For example, the distribution of the college premium for people whose potential wage without college degrees is less than or equal to θ can be written as where Y$_{0}$ and Y$_{1}$ denote the potential wage without and with college degrees, respectively. Prior to considering support restrictions, I briefly discuss bounds on the DTE given marginal distributions without those restrictions. Lemma 1 (Makarov (1981)) Let Then for any δ ∈$_{R}$ , and both F L Δ ( δ ) and F U Δ ( δ ) are sharp. Henceforth, I call these bounds Makarov bounds. One way to bound the DTE is to use joint distribution bounds since the DTE can be obtained from the joint distribution. When the marginal distributions of Y$_{0}$ and Y$_{1}$ are given, Fr´ echet inequalities provide some information on their unknown joint distribution as follows: for any measurable sets A$_{0}$ and A$_{1}$ in$_{R}$, Consider the event { Y$_{0}$ ∈ A$_{0}$, Y$_{1}$ ∈ A$_{1}$ } for any interval A$_{d}$ = [ a$_{d}$, b$_{d}$ ] with a$_{d}$ < b$_{d}$ and d ∈ { 0 , 1 } . In Figure 3, π ( A$_{0}$ × A$_{1}$ ) corresponds to the probability of the shaded rectangular region in the support space of ( Y$_{0}$, Y$_{1}$ ) . 6 Note that since marginal distributions are defined in the one dimensional space, they are informative on the joint distribution for rectangular regions in the two-dimensional support space of ( Y$_{0}$, Y$_{1}$ ), as illustrated in Figure 3. $^{6}$If A$_{0}$ and A$_{1}$ are given as the unions of multiple intervals, { Y$_{0}$ ∈ A$_{0}$,Y$_{1}$ ∈ A$_{1}$ } would correspond to multiple rectangular regions. 9 max { µ$_{0}$ ( A$_{0}$ ) + µ$_{1}$ ( A$_{1}$ ) − 1 , 0 } ≤ π ( A$_{0}$ × A$_{1}$ ) ≤ min { µ$_{0}$ ( A$_{0}$ ) , µ$_{1}$ ( A$_{1}$ ) } . F L Δ ( δ ) = sup y max ( F$_{1}$ ( y ) − F$_{0}$ ( y − δ ) , 0) , F U Δ ( δ ) = 1 + inf y min ( F$_{1}$ ( y ) − F$_{0}$ ( y − δ ) , 0) . F L Δ ( δ ) ≤ F$_{Δ}$ ( δ ) ≤ F U Δ ( δ ) , Pr { Y$_{1}$ − Y$_{0}$ ≤ δ | Y$_{0}$ ≤ θ } , 2.2 DTE Bounds without Support Restrictions