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14 votes

What is the difference between GARCH and ARMA?

ARMA Consider $y_t$ that follows an ARMA($p,q$) process. Suppose for simplicity it has zero mean and constant variance. Conditionally on information $I_{t-1}$, $y_t$ can be partitioned into a known (...
Richard Hardy's user avatar
12 votes
Accepted

Maximum likelihood in the GJR-GARCH(1,1) model

A conditional volatility model such as the GARCH model is defined by the mean equation \begin{equation} r_t = \mu + \sigma_t z_t = \mu + \varepsilon_t \end{equation} and the GARCH equation (this is ...
Johan Stax Jakobsen's user avatar
10 votes
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Why is a GARCH model useful?

GARCH can be used for what you call predictions. The question is: predictions of what? Predictions of volatility. The reason why GARCH is useful is because it may better explain the volatility of ...
Aksakal's user avatar
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10 votes
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Accuracy of Volatility Forecast

The point of volatility forecasting is to forecast the full predictive density. For instance, you might assume a normal future density with mean zero, and forecast the one free parameter, which ...
Stephan Kolassa's user avatar
10 votes
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R and Time Series Analysis; Suggestions for forecasting a series with a shock

This is not a time series question. ARIMA or GARCH are completely irrelevant. There is exactly one question you need to consider, and we can't tell you the answer: what happened at the end of the ...
Stephan Kolassa's user avatar
9 votes
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Forecasting time series using ARMA-GARCH in R

Obtaining accurate point forecasts for financial time series is notoriously hard. That has to do with the nature of the financial markets; actors look for opportunities to exploit any predictability, ...
Richard Hardy's user avatar
9 votes
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Fitting a GARCH(1, 1) model

I explain how to get the log-likelihood function for the GARCH(1,1) model in the answer to this question. The GARCH model is specified in a particular way, but notation may differ between papers and ...
Johan Stax Jakobsen's user avatar
9 votes
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Should log-likelihood values increase when the sample size of a simulation increases?

It depends. More importantly though, it doesn't really matter. Remember, in an iid setting, the Likelihood is the product of PDFs (or PMFs) as a function of $\theta$. If each $f(x_i|\theta) < 1$ ...
knrumsey's user avatar
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8 votes

Modeling timeseries with strong seasonality

Please look at the $y$ axes of your decomposition plots, which are on vastly different scales. Your data is almost perfectly nonseasonal. Compare this answer. As to what I would do: I would use R's <...
Stephan Kolassa's user avatar
7 votes
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Testing if the volatility of single stocks and/or indices have risen in the past

You don't need a model to show that volatilities are changing. Simply show the time series of squared returns, you'll be able to spot the clusters of high and low volatilities easily. If you want to ...
Aksakal's user avatar
  • 61.6k
7 votes

Accuracy of Volatility Forecast

Speaking about evaluating volatility forecasts in general (not GARCH in specific), I will mention an alternative to Stephan Kolassa's answer. One can also study proper scoring rules for statistics or &...
usul's user avatar
  • 884
7 votes
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How to forecast from GARCH-copula model?

How to fit a copula GARCH model? For each series (margins): (a) fit a univariate GARCH model (e.g. using ugarchspec followed by ...
Richard Hardy's user avatar
7 votes

White's test interpretation

I wouldn't base my choice of model on a test for heteroscedasticity. And I'm not the only one. Here is a quote from the great George Box: To make the preliminary test on variances is rather like ...
Peter Flom's user avatar
  • 124k
6 votes

For intuition, what are some real life examples of uncorrelated but dependent random variables?

I found the following figure from wiki is very useful for intuition. In particular, the bottom row show examples of uncorrelated but dependent distributions. Caption of the above plot in wiki: ...
yuqian's user avatar
  • 1,272
6 votes

Initial value of the conditional variance in the GARCH process

I know of at least five ways of initializing the volatility process: 1) Set it equal to $\varepsilon_{t-1}^2$, 2) The sample variance, 3) Unconditional variance of the model ($\alpha_0/(1-\...
Johan Stax Jakobsen's user avatar
6 votes

Estimating ARCH model using ML or OLS

I really felt i needed to clarify this question even though it is fairly old, because there are a lot of misconceptions in this thread. The MLE (Maximum likelihood estimator) & QMLE (Quasi ...
Lovecraft's user avatar
6 votes
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Selecting between ARMA, GARCH and ARMA-GARCH models

the p-value is greater than 0.05 and as such we CAN say that the residuals are a realisation of discrete white noise. Strictly speaking, no. Failure to reject a null hypothesis (here: absence of ...
Richard Hardy's user avatar
6 votes
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Understanding the GARCH(1,1) model: the constant, the ARCH term and the GARCH term

A GARCH(1,1) model is \begin{aligned} y_t &= \mu_t + u_t, \\ \mu_t &= \dots \text{(e.g. a constant or an ARMA equation without the term $u_t$)}, \\ u_t &= \sigma_t \varepsilon_t, \\ \...
Richard Hardy's user avatar
6 votes
Accepted

Bootstrap sample with size greater than the original sample

The objective of bootstrapping is (usually) to get some idea of the distribution of the parameter estimate(s). Since the parameter estimates were formed on the basis of a sample of size $N$, their ...
jbowman's user avatar
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6 votes
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Why do we fit (G)ARCH model?

The purpose of GARCH models is not typically to make point forecasts. When it is, even if the point forecast doesn't change over time, the width of the prediction interval will, which is generally of ...
Chris Haug's user avatar
  • 5,850
6 votes

Why is GARCH offering no predictive value?

First of all, your results look a bit strange. I would advise you to check your code. Nevertheless, I will describe a method that you can use to obtain one-step-ahead forecasts for the conditional ...
Count's user avatar
  • 1,369
5 votes
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GARCH vs GJR-GARCH

GJR-GARCH offers what vanilla GARCH has to offer, plus the leverage effect. In general, a richer model (e.g. GJR-GARCH) will fit the sample data better (at least not worse) than a simpler model (e.g. ...
Richard Hardy's user avatar
5 votes
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Forecasting with ARIMA and GARCH: does my plan look alright?

When modelling stock prices, it is quite common to transform the original prices $P_{t}$ to logarithmic returns $r_t:=\ln(P_{t})-\ln(P_{t-1})$ and then employ a GARCH model. Logarithmic returns ...
Richard Hardy's user avatar
5 votes

Testing if the volatility of single stocks and/or indices have risen in the past

ARCH/GARCH models are appropriate if there is autoregressive conditional heteroskedasticity in the data. So if there is, and if you also know the shape of change in volatility you want to test for, ...
Richard Hardy's user avatar
5 votes
Accepted

Specifying an ARMA-GARCH model without rugarch

You touch upon two main issues: estimation and model selection. Estimation For a given model specification, you may Either write down the likelihood function and feed it into a generic optimizer ...
Richard Hardy's user avatar
5 votes

Bootstrap sample with size greater than the original sample

It is perfectly fine to sample more than 500 draws from the empirical distribution. The 500 standardized residuals make up the empirical distribution from which you sample your realizations of $z_{t+...
Johan Stax Jakobsen's user avatar
5 votes
Accepted

Autocorrelation function of ARCH models

For an ARCH(1) process given by $Z_t = \varepsilon_t (\omega + \alpha Z_{t-1}^2)^{1/2}$, let $\mathbb{H}_{t}$ be the history up to time $t$, then using the law of total expectation: $\text{cov}(Z_t, ...
nestor556's user avatar
  • 280
5 votes
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GARCH specification - why are $\sigma_t^2$ and $\epsilon_t^2$ not the same?

In a GARCH model for a time series $x_t$ we have \begin{aligned} x_t&\sim i.i.D(\mu_t,\sigma_t^2), \\ \mu_t&=... \text{ (conditional mean of } x_t \text{ given past information)} \\ \sigma_t^...
Richard Hardy's user avatar
5 votes
Accepted

Are ARCH and GARCH linear or non-linear models?

In the context of the paper you are quoting, it seems the authors define a GARCH model to be linear if the past squared errors enter linearly1 into the conditional variance equation. E.g. in GARCH(1,1)...
Richard Hardy's user avatar
4 votes

Why do I get very different results estimating GARCH-M model in EViews and R (rugarch)?

This is quiet old thread and screenshots are not available anymore, but probably somebody will find my comment usefull. According to the mean equation we're dealing with GARCH-M which has $σ^2_t$ in ...
Kirill Shilov's user avatar

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