Empirical
FinanceLecture 7: Analysis of long memory
and non-stationary processes: part I.
Module Leader: Dr Stuart Fraser
stuart.fraser@wbs.ac.uk
Room D1.18 (
Social Studies)Warwick Business School 2
Today
Long Memory Processes (Mills Chp3.4)
Testing for long memory in the forward premium.
Analysis of Non-stationary Processes: Part I (Brooks Chp7.1-7.2)
Testing for autoregressive unit roots (‘unit roots’) in economic/finance data.
Seminar 6: Testing for long memory and unit roots in the real exchange rate.Warwick Business School 3
()()()()Σ∞=−−=⎟⎠⎞⎜⎝⎛+++++++=−=032...!321!2111kktkttdtLdddLdddLLyεψεε
Long memory processes (Mills Chp3.4)Example of a long memory process (Fractional White Noise)The ψweights (Woldform coefficients) will only decay if d<1The process will display mean reversionfor d<1.Binomial Expansion (see Appendix 1). of level on theeffect permanent a have Shocksmodel) ingale walk/Mart(Random 1d If0yykktt⇒=⇒=Σ∞=−εFractional sum/integral ofa white noise process.d is a real number -it can take fractional valuesWarwick Business School 3
()()()()Σ∞=−−=⎟⎠⎞⎜⎝⎛+++++++=−=032...!321!2111kktkttdtLdddLdddLLyεψεε
Long memory processes (Mills Chp3.4)Example of a long memory process (Fractional White Noise)The ψweights (Woldform coefficients) will only decay if d<1The process will display mean reversionfor d<1.Binomial Expansion (see Appendix 1). of level on theeffect permanent a have Shocksmodel) ingale walk/Mart(Random 1d If0yykktt⇒=⇒=Σ∞=−εFractional sum/integral ofa white noise process.d is a real number -it can take fractional valuesWarwick Business School 4
Long memory processes
However the process is only covariance (weakly) stationary if d<0.5.
The ACF of FWN is given by:
If d<0.5 the ACF decays hyperbolically(slowly) to zero.
⇒Possible to have a FWN process which is both mean reverting (d<1) andnon-stationary (d≥0.5)!
Compare this with the fast geometric/exponential decay of the ACF for stationary ARMA models.
For example the ACF of an AR(1) process is:
12−=dkckρkkφρ=The stationaritycondition is:(see lecture 5)1<φWarwick Business School 5
ACF of AR(1) and FWN processes (geometric vshyperbolic decay)00.10.20.30.40.50.60.70.80.911357911131517192123252729313335373941434547495153555759AR(1) φ=0.98Long Memory Model d=0.45Warwick Business School 6
Frequency domain/spectral analysisAuto-covariance Generating FunctionPopulation Spectrum()Σ∞−∞==kkkyzzgγ()()()()[]()⎭⎬⎫⎩⎨⎧+=−===ΣΣΣ∞=∞−∞=∞−∞=−−10cos221sincos212121kkkkkkikiyykkikeegfλγγπλλγπγππλλλThe AGF summarizes the auto-covariances/memoryof a time-series:The AGF is finite if the autocovariancesare absolutely summable(roughly this equates to stationary processes).()kttkyy−=,covγSee Appendix 2 for a derivation of the last line.Time series are made up of cyclical/periodic components with different frequencies λ:For example…Seasonal componentsin a time-series have a high frequency(they repeat over a short period).Long-run trend componentshave a low frequency(they repeat over very long periods).To examine the importance of cyclical components at differentfrequencies we need to analyze the spectrumof the process.Warwick Business School 7
Frequency domain/spectral analysis
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