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B.2 In “Does Hospital Crowding Matter? Evidence from Trauma and Orthopedics in England") (American Economic Journal: Economic Policy. forthcoming). Thomas Hoe examines the in- pact of hospital crowding on medical treatment outcomes exploiting variation in emergency admissions. For simplicity, we abstract from some of the details examined in the paper. One possible outcome of interest is the length of the illness (measured in days) for a particular patient i. Let this be denoted by yi. Suppose one focusses on the following model relating this outcome for an individual i admitted to a particular hospital: In(x) = 30 + x3, +ti. where X, comprises variables such as the individual's age, race and disease stage at the time of admission. As indicated in the article, other elements that might affect the outcome of interest relate to patient composition and hospital operation details (such as capacity constraints and utilisation) at the hospital where the individual is admitted, among other factors. Note that the unit of observation in items (a)-(c) below are the individual whereas in items (d)-(e) relates to a time period (day) for a particular hospital. (a) Suppose that one is interested in estimating (7) with data from a particular hospital. Let e > 0 denote the number of days an individual is at the hospital. This variable and x; is observed for every patient in the hospital. If individual is discharged after recovering from the illness (i.e., cyı), y is known, but otherwise we only know ci and that y > What additional assumptions would one need to estimate (7) by maximum likelihood? Write down the log-likelihood function for this regression and explain your answer. (b) Suppose that individual i opts to go to a hospital according to the following choice model: h = 1(90 + rad, +, > 0) (8) where hi = 1 if individual i goes to the hospital and = 0, otherwise. The variable de records i's distance to the closest hospital and u marks idiosyncratic unobservable factors informing this decision. Assume that t; follows a standard normal distribution. One is interested in estimating (7) and information on yi is only available for those who go to the hospital. Assume that OLS estimates are obtained for those observations. First, if e, and w are not necessarily independent, but n = 0, would the OLS estimator above be consistent? Explain. What if yd is not necessarily zero, but d, and x, are independent? Explain.

(c) Suppose that going to the hospital depends not only on d, but also on x. In other words. consider now the following extended version of equation (8): h; = 1(70+ 14d: +1.x+ 420) (9) How would you estimate the parameters in (7) consistently? Explain your answer. (d) Suppose you have time series data on daily admissions to a particular hospital. denoted by 9t where t is a particular date, and consider for simplicity a linear regression of q: on week- of-the-year dummy variables Se recording which week of the year t pertains to. Using the number of admissions per hospital. Hoe notes that there is no evidence of serial correlation in the residuals of a regression of (emergency) admissions on seasonal dummies once one examines the estimated AR(1) coefficients for a regression of residuals on lagged residuals. Under what conditions does this residual regression offer a valid test for the absence of serial correlation? Describe the test in detail. (e) For the regression above, under what conditions would the OLS estimator be consistent? Explain your answer.