The upper bound is obtained as Fr´echet-Hoeffing upper bound as follows: The lower bound is obtained when ε$_{0}$ and ε$_{1}$ are independent conditionally on U , while the upper bound is obtained when ε$_{0}$ and ε$_{1}$ are perfectly dependent conditionally on U . Thus they are sharp. To show that CPQD has no additional identifying power on the DTE, I use the following Lemma which has been presented by ? and ? . Lemma B.1 Let C denote a lower bound on the copula of X and Y , and F$_{X}$$_{+}$$_{Y}$ denote the distribu- tion function of X + Y. If support of ( X, Y ) , supp ( X, Y ) satisfies supp ( X, Y ) = supp ( X ) × supp ( Y ) , Part 2. Sharp bounds on the DTE where C $^{d}$( u, v ) = u + v − C ( u, v ) . Let Y$_{1}$ = X and Y$_{0}$ = − Y . By Lemma B.1, sharp bounds on the DTE are affected by only the upper bound on the copula of Y$_{0}$ and Y$_{1}$. Since CPQD improves only the lower bound on the copula if Y$_{0}$ and Y$_{1}$ , the DTE bounds do not improve by CPQD. Theorem 3 Under M. 1 − M. 4 and MTR, sharp bounds on F ( y$_{0}$, y$_{1}$ ) , and F$_{Δ}$ ( δ ) are given as follows: for d ∈ { 0 , 1 } , y ∈$_{R}$ , δ ∈$_{R}$ , and ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ ×$_{R}$ , Theorem 3 41 F ( y$_{0}$, y$_{1}$ ) ∈ $^{[}$F $^{L}$( y$_{0}$, y$_{1}$ ) , F U ( y$_{0}$, y$_{1}$ ) $^{]}$, F$_{Δ}$ ( δ ) ∈ $^{[}$F L $_{Δ}$( δ ) , F U Δ ( δ ) $^{]}$, sup x + y = z C ( F$_{X}$ ( x ) , F$_{Y}$ ( y )) ≤ F$_{X}$$_{+}$$_{Y}$ ( z ) ≤ inf x + y = $_{z}$C$^{d}$( F$_{X}$ ( x ) , F$_{Y}$ ( y )) P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ ) ≤ inf z ∈ $_{Ξ}${ min { P ( y$_{0}$ | 0 , z ) , P$_{1}$ ( y$_{1}$ | 0 , z ) } (1 − p ( z )) + min { P$_{0}$ ( y$_{0}$ | 1 , z ) , P ( y$_{1}$ | 1 , z ) } p ( z ) } .