Choosing Log or Simple Returns for ARCH and GARCH Volatility Models
Summary
The document addresses whether squared simple returns or squared log returns should be used when studying volatility and applying ARCH or GARCH models. It defines log returns as the difference between the logarithms of consecutive prices and says this is the usual convention for fitting these models. An example points to an EGARCH specification with Student’s t errors, illustrating how log-return data can be used in practice.
The response emphasizes that taking logarithms is a modeling choice rather than a universally correct rule. The suitable return definition depends on assumptions about the data or on the researcher’s model-selection approach. Consequently, squared simple returns should not be described as identical to the log-return inputs commonly used in GARCH work. The document does not compare their empirical forecasting performance or provide a decision procedure; it offers a general modeling principle and a conventional starting point.
Key ideas
- Log returns are calculated as the difference between consecutive log prices.
- Log returns are the usual inputs for ARCH and GARCH volatility models.
- Choosing simple or log returns depends on assumptions about the data and the modeling objective.
- Squared simple returns and squared log returns are distinct quantities and should not be presented as interchangeable.
Tags
Full text
# Squared returns and volatility
# Squared returns and volatility
Squared returns are considered pillars of GARCH/ARCH modelling and most used method for forecasting or studying volatility.
Can you tell me how to calculate it from simple stock price. Is it better it to calculate based on logs or just simple prices?
UPDATE from the Original Poster (posted as an answer)
My question is that do taking log or not taking logs and getting squared returns, would reveal the same volatility. I actually did not take log as I saw on Youtube a method to calculate returns. Now when I find justification for using squared returns, can I quote squared returns as same as logaritmic squared returns and say that I follow the same squared returns as used for GARCH modelling? I want to say that I used squared returns because it is used in volatility models and is a convention in finance but problem is that it uses logrithmic squared returns. Can I still quote that I follow the tradition of GARCH / ARCH modelling while using squared returns (squared returns calculated as simple formula but not taking log). My problem is whether I can still say that I follow the same pattern as GARCH models do for squared returns despite I do not use logarithmic squared returns?
## Answer by drumath (score 1)
https://quant.stackexchange.com/a/46403
There is no single best way of modeling time series. Taking or not taking logs is a modeling question. The answer in general depends on either your belief about the structure of the data set or some model selection method (the one that you believe in) or maybe something else.
That being said, the usual way to go is to take logs and apply (G)ARCH on the log-returns $r_t$, defined as $$r_t=\log p_t-\log p_{t-1},$$ where $p_t$ is the price at time $t$.
Sample code using the `R` package `rugarch` is below. The code fits an EGARCH(1,1) model with Student's $t$ errors to the S&P 500 returns in some period and then plots the standardized residuals.
```
library(rugarch)
data(sp500ret)
spec <- ugarchspec(variance.model = list(model = 'eGARCH', garchOrder = c(1, 1)), distribution = 'std')
fit <- ugarchfit(spec, sp500ret[1:1000, , drop = FALSE], solver = 'hybrid')
plot(fit, which=9)
```
For a tutorial on `rugarch` see for example this tutorial.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.