# cointReg

## Install cointReg

install.packages("cointReg")

If you like to use the development version, you can install the package directly from GitHub:

devtools::install_github("aschersleben/cointReg", build_vignettes = TRUE)

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(...).

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:

deter <- rep(1, 200)
test <- cointRegFM(x = x, y = y, deter = deter)

Print the results:

print(test)
##
## ### FM-OLS model ###
##
## Model:       y ~ deter + x
##
## Parameters:  Kernel = "ba"  //  Bandwidth = 1.40497 ("Andrews")
##
## Coefficients:
##         Estimate  Std.Err t value Pr(|t|>0)
## deter   2.327403 0.576897  4.0344 7.816e-05 ***
## x.coint 0.967884 0.057845 16.7324 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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

Finally, you can plot the residuals:

plot(test, main = "Residuals of the Cointegration Model")

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

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")

deter <- cbind(level = 1, trend = 1:100)
test <- cointRegFM(x, y, deter, kernel = "ba", bandwidth = "and")
print(test)
##
## ### FM-OLS model ###
##
## Model:       y ~ deter + x
##
## Parameters:  Kernel = "ba"  //  Bandwidth = 1.940012 ("Andrews")
##
## Coefficients:
##         Estimate    Std.Err t value Pr(|t|>0)
## level  1.2204577  0.1657569  7.3629 6.462e-11 ***
## trend -0.0076663  0.0077114 -0.9942    0.3227
## x1     1.0852324  0.1159726  9.3577 3.901e-15 ***
## x2     0.9027264  0.0857958 10.5218 < 2.2e-16 ***
## x3     0.9286189  0.0626775 14.8158 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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:

set.seed(26)
x <- cumsum(rnorm(200))
y <- cumsum(rnorm(200))
summary(lm(y ~ x))
##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -10.7889  -3.3236   0.6175   2.8696   8.9689
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   8.7016     0.3899   22.32  < 2e-16 ***
## x            -0.3811     0.0590   -6.46 7.94e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.196 on 198 degrees of freedom
## Multiple R-squared:  0.1741, Adjusted R-squared:  0.1699
## F-statistic: 41.73 on 1 and 198 DF,  p-value: 7.943e-10

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:

cointRegFM(x = x, y = y, deter = rep(1, 200))
##
## ### FM-OLS model ###
##
## Model:       y ~ rep(1, 200) + x
##
## Parameters:  Kernel = "ba"  //  Bandwidth = 51.01288 ("Andrews")
##
## Coefficients:
##         Estimate  Std.Err t value Pr(|t|>0)
## deter    8.11065  1.50333  5.3951 1.943e-07 ***
## x.coint -0.42580  0.22696 -1.8761   0.06211 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

So the x variable doesn’t have an influence on y – which makes sense because they were generated independently.