How to analyze an inflection point in univariate time series? I have a univariate time series with 36 data points (monthly data for 3 years). The general trend of this time series is relatively steep decline, but for the most recent one year, it is almost flat and looks even increasing with a close look. So the goal is to confirm statistically that we have an inflection point. What is a good way to test this?
 A: Try Chow test. The idea is to estimate the model on first 32 points, then forecast using the estimated model and compare to 4 points at the end. When you estimated the model, you got the error variance. You can compare the forecast error to the error variance. If it's too big, then you can claim that the model parameters have changed. Unless you have extremely tight fit in training sample (32 points), the power of this test is going to be very weak. 
The same will go for any other statistical test. You are talking about showing the difference of the trend of only 4 observations from previous 32 observations. I'm afraid that you won't be able to accomplish your goal based solely on statistical tests. You'll need some support from business domain, understanding the underlying phenomenon.
A: A point of clarification : the Chow test was initially suggested to test the hypothesis that a set of coefficients from 1 group was the same as those from a second group. In AUTOBOX a piece of software that I have helped develop we implemented the Chow Test scheme to test for alternative break points which is of course is unknown a priori and is to be possibly found. Finding the break point that yields the maximum contrast (F value) leads then to testing the significance of this particular break point. This then leads naturally to conclude about the possible presence of parameter changes ( i.e. non-constant values ) from time section 1 to time section 2. This test requires that any abnormal data points have been treated with Intervention Detection schemes and a suitable model has been identified. We have found that treating this issue often leads to error variance constancy and is then a less intrusive dissection of the data. Now with respect to your question : If one can detect a time trend change then one may have an answer to the question. A time trend is a wo/[(1-B)(1-B)] intervention variable  where wo is a pulse intervention. Note that a pulse is a difference of a step and a step is a difference of a trend thus a pulse is second differences of a trend .
