# Tag Info

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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 ...

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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 ...

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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 $$r_{2,t} \sim N\left(0,\sigma_{2,t}^{2}\right)$$ $$\sigma_{2,t}^{2} = \beta_{... 10 Okay just to wind things down here, I think an important clarification is needed if readers might come and seek to a similar solution. The Geometric Brownian Motion (GBM) is a model of asset prices dynamics which is usually given as follows:$$ dS_t = \mu S_t dt + \sigma S_t dB_twhere B_t is a standard brownian motion which has several important ... 7 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 ... 6 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 ... 6 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 ... 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 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(... 6 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 ... 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 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 ... 5 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).... 4 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. 4 You need to find the values of the GARCH parameters which fit best your data. To do so, you usually create a function simulating a GARCH simulation taking, as input the parameters, and you run it through an optimizer to that the sum of the squares of the differences of the simulations points and the sample points are minimal. Note that it will not give you ... 4 Squaring normally distributed variables results chi-square distributions, which (as you imply) is why the chi-square distribution is used in hypothesis tests for the variance. If you estimate a Garch model and obtain the conditional variance at every point in time, you could use a chi-squared hypothesis test to ask a question like is the variance in a ... 4 There is no one right answer to this question, but a common starting place is to compare the bias and variance of the forecast vs. the realized variance. Take your forecasted variance \hat y and regress them against the realized variance: y = \beta_0 + \beta_1 \hat y + \epsilon A few things that you want to see: The forecast should be unbiased, ... 4 Fitting a time series on a given stock is really trade off between statistical risk and model error. If your time series is too short then your statistical error will be high. If your time series is too long, then the distribution of the market will probably have changed, and the your model error will be high. 5000 days is about 20 trading years. There is ... 4 Basically he's just saying that you don't have to estimate parameters assuming they're the same in every period. Arch and Garch parameters are typically estimated via maximum likelihood. In MLE, parameters are estimated by$$ \theta \equiv argmax\left\{ \sum_{t=1}^{T}ln\left(f\left(x_{t}|\theta\right)\right)\right\} $$where \theta are some parameters ... 4 The return equation is just an econometric equation that models stock returns (or other asset returns) as a function of: (i) intercept (i.e. the average return), (ii) some independent variables/features, (iii) noise that has zero mean and time-varying variance. There are sometimes other things in the return equation too that form more advanced models. The ... 4 Ah, this is becoming a common question, just in R now. Please look at this [question] (GARCH model and prediction), it has R code to do the prediction. In brief, you keep predicting one day ahead. \sigma_{t+k}^2 =w+\alpha u_{t+k-1}^2+\beta \sigma_{t+k-1}^2. You already know  w,\space \alpha \space and \space \beta  the starting values are the last ... 4 The mean could be the long run variance which is sig2 = fit.Constant/(1-fit.GARCH{1}-fit.ARCH{1}); I hope this explains. If not, note I ran this model through Matlab, I get different values. you can paste your m1 and m2 values and some other intermediate results so I can see why Matlab differs. EDIT: The question refers to forecasting the returns. ... 4$$ 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}}} \right] = {\mathop{\rm var}} \left[ {{\sigma _t}{z_t}|{{\cal F}_{t - 1}}} \right] = \sigma _t^2{\mathop{\rm var}} \left[ {{z_t}} \right] = \sigma _t^2  ...

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You can use Matlab too, that, in my humble opinion, is simpler than R from a syntax point of view. The model you need for is run by the Matlab function arima that can be used with seasonality option to do what you have to do. Here you can find an example and a brief explanation of the model. Type ctrl + F and search for: "Specify a seasonal ARIMA model" ...

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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 ...

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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 ...

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GARCH models are essentially white noise models with some time dependency. The reason GARCH models are used is because they have a lot of nice properties. The main being that the Conditional Volatility is time-dependent. This means that volatility can cluster. It's true that conditional vol will regress towards "normality" as a random walk process with ...

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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) ...

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