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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 2 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 3 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 4 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 5 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 6 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 7 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 8 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 9 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 10 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 11 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 12 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 13 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 14 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 15 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 16 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 17 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 18 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 19 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 20 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 21 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 22 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 23 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 24 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 25 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 26 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 27 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 28 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 29 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 30 2011, Dr. Tufte for Macroeconomics

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