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Amsterdam School of Economics
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Churn behaviour of charity donors
Rikke Logeman (11325429)
July 14, 2017
MSc in Econometrics Free Track
Supervisor: dr. J.C.M. van Ophem Second reader: mw. E. Aristodemou
Abstract
In this study, we investigate churn determinants of charity donors having a subscription to donate monthly to the Dutch charity War Child. We perform duration analyses by estimating several proportional hazard models without imposing restrictive assumptions on the distributional form of the baseline hazard. Our models include the size of the donations to which the donor committed as an explanatory variable, which we allow to be endogenous by building a model that simultaneously estimates the duration of donating and the size of the donations. When ignoring the possible endogeneity of this variable, we find the size of the donations to negatively affect the duration of donating. However, when we do take the possible endogeneity of this variable into account, we find that there are indications that there is no significant effect of the size of the donations on the duration of donating.
Statement of Originality
This document is written by Rikke Logeman who declares to take full responsi-bility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those men-tioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
1 Introduction 1 2 Literature Review 2 2.1 Charitable Donations . . . 2 2.2 Duration Analysis . . . 4 3 Methodology 6 3.1 Duration Data and Censoring . . . 63.2 Kaplan-Meier Estimator . . . 6
3.3 Proportional Hazard Model . . . 7
3.3.1 Cox Proportional Hazard Model . . . 8
3.3.2 Proportional Hazard Model with Piecewise Constant Base-line Hazard . . . 9
3.3.3 Unobserved heterogeneity . . . 11
3.4 Simultaneous estimation of duration and the size of the donations 13 4 Data 14 4.1 Data description and summary statistics . . . 14
4.2 Kaplan-Meier estimates . . . 17
5 Results 18 5.1 Estimation of the duration of donating . . . 18
5.1.1 Effects of time-invariant covariates on the duration of donating 19 5.1.2 Effects of time-varying covariates on the duration of donating 20 5.1.3 Unobserved heterogeneity . . . 21
5.2 Simultaneous estimation of the duration of donating and the amount of money donated . . . 27
6 Conclusion 30
7 Limitations and Recommendations 31
1
Introduction
Giving to the needy and donating to charitable organizations are common prac-tices in many societies of the world today (Mohanty, 2011). However, nowadays it is difficult for charities to raise donations, because changes in social, political and economic environments in many countries around the world have resulted in declining support to charities by governments. Because governments are saving on expenditures to non-profit organizations, charities depend more on individual donors for fundraising. Due to this new development, it is of strong interest to non-profit marketers to understand the individual donor and what motivates them to contribute to charities.
Most research on charitable donations done so far is focused on why people do-nate and what the impact is of charitable giving on both the society and on the people living in it. While most studies focus on the frequency of doing individual donations, in this study duration is at the center. We focus on the determinants of the decision to stop donating by estimating several proportional hazard models without imposing restrictive assumptions on the distributional form of the base-line hazard. Instead of investing in targeting new potential donors, it might be profitable for charities to invest in preventing their current donors from stopping with donating. The proportional hazard models we estimate include the size of the donations to which the donor committed as an explanatory variable. We allow that this variable is endogenous if there are unobservable individual characteris-tics affecting the size of the donations that also affect the duration of donating. To account for possible endogeneity of this variable, we build a model that si-multaneously estimates the duration of donating and the size of the donations. The model is similar in spirit to the model of Bratti and Miranda (2011), who propose an estimator for models in which an endogenous dichotomous treatment affects a count outcome in the presence of either sample selection or endogenous participation. In addition to the different types of both the dependent variable and the possibly endogenous explanatory variable, our approach differs from that of Bratti and Miranda in the sense that we integrate out the common element numerically, whereas they use maximum simulated likelihood.
We apply our models to a data set of 10,000 donors of the charity War Child. This is a Dutch charity committed to children who live - or have lived - in war zones. In 2016 they raised over e28 million to be able to help over 380,000 children in
14 Third World countries.
When ignoring the possible endogeneity of the size of the donations in the du-ration analysis, we find the relation between this variable and the instantaneous probability of unsubscribing to be significantly positive. This implies that donors who committed to donate a higher monthly amount of money are expected to have a shorter duration of donating. However, when we do take the possible endogene-ity of this variable into account, the results report that there are indications that there is no significant effect of the size of the donations on the duration of do-nating. Hence, War Child can not improve the duration of donating by changing their policy regarding the amount of money they ask for. Since there are indi-cations that the size of the donations has no effect on the duration of donating, donors who give a high amount of money have a greater expected lifetime value than donors who give a low amount of money, all else being equal1.
The structure of the paper is as follows. In the next section we report an overview of the existing literature that is relevant for this study. In section 3 the method-ology is presented and section 4 gives an overview of the data. We present our results in section 5, discuss the limitations of this research in section 7 and draw conclusions in section 6.
2
Literature Review
2.1
Charitable Donations
Over the past decade, research on charitable donations has focused on planned giv-ing, donor characteristics, anticipation of intrinsic benefits and giving behaviour. This literature review focuses on giving behaviour and donor characterstics as these are most relevant for this study.
Economists have offered several theories to explain giving behaviour. The three most discussed models are the private consumption model, the impure altruist model and the investment model. The private consumption model states that individuals donate for intrinsic rewards and for the ‘warm glow’ they experience from their act of charitable donations (Adreoni, 1990). Based on the impure al-truist model, the simple act of giving is accredited to the human helping behavior
1Assuming the expected lifetime value of a donor is the total sum of the donations done during the subscription.
(Harbaugh, 1998). Besides donating money, this model also considers another way to contribute to charities: by donating time, i.e. volunteering. This is also the case with the investment model. In this model, the attributes gained from giving by donors and/or volunteers are, among others: status, skills and labor market experience. This is especially the case for giving time instead of money. Handy and Katz (2008) try to solve the puzzle associated with the observation that simultaneous volunteering and donating money is extremely common: if the purpose of a giving individual is to maximize the effect of the donations, then the donations should be done as effectively as possible, which implies that an indi-vidual should donate either time or money, but not both. Based on the private consumption model, they explain this by arguing that volunteering may give in-dividuals a ‘hands-on warm glow’ that is not present or is of a different dimension than the donation of money. However, they do not formally test this hypothesis. Next to economic studies, giving behaviour is also a subject of social-psychology studies. This field of research has shown that people are not always self-seeking, but may be driven by the empathy to help out others (Eveland and Crutchfield, 2007).
Besides giving behaviour, research has focused on donor characteristics. Schlegel-milch, Diamantopoulos and Love (1997) find that donors tend to think that they are more generous and are more likely to be volunteers than non-donors. More-over, they find that people donate more as they become older up until the age of 65, at which point the donating behaviour starts to decrease. However, this could be driven by the level of income, since people generally stop working around that age. Sze, Gyurak, Goodkind and Levenson (2012) report that as people become older, their emotional empathy and altruistic behaviour increase and they argue this is likely due to a greater reactivity to the need to help others.
The findings of previous research regarding the gender of donors are not com-pletely consistent. Dvorak and Toubman (2013) report that women are more likely to be donors and that among donors, women tend to give more frequently and donate more of both time and money. On the other hand, Braus (1994) finds that men make larger donations than women. Lwin, Phau and Lim (2014) con-clude that gender does not influence an individual’s tendency to donate.
Other donor characteristics that are discussed in the literature are income, the level of education and whether a donor has children. The majority of the literature
reports that donors are more likely to have a higher income than non-donors (e.g. Schlegelmilch et al., 1997). On the other hand, Bennet (2003) reports that people with a lower income donate more to the needy people, since they empathize more with them. The level of education and whether a donor has children are often found not to influence charitable donations (e.g. Schlegelmilch et al., 1997; Lwin et al., 2014).
This study adds to the existing literature by investigating why people stop do-nating - rather than why people start dodo-nating - using subscription data from the Dutch charity War Child. Moreover, to the best of our knowledge, no literature has been published in which the duration of donating and the size of the donations are simultaneously estimated.
2.2
Duration Analysis
There are several ways to analyze churn, which is stopping with donating in this study. In binary duration event models, the outcome variable is whether or not someone has churned by some prespecified cutoff duration time Tc. However, these
models do not incorporate all of the information available to the analyst. This issue restricts the appropriateness of binary duration event models to a limited set of scenarios: namely to the case where the only event of interest is whether an individual exceeds the prespecified duration Tc. Another way to model churn
is to perform a simple linear regression on the duration time. A drawback of this method is that it does not incorporate censoring. The most convenient way to analyze churn is by duration models, which do take account of censoring.
Helsen and Schmittlein (1993) compare the different ways to analyze churn men-tioned above, using a data set of weeks during which someone keeps buying a cer-tain product. The duration model they estimate is a proportional hazard model for which they assume a parameteric distribution of the baseline hazard: the Weibull distribution. In this model the duration is modeled via the hazard rate, which is the instantaneous probability that a person churns, given that this per-son has not churned up until now. As benchmarks they estimate a probit model and regression model that consider completed spells only. They report that the proportional hazard model gives more reliable estimates, with greater face valid-ity, than the other two methods. Moreover, they find evidence that the forecasts generated by the proportional hazards model are more accurate.
done by using the proportional hazard model. It is often preferred not to impose restrictive assumptions on the distributional form of the baseline hazard. Dekimpe and Degraeve (1997) apply a proportional hazard model with a semi-parametric baseline hazard: the only assumption they make is that the baseline hazard is constant within certain intervals. Another model that does not impose restrictive assumptions on the distributional form of the baseline hazard, is the Cox pro-portional hazard model (Cox, 1972). Estimating this model results in consistent estimates of the coefficients of the included covariates on the hazard rate, without having to specify the baseline hazard. These proportional hazard models will be further explained in section 3.
The proportional hazard model relies on the assumption that the covariates have a proportional effect on the duration. An alternative model, which does not rely on this assumption, is the accelerated failure time model. The accelerated failure time model is mostly used in the fields of engineering and medicine (e.g. Lam-bert, Collett, Kimber and Johnson, 2004). In this model, the natural logarithm of the survival time is regressed over the covariates. Wei (1992) argues that the accelerated failure time model could be a useful alternative for the Cox propor-tional hazard model. His argument is that the estimates of the coefficients of the included covariates of the Cox proportional hazard model are hard to interpret, because the estimates have an effect on the hazard rate and not directly on the duration2. Since in the accelerated failure time model the outcome variable is the natural logarithm of the duration, the coefficients of this model are easier to interpret.
In order to be able to distinguish true state dependence from heterogeneity across individuals, proportional hazard models should be corrected for unobserved het-erogeneity (Lancaster, 1979; Heckman and Singer, 1984). Ignoring unobserved heterogeneity may cause the coefficients of the included covariates to be biased and inconsistent (Manton, Singer and Woodbury, 1992). Jain and Vilcassim (1991) model the unobserved heterogeneity component semi-parametrically. This method was earlier proposed by Heckman and Singer (1984), who conclude that for a given parametric form of the baseline hazard, the results tend to be very sensitive to
2This argument is true for the Cox proportional hazard model. However, if one is willing to estimate the baseline hazards semi-parametrically when using a proportional hazard model, e.g. by assuming the baseline hazards are piecewise constant, it is possible to calculate average marginal effects on the expected duration, which are also easy to interpret. Moreover, accelerated failure time models are predominantly fully parametric: a probability distribution is specified for the natural logarithm of the duration. The Cox proportional hazard model does not make such an assumption.
the form of the distribution of the unobserved heterogeneity component. Other research indicates that the specification of the distribution of the unobserved het-erogeneity component is not as crucial as a flexible specification of the baseline hazard (e.g. Han and Hausman, 1990; Manton, Stallard and Vaupel, 1986). Based on these findings, Dekimpe and Degraeve (1997) model the baseline hazard semi-parametrically and the unobserved heterogeneity component semi-parametrically. In this study, among other models, we estimate a model that corrects for unobserved heterogeneity, following the approach of Dekimpe and Degraeve (1997).
3
Methodology
3.1
Duration Data and Censoring
We start by modelling the duration of donating, the time in days from the day of the first donation until no longer donating, i.e. defecting, denoted by the non-negative continuous random variable T .
Theoretically there are n independent realisations of T : t∗1, t∗2, . . ., t∗n. However, after a certain time C we do not observe observe these duration times anymore, because the data is right-censored. Some spells will be completed by this time, but others will not be completed yet and in that case all we know is that the length of the spell is at least the censoring time. For each individual we observe ti = min(t∗i, ci) and δi = I(t∗i ≤ ci), indicating whether the observation is censored
or not: δi = 1 if the observation is not censored and δi = 0 otherwise.
3.2
Kaplan-Meier Estimator
The survival distribution can be estimated non-parametrically by the Kaplan-Meier estimator. First, the defection times of the n observations are ordered: t1 ≤ t2 ≤ . . . ≤ tn. The chance of survival is then estimated by
ˆ S(t) = Y j|tj≤t rj− dj rj (3.1)
where dj is the number of spells that end in the interval [tj−1, tj] and rj is the
number of people at risk at time tj−1. However, this method does not allow for
3.3
Proportional Hazard Model
The proportional hazard model assumes that the hazard function, which is a funcion of time and explanatory variables, can be split up into two parts: the baseline hazard function, not dependent on explanatory variables, and a scaling factor, dependent on explanatory variables:
λ(t|x, α, β) = λ0(t, α)φ(x, β) (3.2)
where we choose the usual exponential form for the relative risk function φ(x, β) = exp(x0β), so that the model becomes
λ(t|x, α, β) = λ0(t, α)exp(x0β) (3.3)
The survival distribution is then given by Cameron and Trivedi (2005):
S(t|x, α, β) = S0(t, α)exp(x 0β)
(3.4) Parametric models impose a specific distribution on the baseline hazard λ0(t, α).
If one is willing to make such an assumption, estimates of β can be obtained by maximizing the likelihood. The likelihood is:
L(α, β) = n Y i=1 λ(ti|xi, α, β)δiS(ti|xi, α, β) = n Y i=1 λ(ti|xi, α, β)δiexp(−Λ(ti|xi, α, β)) (3.5) l(α, β) = n X i=1 (δilog(λ(ti|xi, α, β)) − Λ(ti|xi, α, β)) (3.6) where Λ(t) = t R 0
λ(s)ds is the integrated hazard function. However, misspecifica-tion of the distribumisspecifica-tion of the baseline hazard quite likely leads to inconsistent estimates of β.
3.3.1 Cox Proportional Hazard Model
Cox (1972) proposed a semi-parametric method in which β is estimated without the need to specify λ0(t, α). Hence, no assumption on the distribution of the
baseline hazard function is necessary. Since equation (3.6) cannot be estimated without this assumption, Cox (1972) introduced the partial likelihood method to estimate β. Ignoring ties for the moment, the k uncensored churn times are ordered: t1 < . . . < tk, the remaining n − k individuals are right-censored. Let
j denote the individual churning at tj, xl denote the covariate vector for the lth
individual and R(tj) the set of individuals at risk just before the j-th defection
time. Cox (1972) shows that:
L(β) = k Y j=1 exp(xj0β) P l∈R(tj)exp(xl0β) = n Y i=1 exp(xi0β) P l∈R(ti)exp(xl0β) !δi l(β) = n X i=1 δi xi 0 β − log X l∈R(ti) exp(xl0β) (3.7)
Maximizing this likelihood leads to consistent estimates of β, without having to specify λ0(t, α). As can be seen from equation 3.7 the likelihood only considers
uncensored durations, but according to Cox (1972) this is no problem, since the censored observations are included in the risk set.
When there are ties among the uncensored defection times, i.e. multiple individ-uals stop donating at the same time, the partial likelihood is adjusted by using the average likelihood that arises through breaking the ties in all possible ways. This is done as follows. Suppose that t1 < . . . < tk are the distinct defection
times and that dj items defect, i.e. stop donating, at time tj, j = 1, . . . , k. Let
D(tj) = {j1, . . . , jdj} be the set of labels of individuals that defect at tj and let
Qj be the set of dj! permutations of the labels {j1, . . . , jdj}. Let P = (p1, . . . , pdj)
be an element in Qj and R(tj, P, r) = R(tj) − {p1, . . . , pr−1}. Kalbfleisch and
Prentice (2002) show that the average partial likelihood contribution at tj which
arises from breaking the ties in all possible ways is then
1 dj! exp(sj0β) X P ∈Qj dj Y r=1 X l∈R(tj,P,r) exp(xl0β) −1 (3.8)
where sj = dj P
i=1
xi is the sum of the covariates of individuals observed to stop
donating at tj. The corresponding average partial likelihood is proportional to
k Y j=1 exp(sj 0 β) X P ∈Qj dj Y r=1 X l∈R(tj,P,r) exp(xj0β) −1 (3.9)
which we approximate using the Breslow method (Peto, 1972; Breslow and Crow-ley, 1974), by L(β) = k Y j=1 exp(sj0β) P l∈R(tj)exp(xl0β) dj (3.10)
In addition to the effect of covariates that do not change over time, we are inter-ested in the effect of time-varying covariates. We extend the model as follows:
λ(t|x(t), α, β) = λ0(t, α)exp(x(t)0β) (3.11)
In order to estimate this model, each spell is split up into multiple subspells at the points where a covariate changed.
3.3.2 Proportional Hazard Model with Piecewise Constant Baseline Hazard
Like the Cox proportional hazard model (Cox PH model, hereafter), the pro-portional hazard model with piecewise constant baseline hazard (PCH model, hereafter) is of the form
λ(t|x, α, β) = λ0(t, α)exp(x0β) (3.12)
Whereas the Cox PH model estimates β without specifying the baseline hazard, the PCH model does estimate the baseline hazard, but without assuming a distri-butional form on this baseline hazard. Hence, both models assume an exponential form of the relative risk function and do not make restrictive assumptions on the distributional form of the baseline hazard, from which we expect the estimated coefficients of these models to be very similar.
We partition duration into J intervals with cutpoints 0 = τ0 < τ1 < . . . < τJ = ∞.
In equation (3.12), λ0(t, α) is assumed to be constant in an interval, but is allowed
to vary across intervals:
λ0(t, α) = α1 for t ∈ (0, τ1] α2 for t ∈ (τ1, τ2] .. . αJ for t ∈ (τJ −1, τJ] (3.13)
Thus, we model the baseline hazard λ0(t, α) using J parameters α1, . . . , αJ, each
representing the baseline hazard in one particular interval.
We split the spell of each individual into multiple subspells, one for each of the intervals defined in (3.13), up to and including the interval in which the individual churned. Let tij denote the time spent donating by the ith individual in the
jth interval, δij indicate whether an individual i churned in interval j and j(i)
indicate the interval where individual i churned3. We can write the hazard rate
of individual i in interval j as
λij = αjexp(xi0β) (3.14)
where αj is defined as above and exp(xi0β) is the relative risk for an individual
with covariate values xi. It follows that the loglikelihood of the PCH model is
l(α, β) = n X i=1 j(i) X j=1 (dij(logαj+ xi0β) − tijαjexp(xi0β)) (3.15)
When maximizing the loglikelihood, we replace αj = exp(γj) to ensure positivity
of the baseline hazards and maximize across γ and β.
To obtain a more convenient interpretation of the estimated coefficients, we cal-culate the average marginal effects on the expected duration of donating for the PCH model. The expected duration of donating is estimated by
3If an individual did not churn, i.e. the observation is right-censored, j(i) indicates the last interval that we observe for individual i.
ˆ E[T |x, α, β] = ∞ Z 0 t ˆf (t|x, α, β)dt (3.16)
where ˆf (t|x, α, β) = ˆλ(t|x, α, β) ˆS(t|x, α, β). Using equations (3.4), (3.12) and (3.13), we find that ˆ f (t|x, α, β) = ˆ
α1exp(x0β − ˆˆ α1t · exp(x0β))ˆ for t ∈ (0, τ1]
ˆ
α2exp(x0β − exp(xˆ 0β)( ˆˆ α1τ1+ ˆα2(t − τ1))) for t ∈ (τ1, τ2]
.. . ˆ αJexp(x0β − exp(xˆ 0β)( ˆˆ α1τ1+ ˆα2(τ2− τ1) + . . . + ˆαJ(t − τJ −1))) for t ∈ (τJ −1, τJ] (3.17) The average marginal effect on the expected duration of donating of covariate k is calculated as 1 n n X i=1 ∂ ˆE[Ti|xi, α, β] ∂xk (3.18) 3.3.3 Unobserved heterogeneity
Unobserved heterogeneity could arise when there are variables, which are not included in the dataset, that have a significant effect on the hazard rate. We extend the PCH model given by equation (3.12) as follows:
λ(t|x, α, β, v) = λ0(t, α)exp(x0β)v, v > 0 (3.19)
λ0(t, α) = λ(t|x, α, β, v)exp(−x0β)v−1 Λ(t|x, α, β, v) = t Z 0 λ0(u|x, α, β, v)du = exp(−x0β)v−1 t Z 0 λ(u|x, α, β, v)du log [Λ(t|x, α, β, v)] = −x0β − logv + (3.20) where = logRt 0
λ(u|x, α, β, v)du, and v is assumed to be independent of the re-gressors and of censoring time. The multiplicative heterogeneity assumption is rather special, but it is mathematically convenient and more attractive than an additive error, which could violate nonnegativity of t.
We estimate the model by integrating out the heterogeneity term, v, for which we specify a parametric distribution f (v) with a support on the positive line, to respect the property that v > 0: the lognormal distribution4. Starting from
equation (3.5), we obtain L(α, β) = n Y i=1 ∞ Z 0 (λ(ti|xi, α, β, vi))δiexp(−Λ(ti|xi, α, β, vi)) f (vi)dvi (3.21)
Following the same steps as in the previous section, we find that the loglikelihood of the PCH model with unobserved heterogeneity is
l(α, β) = n X i=1 j(i) X j=1 log ∞ Z 0
(αjviexp(xi0β))dijexp(−tijαjviexp(xi0β))
f (vi)dvi
(3.22) where we replace αj = exp(γj) and σv = exp(sv), and maximize across γ, β and sv.
4A flexible specification of the unobserved heterogeneity component based on the method of Heckman and Singer (1984) is to some extent less restrictive, since this method allows for different values of the unobserved heterogeneity component without specifying a parametric distribution. However, in practice, this is often limited to two or three different values. Based on this and due to the complexity of the final model, we decide to follow the approach of Dekimpe and Degraeve (1997).
3.4
Simultaneous estimation of duration and the size of
the donations
We aim to develop a model that simultaneously estimates the duration of donating and the size of the donations. The model is based on the model of Bratti and Miranda (2011) and combines the model of section 3.3.3 with a linear regression on the size of the donations5:
λ(t|x, α, β, w) = λ0(t, α)exp(x0β + w) (3.23)
y = z0θ + q (3.24)
where the elements of equation (3.23) are as defined before, y is the natural log-arithm of the monthly amount of money donated, z is a vector of covariates, θ is a conformable vector of coefficients and q is a residual term.
Correlation among λ(t|x, α, β, w) and y is allowed by imposing some structure on the residuals of equations (3.23) and (3.24),
q = ψw + ξ (3.25)
where we assume w ∼ N (0, σ2
w) and ξ ∼ N (0, σ2ξ). The parameters ψ ∈ R, σ2w,
and σ2
ξ are to be estimated along the other parameters.
We require the covariates to be all exogenous and impose the distributional con-dition
D(w|x, z, ξ) = D(w) (3.26)
where D(·) stands for ‘distribution of’.
The correlation between the error terms in models (3.23) and (3.24) is a function of
ψ, σ2
w and σ2ξ. In particular, the model implies the following correlation structure:
ρq,w = ψσ2 w q σ2 w(ψ2σw2 + σ2ξ) (3.27)
Let Py(yi|wi) denote the conditional probability of y = yi given wi. To simplify
notation, we do not explicitly write the conditioning on observable variables. The loglikelihood of this model is given by
(3.28) l(α, β, θ, ψ, σw, σξ) = n X i=1 j(i) X j=1 log ∞ Z −∞ (αjexp(xi0β + wi))dij · exp(−tijαjexp(xi0β + wi))Py(yi|wi) f (wi)dwi
where we replace αj = exp(γj), σw = 0.0001 + exp(sw) and σξ = 0.0001 + exp(sξ),
and maximize across γ, β, θ, ψ, sw and sξ.
4
Data
4.1
Data description and summary statistics
The data set is constructed from the Customer Relationship Management sys-tem of the Dutch charity War Child. This charity works to ensure that children affected by conflict and violence can realise their fundamental rights and build a better future, both for themselves and their communities. Their programmes, methodologies and fundraising activities are all designed to deliver a meaning-ful positive change in the lives of these children by providing them protection, psychosocial support and education. Together with partners, caregivers and com-munities they improve both the resilience and well-being of these children. With their evidence-based interventions they are the expert in their field, while they inspire their partners and host countries to expand their interventions. Together, they create a lasting impact in order to reach as many children as they can. In 2016 War Child made a difference in the lives of 381,830 children and young peo-ple forced to grow up with the effects of violent conflict, operating in 14 countries. They have developed a ‘Theory of Change’ which guides the development and implementation of a comprehensive care system in the countries where they work. Their R&D activities help them to build the necessary evidence to improve their interventions and increase the scale of their impact.
In 2016 War Child raised a total of e28,668,902 coming from several income sources, e.g. individual donors who do either separate or structural gifts, compa-nies who do either separate or structural gifts, individuals who organize actions, government grants, etc.6. This research focuses on their ‘Friends’, which are
indi-vidual donors who have a subscription to do monthly donations to War Child. The complete data set consists of 224,281 donors with a total of 234,560 spells ranging from 1997 to 2017. Hence, about 4.6% of the donors ended their subscrip-tions and started donating again later, which results in multiple spells belonging to the same donor. Since we do not want to ignore dependence between multiple spells of the same donor, we decide to only consider first spells of donors. More-over, since less than 0.1% of the donors changed the size of the donations during the spell, we decide to exclude those donors from the analysis and assume the size of the donations to be time-invariant.
The first response variable in this study is the duration or time until defection, which is the time until a donor ends his or her subscription, i.e. churns. Donors can defect on a continuous basis and the duration is defined as the number of days from the day of the first donation until a donor churns. The size of the monthly donations, measurd in e, is included as an explanatory variable in the duration models. We account for possible endogeneity of this variable by simultaneously estimating the duration of donating and the size of the donations.
Variables which do not vary during a subscription contained in the data set are age, gender, channel of acquisition and whether the donor had already done a do-nation to War Child before starting his/her subscription7. Moreover, we include
the season in which the donor started his/her subscription.
Besides time-invariant variables, the data set contains variables that vary over time. War Child sometimes calls donors who have a subscription to do monthly donations to ask them to do an extra separate gift on top of their structural do-nations, for example when they have organized an action. Telemarketing is not the only channel through which they ask for extra separate gifts, they also send their donors letters in which they ask for extra gifts now and then, i.e. via direct mailing. It is hypothesized that asking for more money triggers a reaction that is undesired: that donors end their structural donations. In order to test this
6For more information, go to https://www.warchild.nl/.
7Since we only consider first spells, the donation done before starting the subscription must have been a separate donation.
hypothesis, we include these variables in the Cox PH model. A limitation of the data set is that we do not have information on the responses of donors on these requests, i.e. whether they did an extra gift or not after being asked for this. However, the data set does contain points in time at which donors did extra gifts on top of their structural donations, but only extra gifts which donors did on their own initiative, i.e. without being requested to do so. We expect these to positively affect the duration of donating: donors who do extra gifts on their own initiative are generally more committed to War Child and hence are expected to have a longer duration of donating compared to donors who do not do extra gifts on top of their structural donations. Moreover, it is hypothesized that the effect of the time-varying variables depends on the history of these variables. For ex-ample, being called for the first time might have a different effect on the duration of donating than being called for the fourth time. Hence, we include cumulative values of the time-varying variables in the Cox PH model.
Unfortunately, other variables that are expected to affect the duration of donat-ing, such as income, education level, marital status and the number of children are not available. Hence, it is hypothesized that unobserved heterogeneity is present in the data. Moreover, since we expect those unobserved factors to also influence the size of the donations, we expect the size of the donations to be an endogenous variable in the duration models.
Due to computer time limitations, we decide to draw a random subset of 10,000 spells. Based on the Likelihood Ratio tests performed in Appendix A, we conclude that the random subset is a good representation for the complete data set. From this point on, all descriptive statistics and results are based on the random subset of 10,000 spells. We assume the baseline hazard to be piecewise constant in intervals of 3 months for the first 5 years and to be constant from that point on8.
Tables 1 and 2 report the definitions and descriptive statistics of all variables. The censoring percentage is 38.05%.
8We also estimated the PCH model making different assumptions on the baseline hazard, e.g. to be piecewise constant in intervals of 3 months for the first 15 years. However, from 5 years on the estimates of the baseline hazard were equal up to 4 decimals behind the komma, from which we assume the baseline hazard to be constant after 5 years.
Table 1: Variable definitions and descriptive statistics of time-invariant variables
Variable Definition Mean SD
Dependent variables
Duration Time in days until defection 1561.731 1407.758 Amount Size of the monthly donations ine 6.501 3.473
Explanatory variables
Female A dummy indicating gender 0.596
-Age18.29 18≤age≤29 0.185 -Age30.39 30≤age≤39 0.228 -Age40.49 40≤age≤49 0.236 -Age50.59 50≤age≤59 0.176 -Age60.69 60≤age≤69 0.099 -Age70.older age≥70 0.077
-Acq.TM Donor is acquired via telemarketing 0.209 -Acq.DD Donor is acquired via door to door 0.430 -Acq.Online Donor is acquired online 0.089 -Acq.TV Donor is acquired via television 0.151 -Acq.Concert Donor is acquired via a concert organized by War Child 0.066 -Acq.Else Donor is acquired else 0.055 -Donated.before Indicating whether the donor had already done a
dona-tion to War Child before starting his/her subscripdona-tion.
0.253 -Season.start.spring Donor started his/her subscription in March, April or
May
0.276 -Season.start.summer Donor started his/her subscription in June, July or
Au-gust
0.248 -Season.start.autumn Donor started his/her subscription in September,
Octo-ber or NovemOcto-ber
0.194 -Season.start.winter Donor started his/her subscription in December,
Jan-uary or FebrJan-uary
0.282
-Table 2: Variable definitions and descriptive statistics of time-varying variables
Variable Definition Minimum
per spell Maximum per spell Average per spell Explanatory variables
TM.request Points in time at which a donor was called by War Child and requested to do an extra gift on top of the structural donations
0 8 1.280
DM.request Points in time at which a donor was sent post by War Child and requested to do an extra gift on top of the structural donations
0 25 4.294
Extra.gift Points in time at which a donor did an extra gift to War Child on top of the structural donations, without being asked for this
0 15 0.111
4.2
Kaplan-Meier estimates
By plotting the Kaplan-Meier curves, the chance of survival of the donors can be displayed over time. The Kaplan-Meier curves are shown in figure 1. These estimates suggest that after 5 years approximately 45% is still donating. It seems that women are more likely to churn than men (figure 1b), which also holds for the group acquired via door to door compared to the other acquisition channels
(figure 1c). Furthermore, figure 1d suggests that donors who had already donated to War Child before starting their subscription are less likely to churn than donors who had not. Since the Kaplan-Meier curves are purely descriptive statistics, the formal models described in section 3 have to be estimated in order to test whether these statements hold when other explanatory variables are allowed for.
Figure 1: Kaplan-Meier estimates of the data for the first 5 years with 95% confidence intervals
(a) Kaplan-Meier curve of all donors (b) Kaplan-Meier curves per gender
(c) Kaplan-Meier curves per acquisition channel
(d) Kaplan-Meier curves of donors who donated before vs. who did not
5
Results
5.1
Estimation of the duration of donating
In this section, we focus on the determinants of the duration of donating. Table 3 reports the results of models (1) and (2), which are Cox PH models without and with time-varying covariates, respectively. Additionally, we estimate a PCH
model: model (3). This model is extended to allow for unobserved heterogeneity, which is model (4). The results of models (3) and (4) are presented in tables 4 and 5: table 4 reports the estimates of the baseline hazards and table 5 reports the effects of the covariates on the duration of donating. We first discuss the results of the models not allowing for unobserved heterogeneity and then compare the results of these models to the results of model (4).
As expected, the results of models (1) and (3) are very similar. Moreover, when comparing these models to model (2), we conclude that the inclusion of time-varying covariates does not lead to remarkable changes in the coefficients corre-sponding to the time-invariant covariates.
For the PCH models, we calculated the average marginal effects (AME) of the covariates on the expected duration of donating. The Cox PH models do not estimate the baseline hazards, so we are unable to calculate the average marginal effects of the covariates on the expected duration of donating for these models. Since the coefficients corresponding to the time-invariant covariates are very sim-ilar for models (1)-(3) and we are mainly interested in the average marginal effect on the expected duraion of donating, rather than in the effects on the hazard rate, we discuss the effects of time-invariant covariates based on model (3).
When looking at the estimates of the baseline hazards in model (3), there seems to be a downward trend over time. Hence, the probability that a donor stops donating decreases over time.
5.1.1 Effects of time-invariant covariates on the duration of donating The estimated average marginal effect of the size of the donations on the expected duration of donating is -124 days and significant at the 1% level9. Hence,
accord-ing to the results of this model, an increase ofe1 in the amount of money donated per month results in an estimated average decrease of the expected duration of donating of approximately 4 months. However, this model does not account for possible endogeneity of this variable. This will be further investigated later. The results show that older donors have on average a longer expected duration of donating, except for donors of 70 and older. Moreover, women have on average a shorter expected duration of donating than men. The coefficients
correspond-9This effect is calculated by equation 3.18, with a deviation of the size of the donations of 1 unit.
ing to the seasonality dummies report that donors who started their subscription during spring have on average a lower expected duration of donating than donors who started their subscription in one of the other seasons.
The models include interaction effects between the acquisition channel and whether someone had donated to War Child before starting his/her subscription. We first consider donors who had not donated to War Child before. For those donors, the estimated average marginal effect of being acquired via door to door compared to being acquired via telemarketing on the expected duration of donating is -1328 days and significant at the 1% level. An explanation is that donors acquired via door to door are talked into subscribing without really wanting this and hence change their mind and unsubscribe. Except for the category concert, all other average marginal effects regarding the acquisition channel of the donor on the expected duration of donating are significantly positive. Considering donors who had donated to War Child before, the effect of being acquired via a concert on the duration of donating is negative and significant at the 1% level. Moreover, com-pared to a donor acquired via telemarketing who had not donated to War Child before, a donor acquired via television who had donated to War Child before has an expected duration of donating which is on average 1633+2640-2014=2259 days longer.
5.1.2 Effects of time-varying covariates on the duration of donating Model (2) of table 3 shows the results of the Cox PH model including time-varying covariates. The results show that the effect of asking donors to do extra separate donations on top of their structural donations depends on the channel through which this is done. Calling donors with such a request, i.e. via telemarketing, has a significant negative effect on the hazard rate, so this increases the expected duration of donating. Sending donors post with such a request, i.e. via direct mailing, has a significant positive effect on the hazard rate, so this decreases the expected duration of donating. As explained before, there are donors who do extra gifts on top of their structrual donations without being requested to do so. The results report that donors who do this have a significantly lower hazard rate than donors who do not do extra gifts. This result is intuitive: donors who do extra gifts on top of their structural donations are more commmitted to War Child than donors who do not and hence they have a longer expected duration of donating. As mentioned before, comparing the estimated coefficients of the time-invariant
covariates of this model with the coefficients of the time-invariant covariates es-timated by models (1) and (3), we conclude that these are very similar. Figure 2 displays the plots of the Cox-Snell residuals (Cox and Snell, 1968) of the Cox PH models without and with time-varying covariates. When the residuals are close to the diagonal, this implies that the model has a good fit. Eyeballing, we cannot conclude that one of the models has a better fit than the other based on these plots. However, the inclusion of time-varying covariates did decrease both the Akaike Information Criterion and the Bayesian Information Criterion of the model, which suggests that the Cox PH model including time-varying covariates has a slightly better fit.
Figure 2: Cox-Snell residuals of the Cox PH models with and without time-varying covariates
(a) Without time-varying covariates (b) With time-varying covariates
5.1.3 Unobserved heterogeneity
We extend the PCH model model (3) to allow for unobserved heterogeneity -model (4). Due to the high computational burden, we do not estimate these mod-els including time-varying covariates. In the previous section, we concluded that the coefficients of the time-invariant covariates are very similar for models (1)-(3). In this section we compare the results of model (4) to the results of model (3). The estimates of the baseline hazards are presented in table 4 and the estimates of the effects of the covariates on the duration of donating are presented in table 5. Comparing model (4) to model (3), we conclude that the estimates of the baseline hazards of model (4) are slightly lower than those of model (3). Moreover, the average marginal effects of the covariates on the expected duration of donating of model (4) are all greater in absolute value than those of model (3). For example,
based on model (3) the average marginal effect of being female on the expected duration of donating is -472 days, whereas based on model (4), this effect is -616 days. Hence, based on model (4) the covariates have a greater effect on the ex-pected duration of donating than based on model (3).
The estimate of the standard deviation of the unobserved heterogeneity term is ˆ
σv = exp(ˆsv) = 0.882. The standard deviation of this estimate is calculated using
the Delta Method and is found to be 0.09710. Hence, the estimate of the standard deviation of the unobserved heterogeneity term is significant at the 1% level. From this, we conclude that unobserved heterogeneity is present in the data.
Table 3: Results Cox PH models Cox PH
Model (1) Model (2) Without time-varying covariates With time-varying covariates
Coef Exp(coef) Coef Exp(coef) Age (ref. Age18.29) Age30.39 −0.241∗∗∗ 0.786 −0.236∗∗∗ 0.790 (0.038) (0.040) Age40.49 −0.347∗∗∗ 0.707 −0.342∗∗∗ 0.710 (0.039) (0.041) Age50.59 −0.344∗∗∗ 0.709 −0.338∗∗∗ 0.713 (0.044) (0.045) Age60.69 −0.359∗∗∗ 0.698 −0.344∗∗∗ 0.709 (0.054) (0.056) Age70.older -0.0589 0.943 -0.0711 0.931 (0.056) (0.057) Acquisitionchannel (ref: Acq.TM) Acq.DD 0.502∗∗∗ 1.652 0.521∗∗∗ 1.683 (0.036) (0.037) Acq.Online −0.580∗∗∗ 0.560 −0.614∗∗∗ 0.541 (0.059) (0.056) Acq.TV −0.389∗∗∗ 0.678 −0.425∗∗∗ 0.653 (0.048) (0.047) Acq.Concert 0.00811 1.008 -0.046 0.955 (0.059) (0.060) Acq.Else −0.660∗∗∗ 0.517 −0.689∗∗∗ 0.502 (0.068) (0.067) Log(amount) 0.187∗∗∗ 1.205 0.199∗∗∗ 1.221 (0.025) (0.025) Female 0.134∗∗∗ 1.143 0.135∗∗∗ 1.145 (0.026) (0.027) Donated.before −0.572∗∗∗ 0.564 −0.603∗∗∗ 0.547 (0.147) (0.138) Season.start (ref: spring) Season.start.summer −0.148∗∗∗ 0.862 −0.151∗∗∗ 0.860 (0.037) (0.039) Season.start.autumn −0.120∗∗∗ 0.887 −0.129∗∗∗ 0.879 (0.038) (0.039) Season.start.winter −0.108∗∗∗ 0.898 −0.117∗∗∗ 0.890 (0.035) (0.036)
Acquisition channel × donated before (ref: Acq.TM × did not donate before)
Acq.DD × Donated.before 0.516∗ 1.675 0.527∗ 1.694 (0.291) (0.301) Acq.Online × Donated.before 0.485 1.624 0.556∗ 1.743 (0.318) (0.311) Acq.TV × Donated.before 0.881∗∗∗ 2.414 0.952∗∗∗ 2.590 (0.326) (0.281) Acq.Concert × Donated.before 1.166∗∗∗ 3.211 1.251∗∗∗ 3.495 (0.277) (0.212) Acq.Else × Donated.before 0.561∗∗ 1.752 0.619∗∗ 1.856 (0.270) (0.256)
...continued from previous page Cox PH
Model (1) Model (2) Without time-varying covariates With time-varying covariates Coef Exp(coef) Coef Exp(coef)
TM.request −0.0474∗∗∗ 0.954 (0.008) DM.request 0.0348∗∗∗ 1.035 (0.007) Extra.gift −0.0332∗∗∗ 0.967 (0.011) AIC 105297.8 105254.8 BIC 105439.2 105416.4 LogLikelihood -52627.92 -52603.42 Number of observations 10,000 10,000
Table 4: Results PCH models - estimates of the baseline hazards PCH
Model (3) Model (4)
Without unobserved heterogeneity With unobserved heterogeneity Coef Exp(coef) Coef Exp(coef)
γ1 −6.91∗∗∗ 0.0010 −7.23∗∗∗ 0.0007 (0.068) (0.098) γ2 −7.54∗∗∗ 0.0005 −7.90∗∗∗ 0.0004 (0.075) (0.120) γ3 −7.63∗∗∗ 0.0005 −7.98∗∗∗ 0.0003 (0.077) (0.112) γ4 −7.74∗∗∗ 0.0004 −8.10∗∗∗ 0.0003 (0.080) (0.114) γ5 −7.81∗∗∗ 0.0004 −8.17∗∗∗ 0.0003 (0.083) (0.116) γ6 −7.93∗∗∗ 0.0004 −8.30∗∗∗ 0.0002 (0.086) (0.119) γ7 −7, 99∗∗∗ 0.0003 −8.35∗∗∗ 0.0002 (0.089) (0.121) γ8 −7.96∗∗∗ 0.0003 −8.33∗∗∗ 0.0002 (0.090) (0.122) γ9 −8.08∗∗∗ 0.0003 −8.45∗∗∗ 0.0002 (0.094) (0.126) γ10 −8.07∗∗∗ 0.0003 −8.44∗∗∗ 0.0002 (0.095) (0.126) γ11 −8.13∗∗∗ 0.0003 −8.50∗∗∗ 0.0002 (0.098) (0.129) γ12 −8.10∗∗∗ 0.0003 −8.47∗∗∗ 0.0002 (0.098) (0.129) γ13 −8.09∗∗∗ 0.0003 −8.46∗∗∗ 0.0002 (0.099) (0.129) γ14 −8.25∗∗∗ 0.0003 −8.62∗∗∗ 0.0002 (0.105) (0.135) γ15 −8.24∗∗∗ 0.0003 −8.61∗∗∗ 0.0002 (0.106) (0.136) γ16 −8.32∗∗∗ 0.0002 −8.69∗∗∗ 0.0002 (0.110) (0.140) γ17 −8.10∗∗∗ 0.0003 −8.46∗∗∗ 0.0002 (0.106) (0.135) γ18 −8.26∗∗∗ 0.0003 −8.63∗∗∗ 0.0002 (0.114) (0.142) γ19 −8.24∗∗∗ 0.0003 −8.61∗∗∗ 0.0002 (0.115) (0.143) γ20 −8.30∗∗∗ 0.0002 −8.67∗∗∗ 0.0002 (0.119) (0.146) γ21 −8.21∗∗∗ 0.0003 −8.38∗∗∗ 0.0002 (0.066) (0.081)
∗∗∗Significant at 1%;∗∗significant at 5%;∗significant at 10%.
Table 5: Results PCH models - effects of covariates on the hazard rate PCH
Model (3)
Without unobserved heterogeneity
Model (4)
With unobserved heterogeneity Coef Exp(coef) AME Coef Exp(coef) AME Age (ref. Age18.29) Age30.39 −0.240∗∗∗ 0.786 736 −0.257∗∗∗ 0.773 969 (0.038) (0.041) Age40.49 −0.347∗∗∗ 0.707 1125 −0.375∗∗∗ 0.687 1506 (0.039) (0.042) Age50.59 −0.344∗∗∗ 0.709 1114 −0.369∗∗∗ 0.691 1477 (0.044) (0.046) Age60.69 −0.359∗∗∗ 0.699 1171 −0.388∗∗∗ 0.678 1570 (0.054) (0.057) Age70.older -0.0583 0.943 162 -0.088 0.916 304 (0.055) (0.059) Acquisitionchannel (ref: Acq.TM) Acq.DD 0.506∗∗∗ 1.660 -1328 0.521∗∗∗ 1.684 -1692 (0.036) (0.038) Acq.Online −0.583∗∗∗ 0.558 2709 −0.621∗∗∗ 0.538 3574 (0.058) (0.062) Acq.TV −0.391∗∗∗ 0.676 1633 −0.420∗∗∗ 0.657 2169 (0.048) (0.051) Acq.Concert 0.00365 1.004 -12 0.028 1.028 -114 (0.059) (0.063) Acq.Else −0.669∗∗∗ 0.512 3261 −0.693∗∗∗ 0.500 4157 (0.067) (0.072) Log(amount) 0.187∗∗∗ 1.206 -124 0.196∗∗∗ 1.217 -161 (0.025) (0.026) Female 0.134∗∗∗ 1.144 -472 0.141∗∗∗ 1.152 -616 (0.026) (0.028) Donated.before −0.569∗∗∗ 0.566 2640 −0.615∗∗∗ 0.541 3585 (0.147) (0.156) Season.start (ref: spring) Season.start.summer −0.149∗∗∗ 0.8612 507 −0.158∗∗∗ 0.854 665 (0.037) (0.040) Season.start.autumn −0.120∗∗∗ 0.8867 402 −0.124∗∗∗ 0.883 515 (0.038) (0.040) Season.start.winter −0.108∗∗∗ 0.8976 358 −0.110∗∗∗ 0.896 454 (0.035) (0.037)
Acquisition channel × donated before (ref: Acq.TM × did not donate before)
Acq.DD × Donated.before 0.512∗ 1.668 -1396 0.551∗ 1.735 -1859 (0.291) (0.304) Acq.Online × Donated.before 0.482 1.620 -1335 0.551∗ 1.736 -1859 (0.318) (0.331) Acq.TV × Donated.before 0.877∗∗∗ 2.404 -2014 0.951∗∗∗ 2.588 -2689 (0.327) (0.343) Acq.Concert × Donated.before 1.17∗∗∗ 3.221 -2353 1.22∗∗∗ 3.378 -3080 (0.276) (0.299) Acq.Else × Donated.before 0.563∗∗ 1.756 -1498 0.618∗∗ 1.855 -2022 (0.270) (0.286) ˆ sv -0.125 0.882 (note: ˆσv= exp(ˆsv)) (0.110) AIC 106657.9 106635.2 BIC 107066.7 107053.7 Log Likelihood -53286.95 -53274.59 Number of observations 10,000 10,000
∗∗∗Significant at 1%;∗∗significant at 5%;∗significant at 10%. Standard errors in parentheses.
5.2
Simultaneous estimation of the duration of donating
and the amount of money donated
In this section we present the results of the final model, which simultaneously estimates the duration of donating and the size of the donations. Remarkably, optimizing the loglikelihood given by equation 3.28 results in a negative variance of the estimate ˆψ. To overcome this problem, taking into account we only have a limited amount of time, we decide to fix this parameter at the found value, which is approximately 1.79, and optimize the loglikelihood again with respect to the remaining parameters. Unfortunately, due to this problem, we are unable to draw conclusions regarding the significance of ˆψ. The results of the final model are presented in table 6.
The estimates of the standard deviations of w and ξ are found to be ˆσw =
0.0001 + exp(ˆsw) = 0.228 and ˆσξ = 0.0001 + exp(ˆsξ) = 0.369. Using the Delta
Method, the standard deviations of these estimates are found to be 0.004 and 0.073, respectively. Hence, both estimates are significant at the 1% level. The estimate ˆρq,w can be recovered from ˆψ, ˆσw and ˆσξ using equation (3.27) and is
found to be 0.740. However, due to the missing standard deviation of ˆψ, we can-not draw conclusions regarding the significance of this correlation estimate. In the models of the previous section, the coefficient on the variable representing the natural logarithm of the amount of money donated is positive and significant at the 1% level. Based on these models, we conclude that donors who give a higher amount of money have a lower expected duration of donating. The final results show that there are indications that after controlling for endogeneity of the variable, the amount of money donated does not have a significant effect on the duration of donating. Hence, War Child can not improve the duration of donat-ing by changdonat-ing their policy regarddonat-ing the amount of money they ask for. Since there are indications that the size of the donations does not affect the duration of donating, donors who give a high amount of money have a greater expected lifetime value than donors who give a low amount of money, all else being equal. Considering the age of donors, we conclude that compared to donors between age 18 and 29, older donors have on average a higher expected duration of donating and donate a higher amount of money. Hence, older donors have a greater ex-pected lifetime value than younger donors. It should be noticed that these effects could also be driven indirectly by the level of income, since older people generally have more money to spend than younger people. Moreover, women have on
aver-age a lower expected duration of donating and donate a lower amount of money then men, which contradicts the findings of Dvorak and Toubman (2013).
Considering donors who had not donated to War Child before starting their sub-scription, donors acquired either online, via television, or else all have a greater expected lifetime value than donors acquired via telemarketing: they have on av-erage a higher expected duration of donating and the size of their donations is bigger. Donors acquired via door to door, regardless of whether they had do-nated to War Child before, give a higher amount of money, but have on average a lower expected duration of donating than donors acquired via telemarketing11.
This result is in line with the explanation given before: donors who are talked into subscribing without really wanting this are probably also easily talked into donating a high amount of money.
The effect of having done a donation to War Child before starting the subscription on the expected duration of donating depends on the channel through which the donor was acquired and is positive for all acquisition channels12. Moreover, those
donors give a higher amount of money. This is intuitive, since donors who had donated to War Child before are expected to be more commited to War Child and hence have on average a longer expected duration of donating. Hence, when acquiring donors to subscribe for structural donations, it is most profitable for War Child to target individuals who have already donated to them in the past and approach them online13.
11This holds regardless whether someone had donated to War Child before, since the average marginal effect of being acquired via door to door is -1739 for donors who had not donated to War Child before, -1418 for donors who had donated to War Child before, and compared to a donor acquired via telemarketing who had not donated to War Child before, a donor acquired via door to door who had donated to War Child before has on average an expected duration which is -1739+2579-1418=-568, i.e. 568 days shorter. It should be noticed that the interaction dummy between the acquisition channel door to door and whether someone had donated to War Child before is only significant at the 10% level.
12All average marginal effects on the expected duration of donating corresponding to the interaction dummies corresponding to the acquisition channel and whether someone had donated to War Child before are negative, but smaller in absolute value than 2579.
13Note that since the interaction dummy between the acquisition channel online and whether someone had donated to War Child before is insignificant. Hence, compared to donors ac-quired via telemarketing who had not donated to War Child before, donors acac-quired online who had donated to War Child before have on average an expected duration of donating which is 2156+2579=4735 days longer
Table 6: Simultaneous estimation of the duration and the amount of money donated
Model (5)
Baseline hazards Effects of covariates on the hazard rate Effects of covariates on log(amount)
Remaining parameters
Coef Exp(coef) Coef Exp(coef) AME Coef Coef
γ1 −6.51∗∗∗ 0.0015 Age Age ψˆ 1.79
(0.123) (ref. Age18.29) (ref. Age18.29)
γ2 −7.15∗∗∗ 0.0008 Age30.39 −0.205∗∗∗ 0.814 645 Age30.39 0.118∗∗∗ ˆsw −1.48∗∗∗ (0.128) (0.040) (0.005) (0.019) γ3 −7.23∗∗∗ 0.0007 Age40.49 −0.306∗∗∗ 0.737 1016 Age40.49 0.137∗∗∗ ˆsξ −0.996∗∗∗ (0.129) (0.041) (0.005) (0.198) γ4 −7.34∗∗∗ 0.0006 Age50.59 −0.300∗∗∗ 0.741 992 Age50.59 0.148∗∗∗ (0.131) (0.045) (0.005) γ5 −7.41∗∗∗ 0.0006 Age60.69 −0.305∗∗∗ 0.737 1014 Age60.69 0.179∗∗∗ (0.132) (0.056) (0.006) γ6 −7.53∗∗∗ 0.0005 Age70.older -0.0403 0.961 116 Age70.older 0.063∗∗∗ (0.135) (0.056) (0.007) γ7 −7.59∗∗∗ 0.0005 Acquisitionchannel Acquisitionchannel
(0.136) (ref: Acq.TM) (ref: Acq.TM)
γ8 −7.56∗∗∗ 0.0005 Acq.DD 0.623∗∗∗ 1.86 -1739 Acq.DD 0.383∗∗∗ (0.137) (0.046) (0.004) γ9 −7.68∗∗∗ 0.0005 Acq.Online −0.447∗∗∗ 0.639 2156 Acq.Online 0.448∗∗∗ (0.140) (0.068) (0.006) γ10 −7.67∗∗∗ 0.0005 Acq.TV −0.274∗∗∗ 0.760 1204 Acq.TV 0.384∗∗∗ (0.140) (0.056) (0.005) γ11 −7.73∗∗∗ 0.0004 Acq.Concert 0.206∗∗∗ 1.23 -704 Acq.Concert 0.655∗∗∗ (0.142) (0.076) (0.007) γ12 −7.70∗∗∗ 0.0005 Acq.Else −0.579∗∗∗ 0.561 2998 Acq.Else 0.296∗∗∗ (0.142) (0.071) (0.006) γ13 −7.70∗∗∗ 0.0005 (0.143) Female 0.109∗∗∗ 1.12 -389 Female −0.0816∗∗∗ γ14 −7.85∗∗∗ 0.0004 (0.027) (0.003) (0.147) Donated.before −0.555∗∗∗ 0.574 2579 Donated.before 0.0565∗∗∗ γ15 −7.84∗∗∗ 0.0004 (0.148) (0.009) (0.148) Log(amount) -0.119 0.888 75 Constant 1.33∗∗∗ γ16 −7.92∗∗∗ 0.0004 (0.079) (0.006) (0.151) Season.start γ17 −7.70∗∗∗ 0.0005 (ref: spring) (0.148) Season.start.summer −0.150∗∗∗ 0.861 515 γ18 −7.86∗∗∗ 0.0004 (0.037) (0.154) Season.start.autumn −0.120∗∗∗ 0.887 407 γ19 −7.84∗∗∗ 0.0004 (0.038) (0.154) Season.start.winter −0.108∗∗∗ 0.898 363 γ20 −7.90∗∗∗ 0.0004 (0.035)
(0.158) Acquisition channel × donated before
γ21 −7.80∗∗∗ 0.0004 (ref: Acq.TM × did not donate before)
(0.121) Acq.DD × Donated.before 0.514∗ 1.672 -1418 (0.291) Acq.Online × Donated.before 0.486 1.626 -1360 (0.318) Acq.TV × Donated.before 0.881∗∗∗ 2.413 -2045 (0.327)
Acq.Concert × Donated.before 1.17∗∗∗ 3.220 -2385 AIC 311399.1
(0.277) BIC 310602.6
Acq.Else × Donated.before 0.565∗∗ 1.760 -1522 Log Likelihood -155641.6
(0.271) Number of observations 10,000
∗∗∗Significant at 1%;∗∗significant at 5%;∗significant at 10%. Standard errors in parentheses. Average marginal effects on the expected duration of
6
Conclusion
This study focuses on the duration of donating of individuals who have a sub-scription to donate monthly to the Dutch charity War Child. We estimate several proportional hazard models without imposing restrictive assumptions on the dis-tributional form of the baseline hazard. We find that age, gender, the season in which the donor started his/her subscription, the channel through which the donor was acquired and whether the donor had donated to War Child before starting his/her subscription have a significant effect on the duration of donat-ing. Moreover, our proportional hazard models include interaction terms between the acquisition channel and whether the donor had donated to War Child before starting his/her subscription.
Additionally, the proportional hazard models include the size of the monthly do-nations to which the donor committed as an explanatory variable. To account for possible endogeneity of this variable, we build a model that simultaneously esti-mates the duration of donating and the size of the donations. When not allowing this variable to be endogenous, we find the relation between the size of the dona-tions and the duration of donating to be significantly negative. However, when we do allow this variable to be endogenous, we find that there are indications that there is no significant effect of the size of the donations on the duration of do-nating. Hence, War Child can not improve the duration of donating by changing their policy regarding the size of the donations they ask their donors to commit to. Since there are indications that the size of the donations has no effect on the duration of donating, donors who give a high amount of money have a greater expected lifetime value than donors who give a low amount of money.
We find that older individuals have a longer duration of donating and make larger monthly donations. Moreover, our results report that women are less valuable to War Child than men: they make smaller donations and donate for a shorter period of time. Additionally, we find that it is profitable for War Child to target individuals who have already donated to War Child in the past and that these donors should be approached online.
