Taking logs is not a linear transformation. Generally, values that have been very large or very small before the transformation become more similar after the transformation. The difference of the logs therefore tends to be smaller than the simple difference. The simple difference furthermore is not a percentage but an absolute difference so that scaling matters. From what I know, most people would want to take ret and not ret1 since ret can be directly interpreted as continuous returns.

For your second question: From what I know, in an ARIMA model the coefficients are not estimated on the nonstationary series but always on the stationary series so that the I in ARIMA just tells you how many times you had to differences the series to estimate the ARMA model. I dont know if I understand you correctly but the reverse transformation is easy to do: After estimating the model on the stationary returns (ret) you generate the forecast of the returns and then transform this back to prices using

x_t = exp[ret_t + x_t-1]

where x_0 is the is the beginning of your price series. I use a loop for the transformation although there is probably a more elegant vector function for that.


-----Original Message-----
From: r-sig-finance-bounces@stat.math.ethz.ch on behalf of babel@centrum.sk
Sent: Sun 4/12/2009 12:46 AM
To: R-SIG-Finance
Subject: [R-SIG-Finance] ARIMA,GARCH and differences
 
Dear friends

This is maybe trivial question for you, but why I get different results when I use return series and the original price series with 1.differences speciefied in ARIMA model? For example
x<-priceSeries
ret<-diff(log(x))
ret1<-diff(x)

fit_arma = armaFit(~ arma(5,0), data = ret)
fit_arma1 = armaFit(~ arma(5,0), data = ret1)
fit_arima = armaFit(~ arima(5,1,0), data = x)

The first model has the smallest AIC. AR parameters are very similar but not the same. 

The second question is more critical for me. Is there any possibility to enlarge  the package fGarch for the joint estiamtion of ARIMA+GARCH? Because in mean equation you can specifie only AR, MA, or ARMA model, not ARIMA. This extension I consider very important, because using the differences biult-in model,( I mean ARIMA) simplify the entire fitting and forecasting proces. If you use ARIMA, you dont need to make reverse transformation (from return to price series - from diff(log) to original scale). For evauation part it is not a big problem but for prediction part I find it difficult, therefore, I will prefer to use ARIMA so the garchFit function would produce the prediction and make the reverse transformation for me. (like the armaFit does)

Thank you very much for any suggestion you provide

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