As a first attempt for your question 1, I would try recurrence analysis (N. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329, DOI:10.1016/j.physrep.2006.11.001, 2007) and related methods. In the basic approach, once you have computed a recurrence plot from a time series, you can measure the number of recurrences as a function of their distances from the main diagonal of the plot. If when moving far away from this line the number of recurrences decreases, then the process underlying your time-series is not stationary.Other variants are discussed in this short dissertation that I found on line: http://wiredspace.wits.ac.za/bitstream/handle/10539/5993/Dissertation_0311898E.pdf;jsessionid=0976E0756D5DB25E065D645CAC38F640?sequence=1

Hth,

Leonardo

Hello Bidyut,

Stationarity just means that the statistical properties of a time-series don't change over time.  If you want to check for any trends in your data, then test it for stationarity.  Probably the simplest way to check for stationarity is to split your total timeseries into 2, 4, or 10 (say N) sections (the more the better), and compute the mean and variance within each section.  If there is an obvious trend in either the mean or variance over the N sections, then your series is not stationary.  If not, then is may well be stationary.

I took your data file (Book1.txt), and computed the mean and std. deviation within each consecutive set of 600 points; results are in the file attached.  The mean certainly does look stationary, and variability of the 600-point mean is much less than the std. deviation within each 600-point section, which is good evidence for the stationarity of the overall series.  The std. deviation within each section does jump around a bit, but at least doesn't show any clear trend. 

For calculations of the PSD (power spectral density), see, e.g., this page:  www.ldeo.columbia.edu/users/menke/edawm/eda_lectures/lec12.pptx .  You will need to run a Fourier transform on your data.  The PSD is a function of frequency; the dominant periodicities in your time-series show up as peaks in the PSD curve (plotted as function of frequency).

As for "confidence" in your PSD calculations, again, split the total series into sections, compute the PSDs for each; if they are all very similar, you should have high confidence that they are "representative"; if not, then you should have low confidence in any one of them.  You can use Chi^2 statistics to quantify these "confidence" levels more formally.  See https://www.vibrationresearch.com/public_pdf/StatisticalPropertiesOfRandomPSD.pdf for a good explanation of the theory behind this.

Also, See https://cran.r-project.org/web/packages/psd/vignettes/psd_overview.pdf for info. on how to do all these calculations in R. 

I hope the above helps a bit!

-Enda

www.ldeo.columbia.edu/users/menke/edawm/eda_lectures/lec12.pptx

https://cran.r-project.org/web/packages/psd/vignettes/psd_overview.pdf

https://www.vibrationresearch.com/public_pdf/StatisticalPropertiesOfRandomPSD.pdf

Hello,

To use an example to answer your first question:

Consider the example: 

GDP in USA is trending upwards, while GDP in North Korea is also trending upward every year. (However, due to the high sanction imposed on North Korea by USA and vice versa- USA-North Korea: has totally no direct economic interaction).

If you were to regress USA GDP on North Korea GDP, you would find a positive relationship because both variables are trending upwards. However, this result is what most economist consider as spurious regression (nonsense result). 

Thus, we would need to de-trend the series before carrying out any econometric analysis.

Some of the unit root test: Augmented Dickey-fuller Test, PP test, KPSS test etc...

Hope this help you understand further...

Best of Luck on your research!

Dan

Thank you, Dr O'Brien. Thank you, Dr Lancia and Danny L.

Hi, in addition to the aforementioned considerations; it is convenient for you, consider the following: the are two “types” of  stationarity, strong and weak.  

The strong version implies

a)E(Yt)= E( Yt+s)      

b)Var(Yt)= Var( Yt+s)= σ^2

c)Cov(Yt; Yt+s)= Cov(Yt+k; Yt+k+s)= γ^2

In other words, time series is “horizontal” (a);  the variability (can see it like amplitude) doesn’t change over time (b), and the covariance between two values depending only of the distance  between them by not depend of the moment or period (c). These are the statistical propierties.

The weak approach  states only the requirement over mean and covariance

The first judgment is informal and visual; and as the other professor says , exists formal test, for example the τ  (“tau”) test or Dickey-Fuller test (one of the most cited). Dickey-Fuller required that you consider three functional forms

(1) ∆y= δyt-1+εt

(2) ∆y= β1+δyt-1+εt

(3)∆y= β1+β2T+δyt-1+εt

And,  the critical value is different according the functional form. Ho is δ=0 (time series is no stationary) and H1 Is  δ

The stationarity of data can be checked with ADF test or the PP test