I can test for the class. It's close, the curly brackets aren't used for grouping in MathType:
https://i.imgur.com/T9JxuH5.png
Fixing that for the summation terms and t-p subscript produces:
\Delta P_t = \mu + \sum_(j=1)^J \lambda^j S^j_t + \sum_(p=1)^P \delta_p \Delta P_(t-p) + \varepsilon_t
https://i.imgur.com/Imix6GU.pngwhere Word doesn't realize that the terms after the sum should be the content of the sum. Word Help and the equation editor are thoroughly unhelpful in deducing the appropriate syntax of \sum.
What I eventually divined was that the syntax is state-based: If you type it out manually, it will put your cursor into the right position, and once you're done with _t you need to hit the right arrow key before typing the space and plus symbol. Copy-pasting the whole thing is different and does not work.
I mean, it's great that it supports an entry mode that's faster than digging through menus and clicking on symbols with your mouse. But it's still fundamentally different and, dare I say, inferior when the appearance of the equation isn't defined by the sequence of characters you enter (or copy-paste) but by the sequence of arrow keys and escape keys you hit while you type it in. This is exactly the problem with Office vs. LaTeX.