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 ...

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 ...

Its is correct as stated in the previous answer that the difference of two independent normal random variables $X\sim N(a,c^2)$ and $X\sim N(b,d^2)$ is distributed as \begin{equation} X-Y \sim N(a-b,...

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 $z_{t+... View answer 4 votes I know of at least five ways of initializing the volatility process: 1) Set it equal to$\varepsilon_{t-1}^2$, 2) The sample variance, 3) Unconditional variance of the model ($\alpha_0/(1-\...

It is correct that, we can obtain an AR(p) representation for $X_t^2$ if $X_t$ follows an ARCH(p) process and an ARMA(max(p,q),p) representation for $X_t^2$ if $X_t$ follows a GARCH(p,q) process (see ...

This "bias" is simply a result of comparing open-to-close variance (realized measures of volatility) with close-to-close variance (GARCH model). You could try to: Use open-to-close returns in the ...

At least, we have the original paper of Engle (1982, Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation), which includes an application that looks ...

This is a common observations for daily returns series. The level is often found to be unpredictable (if not, then we would be able to make a lot of money with a simple ARMA model), while we are able ...

The first important point is to understand where the "persistence parameter" comes from in the GJR-GARCH model. The specification depends on a normality assumption for the innovations $\... View answer Accepted answer 2 votes The way to show that the correlation is zero: $$C(Y_t,Y_{t-1}) = E[\varepsilon_t (\alpha_0 + \alpha_1 Y_{t-1}^2)^{1/2}\varepsilon_{t-1} (\alpha_0 + \alpha_1 Y_{t-2}^2)^{1/2}] - E[\varepsilon_t (\... View answer Accepted answer 2 votes The moment structure of the first-order Exponential GARCH model is derived by He, Teräsvirta and Malmsten in "Moment Structure of a Family of First-Order Exponential GARCH Models". The matlab ... View answer 1 votes We have the general rules$$ E[a + bX] = a + bE[X] $$and$$ Var[a + bX] = b^2 Var[X] $$So in this case,$$ E[-1X] = -1 \cdot 100 = -100 $$and$$ Var[-1X] = (-1)^2 \cdot 0.1^2 = 0.1^2 $$View answer 1 votes For GARCH modelling with a t-distribution, we want y_t to be t-distributed with mean \mu and variance \sigma_t^2. One way to obtain this is to consider$$ y_t = \mu + \sigma_t \frac{1}{\sqrt{\... View answer 1 votes Ad 1) Yes, when creating$\sigma_t^2$you use$y_t$in the equation $$\sigma_t^2 = \omega + \beta \sigma_{t-1}^2 + \alpha y_{t-1}^2$$ You have made the assumption$E[y_t] = 0$. You have an error ... View answer 1 votes Define $$v_t = r_{t}^2 - E_{t-1}[r_{t}^2] = r_{t}^2 - \sigma_{t}^2$$ Plug this into the ARCH equation $$r_t^2 - v_t = w + \sum_{i=1}^p \alpha_i r_{t-i}^2$$ Rearranging yields the AR(p) model$...

Typically, we will assume that $X_t = \sigma_t Z_t$ where $Z_t$ is iid(0,1). How to find the log-likelihood is described in Maximum likelihood in the GJR-GARCH(1,1) model and how to implement the ...

In my paper Low-frequency Modeling for Capturing Volatility Persistence: A Dynamically Complete Realized EGARCH-MIDAS Model, it was also the case that the log-likelihood was increasing with the ...

I believe that Kroner and Ng (1998) - Modelling asymmetric comovements of asset returns is the relevant reference. You should be able to write an assymmetric BEKK model by writting $$H_t = C^* {C^*... View answer 1 votes The APARCH model is defined as (see vlab) \begin{equation} \sigma_t^\delta = \omega + \sum_{i=1}^p \alpha_i (\vert \epsilon_{t-i} \vert - \gamma_i \epsilon_{t-i})^\delta + \sum_{j=1}^q \beta_j \... View answer 1 votes Yes, it is possible to add exogenous variables directly to the "GARCH" equation in both GARCH, GJR-GARCH, EGARCH, and other volatility models. However, the asymptotic theory is not established yet in ... View answer 1 votes Research in linking financial market volatility and macroeconomic fundamentals has been a widely studied topic. Different interest rates and related variables have been included in papers from the ... View answer 1 votes I believe that one way to go about the problem is to consider a Wald–Wolfowitz runs test or a similar non-parametric test. If you define the two events A and not A, then the test should be ... View answer 1 votes If it is reasonable to model the relationship with a linear regression, then an easy way to test for a structural break is the chow test. see the wiki article here View answer 1 votes In the volatility forecasting literature most apply a rolling window approach. This is motivated by the literature on forecasting evaluation that mostly allows for fixed or rolling windows for the ... View answer 1 votes Assume that we want to model the volatility of SP500 returns, r_t. A GARCH type model consists of both a mean and GARCH equation. The mean equation can be defined as$$ r_t = \mu_t + \sigma_t \...

Yes. That is exactly the idea. Using the independence assumption it is easy to see that $Cov(X_t,X_{t-h})$ for $h>1$ is zero. Cov(X_t,X_{t-1}) = Cov(0.8 \varepsilon_{t-1}^2/(1+\varepsilon_{t-1}...

First of all, we note that \begin{align} \alpha_{1,+}^2 E[z_t^2]+\alpha_{1,-}^2E[z_t^2]-2\alpha_{1,+}\alpha_{1,-}E[z_t^+z_t^-] \end{align} should be \begin{align} \alpha_{1,+}^2 E[(z_t^+)^2]+\...

The distributional forecast of $X_{t+1,n}$ is a non-standardized Student's t-distribution with location parameter $\mu = 0$ and scale parameter $\sigma_{t+1,n}$.