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Discontinuities due to survey redesigns : a structural time series approach

by Goesjenov, Ibragim, MS


Page 41

Table 8 – Estimated discontinuities for ”Victimization”, M1 and M2

Model
M1

M1

M1

M1

M1

M1

M2

M2

M2

Transformation/ Breakdowns
Intervention# Violence Property Vandalism No stop* Other Total

None
Intervention 1 0.31(0.45) 0.13(0.41) 1.74(1.02) -0.06(0.15) -0.13(0.12) 1.68(1.28)
Intervention 2 -0.03(0.70) 1.94(0.77) 2.71(1.63) 0.23(0.23) 0.02(0.16) 3.40(2.06)
Intervention 3 0.22(0.80) 2.39(0.80) 4.66(1.87) 0.00(0.00) 0.17(0.18) 3.45(2.37)
Log
Intervention 1 0.06(0.08) 0.01(0.03) 0.17(0.08) -0.02(0.10) -0.13(0.15) 0.07(0.05)
Intervention 2 0.00(0.13) 0.15(0.06) 0.25(0.13) 0.12(0.15) 0.00(0.23) 0.13(0.08)
Intervention 3 0.05(0.15) 0.19(0.06) 0.40(0.15) 0.00(0.00) 0.01(0.27) 0.14(0.09)
Logit
Intervention 1 0.06(0.08) 0.01(0.03) 0.19(0.09) -0.02(0.10) -0.13(0.15) 0.09(0.06)
Intervention 2 0.03(0.13) 0.18(0.07) 0.28(0.15) 0.12(0.16) 0.00(0.24) 0.18(0.10)
Intervention 3 0.05(0.16) 0.21(0.06) 0.45(0.17) 0.00(0.00) 0.01(0.27) 0.18(0.12)
None
Intervention 1 0.37(0.42) 0.15(0.41) 1.61(0.75) -0.06(0.16) -0.13(0.11) 1.73(1.06)
Intervention 2 0.10(0.66) 1.98(0.74) 2.48(1.20) 0.23(0.23) 0.03(0.15) 3.49(1.72)
Intervention 3 0.31(0.78) 2.28(0.85) 4.01(1.43) 0.00(0.00) 0.17(0.19) 3.18(2.03)
Log
Intervention 1 0.07(0.08) 0.01(0.03) 0.15(0.06) -0.02(0.10) -0.19(0.14) 0.06(0.04)
Intervention 2 0.03(0.13) 0.16(0.06) 0.23(0.10) 0.12(0.16) -0.10(0.22) 0.14(0.06)
Intervention 3 0.07(0.15) 0.18(0.06) 0.34(0.12) 0.00(0.00) -0.13(0.26) 0.12(0.08)
Logit
Intervention 1 0.08(0.08) 0.01(0.03) 0.17(0.07) -0.02(0.10) -0.19(0.14) 0.09(0.05)
Intervention 2 0.03(0.13) 0.17(0.06) 0.25(0.11) 0.12(0.16) -0.10(0.22) 0.18(0.08)
Intervention 3 0.07(0.15) 0.20(0.07) 0.38(0.13) 0.00(0.00) -0.13(0.26) 0.16(0.10)
None
Intervention 1 0.42(0.42) 0.14(0.42) 1.50(0.67) -0.06(0.15) -0.13(0.11) 1.71(1.02)
Intervention 2 0.21(0.66) 1.97(0.73) 2.27(1.08) 0.27(0.21) 0.03(0.15) 3.48(1.65)
Intervention 3 -0.27(0.82) -0.82(0.95) 0.60(1.33) 0.09(0.28) -0.08(0.21) -0.63(2.03)
Log
Intervention 1 0.09(0.08) 0.02(0.04) 0.14(0.07) -0.02(0.10) -0.22(0.14) 0.06(0.04)
Intervention 2 0.06(0.13) 0.17(0.06) 0.22(0.10) 0.15(0.15) -0.15(0.22) 0.15(0.06)
Intervention 3 -0.03(0.16) -0.06(0.09) 0.09(0.13) -0.04(0.19) -0.46(0.27) 0.02(0.08)
Logit
Intervention 1 0.09(0.08) 0.02(0.04) 0.16(0.07) -0.02(0.10) -0.22(0.14) 0.07(0.06)
Intervention 2 0.06(0.13) 0.19(0.07) 0.25(0.11) 0.15(0.15) -0.15(0.22) 0.20(0.08)
Intervention 3 -0.02(0.17) -0.06(0.09) 0.11(0.15) -0.02(0.19) -0.44(0.27) 0.03(0.11)

* - ”Failure to stop after an accident”, - Sample size of ISM is adjusted to 19.000.
Bold value - statistically significant with p-val.<0.05, standard errors in brackets.

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Model
M3

M3

M3

M3

M4

M4

M4

M4

Table 9 – Estimated discontinuities for ”Victimization”, M3 and M4
Transformation/ Breakdowns
Intervention# Violence Property Vandalism No stop* Other Total
None
Intervention 1 0.36(0.24) 0.15(0.38) 1.69(0.88) -0.06(0.16) -0.17(0.09) 1.31(0.65)
Intervention 2 0.13(0.29) 1.99(0.68) 2.73(1.40) 0.23(0.23) -0.02(0.08) 2.87(0.90)
Intervention 3 0.58(0.35) 0.75(0.22) 1.45(0.50) 0.00(0.00) 0.52(0.28) 0.67(0.21)
Log
Intervention 1 0.07(0.06) 0.02(0.04) 0.07(0.09) -0.04(0.10) -0.16(0.10) 0.02(0.04)
Intervention 2 0.02(0.06) 0.17(0.06) 0.14(0.14) 0.14(0.16) 0.00(0.11) 0.09(0.03)
Intervention 3 0.64(0.41) 0.85(0.26) 1.23(0.66) 0.00(0.00) 0.23(0.36) 0.79(0.21)
None
Intervention 1 0.36(0.29) 0.15(0.40) 1.45(0.77) -0.06(0.15) -0.14(0.09) 1.49(0.81)
Intervention 2 0.10(0.41) 1.99(0.71) 2.22(1.24) 0.23(0.23) 0.00(0.11) 3.10(1.25)
Intervention 3 0.44(0.61) 0.74(0.26) 1.16(0.47) 0.00(0.00) 0.58(0.50) 0.71(0.37)
Log
Intervention 1 0.07(0.06) 0.02(0.04) 0.09(0.07) -0.03(0.10) -0.15(0.11) 0.04(0.04)
Intervention 2 0.02(0.08) 0.16(0.06) 0.13(0.11) 0.13(0.16) -0.02(0.16) 0.12(0.05)
Intervention 3 0.42(0.61) 0.74(0.27) 0.90(0.54) 0.00(0.00) -0.09(0.66) 0.79(0.36)

None
Intervention 1 0.36(0.24) 0.15(0.38) 1.69(0.88) -0.06(0.16) -0.17(0.09) 1.31(0.63)
Intervention 2 0.13(0.29) 1.99(0.68) 2.73(1.41) 0.24(0.23) -0.03(0.08) 2.88(0.89)
Intervention 3 0.58(0.36) 0.75(0.22) 1.45(0.50) 0.00(0.00) 0.52(0.28) 0.67(0.21)
Intervention - - - - - 2.89(2.43)
Police series
Log
Intervention 1 0.07(0.07) 0.02(0.04) 0.09(0.09) -0.04(0.11) -0.19(0.10) 0.03(0.02)
Intervention 2 0.02(0.06) 0.19(0.06) 0.19(0.13) 0.15(0.16) -0.02(0.11) 0.11(0.03)
Intervention 3 0.73(0.45) 0.93(0.28) 1.67(0.67) 0.00(0.00) 0.29(0.34) 0.73(0.19)
Intervention - - - - - 0.03(0.03)
Police series
None
Intervention 1 0.36(0.29) 0.15(0.40) 1.45(0.77) -0.06(0.16) -0.14(0.10) 1.49(0.79)
Intervention 2 0.10(0.41) 1.99(0.71) 2.22(1.24) 0.23(0.23) 0.00(0.11) 3.14(1.24)
Intervention 3 0.44(0.61) 0.74(0.26) 1.17(0.47) 0.00(0.00) 0.59(0.50) 0.72(0.37)
Intervention - - - - - 2.92(2.44)
Police series
Log
Intervention 1 0.07(0.07) 0.02(0.04) 0.09(0.08) -0.04(0.11) -0.17(0.12) 0.05(0.03)
Intervention 2 0.03(0.08) 0.19(0.06) 0.16(0.12) 0.13(0.16) -0.03(0.16) 0.14(0.05)
Intervention 3 0.68(0.07) 0.91(0.33) 1.23(0.58) 0.00(0.00) 0.00(0.65) 0.89(0.32)
Intervention - - - - - 0.03(0.03)
Police series
Correlations between the variances of the trends of the police series and ”Total” Victimization series for M4
M4 None 1
M4 Log 1
M4None 1
M4Log 1
* - ”Failure to stop after an accident”, - Sample size of ISM is adjusted to 19.000.
Bold value - statistically significant with p-val.<0.05, standard errors in brackets.

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did not cause severe discontinuities. For M1, M3 and M4 the Property and Vandalism breakdowns have
relatively large significant discontinuities in Intervention 3. The main redesign with the ISM for these
breakdowns was the questioning on crimes related to car theft and car vandalism. These subjects were
severely redesigned and most probably are the main reason for the discontinuities.

Following the results for M1 and M1 the adjustment of the sample size of the ISM to 19000 in
M1 results in a larger amount of significant discontinuities. Note that in theory the coefficients of the
intervention variables for Intervention 3 in M2 should be more or less of the same magnitude as the
coefficients for Intervention 2. In that case the model would point that there is no discontinuity between
the SM and the SMIV . However, these coefficients in table 8 differ substantially. Recall from section 4.2.2
that by including Intervention 3, it is assumed that there is no change in the stochastic trend, such that
all discontinuities can only be due to the introduction of SMIV . The smoothed trend ˆµt = (ˆµt,1, . . . , ˆµt,k)
which is contained in ˆαt, and the smoothed signal Zt ˆαt for M2 None are shown in figure 7 in Appendix
C. From this figure it seems that the model erroneously assigns a change in the stochastic trend to the
coefficient of the intervention variable. In fact, the stochastic trend in figure 7(a) should be more flexible.
This is the risk when applying severe survey redesigns to surveys of such short time span. Recall that
both the SM and SMIV were conducted only for 3 years. When leaving out Intervention 3, such that the
SM and the SMIV are considered as one consecutive survey, the smoothed trend and signal are given in
figure 8. In this figure the development from the SM to the SMIV is more plausible.

In M3, where the difference ˆyISM,t ˆySMIV ,t is implemented in the intervention variable, Intervention
3 shows significant discontinuities. This indicates that the difference ˆyISM,t ˆySMIV ,t explains a part of
the discontinuities. The inclusion of the police series as an explanatory variable in M4 and M4 does
not have a large effect on the estimated coefficients for Intervention 1, 2 and 3 compared to M3 and M3
. However, recall from section 4.2.2 that by including the police series as an explanatory variable, the
model is more robust for a change in the stochastic trend. Hence, M4 is less likely to contain such model
violations as it was the case with M2. The discontinuity in the police series in M4 is not significant in
any model. The correlation between the police series and total victimization in M4 series is 1, justifying
its addition to the model.

5.3.2 Discontinuities in subpopulations

The estimated discontinuities for M5 are provided in table 10. This model differs from previous models
in the sense that its focus is on the gender subpopulations. Note that since the constraint as discussed
in section 4.2.4 is only defined for percentages, no log or logit transformations could be applied for M5.

The breakdown Violence is not significant for any intervention. For the subpopulations the breakdown
Failure to stop after an accident is only significant for Intervention 2 for Men in M5 and M5 , all the
other coefficients are not significant.

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Model
M5

M5

Table 10 – Estimated discontinuities for ”Victimization”, M5
Transformation/ Breakdowns
Intervention# Violence Property Vandalism No stop* Total

None
Intervention 1
-National 0.38(0.29) 0.14(0.27) 1.11(0.43) 0.03(0.09) 1.45(0.66)
-Men 0.53(0.51) -0.48(0.48) 0.42(0.75) 0.18(0.15) 0.55(1.16)
-Women 0.23(0.50) 0.74(0.47) 1.74(0.74) -0.11(0.15) 2.31(1.15)
Intervention 2
-National -0.08(0.47) 1.80(0.50) 1.46(0.70) 0.15(0.14) 2.51(1.08)
-Men -0.63(0.82) 0.82(0.87) 0.54(1.22) 0.47(0.24) 0.54(1.89)
-Women 0.44(0.81) 2.73(0.86) 2.35(1.20) -0.15(0.24) 4.40(1.87)
Intervention 3
-National 0.16(0.54) 2.24(0.53) 3.34(0.80) 0.00(0.00) 2.51(1.24)
-Men -0.71(0.95) 0.79(0.92) 2.03(1.40) 0.00(0.00) -0.40(2.17)
-Women 1.01(0.93) 3.65(0.90) 4.61(1.38) 0.00(0.00) 5.32(2.14)
None
Intervention 1
-National 0.51(0.27) 0.19(0.27) 1.37(0.38) 0.03(0.09) 1.85(0.60)
-Men 0.81(0.47) -0.41(0.48) 0.91(0.66) 0.18(0.15) 1.22(1.05)
-Women 0.21(0.47) 0.76(0.48) 1.80(0.65) -0.11(0.15) 2.47(1.03)
Intervention 2
-National 0.16(0.45) 1.87(0.49) 1.99(0.62) 0.15(0.14) 3.28(0.99)
-Men -0.09(0.78) 0.95(0.86) 1.47(1.08) 0.47(0.24) 1.81(1.72)
-Women 0.40(0.77) 2.77(0.85) 2.48(1.06) -0.15(0.24) 4.69(1.70)
Intervention 3
-National 0.39(0.52) 2.27(0.57) 3.82(0.72) 0.00(0.00) 3.22(1.14)
-Men -0.22(0.90) 0.81(0.99) 2.88(1.25) 0.00(0.00) 0.75(1.99)
-Women 0.97(0.89) 3.67(0.97) 4.73(1.23) 0.00(0.00) 5.58(1.96)
* - ”Failure to stop after an accident”, - Sample size of ISM is adjusted to 19.000.
Bold value - statistically significant with p-val.<0.05, standard errors in brackets.

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More interesting, it is immediately apparent that per intervention there are differences in significance
between the estimates for the two subpopulations and the national estimates. Women appear to be
more sensitive to survey redesigns than men. Three regression coefficients of the intervention variable
were significant for men, for Failure to stop after an accident in Intervention 2 for M5 and M5 and
in Intervention 3 for Vandalism in M5. On the contrary, the regression coefficients of the intervention
variable for ”Vandalism” and ”Total” series are significant for all interventions in both M5 and M5 . For
the ”Property” series the female subpopulation had a significant regression coefficient for Interventions
2 and 3.

It is also noteworthy that the addition of the subpopulations resulted in lower standard errors for all
series, when comparing the national levels of M5 and M5 to the untransformed series in M1 and M1
. This is because the inclusion of the male and female subpopulations added extra information via the
increased the number of observations.

5.4 Diagnostic tests

The core assumptions of the state space model as discussed in section 4 are that the disturbances are
normally distributed and serially independent with constant variances. From these assumptions it follows
that the standardized innovations should also be normal and independent and identically distributed.
Therefore, the tests as described in section 4.3.6 are applied to the standardized innovations. Several
graphical plots are used to check for these model assumptions, including ACF plots, Normal density plots,
plots of the innovations and QQ plots as described in section 4.3.6. To improve the readability of this
paper not all graphs are provided. The statistical tests described in section 4.3.6 have been implemented
and the results are presented in tables 13, 14 and 15 in appendix A. First of all the discussion addresses
to the results for M1-M4, the results for the analysis of the gender subpopulations are discussed in the
last subsection of this paragraph.

5.4.1 Normality tests

Ideally, both skewness and excess kurtosis should have value 0. Hence, a model with values closer to 0
is considered to meet the normality assumption better than a model with higher values. The confidence
intervals for both skewness and kurtosis as provided in (4.24) are in this paper equal to :

CIS =

CIEK =

[

[

6
1.96 ×

, 1.96 ×
19
24
1.96 × , 1.96 ×
19

]
6

= [1.10, 1.10],

19
]

24

= [2.20, 2.20].
19
In tables 13 and 14 no series showed signs of non normality. The QQ plots in figure 3 in appendix B
also do not point at severe non normality. However, for M1, M3 and M4, the models using the adjusted

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sample size of the ISM show slightly better fits in the QQ plot compared to the models using the
original sample size of the ISM in the variance structure.

5.4.2 Heteroskedasticity

Turning to the heteroskedasticity test as discussed in 4.3.6, values above 1 imply increasing variance
in time, and values below 1 imply decreasing variance in time. In this paper, b = 7, such that the
confidence interval (4.25) becomes

CIHSK = [F 7

7( 0.05
2

), F

7
7(10.05
2

)] = [0.20, 5.00].

From tables 13 and 14 it appears that, as aforementioned, the models M1-M4 do not suffer from severe
heteroskedasticity. Only the Vandalism series for M1 None has a slightly significant sign for heteroskedasticity.
Also for heteroskedasticity it holds that the models which used the original sample size for the ISM
resulted in higher heteroskedasticity statistics compared to the models which used the adjusted sample
size of 19000. This indicates that this adjustment is proper and results in a better model fit.
In figure 4 in appendix B some innovation plots are shown, which also confirm the test results of no heteroskedasticity.
The models are within the acceptable range [-2,2] of the innovation plots, as discussed
in section 4.3.6. However, the models using the original sample size of the ISM in the variance structure
systematically exceed this range in the last part of the series.

5.4.3 Serial correlation

The confidence interval (4.26) for the Durbin Watson statistic is in this paper the range
[ √ √ ]
4 4
CIDW = 2 1, 96 , 2 + 1, 96 = [1.10, 2.90].
19 19
Accoring to the Durbin Watson statistics in table 13 and 14, there were no signs for serial correlation
or seasonality. The ACF plots lead to the same conclusion. For illustrative purposes, the ACFs of the
untransformed models are shown in figure 5 in appendix B. Recall from section 4.3.6 that the confidence
intervals for the ACF plots are given in (4.27).

5.4.4 Diagnostic tests for the subpopulations

The results for the diagnostic tests of M5 are reported in table 15. The normality tests do not show
sings of severe non normality. The heteroskedasticity statistics, show only one sign of heteroskedasticity
for the Total victimization series for M5 None Men. The remaining series do not show signs of heteroskedasticity.
Also for the subpopulations it holds that M5 , which has an adjusted sample size of the

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ISM, shows less heteroskedasticity than M5. The Durbin Watson tests do not point at serial correlation
for any series.

5.5 Model choice

Based on the results presented and evaluated in sections 5.3 and 5.4, in this section a model choice will
be made. From section 5.4 it can be concluded that the models with the adjusted sample size of the ISM,
denoted by , appear to show better results for diagnostic tests than the models with no adjustment.
This was already expected, since the oversampling by some local authorities went up to 180000, a big
increase compared to the other years. Therefore, it has been decided to disregard all models which used
the original sample size of the ISM in the variance structure.
The hyperparameter estimates for the remaining untransformed models are provided in table 11. Because
in the parameter estimation ˆσ2
r was assumed to be equal for all breakdowns, there is no need to add
a subscript here. The variance of the transition equation for the police series is estimated separately,
and is therefore also reported separately. As discussed in sections 4.2.3 and 4.2.4, the variance of the
measurement equation is time dependent since it is inversely proportional to the sample size, except for
the police series.

Table 11 – Hyperparameter estimates
Hyperparameter M1 None M2 None M3 None M4 None M5 None
ˆσ2
r 0.00 0.00 0.00 0.00 0.00
ˆσ2
r,expl - - - 0.00 -
ˆσ2
ϕ,violence 0.35 0.34 0.34 0.34 0.33
ˆσ2
ϕ,property1 0.34 0.34 0.34 0.34 0.33
ˆσ2
ϕ,property2 1.70 1.46 1.53 1.53 1.32
ˆσ2
ϕ,vandalism 0.64 0.56 0.67 0.67 0.46
ˆσ2
ϕ,no stop0.11 0.10 0.11 0.11 0.10
ˆσ2
ϕ,other 0.07 0.07 0.07 0.07 n.a.
ˆσ2
ϕ,total 0.92 0.87 0.88 0.88 0.73
ˆσ2
ɛ,expl - - - 0.01 -
* - ”Failure to stop after an accident”

The estimates in table 11 are more or less of the same magnitude between the models, which is a
sign for robustness.
M1-M4
Recall from section 5.3 that the coefficients of the intervention variable for Intervention 3 in M2 were
against the expectations. This means that the coefficients for Intervention 3 of other models should be
treated carefully. M2 will not be considered as a candidate for the best model since its only meaning
was to check for the coefficients of the intervention variable in Intervention 3. A further investigation
of M3 and M4 shows that the estimated coefficients in table 9 are of the same magnitude for all

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interventions. The hyperparameters of M3 and M4 in table 11 are almost equal to each other, with
the exception of the police series. Recall that the only difference between M3 and M4 is that in M4
the police series is included as an explanatory variable. As discussed in section 4.2.2, this might be a
sign that there is no structural break in neither stochastic trend, such that it is unlikely that this model
assumption is violated. This is a sign that M4 does not suffer from the problems with M2. Hence, due
to the information from the police series M4 is more robust, such that M4 is preferred to M3 .
Hence, M3 will be excluded and is not considered as a candidate any longer. The remaining models
are M1 and M4 .
Concerning the transformations, nor the log nor the logit transformations seem to improve the model fits.
The diagnostic tests did not show any significant improvements for those transformations. Additionally,
note that for the log and logit transformations one has to transform the results after applying (4.10)
to obtain the series with eliminated discontinuities. Hence, because the untransformed models provide
plausible results and are easier to interpret, it has been decided to exclude the transformed models from
consideration as well.
The remaining models are M1 None and M4 None. The estimated discontinuities in tables 8 and 9
for Interventions 1 and 2 for these models have almost the same magnitude. This again points out that
the estimates are robust. Recall from section 5.1 that since for M1 None and M4 None different
surveys have been used under Intervention 3, these coefficients cannot be compared.
The smoothed trend ˆµt = (ˆµt,1, . . . , ˆµt,k) which is contained in ˆαt, and the smoothed signal Zt ˆαt are
shown in figures 6 - 10 in Appendix C. Comparing the graphs of the smoothed signals of M1 None
and M4 None, the latter one fits the original data ˆyt more closely. Therefore, M4 None is preferred
to M1 None. This finding, in combination with the robustness obtained from the correlation with the
police series, leads us to conclude that the model with the best fit to the victimization data is M4
None.
Recall that the estimated discontinuities ˆ
βk are presented in tables 8 and 9, and represent the discontinuity
of the survey considered with the LPSS. Then, as discussed in section 4.2.3, using (4.10) it is
possible to obtain the series with the eliminated discontinuities. Note that here the LPSS is seen as the
starting point. Hence, the numbers obtained under the LPSS survey remain unchanged, and the values
obtained under the other surveys are recalculated using (4.10) such that they are comparable with the
LPSS numbers.
M5
For M5 and M5 the constraint (4.14) for the gender subpopulations was introduced in section 4.2.4 in
order to preserve the consistency of the series for the total population. This approach reduces the number
of hyperparameters to be estimated, and makes the model more parsimonious. Also for the gender
subpopulations it was assumed that the variances of the irregular terms of the stochastic trend are equal

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among the k breakdowns.
M5 shows better results than M5 in the diagnostic tests in section 5.4. Therefore, M5 is considered as
the model with the best fit for the subpopulations. The estimated discontinuities for M5 are provided
in table 10, the estimates for the variances of the transition and measurement equations in table 11, and
the graphs of the smoothed trends and smoothed signals are given in figures 11 and 12 in Appendix C.
It is possible to extend M5 by including the difference ˆyISM,t ˆySMIV ,t and the police series. In such
one obtains the numbers for the gender subpopulations for the model with the best fit on national level.

6 Conclusion

Continuous time series are of great importance for national statistical offices. Several parties, including
politicians and policy makers, are interested in the development of a series over time. A survey
redesign might lead to discontinuities after an intervention has occurred, and therefore it is no longer
possible to compare survey outcomes over time. This paper applies this issue to a real life application by
using the victimization dataset provided by Statistics Netherlands. A structural time series framework
with an intervention variable is used to model the series. After this framework is put in a state space
representation, the Kalman filter is used to estimate the discontinuities.
Several extensions have been considered in this paper. First, the series has been extended with explanatory
information available from a parallel run of the SMIV survey, after that the police series has been
added as a source from outside Statistics Netherlands. Additionally, the different gender effects have
been discussed.
However, more extensions are available. First, the structural time series framework can be applied to
a different dataset, e.g. the number of offenses to the Dutch citizens. This dataset is different from the
victimization dataset in the sense that the number of breakdowns sums up to the total value. In such,
one could investigate whether there are any differences in the development of the series over years. For
example, when the number of offenses is going up with a larger percentage than the victimization series
this would mean that the people that are are suffering from offenses are suffering increasingly from multiple
offenses. On top of that, one could consider a larger time span compared to the victimization series.
An other extension is to consider more subpopulations. In this paper it was only possible to consider the
gender subpopulations due to constraints in data availability. However, one could also investigate the
effects of age, or living in an urban area vs. living on the countryside. It is also desirable to implement
the constraint for gender subpopulations for the model with the best fit for the series at national level,
M4 None. Finally, as proposed by Harvey and Durbin (1986), an extension to an ARIMA model can
be made, which is seen as the main alternative to the state space models.
The estimates for the discontinuities in this paper were generally significant for series with the highest

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value of victimization - ”Property”, ”Vandalism” and ”Total”. Apparently, these are the most sensible
series for survey redesigns and should be treated with care when adjusting surveys. Then, one should
be careful when there are several survey redesigns within a short time span. In these cases it is more
likely that the model assigns developments in the trend erroneously to the coefficient of the intervention
variable. An other interesting result was that the male population was much less sensitive for survey
redesigns than women, at least for the dataset considered in this paper. This in turn creates new topics
to be investigated, making it an interesting field of research. This paper can be used as a start up for
investigating these findings more extensively.

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