Since it’s not available on CRAN yet, you have to download and install the package directly from GitHub. Also, the latest version of our `cointReg`

package has to be installed. It provides functions for parameter estimation and inference in cointegrating regressions and the `cointmonitoR`

package depends on it. See its CRAN page for further information.

```
install.packages("cointReg")
devtools::install_github("aschersleben/cointmonitoR", build_vignettes = TRUE)
library("cointmonitoR")
```

Generate a time series to monitor. Since it’s the easiest way to generate a stationary time series, we get `x1`

as a vector of random number from a standard normal distribution.

```
set.seed(1909)
x1 <- rnorm(200)
```

The use of the monitoring procedure is really simple, you just have to choose the length of the calibration period. Here we take `m`

as the first 25 % of the total length of `x1`

(i.e. the first 50 observations).

`test1 <- monitorStationarity(x1, m = 0.25)`

Now you can print the results:

`print(test1)`

```
##
## ### Monitoring Procedure for Level Stationarity ###
##
## Model: x1
##
## Parameters: m = 0.25 (last observation used for calibration: 50)
## Kernel = "ba" // Bandwidth = 1.104631 (Andrews)
##
## Results: Test Statistic (Hsm) = 0.4437515
## p-Value >= 0.1 (not significant to 0.05 level)
```

…and also plot it. Tell the `plot`

method `what`

you want to see – the `"test"`

statistics (default) or the `"values"`

/ `"residuals"`

of the tested time series or `"both"`

(which will generate two plots at once).

`plot(test1, what = "test")`

Let’s also have a look at an example, where (obviously) a structural break occurs and the stationarity ends after 100 observations. You may want exactly the first 93 observations to be part of the calibration period – which is no problem, since you can just specify it via the `m`

argument in both ways (fraction of length *and* number of observations).

```
x2 <- c(x1[1:100], cumsum(x1[101:200]) / 2)
test2 <- monitorStationarity(x2, m = 93)
print(test2)
```

```
##
## ### Monitoring Procedure for Level Stationarity ###
##
## Model: x2
##
## Parameters: m = 0.465 (last observation used for calibration: 93)
## Kernel = "ba" // Bandwidth = 1.552963 (Andrews)
##
## Results: Test Statistic (Hsm) = 5.026546
## p-Value <= 0.01 (significant to 0.05 level)
## Rejection Time = 126
```

As already mentioned above, you can set the `what`

argument to `"both"`

to get to plots from the model:

```
oldpar <- par(mfrow = c(2, 1))
plot(test2, what = "both", legend = FALSE)
```

`par(oldpar)`

We generate a cointegration model with three regressors and get the response variable as the sum of them plus an intercept term of 1 and a random error.

```
set.seed(1909)
x1 <- cumsum(rnorm(100, mean = 0.05, sd = 0.1))
x2 <- cumsum(rnorm(100, sd = 0.1)) + 1
x3 <- cumsum(rnorm(100, sd = 0.2)) + 2
x <- cbind(x1, x2, x3)
y <- 1 + x1 + x2 + x3 + rnorm(100, sd = 0.2)
matplot(1:100, cbind(y, x), type = "l", main = "Cointegration Model",
xlab = "Observation Number", ylab = "")
```

Now, you want to monitor the cointegrating relationship and set the calibration period to 40 observations. There are (at least) two additional questions that you have to think about:

Which method to use for estimating the model parameters?

This is done with the`cointReg`

package, so you first of all may have a look at its documentation. There are three possible choices:`"FM"`

,`"D"`

and`"IM"`

.Include a linear trend or only use an intercept?

For this question it may be helpful to see the results of a “full”`cointReg`

model:

```
m <- 40
cointRegFM(x[1:m, ], y[1:m], deter = cbind(level = 1, trend = 1:m))
```

```
##
## ### FM-OLS model ###
##
## Model: y[1:m] ~ cbind(level = 1, trend = 1:m) + x[1:m, ]
##
## Parameters: Kernel = "ba" // Bandwidth = 2.053036 ("Andrews")
##
## Coefficients:
## Estimate Std.Err t value Pr(|t|>0)
## level 1.0602145 0.4445214 2.3851 0.0226240 *
## trend 0.0069304 0.0153027 0.4529 0.6534249
## x1 0.8707882 0.2105639 4.1355 0.0002106 ***
## x2 0.9704294 0.2608974 3.7196 0.0006967 ***
## x3 0.9829914 0.1061508 9.2603 6.087e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

In the calibration period, the linear trend component is not significantly different from 0, so it should be sufficient to set `trend = FALSE`

.

```
test3 <- monitorCointegration(x = x, y = y, m = m, model = "FM", trend = FALSE)
print(test3)
```

```
##
## ### Monitoring Procedure for Level Cointegration ###
##
## Model: y ~ x
##
## Parameters: m = 0.4 (last observation used for calibration: 40)
## Model = FM-OLS // Kernel = "ba" // Bandwidth = 2.070899 (Andrews)
##
## Results: Test Statistic (Hsm) = 0.1476303
## p-Value >= 0.1 (not significant to 0.05 level)
```

Again, this can be plotted to get a visual understanding of the results:

```
oldpar <- par(mfrow = c(2, 1))
plot(test3, what = "both", legend = FALSE)
```

`par(oldpar)`

Obviously, the test statistics are always on a very low level and the residuals in the post-calibration period, determined by using the calibration period FM-OLS model, do not differ from its former behaviour.

Finally, we can investigate a “broken” cointegration model, like the following one. It’s with the same `x`

and `y`

values as above, but with an additional random-walk term beginning with observation no. 61.

```
y2 <- y + c(rep(0, 60), cumsum(rnorm(40, sd = 0.5)))
matplot(1:100, cbind(y2, x), type = "l", main = "Cointegration Model",
xlab = "Observation Number", ylab = "")
abline(v = 60.5, col = 2)
```

Again, we set the argument `trend = FALSE`

and have a look at the monitoring results:

```
test4 <- monitorCointegration(x = x, y = y2, m = m, model = "FM", trend = FALSE)
print(test4)
```

```
##
## ### Monitoring Procedure for Level Cointegration ###
##
## Model: y2 ~ x
##
## Parameters: m = 0.4 (last observation used for calibration: 40)
## Model = FM-OLS // Kernel = "ba" // Bandwidth = 2.070899 (Andrews)
##
## Results: Test Statistic (Hsm) = 156.1611
## p-Value <= 0.01 (significant to 0.05 level)
## Rejection Time = 80
```

We can see, that the procedure is able to detect this structural break on a significant level (p < 0.01).

The plot shows that the test statistic is increasing and crosses the critical value (grey dashed line) at observation no. `80`

.

```
oldpar <- par(mfrow = c(2, 1))
plot(test4, what = "both", legend = FALSE)
```

`par(oldpar)`

For a detailed view on the test statistics, you can use the base plot argument `log`

. It reveals that the test statistic is starting to grow some observations after no. 61, which is the first one after the structural break.

`plot(test4, what = "test", legend = FALSE, log = "y")`