Periodogram Analysis of the $\nabla \log_{10}$ Dow-Jones Series.

  Now we compute the periodogram of the differenced logg Dow-Jones series and look at that. We also test to see if there might be a periodic effect of period 5 from day of the week (e.g, Mondays would be five trading days apart, so there might be an effect from Mondays). Actually, we know from the autocorrelation that there isn't any such statistically significant effect (look at the lag 5 value of the autocorrelation! ... it is negative and about 1/2 of what it would need to be to stick outside of the error bars for white noise), but we do it for pedagogical purposes.

First of all, let's see if Splus 5.0 will give us help when we are not sure of what we are looking for:

> help(periodogram,window=T)
No documentation for topic "periodogram"

The answer is, ``NO.'' I think this sucks. So I'll just go back to the old version of Splus. Remember on the owlnet machines that you can get that by typing

/net/owlnet-d/splus-3.4/sunos5/Splus

Anyway, in the old version

S-PLUS : Copyright (c) 1988, 1996 MathSoft, Inc.
S : Copyright AT&T.
Version 3.4 Release 1 for Sun SPARC, SunOS 5.3 : 1996 
Working data will be in .Data 
> help.start()
> #Ah, my old friend the help window.  I type "periodogram" in "Topic" entry
> #gives me "spec.pgram".  I click on this and get a little window.
> #boy there's a lot of options for this guy.  Luckily, I know what
> #  most of them are.  To get the raw periodogram
> dldj.pg_spec.pgram(dldj,taper=0,detrend=F)
> dldj.pg.f_dldj.pg$freq
> dldj.pg.I_10^(.1*dldj.pg$spec)
> #I'm still not sure I have the periodogram.
> #Let's start with a plot.
> X11() #have to popup a graphics window in old Splus
> plot(dldj.pg.f,dldj.pg.I,xlab="freq",ylab="Periodogram",
+ main="Plot of Periodogram (?) for dldl")
> #I meant "dldj" in the last word; oh well.
> #Not that great, try again with a line plot
> plot(dldj.pg.f,dldj.pg.I,xlab="freq",ylab="Periodogram",
+ main="Plot of Periodogram (?) for dldj",type="l")
> #OK.  Well, looks like a lot of noise.
> #Let's see how many freqs there are
> length(dldj.pg.f)
[1] 148
> length(dldj)
[1] 291
> 291/2
[1] 145.5
> #Damn.  It padded on me even though I told it not to!!!
> #Basically, it added some zeroes at the end to get a sample size
> #   easy for the FFT to use.  I'll try again.
> dldj.pg2_spec.pgram(dldj,taper=0,detrend=F,pad=0)
> length(dldj.pg2$freq)
[1] 148
> #I guess I'm stuck with it.  That makes all the ANOVA that Box-Jenkins
> # talks about basically not valid.  But we'll try it anyway.
> > 1/dldj.pg.f[2]
[1] 294
> #So it padded 3 zeroes on the end to artificially make the series have
> #   294 values.  Remember that the index for the Fourier frequencies of
> #   a series with an even number N of observations is 0 to N/2, so there
> #   are a total of 1+N/2 Fourier frequencies.
> #Let's see what the I(f) values
> #   look like for f > 0 with a histogram:
> length(dldj.pg.I)
[1] 148
> hist(dldj.pg.I[2:148])
> #Could be an exponential distribution -- looks like it
> title(main="Histogram of Periodogram Values")
> #the last check is with the variance: them expected value of the
> #  periodogram ordinates is 2 sigma^2 if it is a white noise (very
> #  good approximation here, I'm sure).
> mean(dldj.pg.I)
[1] 6.688483e-06
> var(dldj)
[1] 6.741311e-06
> #Slight difference is due to padding, I'm sure.  But we are getting
> #   the average of the "periodogram" equal to the variance (basically).
> #So what we must have is 1/2 the periodogram.
> #But this won't affect the ANOVA, because it is a ratio of values.
> #So let's pretend we have the real periodogram (and not one with
> #   padding) and compute the ANOVA table.  We want to look for
> #   a sinusoid of period 5 -- frequency 1/5 = .2.  This isn't going
> #   to be exactly a Fourier frequency.  See the paragraph beginning
> #   at the bottom of p. 36.  So let's test the 3 frequencies closest
> #   to .2
> .4*147
[1] 58.8
> dldj.pg.f[59:61]
[1] 0.1972789 0.2006803 0.2040816
> fstat.num_sum(dldj.pg.f[59:61])/6
> # 6 d.f. in the numerator of the F-statistic (2 per periodogram ordinate)
> fstat.den_sum(dldj.pg.f[-(59:61)])/284
> # 1 d.f. lost from 291 for estimating the mean; 291-1-6 remain
> #Note the negative subscripts gives everything but
> fstat_fstat.num/fstat.den
> fstat
[1] 0.7829175
> #Clearly no evidence of a period 5 effect as F-statistics have to be
> #  somewhat larger than 1 to be significant.  Just to get a p-value:
> 1-pf(0.7829175,6,284)
[1] 0.5839091
> #p-value of 0.58 not significant.  Need a p-value less than 0.05 to
> #   be significant.

The periodogram line plot and the histogram of the periodogram ordinate values are shown below.

  

Figure 4: Line plot of the $\nabla \log_{10}$ Dow-Jones periodogram. Looks pretty random.

  

Figure 5: Histogram of the ordinate values of the previous periodogram.