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Empirical Finance：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)
代写留学生论文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 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 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
The population spectrum measures the portion of the variance of y which is attributable to periodic components with frequency λ:
–Analysis of the spectrum is referred to as frequency domain analysis.
Analysis of autocovariances/autocorrelations is referred to as time domainanalysis.
The spectrum and autocovariancesare simply ‘two-sides of the same coin’:
•The spectrum is just a function of the autocovariances(and vice-versa): they contain the same information (albeit expressed differently).
•Whether you analyze the spectrum or the autocovariancesis simply a matter of context.Warwick Business School 8
Frequency domain/spectral analysis
λmeasures the frequency of the periodic components in radians: it can take any value in the range [-π, π] .
•But the spectrum is symmetric ab本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。