# ARIMA Models Identification: Correlograms

Last Update: June 20, 2022

ARIMA Models Identification: Correlograms are used to identify ARIMA models autoregressive and moving average orders.

As example, we can delimit univariate time series  into training range  for model fitting and testing range  for model forecasting.

Then, we can do training range univariate time series  model autoregressive  and moving average  orders identification by estimating sample autocorrelation and partial autocorrelation functions correlograms. Notice that we need to evaluate whether level or  order differentiated training range univariate time series is needed for ARIMA model integration  order.

Next, we estimate training range univariate time series sample autocorrelation function correlogram with formula . Training range univariate time series estimated lag  sample autocorrelation function  is the estimated sample covariance between univariate time series  and its lag  univariate time series  divided by univariate time series estimated sample variance  . Training range autocorrelation function is the linear dependence between current  and lagged  univariate time series data.

After that, we estimate training range sample autocorrelation function correlogram confidence intervals with formula . Training range sample autocorrelation function confidence intervals  are the inverse of the standard normal cumulative distribution  with probability equal to one minus statistical significance level  divided by two and this result multiplied by the square root of one plus two times the sum of sample autocorrelation function square values  divided by number of observations . Notice that we need to evaluate whether Bartlett formula is needed for sample autocorrelation function correlogram confidence intervals estimation.

Later, we estimate training range univariate time series sample partial autocorrelation function correlogram with formula . Training range univariate time series lag  sample partial autocorrelation function  can be estimated through lag  linear regression estimated coefficient  with formula . Training range partial autocorrelation function is the linear dependence between current  and lagged  univariate time series data after removing any linear dependence on . Notice that we can also estimate sample partial autocorrelation function using Yule-Walker, Levison-Durbin or adjusted linear regression methods.

Then, we estimate training range sample partial autocorrelation function correlogram confidence intervals with formula . Training range sample partial autocorrelation function confidence intervals  are the inverse of the standard normal cumulative distribution  with probability equal to one minus statistical significance level  divided by two and this result multiplied by one divided by the square root of number of observations .

Next, we can do training range univariate time series  model autoregressive  and moving average  orders identification.

• If sample autocorrelation function correlogram  tails of gradually and sample partial autocorrelation function correlogram  drops after  statistically significant lags then we can observe the potential need of an autoregressive model  of order .
• Alternatively, if sample autocorrelation function correlogram  drops after  statistically significant lags and sample partial autocorrelation function correlogram  tails off gradually then we can observe the potential need of a moving average model  of order .
• Otherwise, if sample autocorrelation function correlogram  tails of gradually after  statistically significant lags and sample partial autocorrelation function correlogram  tails off gradually after  statistically significant lags then we can observe the potential need of an autoregressive moving average model  of orders  and .

Below, we find example of training range univariate time series  model autoregressive  and moving average orders identification correlograms using airline passengers data . Training range as first ten years and testing range as last two years of data. Correlograms confidence intervals with statistical significance level. Notice that correlograms confidence intervals statistical significance level was only included as an educational example which can be modified according to your needs. Figure 1. Microsoft Excel® training range univariate time series ARIMA(p,d,q) model autoregressive p and moving average q orders identification correlograms using airline passengers data.

 Data Description: Monthly international airline passenger numbers in thousands from 1949 to 1960.

Original Source: Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (1976). “Time Series Analysis, Forecasting and Control”. Third Edition. Holden-Day. Series G.

Source: datasets R Package AirPassengers Object. R Core Team (2021). “R: A language and environment for statistical computing”. R Foundation for Statistical Computing, Vienna, Austria.

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