In this section we examine the properties of the ACD model in a small sample. It has already been shown by Grammig and Maurer (2000) that a misspecification of the distribution of the durations has serious consequences for the consistency of the estimation. We therefore focus on the case in which we know the true form of the distribution.
The expected durations (ψi,n) in the ACH model are not updated as a period progresses, they are therefore essentially only identified by the information that is also used in the pure ACD model. As the ACD model has originally been developed for high frequency data and as our durations are measured in terms of months, we have a rather small amount of observations to estimate the expected durations. To cope with the finite sample we pool the series of all countries. A large drawback of pooling is that one implicitly assumes homogeneity among the countries. As a compromise between having a larger number of observations and still having a panel that does not violate the homogeneity assumption, Van den Berg et al. (2008) propose to pool only clusters of similar countries. In our sample, this means that the number of durations in such a cluster would be around 50 or even lower. As we will find from the simulation study below, at this amount of observations the ACD estimations do not give reliable results even under the strongly simplifying assumptions of homogeneity and independence of the series.
4.4.1 Setup of the Simulations
In our simulation we mimic a LACD(1,1) model. In the LACD(1,1) we have from equation (4.3) the following relationships between the actual durations zi,n, the
6Specifically, for the Exponential-LACH, Θ = (δ′, θ′)′, and for the Weibull-LACH, Θ = (δ′, θ′, γ)′,
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expected durations ψi,n, and the innovations ǫi,n:
ln(ψi,n) = ω + α ln(zi,n−1) + β ln(ψi,n−1), (4.8)
zi,n= ψi,nǫi,n. (4.9)
In our simulations we assume the ǫi,n i.i.d. and distributed either Exponential(1) or Weibull(1,γ). The errors are standardised such that their asymptotical mean equals 1. For given individual-specific starting values ψi,0, the durations zi,n and expected durations ψi,n can then be built iteratively.
We focus on two different data generating processes (DGP) based on equations (4.8)-(4.9). The first DGP is an Exponential-LACD(1,1) process. The parameters ω, α and β are chosen such that they roughly mark the process of our empirical application. The second DGP is a Weibull-LACD(1,1) where the parameters ω, α and β are chosen similarly and γ = 2. Notice that for the Weibull(1,2) density the first moment of the innovations ǫi,n is not equal to one. In order to keep the expected value of the simulated durations xi,n at the intended value ψi,n, we need to rescale the innovations such that their first moment is equal to one. For each of the DGPs we simulate an N ∗ K panel of durations and expected durations with four different sample sizes. The number of individuals is kept constant at K = 10, whereas the number of durations per individual is set at N = 5, 10, 100 or 1000.
The simulation study is based on 1000 replications per series.
4.4.2 Simulation Results
Figures 4.1 and 4.2 show the distribution of the estimated parameters. The tables containing the numerical results corresponding to the figures can be found in the appendix to this chapter.
Figure 4.1 contains the results of the Exponential-LACD for the varying sizes of N . We can see that the estimation works fine if the sample is large enough, i.e. N = 100 or N = 1000 per individual. Also, the estimation of the α coeffi-cient appears remain consistent even if the sample size decreases. The two other coefficients however, suffer from consistency problems when N becomes smaller.
At N = 10 (i.e. 100 observations in total), the estimation of ω is slightly skewed to the left, whereas the for the β coefficient it is skewed to the right. When the sample decreases further, to N = 5, both distributions even have a second local maximum, ω at around 0 and β slightly below 1. The observations that are located in these hump are from the same replications. In these replications the sum of the α and β coefficient is very close to one, a sign of a high degree of persistence in ln(ψi,n). It follows that the value of ω needs to be close to zero in order to keep the expected value at a reasonable level.
For the Weibull-LACD(1,1) model (Figure 4.2), the results in the very small samples are very similar to those of the ELACD model. Additionally, the shape coefficient of the distribution, γ, shows a right tail that is a little too heavy,
re-sulting in an upward bias that disappears as the sample size grows.
From this small simulation exercise we may conclude that even under the sim-plifying assumptions of homogeneity, knowing the correct distribution as well as independence of the innovations with respect to both dimensions of the panel, the LACD model suffers from convergence problems in small samples (50 obser-vations). When the sample size is doubled however, the problems in the ELACD and the WLACD are within reasonable bounds. These results indicate that one must be very careful when estimating an ACD in small samples, especially when the number of observations is below 100.
(a) ω, value = 5.6 (b) α, value = 0.2
(c) β, value = -0.6
Note: K = 10. The vertical black line represents the true value of the parameter.
Figure 4.1: Simulated ELACD estimation results when the innovations are Expo-nential.
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(a) ω, value = 5.5 (b) α, value = 0.2
(c) β, value = -0.6 (d) γ, value = 2.0
Note: K = 10. The vertical black line represents the true value of the parameter.
Figure 4.2: Simulated WLACD estimation results when the innovations are Weibull.