14
votes
How to calculate the conditional variance of a time series?
Let’s take a simple example to answer a broad but interesting question:
Imagine that we have a daily return serie denoted $r_{t}$ ( which is assumed to be stationary) and let's take a little time to ...
- 2,552
14
votes
Why is GARCH(1,1) so popular, especially in academia?
Let me start with a disclaimer that I have no interest in promoting GARCH models. However, I am aware of their history, their capabilities and some practical aspects of using them. That helps me come ...
- 2,706
10
votes
Accepted
Kurtosis in GARCH
You've found parameterizations where fantastically long samples are required for sample 4th moments to converge on population 4th moments.
Quick evidence of imprecise estimation
Let $k_i$ denote ...
- 6,924
10
votes
Accepted
Realized Variance (realized volatility)
The TLDR; to your question:
How can one use realized volatility as a volatility model to do out-of-sample prediction? You extend known models to incorporate additional information procured from high-...
- 4,176
9
votes
What are the significant implications of the long-run average variance rate and why Engle won the Nobel Prize for ARCH model development?
The best answer to your question is probably given by the Nobel prize committee itself in "The Prize in Economic Sciences 2003 - Advanced Information" document. You should read it in full. Below is an ...
- 1,704
9
votes
Accepted
Is there a HAR that deals with the leverage effect?
There exists a modification of the HAR model that accounts for leverage effect (á la GJR-GARCH) in a high-frequency setting.
The semi-variance HAR model, termed the SHAR model of Patton and Sheppard (...
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7
votes
What is the preferred GARCH method in practice?
I personally use the simple Garch(1,1) for volatility filtering in the risk management area.
In fact in most cases I don't even estimate the parameters, I stick 0.94 for mean reversion, 0.04 for the ...
- 4,307
7
votes
Accepted
Does the unconditional variance implied by a GARCH equal the sample variance?
In this context, unconditional variance refers to the stationary variance level predicted by your GARCH model. This quantity need not coincide with the sample variance of the data on which the latter ...
- 14.5k
7
votes
Accepted
Why is the GARCH intercept supposed to be strictly positive?
Consider the GARCH(1,1) process
\begin{align}
r_{t+1} &= \sigma_{t+1} z_{t+1} \\
\sigma^2_{t+1} &= \omega+\alpha r^2_t +\beta \sigma^2_{t}
\end{align}
for the returns $r_t$, with ${z_t} \sim ...
- 14.5k
7
votes
Accepted
GARCH models vs VIX
These are 2 completely different ways of estimating volatility.
GARCH models are calibrated on historical time series i.e. information provided under the real-world measure $\mathbb{P}$. Although you ...
- 14.5k
6
votes
Accepted
Markov-Switching E-GARCH with R
There is now a package for that: The MSGARCH package, you can find it on CRAN.
You can find an exhaustive vignette here:
David Ardia, Keven Bluteau, Kris Boudt, Denis-Alexandre Trottier: Markov-...
- 27.3k
6
votes
Accepted
Fractionally Integrated GARCH
The ARMA(m,p) representation of GARCH(p,q) is :
\begin{align*}
\left[1-\alpha(L)-\beta(L)\right]r_{t}^{2} = w + [1- \beta(L)] v_{i}
\end{align*}
where
\begin{align}
&\alpha (L) =\sum_{i=1}^...
- 2,552
6
votes
Accepted
When modelling ARCH/GARCH effects, do we use excess returns?
GARCH models have little to do with the economics of the data generating process of the series you model, so both returns and excess returns (and log-returns, and inflation-adjusted ones, even ones ...
- 932
6
votes
Accepted
Modelling Geometric Browian Motion price model with stochastic volatility
Let me try to answer, this topic is much deeper than my answer
1. Why are these models like this unpopular?
First, these models produce marginal distributions that does not fit the market, which ...
- 401
6
votes
negative gamma value for gjr-garch output
Understanding negative gamma value for the GJR-GARCH model:
$\gamma > 0$ is not a required condition to ensure a "valid" GJR-GARCH model. Let me explain why:
As you probably know, we need ...
- 4,176
5
votes
Accepted
How do I evaluate the suitability of a GARCH model?
Which model to choose from a pool of candidate models depends on what you want to do with it.
If you want to do forecasting, you should select a model that would be expected to deliver the most ...
- 2,706
5
votes
Is a linear combination of GARCH processes also a GARCH process?
No, a sum of two GARCH processes is generally not a GARCH process.
(I am not even sure whether there exists a nontrivial special case where the opposite holds.)
By GARCH I mean the classic ...
- 2,706
5
votes
Accepted
ruGarch - Interpret test results
To test for model misspeicfication:
First ensure that auto correlation of standardized residuals resulted from the ARMA-GARCH model are not significant. Further, you can use Box-Ljung test. It test ...
- 2,218
5
votes
Accepted
GARCH volatility modeling, squared returns, and convergence
Assume that your stationary time series (here a daily close-to-close log-returns' series) is modelled as follows $\forall t \in \mathcal{T}=\{1,...,N\}$
\begin{align}
r_t &= E_{t-1}[r_t] + \...
- 14.5k
5
votes
What is the difference between conditional volatility and realized volatility?
Conditional volatility is the volatility of a random variable given (i.e. conditioning on) some extra information. E.g. in the GARCH model the conditional volatility is conditioned on past values of ...
- 2,706
5
votes
Accepted
How to account for intraday seasonality in GARCH model?
The traditional way is to pre-filter the returns thanks to the a relation similar to : $r^{f}_{t} = r_{t} /\phi_{t}$ where $r_{t}$ are the squared log returns, $r^{f}_{t}$ the filtered squared ...
- 2,552
5
votes
Accepted
EGARCH(1,1) mean
This is because $|z_t|$ is a standard half-normal random variable and have expectation $\sqrt{\frac{2}{\pi}}$.
The expectation, $\mathbb{E}\left[|z_t|\right] = \sqrt{\frac{2}{\pi}}$ is true, when $z_t ...
- 4,176
5
votes
Accepted
GARCH on returns or on log-returns?
What is usually used in practice to forecast volatility?
I believe it is log-returns.
Is it more appropriate, in general, to fit a GARCH on returns or on log-returns to estimate volatility?
The ...
- 2,706
4
votes
Filtering out AR(1) effects before using stochastic volatility model
Even though it's a straightforward extension, it took me a while (a year? yikes!); but now you can easily incorporate Bayesian ar(1) (or more generally, Bayesian regression) in joint estimation by ...
4
votes
Accepted
Density forecast of a GARCH model
EDIT :
I read more about it and I get some help with someone else, here is the correct answer :
The density forecast is the predictive likelihood value of the process
estimated at the realized ...
- 2,552
4
votes
2-step estimation of DCC GARCH model in Python
If $\log{(|R_t|)}$ is your first term, I'm not sure why this is a matrix. Modulus (determinant herein) applied to a matrix $R_t$ gives a scalar. If your implementation in python produces a matrix, ...
- 239
4
votes
Accepted
GARCH variance vs standard deviation for volatility
If your question is: "Given all the information available up to time $t$, if I compute the 1 period ahead forecast $r_{t+1}$, is the conditional volatility over $[t,t+1[$ given by $\sqrt{r_{t+1}}$?", ...
- 14.5k
4
votes
Accepted
Suggestions for a Master thesis in option pricing models
In option pricing, the entire game is fitting the skew with a fairly robust model. All the research right now is in LSV (Local Stochastic Vol) Models. Fitting these is a challenge (with PDE or ...
- 871
4
votes
Accepted
How can I compare 30 day implied volatility forecasts with GARCH forecasts?
This is a partial answer to your 2. statement. The main points are,
the conditional (on information up to time $t-1$) variance of the price $P_t$ is the same as the conditional variance of the "...
- 2,706
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