LIFE BEGAN for me in July 1923, in New York City. When I was a young actress, I once refused to give the year of my birth to a magazine writer. All young actresses are 22 . . ."

That's Nancy Reagan talking, in her tape-recorded autobiography, "Nancy," published during the 1980 campaign by William Morrow and Company, Inc.

Her collaborator, Bill Libby, is the book's introduction, says also that the future First Lady "was born Anne Frances Robbins on July 6, 1923."

But a spokesman at her alma mater, Smith College, said last week that the school's records give her birthdate as July 6, 1921.

Mrs. Reagan enrolled at Smith in 1939 according to alumnae records, at the age of 18 and graduated in 1943.

When she was a young actress, giving the interview she describes in her official memoirs, she would have been 24 by Smith's accounting, not 22.

The magazine writer apparently suspected at the time she might be fibbing.

"When the story was published, he had guessed my age and made me five years older than I was," she writes. "I wrote the magazine asking them to give me back my five years because I had plans for them. They printed my letter but I learned a good lesson.When you're asked a question, it's better to give an answer, or someone will give one for you and it may not be correct or what you want!"

Mrs. Reagan is still giving interviewers the answer she wants to see in print.

According to her, it's 1923, not 1921," a press aide said last week. "I have to go by that."

It was pointed out that the former Nancy Davis would have been a precocious 16-year-old when she entered Smith if her date is right and the school is in error.

"I'll go back and ask again," the aide said. "But I'm sure it's still going to be 1923 and not 1921."

One person who is probably going to be surprised to learn that the new First Lady may be nearer 60 than she has admitted is Barbara Bush, wife of Vice-President-elect George Bush.

Barbara Bush, who is 55, also attended Smith. She has been very candid with reporters, telling them that she knows she looks older than Nancy Reagan, instead of "two years younger."

If Smith is right, Nancy Reagan is four years older than Barbara Bush and will be 60 on the first birthday she celebrates in the White House next July 6.