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20 votes

What is the difference between GARCH and ARMA?

Edit: I realized the answer was lacking and have thus provided a more precise answer (see below -- or maybe above). I have edited this one for factual mistakes and am leaving it for the record. ...
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18 votes
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For intuition, what are some real life examples of uncorrelated but dependent random variables?

In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns $r_t:=(P_t-P_{t-1})/P_{t-1}$, with $P_t$ the price at time $t$, themselves ...
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12 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 (...
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12 votes
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Fit a GARCH (1,1) - model with covariates in R

Here is an example of implementation using the rugarch package and with to some fake data. The function ugarchfit allows for the ...
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  • 3,880
10 votes
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Time series analysis: since volatility depends on time, why are returns stationary?

I think your problem is that you confuse the UNconditional variance and the conditional variance. Indeed, you can have a time-varying conditional volatility but a constant unconditional variance. ...
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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 ...
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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 ...
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9 votes
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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 ...
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9 votes

Time series analysis: since volatility depends on time, why are returns stationary?

The augmented Dickey-Fuller test assesses whether the time series under inspection has a unit root or not. The test is designed specifically for that purpose. It either rejects the null of unit root ...
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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 ...
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8 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, ...
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7 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 $f(x_i|\theta) < 1$ ...
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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 &...
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  • 610
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-\...
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6 votes
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Estimating ARCH model using ML or OLS

It seems that you did not get the equations (nor the idea) quite right. The formula for the AR($p$) model should be $$y_t=\beta_0+\beta_1 y_{t-1}+\dotsb+\beta_p y_{t-p}+\varepsilon_t;$$ the formula ...
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6 votes

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

A simple example is a bivariate distribution that is uniform on a doughnut-shaped area. The variables are uncorrelated, but clearly dependent - for example, if you know one variable is near its mean, ...
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  • 15.6k
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: ...
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  • 1,254
6 votes
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Should I keep a non-significant intercept in a GARCH model?

The intercept of a GARCH model should be kept in the model for the following reasons. If you force the intercept to be zero AND the sum of ARCH and GARCH coefficients is less than one (which will ...
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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 ...
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6 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 ...
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6 votes
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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 ...
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  • 32.7k
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 ...
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  • 1,103
5 votes
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ARMA/GARCH estimation in sequence

Doing joint estimation is the preferred way. If you do estimation in two stages, a logical inconsistency arises. In the first stage of seasonal ARIMA estimation there is an effective assumption of ...
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5 votes

Procedure for fitting an ARMA/GARCH Model

You may look at ACF/PACF plots but this will only be practical if the data generating process is very simple, like AR($p$) or MA($q$). Once you encounter an ARMA($p,q$)+GARCH($s,r$) process where $p,q,...
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5 votes
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Cannot replicate the AIC in a GARCH model

The formula for the AIC can be found on page 23 in here. See below: > (-2*l1)/length(dat)+2*(length(garch1@fit$coef))/length(dat) [1] -3.32639
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  • 7,108
5 votes

The use of GARCH

Consider the data generating process (DGP): $y_{t}= x_{t}^{\prime}\beta+\varepsilon_{t}$ (1) $\varepsilon_{t}= \sigma_{t}z_{t},\quad z_{t}\sim i.i.d.\, N\left(0,\,1\right)$ (2) $\sigma_{t}^{2}= \...
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5 votes
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ARMA GARCH estimation process in practice

I suggest you should determine both the ARMA and the GARCH parts simultaneously. If you determine the ARMA part first by temporarily ignoring GARCH, this will lead to inconsistent ARMA parameter ...
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5 votes
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Which econometric models can be used to forecast security returns + ARIMA/GARCH questions

My goal is simply to ... find statistically significant predictive results. Also, is there a particular market you would look at (energy, rates, equities)? Most if not all the established and liquid ...
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5 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 ...
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5 votes
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GARCH vs SV for Forecasting

This is more like an extended comment than an answer as the answer really depends on the data series at hand. 1) For the purpose of probabilistic forecasting of financial and macro-economic ...
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