Rolling vs Recursive vs Fixed Window Regression

Question

What precisely are the differences between rolling, recursive and fixed window regression?

As far as I understand,

• recursive: we train on a period $y(0)$ to $y(n)$ then predict $\hat{y}(n+1)$. Then we train on $y(0)$ to $y(n+1)$ and predict $\hat{y}(n+2)$ and so forth. The window we train on gets bigger, and we do one-step ahead predictions.

• rolling: we train on a period $y(0)$ to $y(n)$ then predict $\hat{y}(n+1)$. Then we train on $y(1)$ to $y(n+1)$ and predict $\hat{y}(n+2)$ and so forth. The size of the window we train on stays the same size, and we do one-step ahead predictions.

• fixed: here is where I am confused. I was thinking that fixed was that we train on $y(0)$ to $y(n)$, then predict $\hat{y}(n+1)$ to $\hat{y}(n+m)$. But i have the feeling that this is actually multi-step forecasting. I also find some sources on the internet that claim that rolling and fixed are the same. I am confused... Is rolling different from fixed? If yes how?

Some conflicting/confusing sources:

• This paper basically differentiates between fixed, rolling and recusrive, however i could not find where they explain the difference. They also mention that they focus on one-step ahead forecasting, so I am confused how they differentiate between fixed and rolling
• This source (and many others) only explain rolling vs recursive, dont mention fixed..

I am relatively confused at this point and I am very happy for any answers explaining the difference or pointing me towards some relevant sources...

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Update

I found a source giving some explanantion:

Source:

Regression-Based Tests of Predictive Ability, Kenneth D. West and Michael W. McCracken, https://www.jstor.org/stable/2527340?seq=3#metadata_info_tab_contents

However I'm still somewhat confused between multi-step ahead forecasting and fixed windows...

Answer 1

Your definitions are correct. Economic Forecastsing - Elliot and Timmerman (2016) is a good ref for that. See the image therein:

However I'm still somewhat confused between multi-step ahead forecasting and fixed windows..

Recursive (expanding windows), rolling windows and fixed windows, deal with parameters estimation. Multi-step forecasting is another problem. You can make one step ahead or multi step ahead forecasts with any of the three estimation procedure you mention.