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out zero (for real valued time series) so usually we only need to consider the range [0, π] .
The frequenciesare calculated as follows:
The periodof the cycles are given by T/j:
•The shortestcycles repeat every 2 periods (period=T/(T/2)=2).
•The longestcycles repeat every T periods (period=T/1=T)
•As T→∞these cycles neverrepeat ⇒long-run trend component.
•We look at the ‘frequency zero’part of the spectrum to analyze the importance of long-run trends in the data. odd). ( 21,...,1even) ( 2,...,1 where,2TTjTTjTjj−===πλ
Warwick Business School 13
GPH test for long memory
Need to restrict the frequencies used in estimation to low frequencies –otherwise estimate of dwill be biased by higher frequency cycles in the series.Therefore need to choose a cut-off number of frequencies g(T) in the GPH regression.such that:A common choice for g(T) is:()TgjTjj,...,1 ,2==πλ()()0limlim=∞=∞→∞→TTgTgTT⇒Number of frequencies increases with TBandwidth = ⇒estimator becomes increasingly ‘tuned’to frequency zero (long run component) as Tincreases.()10 ,<<=μμTTg()0→Tgλμ=0.5 is typically used.Warwick Business School 14
Application: Testing for long memory in the £/$ forward premium (see Seminar 4).0000.0004.0008.0012.0016.0020.0024.00280.00.40.81.21.62.02.42.83.2LAMBDASPECTRUM
Sample spectrum of the
forward premium00.10.20.30.40.50.60.70.80.911234567891011121314151617181920212223242526272829303132333435363738394041There is evidence for long memory in the forward premiumin both the time domain(sample ACF) and frequency domain(sample spectrum). The analysis in Seminar 4 suggested thepresence of a unit rootin the forward premium (⇒non-stationary process) -incompatiblewith finance theory.
Sample ACF of the forward premiumWarwick Business School 15
GPH estimates of the long memory parameter
()()1147 345.0===TTTg636.0ˆˆ=−=βdBased on a 95% confidence interval: the long memoryparameter lies between 0.461 and 0.810 ⇒cannot reject the hypothesis that the forward premium is stationary (d<0.5).ddσˆ96.1ˆ±Dependent Variable: LOG(SPECTRUM)Method: Least SquaresSample: 1 34Included observations: 34VariableCoefficientStd. Errort-StatisticProb. C-11.05090.487439-22.67130LOG(4*SIN(LAMBDA/2)^2)-0.635590.089053-7.137230R-squared0.614179 Mean dependent va-7.74298Adjusted R-squared0.602122 S.D. dependent var1.395653S.E. of regression0.880344 Akaike info criterion2.640014Sum squared resid24.80016 Schwarz criterion2.7298Log likelihood-42.8802 F-statistic50.94001Durbin-Watson stat0.068861 Prob(F-statistic)0Warwick Business School 16
Auto-regressive Fractionally Integrated Moving Average (ARFIMA) processes
More generally a process is ARFIMA if:
The process is stationary if d<0.5(and all the remaining roots of the AR characteristic polynomial lie outside of the unit circle: see Lecture 5).
The process is invertible if d>-1(and all the remaining roots of the MA characteristic polynomial lie outside of the unit circle: see Lecture 5).
ARFIMA can model a rich variety of short-runand long-runbehaviour of a time-series.
They are now used quite often in empirical finance along with standard ARMA models (see Baillie, 1996, for applications in finance).
()()()ttdLyLLεθφ=−1Warwick Business School 17
Non-stationary processes (Analysis of Price Series)
So far our analysis has involved weakly stationary processes:
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