Non-stationary process -> Stationary process

[Ari]

This may be a naive question but here goes:

I'm reading about all the fancy ways one can convert a non-stationary time series into a stationary one, and then go to town on the ARIMA models to make predictions about the future. It seems a bit unsettling to me though that one can just e.g. log the entire time series and then claim that this new, somewhat stationary time series has anything to do with the original one? As an extreme example, if you do a coordinate transformation from a flat line into a parabola, and then have some revelations about that parabola, those revelations don't really translate back to the original flat line. Returning back to the log(time series) example, ARIMA (or any other stationary regression tool) has no idea that you're using a log'd version of some other time series, as far as it's concerned it's a completely different beast.

Can someone convince me, or show me some good papers/arguments as to why non-stationary -> stationary is legit?

Thanks in advance,
Ari

 

[DavidHilbert]

What's your background?

As is, your question is very vague. Simply taking log of time series doesn't make much sense in the first place, as in general you can have negative values (I don't think anyone in finance wants to deal with complex Log) and in the second place, simply taking log (assuming positive inputs) will create stationary time series from non-stationary only very scarcely.

What you mean is probably taking log differences or differences in genral (ARIMA case). Log difference is used to approximate cumulative returns (if the underlying time series is price) and the resulting time series doesn't have to be stationary, but often so it turns out to be (check out ADF test).
And simply taking differences may result in stationary time series as you might be lucky enough to get rid of trend or seasonality. If you actually write down the equations and take their difference, you will see that the coefficients in which you are interested remains in the same place while the trend/seasonal ones disappear.

So in general no, taking log, log difference or difference of time series does not need to result in stationary ts. However, given the way you asked this question, I would recommend to spend more time studying the math behind it. And no, no one who puts their money on it really believes simple ARIMA predictions are anywhere close to the actual future realizations, so don't get so hyped about it.

 

[Ari]

I'm a physicist.

Yeah, I was pretty sure that my question was vague, I'm just trying to get a handle of this stuff.

Thanks for the reply.

 

[yetanotherquant]

You cannot convert an arbitrary non-stationary process into a stationary one
(btw, ARIMA is NON-Starionary, but accounting for I-Part, you can reduce it to ARMA, which is stationary).
There is a wide class of locally-stationary processes, i.e. though they are non-stationary, their parameters evolve slowly enough so that one realization of a stochastic process is sufficient to infer the process parameters.
You may have a look how I used LSW (locally stationaly wavelet) processes to analyze the volatility of stocks.

 

[Travis Gipson]

Regression analysis does not care whether or not some time series has been transformed via the log function. The least squares estimates and computation will still be BLUE. What changes in a log transformation is the interpretation of our model. For instance, we could have a log-linear model (y is logged x is not). We could have a log-log model (both y and x are logged). Or we could have a linear-log model (only x is logged).

As an example for interpreting the coefficients, I'll present two of the cases.

Log-Linear: The beta coefficient on x represents the percentage change in y given a unit change in x, all else constant.

Linear-Linear: The beta coefficient on x represents the unit change in y given a unit change in x, all else constant.

Also, stationarity by means of differencing has much to do with the original time series, as it represents the changes in some variable, rather than the variable itself. We simply just model "delta" y or "delta" x as you will see in the time series literature. The problem with modeling a stochastic or stochastic trending time series is that the least squares estimates will no longer be unbiased. By differencing a time series and it becoming stationary, we can model some process as "white noise" with mean zero most commonly. This transformation allows us to gather information about the relationship of a set of variables without bearing all of the burden of their stochastic-ness, trend-ness, or lack-there-of.

 

[rajanS]

log transformations are generally done to stabilize the variance of the time series whose conditional variance is proportional to its conditional mean. This is generally done so that the data can have less skewness. If you plot log of a series against the lagged value, you will notice there is generally less variability when compared to differences vs lagged values. Hence, log transformations allows this transformation to be less heteroscedastic and hence easier to model compared to a non logged data. At the end of the day, it depends what your underlying data is and why you are doing this. Log transformations don't render a series stationary. As a proof, If it's a price series, then diff(log(x)) gives you the returns over a period of time, which is stationary, And can be fit by an AR model. Or in which case, an ARIMA on the log of the series.