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We also have that the ex -ante value function is given by: <br />Vt (Bt -1, wt) = It (Bt -1) + F, (5) <br />where F is Euler's gamma. <br />5.2 Renewal Actions <br />The key challenge in identifying dynamic discrete choice models is dealing with the ex- <br />pected continuation values in the Bellman equation. To be able to calculate the expected <br />continuation values, one generally must make assumptions about exactly how agents form <br />expectations, including exactly what information is known to the agent and how they ex- <br />pect market -level state variables to evolve. This normally requires assuming all market state <br />variables (e.g. rents and amenities) are observed and follow assumed transition dynamics. <br />We build on Scott (2013) and make no assumptions about how amenities evolve. We also do <br />not assume how agents form expectations about future market states, other than that they <br />are on average rational. Following work by Arcidiacono and Ellickson (2011) and Arcidia- <br />cono and Miller (2011), we make extensive use of renewal actions, or action(s) which, given <br />current states Bt_r and 0't_1, lead to the same state in the next period. This will allow us <br />to difference out much of the long-term continuation values in the Bellman equation, which <br />are impossible to estimate without strong assumptions. <br />5.2.1 Immediate Renewals <br />Suppose we have two households in states Bt_r and 0t_1. In period t, these two households <br />take the actions x and x' respectively. Using equation (9) and differencing we find that: <br />vt (x; Bt -r) - vt (x', 0' <br />t-1) = In pt (xI B t-1) + It (Bt -1) - It (Bi -r) <br />24 <br />