I am running ARIMA model with the natural log values of 10 years daily frequency data of a stock market index i.e., Nifty
the variable is integrated at first differencing .
the lag lengths of AR and MA are decided based on partial autocorrelation and autocorrelation functions. the decided lag length is '1' for both AR and MA terms.
my doubt is whether the dependent variable should be level form(i.e., LN_Nifty) or in first differenced Nifty while running ARIMA . Because the variable is integrated at first order.
if i run the ARIMA with the level form of Ln_Nifty, the AR and MA terms are significant.
if the first differenced form of LN_Nifty is used, the AR and MA terms are not signficant .
please anyboday clarify my doubt
thanks in advance
i have got one more problem.
is R-squared value important in testing the forecast performance of ARIMA
In order to test for forecasting performance you should use the MSPE and related measures, while for comparisons among different models in terms of fit with data you should see AIC, BIC, HQ and so forth.
Now, why are you differencing the data? Does the logharitm transformation give you a non-stationary time series? If yes, let's differentiate the series, otherwise use the ln_nifty.
If the data are I(1) you should use first differences. Typically Schwarz or Akiake information criteria are used to asses ARIMA models' relative performance. They trade fit against parsimony in the models. You can use these as model selection criteria in a trial and error fashion while experimenting with a few different ARIMA specifications. You might also want to consider seasonal AR and MA terms at lags 5, 10, 15, 20 (or 7, 14, 21 and 28 if 7 days in the week is more appropriate). There may also be monthly seasonal cycles, although they may be difficult to model with simple lags of AR and MA terms with different numbers of days in the month (unless you have set up the data to accommodate this).
I fear you should have specified the software being used. Given your data, I am not at all surprised by your sentence "if the first differenced form of LN_Nifty is used, the AR and MA terms are not signficant ." This is coherent with finance theory. Anyway I would suggest using AR or MA, not both, except if really necessary. But I suppose that you will also consider conditional heteroskedasticity. No, R squared has no meaning. If you consider the model without logarithms and/or without differences, you will get something very different. The same for the information criteria in the case of with/without transformation (except if you correct for the Jacobian of the transformation). Good luck.
I would not be surprised at what you have observed in your analysis of your equity price index. The log of an equity price index is often an I(1) process or can be approximated by an I(1) process. The first difference of this log is the return on the indexx and is then I(0).
It is common to find that these returns are uncorrelated (but not independent due to volatility clustering - GARCH effets).
As the log price indices are an integrated version of the returns they will have a correlation of 1. I don't have an explanation of your MA term but it may be some effect of trying to identify a non-stationary process as an ARMA process.
An ARMA process is a stationary process.
An ARIMA process is one that can be made stationary by differencing and the ARMA analysis applied to the stationary version of the process. How you run the ARIMA depends on the software that you are using. With some software the ARIMA program will accept the log price index and determine that it is non stationary (or trended) and first difference it. It will then estimate the ARMA system for the stationary series forecast it and reverse the first differencing to give you forecasts of your series. The manual for your software should explain this.
I presume that the R squared is some measure of the internal fit of you estimated process. This R squared is of no value in evaluating the appropriateness of a forecasting model. You may increase it by adding extra AR or MA terms. The evaluation of a forecast model must be done in terms of out of sample evaluation of the forecast. For example if you are interested in one step ahead forecasts dtake the first m of your n observations estimate the model and compare the one step ahead with the actual forecast. repeat the process for m+1, m+2,... n-1 observations and evaluate the forecast using these n-m one step ahead errors.
As equity price indices are close to random walks you may find it difficult to forecast them.
ARIMA forecasts are often taken as a benchmark against which other forecasts may be evaluated
Try to use auto.arima function in R package "forecast" to help you in identifying the best model, and also be aware that stock markets data sometimes have volatility clustering so you might need to use ARIMA-GARCH models, not only ARIMA models.
I suggest that you use expert modeler in SPSS. The software will select the best fit model automatically based on you data.
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