# Why non-stationary data cannot be analyzed?

Searching online, i found out that non-stationary cannot be analyzed with traditional econometric techniques as in case of non-stationarity some basic model assupmtions are not met and correct reasoning on relationships between non-stationary time-series is impossible.

Is there anyone who can enlighten me what are the basic model assumptions and the correct reasoning on relationships?

Furthermore, how does techniques such as detrending, detrending, seasonal adjustment, and transformation test helps to make those non-stationary data into useful stationary data for analysis?

(My thinking is that for example if the non-stationary stock data are gibberish, no amount of effort would be able to make them useful for predicting the stock market.)

• Even though assumptions are violated, it doesn't mean you can't analyze it. If you're forecasting it may be okay to have biased parameters resulting from nonstatinoary. However if your goal is to test an empirical hypothesis by analysing standard errors, then yes it is unacceptable. – user2763361 Nov 5 '13 at 13:29

There is a lot of ways to understand why stationarity allows to apply usual time series analysis. Here is one more.

Very often, the theoretical justification of what you do in time series need to be able to identify the mean formula and the expectation: $$\frac{1}{N}\sum_{n=1}^N X_n \underset{N\rightarrow +\infty}{\longrightarrow} \mathbb{E} X,$$ where the $X_n$ are drawn from the distribution of $X$.

Or at least you need something like that: $$\mathbb{E}\left( \frac{1}{N}\sum_{n=1}^N X_n \right)= \mathbb{E} X.$$

There two equalities are false when the $X_n$ are not i.i.d. And when you speak about non stationarity, it is about facing a stochastic process $X_n$ that would better written $X_t$ and not occurrences of the same random variable. For more details about what stationary means, see the excellent Azencott, R. and D. Dacunha-Castelle (1986, June). Series of Irregular Observations: Forecasting and Model Building (Applied Probability) (1 ed.). Springer.

It means that:

1. it will be a little harder to define your $X_n$ (filtration, measurable, adapted, etc -- you know all these words...)
2. in fact $X_n+X_m$ has potentially not the same behaviour that two times $X_n$. And that is really an issue to use classical time series results.

That being said, what can we do with financial time series? "simply" consider them as stochastic processes, and by chance we have an enormous literature about such processes (the best reference: Shiryaev, A. N. (1999, April). Essentials of Stochastic Finance: Facts, Models, Theory (1st ed.). World Scientific Publishing Company.):

1. first of all we know (from a theoretical perspective), that results on Brownian motions can be (more or less) used on the smooth ones. Thanks to the Doob-Meyer theorem (of course there can be an ugly time change in between, but nevertheless it is convenient for theoretical needs).
2. for practical use (as it have been answered already) the best approach is to try to reduce your process to a stationary one... It is not always easy. That for you need to guess a change of variable and to check that your succeeded into splitting your process in two parts: one is easy to deal with (like a deterministic function of time or of external information), the other (potentially multi dimensional) should be stationary.
3. of course you can deal with non stationary processes if needed, but it is not that easy. A typical example are Hawkes processes; they are not stationary but you can deal with them (it is somehow an extreme example since they are not smooth at all neither, but it is the first example I have in mind).

Saying that you can't analyze something as is does not make it garbage. You can't eat flour "as-is", but that doesn't mean you throw it out.

In order to use "standard" analysis tools, you must first transform the series into something compatible. Some examples of such a transformation include k-th order differences or a log transformation. These transformations allow one to analyze the data, while not losing the essence of the data.

• All relevant information can be lost upon first differencing. – user2763361 Nov 8 '13 at 13:46

You can check the wikipedia page to find out "the the basic model assumptions" for the a stationary random process, and I assume "the correct reasoning on relationships" are the model that describe a random process.

But intuitively speaking, if the data are sampled from a stationary random process, then you can predict the future by deductively extrapolate the past data with more or less confidence (e.g. AR, which assumes stationary processes). Or if you have a model to describe the random process, then we can deduce the future based on the model.

If the process is not stationary and we don't have an appropriate model, it's much harder to do prediction. Techniques such as detrending can somewhat circumvent this problem under certain assumptions. For example, detrending by different assumes that trend is modeled by a function with finitely many non-zero derivatives, otherwise the trend cannot be removed by differencing. Detrending by different also assumes the fluctuation or noise (the remainder after detrending) is stationary, so we can predict based on the detrended data.

An excellent introduction can be found here (see 25.6).

The standard errors of non-stationary processes doesn't have the same properties of stationary times series, that is the F-tests and t-values are not reliable for hypothesis testing. So for example if you try to estimate the relationship between two non-stationary variables, and the t-statistic says that the relationship with them are highly significant, this might as well be spurious. However, if you first difference the the two variables the true relationship between the two variables are revealed.

One way to deal with non-stationary time series is to work with the first differences (if they are integrated of order 1, which they often are). Some data might just need detrending, and some just needs to be seasonally adjusted.

When it comes the to stock prices, the non-stationarity is not the reason they "can't" be predicted. Their pricing are simply just too complicated mechanisms which are too hard to model.

• I think the (almost) always difference advice can be misleading. It's important to know why and when they should difference or not. For instance, if they were to difference interest rates, then you would be ignoring the longer-run mean-reversion. Estimating an AR model in differences wouldn't help matters. From the perspective of an AR model, it is only the dependent variables that would need to be differenced for the t statistics to make sense. Lags could still be in levels, allowing for mean-reversion effects. – John Sep 3 '13 at 14:14
• "standard errors of non-stationary processes" doesn't mean much. Your comment applies to what is known as 'mean stationarity'. – Ryogi Sep 8 '13 at 12:08
• @John Very, very true. Some information in a levels process is effectively lost by differencing. You can recover this information through a cumulative sum but from the perspective of chucking it into a model as a predictor, the information is gone. – user2763361 Nov 5 '13 at 13:30

Non-Stationary process can be analyzed and there are various models available that can be used . For example, Autoregressive Integrated Moving Average model (ARIMA) models are used to explain homogeneous non-stationary models as well as random walk with drift can be used for explaining several such series.

have a look at this link : http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/tutorials/xegbohtmlnode37.html

did you try to study the cointegration of your data with other data? if cointegration exist than you may make benefit from research such: Risk Management of Co-integrated Commodities Karl Larsson and Marcus Nossman (2011)