Identifying AR and MA terms using ACF

In this article, primarily I share my experience in understanding the ACF, PACF plots, and their significance in selecting the order of ARMA models.

Selecting candidate Auto Regressive Moving Average *(ARMA) models for time series analysis and forecasting, understanding *Autocorrelation function (ACF), and Partial autocorrelation function (PACF) plots of the series are necessary to determine the order of AR and/ or MA terms. Though ACF and PACF do not directly dictate the order of the ARMA model, the plots can facilitate understanding the order and provide an idea of which model can be a good fit for the time-series data. In this article, primarily I share my experience in understanding the ACF, PACF plots, and their significance in selecting the order of ARMA models.

ACF plot is a bar chart of coefficients of correlation between a time series and it lagged values. Simply stated: ACF explains how the present value of a given time series is correlated with the past (1-unit past, 2-unit past, …, n-unit past) values. In the ACF plot, the x-axis expresses the correlation coefficient whereas the y-axis mentions the number of lags. Assume that, y(t-1)

y(t), y(t-1),….y(t-n) are values of a time series at time t, t-1,…,t-n, then the lag-1 value is the correlation coefficient between y(t) and y(t-1), lag-2 is the correlation coefficient between y(t) and y(t-2) and so on.

PACF  is the partial autocorrelation function that explains the partial correlation between the series and lags of itself. In simple terms, PACF can be explained using a linear regression where we predict y(t)  from y(t-1), y(t-2),  and y(t-3)  [2]. In PACF, we correlate the “parts” of y(t)  and y(t-3)  that are not predicted by y(t-1)  and y(t-2) .

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