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$$E\left[ {{y_t}|{{\cal F}_{t - 1}}} \right] = E\left[ {{\sigma _t}{z_t}|{{\cal F}_{t - 1}}} \right] = {\sigma _t}E\left[ {{z_t}} \right] = 0$$ $${\mathop{\rm var}} \left[ {{y_t}|{{\cal F}_{t - 1}}} \right] = {\mathop{\rm var}} \left[ {{\sigma _t}{z_t}|{{\cal F}_{t - 1}}} \right] = \sigma _t^2{\mathop{\rm var}} \left[ {{z_t}} \right] = \sigma _t^2$$ $$... 3 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 ... 3 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" ... 2 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. ... 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 find R code for seasonal ARIMA models again in the book mentioned (this chapter). Do you really need the GARCH errors? 1 You are right - GARCH model models volatility. They write: " The GARCH [27] can be used to model changes in the variance of the errors as a function of time." What people often do is to fit an ARIMA model (that can be used to forecast a time series) and apply a GARCH model to the errors (which gives you a feeling for the forecast error). See Hyndman and ... 1 You first fit a ARIMA model to the returns data and then a GARCH model to the residuals. 1 alpha + beta < 1 is the stationary condition for GARCH. If alpha and beta are low that means volatility of the stock does not have clustering behaviors. I think you can have a look at ADF and PACF of Return^2 time series first. If the first order autocorrelation is very significant but alpha is not, then perhaps you can check on the parameter calibration. ... 1 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 ... 1 The mean equation specification for ARIMAX(8,0,0)(5,0,1)[7] (as in the R code above):$$ (1 - \phi_1L^1 - \ldots - \phi_8L^8)(1-\Phi_1L^7 - \Phi_2L^{14} - \ldots - \Phi_5L^{35})y_t = \beta x_t + (1 + \Theta_1L^7)\varepsilon_t  where $x_t$ is the holiday dummy variable. Equivalent ARIMA fit in Matlab (+ GARCH and forecasting): % specify seasonal ...

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Some models do use ln(r_t), like Black–Derman–Toy and the Black–Karasinski models. Mainly to avoid negative interest rates in low rates / high volatility environments through the use of the log-normal distribution. Negative rates can wreak havoc in option premiums for example. They are interest rates indeed, that we call short rates, not yield on ...

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These are not yield. They are instantaneous short rates which are not directly observable in the market.

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I have the same problem as you. Up to my knowledge, there is no package allowing to combine seasonal ARIMA process with GARCH effects.

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If log returns have a symmetric distribution, prices will have a positively skewed distribution, since exponentiating induces positive skew.

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Did you try rmgarch package of R ? http://cran.r-project.org/web/packages/rmgarch/index.html http://unstarched.net/r-examples/rmgarch/mgarch-comparison-using-the-hong-li-misspecification-test/

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You can pass in the parameters are you estimating with EWMA or GARCH using the mu (mean), sigma (co/variance) m3 (co/skewness) and m4(co/kurtosis) arguments. e.g. blahblah = EWMA(my_time_series) VaR(my_time_series,mu=blahblah)

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Concidering 22 days of trading per month you have approximatly 132 days of trading. I highly doubt that this will be sufficient for any forecasting. The sample might be too small. Have a look here: http://research.stlouisfed.org/wp/2012/2012-008.pdf Erdemlioglu, Laurent and Neely used the data of ~10 years to conduct their survey.

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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.

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