# Question on Levinson's proposed discrete form of Wiener filter

The whole foundation of Levinson's discrete version of Wiener filter is based on the assumption of stationarity of a time series, and aims to predict a value based on the past observed values. Now, if I try to use the theory to interpolate/predict values in a time series comprising financial data or other practical time series like the GDP of a country over some years, is it justified, since the financial data series is not necessarily stationary? Does the whole theory crumble due to this when applied for financial data, or do I still get adequate results, depending on certain properties of the data?

Any insight in this regard would be helpful.

• I use the "Allan Deviation" (Or "Allan Variance" otherwise known as a 2 sample variance) at work in the characterization of Atomic Clocks and specifically I find it useful to show the shorter-term time durations of signals over which stationarity can be assumed. The Allan Deviation (ADEV) is used to characterize the output frequency stability over various averaging times but can be applied to any time domain waveform for the same purpose. Look into using it for your financial time-domain data-- The horizontal axis is averaging time (tau) and the vertical axis is an rms accuracy metric. – Dan Boschen Sep 23 '18 at 23:50
• For low taus (small time offsets) the vertical axis metric will decrease as tau increases, implying stationarity. What you will find is eventually a floor will be reached, and then will actually start to increase as tau increases (drift). – Dan Boschen Sep 23 '18 at 23:52

Not specific to statistical Wiener filtering but also true in general, any algorithm will fail to work, if its assumptions are not met. However, it's also known that most algorithms will perform quite acceptable when their assumptions are sufficiently holding.

In this sense, if a statistical algorithm relies on the stationarity of its inputs, then it will be acceptably performing fine when the inputs are sufficiently stationary. Hence the keyword is sufficiency...

One might therefore argue that, a slowly changing (in statistical characteristics) non-stationary input would still allow the short term predictions possible , albeit with more erros than the ideal expectation of a strictly stationary input.

Taken the other way, many (if not all) nonstationary processes can be considered to be approximately stationary over a short enough period of time. Indeed this is the whole basis of applicability of most statistical signal processing algorithms that makes any use of real world data (which is, as you have outlined, nonstationary by its very nature, except the exceptional cases). Divide the input into pieces, assume stationarity over that period, apply the algorithm.

That being said for certain engineering signals of the type: speech, music, image, video, radar, sonar, seismology, you should make it clear by yourself that those financial data are also approximately stationary over some short periods of time. Otherwise you will have trouble in applying those algorithms that rely on stationary inputs to work.

• Thanks. Just as an extension, I am taking the data of 58 years for a country's GDP and am trying to extrapolate the GDP for 59th year. The data, from observation, is linearly increasing. I know its not possible to answer precisely, yet would you consider the series to be stationary in general? That is, the extrapolation does yield a reasonably good estimate as per the trend. But, how to claim that applying the stationary assumption on the GDP data is valid? – Curiosity Sep 23 '18 at 16:20
• Also, for finite data (data of finite length), in general, what trend of the data will assure me that stationarity is definitely not applicable to the data? – Curiosity Sep 23 '18 at 16:22
• Financial data has a less predictable nature than physical signals bound by physical laws. Note that, when a process is known to be non-stationary, then going back for long periods of observations does not help much in predicting the future. So it does not much matter whether you take 58 years of data or 5000 years of data, if it's nonstationary. Indeed it's about the correlation between data samples that determines the predictability. If the process has uncorrelated samples (white noise) then it's not possible to predict it even if it's (WS) stationary. So look for correlation... – Fat32 Sep 23 '18 at 16:51
• So, is it logical to apply the discrete Wiener interpolation method for such financial data, with the stationarity assumption? – Curiosity Sep 23 '18 at 18:06
• That is, the data samples of GDP that I considered are indeed correlated, but I can't guarantee that the data is stationary. So, what do I conclude from that? Can discrete Wiener estimation theory still be applied? – Curiosity Sep 23 '18 at 18:10