Hi Lars,

I agree with Muhammad that you may want to study the two periods separately, or at least partition them out. There is likely a very good reason why the trend changes at a certain point in time, and the question is "why"?

One approach you can take to modelling this is to treat it as an interrupted time series, where you, in fact, partition the periods into two (or more) periods. You can then investigate the changes in these periods (see the attached paper on this approach in Stata).

Of course, a lot of this will rely on content expertise, so you should bring that to bear in this analysis.

I hope this helps

Ariel

Article Conducting Interrupted Time-series Analysis for Single- and ...

Sir, I think that you should use a mode decomposition method such as wavelets or EMD to decompose the signal and then look at the different modes separately so that you can either de-trend It if you wish or study it at different scales as is common in many economic applications. 

Hi Lars,

I agree with Muhammad that you may want to study the two periods separately, or at least partition them out. There is likely a very good reason why the trend changes at a certain point in time, and the question is "why"?

One approach you can take to modelling this is to treat it as an interrupted time series, where you, in fact, partition the periods into two (or more) periods. You can then investigate the changes in these periods (see the attached paper on this approach in Stata).

Of course, a lot of this will rely on content expertise, so you should bring that to bear in this analysis.

I hope this helps

Ariel

Article Conducting Interrupted Time-series Analysis for Single- and ...

Dear Lars Ahnland Nordfors

I think your time series has the threshold that why you should consider some threshold model such as TVAR

Or you should consider the non-linear model

i hope it helpful

Best

Dear colleagues, I agree with all answers. but I just want to make a reflection about the assumption : '' the profit share should theoretically return to an equilibrium''? perhaps you can find other result than that supposed by theory? it is just a reflexion? thank you

 In my opinion, non-stationarity in the profit share can not be ignored. I agreed with all possible answers given above. I want to add one more thing please. Why don't you try heteroscedastic models (ARCH/GARCH) rather than linear time series models? Although, they will model the volatility, but can be used for modeling the mean as well. 

Hi dear! the answers given are all good for understanding. but I would like to add more into it. if u ignore the non-stationarity, then u cant get the desire results and  obtained results would be non-sense. so u either split the data into different periods or model the relationship by applying non.linear models or go to use non-parametric approach. See the papers of Prof Aman Ullah on non-parametric approach.

Hi Lars,

for decomposing the signal it might be helpful to use an appropriate model that reflects the different components, e.g. by combining an exponential term for the growth with a periodic term for the seasonal changes, like s(t) = a*exp(kt) + b*sin(t/T) or

s(t) = exp( kt + b*sin(t/T) ). The procedure of decomposition then depends on the mathematical structure of the model. In the case of the latter model, for example, one could follow these steps:

1) Take the logarithm of the time series s(t), i.e. s'(t) = log( s(t) )

2) Calculate the linear regression of s'(t) (this will yield the parameter k)

3) Subtract the regression line k*t from s'(t)

The result is the seasonal (periodic) time series that follows the partial model b*sin(t/T), which is bounded (and in this case stationary), although the original signal was unbounded.

However, as Ariel stated already, content expertise is important for developing the right model.

Good luck, Ralf

Hi Lars,

In general, run the regression for non-stationary variables provides spurious results. The only way to prevent that is to eliminate the stochastic trends to produce stationary residuals or cointegration. There are some methods to treat a spurious regression problem such as  Fractional integration,… or nonlinear time series models  such as regime switching model. ( see Hamilton, chapter 18, p560).

If you are specifically looking for linear models with the trend, then some candidate might be:

Models with deterministic trends

Integrated processes (stochastic trend models ) and more… .

If you willing to the nonlinear models with trends

Model with breaking trends (See Hansen, 2001 for a survey).

Best,

Afsaneh

Hi Lars,

May be you can have a look at FARIMA or ARFIMA model. 

I hope it helpful.

Good luck,

Muzi

 I agree with Ariel suggestion to split the series into two parts and treat each part separately. 

good luck

I propose to study the outlier before you divide the period. and I ask about the number of year for to make separation in two periods. and you can use ''log'' . check if the series is stationary in second difference. or to seprate trend and cycle using filter... there is differ manner to do it. i hope it is useful

Hi Lars

The nonstationarity in time series has a long history. There is several features of nonstationary in datasets. The first one is due to Box & Jenkins ( ARIMA, SARIMA, …) which can be make stationary via some filtrations techniques. The second one is characterized by the dependence of its first and second moments (among others) on the time and hence  the differentiation technique dont work.

Box-Jenkins's approach (now developed into many directions) addresses this point.

As much as I enjoy using time-series modeling in forecasting, they have limitations. It all depends on what your end goal is.  For example, you may have a runaway process limited by the supply, in which case you need to address it as a resource management / capacity planning problem.  Or you may have a situation where the pattern you are describing is just the "nature of the beast", like in the use cases for Bass diffusion model.  There is no universal truth when it comes to applicability of forecasting models to data.  Every case is special.