# Structural change or GARCH model

I have GDP Time series, that has a positive stochastic trend trend:

> CADFtest(logGDP, type= "trend", criterion= "BIC", max.lag.y=max.lag)


The first differenced log GDP time series removes the trend and looks like:

and is stationary:

> CADFtest(dlogGDP, type = "drift", criterion= "BIC", max.lag.y=max.lag)


The problem: There is some obvious heteroscedasticity in the data. I have removed the data from 19670-1992, reasoning that there has been structural change. I have fit ARIMA (1,1,1) model and used the Q-test to validate it - the residuals are white noise. Is this correct?

Alternatively, I have tried to fit the ARCH and GARCH model on the entire time seties (1972 - 2015). ARCH has not yield an parsimonious model, based on the the correlogram of squared errors. I then fit a GARCH model and validated it.

QUESTION

1. Which procedure is better/ more correct?
2. Is there a way to compare the ARIMA model and GARCH model? How can I compare their performance?
3. Are both models equally good for prediction?
• What exactly is your goal? ARMA and GARCH model different things. Check out this post – Numb3rs Nov 17 '17 at 17:43
• @Numbe3rs I want to use this data for forcasting and multicariate time series anaylsis – user1607 Nov 19 '17 at 10:09

I have fit ARIMA (1,1,1) model and used the Q-test to validate it - the residuals are white noise. Is this correct?

According to Alecos' answer to this question, Ljung-Box test is not appropriate for residuals of ARMA models, even though such use of the test is widespread. Breusch-Godfrey test is a valid substitute.

1. Which procedure is better/ more correct?

• If you find an adequate ARIMA model, you may stick to it. However, in presence of ARCH patterns in the residuals, serial correlation tests may not work properly (they depend on conditional homoskedasticity). Therefore, you should also check whether the residuals of the ARMA model display ARCH patterns.
2. Is there a way to compare the ARIMA model and GARCH model? How can I compare their performance?

• Yes, there is. First, you can check model adequacy by verifying that the model errors (standardized errors, in case of GARCH) are indistinguishable from an i.i.d. process. Second, you can compare model fit adjusted for its complexity by information criteria such as AIC or BIC. Third, if your time series is sufficiently long, you can split it into a training and test subsamples and see how well the model fit on the training subsample predicts the data in the test subsample. Whichever model delivers better forecasts would be preferred.
3. Are both models equally good for prediction?

• There is no definite answer in the general case because different models may be preferred for different data generating processes. But in your case, you may assess this by pseudo out-of-sample forecasting as suggested above, or by comparing AIC values, which is roughly in the same spirit.