Building ARIMA Models

The simple ARIMA modelAutoRegressive Integrated Moving Average (ARIMA) models intend to describe the current behavior of variables in terms of linear relationships with their past values. These models are also called Box-Jenkins (1984) models on the basis of these authors pioneering work regarding time-series forecasting techniques. An ARIMA model can be decomposed in two parts. First, it has an Integrated (I) component (d), which represents the amount of differencing to be performed on the series to make it stationary. The second component of an ARIMA consists of an ARMA model for the series rendered stationary through differentiation. The ARMA component is further decomposed into AR and MA components. The autoregressive (AR) component captures the correlation between the current value of the time series and some of its past values. For example, AR(1) means that the current observation is correlated with its immediate past value at time t-1. The Moving Average (MA) component represents the duration of the influence of a random (unexplained) shock. For example, MA(1) means that a shock on the value of the series at time t is correlated with the shock at t-1. The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PCF) are used to estimate the values of p and q, using the rules reported in Table 1. In the next section, we provide an example of a simple ARIMA model.

ARIMA models have three parts, although not all parts are always necessary. The three parts are the autoregression part (AR), the integration part (I) and the moving average part (MA).The main assumption surrounding the AR part of a time series is that the observed value depends on some linear combination of previous observed values up to a defined maximum lag (denoted p), plus a random error term t. The main assumption surrounding the MA part of a time series is that the observed value is a random error term plus some linear combination of previous random error terms up to a defined maximum lag (denoted q).To analyse a time series we require that all of the observations are independently identifiable. Hence there should be no autocorrelation in the series and the series should have zero mean. In order for these requirements to be met all of the signal (the trend and seasonal components of the series being modelled) must have been removed from the series so that we are left with only noise. Therefore it is only the irregular component of the series which is being modelled, not the trend or seasonal components. If the series has zero mean and other moments such as the variance and covariance do not depend on the passage of time, then the series is said to be stationary. In order to achieve stationarity the series must be differenced (unless it is stationary to begin with). This means taking the differences between successive observations and then analysing these differences instead of the actual observations. This process of differencing is known as integration and the order of differencing is denoted d.

Definition of 'Autocorrelation'

A mathematical representation of the degree of similarity between a given time series and a lagged version of itself over successive time intervals. It is the same as calculating the correlation between two different time series, except that the same time series is used twice - once in its original form and once lagged one or more time periods.

The term can also be referred to as "lagged correlation" or "serial correlation".

Investopedia explains 'Autocorrelation'

When computed, the resulting number can range from +1 to -1. An autocorrelation of +1 represents perfect positive correlation (i.e. an increase seen in one time series will lead to a proportionate increase in the other time series), while a value of -1 represents perfect negative correlation (i.e. an increase seen in one time series results in a proportionate decrease in the other time series).

statistics, the occurrence of several independent variables in a multiple regression model are closely correlated to one another. Multicollinearity can cause strange results when attempting to study how well individual independent variables contribute to an understanding of the dependent variable. In general, multicollinearity can cause wide confidence intervals and strange P values for independent variablesIn statistics, when the standard deviations of a variable, monitored over a specific amount of time, are non-constant. Heteroskedasticity often arises in two forms, conditional and unconditional. Conditional heteroskedasticity identifies non-constant volatility when future periods of high and low volatility cannot be identified. Unconditional heteroskedasticity is used when futures periods of high and low volatility can be identifiedIn an ARIMA model, we do not have a priori for forecasting model before model identification takes place. ARIMA helps us to choose a right model to best fit the time series. Put it inaflow chart:

Demonstration to find "right ARIMA model (p, d, q)" to fit the time series throughtrial and error:Firstly, download the excel file called "exchange_rate" from the "Sample Data" of Econ3600 homepage.Second, openEVIEWSprogram in this way: click "File", "New", "Workfile" commands, then in the "Workfile Range", choose "Monthly" and type "1990.01"for the "Start observation" and "2000.07"for "End observation" in the dialogue box.Then, we will get a workfile.Next, import the data from the excel file to generate the following result:(Remember to change "B8" for upper left data cell.)

Double click the variable "yen" to check its data whether it is consistent with the Excel file and choose "View", "Line" to get a general idea about the time series is stationary or no. Also, choose "View", "Correlogram" to get the tentatively identify patterns and model components (i.e. the degree of p, d, q of ARIMA) The resulting graphs are:

From the above graphs, you can see that the time series is likely to have random walk pattern, which random walk up and down in the line graph. Also, in correlogram, the ACFs are suffered from linear decline and there is only one significant spike for PACFs. The graph of correlogram suggests thatARIMA(1, 0, 0)may be an appropriate model. Then, we take the first-difference of "Yen" to see whether the time series becomes stationary before further finding AR(p) and MA(q).(Remember that I(d) is used to get stationary series if necessary.)To see whether first difference can get level-stationary time series or not, you need to generate it by choosing "GENR", type "dyen=d(yen)". Then, you will get "dyen" item in the "Workfile", and use it to draw a line graph and also get a correlogram graph. the results are:

Now, the first-difference series "DYEN" becomes stationary as showing in line graph and is white noise as shown no significant patterns in the graph of correlgram. And the unit root test also confirms the first-difference becomes stationary. The strong evidents support that theARIMA(0,1,0)is suitable for the time series. Then, we can construct the ARIMA model as following steps:Step 1. Choosing "Quick", "Estimate Equation", then specify the mode and type "yen c ar(1)",

click "OK", the result is:

Step 2. choosing "View", "Residual tests", "Correlogram-Q- Statistic" the result is:

(Since there is no significant spikes of ACFs and PACFs, it means that the resduals of this selected ARIMA model are white noise, so that there is no other significant patterns left in the time series, then we can stop at here and don't need to further consider another AR(p) and MA(q))The criterions to judge for the best model are as follows: Relatively small of BIC (Schwarz criterion which is measured by nLog(SEE)+kLog(n)) Relativelysmall of SEE Relatively high adjust R2 Q- statistics and correlogram show that there is no significant pattern left in the ACFs and PACFs of the residuals, it means the residuals of the selected model are white noise.You may try another ARIMAs and compare the statistical results as in the following table:ARIMA modelBICAdjusted R2SEE

(1, 0 , 0)5.7080.934764.075

(1, 0, 1)5.7340.935034.067

(2, 0, 0)5.7250.934254.047

(0, 0, 1)7.3840.655989.422

(0, 0, 2)6.8880.797547.220

(1, 1, 0)5.7240.00194.108

(0, 1, 0)5.7080.93474.075

As you can see thatARIMA(1,0,0)is a relatively best model,Remark: TheARIMA (1, 0, 0)is same asARIMA (0, 1, 0). The result of ARIMA(0,1,0) is:

In our several trial and error procedures, the ARIMA(1,0,0) or ARIMA(0,1,0) is selected as the best model.Now, we can express this selected best model as

Students are encouraged to try to find the best ARIMA model for the series of "pound".Now, let's try another complicated time series.Click Hereto continue.