arima arch garch(2)
TRANSCRIPT
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EKONOMETRITIME SERIES
SANJOYO
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TOPIK - TOPIK
1.Pengertian Dasar2.Pengujian Stasioneritas3.ARMA & ARIMA4.ARCH & GARCH5.VAR6.COINTEGRATION & ECM7.SIMULTAN EQUATION
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ARMA & ARIMA(1)
• Metodologi Box Jenkin:o Identifikasi stasioner ? ; jika diperlukan ➔
tranformasi. o Berdasarkan property dr autocorelasi suatu series
(yg sdh ditransformasikan) pilih model ARMA / ➔ARIMA estimasi & uji model yang cocok ➔ ➔dimana residual bersifat white noise
o Autocorrelation mengukur korelasi antara suatu series dg beberapa lag sebelumnya. Misalnya: antara Zt dg Zt-1, untuk seluruh pasangan (jumlah observasi ada n-1 pasangan).
o Lakukan forecast berdasarkan kurun waktu pengamatan yg sesuai.
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ARMA & ARIMA(2)
• ARIMA dibangun berdasarkan bahwa suatu proses stokastik dr data series memiliki struktur yg berkaitan dg:o Trend jangka panjango Nilai pd waktu sebelumnya (AR struktur)o Nilai disturban pd periode sebelumnya (MA).
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ARMA & ARIMA(3)
• Autoregresive (AR):o AR(p) Z➔ t=m+1Zt-1 +2Zt-
2+...+pZt-p+t dimana m=konstanta, t = white noise proses.
• Contoh:o AR(1) •Yt=0,8Yt-1+ t
Lat7: AR Proses
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ARMA & ARIMA(2)
• Contoh:o AR(2) •Zt=0,5Zt-1 +0,3Zt-
2+ t
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ARMA & ARIMA(2)
• Moving Average (MA) :o Zt=+0t+1t-1+...+qt-q ;=konstanta, t =white noise
proses. o Contoh:
MA(1) • Yt=2+ t+t-1
Lat8: MA Proses
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ARMA & ARIMA(2)
• Contoh:o MA(2) • Zt=2+ t+t-1 +t-2
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ARMA & ARIMA(2)
• ARMA:o Zt=+1Zt-1+0t+1t-1 N ARMA(p=1,q=1) ➔o Jika series sdh first dif• I(1) ARIMA(1,1,1)➔
Lat9: ARMA Proses
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ARMA & ARIMA(3)
• Identifikasi struktur series:
Exponentially decayExponentially decayARMA
Terputus/ terpotong setelah Lag q
MA (q)
Finete: terputus sesudah lag p
Decay exponentiallyAR (p)
Pola PACFPola ACT
• Pemilihan Lag:o ACF• max q (lag MA)o PACF • max p(lag AR) Bila model cenderung MA atau
AR saja
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ARMA & ARIMA(5)
• Kriteria pemilihan model terbaik:o Error random Q statistik (correlogram)➔o Signifikasi Veriabel t statistik➔o SE of Regresi / R2
Berkaitan dengan forecasting:• Root Mean square error (RMSE)• Mean Absolut Error (MAE)• Mean Absolut Percent Error (MAPE)
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ARMA & ARIMA(4)
CARA PERTAMA: forward regresion (basis residual white noise):• Lihat PACF untuk menentukan Lag AR(p), kemudian:• lihat correlogram apakah model sdh White Noise.• Bila belum • modelkan sebagai lag MA(q)
o Mis dari data file :univariato CPI non-stat • first diff (d=1) • ΔCPI Stasionero ΔCPI • lihat correlogram • ar(1)o OLS Δcpi c ar(1) • significant ? Ya. Lihat residual correlogram •
White noise? • tambah ma(5)o OLS Δcpi ar(1) ma(5) • significant ? Ya. Lihat residual correlogram
• White noise? • tambah ma(8) atau ma(6)o Dan seterus nyao Model Akhir:
(1). Δcpi ar(1) ma(5) ma(6) (2). Δcpi ar(1) ma(5) ma(8)
Lat9a: ARMA Proses
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ARMA & ARIMA(4)
CARA KEDUA: Backward Regression (basis Signifikansi koef):• Mis dari data file :univariat • CPI• CPI • Non-stasioner • first diff (d=1) • ΔCPI
Stasioner • Dari correlogram (ΔCPI ): ar(1) ar(7) ma(1-6)• yg
penting significant individual koefisien• OLS: ΔCPI c ar(1) ar(7) ma(1-6) • hilangkan yang
tidak significant mulai dari ma(6) dg redundant test dan seterusnya.
• Model Akhir: ΔCPI c ar(1) ar(7) ma(2) ma(5)
Lat9b: ARMA Proses
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Kriteria Pemilihan Model
1.746126 Error • WN0.0356 0.0000 0.0458 0.0069 0.0004
1.326451 0.842224 -0.119512 -0.274655 0.344174
CONSTANTAR(1)AR(7)MA(2)MA(5)
1.706071 Error • WN0.0673 0.0000 0.0004 0.0070
1.258723484 0.7521177939,0.3430463389,-0.2584077005
CONSTANTAR(1)MA(5) MA(8)
1.702521 Error • WN0.1202 0.0000 0.0005 0.0050
1.302799 0.707553 0.327614 0.259056
CONSTANTAR(1)MA(5)MA(6)
SE of RegProb Qp-valuet-ratio
EstimasiParameter
ParameterModel ARIMA
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ARCH & GARCH(1)
• GARCH (Geneneralized Autoregresive Conditional Heteroscedasticity):o Model time series dg varian tidak konstan, t. o Varian tidak konstan:
Adanya heteroscedasticity Asumsi OLS tak terpenuhi Parameter masih tak bias Estimasi standar error & confident interval terlalu narrow ➔ ➔
a false sense of percision.• Mendeteksi GARCH: secara visual ditandai
volatility clustering (adanya varian meningkat interval tertentu).
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ARCH & GARCH(2)
• Varian t dimodelkan bergantung pada :o Rata2 series ()o Volatility data yg terjadi periode sebelumnya (diukur
dg lag kuadrat residual t-12)- ARCH termo Varian forecast periode sebelumnya, t-12
• Model ARCH(q): o Yt=φ0+φ0Yt-1+t dimana t ~ N(0, t) heterosedastic
t= t t ; dimana t adalah white noise N(0,1) t2= α0+ α12t-1 +…..+ αq2t-q
Bila ARCH(1) maka: t2= α0+ α12t-1
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ARCH & GARCH(3)
• Model GARCH(p,q)o Yt=φ0+φ0Yt-1+t
dimana t ~ N(0, t) heterosedastic t= t t dimana t adalah white noise N(0,1) . . t = conditional varians Bila GARCH(1,1) maka: t2= α0+ α12t-1 +t-1 2t-1
• Pengujian Model ARCHo Engle (1982) Lagrange Multiplier test utk ARCH, dg step:
Ettimasi AR(n) (regressi) dg OLS: yt= α0+ α1yt-1 +…+ αnyt-n + t
Hitung Bila tak ada ARCH/ GARCH maka α0= α2=…= αq=0
• Moving average q lag of ε2t-1-ARCH term
• Autoregresive p Lag of σ2t-GARCH term
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ARCH & GARCH(4)
• Pengujian Model ARCHo Hipotesis:
Ho: α0= α2=…= αq=0 •tidak ada ARCH error s/d order q H1: ada ARCH
o Test Statistik TR2 ~χ2q
o Keputusan : Tolak Ho bila TR2 >χ2q
Lat10: ARCH Proses
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Threshold ARCH/ GARCH (1)
• Model T-GARCH(p,q)o Yt=φ0+φ0Yt-1+t
dimana t ~ N(0, t) heterosedastic t= t t dimana t adalah white noise N(0,1) . . t = conditional varians dimana: It-k=1 jika εt <0, lainnya =0 Good News εt-i >0; bad news εt-1 <0; mempunyai dampak pada
conditional variance. Good News berdampak αi dan bad news berdampak αi+γi
• Moving average q lag of ε2t-1-ARCH term
• Autoregresive p Lag of σ2t-GARCH term
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Threshold ARCH/ GARCH (2)
• Hipotesiso H0: γi=0 (bad news tak berdampak pada cond. Variance)o H1: γi≠0 (bad news berdampak pada cond. Variance)
• Statistik Uji : z-test : γi/SE(γi)• Keputusan: p-value < 5% • H0 ditolak