑚𝑒𝑘𝑡𝑒𝑑𝑖𝑟.
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