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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.)

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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
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3 Answers

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.

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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
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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).

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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.

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