Johan Stax Jakobsen
  • Member for 4 years, 5 months
  • Last seen this week
  • Copenhagen, Denmark
Fitting a GARCH(1, 1) model
Accepted answer
8 votes

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 ...

View answer
Maximum likelihood in the GJR-GARCH(1,1) model
Accepted answer
7 votes

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 ...

View answer
Distribution of normal variable subtracted from another normal random variable?
4 votes

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,...

View answer
Bootstrap sample with size greater than the original sample
4 votes

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
Initial value of the conditional variance in the GARCH process
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-\...

View answer
What's the point of (G)ARCH when you can square the residual and use ARMA?
3 votes

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 ...

View answer
GARCH(1,1) volatility forecast looks biased, it is consistently higher than Parkinson's HL vol
Accepted answer
3 votes

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 ...

View answer
Applications of ARCH models
3 votes

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 ...

View answer
Serially Uncorrelated but dependence in ARCH model
3 votes

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 ...

View answer
Persistence in TGARCH
Accepted answer
2 votes

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
Mean and Correlation of a First-Order ARCH(1) Process
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
Expression for the unconditional variance in the EGARCH model
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
The expected value and variance of E(-1X)?
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
Derivation of GARCH Student-$t$ log-likelihood
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
GARCH Model Estimation
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
Parameters in Autoregressive representation of an ARCH model
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 $...

View answer
Is my understanding on how to estimate the parameters in a GARCH model correct?
Accepted answer
1 votes

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 ...

View answer
How can i choose the optimal lag in GARCH-MIDAS?
1 votes

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 ...

View answer
How can I write an asymmetric-BEKK(1,1,1) model
Accepted answer
1 votes

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
The exponent delta of the variance recursion in a GARCH model
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
Macroeconomic variables in GARCH
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
Interest rate control variable GARCH
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
Hypothesis Test For Independence in Trials?
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
How to test if the process that generated a time-series has changed over time
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
Markov Switching GARCH - Expanding or Rolling window forecasting?
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
Newcomer question: How does the GARCH recursive formula actually work?
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 \...

View answer
Time series - autocorrelation
1 votes

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}...

View answer
Stationarity of the TGARCH
1 votes

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]+\...

View answer
Estimating the confidence interval for the volatility of a GARCH model
1 votes

Due to the fact that I am not an avid R user, I will only contribute to question 1). The GARCH process is defined by the mean equation \begin{equation} r_t = \mu + \sigma_t z_t \end{equation} and ...

View answer
Distributional forecast for $X_{t+1}$ in GARCH(1,1) with residuals student t distributed
1 votes

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

View answer