A Flight Instructor reported taking the runway with landing traffic and continuing takeoff resulted in a critical ground conflict and NMAC on climbout.

Narrative

Student and I were training at an airport with a large building to the NE. Getting out of our starting airport had taken a longer than usual time; so I was already feeling pressure to get my student some valuable training time instead of sitting at the hold short line.The student positioned the plane at the hold short line in a way that partially blocked sight to base and final. I relied on my iPad with ADS-B in for traffic avoidance. I saw an aircraft that was showing as a ~2 mile base. The other pilot called a right base for the active. I instructed my student to take the runway and depart. The other aircraft called a short final and I made the mistake of instructing my student to continue instead of taking the controls and stopping prior to entering the runway.This forced the other pilot to perform a go around. The similar performance of our airplanes created a situation where we were on the same flight path; climbing at the same rate and with the same speed. At this point I took the controls from my student and turned away from the other aircraft. I departed the pattern and climbed up to let myself decompress. I pride myself on being a safe pilot; and I let my impatience get in the way of safely operating the aircraft. Going forward; I will ensure that I have a good view of base and final; and have visually identified any potential conflicting aircraft instead of relying on the information provided via ADS-B.