the borrower exceeds his or her credit limit in a given month. Penalty pricing also results from over limit transactions. As above, our analysis measures only direct over limit fees 3. Cash Advance Fee: a direct cash advance fee of 3 percent of the amount advanced or S5(whichever is greater) is levied for each cash advance on the credit card. Unlike the first two types of fees, a cash advance fee can be assessed many times per month Cash advances do not invoke penalty pricing. However, the APR on cash advances is typically greater than that on purchases, and is usually 16 percent or more. Our analysis measures only direct cash advance fees(and not subsequent interest charges 2.2 Fee payment by account tenure Figure 1 reports the frequency of each fee type as a function of account tenure. The regression-like all those that follow -controls for time effects, account fixed effects, and time-varying attributes of the borrower(e. g, variables that capture card utilization each pay ycle). The data plotted in Figure 1 is generated by estimating fit= a+oi+vtime Spline(Tenure. +n, Purchase. t +n2 Activei.t +n3 Bille ristit-l +yUtilit-1+Ei. The dependent variable fit is a dummy variable that takes the value l if a fee of type j is paid by account i at tenure t. Note that t indexes account tenure- not calendar time. When we refer to calendar time we use subscript time. Fee categories, j, include late payment fees filate-over limit fees-fouer-and cash advance fees--f Aduance. Parameter a is a con- stant;i is an account fixed effect; vtime is a time fixed-effect; Spline(Tenure. t) is a spline lo that takes account tenure(time since account was opened) as its argument; Purchasi.t is the total quantity of purchases in the current month: Activei. t is a dummy variable that re- The spline has knots every 12 months through month 72. We have also tried replacing the spline with dummies for each month of account tenure; the results are quantitatively similar. We report results with a spline as our baseline specification because the reduction in the number of parameters meaningfully decreases the computation time required to estimate the model; each regression has 4,000,000 observations of credit card statements
the borrower exceeds his or her credit limit in a given month. Penalty pricing also results from over limit transactions. As above, our analysis measures only direct over limit fees. 3. Cash Advance Fee: A direct cash advance fee of 3 percent of the amount advanced or $5 (whichever is greater) is levied for each cash advance on the credit card. Unlike the first two types of fees, a cash advance fee can be assessed many times per month. Cash advances do not invoke penalty pricing. However, the APR on cash advances is typically greater than that on purchases, and is usually 16 percent or more. Our analysis measures only direct cash advance fees (and not subsequent interest charges). 2.2 Fee payment by account tenure Figure 1 reports the frequency of each fee type as a function of account tenure. The regression – like all those that follow – controls for time effects, account fixed effects, and time-varying attributes of the borrower (e.g., variables that capture card utilization each pay cycle). The data plotted in Figure 1 is generated by estimating, (1) = + + + ( ) + 1 + 2 + 3−1 +1 −1 + The dependent variable is a dummy variable that takes the value 1 if a fee of type is paid by account at tenure . Note that indexes account tenure — not calendar time. When we refer to calendar time we use subscript . Fee categories, include late payment fees – – over limit fees – – and cash advance fees – Parameter is a constant; is an account fixed effect; is a time fixed-effect; ( ) is a spline10 that takes account tenure (time since account was opened) as its argument; is the total quantity of purchases in the current month; is a dummy variable that re- 10The spline has knots every 12 months through month 72. We have also tried replacing the spline with dummies for each month of account tenure; the results are quantitatively similar. We report results with a spline as our baseline specification because the reduction in the number of parameters meaningfully decreases the computation time required to estimate the model; each regression has 4,000,000 observations of credit card statements. 6
fects the existence of any account activity in the current month; Bille risti t-1 is a dummy variable that reflects the existence of a bill with a non-zero balance in the previous balance Utili t. for utilization. is debt divided by the credit limit: Eit is an error term. Table 1 provides the regression results Figure 1 plots the expected frequency of fees as a function of account tenure, holding the other control variables fixed at their means. This analysis shows that fee payments are fairly common when accounts are initially opened, but that the frequency of fee payments declines rapidly as account tenure increases. In the first four years of account tenure, the monthly frequency of cash advance fees drops from 57% of all accounts to 13% of all accounts The frequency of late payment fees drops from 36% to 8%. Finally, the frequency of over limit fees drops from 17% to 5% To establish that the estimated pattern is robust to alternative specifications, we estimate several variants. We estimate equation 1 as a conditional logit; the results are qualitatively similar. We also repeat the analysis controlling for behavior and FICO scores, both lagged by three months to reflect the fact that they are only computed quarterly. The coefficients on the spline of tenure are almost unchanged. Finally, to eliminate the possibility that attrition from the sample has distorted the results, we have tried restricting the sample te only those account present for all 36 months; the results are little changed It is also of interest to know how the amount of fees paid vary by account tenure. For the late fee and over limit fees. the fee amount is constant, while for the cash advance fee the fee amount can vary over time. Figure 2 reports the average value of each fee type as a function of account tenure, conditional on other factors that might affect fee payment. The data plotted in Figure 2 is generated by estimating Vit =a+oi+vtime SplineTenure t) +n, Purchase t +n2 Activei. t n3 bille risti t-l +1Util;t-1+∈;t The dependent variable Vit is the value of fees of type j paid by account i at tenure t. All other variables are as before. Table 2 reports the regression results II Tenure in all figures starts at month two since borrowers cannot, by definition, pay late fees or over limit fees in the first month their accounts are op
flects the existence of any account activity in the current month; −1 is a dummy variable that reflects the existence of a bill with a non-zero balance in the previous balance; , for utilization, is debt divided by the credit limit; is an error term. Table 1 provides the regression results. Figure 1 plots the expected frequency of fees as a function of account tenure, holding the other control variables fixed at their means.11 This analysis shows that fee payments are fairly common when accounts are initially opened, but that the frequency of fee payments declines rapidly as account tenure increases. In the first four years of account tenure, the monthly frequency of cash advance fees drops from 57% of all accounts to 13% of all accounts. The frequency of late payment fees drops from 36% to 8%. Finally, the frequency of over limit fees drops from 17% to 5%. To establish that the estimated pattern is robust to alternative specifications, we estimate several variants. We estimate equation 1 as a conditional logit; the results are qualitatively similar. We also repeat the analysis controlling for behavior and FICO scores, both lagged by three months to reflect the fact that they are only computed quarterly. The coefficients on the spline of tenure are almost unchanged. Finally, to eliminate the possibility that attrition from the sample has distorted the results, we have tried restricting the sample to only those account present for all 36 months; the results are little changed. It is also of interest to know how the amount of fees paid vary by account tenure. For the late fee and over limit fees, the fee amount is constant, while for the cash advance fee, the fee amount can vary over time. Figure 2 reports the average value of each fee type as a function of account tenure, conditional on other factors that might affect fee payment. The data plotted in Figure 2 is generated by estimating, (2) = + + + ( ) +1 + 2 + 3−1 +1 −1 + The dependent variable is the value of fees of type paid by account at tenure . All other variables are as before. Table 2 reports the regression results. 11Tenure in all figures starts at month two since borrowers cannot, by definition, pay late fees or over limit fees in the first month their accounts are open. 7
S. Figure 2 shows that, when an account is opened, the card holder pays $6, 65 per month in ash advance fees, $5.63 per month in late fees, and $2.46 per month in over limit fees. These numbers understate the total cost incurred by fee payments, as these numbers do not include interest payments on the cash advances, the effects of penalty pricing (i.e, higher interest rates), or the adverse effects of higher credit scores on other credit card fee structures. Like Figure 1, Figure 2 shows that the average value of fee payments declines rapidly with account Figures 1 and 2 imply that fee payments fall substantially with experience. We next turn to a second pattern in our data 2. 3 The impact of past fee payment on current fee payment There are four reasons to expect fee payments of agent i to be correlated(positively or negatively) at two arbitrary dates t and t+k First, the(cross-sectional) type of the card holder (e.g. forgetful) may influence fee paying behavior. If person i pays a fee in period t then person i is more likely to be of the type that pays fees in general, implying that person i has a higher likelihood of paying a fee in period t+ h relative to other subjects in our sample. As long as the cross-sectional type of the account holder is a fixed characteristic, this first source of intertemporal linkage could be modeled as a fixed effect. If the true fixed effect is added to the model the residual variation is no longer correlated across time. however as we discuss below nickell bias (1981)introduces a wrinkle when the fixed effect needs to be estimated Second, transitory shocks that persist over more than one month(for instance, an un- employment spell) may influence fee paying behavior, causing fees to be positively autocor- hird, transitory shocks that are negatively correlated across months(for instance,an annual summer vacation) will cause fees to be negatively autocorrelated Fourth, fee payments may engender learning, causing fees to be negatively autocorrelated One natural approach to estimating the force of these four effects would be to estimate in autoregressive model, in which current fee payment would be allowed to depend on lagged fee payment, controlling for account fixed effects and time-varying characteristics. However Nickell(1981)showed that including fixed effects in dynamic panel data models causes the autoregressive coefficients to exhibit a bias of order-1/T, where T is the number of time
Figure 2 shows that, when an account is opened, the card holder pays $6.65 per month in cash advance fees, $5.63 per month in late fees, and $2.46 per month in over limit fees. These numbers understate the total cost incurred by fee payments, as these numbers do not include interest payments on the cash advances, the effects of penalty pricing (i.e., higher interest rates), or the adverse effects of higher credit scores on other credit card fee structures. Like Figure 1, Figure 2 shows that the average value of fee payments declines rapidly with account tenure. Figures 1 and 2 imply that fee payments fall substantially with experience. We next turn to a second pattern in our data. 2.3 The impact of past fee payment on current fee payment There are four reasons to expect fee payments of agent to be correlated (positively or negatively) at two arbitrary dates and + . First, the (cross-sectional) type of the card holder (e.g. forgetful) may influence fee paying behavior. If person pays a fee in period then person is more likely to be of the type that pays fees in general, implying that person has a higher likelihood of paying a fee in period + relative to other subjects in our sample. As long as the cross-sectional ‘type’ of the account holder is a fixed characteristic, this first source of intertemporal linkage could be modeled as a fixed effect. If the true fixed effect is added to the model, the residual variation is no longer correlated across time. However, as we discuss below, Nickell bias (1981) introduces a wrinkle when the fixed effect needs to be estimated. Second, transitory shocks that persist over more than one month (for instance, an unemployment spell) may influence fee paying behavior, causing fees to be positively autocorrelated. Third, transitory shocks that are negatively correlated across months (for instance, an annual summer vacation) will cause fees to be negatively autocorrelated. Fourth, fee payments may engender learning, causing fees to be negatively autocorrelated. One natural approach to estimating the force of these four effects would be to estimate an autoregressive model, in which current fee payment would be allowed to depend on lagged fee payment, controlling for account fixed effects and time-varying characteristics. However, Nickell (1981) showed that including fixed effects in dynamic panel data models causes the autoregressive coefficients to exhibit a bias of order -1, where is the number of time 8
eries observations. We thus adopt two approaches to deal with this problem. The first to calculate the following statistic E[f1|f-k=1] k EIf Probability of paying a fee at tenure t given the agent paid a fee k periods ago Probability of paying a fee at tenure t without conditioning on the RHS variables from the previous subsection(most importantly, we do not use person fixed effects to calculate Lt k). Specifically, E It is just the average frequency of fee payments in period t. 2 Likewise, E It I ft-k= l is the average frequency of fee payments in period t among the account holders who paid a fee at time t-k Conditioning on this sparse information, a consumer who paid a fee k periods ago has a probability of paying a fee equal to the baseline probability, E It, multiplied by Ltk.A value of 1 for Lt k indicates that having paid a fee k periods does not change the expected probability of paying a fee this period; a value less than one indicates lagged fee payment is associated with a reduction in the expected probability, and a value greater than one indicates lagged fee payment are associated with an increase in the expected probability For example, if Lt 1=0.6, a consumer who paid a fee last month has a probability of paying a fee this month that is 40% below the baseline probability We report averages of Lt k L Hence, Lk is the average relative likelihood of paying a fee, if the account holder paid a fee k periods ago. The Lk statistic illustrates some important time series properties in our data while avoiding econometric problems associated with estimating probit or logit models with fixed effects for a large n dataset Figure 3 plots Lk for all three types of credit card fees for values of k ranging from 1 to 35. All three lines start below 1, indicating that a fee payment last month is associated with a less than average likelihood of making a fee payment this month. For both cash advance and late fees, having paid a fee one month ago is associated with a 40 percent reduction in the likelihood of paying a fee in the current month. For over limit fees, having paid a fee Observations used for the calculation of Lk are from subjects who are in our sample at both date t and date t-k
series observations. We thus adopt two approaches to deal with this problem. The first is to calculate the following statistic: ≡ [ | − = 1] [] = Probability of paying a fee at tenure given the agent paid a fee periods ago Probability of paying a fee at tenure (3) without conditioning on the RHS variables from the previous subsection (most importantly, we do not use person fixed effects to calculate ). Specifically, [] is just the average frequency of fee payments in period 12 Likewise, [ | − = 1] is the average frequency of fee payments in period among the account holders who paid a fee at time − Conditioning on this sparse information, a consumer who paid a fee periods ago has a probability of paying a fee equal to the baseline probability, [] multiplied by . A value of 1 for indicates that having paid a fee periods does not change the expected probability of paying a fee this period; a value less than one indicates lagged fee payment is associated with a reduction in the expected probability, and a value greater than one indicates lagged fee payment are associated with an increase in the expected probability. For example, if 1 = 06, a consumer who paid a fee last month has a probability of paying a fee this month that is 40% below the baseline probability. We report averages of : ≡ 1 X =1 Hence, is the average relative likelihood of paying a fee, if the account holder paid a fee periods ago. The statistic illustrates some important time series properties in our data while avoiding econometric problems associated with estimating probit or logit models with fixed effects for a large dataset. Figure 3 plots for all three types of credit card fees for values of ranging from 1 to 35. All three lines start below 1, indicating that a fee payment last month is associated with a less than average likelihood of making a fee payment this month. For both cash advance and late fees, having paid a fee one month ago is associated with a 40 percent reduction in the likelihood of paying a fee in the current month. For over limit fees, having paid a fee 12Observations used for the calculation of are from subjects who are in our sample at both date and date − 9
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)=1/0.8
last 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 108=125. 13 2.3.1 Dynamic panel models with fixed effects Our second approach to estimating the impact of past fee payments on current fee payments 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) analytically 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 −136 = −028. This amount is much too small to explain the large reduction in fee frequency this month associated 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 08. So = (08)=108. 10