---
title: "cointReg"
author: "Philipp Aschersleben"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 2
vignette: >
  %\VignetteIndexEntry{cointReg}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
```{R, include = FALSE}
knitr::opts_chunk$set(fig.path = "figures/vignette-", fig.width = 5,
                      message = FALSE)
```


## Install cointReg

```{R, eval = FALSE}
install.packages("cointReg")
```

If you like to use the development version, you can install the package directly
from GitHub:
```{R, eval = FALSE}
devtools::install_github("aschersleben/cointReg", build_vignettes = TRUE)
```

Load the package:
```{R}
library("cointReg")
```



## Basic examples

### Simple test model with one regression variable

Generate a regression variable `x` and a dependant variable `y`. The fastest
and easiest way to plot both time series is `matplot(...)`.
```{R}
set.seed(42)
x <- cumsum(rnorm(200, mean = 0, sd = 0.1)) + 10
y <- x + rnorm(200, sd = 0.4) + 2
matplot(1:200, cbind(y, x), type = "l", main = "Cointegration Model")
```

Now you can estimate the model parameters with the FM-OLS method and include an
intercept in the model via the `deter` variable:
```{R}
deter <- rep(1, 200)
test <- cointRegFM(x = x, y = y, deter = deter)
```

Print the results:
```{R}
print(test)
```

You can see that both the intercept and the regression variable are significant.

Finally, you can plot the residuals:
```{R}
plot(test, main = "Residuals of the Cointegration Model")
```



### Another test model with three regression variables and a linear trend

```{R}
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 <- x1 + x2 + x3 + rnorm(100, sd = 0.2) + 1
matplot(1:100, cbind(y, x), type = "l", main = "Cointegration Model")
```


```{R}
deter <- cbind(level = 1, trend = 1:100)
test <- cointRegFM(x, y, deter, kernel = "ba", bandwidth = "and")
print(test)
```

```{R}
plot(test, main = "Residuals of the Cointegration Model")
```



## Spurious regression example

This is why you should use modified OLS methods instead of a normal OLS model
to estimate parameters of a cointegrating regression:

```{R}
set.seed(26)
x <- cumsum(rnorm(200))
y <- cumsum(rnorm(200))
summary(lm(y ~ x))
```
The independant variable `x` seems to be significant at a very secure level.

And now have a look at the results of an FM-OLS regression:
```{R}
cointRegFM(x = x, y = y, deter = rep(1, 200))
```
So the `x` variable doesn't have an influence on `y` -- which makes sense
because they were generated independently.