Dec 31st.
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If he makes that statement on Jan 1st, he turned 16 the day before, and the day before that he was still 15, later on "this" year he will turn 17, then "next" year he will turn 18.
Greg's birthday is December 31st. On December 31, 2018, he turned 16. The next day, January 1, 2019, he said “two days ago I was 15 years old...", which was true because that was the day before his 16th birthday. He will turn 17 on December 31, 2019. So he also said on January 1, 2019: "...Later next year I will be 18 years old." That's true because he said it in 2019 and "next year" will be 2020, and he will turn 18 on December 31, 2020, which is later next year. Edwin