When you are done with Hillside, which includes exploring the basement, the underground entrance, freeing the first bird for the Bird Cage side quest, and recalling the police patrols, it is time to move on to the next area, Pearl Creek.

Go north on Lansdale ave, then when you reach the edge, go around the house. There is a long hook leaning on the bench in the back yard. This is a good weapon, but it also serves another purpose. Use the hook on the retractable ladder to bring it down. Climb the ladder and turn right to find a barrel with a mystery object on it, a crime scene photo.

Go back and cross the walkway, then enter the next building through the window. You are now in the next district, Pearl Creek. In the first room, with the TV, you can search the drawers to find a butcher's knife you could use as a weapon, and a first aid kit. Open the door to the next room. The radio there is on. DJ Ricks from WLMN FM dedicates another song for Murphy Pendleton. Exit through the second door into the corridor, and keep exploring. Go to the stairway, and climb up. Open a door on the top floor. Go through the kitchen and into the bedroom with the fireplace. Explore the rooms for any weapons or supplies you might need, then go out to the corridor and to the door across. Pick up a note attached to one of those doors:

Dear Shithead.
I'm on to you, you junkie asshole. Next time something goes missing from my apartment, you can expect the cops to come knocking on your door. And if it's not them, it'll be me, and my knock is a 12 gauge if you catch my drift. So if you don't want an ass full of buckshot, I suggest you stay the fuck away from my apartment. Last Warning.

Enter the apartment, A fire is burning in the fireplace. A man there appears to have committed suicide by hanging himself from the ceiling fan. Pull the metal bed and find a box under it with all sorts of stolen goods: A  Money Box, Locket, War Medal, Gold Watch.

This starts a side quest: Stolen Goods:
The thief stole these people's things. Maybe I can set things right by returning them to the rightful owners?

Your objective is to return the stolen items to their rightful owners. First, check your inventory for clues:
Money Box: A child's money bank. Shaped like an elephant
Locket: A picture of a woman is inside.
War Medal: A Purple Heart from WWII
Gold Watch: Solid gold watch with metal bracelet.

Stolen Goods Puzzle Solution

The location which you have to place the stolen goods in are as follows:

Money Box:

Downpour Stolen Goods Side Quest Money Box Location

On the second floor, far end of the hallway, enter a room that looks like a child's room. There is a disco ball on the ceiling there. This room is pretty well lit, in contrast to the rest of the game's setting. As you enter the room, depending how hard you have set puzzles to be, you will hear a child crying and the clinking of coins. Place the piggy bank on the stand near the bed, with the scattered coins. You will now hear a baby laugh. This floor also contains a room with a bullets case in the dresser, so make sure to search the whole floor thoroughly.

Locket:

Downpour Stoen Goods Side Quest Locket Location

Third floor, right across from the thief's apartment.  Locate a small table, with a case on top of it. Depending on puzzle difficulty level, you will hear clues such as a woman crying, indicating that the locket goes here. Place the locket in the case.

War Medal:

Silent Hill Downpour Stolen Goods Puzzle War Medal Location

Search the apartment on the ground floor. As you enter one of the rooms, that has a locking chair. Depending on puzzle difficulty, you will hear some noises as if from a war, including shouts and gunfire. This is a hint that the war medal belongs there. On the stand near the bed, locate the display case. Open your inventory and place the War Medal there. This apartment also has a painting, "Wonder", needed for The Art Collector quest.

Gold Watch: 

Downpour Gold Watch location stolen goods side quest

Third floor, across the room where you found the thief and the stolen goods. As you enter the bedroom, depending on difficulty of the puzzle you have set, you will hear a clock ticking. Place the gold watch on the hand display.

This completes the side quest "Stolen Goods" and also unlocks the following tropy/achievement:

Neighborhood Watch
Completed the "Stolen Goods" Side Quest.

Reward: Go back to the third room, to the apartment where the thief lived in. The body is gone, but you find a new set of clothes on the floor, which Murphy can change into.