SAN DIEGO (AP) — Investigators said Tuesday they believe a couple that abruptly disappeared with their two young boys left for Mexico voluntarily, a statement that closes a chapter on an exhaustive search that began more than three years ago.

The search for Joseph and Summer McStay and their sons started in February 2010 when authorities found rotten eggs on the kitchen counter of the family's suburban home and two dogs in the backyard.

Neither Joseph, then 40, nor Summer, 43, told family or friends they were leaving, and the San Diego County Sheriff's Department responded with what it calls their most extensive missing persons search ever.

Investigators say they are convinced that four people seen on a dimly lit surveillance video walking into Tijuana, Mexico — four days after the disappearance — are the McStays. They had long maintained that possibility but avoided such a definitive statement.

The U.S. Customs and Border Protection video doesn't show anyone seeking help and there were no signs of forced entry at the McStay home in Fallbrook, about 70 miles north of the border, or in their SUV parked near the border. That has led investigators to believe they went to Mexico voluntarily, sheriff's Lt. Glenn Giannantonio said.

"The reasons behind it? Your guess is as good as mine," Giannantonio said. "It's very odd. We really have no idea what happened to them."

Some family members have doubted that the video shows the couple and their 3- and 4-year-old boys. But the Sheriff's Department said it was turning over the case over to the FBI because of that agency's extensive experience working outside the United States.

Investigators phoned relatives Tuesday shortly before the department issued a statement that said the family likely went to Mexico "of their own free will."

A statement posted on a website set up by Joseph's brother, Michael, said the family was satisfied with the investigation.

"We simply want to know what happened to our missing loved ones," the statement read. "We are grateful that there has been so much love and support the past 3 years."

The family hasn't been seen or heard from since Feb. 4, 2010. A neighbor's surveillance tape shows the bottom of what looked like the McStays' Isuzu Trooper.

Less than an hour after the footage was taken, Joseph made his last known phone call to an employee to follow up on a business meeting earlier that day.

Four days later, their SUV was towed from a parking lot steps from the border that has a two-hour limit and is closely watched by guards. Investigators found a Pottery Barn Kids refrigerator and sink for a playhouse, with children's videos like "Franklin Goes to School" inside.

Ten days after the family was last seen, Michael McStay went to the home but only found the family's dogs. He called authorities the next day.

A detective who entered the home found the rotten eggs, as well as two cereal bowls full of popcorn on the living room futon.

There were no signs of a struggle and no arrangements had been made to care for the dogs. All of that suggested to investigators that the McStays left in a hurry.

Giannantonio said tens of thousands of dollars left in business bank accounts have not been claimed. Joseph ran a business from home designing and installing water fountains.

The Sheriff's Department said it has pursued hundreds of tips.

Copyright 2013 The Associated Press.

THIS IS AN UPDATED STORY. For an earlier version, read below.

SAN DIEGO (CBS 8) - The San Diego County Sheriff's Department is reporting Tuesday that they believe the McStay family, missing for more than three years, traveled to Mexico of their own free will. The Sheriff's Deparment has been investigating this case since February 2010, which is when the family was reported missing.

Due to the fact that the family is believed to be out of the country, and because minor children are involved, the case is being transferred to the FBI.

The Fallbrook family disappeared more than three years ago, leaving behind their dogs, $100,000 in the bank and no clues as to where they were going, or why. Their vehicle was found near the San Ysidro border, and on February 8, 2010, surveillance video captured what appeared to be a family of four fitting the description of the McStays crossing into Mexico. However, a family member did not believe the individuals shown in the surveillance video were his family.

"I try to wrap my head around it, but I'm just looking for closure to the case one way or another," brother Michael McStay said.

McStay hopes that closure will come sooner than later.

"They just need that one incredible tip. It can be anonymous -- I really don't care how it comes," he said.

The case was featured on "48 Hours Mystery" and an "America's Most Wanted" episode on the McStays generated about 10 new leads, but none panned out.

Sheriff's officials say the switch in no way indicates foul play is involved. But Rick Baker, who wrote a book on the case, disagrees.

"They didn't break any laws, and so for the FBI to get involved, something had to happen, which would go back to what I believe, one of the parents is dead," Baker said.

Anyone with information about this case is asked to call Crime Stoppers at (888) 580-8477. Callers will remain anonymous and could be eligible for up to a $1,000 reward for information leading to the resolution of the case.