Parametric, semi-parametric and non-parametric models of telecommunications demand

Information Economics and Policy 13 (2001) 311–329 www.elsevier.com / locate / econbase

Parametric, semi-parametric and non-parametric models of telecommunications demand An investigation of residential calling patterns Erik Heitfield a , *, Armando Levy b a

Division of Research and Statistics, Board of Governors of the Federal Reserve System, Washington, DC 20551, USA b Analysis Group /Economics, 1010 El Carrino Real, Suite 310, Menlo Park, CA 94025, USA

Abstract We investigate long distance telephone calling patterns using billing information and demographic data for a large cross-section of residential households. The joint distribution of the number and duration of toll calls is estimated using a non-parametric kernel model, a semi-parametric Cox proportional hazard model, and a fully parametric Poisson–Weibull model. Particular attention is paid to the estimation and analysis of call duration hazard functions. We find that call duration is quite inelastic with respect to price, and that while evening and night / weekend rate calls share vary similar duration characteristics, they differ substantially from peak rate calls. Published by Elsevier Science B.V. Keywords: Telecommunications; Call duration; Non-parametric; Semi-parametric JEL Classification: C14; C4; D12

1. Introduction Modeling demand for telephone services is complicated. Over any billed month, residential households demand a portfolio of local and long distance calls composed of calls to different locations, spanning different distances, at different times of day, and for different lengths of time. Tariffs faced by households vary along those same dimensions. If the aim of demand research is revenue and / or cost forecasting (for rate-making by regulators, for example), then customers’ * Corresponding author. 0167-6245 / 01 / $ – see front matter PII: S0167-6245( 01 )00033-6

Published by Elsevier Science B.V.

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demand cannot be effectively summarized by a single value such as the number of calls or the total minutes of use. Because the mix of calls that make up a household’s aggregate demand has an impact on revenues, network capacity, and switching load, a single statistic is inadequate for evaluating consumers’ responses to new tariff schedules. For example, suppose tariffs are raised and two households respond by shifting their telephone usage patterns. The first household responds by substituting evening calls for day calls, while the second substitutes fewer but longer calls for many shorter calls. These two substitution patterns may have very different effects on revenues and network switching costs while appearing identical when measured in terms of minutes of use. Characterizing the joint distribution of the length and number of toll telephone calls customers choose to make is an important first step in developing accurate cost estimates and effective tariff schedules. A wealth of previous research has investigated the aggregate (across households, or across calls within a household) demand for telephone service,1 however relatively little work has focused on individual customers’ calling patterns.2 While aggregation reduces the need for detailed data and simplifies the task of specifying a statistical model, it precludes inference about the complex relationship between individual calling patterns, tariff schedules, and demographic characteristics. In this paper, we undertake a non-structural analysis of toll calling patterns using highly detailed billing data from PNR Associates. We focus particularly on the estimation of hazard functions which can be used to characterize the rate at which individual telephone conversations are terminated as a function of time. Fully parametric, semi-parametric and non-parametric techniques are employed to investigate calling patterns. The non-parametric analysis allows us to describe the distributions of the number and length of toll calls without making strong assumptions about the functional form of these distributions.. Information gleaned from the non-parametric analysis informs our specification of fully parametric and semi-parametric models. These models, while imposing stronger assumptions on the data, allow us to explore the relationship between calling patterns and demographic covariates. The paper proceeds as follows: Section 2 describes the PNR Database and the variables used in this analysis. Section 3 summarizes some of the statistical theory of duration analysis. In Section 4, we present a non-parametric analysis of call duration based on kernel density estimators. A parametric estimation of the joint distribution of the duration and number of toll calls is described in Section 5, while a semi-parametric Cox proportional hazard model is developed in Section 6. The 1 For US long distance demand see most recently Levy (1999), Duncan and Perry (1994) and Taylor (1994), Wolak (1993) for a review. For an analysis of international telephone demand, see Hackl and Westlund (1995). 2 Two notable exceptions are a study of Train et al. (1987) and a study of Cameron and White (1990). The Cameron and White paper examines individual calls placed between two cities in a 24-hour period. While their results are comparable to ours, we examine complete monthly demand by a household, which allows us to estimate distributions for numbers of calls as well as call durations.

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final section compares the results from our parametric, semi-parametric and non-parametric specifications, draws conclusions, and discusses opportunities for future research.

2. Data PNR Associates’ 1994 Bill Harvesting database contains survey information for 8700 households from across the United States. In this survey each household was asked to describe its demographic characteristics and submit copies of its local and long distance telephone bills. PNR Associates used the billing information provided by respondents to construct a detailed database of individual telephone calls. This database includes information on the originating area code and exchange, the terminating area code and exchange, the price, the distance (in miles), and the duration of each metered telephone call (in minutes). We have used this database to determine the total number of calls made by each household over the course of a month, as well as the number of calls made by each household to each foreign area code / exchange. For reasonably distant calls, we believe that individual area code / exchange numbers probably correspond to unique destinations. Our analysis is limited to standard land line toll calls billed to the originating telephone number. To focus on metered long distance calls, we consider only calls made to destinations at least ten miles away from the call originator. In order to properly identify the marginal costs for each call, a small number of households with calling plans are excluded. Table 1 provides summary statistics for the observed household calling patterns. Responses to survey questions provide values for household income 3 and household size, as well as dummy variables for the presence of an adolescent or Table 1 Summary statistics Variable

Mean

Std. Dev.

Median

Calls / hsld Price / min Network a Duration Income Distance Household size

21.45 0.20 8.02 7.62 8.71 318 2.86

27.43 0.18 7.90 11.91 4.01 524 1.47

13 0.17 6 3 8 69 3

a

The number of calls placed to unique locations.

3 In this survey income was recorded as an ordered categorical variable. Categories include bands of $2,500 from $7,500 to $200,000. We treat the variable as approximately continuous (hence a unit of income equals $2,500).

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teenager, and whether the household owns a modem or a cellular phone. For reference, 16% of survey households own a modem, 8% own a cellular phone, 23% have an adolescent present, and 16% have a teenager present.

3. Framework for Analysis Let i be an index of customers in our sample and let n i be some measure of the number of calls made by customer i. We investigate two possible definitions of n i , one in which n i is the total number of calls made by customer i over the course of a month, and one in which n i is a vector that records the number of calls placed to each destination called by customer i. Define j as an index of customer i’s calls so that t ij records the duration of the jth call placed by customer i. Finally let w ij be a vector of call-specific attributes such as time of day, distance, and price, and let z i be a vector of customer-specific demographic characteristics. We are interested in characterizing the distribution of n i and t i 5 (t i 1 , . . . ,t in i ) as a function of z i and w ij . Assuming that durations of individual calls are conditionally independent of one another, this joint distribution can be factored into the product of the marginal distribution of n i and the conditional distribution of the t ij ’s given n i :

SP ni

f(t i ,n i uz i ,w ij ) 5

j 51

D

ft (t ij un i ,z i ,w ij ) fn (n i uz i ).

(1)

where ft and fn denote the unknown conditional densities of calls and number of calls in a month respectively.4 Our strategy is to estimate the conditional distribution of call durations and the marginal distribution of call numbers separately. We use this two-stage approach rather than a more direct bivariate approach because of the different character of the two variables. Call durations, while reported in minutes, can actually take on any positive real value. The numbers of calls, on the other hand, are fundamentally discrete. The units of analysis for these two variables differ as well. For durations, observations are individual calls. For call quantities, observation are either customers or customer-destination pairs. Count data such as the number of calls can be analyzed by tabulating the probabilities associated with each discrete integer event. In a non-parametric setting, this tabulation corresponds to a simple histogram. In a parametric framework, a stochastic process that generates only integer valued data must be specified. 4 There are many reasons to believe the distribution of call durations are not independent of one another. For example, a household may prefer to make only long calls to one location and only short calls to another. However, in our investigation the correlation structure is left to future research.

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The distribution of call durations is best described using two related functions of the probability density: the survival and hazard functions. If Ft (t) is the cumulative density function for duration, then the survival function is defined as S(t) 5 1 2 Ft (t). As its name suggests, the survival function tells us the probability that a given telephone conversation will survive beyond time t. The hazard function is u (t) 5 ft (t) /S(t). This function measures how likely a conversation is to end at time t given that it has survived up to time t. The value of the hazard function evaluated at a point in time is called the hazard rate.5 Understanding the difference between the information embodied in the hazard function and the survival function is important for understanding the analysis that follows. The survival function can tell us the probability that a call will last longer than 20 minutes. The hazard function tells us the probability that a 20-minute phone call will end before the 21st minute. Comparing survival functions gives us information about which types of calls are longer; comparing hazard functions tells us the probability that different types of calls of a given length will be terminated.

4. Non-parametric analysis The principle advantage of non-parametric statistical methods is that they allow us to describe data without making strong assumptions about the distribution from which they are drawn. Characteristics of the data that might be missed in a fully parametric analysis can be uncovered. In addition, information from a nonparametric analysis can inform our choice of parametric specifications. In this section, we analyze the distributions of the number and duration of calls using non-parametric techniques.

4.1. Number of calls To estimate the unconditional marginal distribution of the number of calls, we use a variable bin-width histogram. This estimator is similar to a standard histogram, but the widths of bins are allowed to vary depending on the sparseness of the data. For regions of the distribution with many observations, bin widths are small, generally covering single integer values. For regions where observations are more sparse, bin widths are set wider to allow more smoothing. Variable bin-width histograms for the number of calls for each household and the number of calls between each source–destination pair are presented in Fig. 1. Both histograms show significant clustering of the data at lower values. Of those customers who made at least one call, nearly 10% made only one. Clustering for source–destination pairs is even greater. More than half the destinations called by customers were called only once over the course of a month. Hence, while a 5

A detailed survey of duration analysis techniques is presented in Lancaster (1990).

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Fig. 1. Histogram of number of calls.

portion of the households make a large number of long distance calls, they do not generally call the same destinations repeatedly.

4.2. Call duration To construct the survival and hazard functions for call duration, we use an adaptive kernel technique to estimate the conditional probability distribution of toll call durations. In its simplest form, a kernel density estimator involves attaching weights to each observation in the sample. These weights generally decline with distance, and the rate of decline controls the smoothness of the resulting density estimator. If our data is given by x 1 . . . x n , then the kernel estimate of the density at x is x2x 1 O] KS]]D nw w n

i

ˆ 5 f(x)

(2)

i 51

where K( ? ) is some general weighting function, and w is a parameter that must be

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chosen by the analyst.6 The window width w determines the rate at which weights decline with distance from x. A large value for w ensures that the weights decline slowly, resulting in a smoother density estimator. Smaller values for w produce density estimators that exhibit a large amount of local variation. In general, the optimal smoothing parameter depends on the degree of sparseness of the data. For regions of the distribution where many observations lie, less smoothing is needed. Thus, for a fixed w, the kernel estimator tends to over-smooth the data in regions with many observations and under-smooth in tail regions. To solve this problem, we use an extension of the simple kernel estimator described by Silverman (1986). This technique, known as the adaptive kernel estimator, involves first estimating a pilot kernel density to measure the sparseness of the data at each observation. A second estimator is than constructed which allows the window width to vary depending on the results of the initial estimate. If we let fˆ1 ( ? ) denote the pilot density estimate in (2), than the adaptive kernel estimator fˆ2 ( ? ) is, x2x 1 O ]]]] KS]]]D nh(w,fˆ (x )) h(w,fˆ (x )) n

fˆ2 (x) 5

i

i 51

1

i

1

(3)

i

where h( ? , ? ) is a function that depends on the smoothing parameter and the pilot density.7 Allowing the degree of smoothing to vary across the distribution does not eliminate the need to choose a window width. Picking a smoothing parameter requires a tradeoff between bias and variance. If the window width is too narrow, the estimated density function will have small bias, but will fluctuate rapidly across its support. Such an under-smoothed estimate is difficult to interpret. If the window width is too wide, the estimated density will appear to be much smoother than the true underlying density of the data. Such estimators cover up local

6

We use a tent shaped weighting function of the form

K(t) 5

7

H

1 2 utu if utu , 1

.

0 if utu $ 1

We use

S

fˆ1 (x i ) h(w,fˆ1 ) 5 ]]]]]] 21 exp n log( fˆ1 (x j ))

F O j

G

D

21 / 2

(4)

This function imposes more smoothing when the pilot density suggests sparsely distributed data, and possesses some important optimality properties. See Abramson (1982) or an analysis of this adaptive function.

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features of the distribution, leading to the same kinds of bias present in parametric models. Unfortunately, the optimal window width for any given compromise between bias and variance depends on the characteristics of the underlying distribution being estimated. This is, of course, unobservable. Though many approaches have been proposed for choosing w under this limitation, they are all fundamentally ad hoc. In our analysis several techniques were considered, but we eventually simply chose a window width that appeared to eliminate excess volatility in the estimated density function without smoothing over important characteristics. Our main interest in exploring call duration is to uncover the conditional distribution given covariates such as the price per minute of the call and the time of day. To model duration conditional on the number of calls, individual call data were initially stratified into subsamples corresponding to each of the bins constructed in the histogram analysis of the number of calls. Separate density functions for call duration were then estimated for each subsample. We found relatively little variation in these distribution functions, so for the sake of brevity we present results only for two larger subsamples that combine data across several of the histogram bins. To analyze the effects of distance and time of day on call duration, we further segmented our samples. Each call was classified as short-haul (10–50 miles) or long-haul (more than 50 miles) toll calls, and by whether it was placed during day, evening, or night / weekend billing hours. The hazard and survival functions for each of these subsamples are presented in Figs. 2 and 3. Several observations can be drawn from an examination of the estimated hazard functions. First, in every category considered, call lengths exhibit negative duration dependence. That is, the longer a call has lasted, the smaller is the probability that that call will end in the next minute. The magnitude of this duration dependence seems to differ substantially depending on whether calls are made to nearby or distant destination. For calls to locations less than 50 miles away, the hazard functions fall monotonically over time. However the hazard function for evening and night / weekend calls to destinations greater than 50 miles away appear to level off after falling precipitously for the first 5 minutes. This suggests, for example, that a 10-minute evening long haul call is just as likely to end in the next minute as a similar 45-minute call. The hazard functions for both long and short haul day calls lay well above those for evening and night / weekend calls through the first 25 minutes. Relatively short day calls appear more likely to end than short evening or night calls. However hazard rates for longer day calls are equal to or smaller than those of off-peak calls. Intuitively, this is what we might expect since most day calls are presumably business calls which tend to be relatively short. Once a day call has lasted for more than 30 minutes, we can infer that it is most likely a personal call and therefore has a similar hazard to other personal calls made during evening and night / weekend hours. We can also observe some important similarities across categories of calls. The hazard functions for evening and night / weekend calls appear quite similar to one

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Fig. 2. Hazard functions.

another. Furthermore, the hazard functions do not appear to change much as we condition on different numbers of calls between source–destination pairs. The survival functions presented in Fig. 3 verify one stylized fact about call durations: day calls tend to be substantially shorter than off-peak calls. As expected, infrequent long-haul calls appear to be longer than infrequent short haul calls.

5. A parametric model Clearly, the distance, time of day, and the number of calls to a destination are not the only factors that influence call duration. In this section we estimate the distributions of n i and t ij conditional on explanatory variables to better understand how calling patterns vary with price levels and household demographic characteristics. To accomplish this we must first postulate a probabilistic model of toll demand. Recall that the joint distribution of the number and durations of calls can be factored into fn , the marginal distribution of n i , and ft , the conditional distribution of t ij given n i . The information gained from our non-parametric

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Fig. 3. Survivor functions.

analysis helps us specify reasonable functional forms ft and fn . Given these specifications, model parameters are estimated using maximum likelihood.

5.1. A model for fn Toll calls are initiated in order to communicate information. To a great extent, the events that necessitate communication occur stochastically, and so the need to communicate can be viewed as a function of random events. We model the occurrence of such events as arriving according to a Poisson distribution with parameter m 8 . Of course, we do not expect m to be constant across all residential households, but related to characteristics such as the size of a household’s network of family and friends, the time of day, and most certainly the cost of the call. Specifically, we assume: 8 In reality events which might require communication come in different sizes (e.g remembering to call a friend versus. a death in the family), and we will model those events whose benefit from communication exceeds the connection charge as Poisson ( mi ).

E. Heitfield, A. Levy / Information Economics and Policy 13 (2001) 311 – 329

Og z ,

321

k

lnmi 5 z i g 5 g0 1

k ik

(5)

k 51

where the variables z i include average price per minute for the household’s portfolio, household income, presence of adolescents or teenagers, household size and the presence of a modem. The Poisson model for the marginal distribution of n i can be written as e mi ( mi )n fn (nuz i ) 5 ]]]. n!

(6)

5.2. A model for ft Let u (tuw,n) 5 ft (tuw,n) /St (tuw,n) be the hazard function of a call characterized by w and total calls n so that we can write ft (tuw,n) 5 u (tuw,n)(St (tuw,n)) 5 u (tuw,n) e 2 E0 u (t uw,n) dt t

(7)

to characterize the density in terms of the product of the hazard and the integrated hazard functions. The vector w ij contains the price per minute for call ij, the distance traversed by the call (in linear and quadratic forms), the number of calls placed to call ij’s destination (in linear and quadratic forms), and the household network size (i.e. the number of unique destinations called in a month). To specify this model, we must find an appropriate form for u. Once a call is placed, communication continues until the marginal utility of information transmitted falls below the marginal cost of communication (i.e. the price per minute and the opportunity cost of time). Our non-parametric analysis suggest that the hazard functions exhibit negative duration dependence. Moreover, we observe that the estimated hazards decrease rapidly as a function of time, but then fall more slowly as the call becomes longer. This suggests that the function u should decrease monotonically with a smoothly increasing slope. The Weibull function has these characteristics and is given by

u (tux) 5 a e x b t a 21

(8)

where x 5 (w,n). Changes in x covariates shift the hazard function up and down, while a determines the shape of the hazard functions 9 Values of a less than one imply a monotonically decreasing hazard function. Deriving ft (tuw,z) involves integrating u (tuw,z) to obtain

9

See Lancaster (1990) for a detailed analysis of the properties of the Weibull hazard model.

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ft (tuw,n) 5 a e x b t a 21 e 2t

a e xb

.

(9)

5.3. Parametric likelihood The log likelihood of observed data under the parametric specification described by (6) and (9) is

Osln( f (t un ,w )) 1 ln( f (n uz ))d 5 O SO [ln(a ) 1 (a 2 1)ln(t ) 1 x b 2 t e N

L(b,g,a ) 5

t

i

i

ij

n

i

i

i 51 N

ni

i 51

j 51

ij

ij

a x ij b ij

]

D

1 [z ig 1 n i ln(zg ) 2 n i !]

(10)

This likelihood function is globally concave and its global maximum can be found using standard numerical techniques. As in the previous section, separate sub-models are estimated for day, evening, and night / weekend calls. Plots of the hazard and survivor functions for the median household are shown in Fig. 4. Notice that the Weibull hazard and survivor estimates roughly capture the characteristics of our non-parametric estimates. Maximum likelihood parameter estimates are shown in Tables 2 and 3. Care should be taken in interpreting these estimates because the conditional means for both the Poisson model and the Weibull model are nonlinear functions of estimated coefficients. The mean number of calls is e x b , and the mean call duration is

Fig. 4. Weibull baseline hazard and survivor functions.

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Table 2 Parameter estimates Poisson m : number of calls Variable

Coef.

Std. error

Intercept p¯ Income hsldsize Modem Teen Adol Cellular

12.94 25.013 0.524 1.835 3.191 20.796 25.187 6.495

0.164 0.037 0.015 0.055 0.170 0.175 0.162 0.256

(e x b )1 / a G (1 1 1 /a ). However, because both of these relationships are monotonic, we can easily interpret the signs of the estimates.10 As expected, the coefficients on price with respect to number of calls and call duration are negative and significant. Households seem to be insensitive to marginal cost once a call is initiated. A 10% increase in price per minute for all calls reduces the average number of calls by 0.13 (calculated at mean values for all covariates). It decreases the duration of an average day call by only 1.5 seconds, while reducing the durations of evening and night calls by 9 seconds and 7 seconds respectively. The price elasticities of call duration (at the mean) are 20.06

Table 3 Parameter estimates. Weibull a, b : call duration Variable

a Intercept p¯ income n distance (distance)2 num (num)2 network

Day

Evening Std. Error

Coef.

Std. Error

Coef.

Std. Error

0.9069 1.2955 20.1779 0.0036 0.0003 0.6e23 20.2e26 0.0119 20.1e23 20.0069

0.0032 0.0157 0.0052 0.0013 0.2e23 0.2e24 0.1e27 0.9e23 0.1e24 0.6e23

0.8134 1.7285 20.6904 20.8e23 20.0011 0.7e23 20.2e26 0.0044 20.6e24 20.0036

0.0030 0.1694 0.1811 0.0013 0.2e23 0.2e24 0.7e28 0.7e23 0.5e25 0.9e23

0.8018 1.5520 20.5859 0.0023 20.0016 0.7e23 20.2e26 0.0019 20.2e24 20.0023

0.0030 0.0168 0.0162 0.0012 0.2e23 0.2e24 0.6e28 0.8e23 0.7e25 0.9e23

N538 734

10

Night

Coef.

N539 325

N539 258

The conditional mean of the Weibull distribution is monotonically increasing when a , 1.

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for day calls, 20.17 for evening calls, and 20.13 for night / weekend calls. These findings are consistent with the idea that off-peak calls are more likely to be discretionary in nature. The magnitudes of our elasticity estimates correspond roughly with that of Cameron and White (1990) who find an elasticity of approximately 20.2 for durations of calls between British Columbia to the United States.11 Turning now to household characteristics, we see that ownership of a modem is associated with more calls, and that a cellular phone seems to be a complement to land-line long-distance calls. Number of calls and call duration are both normal goods. Interestingly, the presence of teens or adolescents corresponds to fewer calls. Conventional wisdom (or personal experience) may lead one to assume that adolescents and teens tend to drive up the number of calls, but we expect the majority of those calls to be local and hence long-distance calls by adults may be ‘‘crowded out’’. Additionally adolescents and teenagers may obtain their own lines, and hence inflate the household size, but make no use of the sampled line. Off-peak durations fall as a household makes more calls, while on-peak durations increase. Durations fall as the household network size increases. Both distance and the number of calls to the same location initially increase duration but the effects are declining in both variables.

6. A semi-parametric model In the previous section, we assumed a Weibull form for the baseline hazard function in order to capture the basic shape of the non-parametric hazard functions estimated in Section 4. The Weibull model is a special case of a general class of proportional hazards models. A proportional hazards (PH) function is one which can be factored as

u (tux) 5 l(x)u0 (t)

(11)

where u0 (t) is the called the baseline hazard (see Cox, 1972). Under the Weibull specification u0 (tux ij ) 5 a t a 21 and l(x ij ) 5 e 2x ij b . It is possible to estimate b without assigning a functional form to u0 .

6.1. Partial likelihood Denote the joint distribution of call duration and a vector of covariates as (T,X) with realizations ((t 1 ,x 1 ), . . . ,(t N ,x N )). These realizations can be equivalently expressed as ((r 1 ,o 1 ,x 1 ), . . . ,(r N ,o N ,x N )) where r i and o i are the rank and order statistics derived from T. For example, r i tells us the position of the ith call when 11

Their data includes both business and residential calls which may explain their higher elasticity estimate.

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calls are ordered by duration while o i tells us the number of minutes of the ith shortest call. This relationship also works in reverse,– from the rank and order data we can derive T. If we denote x ( j ) 5 (x 1 , . . . ,x j ) and similarly define r ( j ) and o ( j ) (x ( 0 ) ,o ( 0 ) ,r ( 0 ) we will define as null), then we can factor the full likelihood as f((t 1 , . . . ,t N )u(x 1 , . . . ,x N )) 5 f((r 1 ,o 1 ), . . . ,(r N ,o N )u(x 1 , . . . ,x N ))

P f(r ,o ux ,r 5 P f(o ux ,o N

5

i 51

i

i

(i21)

i

,o (i 21))

N

(i 21)

i 51

i

i

P f(r ux ,o ,r N

,r (i 21 ))

i 51

i

i

i

(i 21)

)

(12)

using Bayes’ rule. The final term in the last equality is the partial likelihood of T conditional on X. When the hazard function of T is of the proportional hazard type, we can factor the partial likelihood in a convenient way. Suppose we index in ascending order and consider the first (and smallest) call duration o 1 :

u (o( 1 ) ux 1 ) f(r 1 uX,o 1 ,r (0 )) 5 ]]]] N u (o( 1 ) uxi )

O

i 51 x1b

e u0 (o 1 ) 5 ]]]] N e x i bu0 (o 1 )

O

(13)

i 51 x1b

e 5 ]] N exi b

O

i 51

which is simply the probability that the first call is the shortest given that a call of length o 1 is made. We then specify the probability of any given rank recursively as: exi b f(r i uX,o i ,r (i 21)) 5 ]] exj b

O

(14)

j ,i

Note that we have cancelled the baseline hazard terms in the likelihood by focusing on the order information. Hence we can maximize the partial likelihood function based on Eq. (14) and estimate b without specifying a parametric form for u0 . Cox (1975) demonstrates that, like traditional maximum likelihood estimators, estimators based on the partial likelihood function are consistent and asymptotically normal. Given parameter estimates for the partial likelihood, the baseline hazard function can be estimated non-parametrically.

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Table 4 Parameter estimates. Cox proportional hazard b : call duration Variable

p¯ income n distance (distance)2 num (num)2 network

Day

Evening

Night

Coef.

Std. error

Coef.

Std. error

Coef.

Std. error

20.186 0.0023 0.8e25 0.228 20.0381 0.0079 20.7e24 20.0046

0.0049 0.0013 0.2e23 0.0195 0.0100 0.9e23 0.1e24 0.6e23

20.625 0.2e23 20.0011 0.318 20.068 0.0033 20.4e24 20.0033

0.0188 0.0013 0.2e23 0.0157 0.0074 0.7e23 0.5e25 0.9e23

20.529 0.0036 20.0015 0.374 20.0728 0.7e23 20.1e24 20.0023

0.0175 0.0013 0.2e23 0.0149 0.0067 0.8e23 0.7e25 0.9e23

N538 962

N539 530

N539 499

Estimates for the parameters of Cox proportional hazard model are reported in Table 4. Qualitatively, these estimates are similar to those of the previous section. Both sets of estimates have the same signs, and are generally of comparable magnitudes. Price and income coefficients show particularly close agreement. The most noticeable difference between the parametric and the semi-parametric estimates is that the semi-parametric estimates indicate a much stronger effect for the call distance variables. Estimated baseline survivor functions for day, evening, and night calls are shown in Fig. 5. While these functions closely resemble the parametric survivor functions in Fig. 4, they suggest greater differences between the conditional distributions of day calls and evening or night / weekend calls than is implied by the parametric analysis. Overall, the semi-parametric analysis largely confirms the findings of our parametric analysis, particularly with regard to the effects of price and household income on call durations. However, discrepancies between the two approaches suggest that relaxing the strong parametric assumptions of the Weibull specifications permits a more accurate representation of the conditional distribution of call durations.

7. Conclusion In this investigation of residential calling patterns, we leveraged the advantages of both parametric and nonparametric statistical methods. Non-parametric models allow robust estimation of call number and duration distributions, but are limited as a means of describing conditional distributions with even a moderate number of covariates. In contrast, parametric models are well suited to describing conditional distributions, but are not robust to specification error. We used non-parametric

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Fig. 5. Non-parametric survivor functions: Cox PH model.

analysis to explore broad characteristics of residential calling patterns, and then used the information learned from this analysis to specify a parametric model. We then verified the accuracy of our parametric specification by comparing it to a semi-parametric Cox proportional hazard model. From the non-parametric analysis, we were able to examine the effects of general attributes such as the time of day and distance category of calls by estimating conditional distributions. However, we did not use non-parametric techniques to explore the effects of demographic covariates. To investigate the relationship between calling patterns and demographic and price variables, we developed a parametric statistical model. The specification of this model was driven largely by the results of our non-parametric analysis. In particular, our decisions to use a Weibull hazard function, and to estimate separate duration models for day, evening, and night / weekend calls grew out of our experience estimating non-parametric hazard functions. In order to test the robustness of the Weibull specification of the baseline hazard rate, we estimated a semi-parametric Cox proportional hazard model. We found that although the Weibull specification for call duration captures the qualitative features of the data and appears to accurately capture the effects of economic variables on call durations, it imposes

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strong functional form restrictions on the shape of the hazard function that may not be justified. Chief among our empirical findings is the qualitatively similar characteristics of night and evening calls, the inelastic response of call duration to price, and the lack of evidence for repeated calls by a household to the same location. We also find that call duration is modestly increasing with distance and that adolescents and teenagers in a household correspond to a lower number of long distance calls. An important limitation is this analysis is that it is fundamentally descriptive in the sense that it is only able to illuminate the relationship between economic and demographic variables and calling patterns, rather than explain how household calling decisions are made. A more structural analysis of the way households allocated telephone calls across time could go a long way toward increasing the utility of empirical models of residential calling patterns, and could provide important insights into the differing characters of peak and off-peak calling patterns.

Acknowledgements We thank PNR Associates, and Paul Rappaport in particular for making the Bill Harvesting dataset available to us.

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