key: cord-0232169-a4h4lau5
authors: Berrebi, Simon; Gibbs, Taylor; Joshi, Sanskruti; Watkins, Kari E
title: On Ridership and Frequency
date: 2020-02-06
journal: nan
DOI: nan
sha: 2285f5f71b479cf19b778e10188ede84495d895e
doc_id: 232169
cord_uid: a4h4lau5

In 2018, bus ridership attained its lowest level since 1973. If transit agencies hope to reverse this trend, they must understand how their service allocation policies affect ridership. This paper is among the first to model ridership trends on a hyper-local level over time. A Poisson fixed-effects model is developed to evaluate the ridership elasticity to frequency using passenger count data from Portland, Miami, Minneapolis/St-Paul, and Atlanta between 2012 and 2018. In every agency, ridership is found to be elastic to frequency when observing the variation between individual route-segments at one point in time. In other words, the most frequent routes are already the most productive. When observing the variation within each route-segment over time, however, ridership is inelastic; each additional vehicle-trip is expected to generate less ridership than the average bus already on the route. In three of the four agencies, the elasticity is a decreasing function of prior frequency, meaning that low-frequency routes are the most sensitive to frequency change. This paper can help transit agencies anticipate the marginal effect of shifting service throughout the network. As the quality and availability of passenger count data improve, this paper can serve as the methodological basis to explore the dynamics of bus ridership.

Following six years of consecutive decline, 2018 bus ridership in the United States was the third lowest year since World War II (Dickens and MacPherson, 2019) . Each transit trip lost to private cars contributes to traffic congestion, pollution, and road fatalities. The revenue lost from declining ridership also impedes the ability of transit agencies to provide service, which hurts ridership further in a downward cycle. Transit agencies, therefore, need to understand how this trend can be reversed and at what cost.

The main tools transit agencies have available to influence ridership are service allocation policies. Transit planners are tasked with setting frequencies throughout the network under constrained resources. They must balance ridership with other, sometimes conflicting, objectives including eq- * Corresponding author uitable access, connection to places of strategic importance, and reliability. In particular, agencies must decide whether to spread or concentrate service. In order to allocate service in a transparent manner that maximizes total welfare, the effect of frequency on ridership should be quantified. This effect may not be linear; each vehicle-trip added to a route may not produce as much (or as little) ridership as the current route productivity, measured in passengers per vehicle-trip. The elasticity of ridership with respect to frequency measures the percentage change in ridership resulting from a 1% increase in frequency. When this elasticity is greater than one, adding more service increases the route productivity. Because this elasticity measures the sensitivity of demand to frequency, it varies across routes based on frequency and across stops based on local characteristics. Automated Passenger Count data (APC) can be used to empirically estimate a model of ridership on a hyper-local level over time. Although transit agencies started deploying APCs in the mid-1970's (Attanucci and Vozzolo, 1983) , the technology has not yet been used to conclusively explain disaggregated ridership change. The first reason is that passenger count data are rife with errors and inconsistencies, which can quickly overwhelm sensitive ridership models. The second reason is that complex models are required to capture the variation happening on different spatial and temporal scales. This paper presents a new method to scrub, process, and model ridership data over time and space. Passenger counts are cross-checked with the General Transit Feed Specification (GTFS), a schedule meta-data standard. Passenger counts are then aggregated by route-segments (groups of seven stops on the same route and direction) and combined with data sources on population and jobs. The change in ridership is modeled over time through panel regression. Fixed-effects models avoid unobserved heterogeneity and endogeneity biases that plague cross-sectional models by using each individual as its own control over time. Fixed-effects models therefore control for variation between individual locations to capture the variation within each. In this paper, ridership is modeled using Poisson fixed-effects, which was developed by Hausman et al. (1984) for count data such as passenger boardings and alightings.

The paper proceeds as follows: Section 2 organizes the main studies from the literature by level of aggregation and identifies panel models of hyper-local ridership trends as the gap in research.

Section 3 presents the four case studies. Section 4 describes the process of cleaning, aggregating, and combining multiple relevant datasets. In Section 5, the fixed-effects Poisson regression model is described. Section 6 presents the modeling results. Section 7 discusses their implications and identifies future research questions.

Ridership elasticity to frequency on a local-level and over time remains largely unaddressed in the literature. Table 1 classifies the main studies on transit ridership by level of spatial aggregation (rows) and whether the sample is observed once or at multiple time periods (columns). All the references in the top row evaluate ridership at the transit agency or metropolitan area level. These Kain and Liu (1999) ; Kohn et al. (2000) ; Brown and Thompson (2008) ; Lane (2010) (2018) found that the service quantity is among the best predictors of ridership. This relationship, however, seems to have deteriorated in recent years. Following nation-wide cuts in bus service between 2009 and 2012, bus ridership kept falling even as vehicle revenue miles continually increased . While these studies shed light on the impact of overall operating expenses on overall ridership, the level of aggregation may drown some of the local dynamics. For example, shifting resources from one route to another can impact overall ridership without even affecting the total service provided. Likewise, the local effects of population, jobs, and demographic factors are diluted in system-level studies.

The bottom left quadrant of Table 1 contains studies modeling bus ridership at a local level at a single point in time. These studies capture the local variation between individual stops, routes, or route-segments. These models are formulated as if each individual location was similar in every aspect except for the variation captured by explanatory variables. In reality, individual locations are different from each other in ways that cannot be measured and that may affect ridership. Ridership itself is likely to affect frequency directly; when allocating service throughout the network, transit planners strive to maximize ridership. Therefore, estimates for frequency, population, and jobs are likely inflated by the two-way causality. While these estimates inform the relationship between these variables at one point in time, they cannot serve as true elasticities.

The bottom right quadrant contains studies modeling bus ridership on a local level over time.

Results from Maloney et al. (1964) and Kyte et al. (1988) still serve as the main reference on ridership elasticity to frequency (Evans et al., 2004; Litman, 2017) . Following service changes in Boston, MA, and Portland, OR, these studies computed separate ridership elasticities for each route. With values ranging from zero to 3.77 (Kyte et al., 1988) , route-specific elasticities are not overly meaningful. Fixed-effects models harness the combined explanatory powers of all individual locations in the panel to explain the change in ridership. Tang and Thakuriah (2012) and Brakewood et al. (2015) use fixed-effects models at the route level to evaluate the impact of real-time passenger information on ridership. The regression structure in both papers is linear, which can be suitable for route-level ridership, where counts are approximately normal.

In more disaggregated analysis, however, ridership data is skewed to the right. Frei and Mahmassani (2013) and Kerkman et al. (2015) model ridership change at the stop-level over a one year period. They apply a log-transform to reduce the asymmetry in the ridership distribution. Several studies have shown by simulation that log-transformed count data tends to yield biased estimates in OLS in the presence of heteroskedasticity (King, 1988; Manning and Mullahy, 2001; Silva and Tenreyro, 2006) . The magnitude of logged-variation is typically far greater at low-ridership stops, which are correlated with explanatory variables by design. In addition, zero-values in the original data must be either truncated or a constant must be added to each observation, which can affect the distribution of variation and produce further heteroskedasticity.

Modeling ridership change at a disaggregated level is complicated by the missing data, endogeneity, and multiple levels of interaction. The models are highly sensitive to even slight misspecification.

This paper presents a framework to analyze the causes of ridership change on a disaggregated level.

A step-by-step methodology to clean, process, and model APC data over multiple periods is described in detail to facilitate the development of this emergent field of research. The effects of service changes on ridership are estimated. Retrospectively, this analysis can be used to control for frequency, and identify underlying trends based on a host of other factors. More broadly, this study opens the door to a wide range of research topics on the sensitivity of transit ridership.

To evaluate the relationship between transit ridership and frequency, four transit agencies were selected:

• Table 2 shows the first and last markup for each agency. Every markup in between was used in this analysis. The transit agencies in this study are similar in size but they vary widely in other aspects.

Ridership and revenue hours in each of the four agencies are all within 15% of each other. Dividing passenger trips by revenue hours gives operational efficiencies. TriMet is the most efficient agency with 28.5 passengers per revenue hour and MARTA is the least efficient with 21.9 passengers per revenue hour. While TriMet serves more passenger trips than Miami-Dade, the Miami region has 2.5 times more population than the Portland area. The Miami region is the densest, followed by Portland, Minneapolis/St-Paul, and Atlanta, where only 11% of the population lives at transitsupportive densities. The case-studies therefore represent a wide cross-section of mid-sized transit agencies, forming a basis of comparison that can be useful to their peers.

1 Contracted service is not considered in this study because the passenger count data differs from directly operated service in accuracy, format, and availability.

Transit agencies in this study provided the research team with ridership data at the stop-routedirection-trip level. Passenger count data were then scrubbed, aggregated, and combined with other data-sets through a process illustrated in Figure 1 . Each step of this process is described in this section. The outcome is a dataset that serves as input to our models. For a detailed description, see Joshi (2019) .

Combine ridership, frequency, population and jobs data at segment level Aggregate daily frequency & ridership by SRD in each mark-up

Comparison of frequency between APC and GTFS GTFS et al., 2005) . The exact steps in the filtering process, however, are unknown. With the exception of TriMet, all agencies in this study rely on di↵erent vendors, which have their own proprietary filtering algorithms. At the end of the markup, passenger boardings and alightings are averaged 210 across days of the markup. Transit agencies in this study provided the research team with ridership data at the stop-route-direction-trip level. Passenger count data were then scrubbed, aggregated, and combined with other data-sets through a process illustrated in Figure 2 . Each step of this process is described in this section. The outcome is a dataset that serves as input to our models.

For a detailed outlook, see Joshi (2019). 

In order to understand the relationship between the change in ridership and the change in frequency, only stop-route-directions that remained constant over the entire study period were considered. Any stop that was either added or removed between the first and the last markup was disregarded. Table 4 shows the percentage of constant stop-route-directions in the last markup and the ridership they represent. In Portland, Miami, and Minneapolis/St-Paul, the vast majority of stop-route-directions were already there in the first markup. The share of stop-route-directions that remained constant and the ridership they represent are close in all three agencies, which indicates that they are somewhat representative of all stop-route-directions. In Atlanta, however, almost half of all stop-route-directions in the last markup had been added since the first markup. The agency underwent a comprehensive operational analysis in which many routes were altered. In any case, the purpose of this analysis is to evaluate the sensitivity of ridership to frequency, not to explain overall ridership change. To avoid this issue, the number of observed daily trips in APC was compared with the number of scheduled trips for each stop-route direction. Historical schedule data published by transit agencies in GTFS format were obtained from third-party websites 2 . The number of daily trips in APC and GTFS was then compared. All segments with more than one missing trip in the first markup were filtered out of the analysis entirely. Route-segments with more than one missing trip in subsequent years are removed from the analysis only for the offending years. The sensitivity of the results to missing trips is analyzed in Appendix A.

While the main objective of this paper is to understand the relationship between frequency and ridership, population and jobs also affect transit ridership on a local level. To extract meaningful results, these variables should be measured on the scale of their variation. While the service coverage area (i.e. accessible walking distance) for a bus stop is defined by the Transit Capacity and Quality of Service with a 1 4 mi radius (Kittelson Associates, 2013, §5-10), typical stop spacing in urban areas is only 1 8 mi according to the TCRP Report 19 -Location and Design of Bus Stops (Fitzpatrick et al., 1996) . Hyper-local variations in ridership are more likely to be explained by walkability, which is determined by connectivity, land-use patterns, quality of path, and context, on which no data is available (Southworth, 2005) . Therefore, what accounts for differences in ridership between adjacent stops on the same route-direction is unlikely to be captured by our explanatory variables. Modeling ridership at the stop-route-direction level would reduce explanatory power.

Furthermore, the passenger's choice to use one stop over the next one introduces serial correlation, which may affect estimated variances.

To address these issues, we define route-segments, clusters of seven adjacent stops on the same route-direction, as the spatial unit of analysis. To create route-segments, the stop sequence (i.e. order on the route) of constant stop-route-directions was obtained from GTFS. Since a stop can have different sequences on the same route-direction across different trips, the stops, stop times, trips, and routes tables were joined and the sequence index common to the most trips was recorded.

Spatial coordinates for each stop were also obtained from the GTFS stops table. Each routedirection was then divided into segments of seven stops in sequential order. The last remaining stops at the end of each route were merged with the upstream segment. Ridership and frequency were then averaged by segment across stops.

Population and job data were obtained from Longitudinal Employer Household Dynamics (LEHD). The LEHD are data products compiled by states using Unemployment Insurance earnings and published by the US Census Bureau. The number of jobs is provided at the Census Block level, by year. These data, however, are only available between 2011 and 2015. We therefore linearly extrapolated the LEHD data to match the APC time-frame. The underlying assumption is that population and job trends between 2011 and 2015 continued their course until 2018 in Atlanta and until 2017 everywhere else. In other words, a Census Block that gained five residents per year until 2015 was assumed to keep growing at the same rate.

The LEHD data were first cleaned and prepared for import into ArcMap. Block Group shapefiles were joined using a common GEOID field. A dissolved 1 4 mi buffer was applied to the stops based on the common route-segment field. This buffer was overlaid with LEHD data using a pairwise intersection tool, which compares the input features of overlapping layers. Features common to both input layers were sent to the output feature class. The output mirrors the geometric intersections of the two layers while considering which layers they are derived from. Therefore, the overlapping service coverage areas of bus stops within the same segment were only counted once. Note, however, that service areas falling in the overlap of multiple segments were assigned to each of the segments.

Under the assumption that people and jobs are uniformly distributed within their geographic unit, LEHD data were weighted by the proportion of each Census Block in the segment buffer. 

Ridership data is count data 3 . These data violate the Ordinary Least Square (OLS) assumption of normally distributed errors. While the Normal distribution is defined from -∞ to ∞ and has constant variance, count data are necessarily non-negative and their variance grows with their mean. Skewed data, however, can be transformed by taking the natural logarithm (log) to fit OLS as long as they have constant variance-to-mean ratio (Nelder and Wedderburn, 1972) . The log-linear model for ridership is given in Equation (1). Variables are defined in Table 5 .

In Equation (1) Year ∈ (0, ..., T ) i

Route-segment ∈ (0, ..., n)

homes and job locations within 1 4 mi of bus stops are added to form a potential trip generation variable.

Taking the exponent of both side sides of (1) yields the more intuitive formulation (2), in which, explanatory variables have a multiplicative relationship. The model can be summarized as follows: when a bus passes by a route-segment, each residential home and job has the potential to generate a trip. This formulation is consistent with the literature: the effect of frequency on ridership is dependent on population and jobs (Badoe and Miller, 2000) . A route running through areas with low population and few jobs is unlikely to attract ridership, even if it has high frequency. Conversely, a route-segment passing through a high concentration of population and jobs still needs frequency to attain high ridership.

While the log-linear OLS regression of Equation (1) gives an estimate of E(log(Rid)), we are primarily interested in log(E(Rid)). A model that estimates log(E(Rid)) directly and deals with zero-values adequately is the Poisson regression. In the Poisson regression, the expected ridership is estimated directly as per Equation (3) 

Given a (possibly non-linear) model specification, MLEs estimate the distribution parameters that maximize the probability of obtaining the observed data. In our model, the Poisson regression is computed with a robust covariance matrix (also known as quasi-Poisson regression) to relax the assumption that the variance must equal the mean. The quasi-Poisson model returns the same coefficients as the regular Poisson regression, but standard errors are better calibrated for potential over-dispersion (Wooldridge, 2002, §19.2) . Note that our standard errors may still be inflated due to spatial correlation between route-segments. This effect, however, is limited by the clustering of adjacent stops, which leaves little overlap between segments on the same route and direction.

ways that are not considered in this paper.

Equation (4) shows the derivative of E(Rid|x) with respect to frequency. This derivative is the expected ridership gained from adding a single vehicle-trip to the route-segment, henceforth referred to as "marginal productivity". In the original model, β 1 is the elasticity of ridership to frequency.

In the derivative of Equation (4), we see that the marginal productivity is proportional to Freq β 1 −1 . Therefore, when β 1 > 1 (β 1 < 1), then the marginal productivity is a monotone increasing (decreasing) function of Freq, i.e. each additional trip yields increasing (decreasing) ridership returns. Identifying the sign of (β 1 − 1) can help inform the trade-off between service allocation policies.

Let us consider the observed ridership, Rid it , on route-segment i and year t, where i ∈ {1, ..., n} and t ∈ {1, ..., T }. The standard formulation for the fixed-effects model is shown in Equation (5).

The linear equation βx it is the effect of explanatory variables x it . The terms α i represents the individual specific effect, µ is a linear time trend, and it is the error.

The α i term allows each individual route-segment to have its own intercept, while the β terms force the relationship between x it and Rid it to be the same for all. The term µ represents the linear effect of time, which is not explained by other variables. It takes into account conjunctural phenomena that may be affecting all route-segments equally.

Similarly to the cross-sectional case in the last section, panel ridership data can be modeled with a Poisson fixed effects model 4 . As in Equation (3), the expected ridership is modeled as follows:

4 As in the cross-sectional case, the fixed-effects Poisson model assumes that the variance, V (Ridit|xi), is equal to its mean, E(Ridit|xi). To relax this assumption, Hausman et al. (1984) introduced the fixed-effects Negative-Binomial model, which estimates a dispersion parameter. Like Hausman et al. (1984) 's Poisson fixed-effects, their Negative Binomial fixed-effects model is conditioned on the sum of outcomes. This model, however, was shown to be misspecified by Allison and Waterman (2002) and later in more detail by Guimaraes (2008) . The Negative-Binomial fixed-effects can also be estimated unconditionally (i.e. by fitting dummy variables, called incidental parameters, for each individual except for one). However, there is, to the authors' knowledge, no proof that it does not (or does) cause an incidental parameter problem. In the unconditional Negative Binomial fixed-effects model, the incidental parameter is not divided out. In cases, such as ours, where T is fixed and N is large, this parameter can be inconsistent E[Rid it |x it ] = e βx it +α i +µt (6) The β parameters are estimated conditionally on the sum of outcomes at the individual level over the years (Hausman et al., 1984) . The resulting conditional likelihood is proportional to the right-hand side of Equation (7) as per (Cameron and Trivedi, 1998, §9.3) . Estimates are obtained by setting the partial derivative of the log-likelihood with respect to beta to zero as in Equation (8).

Notice that any constant added to the observed explanatory variables, x it , gets canceled out in both equations. Therefore, only the variation taking place over time within each individual route-segment is modeled. Any heterogeneity and endogeneity disappear as long as it does not vary over time. Because spatial demand and the resulting allocation of service change at the pace of land-use, we are not concerned about these effects over a five to six year panel.

Time-invariant explanatory variables cannot be considered in fixed-effects models. However, some of the research questions raised in the introduction compel us to evaluate the relationship between prior frequency (frequency at the beginning of the panel, when t = t 0 ) and the change in transit ridership. Time-invariant variables, in this case prior frequency, can be interacted with time-varying variables, in the case frequency at time t. The Poisson fixed-effects model considered in this paper is shown in Equation (9). Noticing that logs cancel their exponents, we obtain the more intuitive formulation of Equation (10).

Frequency (Freq it ), Population (Pop it ), Jobs (Job it ), and time (t) are time-varying, while prior and bias the β estimates (Hsiao, 2003, §7.3) . Fortunately, Wooldridge (1999) showed that the Poisson fixed-effects estimates are robust to overdispersion.

frequency is time-invariant (Freq it 0 ). The interpretation of β 1 is the elasticity of ridership to frequency, as in the cross-section model. However, sensitivity is measured within individual routesegments instead of between. The second term is the interaction of the frequency at t = t 0 with the time-varying frequency. The coefficient β 2 tells us whether ridership is more elastic to frequency on frequent routes (positive) or on infrequent routes (negative). Finally, the fourth term, β 3 , gives the within elasticity to change in population and jobs.

Equation (10) can be used to predict how changing frequency on a route-segment would affect ridership. However, this relationship is dependent on the individual-specific error, α i . This term is divided out in Equation (11).

As in (4) 

Equation (12) enables schedule planners to anticipate the effects of service allocation policies just by knowing the prior ridership and frequency. While the static E[Rid it 0 |x it 0 ] is modeled in Subsection 6.1, using observed values of prior ridership, Rid it 0 , instead allows to capture the unobserved heterogeneity. This formulation enables planners to assess how much ridership will be lost by removing one vehicle from a specific route and how much will be gained by adding it to a different route.

This section presents the results from the Poisson cross-section and fixed-effects models. These models were run in the software program R using the stats and pglm packages, respectively (R Core Team, 2017; Croissant, 2017). Table 6 shows the results of the cross-sectional model in the first markup (2012/2013/2014) presented in Equation (2). McFadden's pseudo-R 2 is shown for each agency at the bottom of For all four agencies, elasticity is significantly greater than one, and hence elastic. In other words, for two route-segments with the same population and jobs, the one with the greater frequency is likely to have more passengers per trip. The second row shows the sensitivity of ridership to the total population and jobs. In all four agencies, the parameter estimate is significantly below one. For the same level of frequency, a segment surrounded by more development than another is expected to have less ridership per capita. In order to verify the model results of Table 6 , we need to represent the relationship between ridership and frequency in the first markup (2012/2013/2014) graphically. Figure 3 shows route-level productivity in weekday passenger boardings per trip against frequency in daily trips. Productivity allows us to compare the ridership contribution of each vehicle-trip on routes with different frequencies. The size of each dot corresponds to the number of stops on the route. One would expect routes with more stops to connect more places, and therefore yield more passengers per trip. Express routes in all agencies are excluded because their stop densities are so much lower than the rest. A trend line is shown in red. The gray band represents its standard error. Both the cross-sectional model and the graphical representation show that, when comparing the variation between route-segments or routes, productivity is positively correlated with frequency.

As discussed in the literature review, this does not necessarily mean that increasing frequency on a route-segment will produce increasing returns. High-frequency segments may be more productive due to unobserved heterogeneity or endogeneity. The effect of frequency change on ridership can, therefore, only be tested by comparing the variation within each route-segment over time. The next subsection does just that. Table 7 shows the results of the Poisson fixed-effects model. For each agency, the elasticity of ridership to frequency and to population and jobs is presented. In this model, the interaction between frequency and prior frequency is omitted to enable a comparison with the cross-sectional results. The log-likelihood, total number of observations, individual segments, and time periods are shown at the bottom of the table. Unfortunately, there is no equivalent to the pseudo-R 2 for the fixed-effects Poisson model. However, the good fit of the cross-sectional models indicates that the fixed-effects model is also well specified. Finally, note that the linear time trend is significantly negative for all agencies. This indicates that, even when controlling for frequency, population, and jobs, ridership is still declining over time.

section. While the between elasticity in the previous model ranged from 1.21 to 1.50, the within elasticity shown in Table 7 ranges from 0.66 to 0.78. For all agencies studied, ridership is inelastic to frequency. In other words, each vehicle-trip added to a route-segment generates diminishing productivity returns. is not dependent on frequency change. However, if frequency had no impact on productivity (i.e. elasticity = 1), then all routes would lose the same relative productivity. In all agencies, routes in which frequency increased seemed to experience more relative decline in productivity than expected and vice versa. This is particularly true in Portland, where long arrows pointing towards the bottom-right contrast with short arrows spread in every direction. The trends in Figure 4 also show the connection between frequency change and prior frequency.

For example, in Portland, frequent routes gained frequency, perhaps in an attempt to combat overcrowding, while in Miami, frequent routes lost the most frequency, perhaps in an attempt to maintain coverage. Adding the interaction term between frequency and prior frequency can help determine whether the elasticity differs between previously frequent and infrequent routes. Table 8 shows the fixed-effects model results where frequency is interacted with prior frequency.

The log(Pop it + Job it ) and linear time coefficients are almost identical to the previous model. The elasticity is the sum of its two first terms, β 1 + β 2 log(Freq it 0 ). The estimate of β 1 is far off from the estimate in Table 7 because the term is not centered. Its interpretation should assume that log(Freq it 0 ) = 0, i.e. that Freq it 0 = 1. Figure 5 shows the elasticity of ridership to frequency as a function of prior frequency. Confidence bands of one standard deviation surround the estimated elasticity. Since the elasticity term includes two components, one fixed, β 1 , and one based on frequency in the first markup, β 2 log(Freq it 0 ), the combined standard deviation was obtained using the Delta Method (Oehlert, 1992) . In Portland, Miami, and Atlanta, elasticity is greatest on low-frequency routes, while in Minneapolis/St-Paul elasticity is greatest on high-frequency routes. These results indicate that in all agencies besides Minneapolis, each percentage increase in frequency will produce greater percentage increase in ridership on routes that were previously infrequent than on routes that already had high-frequency. 

This study reveals the relationship between bus ridership and frequency on a highly disaggregated level. In a cross-sectional analysis we found that the most frequent routes at one point-in-time are also the most productive (passengers per vehicle-trip) in all four transit agencies. This relation-ship, however, contains both the exogenous effect of ridership on frequency, which we are interested in, and the reverse causality. Through decades of service planning, the bus network reflects the spatial distribution of travel demand. To control for time-invariant endogeneity and other unobserved heterogeneity between route-segments, we developed a Poisson fixed-effects model, which captures the variation in ridership within each route-segment over time. The results indicate that the ridership elasticity to frequency is in the range of 0.66 to 0.78 for all agencies. We also find that in Portland, Miami, and Atlanta, the most frequent routes are the least elastic while the opposite is true in Minneapolis/St-Paul. Overall, three broad conclusions can be drawn from the analysis presented herein:

• The wide difference in estimated elasticity between the cross-section and fixed-effects models, which accounts for the endogeneity and unobserved heterogeneity, indicates that the demand for bus service is more concentrated than the supply. As shown in the cross-sectional analysis of Subsection 6.1, frequent routes serve more passengers per trip than lower-frequency routes.

• For all agencies studied, ridership is inelastic to frequency, as shown in Subsection 6.2.

Therefore, each marginal trip added to a route will typically generate less ridership than the average vehicle already on the route. In other words, the expected ridership returns from frequency are diminishing.

• If transit agencies were to increase frequency on all routes by the same proportion, the productivity advantage of frequent routes would decline in Portland, Miami, and Atlanta and increase in Minneapolis/St-Paul as shown in Subsection 6.3.

The main limitation of this study is that we do not account for potential time-varying endogeneity. If transit planners readjusted service in response to the organic change in travel demand between 2012 and 2018 that happened independently of service frequency, population, and jobs, then elasticity estimates may be inflated in the fixed-effects models. We expect, however, this time-varying endogeneity to be relatively small. Transit planners allocate service through a process that involves stakeholders with conflicting objectives, including coverage, access, and connectivity.

The consensus emerging from this process is often not to adjust service according to recent changes in travel demand. Even when planners do allocate service in response to changes in demand, the potential time-varying endogeneity does not undermine the main conclusions of this paper. If anything, inflated elasticity estimates would bolster the conclusion that ridership is inelastic to frequency.

Service allocation is by far the most important lever transit agencies have available to affect ridership without changes in operating budgets. The results in this paper can be used to anticipate the expected ridership change from cutting service on one route to prioritize another solely based on prior frequency and ridership of both routes, as shown in Section 5.3. The decision to prioritize service coverage or concentration should be made on a case by case basis to attain the best possible compromise.

The service allocation problem consists in setting frequencies throughout the bus network with the objective of minimizing a combination of waiting costs for passengers and operating costs for the agency (Mohring, 1972) . The method was extended by Furth and Wilson (1981) to consider the societal benefit of transit ridership in the objective function based on elasticities from Mayworm et al. (1980) . More recent research has used ridership elasticity to frequency as an input parameter (Verbas and Mahmassani, 2013) . The optimal service allocation policy was found to be sensitive to the assumed elasticity. The elasticities modeled in this paper can support these types of service allocation optimization tools. Future research could take the relationship between elasticity and prior frequency into account to attain even greater efficiency.

Historical APC data are expected to become increasingly available (Porter et al., 2018) . The methodology applied to this problem for the first time can serve as a framework to support future research in transit demand using observational data. While frequency is not the only factor affecting ridership, it determines the feasibility, travel time, and reliability of transit trips. Frequency changes every three to four months, bringing sudden jolts to the ridership level. The presence of unobserved heterogeneity suggests that the geographical unit of analysis should be linked to service provision, such as the stop, segment, or route-level to capture the exogenous effect of ridership on frequency.

Changes in service level may even affect ridership on adjacent lines. The model presented in this paper could be extended in future research to consider the ridership impact of nearby service changes, from slight bus schedule adjustment to heavy rail station opening. Future research should also explore the sensitivity of demand to service quantity by time of day and day of the week.

Several research questions remain unaddressed. In particular, service changes do not explain the current nation-wide ridership crisis. Other factors such as changing travel behaviors, demographics shifts, and competition from dynamic mobility companies may also affect the demand for buses on a local level. They may ultimately explain the ridership change at the regional and national levels.

But their effects are necessarily lower order as they tend to drift slowly over time. Therefore, the elasticity to frequency is necessary to understand the underlying causes of ridership change. The approach presented in this paper will allow future research to understand the causes of ridership decline and identify strategies to reverse the trend.

The authors wish to extend heartfelt gratitude for staff members at TriMet, Miami-Dade Transit, Metro Transit, and MARTA for providing us with the data and helping us understand their transit systems. In particular, we thank Nathan Banks at Trimet, Eric Lind, Janet Hopper, and John Levin at Metro Transit, Esther Frometa-Spring at Miami-Dade Transit, and Charlotte Owusu-Smart, Robin Salter, and Robert Goodwin at MARTA. Finally, we thank Thomas Kowalsky from Urban Transportation Associates for his help navigating the data and understanding the technology.

The results of Section 6 are revisited in this Appendix based on the number of missing trips from GTFS in APC. Figure A .6 shows (a) the sample size and (b) the estimated ridership elasticity to frequency based on the maximum number of missing trips. In all four agencies, allowing segments with missing trips in the analysis marginally increases the same size and causes slight fluctuations in the elasticity coefficient estimate. This analysis confirms that excluding segments with missing trips helps avoid spurious variation in the coefficients.

(a) Sample size (b) Estimated ridership elasticity to frequency 

The impact of real-time information on bus ridership in

The relationship between transit ridership and urban decentralization: insights from Atlanta

Regression analysis of count data

Examining the influence of stop level infrastructure and built environment on bus ridership in montreal

Does service reliability determine transit patronage? Insights from the Los Angeles Metro bus system

Out of service: Identifying route-level determinants of bus ridership over time in Montréal

What affects transit ridership? a dynamic analysis involving multiple factors, lags and asymmetric behaviour

pglm: Panel generalized linear Models

APTA 2019 Public Transportation FactBook, Appendix A

Predicting transit ridership at the stop level: The role of service and urban form

The effect of demographic changes on transit ridership trends

Comparing transit agency peer groups using cluster analysis

The relationship between urban form and station boardings for Bogota's BRT

Traveler response to transportation system changes

National transit database

Guidelines for the location and design of bus stops

Riding more frequently: Estimating disaggregate ridership elasticity for a large urban bus transit network

Setting frequencies on bus routes: Theory and practice

Understanding the recent transit ridership decline in major US cities: Service cuts or emerging modes

The fixed effects negative binomial model revisited

Transit ridership forecasting at station level: an approach based on distance-decay weighted regression

Is Uber a substitute or complement for public transit

Econometric models for count data with an application to the patents-r&d relationship

Analysis of panel data

Impacts of land use and amenities on public transport use, urban planning and design

How urban density, network topology and socio-economy influence public transport ridership: Empirical evidence from 48 European metropolitan areas

Fixed-effects panel data analysis of gasoline prices, fare, service supply, and service frequency on transit ridership in 10 US urbanized areas

Using automated passenger count data to understand ridership change on disaggregated level

Secrets of success: assessing the large increases in transit ridership achieved by houston and San Diego transit providers

Factors influencing stop-level transit ridership in Arnhem-Nijmegen city region

Using GIS to measure the effect of overlapping service areas on passenger boardings at bus stops. Urban and Regional Information Systems Association Journal

Statistical models for political science event counts: Bias in conventional procedures and evidence for the exponential Poisson regression model

Transit capacity and quality of service manual

Determinants of bus rapid transit ridership: System-level analysis

Factors affecting urban transit ridership

A time-series analysis of public transit ridership in Portland

The relationship between recent gasoline price fluctuations and transit ridership in major US cities

Understanding transport demands and elasticities. Victoria Transport Policy Institute

A geographically and temporally weighted regression model to explore the spatiotemporal influence of built environment on transit ridership

Mass transportation in Massachusetts. Final Report on a Mass Transportation Demonstration Project

Estimating log models: to transform or not to transform?

Patronage impacts of changes in transit fares and services

Optimization and scale economies in urban bus transportation

Evaluating the ability of transit direct ridership models to forecast medium-term ridership changes: Evidence from San Francisco

Generalized linear models

A note on the delta method

A simultaneous route-level transit patronage model: demand, supply, and inter-route relationship

Statewide data standards to support current and future strategic public transit investment

Assessment of models to estimate bus-stop level transit ridership using spatial modeling methods

R: A language and environment for statistical computing. R Foundation for Statistical Computing

Pedestrian environments and transit ridership

The log of gravity

Designing the walkable city

Estimation of transit ridership based on spatial analysis and precise land use data

Ridership effects of real-time bus information system: A case study in the city of Chicago

Why is public transit use declining? Evidence from California

Nature and/or nurture? analyzing the determinants of transit ridership across US urbanized areas

American community survey

Optimal allocation of service frequencies over transit network routes and time periods: formulation, solution, and implementation using bus route patterns

Analysis of recent public transit ridership trends

Distribution-free estimation of some nonlinear panel data models

Econometric analysis of cross section and panel data

How transit scaling shapes cities