Structural breaks, stationarity and time series modelling - Cross Validated most recent 30 from stats.stackexchange.com 2019-07-19T21:34:57Z https://stats.stackexchange.com/feeds/question/161224 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/161224 2 Structural breaks, stationarity and time series modelling arroba https://stats.stackexchange.com/users/81670 2015-07-13T16:04:14Z 2017-08-04T16:11:02Z <p>This is a simplified version of my problem... Say I have two time series ($X$ and $Y$) and I know that $Y_t$ is somehow dependent on $X_t$ but not on $X_{t-k}$ for any $k &gt; 1$.</p> <p>Ultimately I want to have a model describing the relationship between $Y$ and $X$. My objective with this model is to describe past values of the system, not to do forecasting.</p> <p>It seems however that the series have a structural break. Following the work of <a href="http://www.sciencedirect.com/science/article/pii/S0304407608000961" rel="nofollow">Kim and Perron</a> I tested each series for unit roots and found none - but I did find breaks. </p> <p>Just for clarification, in this test I'm assuming each series can be described as: $$\text{SERIES}_t = \begin{cases} a + b*t + u_t \;,\;t \leq t_{break} \\ (a + a_{break}) + (b+b_{break})*t + u_t \;,\; t &gt; t_{break} \end{cases}$$ where $a, a_{break}, b, b_{break}$ are constants (which can be $=0$) and $u_t$ is a (potentially ARIMA) noise term. The test checks if the noise $u_t$ has an unit root. Say the results are that $X$ has a break in the mean, $Y$ has a break in the trend and both series are stationary.</p> <p>My question is, how should I model / regress time series that have structural breaks? Since I found breaks while testing for unit roots, should this somehow be included in the model? The time of the break is different for each series, and I have no idea of how to check the validity / significance of such model anyway. Would it make any sense to try a regression with autocorrelated errors (such as in <a href="https://www.otexts.org/fpp/9/1" rel="nofollow">Hyndman and Athanasopoulos</a>) even though there is evidence of breaks in the series?</p> <p>(just out of curiosity... if I don't assume the possibility of structural breaks in the series and use standard KPSS or ADF-GLS tests to check for unit roots, the results are quite confusing: both tests reject the null - so KPSS results in an unit root while ADF-GLS results in stationarity)</p> <p>(also, I know there's a lot of discussion out there about whether structural breaks make sense or not after all, but assume that in this case there's strong visual evidence of a change... you can imagine $X$ is a step function + noise and $Y$ follows an increasing linear trend before the break and switches to fluctuations around a constant after the break)</p> https://stats.stackexchange.com/questions/161224/-/161398#161398 1 Answer by Georg M. Goerg for Structural breaks, stationarity and time series modelling Georg M. Goerg https://stats.stackexchange.com/users/11476 2015-07-14T14:25:48Z 2015-07-14T14:25:48Z <p>You can estimate this using the <a href="http://cran.r-project.org/web/packages/strucchange/index.html" rel="nofollow">strucchange</a> R package with a simple linear regression of y given x. In your case the slope coefficient equals $b$ before the break, and $b + b_{break}$ after the break. </p> <p>Using the breakpoints() fct in <strong>strucchange</strong> this will be something like</p> <pre><code>bp.mod &lt;- breakpoints(y ~ x, breaks = 1) # specify 1, also automated possible summary(bp.mod) plot(bp.mod) </code></pre> <p>See also the very useful <a href="http://cran.r-project.org/web/packages/strucchange/vignettes/strucchange-intro.pdf" rel="nofollow">strucchange vignette</a>.</p>