BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

20 ACTA WASAENSIAstocks.Engle (1982) suggest a Lagrange Multiplier test to test the assumption of Gaussianwhite noise t |I t−1 ∼ N (0, σ 2 ) in the dynamic regression model y t = x t β + t againstthetimevaryingalternativep t |I t−1 ∼ N (0, σt 2 ), σ2 t = α 0 + α j 2 t−j ≡ α 0 + α(L) 2 t (2.15)where t |I t−1 denotes the residuals conditional on the information set I t−1 , N (0, σt 2 )denotes the normal distribution with mean 0 and time-varying variance σt 2 ,theα j ’s arenon-negative parameters not to be mixed up with the tail index, L is the back-shift operatorand α(L) is the correlsponding polynomial in L with coefficients α j . Engle namedthis alternative ARCH for AutoRegressive C onditional H eteroscedasticity. ARCH effectshave been been extensively documented for a wide range of financial time series,including stock and stock index returns 15 .j=1ARCH effects provide a potential explanation for leptokursis of returns by application ofJensen’s inequality to (σt 2)2 in (2.15). Assuming the returns R t to be ARCH-distributedimplies for the standardized returnz t ≡ R tσ t I t−1 ∼ N (0, 1)which yields for the kurtosis of the return process:E(R 4 t )E(R 2 t ) 2 = E(z4 t ) · E(σ4 t )E(z 2 t ) 2 · E(σ 2 t ) 2 ≥ E(z4 t )E(z 2 t ) 2 = 3 (2.16)Note that the reasoning above is not confined to ARCH but may be applied to anyheteroscedastic volatility process. As such, heteroscedasticity will always increase kurtosis,no matter whether the underlying volatility process is specified as ARCH ornot.15 see e.g. Bollerslev (1987); French, Schwert & Stambaugh (1987); Lamoureux & Lastrapes (1990);Koutmos, Lee & Theodossiou (1994) and the reviews in Bollerslev, Chou & Kroner (1992); Gouriéroux(1997) and Degiannakis & Xekalaki (2004).