# Differencing vs Detrending financial time series

I'm quite newbie to time series analysis and I have to understand what's the difference between differencing time series (i.e considering $$Y_t= X_t-X_{t-1}$$) and detrending (using linear regression for example) the series to make a time series stationary. I've read in my book that these are two diffent approaches but I don't understand which is better in which context.

Hi: It depends on what the DGP of the original process is. Is the process trend stationary or difference stationary ? If it's trend stationary then de-trending is the way to go. If it's difference stationary, then differencing is the way to go.

The two models are quite different:

Trend Stationary: $$y_t = \beta_{0} + \beta_1 \times t + \epsilon_t$$

Difference Stationary:

$$y_t = u_{t} + \epsilon_t$$
$$u_t = u_{t-1} + \omega_t$$

In the early 1980's, Nelson and Plosser (link to paper below) found that a lot of econometric series that were though to be trend-stationary were actually difference stationary and this caused an explosion of research on the question of difference versus trend stationary.

http://schwert.ssb.rochester.edu/a425/jme82_NP.pdf

• Thank you, maybe i get it. But from a statistic point of view, how I can distinguish the real DGP? I'm trying to model netflix (NFLX) Adjusted Close Price and related log-return but I really can't find it. – perseo Dec 24 '19 at 18:26
• More details: I was trying to model the time series with an ARIMA(p,1,q)+GARCH(m,n) in R but i got no stable model for the ARIMA part; in fact i got many NAs in fitted value, probably because of Hessian troubles. Now i just de-trended the ts with a simple linear model and I got an AR(1). I guess ADF test can relate but i can't understand how to distinguish the two models. – perseo Dec 24 '19 at 18:42
• There are tests that can be used but your project sounds large. If your trying to model Netflix, I would use returns rather than prices. That might get rid of a lot of the non-stationarity, be it trend or difference. See how that goes because using prices for prediction is not a good idea. Prices are EXTREMELY non-stationary. Returns might be but not as much. I would also do things in steps. Don't just fit arima-garch. Try to see what kind of arima model looks reasonable and then worry about adding garch in after that. – mark leeds Dec 24 '19 at 20:22

Let me try to write formulae to explain the differences:

• When $$X_t=a+b\,t + c\,\xi_t$$, where $$\xi_t$$ is an iid centered and reduced noise (ie $$\mathbb{E}\xi=0$$ and $$\mathbb{E}\xi^2=1$$.

With $$X_(t+1)-X_t=b + c\Delta\xi$$, you read that you increased the amplitude of the noise $$\xi$$ by a factor $$\sqrt{2}$$, you removed $$a$$ and you have no more time dependent.

• When $$X_t=a+b\,t + c\,W_t$$ where $$W$$ is a Wiener process, ie $$dX=b\, dt + c\, dW$$ and $$dW\sim {\cal N}(0,1)$$.

Here it is more natural to immediately look at $$dX$$: you removed the constant, and this time you reduced the amplitude of the noise. Moreover, since $$\mathbb{E}W_t^2=t$$ you removed the heteroskedasticity of the process (see Bollerslev's papers).

For stock returns, just take $$dX=\frac{dP}{P}$$ and you are closer to the second case. If you consider daily prices, we know the model should be more sophisticated:

• you should have a jump component because $$dW$$ is too regular (see Cont and Tankov's book)
• you should write $$c_t$$ because the volatility is time dependent, and especially it is clustered (have a look at Rob Engle's Nobel lecture).