# Looking at Time-Series Using Waves

Author: Andrew McCulloch

The term time-series is used to refer to observations made over a period of time, such as the maximum monthly temperature or quarterly GDP. In many cases, the focus of analysis is on the variation in observations as a function of time. For some problems, such as modelling seasonality, it is more useful to study the variation in a time-series as a function of wavelength or frequency. It is perhaps simplest to illustrate this idea with an example and in the following I’ll use a time-series for the maximum monthly temperature in England from 1910 to 2012.

In analysing this data the main question concerns whether the average temperature is rising over time, but we know that any rise in temperature over this period is going to be smaller in magnitude than the month-to-month variation in temperature within a year. In order to remove the seasonal variation in temperature, one approach is to use a model which includes a set of month indicators to capture the month-to-month variation in temperature. Using this approach the figure below shows the difference in mean temperature in each month relative to the temperature in January, the coldest month of the year. The figure shows that mean monthly temperatures peak in July and August when the maximum temperature tends to be around 14 degrees higher than in January.

In the frequency domain the main building block of time-series analysis is a wave. The two main parameters (or unknowns) which characterise a wave are how fast it goes up-and-down (termed the frequency) and the height of the wave (or amplitude). In modelling seasonal variation we are fortunate in that we know that the pattern of seasonal variation repeats on a 12 month cycle. Rather than a set of month indicators, we include indicators with the characteristics of a wave having a period of 12 months. Using this approach the figure below plots the predicted maximum temperature for the period from 2000 to 2012 together with the observed maximum temperature in each month. The advantages of using a wave to model seasonal variation rather than a set of monthly indicators are that we have to estimate fewer parameters and we get a nice smooth trend rather than discrete jumps from month-to-month.

The idea that we can model variation in a time-series as a function of frequency can be extended to situations where there is more than one cycle in the data. For example, the figure on the left below shows two waves. The first has an amplitude of 2 and a period of 8 years so that over a 12 year interval we observe 1.5 cycles (the blue one). The second has an amplitude of 1 and a period of 1.5 years so that over a period of 12 years we observe 8 cycles (the red one). The figure on the right plots the series produced by adding the two waves in the figure on the left and we can see that even though we only have two components of variability the resulting series is starting to have characteristics similar to that of many economic series.

There is a snag, however, in that we usually don't know what the characteristics of the cycles are that are responsible for the variation we see in a time-series. It is possible, however, to estimate how the variation in a time-series is distributed over frequencies using a technique called spectral analysis. An area where this technique is useful is in the estimation of business cycles which I talked about in my last article. In the figure above I chose to plot waves with a period of 1.5 and 8 years because variation in macroeconomic series between these frequencies is commonly used to define the business cycle. At shorter periods or higher frequencies variation in macroeconomic series is usually considered to be noise while at longer periods or lower frequencies variation is part of the trend.

My last article looked at how we can use smoothing to remove a trend from GDP leaving an estimate of the business cycle. Another way to produce an estimate of business cycle fluctuations is to decompose the variation in GDP into cycles with different frequencies and then add-up the cycles with periods of between 1.5 and 8 years. In the time-series literature this approach is termed filtering. The figure below shows one such estimate of the UK business cycle using what is termed the Christiano-Fitzgerald filter.

The results are fairly similarly to those in my previous article with economic downturns taking place around 1973, 1979-1980, 1986-1987, 1992-1993, 2009. I'm not sure why the period 1986-87 shows-up as a period of below trend growth as this is generally considered to have been a time of fairly strong growth in the UK. I suspect that judgement plays a key role when using filtering techniques to analyse time-series. Perhaps not one to try at home.

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