# Correlogram and ACF/PACF applied to US index of unemployment rate

This eviews workfile contains US index of unemployment from 1960 to 2008 quarterly.

I'm trying to understand ACF and PACF. Below is a correlogram for the first 24 lags:

What can the correlogram and ACF/PACF tell me about the data? To my understanding, the ACF applies to Autoregressive model while PACF is for Moving Average?

Stationarity First of all, you should conclude if your series is stationary or not. Sometimes this is evident from a plot of the series, while other times you will rely on hypothesis testing. ACF patterns may reveal the need to difference if, for example, the autocorrelations decay linearly.

Regular AR and MA processes are fully specified by the autocorrelation and partial autocorrelation functions. Consequently, Sample ACF and PACF are compared to theoretical ACF and PACF for ARMA processes of different orders to guide you through the identification process.

• $AR(p)$: ACF with an exponential (real roots) or sinusoidal (complex roots) decay starting from lag $p$ and a PACF with a sudden stop after the $p$th lag
• $MA(q)$: sudden stop in the PACF at the $q$th lag and an exponential (real roots) or sinusoidal (complex roots) decay starting from lag $q$ in the ACF.
• $ARMA(p, q)$ are not easily identified from the plots. ACF decays after the $q-p$th lag and PACF decays after the $p-q$th lag.

See Hyndman for a very brief introduction.

Seasonality After you have chosen your model for the regular part of the series, you should check that the error term is white noise. Given that the frequency of your series seems to be quarterly, you may find that the errors are significantly autocorrelated for some lags (i.e. 4th lag). Sometimes, the seasonality is so evident that it will show in the ACF/PACF of the original series. If any of the two is true, you would want to try a seasonal ARIMA model to capture both the autocorrelation from the regular and the seasonal part of the series. See, for example, the European retail trade index example by Hyndman.

References I would personally start with the first two books listed below.

• Hyndman, R. J., & Athanasopoulos, G. (2014). Forecasting: principles and practice. OTexts. See Chapter 8.
• Wei, W. W. S. (1994). Time series analysis. Addison-Wesley publ. See Chapter 6.
• Shumway, R. H., & Stoffer, D. S. (2010). Time series analysis and its applications: with R examples. Springer Science & Business Media. See Section 3.4.
• Franses, P. H. (2014). Time series models for business and economic forecasting. Cambridge university press. See Section 3.2.

Example For illustrative purposes, I have plotted the true ACF and PACF for some simple non seasonal models.

Bear in mind that ACF and PACF are not a perfect tool for model identification and hence are used only to pick the order of initial models.

P.S. The code for the plots in R (quick code and not so much polished)

myplot<- function(y, x, title) {
plot(y = y, x = 0:20, type = "h", ylim = c(-1,1), xlab = "h", ylab = expression(rho[h]), main = paste("True ACF for", title))
abline(h = 0)
plot(x = x, type = "h", ylim = c(-1,1), xlab = "h", ylab = expression(phi[hh]), main = paste("True PACF for", title))
abline(h = 0)
}

par(mfrow=c(3, 2))

myplot(ARMAacf(ar = 0.9, lag.max = 20), ARMAacf(ar = 0.9, lag.max = 20, pacf=TRUE), "ARMA(1, 0)")
myplot(ARMAacf(ma = -0.5, lag.max = 20), ARMAacf(ma = -0.5, lag.max = 20, pacf=TRUE), "ARMA(0, 1)")
myplot(ARMAacf(ar = 0.9, ma = -0.5, lag.max = 20), ARMAacf(ar = 0.9, ma = -0.5, lag.max = 20, pacf=TRUE), "ARMA(1, 1)")