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What are the possible problems/issues of regressing smooth time series with almost no fluctuation? Here is a specific example.

My Example

Is there anything I have to pay attention to when interpreting coefficients? Or is there no proper way to regress time series like these? If so, what are the reasons?

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One thing to consider, is that it looks like you may have a unit root, though not necessarily.

An example of a unit root would be the stochastic process $y_k=y_{k−1}+\epsilon_{k−1}$, where the error term is mean-zero.

Unit roots can cause problems. For one, they are not stationary processes. Using OLS relies on stationarity. A violation can lead to a 'spurious regression': invalid estimates but with a high R-squared.

However, these problems are solvable. The first step is to run a unit-root test, of which there are many.

I suggest looking elsewhere for further material on this, e.g. here. There is lots out there and a full treatment would be outside the scope of one answer.

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    $\begingroup$ +1, although to me it looks like stationary series with break, since the article you linked to does not mention this possibility, but I don't think this warrants separate answer, I will just add that there are also breakpoint unit root tests such as Zivot-Andrews test that might be more appropriate here. $\endgroup$
    – 1muflon1 ♦
    Commented Mar 10, 2021 at 22:30

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