Choosing Differencing or Detrending for Financial Time Series

# Differencing vs Detrending financial time series


# 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.

## Answer by mark leeds (score 6)

https://quant.stackexchange.com/a/50429

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

## Answer by lehalle (score 2)

https://quant.stackexchange.com/a/50432

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).