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 ...
Richard Hardy's user avatar
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 ...
Matthew Gunn's user avatar
  • 6,944
10 votes
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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-...
Pleb's user avatar
  • 4,276
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 ...
zer0hedge's user avatar
  • 1,704
9 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 (รก la GJR-GARCH) in a high-frequency setting. The semi-variance HAR model, termed the SHAR model of Patton and Sheppard (...
Pleb's user avatar
  • 4,276
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 ...
Quantuple's user avatar
  • 14.6k
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 ...
Quantuple's user avatar
  • 14.6k
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 ...
Quantuple's user avatar
  • 14.6k
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-...
vonjd's user avatar
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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}^...
Malick's user avatar
  • 2,572
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 ...
Igor Pozdeev's user avatar
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 ...
ryc's user avatar
  • 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 ...
Pleb's user avatar
  • 4,276
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 ...
Neeraj's user avatar
  • 2,238
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 ...
Richard Hardy's user avatar
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 ...
Richard Hardy's user avatar
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] + \...
Quantuple's user avatar
  • 14.6k
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 ...
Richard Hardy's user avatar
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 ...
Malick's user avatar
  • 2,572
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 ...
Pleb's user avatar
  • 4,276
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 ...
Richard Hardy's user avatar
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 $\sqrt{r_{t+1}}$?", ...
Quantuple's user avatar
  • 14.6k
4 votes
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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 ...
Drew's user avatar
  • 871
4 votes
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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 "...
Richard Hardy's user avatar
4 votes

What are the significant implications of the long-run average variance rate and why Engle won the Nobel Prize for ARCH model development?

$V_L$ is the long-run variance (or the unconditional variance) if and only if $\gamma=1-\sum_{i=1}^n \alpha_i$, because the long-run variance compatible with the model $$ \sigma_n^2 = \gamma V_L + \...
Richard Hardy's user avatar
4 votes
Accepted

GARCH modeling - sliding or expanding window?

If you are just modelling volatility and not stochastic volatility of volatility then it should be better to use a sliding window. The reason is that volatility itself is time-varying and therefore an ...
phdstudent's user avatar
  • 8,306
4 votes
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Simulation of a DCC-GARCH

Simulating a DCC-GARCH(1,1) model Given that you already have a given set of proper defined parameters for the DCC-GARCH model, the standardized residuals $\varepsilon_{t-1}$ are recovered from the ...
Pleb's user avatar
  • 4,276
4 votes

Pros and cons of mean equation equal to zero in a GARCH model

When you model log-returns $(Y_t)$ by $Y_t=\varepsilon_t$ where $\varepsilon_t|\mathcal{F}_{t-1}\sim N(0,\sigma^2_t)$ and a standard GARCH($p,q$) model with $$\sigma_t^2=\omega+\sum_{i=1}^p \alpha_{i}\...
Kevin's user avatar
  • 15.9k
4 votes
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Why is volatility unobservable even ex post?

Let me start from the beginning. What do you observe in financial markets? The data, the information that is given to you in as raw a form as possible, are things like bid prices, ask prices and ...
Stéphane's user avatar
  • 2,476

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