We are concerned with the third order linear difference equation Δ³y(t−1)+p(t)Δy(t) +q(t)y(t) = 0. First we prove some important identities for solutions of this equation. We then use these identities to prove three results. The first result gives criteria for which this third order difference equation is (1, 2)-disconjugate. The second result gives sufficient conditions for the existence of a positive solution of this difference equation. Finally, we give criteria ensuring the disconjugacy of this third order difference equation.