DA40 Instructor Pilot reported while in traffic pattern; took evasive action to avoid traffic resulted in a NMAC.

Narrative

My student and I were flying in the pattern at ZZZ practicing take off - landings/emergency landing procedures. On final we saw a Yak - local to ZZZ holding short of Runway XX. We executed a touch-n-go and were closed traffic at ZZZ. On our crosswind to downwind turn we see a lot of smoke on the runway; thinking he had just crashed and was looking for a downed aircraft. My student then aggressively calls out traffic 9 o' clock; almost same altitude as we were turning downwind. The Yak made no radio calls and had climbed vertical then cut inside of our pattern on the crosswind while we were almost at 1600 ft - pattern altitude. I instantly cut power and dived away in an excessive sink rate going down to 1200 ft very rapidly. Had I continued to climb it would of resulted in a midair. I called him on the radio; he apologized and said it was his fault. We landed with no further incident. However; multiple people had seen what had happened and the dispatch coordinator filed a report as well. The Yak continued pattern work and we cautioned company planes to look out for him.