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Suppose we estimate the regression model $$\triangle y_{t}=\alpha + \beta y_{t-1}+\varepsilon_{t}$$ This is actually quite similar to the Dickey-Fuller test. If $\beta=0$, then the process has a unit root. Let's proceed assuming that $\beta<0$, i.e. that the process is stationary. The first equation is also similar to the continuous time ...

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What you are saying might be correct for discrete time processes. In continuous time the process $$dX_t = X_t^2 dW_t,\quad X_0 > 0$$ is stationary but not mean reverting.

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There are two answers to your question If you want to use the Neston-Nandi model, you can use it directly with the parameters that you already show above: model = list(omega = 0.000001, alpha = 0.5, beta = 0.4) In r, the fOptions package has an HN model that can use them: HNGOption(TypeFlag, model, S, X, Time.inDays, r.daily) If you want to calculate ...

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This is a very broad question and a large number of issues have been discussed in the literature. As such, it's hard to give specific advice except that it is better to model returns instead of prices directly. What I would do if I were you: Read some of the available literature to get a good overview. This is an interesting paper but many more exist. ...

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You can try: daily.fit=ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)), mean.model = list(armaOrder = c(35, 7), include.mean = T, arfima=F), fixed.pars=list(ar9=0,ar10=0,...,ar13=0,ar15=0,...,ar20=0,ar22=0,...,ar27=0,ar29=0,...,ar34=0,ma1=0,...,ma6=0)) from rugarch package.

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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 ...

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Actually you can find a papers talking about some relationship between almost any two types of indicators. But based on my work this what I suggest you add: Commodities futures (continues) (Many paper on the relationship between oil prices and USD, gold/silver and USD) Market indices (DJI, S&P500, DAX, SET, NZ40, ...) I would also suggest that you ...

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Interest rate differentials is the most justifiable fundamental input, also the most explanatory second order input you should use. Another is the slope of the yield curve and the difference in slope between pairs. You can also look at the relative difference in speed of interest rate changes. Another fundamental input is the real IR spread. Good luck.

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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. ...

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