Ricardo y Ramona: campesinos y experimentadores

We estimate the parameters of our model using the maximum likelihood approach. In short, we maximize the likelihood of observing the abandonment and redialing decisions of every caller in the dataset and observing the volume of call strings initiated by every cardholder in the bank. The details of our estimation approach are explained in B.3. To limit the computational burden of estimation we only include calls with waiting times no greater than 600 seconds (120 periods), which reduces the total number of calls by only 1.1%. Finally, for the piecewise

9

Note that we do not model dependence between the call string arrival process and caller history. In other words, caller waiting times and their abandonment and redialing decisions have no effect on how often they initiate new call strings. We recognize that this is a strong assumption; however, with only 2.40 call strings per caller in our dataset, it would difficult to draw meaningful conclusions regarding how caller history affects the call string arrival process.

linear function αa_s(t) in (3.1) we choose 10 line segments with the following 11 endpoints:

t={1,3,6,10,15,21,28,37,48,66,120}.10 _{We choose these endpoints to make the total number}

of abandonment decisions made by callers within each line segment as close as possible. 3.5.2 Estimation Results

We perform our estimation procedure at 50 random starting points, select the estimates that result in the highest likelihood, and then obtain standard errors of the estimates using non- parametric bootstrapping (Horowitz, 2001). While our model can accommodate a large number of segments, for parsimony we assume that there are two segments (s ∈ {1,2}).11 _{We find}

significant differences in the estimated population sizes, call string arrival rates, abandonment behavior, and redialing behavior across the two segments. We discuss the results below:

Estimation Result 1 - Most callers belong to segment 1, but callers from segment 2 call more frequently: In Table 3.2 we show the prior probability that a given caller from the population of cardholders belongs to each segment (πs) and the call string arrival rate (in years) of callers in each segment (λs). While 92.8% of callers belong to segment 1, each caller in this segment initiates a new call string only .044 times per year (about once every 23 years). Conversely, 7.2% of callers belong to segment 2, where each caller initiates a new call string 1.556 times per year. Although the population size of segment 1 is far greater than that of segment 2, we calculate that 73.3% of call strings in the data are initiated by callers in segment 2 due to their higher call string arrival rate. In other words, the number of arriving calls from each segment is large.

10

We find that increasing the number of line segments beyond 10 does not have a substantial effect on our estimation results as this number of segments appears to capture the major patterns of intertemporal variation in caller abandonment behavior.

11

We also ran our estimation procedure under the assumption that there are three latent segments. We found that the estimated population size of the third segment was small and that the parameter estimates of the first two segments did not differ substantially from their estimates under the assumption that there are only two segments.

Table 3.2: Prior Probability of Belonging to each Segment (πs) and Call String Arrival Rate by Segment (λs)

Segment 1 Segment 2

Parameter Description Estimate Standard Error Estimate Standard Error

πs Prior probability of belonging to segment 0.928 (3.98E-04) 0.072 (5.96E-07) λs Call string arrival rate (in years) 0.044 (2.18E-04) 1.556 (6.78E-03)

Estimation Result 2 - Caller history influences their abandonment behavior: Re- call from (3.1) that caller abandonment probabilities depend, in part, on their history covariates and seasonality control (Hijk) and their coefficients (βsa). We briefly discuss the estimated co- efficients for each segment below, which we provide in Table 3.3.

Table 3.3: Estimated Effects of Covariates on Abandonment Probabilities by Segment (β_sa)

Segment 1 Segment 2

Covariate Estimate Standard Error Estimate Standard Error

Lag1ABAN DON 0.098* (0.038) 0.321*** (0.019)

Lag2ABAN DON 0.235 (0.887) 0.396*** (0.032)

Lag1W AIT -0.076 (1.056) -0.054*** (0.004)

Lag2W AIT -0.059 (3.233) -0.030*** (0.004)

Lag1ABANDON×Lag1WAIT 0.058 (1.037) 0.042*** (0.006)

Lag2ABANDON×Lag2WAIT 0.012 (3.096) -0.009 (0.008)

1{k= 2} 0.170 (0.086) 0.123*** (0.027) 1{k≥3} 0.594 (0.850) 0.214*** (0.040) CALL2 0.095 (0.052) -0.001 (0.022) CALL3+ -0.030 (0.183) -0.008 (0.025) BU SY T IM E -0.284*** (0.014) -0.262*** (0.009) *p <0.01 **p <0.001 ***p <0.0001.

• Observation 1 - Past abandonment decreases caller patience: Recall that we showed in Figure 3.3 of §3.3.2 that callers who abandoned their previous call have higher abandon- ment probabilities in their next call, which led us to suggest that abandoning in one call could increase caller abandonment probabilities in their next call. Our estimated coefficients ofLag1 ABANDON confirm this as they are positive and significant for both segments. Addi- tionally, we find that the estimated coefficient ofLag2 ABANDON is positive and significant for segment 2.

§3.3.2 that callers who waited more than 30 seconds to enter service in their previous call had lower abandonment probabilities in their next call, which prompted us to suggest that waiting longer to enter service in one call could decrease caller abandonment probabilities in their next call. The estimated coefficients of Lag1 W AIT andLag2 W AIT are negative for both segments and significant for segment 2, indicating that waiting longer to enter service in one call indeed decreases caller abandonment probabilities (makes callers more patient) in their next call. However, note that the standard errors for segment 1 are high for both of these estimates along with several other estimates for callers in segment 1. This is because segment 1 callers rarely initiate new call strings and we therefore rarely observe their histories in our dataset.

We also find that Lag1 ABANDON × Lag1 WAIT is significant for segment 2 with an esti- mated effect of 0.042. This means that for callers in segment 1 who abandoned their previous call, the per minute effect of their waiting time in their previous call on their abandon- ment probabilities in their next call is −0.012 (−0.054 per minute from Lag1 WAIT + 0.042 per minute from Lag1 ABANDON × Lag1 WAIT). Thus, waiting longer in one call makes callers in segment 2 more patient in their next call even if they abandoned the last call. • Observation 3 - Redialing decreases caller patience: Recall that we included1{k= 2}

and 1{k ≥3} to capture how caller behavior changes as they redial (initiate calls within a call string). We find that all of the estimated coefficients of these indicators are positive and both are significant for segment 1, suggesting that callers become less patient as they redial.

Estimation Result 3 - Caller with no history in segment 1 have higher abandon- ment probabilities (are intrinsically more patient) than callers with no history in segment 2: The statistically significant effects of the history covariates on caller abandonment probabilities demonstrate that caller history influences caller abandonment behavior. However, we are also interested in determining whether callers differ intrinsically in their abandonment behavior, irrespective of their history. We therefore find the estimated abandonment proba- bilities of callers with no history in each segment. To calculate these probabilities we use the abandonment probability formula in (3.1) using only (α_sa(t)) at each period. Note that we provide the estimates of the piecewise linear function αa_s(t) at our chosen endpoints in B.4.

In Figure 3.5 we plot the estimated abandonment probabilities and their confidence intervals over the callers’ first three minutes of waiting (36 periods). While we find similar patterns of variation across time for both segments, the abandonment probabilities are higher in every period for segment 1 callers, which suggests that callers in segment 1 are intrinsically less patient than callers in segment 2.

Figure 3.5: Estimated Abandonment Probabilities over the First 180 Seconds (36 Periods) by Segment

Estimation Result 4 - Caller history influences their redialing behavior: In Table 3.4 we show the estimates of the parameters that determine the callers’ redialing probabilities, which include the intercept term (αr_s) and the coefficients of the covariates (β_sr). We briefly discuss the estimates below:

• Observation 1 - Waiting longer before abandoning increases the probability of redialing: Note that we have included an additional covariate calledWAIT, which we define as how long (in minutes) the caller waited before abandoning the observed call. We find that the estimated effect of WAIT is positive and significant for both segments. This result is expected as we found in §3.3.3 that callers who waited longer than one minute before abandoning have higher redialing probabilities.

Table 3.4: Estimates of Parameters that Determine Caller Redialing Probability by Segment (αr_s andβr_s)

Segment 1 Segment 2

Covariate Estimate Standard Error Estimate Standard Error

Intercept (αrs) -1.195*** (0.064) -0.390*** (0.021)

Lag1ABAN DON -0.049 (0.069) -0.315*** (0.063)

Lag2ABAN DON -0.227 (2.702) -0.158 (0.064)

W AIT 0.071*** (0.005) 0.019** (0.006)

Lag1W AIT -0.019 (2.762) 0.010 (0.007)

Lag2W AIT -1.245 (7.513) -0.002 (0.010)

Lag1ABANDON×Lag1WAIT 0.092 (2.518) 0.019 (0.010)

Lag2ABANDON×Lag2WAIT 1.264 (10.624) 0.033 (0.020)

1{k= 2} 0.672* (0.248) 0.546*** (0.088) 1{k≥3} 0.909 (4.348) 1.046*** (0.147) CALL2 -0.244 (0.188) -0.025 (0.051) CALL3+ -0.017 (0.232) -0.137* (0.046) BU SY T IM E 0.008 (0.038) 0.032 (0.026) *p <0.01 **p <0.001 ***p <0.0001.

that the estimated effect of 1{k= 2} is positive and significant for both segments and that the estimated effect of 1{k≥3} is positive for both segments and significant for segment 2 callers. This indicates that redialing within a call string increases caller redialing probabilities within the same call string and provides an explanation for the trend we observed in §3.3.3 that callers are more likely to redial if the call they abandoned was itself a redial from a previous abandon.

Estimation Result 5 - Callers with no history in segment 1 have intrinsically lower redial probabilities than callers with no history in segment 2: Finally, we are interested in determining whether callers differ intrinsically in their redialing behavior, irrespec- tive of their history. We therefore find the estimated redialing probability for callers in each segment who have no history and who immediately abandon their first call. To calculate these probabilities we use the redialing probability formula from (3.2) using only the the estimated intercept terms (αr

s) in Table 3.4. We find that callers in segment 1 with no history who im- mediately abandon their first call have an estimated redial probability of 23.2% while callers in segment 2 with no history have an estimated probability of 40.4%. This suggests that segment 1 callers are intrinsically less likely to redial than segment 2 callers.

For future reference we summarize the most salient estimation results by comparing caller behavior across the two segments in Figure 3.6. While caller history has a similar influence on caller behavior within both segments, callers differ across segments in their intrinsic behavior. Specifically, callers from segment 1 initiate call strings far less frequently, are intrinsically less patient, and intrinsically less likely to redial after abandoning. Hence, we see the value of using the latent class framework as we are able to capture the effects of caller history and intrinsic caller heterogeneity on caller behavior. Because the most differentiating behavior between the two segments is their call string arrival rate, for the remainder of this paper we refer to the callers in segment 1 asthe infrequents and the callers in segment 2 as the frequents.

Figure 3.6: Comparison of Behavior between Segments

Segment 1 (The Infrequents)

INTRINSIC BEHAVIOR

• Call lessfrequently

• Intrinsically lesspatient

• Intrinsically lesslikely to redial

INFLUENCE OF HISTORY ON BEHAVIOR

• Previous abandoning makes less patient

• Waiting makes more likely to redial

• Redialing within call string makes more likely to redial in same call string

Segment 2 (The Frequents)

INTRINSIC BEHAVIOR

• Call morefrequently

• Intrinsically morepatient

• Intrinsically morelikely to redial

INFLUENCE OF HISTORY ON BEHAVIOR

• Previous abandoning makes less patient

• Previous waiting makes more patient

• Waiting makes more likely to redial

• Redialing within call string makes more likely to redial in same call string

3.5.3 Model Validation: Out of Sample Testing

Finally, to validate that our model can predict caller abandonment and redialing behavior under various policies, we conduct out of sample testing. We find that, despite the fact that the operating conditions in the out of sample data were quite different than those of the sample data, our model provides good out of sample predictions of the performance measures of this call center, with relative errors as low as 0.4% and no greater than 7.9%. See B.5 for a description of our out of sample testing procedure and the results of the test.