# Approach for Trend comparison in a Time Series

I'm a bit confused on how to properly compare a known trend to an estimated one. I've got two sets of data which are of the following format,

$$y_{t_1}=x_{t_1}+WN$$ $$y_{t_2}=x_{t_1}+MA(2)$$

Where $$WN \sim N(0,1)$$, $$MA(1)$$ is a simulated 2nd order moving average and $$x_{t_1}$$ is a known trend function. Now I have gone ahead and estimated the trend function $$\hat{x_{t_1}}$$ using Splines via generalized cross validaiton (GCV). So my two data setups are,

$$y_{t_1}=\hat{x_{t_1}}+WN$$ $$y_{t_2}=\hat{x_{t_2}}+MA(2)$$

Where $$\hat{x_{t_1}}$$ is the modeled trend function by Splines. I now want to compare the modeled trend, $$\hat{x_{t_1}}$$,$$\hat{x_{t_2}}$$, to the actual known trend, $$x_{t_1}$$ and come to the conclusion that

A few things I have already tried,

1. Fixed $$x_{t_1}$$ as an increasing trend.
2. Added WN and MA(2) to create $$y_{t_1}$$, $$y_{t_2}$$
3. Modelled $$x_{t_1}$$, without the error terms, via Splines to get $$\hat{x_{t_1}}$$.
4. I then added $$\hat{x_{t_1}}$$ to the WN and MA(2) separately and calculated the MSE.
5. However the MSE values ended up being the same when compared to the original values, which is what I was not expecting as this concludes there is no difference in the trends.

Note: For #3 I also tried creating two different models, $$m_1$$ and $$m_2$$, with the errors included. I then subtracted the error terms from both models to get two estimated trends, which were very similar to one another. I then planned on comparing these two trends to the original fixed one however after putting some more thought into it, this approach doesn't really make sense.

Question: How can I compare the two trends, $$\hat{x_{t_1}}$$,$$\hat{x_{t_2}}$$, and $$x_{t_1}$$ to either prove or disprove my assumption of that $$\hat{x_{t_1}}$$,$$\hat{x_{t_2}}$$ is a good trend estimate for the setup with White Noise and not a good trend estimate for the setup with an MA(2) process?

1. Fixed $$x_{t_1}$$ as an increasing trend.
2. Added WN and MA(2) to create $$y_{t_1}$$, $$y_{t_2}$$
3. Modelled $$x_{t_1}$$ with both of the error terms separately, via Splines to get $$\hat{x_{t_1}}$$ and $$\hat{x_{t_2}}$$.
4. Plotted $$x_{t_1}$$ with $$\hat{x_{t_1}}$$ on one plot and $$x_{t_1}$$ with $$\hat{x_{t_2}}$$ on another plot.
The result of 3 different simulations was that the MSE of $$\hat{x_{t_1}}$$ estimated tend was lower than the estimated trend $$\hat{x_{t_2}}$$ when compared to the actual known trend of $$x_{t_1}$$. Which confirms the original assumption that estimating a trend via WN error terms is better than estimating a trend with MA(2) error terms. The details are quite vague but for the purpose of the work I was doing it is enough to make this conclusion. More details can always be added if needed or requested.