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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

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 ...


3

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 ...


3

$$ E\left[ {{y_t}|{{\cal F}_{t - 1}}} \right] = E\left[ {{\sigma _t}{z_t}|{{\cal F}_{t - 1}}} \right] = {\sigma _t}E\left[ {{z_t}} \right] = 0 $$ $$ {\mathop{\rm var}} \left[ {{y_t}|{{\cal F}_{t - 1}}} \right] = {\mathop{\rm var}} \left[ {{\sigma _t}{z_t}|{{\cal F}_{t - 1}}} \right] = \sigma _t^2{\mathop{\rm var}} \left[ {{z_t}} \right] = \sigma _t^2 $$ $$ ...


2

work you way from GARCH(4,4) to GARCH(0,0) removing the intercept too. 5*5*2-1 = 49 estimations Make sure your coefficients are all statistically significant at least to 95% confidence. Make sure you have no autocorrelation in your error terms. pacf and acf should be clean. Likelihood ratio tests assess whether you lose explaining power from ...


2

In the case of application in finance, usually, GARCH is used in estimating realized volatility of returns based on the weight we would like to give to each past observation. Ultimately after estimating (calibrating) the parameters of the model to an existing time-series, GARCH is used for forecasting multi-step ahead return (future) volatility. Different ...


1

You can use the known result, that when $X\sim N(0,1)$, then $aX\sim N(0,a^2)$ where $a=\sigma_t$ is conditionally constant.


1

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. ...


1

For the question in your title, The mean reversion of the volatility is due to the Moving Average part of the volatility process. The solution would be to set $\beta = 0$. In other words you have to use an AR process for the volatility (so an ARCH model for price). The restriction in p and q come from the estimation process of the parameters. You test ...


1

Perhaps not the most encouraging answer, but: I would think that it is contingent upon the specific implementation, magnitude, regularity, and transiency of arbitrage available as well as the volatility estimate time-scale. In a very simple case, the existence of arbitrage opportunities would likely result in larger fraction of informed traders (relative to ...


1

If you look at it from a mathematical point of view - presence of arbitrage should not matter for volatility estimates. Absence of arbitrage can be associated with the existence of an equivalent martingale measure for the bank account numeraire. (first fundamental theorem of asset pricing) Let's assume the real world process is something like ...


1

I would confirm it. For time series forecasting, one can use 3 versions of random walk: RW model 1 (basic geometric random walk): stock returns in different periods are statistically independent (uncorrelated) and identically distributed (constant volatility) RW model 2: stock returns in different periods are statistically independent bot not identically ...


1

Since you are talking about using volatility of stocks you could just use the straddle strategy both on long or short. I will answer only with theory about trading strategies. If you are 100% certain (we know this is not possible, but let´s take this as an assumption just for the sake of theory matter) of the volatily you can go two ways: High Volatility: ...


1

My guess would be most people approach this using rolling regressions. My approach would be to generate a matrix using all the lookbacks that you want to predict the present on each row and a corresponding squared return, then subsample the two.


1

I would suggest writing the joint density as the product of the conditional densities then estimate parameters using an optimization package. The joint density is given by $$f(r_0, \ldots, r_T) = f(r_0) \prod_{t=1}^T f(r_t|r_0, \ldots, r_{t-1})$$ then the log likelihood function is $$L = \log(f(r_0)) + \sum_{t=1}^T \log(f(r_t | r_0, \ldots, r_{t-1}) ...


1

There is no guarantee that the optimization method always converges! In an introduction the author of the package recommends using the "hybrid" solver, which starts out with the "solnp" and goes through the other solvers, if it doesn't converge. According to him, this should at least guarantee convergence in 90 % of the cases. ...



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