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How to Estimate ARIMA Models in EViews

Time series forecasting is an incredibly popular and widely used technique in finance, economics, and other data-driven fields. As the demand for accurate predictions of future events continues to grow, so does the need for reliable and efficient methods of forecasting. One such method is Autoregressive Integrated Moving Average (ARIMA). ARIMA is a powerful model that has been widely adopted by practitioners in the field due to its ability to quickly generate accurate forecasts.

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What is an ARIMA Model?

ARIMA stands for Autoregressive Integrated Moving Average and is one of the most popular and widely used techniques for univariate time series forecasting. Some of the variables you can forecast with ARIMA models are: GDP, Consumer Price Index (CPI), and price of stocks or commodities.

Univariate Models

We won’t try to forecast future values of a variable (i.e., inflation) by using many other regressors (i.e., GDP, Money supply, interest rates). Instead, we will rely on past levels of inflation to forecast future levels of inflation. Knowing how the variable behaved in the past will allow us to predict where it will head in the future.

In this tutorial we apply the Box Jenkins Method to select appropriate models and forecast future values of our variable of interest.

What is the Box-Jenkins Methodology?

The Box Jenkins methodology was named after the authors George Box and Gwilym Jenkins, who proposed a three steps method to select appropriate ARIMA models to forecast economics variables. Using the Box-Jenkis methodology, we will try to find a model that fits the data well and can forecast appropriate values. The method consists of three basic steps:

Stage 1: Identification
Stage 2: Estimation
Stage 3: Diagnostics and Forecasting

Some Textbooks indicate that the Box-Jenkins method involves four steps

Don’t be afraid! The original book written by Box and Jenkins entitled “Time Series Analysis: Forecasting and Control” specifies only three steps. However, some textbooks have split the method into four stages. Stage 3 is diagnostics and, Stage 4 is Forecasting. The analysis remains the same.

How to estimate autorregresive integrated average (ARIMA) models in EViews

Box-Jenkins, Stage 1: Identification

ARIMA is written as ARIMA (p,d,q) where “p” is the order of the autoregressive component, “d” is the times we need to differentiate the variable to achieve stationarity, and “q” is the order of the moving average element.

Stage 1 focuses on two aspects. We are first checking for stationarity of our variable of interest. Next, determining the order of our autoregressive and moving average components. In other words, on stage 1 we will determine “p”, “d” and “q”.

In our example, we are trying to fit an ARIMA model for the series “consumer price index — USA“. We have to begin our analysis by checking for stationarity. Why? Our series needs to be stationary in order to forecast it. If our variable is non stationary in levels, we need to apply the appropriate transformations (logs/differences) to make it stationary.

How to check stationarity in EViews

To check for stationarity, we look at :

The Graph
The correlogram
Formal tests: Augmented Dickey Fuller, Phillips-Perron Test and KPSS test.

Please watch my stationarity tutorial if you need further clarification on the procedure.

In our example, we verified that CPI is non stationary in levels, but stationary in first differences. Consequently, we use the variable in first differences.

Note: Since P-Value > 0.05, we cannot reject the null hypothesis (H0: CPI has a unit root).

ARIMA: How to determine the order of p,d and q

To identify the order of the autoregressive and moving average components, we will focus on the correlogram of “CPI” in the first differences. We are displaying the correlogram in the first differences because we have confirmed that “CPI” is stationary in the first differences. The aim of this step is to find all the possible models to estimate.

In order to determine the order of the autoregressive component (“p”), we have to observe the partial autocorrelation column (PACF). In the column, we observe a confidence band on the sides. The values that exceed the band suggest the possible order of the autoregressive component. Looking at the correlogram, the first lag is a highly significant AR(1) component, and then lags 2 and 3 are on the line and could be tested. For the purpose of this example, I will only consider an AR(1) component.

Next, to determine the order of the moving average component (“q”), we have to observe the Autocorrelation column (ACF). We can see that lags 1, and 3 exceed the confidence bands. Consequently, there are two possible moving average components MA(1) and MA(3).

Box-Jenkins, Stage 2: Estimation

Once we have identified possible ARIMA models candidates, we need to estimate them and decide which model is the most appropriate. The two models we decided to estimate are:

ARIMA (1,1,1)
ARIMA(1,1,3)

In Box Jenkins Method, Stage 2 we:

Estimate the models we identified in Stage 1
We select a model based on the significance of the coefficient estimates
and, based on model criterions such as: Schwartz, Akaike and Hannan-Quinn
The model with the smallest values in the model criterions and most significant coefficient will be the most appropriate

Note: Criteria to select the appropriate model.

How to select the appropriate ARIMA model?

To select the most appropriate model, I recommend you to do a table like the one below, and fill the information with the data we obtained in the previous figure (estimated ARIMA models).

We need to ensure the following:

Significance of the ARMA terms : select the model with most significant terms (p-values<0.05)
SigmaSQ: is a measure of volatility. Select the smallest one
Log Likelihood: We need to select the biggest value, since we are maximizing the log-likelihood function. (in our case the biggest is the least negative value).
Model selection criterias: Select the model with smallest Akaike, Schwarz and Hannan-Quinn

Conclusion: Model B has a better fit than model A.

Note: Model B is preferred over Model A.

Box-Jenkins, Stage 3: Diagnostics and Forecasting

We identified possible models and estimated them in stage 2. We also selected the most appropriate model based on diverse criterions. Now it is time to ensure the model satisfies the requirementes to forecast and predict future values!

In Box Jenkins Method, Stage 3 we:

Ensure the model satisfies the stability conditions
There is no autocorrelation

If the above requirements are satisfied, then we can forecast!

How to check for autocorrelation in ARIMA Models

To check for autocorrelation, we display the correlogram of the ARIMA(1,1,3) model and look at the Ljung Box Q statistic, where the null hypothesis is “residuals are white noise”.

As we can see in the figure below, the p-values for the Q-statistics are all over 0.05 which confirms that the residuals are white noise. The last step is to confirm if the inverse AR/MA roots lie inside the unit circle.

Note: All the p-values are bigger than 0.05, therefore, residuals are white noise.

Stability conditions in ARIMA models

The estimated model is covariance stationary: inverse AR roots should lie inside the unit circle
The estimated process is invertible: inverse MA roots should lie inside the unit circle

Note: All roots lie within the circle. The model is stable.

We can see in the figure above that all the inverse roots lie inside the unit circle. Our ARIMA(1,1,3) satisfied the stability conditions and the error terms are white noise. We are in a good spot now to forecast future values of the consumer price index. If the model you had selected did not satisfy the stability condition, you would need to repeat stage 2 and 3 again, and find another suitable possible candidate.