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15 votes
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Correctly applying GARCH in Python

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 ...
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15 votes
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Difference between GARCH and Heston Volatility model

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 ...
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13 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 ...
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  • 2,504
13 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 ...
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10 votes
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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 ...
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  • 6,294
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 ...
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  • 1,624
8 votes
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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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  • 3,273
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 ...
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  • 4,217
7 votes
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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 ...
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7 votes
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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 ...
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7 votes
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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 ...
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6 votes
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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-...
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6 votes

GARCH(1,1) good fit found, how to predict one day volatility ahead?

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. $\...
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  • 1,688
6 votes
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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}^...
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  • 2,504
6 votes
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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 ...
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6 votes
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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 ...
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  • 356
5 votes
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GARCH model and prediction

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 ...
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  • 1,688
5 votes
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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 ...
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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 ...
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5 votes

The use of GARCH

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 ...
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  • 2,310
5 votes
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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 ...
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  • 2,108
5 votes
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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] + \...
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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 ...
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5 votes
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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 ...
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  • 2,504
5 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 ...
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5 votes
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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 ...
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  • 3,273
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 ...
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4 votes
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Understanding the conditioning in a GARCH process

$$ E\left[ {{y_t}|{{\cal F}_{t - 1}}} \right] = E\left[ {{\sigma _t}{z_t}|{{\cal F}_{t - 1}}} \right] = {\sigma _t}E\left[ {{z_t}} \right] = 0 $$ $$ {\mathop{\rm var}} \left[ {{y_t}|{{\cal F}_{t - 1}}...
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  • 604
4 votes
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R ARMA-GARCH rugarch package doesn't always converge

There is no guarantee that the optimization method always converges! In an introduction the author of the package recommends using the "hybrid" solver, which starts out with the "solnp" and goes ...
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