Dogs do miss their owners because they remember anything to which they have attached strong emotional ties. Emotionally speaking, dogs have been shown to be on the level of a 3-year-old child. This means that they possess many basic and simple emotions such as sadness, fear and anger.
Dogs can recognize when their owners are home, or absent, largely by scent. The scent of their owner, or any familiar person, activates a specific part of the brain and elicits an emotional response from the dog.
When the owner of a dog leaves them, the dog experiences actual grief because they do not know that the owner is actually returning to them. They will actually come to feel that they have lost their owner for good once he or she is away from them for a long enough period of time.