# Tag Info

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

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

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 ...

4

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) ...

4

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 model has been calibrated. That being said, in an effort to reduce the complexity of the GARCH parameters' estimation process (nasty non-linear optimisation ...

4

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] + \epsilon_t \\ &= E_{t-1}[r_t] + \sigma_t z_t \end{align} with $z_t \sim N(0,1)$ and $\{z_t\}_{t \in \mathcal{T}}$ are IID. The above equations suggest that, ...

4

If you estimate your model via Maximum Likelihood method, you are forced to re-estimate the full model. This is due to the fact that estimates are values which maximize the full likelihood, the latter being based on a recursive algorithm which use all observations (including the new one) and implies that a new observation may also impact likelihood values of ...

3

To solve for $U_t$, we can proceed as follows. First, note that \begin{align*} d\left(e^{(\theta + \frac{1}{2}\xi^2)t - \xi W_t} U_t \right) &= e^{(\theta + \frac{1}{2}\xi^2)t - \xi W_t} U_t \left((\theta+\xi^2) dt -\xi dW_t\right) \\ &\qquad+ e^{(\theta + \frac{1}{2}\xi^2)t - \xi W_t} dU_t -\xi^2e^{(\theta + \frac{1}{2}\xi^2)t - \xi W_t} U_t dt\\ &...

3

$\alpha=0$ does not imply constant volatility. Consider just a simple Garch(1,1): $$\sigma^2_t = \omega + \alpha \eta_t^2 + \beta \sigma^2_{t-1}$$ Note that: $$\sigma^2_t = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2- \sigma^2_{t-1})$$ Now add $\eta_{t+1}^2$ to both sides: $$\eta_{t+1}^2 = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2-... 3 You should check for autocorrelation. However, its presence does not necessarily mean your model will produce inaccurate figures. The ARCH family of models were developed to help analyze the volatility of a time-series. This data is assumed to display a degree of heteroskedasticity. Using the GARCH model, small amounts of auto-correlation (not of practical ... 3 EDIT : I read more about it and I get some help with someone else, here is the correct answer : The density forecast is the predictive likelihood value of the process estimated at the realized value computed in a one step ahead way. Thus for instance for a standard arma garch process with normal errors: You forecast the mean u^{f}_{t|t-1} and ... 3 Let me use a notation that I am more used to:$$ \sigma^2_{i,t} = \omega_i + \alpha_i\varepsilon^2_{i,t-1} + \beta_i\sigma^2_{i,t-1} $$where i=1,2. Since$$ \text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + \text{Corr}(X,Y)\sqrt{\text{Var}(X)}\sqrt{\text{Var}(Y)} $$and$$ \text{Var}(x_{1,t})=\sigma_{1,t}^2, \ \ \ \text{Var}(x_{2,t})=\sigma_{2,t}^...

3

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 using designmatrix = "ar(1)" as an argument to svsample. It's not well documented yet (except in the help files), but I nevertheless hope easy to use. From the ...

3

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 define main concepts : Mean Process (First moment process) The unconditional mean of $r_{t}$ denoted $u$ is just its expectation $E(r_{t})$. It is not time ...

3

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 $\sqrt{r_{t+1}}$?", the answer is NO. To compute the 1 period ahead conditional variance, you should use your model equations (see this post which might help you better understand ...

2

What is the mathematical basis to say that $u^{2}_{t}/\sigma_{t}^{2}$ will exhibit little auto-correlation in the series? Let's $r_{t}$ be a series of returns and let's assume (Assumption I) it follows a covariance stationary process defined as : $r_{t}=\sigma_{t} z_{t}$ where $z_{t}$ is i.i.d with $E_{t}(z_{t})=0$ and $Var_{t}(z_{t})=1$ ; Then $... 2 To the best of my knowledge there is no public implementation in matlab. However both R and Ox have some packages devoted to this end : -Ox - see G@rch package -R - see rugarch package 2 The standardized error in a GARCH model has unit variance (which is needed for identification) and a zero mean. Whatever the conditional distribution, it is scaled and shifted so as to fit those requirements. The answers to your questions are: No. No, they don't drop the assumption of mean zero and variance one; and Yes, they are using something like a "... 2 PYTHON I have found this class from the statsmodels library for calculating Garch models. Unfortunately, I have not seen MGARCH class/library. Below you can see the basic information about the garch models in mentioned class from the statsmodels. Probably you have to implement it by your own in python, so this class might be used as a starting point. ... 2 As far as I know, technical analysis won't work to predict intraday Forex movement. I've done so many backtest using technical analysis but it doesn't have any predictive power. The best way to predict FOREX is to find the difference of interest rates issued by both government of that currency pair. $$Pn = P_0 . e^{(r_{jpy}-r_{usd}) \Delta t }$$ $$\... 2 Yes, you can use Multivariate GARCH model to estimate the volatility of a portfolio. For example, the Constant Conditional Correlation(CCC) GARCH model. In the CCC GARCH model, it says there is a constant correlation between portfolio and the model is defined as: Once you have estimated the correlation matrix, the the composed volatility can be computed by ... 2 Interesting question, as All the answers (including mine) could not be generalized unfortunately. As far as I am concerned, I use a univariate EGARCH for risk modelling purposes (Filtered Historical Simulation (FHS), etc.). 1 - EGARCH, merely because GARCH models do not take into account so-called leverage effects, which is crucial to me for skewed and ... 2 This question has already been answered on Stack Overflow. As it is important to Quant Finance, so I have added R code here. Others users may add code of other programming software to simulate ARMA(1,0)-GARCH(1,1) model. sim.GARCH <- function( horizon=5, N=1e4, h0 = 2e-4, mu = 0, omega=0, alpha1 = 0.027, beta1 = 0.963 ){ ret <- zt &... 2 GARCH model is used to model persistence in volatility. If you square demean exchange rate and calculate autocorrelation you will find significant autocorrelation upto many lags that indicates the clustering of volatility in data. A simple ARMA(1,0)-GARCH(1,1) model can be written as :$$y_t=\mu + \phi y_{t-1}+e_t e_t \sim N(0, \sigma^2_t)\... 2 First of all I would examine whether the model performs the task it is supposed to perform, i.e. account for the conditional heteroskedasticity in the data. That would amount to testing for remaining ARCH effects in the standardized model residuals by the Li-Mak test. If the model fails the test, there is evidence that it does not do its main task well. ... 2 Before you start asking about the number of dof, how do you know that the finite sample distribution of parameters is student-t? I don't think it is. In linear regression they are student-t because of linearity and under assumption for the residual distribution. In Garch you can just say that if you estimate using max-likelihood then asymptotically (not ... 2$v$should be the total number of parameters (constants + AR + MA + GARCH + ARCH). I disagree with @kiwiakos, the student t(df) distribution is used because we are using standard errors which are estimates of standard deviations (and not true standard deviations) to compute the statistic. That is the reason why we use student t test eventhose the ... 2 1) The answer is yes: you can use the AIC/BIC to select the best model. 2) You can have confidence intervals by:$r_t^{fitted} \pm 2* \sigma_t^{fitted}$so that you have a confidence interval of$\pm 2\$ times the conditional standard deviation. You can see a plot of this by: plot(your_garchFit_object) and typing 3 to select the plot of the series with the ...

2

Information Criteria estimate the quality of a model based on the likelihood / the numbers of parameters (or degree of freedom) and the number of observations. It is a measure of goodness-of-fit and so it may suffers of overfitting problems. You can use it to compare any models (even with different errors distributions) however you may risk to select the ...

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