# How to determine the line which a time-series fluctuating around

Let $$X_t=1+3t+0.5X_{t-1}+ \epsilon_t$$ be a trend-stationary model, where $$\epsilon$$ is a white noise, which has zero expected value and standard deviation.

Which line is the time series fluctuating around? And why?

• Is this a homework question? If so, please add the self-study tag and describe what you’ve tried so far. – The Laconic Dec 31 '18 at 13:09
• I added the tag. I don't know how to start or approach the problem. – nyaki Dec 31 '18 at 13:19
• But you didn’t get this question out of the blue; you must have some existing knowledge about time series etc. So where are you stuck specifically? Make sure you understand what trend-stationary means. Then determine what “fluctuate around” means specifically, in mathematical terms. – The Laconic Dec 31 '18 at 13:31
• What would your answer be if all the $\epsilon_t$ were zero? That might provide a good guess. BTW, that's exactly the case with "zero ... standard deviation" as stated in your question, but I suspect you might have omitted a crucial word, such as "constant standard deviation." – whuber Dec 31 '18 at 16:41
• I gave x(0) a value and calculated the first 20. From this, I got -4+6t. Does this might be a good approach? – nyaki Dec 31 '18 at 18:25