15.4 ARIMA forecasting models

is the unshifted time series. Lag 1 is the time series shifted one position to the left. Lag 2 shifts the time series two positions to the left, and so on. Time series can be lagged using the function lag(ts,k), where ts is the time series and k is the number of lags.

Table 15.5 The Nile time series at various lags

Autocorrelation measures the way observations in a time series relate to each other. ACk is the correlation between a set of observations (Yt) and observations k periods earlier (Yt-k). So AC1 is the correlation between the Lag 1 and Lag 0 time series, AC2 is the correlation between the Lag 2 and Lag 0 time series, and so on. Plotting these correlations (AC1, AC2, …, ACk) produces an autocorrelation function (ACF) plot. The ACF plot is used to select appropriate parameters for the ARIMA model and to assess the fit of the final model.

An ACF plot can be produced with the acf() function in the stats package or the Acf() function in the forecast package. Here, the Acf() function is used because it produces a plot that is somewhat easier to read. The format is Acf(ts), where ts is the original time series. The ACF plot for the Nile time series, with k=1 to 18, is provided a little later, in the top half of figure 15.12.

A partial autocorrelation is the correlation between Yt and Yt-k with the effects of all Y values between the two (Yt-1, Yt-2, …, Yt-k+1) removed. Partial autocorrelations can also be plotted for multiple values of k. The PACF plot can be generated with either the pacf() function in the stats package or the Pacf() function in the forecast package. Again, the Pacf() function is preferred due to its formatting. The function call is Pacf(ts), where ts is the time series to be assessed. The PACF plot is also used to determine the most appropriate parameters for the ARIMA model. The results for the Nile time series are given in the bottom half of figure 15.12.

ARIMA models are designed to fit stationary time series (or time series that can be made stationary). In a stationary time series, the statistical properties of the series don’t change over time. For example, the mean and variance of Yt are constant. Additionally, the autocorrelations for any lag k don’t change with time.

It may be necessary to transform the values of a time series in order to achieve constant variance before proceeding to fitting an ARIMA model. The log transformation is often useful here, as you saw in section 15.1.3. Other transformations, such as the BoxCox transformation described in section 8.5.2, may also be helpful.

Because stationary time series are assumed to have constant means, they can’t have a trend component. Many non-stationary time series can be made stationary through

differencing. In differencing, each value of a time series Yt is replaced with Yt-1 – Yt. Differencing a time series once removes a linear trend. Differencing it a second time removes a quadratic trend. A third time removes a cubic trend. It’s rarely necessary to difference more than twice.

You can difference a time series with the diff() function. The format is diff(ts, differences=d), where d indicates the number of times the time series ts is differenced. The default is d=1. The ndiffs() function in the forecast package can be used to help determine the best value of d. The format is ndiffs(ts).

Stationarity is often evaluated with a visual inspection of a time-series plot. If the variance isn’t constant, the data are transformed. If there are trends, the data are differenced. You can also use a statistical procedure called the Augmented Dickey-Fuller (ADF) test to evaluate the assumption of stationarity. In R, the function adf.test() in the tseries package performs the test. The format is adf.test(ts), where ts is the time series to be evaluated. A significant result suggests stationarity.

To summarize, ACF and PCF plots are used to determine the parameters of ARIMA models. Stationarity is an important assumption, and transformations and differencing are used to help achieve stationarity. With these concepts in hand, we can now turn to fitting models with an autoregressive (AR) component, a moving averages (MA) component, or both components (ARMA). Finally, we’ll examine ARIMA models that include ARMA components and differencing to achieve stationarity (Integration).

15.4.2ARMA and ARIMA models

In an autoregressive model of order p, each value in a time series is predicted from a linear combination of the previous p values

AR(p):Yt = μ + β1Yt−1 + β2Yt−2 + ... + βpYt−p + εt

where Yt is a given value of the series, µ is the mean of the series, the βs are the weights, and εt is the irregular component. In a moving average model of order q, each value in the time series is predicted from a linear combination of q previous errors. In

this case

MA(q):Yt = μ −θ1εt−1 − θ2εt−2 ... − θqεt−q + εt

where the εs are the errors of prediction and the θ s are the weights. (It’s important to note that the moving averages described here aren’t the simple moving averages described in section 15.1.2.)

Combining the two approaches yields an ARMA(p, q) model of the form

Yt = μ + β1Yt−1 + β2Yt−2 + ... + βpYt−p − θ1εt−1 − θ2εt−2 ... − θqεt−q + εt

that predicts each value of the time series from the past p values and q residuals.

An ARIMA(p, d, q) model is a model in which the time series has been differenced d times, and the resulting values are predicted from the previous p actual values and q

The goal is to identify the parameters p, d, and q. You already know that d=1 from the previous section. You get p and q by comparing the ACF and PACF plots with the guidelines given in table 15.6.

Table 15.6 Guidelines for selecting an ARIMA model

The results in table 15.6 are theoretical, and the actual ACF and PACF may not match this exactly. But they can be used to give a rough guide of reasonable models to try. For the Nile time series in figure 15.12, there appears to be one large autocorrelation at lag 1, and the partial autocorrelations trail off to zero as the lags get bigger. This suggests trying an ARIMA(0, 1, 1) model.

FITTING THE MODEL(S)

The ARIMA model is fit with the arima() function. The format is arima(ts, order=c(q, d, q)). The result of fitting an ARIMA(0, 1, 1) model to the Nile time series is given in the following listing.

Listing 15.8 Fitting an ARIMA model

>library(forecast)

>fit <- arima(Nile, order=c(0,1,1))

>fit

Series: Nile

ARIMA(0,1,1)

Coefficients:

ma1 -0.7329 s.e. 0.1143

Note that you apply the model to the original time series. By specifying d=1, it calculates first differences for you. The coefficient for the moving averages (-0.73) is provided along with the AIC. If you fit other models, the AIC can help you choose which one is most reasonable. Smaller AIC values suggest better models. The accuracy