Non-Stationary Processes: Part IV. ARCH(m) (Autoregressive Conditional Heteroskedasticity) Models

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1 Alber-Ludwigs Universiy Freiburg Deparmen of Economics Time Series Analysis, Summer 29 Dr. Sevap Kesel Non-Saionary Processes: Par IV ARCH(m) (Auoregressive Condiional Heeroskedasiciy) Models Saionary and nonsaionary processes presened ill now are assumed o have consan variance (homoskedasiciy)-uncondiional variance. However, some series exhibi periods of unusually large volailiy resuling in non-consan variance. The volailiy in he series is modeled by aking ino accoun he condiional variance. Consider he reurn or relaive gain of a sock a ime is Y Y 1 = where Y is he price of he sock a ime. Y 1 Y = (1 + ) Y 1 Taking he logarihm of boh sides and firs difference ln( Y ) = ln(1 + ) + ln( Y 1) lny = ln( Y ) ln( Y ) = ln(1 + ) + ln( Y ) ln( Y ) [ ] lny = ln(1 + ) If he percen change,, says relaively small in magniude, hen ln(1 + ) p and lny p. Therefore, he highly volaile periods end o be clusered ogeher. ARCH(1) = σ Z and Le 2 2 σ = α + α1 1 where Z Gaussian WN(,1) The condiional disribuion of Represenaion of ARCH(1) as AR(1) = σ Z ( α + α = σ ) given 1 is N(, α + α ) = α + α1 1 + υ, where υ = σ ( Z 1) and Z Chi square(1) The Properies of ARCH process are: Le he series conain = {, 1,... } 1. E[ ] = E[ σ Z ] = E [ σ ] E [ Z ] = byindependence Var = = E σ Z = E σ E Z = E σ 2. [ ]

2 [ ] σ = Var = E σ = α + α1e 1 α σ = α + α σ σ α σ = α σ = α1 [, ] [ ] γ + h = E + h = E E + h + h 1 3. γ [ + h, ] = E E + h + h 1 = 4. If α 1 < 1, he process is Whie Noise and is uncondiional disribuion is symmerical around zero (lepokuric disribuion: see below) 2 5. If 3 1 α < in addiion o propery 4, 1 6. If 3α 1 1, in addiion o propery 5, hen GARCH(m,r) Generalized ARCH model wih order m,r is = σ Z m r = + j j + j j j= 1 j= 1 σ α α β σ GARCH(1,1) is σ = α + α1 1 + β1σ 1; α1 + β1 < 1 The process can be expressed as ARMA(1,1) 2 2 = α + ( α + β ) + σ ( Z 1) β ( Z 1) σ = σ ( Z 1) 2 β ( σ ) = β σ ( Z 1) σ β σ = α + α ρ( ) = α, h > 2 h is a causal AR(1) process wih h 1 2 is sricly saionary wih infinie variance. Conribuion of descripive saisics on Model deerminaion: The skewness and he kurosis of a disribuion are, respecively. If he disribuion is normal, K(y)=3, S(y)=. Thereofre, for any disribuion, K(y) 3 is called he excess kurosis. Under normaliy assumpion, and are disribued asympoically as normal wih zero mean and variances 6/T and 24/T, respecively. Financial daa ofen exhibi lepokurosis, i.e. a kurosis higher han 3 or an excess kurosis higher han. We consider such reurn paern especially for high frequency daa, for example daily daa. For monhly, quarerly or yearly aggregaed daa he disribuion urns more owards a normal disribuion.

3 Figure 1. The forms of Skewness and Kurosis for differen values b) Tes of Normaliy Addiional o Q-Q plo and goodness of fi ess Jarque-Bera es saisic measures he difference of he skewness and kurosis of he series wih hose from he normal disribuion. The saisic is compued as: Under he null hypohesis of a normal disribuion, he Jarque-Bera saisic is disribued as χ 2 wih 2 degrees of freedom. [H : The disribuion is Normal] 1% 9,21 ;5% 5,99 The es is only adequae for large samples, whereas for small samples you have o inerpre i cauiously. Example 1: Dax TR reurns beween (source R.Fuess) DATR Figure 2. Original series

4 ,3,2,1 -,1 -,2 -,3 Jan 65 Jan 67 Jan 69 Jan 71 Jan 73 Jan 75 Jan 77 Jan 79 Jan 81 Jan 83 Jan 85 Jan 87 Jan 89 Jan 91 Jan 93 Jan 95 Jan 97 Jan 99 Jan 1 Jan 3 Figure 3. Differenced series 1 8 Series: DLNDATR Sample 1965:1 23:12 Observaions 468 Mean Median.6365 Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Figure 4. Hisogram of he differenced series, Normaliy es and descripive saisics

5 Example 2: The series conain observaions from Isanbul Sock Exchange ( Naional Defence) daily from 7/3/2 o 31/12/ Figure 5. The plo of he original series DEFENCE

6 Series: DEFENCE Sample 1/2/ /31/27 Observaions 1874 Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy. Figure 6. The hisogram of he original daa (skewed o lef) Tes he saionariy of original daa by using ADF Tes : Tes he saionariy of differenced daa by using ADF Tes : The series becomes saionary afer differencing wih order 1.

7 Series: DIF_DEFENCE Sample 1/2/ /31/27 Observaions 1873 Mean Median. Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy. Figure 3. The hisogram of he difference daa show high kurosis and is symmeric. Normaliy es is done (Jarque-Bera) As he Kurosis is high, his is a sign for ARCH effecs The correlogram of he differenced daa shows ha he reurns are no correlaed.

8 However, he squared reurns are correlaed. Here, he Coefficien for AR and MA erms are no significan. Therefore, we regress differenced daa on consan and he variance.

9 The Model is σ Residual Checks: = σ 1 1 Boh correlogram show ha residuals and squared residuals are whie noise NO ARCH effecs lef Readings: 1. Enders, W., Chaper pp Shumway, R.H., Chaper 5.3, pp