last month is associated with a 50 percent reduction in the likelihood of paying a fee in the current month The Lk plots rise with k indicating that as a given fee payment recedes into the past the negative association between this fixed lagged fee payment and current fee payments is diminished. By the time one year has passed, the association between the lagged fee payment and current fee payment has almost vanished For large values of k, all three graphs rise above 1. This asymptotic property reflects the variation in fee payments that is driven by cross-sectional variation in the(persistent) type of the borrower. Some individuals have a relatively high long- run likelihood of paying fees For instance, imagine that 20 percent of consumers never pay a fee(maybe because they are rery disciplined), while the others have a long- run monthly probability b>0 of paying a fee Then, the long run Lk is 1/ 0.8=1.25. 3 2.3.1 Dynamic panel models with fixed effects Our second approach to estimating the impact of past fee payments on current fee pay ments is to estimate dynamic panel data models with fixed effects for each fee, while using Monte Carlo simulations to bound the size of the bias. As noted above, Nickell(1981)ana- lytically derived that the size of the bias for an ar(1) with fixed effects is of order -1/T Since in our estimation, T=36, the implied bias is approximately -1/36=-.028. This amount is much too small to explain the large reduction in fee frequency this month as- sociated with a fee payment last month. However, we are interested in the impact of fe payments at longer lags than one month on current fee payment; no analytical results exist for determining the size of the bias for higher-order autoregressive models. We thus use Monte Carlo simulations to determine the size of the bias in such cases Specifically, we first draw 10,000 times from a uniform(0, 1) distribution; call each draw ai,where i=1,.,10, 000. For each i, we then draw 36 times from a uniform(0, 1), recoding the results as l if the draw is less than ai and 0 otherwise. This algorithm simulates 36 i.i.d draws from a binomial distribution with probability of success a;(for 10,000 households). On these N=10,000 and T=36 observations, we then estimate an ar(1), AR(12 and AR(18 with fixed effects. We repeat the whole process 5,000 times, and report mean and standa 13The numerator of(3)is b, while the denominator is 0.8b. So Lk=b/(0.8b)=1/0.8last month is associated with a 50 percent reduction in the likelihood of paying a fee in the current month. The  plots rise with , indicating that as a given fee payment recedes into the past, the negative association between this fixed lagged fee payment and current fee payments is diminished. By the time one year has passed, the association between the lagged fee payment and current fee payment has almost vanished. For large values of  all three graphs rise above 1. This asymptotic property reflects the variation in fee payments that is driven by cross-sectional variation in the (persistent) type of the borrower. Some individuals have a relatively high long-run likelihood of paying fees. For instance, imagine that 20 percent of consumers never pay a fee (maybe because they are very disciplined), while the others have a long-run monthly probability   0 of paying a fee. Then, the long run  is 108=125. 13 2.3.1 Dynamic panel models with fixed effects Our second approach to estimating the impact of past fee payments on current fee pay￾ments is to estimate dynamic panel data models with fixed effects for each fee, while using Monte Carlo simulations to bound the size of the bias. As noted above, Nickell (1981) ana￾lytically derived that the size of the bias for an AR(1) with fixed effects is of order −1. Since in our estimation,  = 36, the implied bias is approximately −136 = −028. This amount is much too small to explain the large reduction in fee frequency this month as￾sociated with a fee payment last month. However, we are interested in the impact of fee payments at longer lags than one month on current fee payment; no analytical results exist for determining the size of the bias for higher-order autoregressive models. We thus use Monte Carlo simulations to determine the size of the bias in such cases. Specifically, we first draw 10,000 times from a uniform (0,1) distribution; call each draw , where  = 1  10 000. For each , we then draw 36 times from a uniform (0,1), recoding the results as 1 if the draw is less than  and 0 otherwise. This algorithm simulates 36 i.i.d. draws from a binomial distribution with probability of success  (for 10,000 households). On these  =10,000 and  = 36 observations, we then estimate an AR(1), AR(12) and AR(18) with fixed effects. We repeat the whole process 5,000 times, and report mean and standard 13The numerator of (3) is , while the denominator is 08. So  =  (08)=108. 10