Part III. Emanuel and Harrington (2023)1 study if remote work affects productivity of call center workers at a US Fortune 500 retailer using data on calls per-hour before and during the Covid- 19 pandemic. The retailer hired both remote and on-site call center workers prior to Covid-19 and went entirely remote during the pandemic. To formalize the problem, let the potential outcome Yid(t) be the number of calls per-hour if a worker were to have been remote (t=1) or on-site (t=0), where i indexes the individual callcenter worker and d indexes the pre-Covid (d=0) and lockdown (d=1) periods. Let Zid be the realized treatment in period d, where Zid=1 if a worker is remote and 0 if on-site. Finally, let Ri=1 if i worked remotely before Covid and 0 if she worked on-site. Consider the problem on inferring the average treatment effect for on-site workers in the pre-Covid period, ATT0=E[Y0(1)R=0]E[Y0(0)R=0] using the following observed moments (ignoring sampling variability): E[Y1R=1]=4.0E[Y1R=0]=4.2P[Z1=1]E[Y0R=1]=3.4E[Y0R=0]=3.8=1.0P[Z0=1]=0.2 Note that Y1 is the realized number of calls-per hour during the lockdown period and Y0 is the number of calls per-hour during the pre-Covid period. Assume the number of calls per-hour never exceeds 8 . a. What can you infer about ATT0 from these moments? Briefly interpret this result. b. How would your answer to (a) change if selection is exogenous? That is, E[Y0(t)]=E[Y0(t)R] Briefly interpret this result. c. Show that the observed difference in the pre-Covid call center outcomes, E[Y0R=1]E[Y0R=0] equals the sum of ATT0 and the "selection effect" E[Y0(1)R=1]E[Y0(1)R=0]. d. Assume the common selection effect assumption that E[Y0(1)R=1]E[Y0(1)R=0]=E[Y1(1)R=1]E[Y1(1)R=0] Given this assumption and the population moments, what can you infer about ATT0 ? Can we reject the exogenous selection assumption in (b)