Is it normal to be depressed over pets death?

Three years ago I bought an Indian Ringneck bird named Calais. She was untame and very aggressive because her previous owners abused her, so she would bite me or run away whenever I tried to pet or hold her.

After some years she finally had gotten use to me and would sit on my shoulder and let me pet her wings. She loved to whistle and I even taught her a few words and how to dance.

She lived in the house, however my father found her annoying and he made me put her outside all day and wouldn't let me bring her inside the house to handle her. I kept telling my dad that she would undo her lock on the cage and get out but my dad told me she wouldn't do it.

After feeding Calais some apple and cleaning out her cage I sat down inside to have a break. I then went outside and realized Calais wasn't in her cage.

I panicked and looked in my street for hours, but I couldn't find her. I put up ads everywhere, even on the internet, although I never found her.

I grew really depressed and wouldn't eat and I would just lay in my room all day crying. It's been five months and my dad thinks I've gotten over her but I haven't.

I cry to myself every night and blame myself. Over the months I've been cutting myself as punishment. I feel like I'm a monster and murderer, I don't deserve to live. Calais was like a daughter to me and I loved her so much. Now that Christmas is coming up I feel worse, I was planning on saving up to get Calais a beautiful new cage that would of been bigger then her last one and buying her lots of treats and new toys. I feel like killing myself because I don't deserve Christmas because she can't because I killed her and it's all my fucking fault