If my wife is out of the country for an extended trip, am I required to light Shabbat candles in the house if I am home alone? Does it matter how long she is going to be away? Or that the time difference between her and me is five hours?
Of course, as the comment notes, consult your local Orthodox rabbi.
But see here for a halacha sheet shown before and approved by Rav Shternbuch:
If one’s wife is not home must the husband light the candles?
Seeing that men are obligated to light Shabbos candles as much as women are, if the wife is not present it is up to the husband to light the Shabbos candles.
To proffer a reason: The purpose of lighting candles is kavod shabbat, shalom bayit, and oneg shabbat. (See here.) To cite a definition of the last item:
Tosefos tell us that the Shabbos meal should be in the presence of the candles, for that is oneg Shabbos.
If so, if a man's wife is away, he still has this obligation of lighting the candles.
See also this, from Rabbi Doniel Neustadt:
If one's wife is not home for Shabbos, it is preferable that the husband himself light candles and not one of the daughters (8)
Footnote 8 reads:
8 Oral ruling by Harav M. Feinstein (quoted in The Radiance of Shabbos pg. 7); Shmiras Shabbos K'hilchasa 43:fn46.
No distinction was made between the wife lighting or not lighting elsewhere.