Approach for Trend comparison in a Time Series

Question

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?

Many thanks in advance.

Answer 1

After mangling around with this for a short while, I finally managed to get everything to work properly. Here were the steps I took,

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.
5. The differences were quite hard to see graphically so I decided to calculate the MSE of both plots.
6. Compared the two MSEs with one another to see which one performed better.

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.