ARIMA model transformation

#1
Let an annual series Yt be stationary. However, the series transformed and differentiated Dt = ln(Yt) - ln(Yt-1) is stationary. Moreover, we suppose that it obeys the following theoretical model: Dt = -0.12 + 0.75 Dt-1 + et, in which the error term and is a white noise of variance σ2 = 0.012.

How can I transform this model to get the original one before the transformation?
 

hlsmith

Not a robit
#2
What do you actually have in front of you. Were you given this scenario or do you have an actual data series? Can't recall the rule behind differencing logs but it may be converted to division of something of that nature. Those rules are pretty searchable. However, if I have say the value 5 and you say it is the result of a differencing - I would imagine there are an infinite combination of values that could result in it. I am unsure how you can work back from that. Are you just trying to convert the model equation above, if so, is -0.12 your theta-naught, etc.

Can you provide more information about the origins and context of your question.

Thanks.
 
#3
Thanks for your help.

I have questions based on this scenario. There is no data series, only the model.

I think that I need to resolve the initial model to answer the questions.

Question 9.
Let an annual series Yt be stationary. However, the series transformed and differentiated Dt = ln(Yt) - ln(Yt-1) is stationary. Moreover, we suppose that it obeys the following theoretical model: Dt = -0.12 + 0.75 Dt-1 + et, in which the error term and is a white noise of variance σ2 = 0.012.
a) It is an ARIMA model (p, d, q) with which values for p, d and q?
b) Is this model stationary? Justify.
c) If the most recent observations for 2017 and 2018 are Yt-2 = 207 and Yt-1 = 231, calculate a one-time forecast Ŷt for the year 2019.
d) Build a 95% confidence interval on the Yt value that will be observed in 2019. Suggestion: first calculate an IC on Dt using the normal law and then do the transformations required.
 

hlsmith

Not a robit
#4
Well this is a completely different set up than your first post. We typically ask OP's to show initial due diligence before just giving out answers to possible school work.
a). what terms in the model and was differencing used?
b.) is the Dt-1 a unit root term?
c + d) plug data into calculations.

What does IC stand for?
 

noetsi

Fortran must die
#6
Why would you difference a series that is already stationary. That violates a basic logic in doing ARIMA that you don't transform data that is already not violating the ARIMA assumptions. It would be like making MA 5 when its correct at 2.

This might answer your question I am not sure.

0

Let "xd" denote the differenced data and "x" denote the original data. Then xd[n]=x[n+1]-x[n]. Therefore, x[n+1]=x[n]+xd[n]. If you add the first element of first difference forecast to the real data with the same indice, then you will get the next real data forecast.

This is about forecasting. I am not sure if the formula works with past data.

https://stackoverflow.com/questions/47793868/get-back-to-un-differenced-data-after-forecast
 

Dason

Ambassador to the humans
#7
I think you're misreading. It says that after the transformation and after differencing that the result is stationary.
 

noetsi

Fortran must die
#8
"Let an annual series Yt be stationary. However, the series transformed and differentiated Dt = ln(Yt) - ln(Yt-1) is stationary."

Isn't that saying the original series is stationary?