TRAS 12 MESES DE TRATAMIENTO

The listeners’ demand model is estimated using GMM proposed by Berry et al. (1995) and Nevo (2000). The moments are based on a comparison of the predicted listener ratings for different age-group combinations for station j at time t (see equation (4.8)) with the corresponding observed listener ratings. The error term is defined in the same way as in Berry (1994) and it represents the unobserved station characteristics which is ξ˜_jdt(see equation (4.2) for a definition of ˜ξ_jdt).¹² These unobserved characteristics can be calculated by first solving for the mean valuations δ_jdt which obey the following equations:

S_j|dt=s_j|dt(δ_·dt, θ2), j=1, . . . , Jt, d=1, . . . , D, t=1, . . . , T

where S_j|dtare the observed listener ratings, s_j|dtare the predicted listener rating (see equation (4.8)), and θ₂are the parameters that enter nonlinearly (cf. equation (4.5b)).

For the multinomial logit model without random preferences the solution, δ_jdt(s_·dt, θ2), is equal to ln(S_j|dt) −ln(S_0|dt)while for the random-coefficients logit model this in-version can only be done numerically. We use the contracting mapping proposed by Berry et al. (1995) to numerically solve the system of equations. So, we start with an initial guess of δ⁰_jdt(which are the IV logit estimates in our application) together with an initial guess of the parameter set θ2to calculate s⁰_j|dt. Then, we construct a new guess using the contracting mapping by solving for the exponent of δjdt. If we define

12As we already stated in Subsection 4.2.1, we exploit the panel structure of our data and control for station-specific intercepts ξ_j.

wjmt=exp(δ_jdt)then the contracting mapping routine becomes:

w^h+1_jdt =w^h_jdt S_j|dt

s_j|dt(δ_·dt^h ; θ2)^{, h}=0, . . . , H

and this is repeated H times until for each station j, demographic group d and time t:

|δ^H_jdt(S_·dt, θ2) −δ^H−1_jmt (S_·dt, θ2)|

is arbitrarily close to zero which is 10⁻⁶in our application. The structural error term can then be calculated as: number of instruments). Then the GMM estimate is equal to:

ˆθ=argmin

)⁰and W is a weighting matrix. The weighting matrix is first set to Z⁰Z to obtain a consistent estimate of E[Z⁰ξ ˜˜ξ⁰Z]which is used in the final specification. The main advantage of this estimation routine is that the parameters θ1

enter the GMM objective function linearly. However, the parameters θ2still enter non-linearly and computation time increases exponentially with the number of nonlinear parameters to be estimated. We take 200 draws from the standard Normal distribution for each random utility parameter for each age-group-time combination in order to calculate the integral in equation (4.8) numerically.

Some of the station characteristics X_jtare time invariant. We denote this subset of station characteristics as ˜X_jand the corresponding parameter vector ˜β. Obviously we cannot separately identify the station-fixed effect ξ_j and the parameter vector ˜β: the

time-invariant station characteristics are subsumed in the fixed effect ˜ξ_j:

ξ˜_j=X^˜_j˜β+ξ_j.

In order to identify ˜β we have to make the following assumption: E(ξ_j|X^˜_j) =0. In that case, we first estimate θ2, ˜ξ_j and the remaining parameters of θ₁(i.e. α,Π and the β parameters corresponding to the time-varying station characteristics) by means of the GMM procedure outlined above. In the second step we retrieve the parameter vector

˜β using a Feasible Generalized Least Squares procedure, i.e.:

βˆ˜= (X^˜0Vˆ⁻¹_ˆ˜

ξ

X˜)⁻¹X^˜0Vˆ_ˆ˜⁻¹

ξ

ξˆ˜

where ˆV_ˆ˜

ξ is the estimated covariance matrix of ˆ˜ξ(obtained in the first step).

We have explored many set of instruments starting with the same set of instru-ments as in the previous chapters. These chapters estimate a nested logit model with regional listener rating data and uses the market population, economic climate, and the number of competing radio stations as instruments for advertising and the within-group market share. These instruments do not perform well for national rating data for different demographic groups. First, the regional data allow for variation in the num-ber of competing radio stations because the geographical location of a market mainly determines how many radio stations can be received in a market. This variation is not present with national data for different demographic groups. A similar argument holds for population. The population sizes for the different age-gender combinations do not differ as much as the population as the different provinces of the Netherlands.

Economic climate still is a good candidate as an instrument for advertising which is an index from Statistics Netherlands indicating how individuals perceive the economic situation. It turns out that economic climate does not enter linearly and testing the higher-order effects leads to the inclusion of the second and third-order effect of eco-nomic climate. Furthermore, the average antenna power and total antenna power of competitors are included as instruments. These are valid instruments by assuming that the station characteristics are determined exogenously which is a reasonable as-sumption for antenna power (see Berry et al. (1995)).

Other sets of instruments that are often used in the literature on discrete choice models are the Hausman-type instruments (Hausman, 1996). The identifying assump-tion is that, after controlling for staassump-tion-fixed effects, the age-group-time specific valu-ations for stvalu-ations, ˜ξ_jdt, are independent across demographic groups but are allowed to be correlated within a demographic group over time. Given this assumption, the num-ber of advertisements within other demographic groups would be valid instruments.

However, advertising does not vary across demographic groups due to radio signal distribution technology. Therefore, we cannot use these instruments in our empirical application.

Another strategy is proposed by Sweeting (2011) and Jeziorski (2012). They as-sume that the unobserved station quality parameters ˜ξ_jdtfollow an AR(1) process with an autoregression parameter ρ which does not vary across stations. By introducing autocorrelation of ˜ξ_jdt, the lagged advertising and market shares of other stations be-come valid instruments. On the other hand, the autoregression parameter needs to be estimated jointly with the other parameters. Notice that their estimates will be incon-sistent if the assumption of a station-invariant ρ does not hold. We estimated the θ parameter vector using the strategy proposed by Sweeting (2011). After this estima-tion we investigated the time series properties of ˜ξ_jdt. We indeed find evidence that ξ˜_jdtfollows an AR(1) process. However, we also reject the null hypothesis of a station-invariant autoregression parameter ρ. This finding implies that we have to extend the estimation method of Sweeting (2011) and Jeziorski (2012) in order to obtain a consis-tent estimate of the θ vector. This extension is beyond the scope of this chapter and we therefore do not use lagged advertising and market shares as instruments.