# How find relationship of X on Y if both X and Y have trends?

I want to find an effect of X on Y. However, my Y and X are both have trends and seasonality. The first picture is my X and the second Y. The logic and academic results show that X should positively influence Y. However, when I will do regression I will have a negative coefficient (mainly due to opposing trends). Additionally< i have several different regressors which will help in the model. What can I do to find this effect. I guess the detrending should be done, but by which means? I heard that rolling window regression can help.

Depending on the nature/origin of your data, a simple first way of detrending would be to use a log-linear model for your time-series. For instance, for $$Y$$ the model would be

$$log Y_t = y_t = \alpha + gt + \epsilon_t$$.

where $$\alpha+gt$$ is your trend component and $$\epsilon_t$$ the cyclical component. (Note that $$t$$ indexes time). Below is an example with data from the FED.

If you would take log-differences ($$\Delta y_t = y_t - y_{t-1}$$) you get something equivalent to a growth rate with a constant trend growth $$g$$ and the change in the cyclical component ($$\Delta \epsilon_t$$).

Important caveat with this approach is that there might appear to be a mean-reverting cyclical component, which is not there. Instead you could also use a Hodrick-Prescott filter. Depending on the software you use there is also decompose in R which can help remove the trend and seasonal component.

I am sure others can add more useful suggestions as well.

### Example GDP data US.

library(Quandl)
gdp <- Quandl("FRED/GDPC1",order="asc")          #data
gdp <- ts(gdp\$Value,start=c(1947,1),frequency=4) #time-series: quarterly
gdp<-log(gdp)                                    #log-transform
gdp.dt<-ts(residuals(lm(gdp~index(gdp))), start=c(1947,1),frequency=4) #detrend
plot(gdp.dt)

• Thank you for your help. This approach helps to detrend time series data in Y. Should I also use natural logarithm for my Xs, which have trend? Additionally, interpretation of the coeffcients will be changed using logarithms. I want to get coefficents what say in absolute terms that if X changes by 1 unit, Y changes by .. units, because some time my Y(not aggregated number as in the plot) is null. So it makes confusing in terms of interpretation using logarithms. I know that trend variable can be used as X2, but are there any other solutions to my problem_ May 29, 2020 at 15:29
• Yes if there is a trend it is best to remove it, otherwise you end up with spurious regression (e.g.). For the coefficients you can take the exponential function to transform them back I believe. But I am not 100% certain. May 29, 2020 at 15:44