23
First, Garch models stochastic volatility. Thus its use should be limited to estimating the volatility component. The difference in some of the volatility models is the assumption made of the random variance process components.
I believe it has been popular because it is an extension of the ARCH family of models and it is relatively easy to setup and ...
16
You may want to first broadly categorize volatility models before comparing between them within each class, it does not make sense to compare standard deviation models with an implied vol model.
I would broadly classify as follows:
Historical realized volatility: Those include standard deviation (sum of squared deviations), realized range volatility ...
12
I think there are a lot of different ways to specify this problem. For simplicity, consider independent Garch processes
$$
r_{1,t} \sim N\left(0,\sigma_{1,t}^{2}\right)
$$
$$
\sigma_{1,t}^{2} = \beta_{1,1}+\beta_{1,2}\varepsilon_{1,t-1}^{2}+\beta_{1,3}\sigma_{1,t-1}^{2}
$$
and
$$
r_{2,t} \sim N\left(0,\sigma_{2,t}^{2}\right)
$$
$$
\sigma_{2,t}^{2} = \beta_{...
10
Let me start with a disclaimer that I have no interest in promoting GARCH models. However, I am aware of their history, their capabilities and some practical aspects of using them. That helps me come up with a few points to answer your question (Why is GARCH(1,1) so popular?):
GARCH is inspired by ARMA, a classic time series model, and most likely the most ...
10
Okay just to wind things down here, I think an important clarification is needed if readers might come and seek to a similar solution.
The Geometric Brownian Motion (GBM) is a model of asset prices dynamics which is usually given as follows:
$$ dS_t = \mu S_t dt + \sigma S_t dB_t$$
where $B_t$ is a standard brownian motion which has several important ...
10
Heston gives an expression for the characteristic function, from which option prices can be computed. Therefore it can be calibrated (statically) on a set of vanilla option prices with different strikes and maturities. Hence this produces risk neutral parameters that can be used to price other more exotic products. However, it is a pain to estimate the ...
9
The general procedure is to start out simple, real simple, and build your model up only as necessary. AR(q), q=0 to start with. Test the lagged autocorrelations of the error terms, and increase q until they are no longer significant. Test for ARCH, and if it's significant, you have an ARCH(q) model. Then move on with GARCH(1,q), GARCH(2,q), and when the ...
9
You want to set the parameter n.roll to the number of n.ahead, n.roll rolling forecasts you want. (The n.ahead parameter controls how many steps ahead you want to forecast for each roll date.) Thus by setting n.roll to a number almost equal to your sample size, and critically setting the out.sample parameter almost equal to your sample size, you're telling ...
9
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 define main concepts :
Mean Process (First moment process)
The unconditional mean of $r_{t}$ denoted $u$ is just its expectation $E(r_{t})$. It is not time ...
9
It doesn't matter if you use *100 or just pct_change, as long as you are consistent. However, in practice, due to underlying floating point numerical instabilities in the underlying optimization algorithms/default tolerances used in scipy/arch, having the returns expressed in %, i.e. multiplied by 100, will have a better chance of converging during the ...
9
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 excerpt.
According to the committee:
Financial economists have long since known that volatility in returns tends to
cluster and that the marginal ...
8
You can have a look at rgarch. It's quite versatile. From what I remember, you have to get it explicitly from R-Forge, as it's not available from CRAN.
See the rgarch website for more details.
Last time I checked, usage was something like this:
spec.gjrGARCH = ugarchspec(variance.model=list(model="gjrGARCH", garchOrder=c(1,1)), mean.model=list(armaOrder=c(...
8
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 your estimated kurtosis in simulation $i$. Looking across your $i = (1,\ldots, 1000)$ simulations, your $k_i$ estimates are all over the place. What's your ...
7
I think there is some room for improvement here.
1. GARCH
GARCH models are appropriate for modeling time series that exhibit a heavily-tailed distribution and display some degree of serial correlation.
That's not the case. GARCH is used for modelling series where there is serial correlation in variance, not in actual observations. And heavy tails are ...
7
Ah, this is becoming a common question, just in R now. Please look at this [question] (GARCH model and prediction), it has R code to do the prediction.
In brief, you keep predicting one day ahead. $\sigma_{t+k}^2 =w+\alpha u_{t+k-1}^2+\beta \sigma_{t+k-1}^2$. You already know $ w,\space \alpha \space and \space \beta $ the starting values are the last ...
7
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 squared error and I get the constant by matching the series variance. My experience is that there is no point pretending to finetune parameters when vol is ...
7
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 can obviously use them for forecasting, the core information which is used to build the model is backward-looking in nature (historical behaviour of the stock)....
6
I'm guessing, and correct me if I'm wrong, you want to create a number of possible paths the stock price could follow with the local volatilty given by GARCH depending on the simulated history, or in pseudocode:
N <- numberOfPaths
T <- numberOfSteps
for (i in 1:N) {
newSeries <- pastPrices
for (t in 1:T) {
epsilon <- normrnd(0,1)
...
6
ARCH and GARCH are, by essence heteroskedastic models, that is, with non-constant volatility.
If you fit these models to your sample, it will provide you with a time series of the volatility for each point (you can construct it actually). If the values are not the same for all $t$, then the volatility is not constant, according to these models.
What you ...
6
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 model has been calibrated.
That being said, in an effort to reduce the complexity of the GARCH parameters' estimation process (nasty non-linear optimisation ...
6
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 N (0,1)$ IID.
In what follows, let us distinguish the conditional return variance
$$ V [ r_{t+1} \vert \mathcal{F}_t ] = \sigma^2_{t+1} $$
from the ...
5
I am going to supply an answer that is quite similar to SRKX's (which is very very good) because I want to discuss in more detail a few important things. First, you cannot use a stochastic volatility model for the SDE that you've provided as that's GBM with constant diffusion. However, based on what you've said it's obvious you wish to model a discretized ...
5
I don't use Stata often, but the help() function is typically very good. Try help(garch). It looks like the command is
garch _depvar_ _indepvars_ _options_
Here's the help page on the web.
5
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 definition of GARCH due to Bollerslev (1986), not an arbitrary variation like EGARCH, IGARCH, FIGARCH or whatever else.
Let me provide an example. Take two ...
5
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-Switching GARCH Models in R: The MSGARCH Package (2016)
Abstract
Markov-switching GARCH models have become popular to model the
structural break in the ...
5
SOLUTION:
Let $r_t$ be the log-return at time $t$, and $\hat{r}_t$ be the predicted log-return from the regression model.
Initialize $loglik(0:T)=0$,$\epsilon_1=0$, $\sigma_1 = 0$, $\mu=U(0,1)*0.0001,\phi=U(0,1)*0.01$, $\alpha_0=U(0,1)*0.00002,\alpha_1=U(0,1)*0.01,\beta_1=0.9 + U(0,1)* 0.01$, $B=10,000$
For $b$ = 1 to $B$
$\quad$ For $t$ = 2 to $T$:
$ \...
5
The mean could be the long run variance which is
sig2 = fit.Constant/(1-fit.GARCH{1}-fit.ARCH{1});
I hope this explains.
If not, note I ran this model through Matlab, I get different values. you can paste your m1 and m2 values and some other intermediate results so I can see why Matlab differs.
EDIT: The question refers to forecasting the returns. ...
5
Any ARCH type model always requires an additional model for the mean of the time series. If nothing is said about the mean model, then usually is simply a time average plus residual. So, if $y_t$ is your stationary time series, the mean model would be
$$ y_t = \bar{y} + \epsilon_t $$
where $\bar{y}$ is the average value of $y_t$. And then $\epsilon_t$ would ...
5
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 joint significance of auto correlation upto lag $K$.
Leverage effect is tested by sign bias test. If $p$ value is less than .05 (assumed significance level) ...
5
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] + \epsilon_t \\
&= E_{t-1}[r_t] + \sigma_t z_t
\end{align}
with $z_t \sim N(0,1)$ and $\{z_t\}_{t \in \mathcal{T}}$ are IID.
The above equations suggest that, ...
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