Using AIC to Test ARIMA Models

The Akaike Information Critera (AIC) is a widely used measure of a statistical model. It basically quantifies 1) the goodness of fit, and 2) the simplicity/parsimony, of the model into a single statistic.

When comparing two models, the one with the lower AIC is generally “better”. Now, let us apply this powerful tool in comparing various ARIMA models, often used to model time series.

The dataset we will use is the Dow Jones Industrial Average (DJIA), a stock market index that constitutes 30 of America’s biggest companies, such as Hewlett Packard and Boeing. First, let us perform a time plot of the DJIA data. This massive dataframe comprises almost 32000 records, going back to the index’s founding in 1896. There was an actual lag of 3 seconds between me calling the function and R spitting out the below graph!

DJIA since March 1896

Dow Jones Industrial Average since March 1896

But it immediately becomes apparent that there is a lot more at play here than an ARIMA model. Since 1896, the DJIA has seen several periods of rapid economic growth, the Great Depression, two World Wars, the Oil shock, the early 2000s recession, the current recession, etcetera. Therefore, I opted to narrow the dataset to the period 1988-1989, which saw relative stability. As is clear from the timeplot, and slow decay of the ACF, the DJIA 1988-1989 timeseries is not stationary:

Timeseries (left) and AIC (right): DJIA 1988-1989

Time plot (left) and AIC (right): DJIA 1988-1989

So, we may want to take the first difference of the DJIA 1988-1989 index. This is expressed in the equation below:

\Delta Y = Y_t - Y_{t-1}

The first difference is thus, the difference between an entry and entry preceding it. The timeseries and AIC of the First Difference are shown below. They indicate a stationary time series.

First Difference of DJIA 1988-1989: Timeseries (left) and ACF (right)

First Difference of DJIA 1988-1989: Time plot (left) and ACF (right)

Now, we can test various ARMA models against the DJIA 1988-1989 First Difference. I will test 25 ARMA models: ARMA(1,1); ARMA(1,2), … , ARMA(3,3), … , ARMA(5,5). To compare these 25 models, I will use the AIC.

Table of AICs: ARMA(1,1) through ARMA(5,5)

Table of AICs: ARMA(1,1) through ARMA(5,5)

I have highlighted in green the two models with the lowest AICs. Their low AIC values suggest that these models nicely straddle the requirements of goodness-of-fit and parsimony. I have also highlighted in red the worst two models: i.e. the models with the highest AICs. Since ARMA(2,3) is the best model for the First Difference of DJIA 1988-1989, we use ARIMA(2,1,3) for DJIA 1988-1989.

The AIC works as such: Some models, such as ARIMA(3,1,3), may offer better fit than ARIMA(2,1,3), but that fit is not worth the loss in parsimony imposed by the addition of additional AR and MA lags. Similarly, models such as ARIMA(1,1,1) may be more parsimonious, but they do not explain DJIA 1988-1989 well enough to justify such an austere model.

Note that the AIC has limitations and should be used heuristically. The above is merely an illustration of how the AIC is used. Nonetheless, it suggests that between 1988 and 1989, the DJIA followed the below ARIMA(2,1,3) model:

\Delta Y_t = \phi_2 Y_{t-2} + \phi_1 Y_{t-1} + \theta_{3} \epsilon_{t-3} + \theta_{2} \epsilon_{t-2} + \theta_1 \epsilon_{t-1} + \epsilon_{t}

Next: Determining the above coefficients, and forecasting the DJIA.

Analysis conducted on R. Credits to the St Louis Fed for the DJIA data.

Abbas Keshvani

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How to use the Autocorreation Function (ACF)?

The Autocorrelation function is one of the widest used tools in timeseries analysis. It is used to determine stationarity and seasonality.

Stationarity:

This refers to whether the series is “going anywhere” over time. Stationary series have a constant value over time.

Below is what a non-stationary series looks like. Note the changing mean.

Time series plot of non-stationary series

Time series plot of non-stationary series

And below is what a stationary series looks like. This is the first difference of the above series, FYI. Note the constant mean (long term).

Stationary series: First difference of VWAP

Stationary series: First difference of VWAP

The above time series provide strong indications of (non) stationary, but the ACF helps us ascertain this indication.

If a series is non-stationary (moving), its ACF may look a little like this:

ACF of non-stationary series

ACF of non-stationary series

The above ACF is “decaying”, or decreasing, very slowly, and remains well above the significance range (dotted blue lines). This is indicative of a non-stationary series.

On the other hand, observe the ACF of a stationary (not going anywhere) series:

ACF of nonstationary series

ACF of stationary series

Note that the ACF shows exponential decay. This is indicative of a stationary series.

Consider the case of a simple stationary series, like the process shown below:

Y_t = \epsilon_t

We do not expect the ACF to be above the significance range for lags 1, 2, … This is intuitively satisfactory, because the above  process is purely random, and therefore whether you are looking at a lag of 1 or a lag of 20, the correlation should be theoretically zero, or at least insignificant.

Next: ACF for Seasonality

Abbas Keshvani