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Addendum
The GPH regression for lnr (log real exchange rate) indicated that the series is covariance nonstationary (. 5.0≥dHowever, the GPH regression 代写留学生论文/留学生论文代写 https://www.51lunwen.org
requires 5.0Note that if a series then)(~dIyt())1(~1−−dIyLt(in words: the first difference is integrated of order ). So if 1−d5.15.0<≤d (i.e., is covariance nonstationary) then will be stationary: it will be integrated of some order in the interval . ty()tyL−15.015.0<−≤−d
Based on the above, the GPH test for the real exchange rate returns will provide an estimate of . Importantly, since the returns are stationary 1−d()5.01<−d, the GPH regression will be consistent and asymptotically normal.
Firstly, with the ‘untitled page’ (first page) in the workfile active, create the real exchange rate returns:
On the workfile toolbar click Genr and enter
dlnr=d(lnr)
Now estimate the spectrum for dlnr
Open spectrum.prg and click on the Run button.
Program Arguments: dlnr 20
Execution mode: Quiet
Figure A1: Spectrum of the real exchange rate returns .0004.0006.0008.0010.0012.0014.00160.00.40.81.21.62.02.42.83.2LAMBDASPECTRUM
Compare this spectrum with the previous one for the log level of the series. First differencing has removed the spike at frequency zero. In other words differencing has removed the long-run trend component from the series.
In particular, the spectrum appears to have a finite positive value at frequency zero. This suggests that the returns are I(0) (and hence the log real exchange rate itself is I(1)). See box below: ‘Different shapes of spectra around frequency zero’
Different shapes of spectra around frequency zero
Consider the process. This process is integrated of order: . For example: if this is a ()tdtLyε−−=1d()dI1=drandom walk/martingale process ( ; if this is a 0=dwhite noise process ; if is a fractional value then this is a dfractional white noise process .
The spectrum for this process is given by (see lecture 7): ()πσλλ2122diyef−−−=
• For the spectrum has a spike (0>descapes to infinity) at frequency zero:
o ()∞=−=−πσ211022dyf.
• For the spectrum has a 0=dfinite positive value at frequency zero o .
• For the spectrum is 0d Frequency ()λ ()λyf ()0,>ddI ()0,02>−d for 0Figure A2: Shapes of spectra around frequency zero
For example, over differencing anprocess results in an ()0I()1−Iprocess which has a zero valued spectrum at frequency zero. This observation forms the basis for the KPSS test (see mainhandoutforSeminar6).
Finally run the GPH regression using the returns spectrum to estimate . 1−d
On the main toolbar click Quick/Estimate Equation and enter:
log(spectrum) c log(4*sin(lambda/2)^2)
Sample: 1 20
Dependent Variable: LOG(SPECTRUM)Method: Least SquaresDate: 03/08/07 Time: 19:11Sample: 1 20Included observations: 20VariableCoefficientStd. Errort-StatisticProb. C-7.1527230.032983-216.8620LOG(4*SIN(LAMBDA/2)^2)-0.0079480.0076-1.045760.3095R-squared0.0