The
Limits of the| by Andrew Rowson, supervised by Prof. A.R.Champneys | ||
| Home A Dismal Science Economic TheoryClassical Neoclassical Keynesian Cycles and Crises Measuring cycles Economic Time Series Data Processing Spectral Analysis Cycle Modelling Kaldor's Trade Cycle A Kaldorian model Conclusion Full Report |
Spectral Analysis
The Discrete Fourier
Transform Spectral analysis can help to isolate important frequencies within the signal through the Discrete Fourier Transform (DFT). This is a transformation from the time to the frequency domain and for a sequence x of length N the DFT is complex vector X (also length N) where  In practice the discrete Fourier transform is carried out using the fast Fourier transform (FFT) algorithm which reduces the number of computations significantly. The relative contribution to the variance, or power, of the different frequency components is known as the power spectrum which, in the frequency domain, is given by the square of the amplitude of the Fourier coefficients, |X(k)|2. The spectral plot can be used to determine how many cyclic components are present and whether a dominant frequency exists. A range of spectra for different countries can be seen below, in each case the data was filtered using the Hodrick-Prescott filter.  There is evidence of at least three cycles in these spectra; 3.0 years, 5.3 years and 9.5 years but the result is not conclusive and might be expected among or trading partners. Spectral analysis also has a number of difficulties, one of which is the length of reliable datasets. For reasonable estimates it is suggested that between 100 and 200 observations should be the minimum. It has also been suggested that a series should be at least seven times the length of the cycle under consideration. Beyond this statistical significance should be considered by comparison with similarly composed random series. Also the best estimates of frequency components come from stationary series which is not the case for most real data. |
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