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					                  Some Spectral Analysis

                         And Its Implications for
                           Macroeconomics



UNO, ECON 6204, Summer      Some Spectral Analysis and Its Implications
                                                                          1
2011, Dr. Tufte                      for Macroeconomics
                         What’s This?




UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
           Clearly It’s Musical Notation
• How is the passage of time noted in musical
  notation?
      – Horizontally
      – And then from line to line going down
• How is pitch noted in musical notation?
      – Vertically




UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
 Does Musical Notation Also Represent
           a Time Series?
• Yes
• A time series is just a collection of data in
  some framework in which observations are
  associated with time
      – In Cartesian pairs, triples, or whatever




UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
      What We Hear Is Not the Same As
              What We Play
• Musical notation only shows notes played
      – Called 1st fundamentals
• But we hear a lot more than that
      – If the musical sound doesn’t have all this other
        stuff it will sound hollow and fake
      – This is partly why some computer generated
        sounds don’t sound very “rich” or “thick”



UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
            How Strings Produce Tones
• Most music is created by vibrating strings
• The tone a string generates when plucked
  depends on
      – What it’s made out of
      – Its tension, and
      – How long it is
• The last one is the key point
      – When a musician plays they vary the length of the
        string to produce different tones

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
         What Is the 1st Fundamental?
• Given a certain material, tension, and length, a
  string will produce one tone that is more obvious
  than others: the 1st fundamental
• This tone is produced at a certain frequency that
  we hear
      – That frequency is how many peaks of complete (and
        repetitive) sine waves that are produced by the
        vibration of the string.
      – Complete is the key point. This means there are an
        integer number of complete sine waves coming off
        the string.

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
  But, Now Think About the Arithmetic
• If a string can produce 100 sine waves per
  second (with nothing left over), it can also
  produce 200 sine waves per second
      – And fulfill that key point that the waves be
        complete with nothing left over.
      – Those two tones have different frequencies
• But, since the frequency and period of a wave
  are inversely related, they also have different
  periods

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
        What Other Tones Could a String
                  Produce?
• It turns out that a string will be able to
  produce (a theoretically infinite) number of
  other tones as long as they are integer
  multiples of the 1st fundamental
      – These are called harmonics
      – All their periods will an integer relationship as well




UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
                         An Example
• The second note of the musical piece on slide
  2 is an “A”.
• When you play an “A” you get
      – It’s 1st fundamental at 440 hertz, and a
      – Harmonic at 880, and a
      – Harmonic at 1320, and a
      – Harmonic at 1760, and a
      – Harmonic at 2200, and so on …

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
  What If We Want to Notate All Those
               Tones?
• Musical notation doesn’t notate all those
  harmonics because they’re automatically
  produced
      – It doesn’t have anything to do with strings or the
        player, but more with the properties of vibration
        (and variation, and volatility)
• But, there is a way to look at all those tones,
  used by electrical engineers and music
  producers called spectral analysis
UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
  Two Views of Data Gathered Through
                 Time
• Any set of data collected over time can be
  thought of as either
      – A time series
      – A spectrum
• Music can be, and is, studied as both.
• All time series data, including macroeconomic
  data, can be studied as both.
      – In economics, we do a lot more with time series
      – But the spectrum has a most unusual story to tell
        about macroeconomics.
UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
                         Fourier Transforms
• Fourier stunned mathematicians when he
  asserted that any function could be represented
  as the sum of a finite number of sine waves, plus
  a remainder.
      – Each wave could have a different frequency
      – Each wave could have a different amplitude
      – Each wave could be out of phase (not peaking at the
        same time) with the other waves
• The most important waves for explaining a series
  would be those with the biggest amplitude
      – This is called power

UNO, ECON 6204, Summer      Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                      for Macroeconomics
                  Fast Fourier Transforms
• Today, it’s routine for computers to execute
  fast Fourier transforms for us
• These break down a time series into the
  component sine waves that sum up to it.
• One problem: the sine waves are poorly
  identified in an econometric sense (we can
  substitute one sine wave with another one
  that has a close frequency)

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
          Take this Identification Problem
                      Seriously!
• It doesn’t suggest that spectral analysis isn’t
  worthwhile,
• But it does suggest that you should not read
  too much into where the peak of a particularly
  powerful sine wave falls, because that peak
  isn’t estimated very sharply, nor is the length
  of the time to the next peak.


UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
   What Sine Waves Could You Observe
            In a Time Series?
• In order to know you have a complete wave, it
  would either have to
      – Complete 2 cycles
            • So that you’re sure it cycles
      – Complete 1 cycle in 2 or more observations
            • So that you’re sure you’re picking up a peak and a
              trough
      – Or fall somewhere in between


UNO, ECON 6204, Summer     Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                     for Macroeconomics
                         An Example
• You have 60 observations. You could then isolate
      –   A sine wave with frequency 2 and period 30
      –   A sine wave with frequency 3 and period 20
      –   A sine wave with frequency 4 and period 15
      –   A sine wave with frequency 5 and period 12
      –   A sine wave with frequency 6 and period 10
      –   …
      –   A sine wave with frequency 30 and period 2,
      –   Plus a remainder

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
     Where Do the Harmonics Fit In?
• If you’re trying to figure out something about
  the wave with frequency 4, whatever is
  causing it will also cause waves with frequency
  8, 12, 16, 20, 24, and 28 (that you can
  observe).
• This may overlap a wave of frequency 3 which
  will have harmonics at 6, 9, 12, 15, 18, 21, 24,
  27, and 30.

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
How Do We Proceed If Harmonics May
       Obscure Things a Bit?
• Focus on the lowest frequency of interest
      – Just as in music we focus on the 1st fundamental
• “Tell stories” about our data that focus on the
  lowest frequency of interest
• Pay attention to the harmonics, but keep in
  mind that a particular frequency might be a
  harmonic for more than one thing you’re
  interested in

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
     What Does All This Have to Do with
            Macroeconomics
• Consider U.S. real GDP
      – We have (about) 256 (quarterly) observations that we
        think are pretty good
• We can break that down into sine waves with
      –   Frequency 2 and period 128 (32 years)
      –   Frequency 3 and period 85.3 (21 years)
      –   Frequency 4 and period 64 (16 years)
      –   …
      –   Frequency 128 and period 2
      –   Plus a remainder
UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
                           Granger ‘64
• Granger won a Nobel Prize a few years ago for his
  work in time series
      – In the 1960’s, he pointed out that there is a “typical
        spectral shape” for the power spectrum of economic
        and financial data
            • The power spectrum plots the (important) amplitudes of the
              sine waves
            • Steeply declining as frequency increases (and period
              decreases)
      – This means that the big amplitude sine waves, that
        determine most of the data we observe, are at low
        frequencies (and have long periods)

UNO, ECON 6204, Summer     Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                     for Macroeconomics
          What Do We Find from Spectral
             Analysis of Real GDP?
                     Fact 1
• The remainder is the biggest component, and
  while it is picking up stuff whose frequency is
  too high or too low to measure, it’s dominated
  by the too low frequency data.
• Conclusion: real GDP is dominated by stuff
  that takes longer than 32 years to cycle


UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
          What Do We Find from Spectral
             Analysis of Real GDP?
                     Fact 2
• If we remove the trend from the data, all the
  power of the remainder goes away, revealing
  that the secondary, but still powerful, sine
  waves are still concentrated at low
  frequencies (and long periods)
• Conclusion: growth rates of real GDP are
  dominated by stuff that has fairly long periods
  too
UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
          What Do We Find from Spectral
             Analysis of Real GDP?
                     Fact 3
• The power of periods longer than 4-6 years
  dominates the power of periods in the 2-4
  year range
• Conclusion: politicians do not have as much
  control over the economy as they think they
  do (otherwise it would show up at the
  frequency of elections)

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
          What Do We Find from Spectral
             Analysis of Real GDP?
                     Fact 4
• Seasonal cycles are pretty big, but the power
  for cycles with periods in the range of
  business “cycles” is even higher
• Conclusion: whatever business “cycles” are,
  they’re more powerful than holiday shopping,
  good harvests, summer vacations, and so on

UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
          What Do We Find from Spectral
           Analysis of Real GDP? Fact 5
• Very high frequency cycles, with periods of a
  few months, have very low power.
• Conclusion: “surprises” and “news” that pass
  quickly don’t have much of an effect on the
  macroeconomy.




UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
         What Does All This Tell Us About
               Macroeconomics?
• Political cycles are not strong
      – So elections (and policy) don’t matter much
• Seasonal cycles are stronger
      – Weather matters a little
• Business cycles are even stronger
      – Animal spirits, inventory cycles, and so on, matter
• But something that takes even longer to cycle
  is even stronger
UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
What Could Take a Very Long Time, but
       Be Very Important for
 Macroeconomics (and Human Well-
              Being)?
• Technological change
• Social and cultural change




UNO, ECON 6204, Summer   Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                   for Macroeconomics
         A Note About “Long Waves” and
                Kondratieff Cycles
• Asserting that civilization is driven by very long
  cycles is an old idea
      – This is not really what we’re doing with spectral
        analysis
• Instead, long wave research relies on identifying
  widely separated peaks and troughs first, and
  then asserting that they are part of repeated
  cycles
      – The problem with this is that there simply aren’t a lot
        of observations to make very many complete cycles
            • They might be there, but the evidence can’t be strong until
              several hundred years pass

UNO, ECON 6204, Summer     Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                     for Macroeconomics
    What Is Spectral Analysis Doing that
      Long Wave Research Is Not?
• Spectral analysis doesn’t focus on peaks and
  troughs, but rather on amplitude
      – When we add one observation to the time series, the
        period of the cycle with frequency 2 will change
            • And so must the position of its peaks and troughs
            • All the other cycles will shift too
            • So the peaks and troughs are not sharply identified
      – But, the amplitudes of the constituent waves won’t
        change much
            • The long period waves are always the strongest, no matter
              where they peak and trough.

UNO, ECON 6204, Summer     Some Spectral Analysis and Its Implications
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2011, Dr. Tufte                     for Macroeconomics

				
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