# strucchange package on ARIMA model

Is there a way to use strucchange package in R on ARIMA models? I haven't been able to find any. Thanks a lot.

The package strucchange requires as input the formula of a linear model to be passed to lm. I don't think there is a straightforward way to use the package with function arima. I don't know either any other R packages implementing this but I can give some basic guidelines that may be helpful for your purposes.

You can carry out some diagnostics based on the cumulative sum of squared residuals (CUMSUM) and based on F-tests for the parameters of the model in different subsamples.

Let's take for illustration the following simulated AR process, x. The first 50 observations are generated from an AR(1) model and the next 100 observations from an AR(2) model:

set.seed(135)
x1 <- arima.sim(model = list(order = c(1,0,0), ar = -0.2), n = 50)
x2 <- arima.sim(model = list(order = c(2,0,0), ar = c(0.3, 0.5)), n = 100)
x <- ts(c(x1, x2))


CUMSUM approach: Once an AR model is fitted to the entire series the CUMSUM process can be obtained as follows:

fit <- arima(x, order = c(2,0,0), include.mean = FALSE)
e <- residuals(fit)
sigma <- sqrt(fit$sigma2) n <- length(x) cs <- cumsum(e) / sigma  As a reference, confidence limits can be obtained as done in package strucchange for the OLS-based CUSUM test. For that, we can create an object of class efp and plot it: require(strucchange) retval <- list() retval$coefficients <- coef(fit)
retval$sigma <- sigma retval$process <- cs
retval$type.name <- "OLS-based CUSUM test" retval$lim.process <- "Brownian bridge"
retval$datatsp <- tsp(x) class(retval) <- c("efp") plot(retval)  The confidence limits are just for reference, I'm not sure they are the right values to carry out a formal test in this context. Regardless of this, a sudden change or shift in the sequence cs can be interpreted as a sign that something is going on around that time point, possibly a structural change. In the plot we observe that at around observation 50, where we introduced a change in the data generating process. F-tests: Another approach is based on F-test statistics computed as: $$Fstat = \frac{RSS - USS}{RSS/n}$$ where RSS is the residual sum of squares in the restricted model (the model fitted for the entire data) and USS is the residual sum of squares of models fitted to two subsamples. The statistics can be computed iteratively for the following sequence of subsamples: from observations 1 to 20 and 21 to$n$; then from 1 to 21 and a next subsample from 22 to$n$, and so on as done below: rss <- sum(residuals(fit)^2) sigma2 <- fit$sigma2
stats <- rep(NA, n)
for (i in seq.int(20, n-20))
{
fit1 <- arima(x[seq(1,i)], order = c(2,0,0), include.mean = FALSE)
fit2 <- arima(x[seq(i+1,n)], order = c(2,0,0), include.mean = FALSE)
ess <- sum(c(residuals(fit1), residuals(fit2))^2)
}


Similarly to the CUMSUM plot, a plot of the F-statistics may reveal the presence of a structural change. A 95% confidence limit can be obtained based on the chi-square distribution.

plot(stats)
abline(h = qchisq(0.05, df = length(coef(fit)), lower.tail = FALSE), lty = 2, col = "red")


If the minimum p-value related to each statistic is below a significance level, e.g. 0.05, then we can suspect that there is a structural change at that point. In this simulated series that happens at observation 50, when the AR coefficients changed in the data generating process:

which.min(1 - pchisq(stats, df = 2))
#[1] 50


You may find further details in the vignette of the strucchange package that you probably already know and in the references therein.

• + very nice and thorough answer. – forecaster Jul 2 '14 at 18:41
• @Frank The ideas in this answer gave an overall idea about how to adapt the methods already available in strucchange to the case of ARIMA models. There is literature specific for the detection of structural breaks in time series, see for example the references given here. – javlacalle Apr 8 at 19:45
• @Frank It's interesting to see that the method shows a sensible performance in the simulation exercise. The issue that I see with using segments is that not all the data are used at the same time. The standard deviation of parameters may be higher and this may affect the test if no break is actually present. Anyway, with a relatively large sample as the one in your example, it seems a sensible approach to try. – javlacalle Apr 8 at 19:47
• @Frank Instead of segments like [0-50], [51-100], [101-150],... it could be interesting to use a moving window, e.g.: [0-50], [10-60], [20-70], ... and see the behavior of the statistic for structural break at each point. For example, see if there is a spike when the window enters an observation,... The distribution of the statistic at a given observation in this moving-window-approach may in fact have a distribution under the null of no break, which could be worth exploring by simulations and perhaps analytically, but this would require some further thought. – javlacalle Apr 8 at 19:52
• @Frank Finally, another possibility is an iterative process in which ARIMA models are fitted to different segments as new structural breaks are found. For example, if a break is found at observation 60, then fit an ARIMA model for the first 60 observations an another for the remaining observations. Then reassess the significance of the break and test for another break given the last accepted multi-ARIMA model. If interested in a similar idea, see e.g. the documentation of the software TRAMO or tsoutliers. – javlacalle Apr 8 at 19:56

I have blogged about detecting structural break using the strucchange package in R. It is pretty straight forward - here's the outline:

# assuming you have a 'ts' object in R

# 1. install package 'strucchange'
# 2. Then write down this code:

library(strucchange)

# store the breakdates
bp_ts <- breakpoints(ts)

# this will give you the break dates and their confidence intervals
summary(bp_ts)

# store the confidence intervals
ci_ts <- confint(bp_ts)

## to plot the breakpoints with confidence intervals
plot(ts)
lines(bp_ts)
lines(ci_ts)


The time series data used in my blog happens to be an ARIMA(0,1,1) process. If you want to verify that, check my Github repo regarding the same.

• Your blog post seems neat, however we prefer if the answers on this site are self contained. You could improve your answer be adding a bit about the method (how, and why it works for example). – Repmat Nov 16 '16 at 16:16
• @Repmat I'll surely improve my answer in a while, with more details on the method, the motivation and how it works. – Anirudh Nov 16 '16 at 16:33