An Interpretation in Terms of Scheduled Activities

**PIET RIETVELD***

Vrije Universiteit

In travel surveys, most respondents apply rounding of departure and arrival times to multiples of 5, 15, and 30 minutes; in the annual Dutch travel survey, about 85% to 95% of all reported times are rounded. In this paper, we estimated rounding models for departure and arrival times. The model allowed us to compute the probability that a reported arrival time *m* (say *m* = 9:15 a.m.) means that the actual arrival time equals *n* (say *n* = 9:21 a.m.). Departure times appear to be rounded much more frequently than arrival times. An interpretation of this result is offered by distinguishing between scheduled and nonscheduled activities and by addressing the role of transitory activities.

We argue that explicitly addressing rounding of arrival and departure times will have at least three positive effects. First, it leads to a considerably better treatment of reported travel time variances. Second, biases in the computation of average transport times based on travel surveys can be avoided. Third, it overcomes the problem of erratic patterns that appear in travel survey data for the minute-by-minute records of increases in the number of persons in traffic.

Research on travel behavior is often based on travel times and distances reported by travelers. It is well known that these reported values tend to be rather inaccurate. For distances, this is understandable, because there are many circumstances where travelers do not have instruments to measure distance. In the case of travel time, one might expect a more accurate measurement since most travelers wear watches, and, in particular, must pay attention to time in order to arrive at scheduled activities. Nevertheless, it is clear that inaccuracies occur (see, e.g., Rietveld et al. 1999). Some people take clock time more seriously than others, and there are also notable differences between cultures in the precision of timing activities (Levine 1997). In the present paper, we address the issue of rounding travel timesin particular, the rounding of arrival and departure times.

Consider the example of reported departure times of trips in the annual national transport
survey in the Netherlands (CBS 1998). This survey is based on the travel diaries of about 144,000
randomly drawn Dutch citizens who reported their travel activities during one day in 1997.
Respondents were requested to report the arrival and departure times of all trips on a
certain day. Suppose a respondent *j* indicates that a trip started at departure
time [*h _{j}:m_{j}*], where

When analyzing travel statistics, it is important to be aware of rounding because unreliable data on travel times can result. For example, if departure and arrival times are normally rounded to multiples of 15 minutes, travel time will thus be rounded to multiples of 15, implying inaccurately reported travel time. Analysis of travel behavior will then be based on inaccurate travel time data. A similar conclusion holds for the analysis of travel time budgets (see, e.g., Zahavi 1977) and travel speeds. The rounding problem adds another error to the usual errors in statistical analysis (incomplete data, specification error, fundamental unpredictability of human behavior) and thus leads to larger variances of estimated coefficients.

Rounding does not only affect variances, it may even lead to a systematic bias for averages. As we will demonstrate later in this paper, there is no guarantee that in the case of travel times the probabilities of rounding upward and rounding downward are equal. Thus, rounding not only affects the reliability of individual observations, but it may also have an adverse effect on the reliability of national averages. We will demonstrate that rounding practices provide an explanation of the result reported by Battelle (1997) that the average of reported travel times is higher than the average of actual travel times.

Another example of the problem with rounding is found when departure and arrival time data are used to describe the development of traffic volumes during peak periods. Travel survey data of the type discussed here can be used to find out how many cars are on the Dutch roads from minute to minute (see, e.g., CBS 1996), but rounding can lead to erratic
patterns.^{1}
The simplest way to overcome this would be to present data for time
periods of 30 or 60 minutes, but this would imply that information is lost on how
traffic volumes build up during the shoulders of the peak. This information is
important for public and private decisionmakers who address congestion problems.

The above examples demonstrate how rounding departure and arrival times can affect data quality that influences transport analysis and policymaking. However, the relevance of the topic of rounding of departure and arrival times goes beyond data reliability. We will demonstrate that the rounding phenomenon sheds light on the nature of scheduling of transport-inducing activities. We develop a simple statistical model to analyze the propensity to round departure and arrival times and estimate it in the next section. An interpretation of differences between rounding in departure and arrival times is given in the discussion section in the context of scheduled activities.

As figure 1 shows, rounding departure times seems to take place toward certain anchor points such as:

- multiples of 5 minutes: 0, 5, 10, 15, 20, ,55
- multiples of 15 minutes: 0, 15, 30, 45
- multiples of 30 minutes: 0, 30
- multiples of 60 minutes: 0

Note that according to this approach the high outcome for the [*h*:00] oclock departure time in figure 1 is the joint result of rounding to all multiples of 5, 15, 30, and 60 minutes. Another possibility is that people do not apply rounding but report the exact minute of departure.

Consider in more detail the possibility of rounding to the nearest multiple of 5. Let *m* be the actual minute of departure, and let *d _{m}*

*p _{m}*

The coefficient *a*_{5} is interpreted as a base value for rounding to a multiple of 5 minutes, whereas *b*_{5} indicates the decrease of the probability of rounding as one moves away from a multiple of 5 minutes. We expect *a*_{5} to be positive and *b*_{5} to be negative; there is a tendency to round to the nearest multiple of 5 minutes, but this tendency decreases as one moves away from the nearest multiple of 5. For example, the probability of rounding 11 to 10 is larger than the probability of rounding 12 to 10. Note also that as *p _{m}*

In a similar way we formulate the rounding mechanisms for the other multiples of minutes:

*p _{m}*

*p _{m}*

*p _{m}*

In the case of rounding to a multiple of 30 minutes, there are two nearest multiples when *m* = 15. In this case, the probabilities of rounding to [*h*:00] and [*h*:30] are assumed to be equal, so that the resulting probabilities of rounding are (*a*_{30} + 15 · *b*_{30})/2. A similar case holds for the rounding to a multiple of 60 minutes.

After having defined these rounding probabilities, the probability that rounding of departure time *m* does not take place (*p _{m}*

*p _{m}*

*p _{m}*

*p _{m}*

*p _{m}*

*p _{m}*

Thus, there is only one case where we assume that rounding does not take place, that is, when *m* = 0. The resulting structure of transition probabilities can be found in
table 1.

Example: when the actual time of departure *m* is 8:16, rounding can take
place to 8:15 (via *p*_{16,5}; nearest multiple of 5), another time to 8:15 (via *p*_{16,15}; nearest multiple of 15), to 8:30 (via *p*_{16,30}; nearest multiple of 30), and to 8:00 (via *p*_{16,60}; nearest multiple of 60). The other possibility is that the actual and reported time of departure coincide (last column of table).

Consider now the distribution of actual departure times. Let *g _{m}* denote the probability that a trip made by the respondent actually starts at minute

*q*_{45} = [*g*_{38} · *p*_{38,15 }+
+ *g*_{52} · p_{52,15}] + [*g*_{43} · *p*_{43,5 }+
+ *g*_{47} · *p*_{47,5}].

For the other departure times, similar formulations can be derived. Note that for departure times *m* that are not equal to multiples of 5 we have simply:

*q _{m}* =

We still have to formulate the distribution of actual departure times *g _{m}*. We will assume that all departure times within an hour are equally probable:

*g _{m}*= 1/60.

This assumption has to be made since we have no prior knowledge about the distribution
of the exact minute in the hour during which departures take
place.^{3}
Another assumption we make is that rounding is the only source of error. Thus, we will not consider other sources of error, such as mistakes made when filling out the survey questionnaire, inaccurate watches, etc. The possible implications of these assumptions are discussed at the end of the next section. These assumptions suffice for a specification of the likelihood *q _{m}* for all reported departure times

*lnL* = *N*_{0}* ln q*_{0} + *N*_{1}*ln q*_{1} +
+ *N*_{59}*ln q*_{59}

Under the null hypothesis that reported departure times are equal to the actual departure times, all probabilities in table 1 are equal to zero, except the ones in the last column. This implies that

* ln L*_{0} = *N*_{0}*ln*(1/60) + *N*_{1}*ln*(1/60) +
+
*N*_{59}*ln*(1/60) = *N ln*(1/60)

where *N* equals the total number of observations.

The results of the maximum likelihood estimation for the departure minutes are
reported in
table 2.
The likelihood values indicate strong support for rejection of the null hypothesis.
The test statistic χ^{2} = 2(*lnL* - *1nL*_{0}) is an asymptotically distributed chi-square with degrees of freedom equal to the number of restrictions on the parameters (8). The value of the test statistic corresponding to a 99% probability of rejection of the null hypothesis is 20.1 in this case. We found overwhelming evidence of the importance of rounding to multiples of 5, 15, and 30 minutes: their base values *a*_{5}, *a*_{15}, and *a*_{30} are clearly significant. Only rounding to the whole hour assumes a small value (*a*_{60} is less than 1%). The *b* values were very small, with the exception of *b*_{5}, indicating that the probability of rounding 4 to 5 equals 46.4%, whereas rounding 3 to 5 equals 42.8%. For rounding to multiples of 15, 30, and 60, the *b* values were positive, which was unexpected. Their levels were very small, however. The reason that some of them are significant is that the number of observations is large. Considering the magnitudes they assume, they can be ignored. Thus, we conclude that, with the exception of rounding to multiples of 5 minutes the rounding probabilities hardly depend on the distance to the reference value.

To illustrate the meaning of the estimates, we computed the implications for the rounding probabilities when the actual observation is 19 minutes after the hour. The following rounding possibilities and the corresponding probabilities are:

to 0 minutes after the hour (the nearest multiple of 60): 2.1%

to 15 minutes after the hour (the nearest multiple of 15): 29.0%

to 19 minutes after the hour (no rounding): 4.6%

to 20 minutes after the hour (the nearest multiple of 5): 46.4%

to 30 minutes after the hour (the nearest multiple of 30): 17.9%.

The estimation result in table 2 indicates that rounding to multiples of 5 minutes dominates when we consider an individual observation. Note, however, that rounding to a certain multiple of 5 (say *n*) only takes place for the 4 nearest neighbors (*n*-2, *n*-1, *n* + 1, *n* + 2). With the multiples of 15, 30, and 60, the numbers of these neighbors are 14, 29, and 59, respectively. Thus, the base values for *a*_{5} to *a*_{60} must be multiplied by factors 4 through 59 to calculate the total number of reported departure times. In that case, the 30-minute multiple is used most frequently, and this is confirmed by the original data in table 1.

A similar approach was applied to arrival time data. The raw data are presented in figure 2. It shows a pattern similar to the departure time figures, although the scores are less peaked in multiples of 5. The share of unrounded departure times is clearly higher (about 15% are rounded to a value like 1, 2, 3, 4, 6, 7, etc., as opposed to about 5% for arrival times).

Estimation results are shown in
table 3. The results of the arrival time estimates are to some extent similar to the departure time roundings: the 60-minute rounding was the least important, and the *b* values were negligible, except *b*_{5}. A striking difference between departure and arrival times is that rounding to a multiple of 5 was much more dominant for arrival times. To illustrate, we again computed the rounding probabilities when the actual time of arrival was 19 minutes after the hour:

to 0 minutes after the hour (the nearest multiple of 60): 0.0%

to 15 minutes after the hour (the nearest multiple of 15): 9.3%

to 19 minutes after the hour (no rounding): 10.4%

to 20 minutes after the hour (the nearest multiple of 5): 76.0%

to 30 minutes after the hour (the nearest multiple of 30): 4.3%.

Thus, rounding to multiples of 5 minutes was dominant. Absence of rounding had the next highest shares and rounding to the nearest multiple of 15 was fairly unimportant. Rounding probabilities to multiples of 30 and 60 minutes were small.

We conclude this discussion by noting that we have now derived the distribution of *reported* *departure time, conditional on the actual departure time*. It may also be interesting to derive the reverse: the distribution of the *actual departure time, conditional on the reported departure time*. For example, when the reported time of departure *m* equals 15 minutes, what is the probability that the actual time *n* equals 8, 9, 10, and so forth? This can be achieved by using Bayes formula (Hogg and Craig 1970). Let *p _{m,n}* be the probability of the reported time

*f*(*m,n*) = *p _{m}*

Since we want to determine *k*(*n*|*m*), the distribution of the probability of an actual arrival at *n* given a reported value *m*, we make use of the Bayes formula

* k*(*n*|*m*) = [*p _{m}*

Since we assume that the density of the actual departure time *g*(*n*) is given as

*g _{n}* = 1/60 for

the Bayes formula can be simplified as

*k*(*n*|*m*) = *p _{m,n}* / [

Application of this formula to, for example, *k*(4,4) implies that *k*(4,4) = 1:
when the reported time of departure equals 4, one can be sure that the actual departure
time equals 4. On the other hand, we find the following probabilities
(table 4)
for the actual values underlying the reported observation *m* = 15. The table shows that a reported departure time of *m* = 15 means the probability that the actual departure time is indeed 15 is only 12.5%. The higher probabilities for the actual departure time are found in the range between 13 and 17 minutes, but the share for the remaining departure times is still substantial (43%).

Information of this type can be used in further statistical analyses of travel behavior data to give an adequate representation of errors in variables (see e.g., Johnston 1984). An important implication of our approach is that rounded observations of travel times have a much larger variance than unrounded ones. For example, in our approach, the reported duration of a trip of 32 minutes has a much smaller variance than a trip with a reported duration of 30
minutes.^{4}
Such differences in variance are not well captured in standard econometric methods.

One may wonder why the rounding rules applied to arrival times are more accurate than those for departure times (rounding to multiples of 15 and 30 minutes take place much less frequently). Various explanations exist.

*The structure of the questionnaire*. The question on the times of departure and arrival are posed in an identical way: "At what time did you depart/arrive? .... hour .... min." Note that these questions invite respondents to give an exact specification of the departure/arrival time. We conclude that the difference in the rounding practice for arrivals and departures cannot be explained by the way the questions are phrased.

Another point is that most respondents will fill out the questionnaire at the end of the day. Many of them will have forgotten their exact minute of departure and arrival for trips made 3 to 15 hours earlier. This explains the practice of rounding, but it does not explain why it occurs more often with departures than with arrivals.

*Structure of public transport timetables*. A bias of public transport timetables
toward multiples of 30 minutes as frequently used departure times might influence the
reported departure
times.^{5}
Such a timetabling practice, however, does not exist in The Netherlands. Note also that departure times reported here relate to the whole chain, so that the departure time would not indicate the time of departure of the train, but the time the respondent leaves to make a trip. A final observation is that in developed countries the only collective transport mode that does not use timetables at the one-minute level of precision is aviation (it uses multiples of 5).

As opposed to public transport time tables, most nontransport activities have a scheduled start at multiples of 15, 30, or 60 minutes: examples are hours at school, meetings, appointments, work, church services, sport events, cinema performances, etc. In some cases, both the *start* and *end* times are exactly specified, but often the beginning is more rigid and explicit than the end. This may create the perception that an important share of activities start at multiples of 15, 30, or 60 minutes and that a smaller share end at multiples of 15, 30, or 60 minutes. Consequently, the expectation is that the concentration of reported times at multiples of 15, 30, and 60 minutes is larger for arrivals than for departures. However, the data reveal that the opposite takes place. On the other hand, there are many activities that are not scheduled. For example, the arrival at home after an activity is usually not followed by an activity scheduled at an exact point in time. Thus, the share of scheduled activities in activity patterns must not be exaggerated.

Another point is that the start/end of an activity does not necessarily coincide with the arrival/departure of a trip. In many cases, there are *transitory activities* (e.g., relax, wait, talk to other participants, deposit ones coat at the cloak room, report at the entrance, find ones way to the exact place of the activity, wait for the elevator). The Dutch travel survey (like many other travel surveys) does not specify these transitory activities, so it is left to the respondent whether he considers them as part of the trip or of the activity carried out. Consider the case of a student whose lecture is scheduled to end at 12:45 sharp; in reality it ends at 12:47, the student talks to his classmates until 12:49, and he then leaves the university building at 12:53 to walk to his car, which he starts to drive at 12:56. Then he may answer the question "at what time did you leave" by filling out any of the above-mentioned times, plus rounded times such as 12:45, 12:50, 12:55, and 13:00 oclock. A similar story, of course, holds true for transitory activities before a scheduled activity.

The question remainswhy are people more inclined to round with departure times than with arrival times. Probably, the most important answer is that *scheduled activities force people to plan their trips in advance, which provides them with anchor points for their memory afterward*. At the end of the day, they will still remember whether they arrived long before the scheduled time, or whether they were late. Since, as mentioned above, scheduling takes place more often in terms of the start of an activity rather than the end, people will have more precise memories about the time of arrival and they will therefore also have a tendency to apply rounding less frequently than with departures. This sheds some light on the literature of scheduling. As put forward, for example, by Small (1982; 1992) and Wilson (1989), travelers face the challenge of arriving on time to scheduled activities (e.g., the start of work or the start of a business meeting). Given a high penalty for arriving late, travelers tend to take into account transport systems that are unreliable (congestion caused by nonrecurrent events, delays or missed connections in public transport) and thus plan their trip in such a way that delays can be accommodated. This means that we may expect travelers to arrive early in cases of scheduled activities with penalties and uncertainty in travel times. Because of the penalty for a late arrival, the traveler will have a keen eye on whether he really arrived early or late. When he arrives early, the traveler will have an additional type of transitory activitywaiting time, which is a cushion to avoid being late.

Thus, we arrive at several differences between the start and the end of an activity.
First, the start is more often fixed in time than the end is. Second, the element
of transport system uncertainty is present for the person who needs to meet requirements
of being on time; it does not play a role at the end of the meeting. Third, the penalty
for arriving late may be perceived to be larger than the penalty of leaving
early.^{6}
These three differences imply that on average travelers will be much more concerned about the starting time of activities than the time they end.

We finish this section with a discussion of the possible implications of two assumptions on which the above estimations are based: uniform distribution of actual departure times during an hour and absence of measurement errors. The assumption that departure and arrival minutes are distributed uniformly was made since we have no prior knowledge about the distribution of the exact minute in the hour during which departures take place. One might argue that since scheduled activities usually end at 0, 15, 30, or 45 minutes after the hour, there will be a tendency that the density of actual departure times is higher at those times. This would offer an alternative interpretation for the empirical results. With the given data, this alternative interpretation cannot be falsified. However, it may be argued that it is not a very plausible explanation for several reasons.

First of all, we can make use of other data sources that include both actual and reported
departure times. From a survey done in the United States (Battelle 1997) among car drivers in
Lexington, it appears that the distribution of actual departure times is very close to uniform.
The second reason is that transport statistics show that a considerable portion of human
activities are not strictly scheduled: in the Netherlands more than half of all movement
relates to activities such as shopping, recreation, and social visits (CBS 1998).
Therefore, an outcome of 95% of actual departures taking place at round minutes (i.e.,
at multiples of 5) would be implausible. Another reason is that this explanation ignores
the importance demonstrated above of transitory activities taking place between the end
of an activity and the start of a trip. Another argument concerns trips where scheduled
public transport services are used. The departure times at bus stops and railway stations
tend to be distributed uniformly during the hour, so that one would expect a uniform
distribution of departure times as described in the earlier section on formulation and
estimation of the statistical
model.^{7}
Also, the discussion above of the difference
between the distribution of departure and arrival times strongly supports the view that
the peaks in the distribution of reported times are due to rounding and not to peaks in
actual times. We noted that if an activity is scheduled, the certainty about its starting time is usually higher than about its end time. Therefore, if the distribution of reported departure and arrival times were dictated by the actual start of these activities, one would expect larger peaks in the distribution of arrival times compared with departure times. In reality, however, the opposite is
true.^{8}

We conclude that with the given data we cannot test whether the distribution of actual departure minutes is uniform. It is highly implausible, however, that a non-uniform distribution is the sole reason for the peaks in the reported departure times. One cannot exclude, however, the possibility that there is a tendency for more people to arrive and depart at round minutes rather than at other minutes. If this were true, it would imply that we have overestimated the rounding tendency. Given the above arguments, the possibility of overestimation is probably small.

The second assumption in the Formulation and Estimation section that may need some discussion concerns the premise that rounding is the only source of error when reporting departure and arrival times. In the statistical analysis, we ruled out the possibility that people report wrong departure times because of mistakes, inaccurate watches, or bad memory. Of course such errors will take place frequently in travel surveys and they will in part express themselves in rounding. For example, if a respondent does not remember the exact times at the end of the day, he may use proxies. In cases where these mechanisms do not express themselves via rounding, they contribute to the variance of error in observed data, but there is no reason to expect that they will lead to systematic distortions in the analysis of
rounding.^{9}

Our analysis of departure and arrival times indicates that rounding is a rule, rather than an exception. About 5% to 15% of all reported times assume values that are not multiples of 5, whereas these are 80% of the possible clock times. In the case of scheduled activities, the reported times are probably more precise because scheduling implies the use of anchor points in the timeframe. With fixed schedules, there may be a high penalty for being late so that travelers will be more likely to remember the exact timing of trips. Since scheduling of start times takes place more often than for end times, it is plausible that reported times of arrival are more accurate than reported times of departure.

In the research on travel behavior, data on travel times usually play an important role. These travel times follow as the result of subtracting reported times of arrival and departure. Given the large rounding errors observed here, it is clear that errors in reported travel times (and related variables such as travel speeds) will be large. This "error in data" phenomenon will obviously hamper the analysis of data on individual travel behavior. In the present paper, we developed a method, based on a Bayesian approach to derive the probability that a reported arrival time *m* means that the actual arrival time equals *n*. This method can be used in "errors in variable methods" to give an adequate representation of the measurement error. We demonstrated that the variance of rounded travel times is much larger than that of unrounded times. This approach must be considered superior to the usual approach where all measurement error is supposed to be represented by a common variance.

Rounding has a larger impact than just affecting the variance of travel times, however. Given the large scale at which rounding takes place, it may also affect averages computed on the basis of national surveys when probabilities of rounding upward and downward do not cancel. Consider, for example, the distribution of reported trip duration in the Netherlands. This distribution is skewed: the most frequently reported trip duration (mode) is 10 minutes, the median value is 15 minutes, and the mean value is about 20 minutes. Therefore, the number of trips with an actual duration of between 15 and 30 minutes will be considerably larger than the number of trips with a duration between 30 and 45 minutes. As a result, the probability of rounding upward is considerably higher than the probability of rounding
downward.^{10}
The conclusion is that in this case
rounding of arrival and departure times leads to overestimates of average travel
times.^{11}

Finally, ignoring the rounding problem could lead to erratic patterns when the travel survey data are used to give a minute-by-minute record of the number of travelers in the transport system. Consider, for example, the 24-hour average number of people in transport in each minute for our sample of 550,000 respondents. The departure and arrival data indicate that during the first minute of the hour 120,000 persons enter the transport system, whereas only 55,000 persons leave. This would imply a sudden net increase of 65,000 persons during 1 minute, which is much higher than the small net decreases during subsequent minutes of about 1,500 persons per minute. This obviously hinders a proper assessment of the development of the number of persons in traffic in the course of time. By using the Bayesian approach presented earlier, this problem can be overcome.

In our discussion of rounding, we touched on the importance of *transitory activities* in scheduled activity patterns. These transitory activities are often ignored in the analysis of travel behavior. A main reason for these transitory activities is that they emerge in a response to reduce the penalty of arriving late at a scheduled activity. They also result from infrequent public transport services. Transitory activities are important to reduce bottlenecks in internal and external transport systems. An example of an internal transport system is elevator capacity, which usually will not allow everybody to arrive just in time or leave immediately after a big event. Similarly, parking facilities do not function well under these circumstances. An example of external transport systems concerns the capacity to absorb visitors for large-scale events in stadiums, exhibition centers, etc. Transitory activities do not only keep bottleneck problems manageable, they may also have value per se for the travelers. They deserve more attention in transport behavior than they usually get. To properly analyze their presence and size, detailed questionnaires are needed.

A final point of attention is the possibility of linking reported time data to archived global positional data. The combination of geographic information systems and global positioning systems offers great potential for improving the quality of data on travel time and distance in passenger surveys. This holds true not only for automobile trips, but these systems may also provide useful data on other kinds of trips (Quiraga and Bullock 1998; Uchida et al. 2001).

The author thanks Uty Pang Atjok who provided computational assistance. In addition, David Greene and two anonymous referees gave constructive comments.

Battelle Transportation Division. 1997. *Lexington Area Travel Data Collection Test: Global Positioning Systems for Personal Travel Surveys.* Washington, DC: U.S. Department of Transportation, Federal Highway Administration.

Central Bureau of Statistics (CBS). 1996. *Autos in Nederland (Cars in the Netherlands)*. Deventer, The Netherlands: Kluwer.

_____. 1998. *De Mobiliteit van de Nederlandse Bevolking (The Mobility of the Dutch Population)*. Voorburg, The Netherlands.

Hogg, R.V. and A.T. Craig. 1970. *Introduction to Mathematical Statistics*. London, England: Macmillan.

Johnston, R.J. 1984. *Econometric Methods*. New York, NY: McGraw-Hill.

Levine, R.V. 1997. *A Geography of Time*. New York, NY: Basic Books.

Quiraga, C.A. and D. Bullock. 1998. Travel Time Studies with Global Positioning and Geographic Information Systems: An Integrated Methodology. *Transportation Research C* 6:101-27.

Rietveld, P., B. Zwart, B. van Wee, and T. van den Hoorn. 1999. On the Relationship Between Travel Time and Travel Distance of Commuters: Reported Versus Network Travel Data in the Netherlands. *The Annals of Regional Science* 33:269-88.

Small, K.A. 1982. The Scheduling of Consumer Activities: Work Trips. *American Economic Review* 72:467-79.

_____. 1992. *Urban Transportation Economics*. Luxemburg: Harwood Publications.

Uchida, T., M. Pursula, A. Suzuki, Y.H. Lee, and D. Takehiko. 2001. Monitoring Personal Movement: Development of PEAMON (Personal Activity Monitor) for Automated Trip-Diary. Tohoku University, Japan.

U.S. Department of Transportation, Bureau of Transportation Statistics (BTS). 1997. *Nationwide Personal Transportation Survey*, Washington, DC.

Wilson, P.W. 1989. Scheduling Costs and the Value of Travel Time. *Urban Studies* 26:356-66.

Zahavi, Y. 1977. *The UMOT Model.* Washington, DC: The World Bank.

Piet Rietveld, Vrije Universiteit, Amsterdam, The Netherlands. Email: prietveld@econ.vu.nl.

^{1} Note that if the same level of rounding is used for both departure and arrival times, traffic volumes would be relatively stable from minute to minute. However, when rounding is greater in one of the two processes, irregular patterns will be found in the minute-to-minute records of traffic volume.

^{2} Thus *p _{m}*

^{3} Of course we have fairly accurate information about the distribution of departure times during the 24 hours of the day: during the night, the number of departures is much smaller than during the day. However, very little is known about the distribution between the minutes within the hour.

^{4} For example, in the most extreme case, a 2-minute trip with a departure at 8:14 and arrival at 8:16 may be reported as a 30-minute trip after rounding. The same holds true for a 58-minute trip that started at 8:16 and ended at 9:14. This illustrates the large range on which a trip with a reported duration of 30 minutes may be based. On the other hand, a trip starting at a reported time of 8:16 and ending at 8:48 will just have lasted 32 minutes according to our model, implying a 0 variance (remember that apart from rounding, all other data errors are ignored in our analysis).

^{5} Public transport maintains a 5% share of the total number of trips in the Netherlands. Its share in the total number of kilometers traveled is about 13%.

^{6} We do not go into details about chaining activities with fixed start and end times. Travelers who are able to leave a sufficient amount of time between the end of one activity and the start of a second activity may then have spare time for an additional type of transitory activity. When the time is not sufficient, the traveler reveals which of the two activities will have the higher penalty (leaving early versus arriving late).

^{7} What really matters is not the official departure time of the public transport services, but the departure time of the traveler from his origin, thus taking into account the access time to the public transport node. Thus, even if there is a tendency for public transport timetables to be biased toward departure times of the services in multiples of 5 minutes, the variance in the access times would make this invisible when departure times of travelers are considered.

^{8} Another possibility with arrival times is that the distribution of actual times has high probabilities at times just before round minutes because most people try to arrive on time. However, inspection of the reported arrival times does not reveal such a tendency. For example, the data in figure 2 even demonstrate a slight tendency in the opposite direction: the share of respondents reporting they arrived between 1 and 15 minutes after the hour is somewhat larger than the share reporting they arrived between 45 and 59 minutes before the hour (26% versus 22%).

^{9} Note also that without additional data, adding an error term Ε* _{m}* with mean zero and variance σ

^{10} This implies that the figures of 20 and 15 minutes mentioned in the text for mean and median are biased. The effect on the mean is probably larger than on the median.

^{11} In the Battelle study (1997), a comparison of reported and actual travel times indeed revealed that reported travel times based on recall generally overstate travel time. A similar conclusion was drawn about travel distances.