Does physical aerobic exercise ameliorate depression?
There are different theories about the causes of depression. Three of them assume a cause that could be affected by aerobic exercise.

Neurochemical theory of depression:

It seems that depression might be caused by low serotonin and/or noradrenalin levels. Physical activity releases both neurotransmitters, thus alleviating depressive symptoms.

Cognitive theory of depression:

It seems that negative cognitive schemata cause depression. Sports improves self-efficacy and other schemata.

Diathesis–stress model of depression:

Depression is the result of a predispositional vulnerability combined with stress from adverse life experiences. Aerobic exercise relaxes the sympathetic nervous system thus alleviating stress and, as a consequence, depression.

Maybe someone else will come up with studies showing a causal connection. These (and other) theories might explain the connection.