# Discontinuities due to survey redesigns : a structural time series approach

4.3.3 Missing values

Independent of the type of information set the analysis is based on, either filtering or smoothing, one

has almost always to deal with practical problem of missing data. For several reasons, it is difficult to

obtain complete datasets, generally because of the processing time the statisticians need to publish the

data. Using the Kalman filter, one can easily deal with this problem.

Suppose that the data ˆy_{j}, with j = {t_{m}_{1}, . . . , t_{m}_{2 }− 1} are missing for 1 < t_{m}_{1 }< t_{m}_{2 }≤ t. To get the

equation for smoothing and filtering for these missing values, note that a missing observation implies

that the innovations v_{t }and the corresponding inverse variance matrix F^{−}^{1}

t

this to the Kalman fliter in (4.18) leads to

are equal to zero. Applying

a_{t}_{+1 }= Ta_{t}, P_{t}_{+1 }= TP_{t}T^{′ }+ Q_{t}, t = t_{m}_{1}, . . . , t_{m}_{2 }− 1 ^{, }^{(4.20)}

and similarly the smoothing recursions in box (4.19) become for the missing observations

w_{t}_{−}_{1 }= T^{′}w_{t}, N_{t}_{−}_{1 }= T^{′}N_{t}T, t = t_{m}_{2 }− 1, . . . , t_{m}_{1}

. (4.21)

Other relevant formulas for smoothing do not change and can be applied directly.

4.3.4 Diffuse priors

In the previous sections it was assumed that the distribution parameters a_{1 }and P_{1 }of α_{1 }were

known. In such, the recursive formulas (4.18) and (4.19) could be applied. However, in many real life

applications this is an unrealistic assumption, usually the distribution of α_{1 }is not known. Durbin and

Koopman (2001) proposes to represent α_{1 }as having a diffuse prior density. That means that one should

fix a_{1 }at some value and let P_{1 }→ ∞. Then, the Kalman filter determines the values starting from k = 2

and after that the standard recursive formulas for filtering and smoothing can be applied. This process

is called diffuse initialization of the Kalman filter.

4.3.5 Parameter estimation

The covariance matrices H_{t }and Q_{t }in 4.17 are not known since they cannot be observed. However,

the Kalman filter assumes these parameters, also called hyperparameters, are known and therefore they

have to be estimated using maximum likelihood estimation. Depending on whether the initial conditions

are known or diffuse, several forms of the likelihood function exist. Because of the assumption that the

innovations are Normal and i.i.d. distributed, the likelihood function for the Normal distribution can be

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used. Then, let the classical likelihood be represented by

t_{∏}

L(ˆy) = p(ˆy_{1}, . . . , ˆy_{t}) = p(ˆy_{1}) p(ˆy_{t}| ^{ˆ}Y_{t}_{−}_{1}),

t=2

where ^{ˆ}Y_{t}_{−}_{1}= {y_{1}, . . . , y_{t}_{−}_{1}} is the information set for y up until t − 1. Instead of the original likelihood

usually the loglikelihood is considered :

t_{∑}

log L(ˆy) = log p(ˆy_{t}| ^{ˆ}Y_{t}_{−}_{1}),

t=1

where p(ˆy_{1}| ^{ˆ}Y_{0}) = p(ˆy_{1}).

Relating to the model considered in this paper, first assume that the initial state vector has a known

distribution N(a_{1}, P_{1}). Note that E(ˆy_{t}| ^{ˆ}Y_{t}_{−}_{1})= Z_{t}a_{t}, v_{t }= ˆy_{t }− Z_{t}a_{t }and F_{t }= V ar(ˆy_{t}| ^{ˆ}Y_{t}_{−}_{1}), substitute

N(Z_{t}a_{t}, F_{t}) for p(ˆy_{t}| ^{ˆ}Y_{t}_{−}_{1}) in the log likelihood function above such that it becomes

t × k

log L(ˆy) = − log (2π) −

2

1

2

t_{∑}

t=1

(log |F_{t}| + v^{′}

t^{F}^{−}^{1}_{t }v_{t}), (4.22)

where F_{t }is nonsingular. Because the quantities F_{t }and v_{t }are computed by the Kalman filter, the log

likelihood is easily computed from the Kalman filter output.

In the case not all elements of the initial state vector are known, the log likelihood is changed as well

since it has to be adapted for its missing elements. More specific, log L(v_{t}) becomes log L(v)_{d }to account

for d unknown diffuse hyperparameters :

t × k

log L(ˆy)_{d }= − log (2π) −

2

1

2

t_{∑}

t=d +1

(log |F_{t}| + v^{′}

t^{F}^{−}^{1}

t ^{v}^{t}^{)}^{.}

Hence, Koopman et al. (2008) assumes that for diffuse priors the summation in (4.22), which runs from

1 to t, should start with d +1 since for the first d summations the sum will be approximately zero. Now,

the Kalman filter can be applied since the hyperparameters are estimated. The details of the derivations

and extensions are discussed in chapter 7 of Durbin and Koopman (2001).

4.3.6 Diagnostic Tests

The core of the foregoing Gaussian state space model relies the assumption that the disturbances ϕ_{t}

and η_{t }are Normally distributed and serially independent with constant variances. To check for these

assumptions, this section provides several tests. First, note that the standardised one-step forecast errors,

defined as

u_{t }= ^{v}^{t}

^{√ }, t = 1, . . . , t. (4.23)

f_{t}

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also have to be normally distributed and serially independent. Here, the standard error of v_{t }is an entry

in the matrix F_{t }and denoted by f_{t}. Note that in the diffuse case, (4.23) starts at t = 2 for all tests

considered in this section. The following tests can be applied.

Normality

The first four moments are defined as

m_{1 }= ^{1}

t

t_{∑}

t=1

u_{t}, m_{q }= ^{1}

t

t_{∑}

t=1

(u_{t }− m_{1})^{q }, q = 2, 3, 4.

Then, the standard statistic indicators skewness S and kurtosis K are defined as

S ≡

m_{3}

√_{m}

3

2

, K ≡ ^{m}^{4}

m^{2 }^{.}

2

When these assumptions are valid, S should be around 0 and K around 3, or

S ∼ N

(_{0}_{, }_{6 }) (

, K ∼ N 3,

t

24 ^{)}

t

Usually for convenience instead of the kurtosis, the excess kurtosis is considered which is defined as

EK = K − 3. Standard statistical tests can be performed to check for significance of skewness and

kurtosis. So the confidence intervals for S and EK are :

√ √ √ √

6 6 24 24

CIS = [−1, 96 , 1, 96 ], and CIEK = [−1, 96 , 1, 96 ] (4.24)

t t t t

If the skewness and kurtosis are within the boundaries of their confidence intervals, there is no reason to

suspect them of violating the normality assumption.

Additionally the so-called QQ-plot is a graphical method to compare two distributions. In this paper,

the ordered residuals of the standardized innovations are plotted against their theoretical quantiles of

a Normal distribution. Under perfect circumstances the match is one-to-one and is represented by a

45-degree line. Hence, the further away the QQ-plot from the 45-degree line, the worse is the match of

the ordered residuals.

Heteroscedasticity test

The distribution of the standardized forecast errors u_{t }is assumed to be homoscedastic, or have the same

finite variance. A simple test for heteroskedasticity is defined as

HSK(b) is F ^{b}

b

HSK(b) =

∑_{t}

t=t−b+1 ^{u}^{2}

t

∑_{b}

t=1 ^{u}^{2}

.

t

distributed with b degrees of freedom, under the null of homoscedasticity. b has to be

chose in such a way that the standardized innovations are roughly divided in three parts. In this paper

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b = 7, such that it results in the following confidence interval :

CIHSK = [F ^{7}

7( ^{a}

2 ^{)}^{, F }^{7}

7(1− ^{a}

2 ^{)}^{]}^{. }^{(4.25)}

Here F refers to the F distribution and a is the significance level, which is equal to 0.05 in this paper.

Serial correlation

Durbin-Watson (DW) tests have been performed to check for serial correlation :

DW =

∑

t

_{t}_{=2+}_{d}(u_{t }− u_{t}_{−}_{1})^{2}

∑_{t}

t=1+d ^{u}^{2}

,

t

where d is the number of innovations skipped due to the initialization of the Kalman filter, in this paper

d=1. It should hold that DW ∼ N(2, ^{4}). The confidence interval is computed as

t

√ √

4 4

CIDW = [2 − 1, 96 , 2 + 1, 96 ], (4.26)

t t

with a significance level a = 0.05 (van den Brakel et al., 2011). Additionally, ACF graphs can be used

to check for serial correlation as well. The confidence intervals for the ACF graphs are computed as

[− ^{2}

√_{t }_{, }^{2}

√

_{t }]. (4.27)

One-step forecast errors

Finally, the one-step forecast errors u_{t }can be plotted against the time span, such that graphical outliers

can be detected. Even though it is not a test, it might help in case an extreme observation distorts the

analysis of entire sample. Generally the values of u_{t }should be between −2 and 2.

5 Application and results

In this section, the structural time series approach as described in section 4 is applied to the victimization

data set as described in section 3. The programming language used to build the model is Ox, and

the Kalman filter is implemented using the specialized library of Ox called Ssfpack 3.0. The reference

for this library is Koopman et al. (2008), where all the necessary functions and input requirements are

described. Note that since no information was available on the starting values, diffuse priors have been

used. The system matrices T and Z_{t }as described in the previous section are stacked in one matrix Φ and

the covariance matrices H_{t }and Q_{t}, are stacked in one matrix called Ω. If the target parameters are not

correlated, Ω is a diagonal matrix, if they correlate there are coefficients at the respective off-diagonal

entries. The missing values such as the values for Failure to stop after an accident for ISM, can be

modeled by leaving the entries empty in the original dataset.

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The maximization procedure as described in 4.3.5 is already implemented as the numerical maximization

function MaxBFGS in the standard Ox library maximize, and therefore can be applied directly.

The MaxBFGS function, named after Broyden-Fletcher-Goldfarb-Shannon, also returns the degree of

convergence ranging from ”no convergence” to ”strong convergence”.

This section is set up as follows. First, the basic models used in this section will be described in section

5.1. Some improvements for these models are presented in section 5.2. Then, section 5.3 will present the

estimated discontinuities obtained using the Kalman filtering and smoothing recursions. Section 5.4 in

turn discusses the diagnostic tests performed on these models, whereas section 5.5 will present the final

model choice.

5.1 Description of the models

For the victimization time series four models have been developed, starting from a basic one and

building up to more complex ones. As aforementioned, it is assumed that the series can be decomposed in

a stochastic trend, a level intervention variable and an irregular term. Additionally, explanatory variables

can be added. No seasonal or cyclical components are included since the data is observed yearly, as it

has been discussed in section 4.2.2. Next to these four models, the subpopulations have been investigated

as well, as discussed in section 4.2.4. The dataset consists of the surveys LPSS, PSLC, SM and ISM,

conducted in the years 1992 to 2010. In such, there have been three survey redesigns which are denoted

in this section by

– Intervention 1 : LPSS to PSLC,

– Intervention 2 : PSLC to SM,

– Intervention 3 : SM to ISM.

The estimated coefficient ^{ˆ}

β_{k }of the intervention variable models the difference between the introduced

survey, hence the PSLC, the SM or the ISM, and the LPSS.

The first model called M1, consists of the original victimization series. Its time span is from 1992 to

2010, containing the surveys LPSS, PSLC, SM and ISM. The state space representation of M1 is given

in (4.7), with its entries as defined in (4.9). The total victimization series is divided in five breakdowns.

As a result, ˆy_{t }contains six series, such that k = 6.

The second model M2 coincides with M1, except that in the period 2008 - 2010 instead of the ISM the

SM_{IV }is used. Recall from section 3.1 that the SM_{IV }only differs from the SM by a lower sample size.

Therefore, M2 can be seen as a model check since it is expected that there are no discontinuities between

SM and SM_{IV }. Also for M2, k = 6.

The third model M3 again contains the surveys LPSS, PSLC, SM and ISM. Additionally, in M3 an

explanatory series is introduced in the form of a time varying intervention variable as defined in (4.11).

The state space representation of M3 is given in (4.7), with its entries as defined in (4.9) except for

34

Intervention 3 where δ^{i}_{t }is defined in (4.11) with ˆy_{dif,t }= ˆy_{ISM,t }− ˆy_{SM}_{IV }_{,t}. Here, the SM_{IV }is defined in

section 3.1. The reason for choosing for the difference between the ISM and the SM_{IV }as an explanatory

variable in the intervention variable, is because this difference denotes the discontinuity between ISM

and SM_{IV }. This might result in better model fits. Also for M3, k = 6.

Model four, denoted as M4, expands M3 by adding the police series as described in section 3.2.2 to the

model. The state space representation of M4 is given in (4.7), with its entries as defined in (4.12), where

there is correlation between the irregular terms of the trends of total victimization series and police series.

Similar to M3 the difference ˆy_{dif,t }= ˆy_{ISM,t }− ˆy_{SM}_{IV }_{,t }is also implemented in the intervention variable

for Intervention 3 as defined in (4.11). Note that for M4 the police series is treated as an explanatory

variable such that k = 7. The implementation of M4 has been discussed in detail in section 4.2.4.

At last an extension to subpopulations has been implemented in M5. The subpopulations classified on

age were not comparable due to different age classes between surveys, and the data for urbanization

were not available. Therefore, the gender subpopulations male and female have been implemented, to

check for differences in discontinuities in subpopulations. Recall from section 3.2.1 that the data for the

subpopulations ranges from 1992 to 2009, since the data for 2010 is not available. The model applied to

M5 is (4.7), with its entries as defined in (4.15). As discussed in section 3.2.1, series for gender differed

with the original victimization series since instead of five there are only four breakdowns, the breakdown

”Other offences” is left out. For the gender subpopulations H = 2, such that as discussed in section

4.2.4, k = 5 × 3 = 15.

Several transformations have been applied to the original victimization series as provided by Statistics

Netherlands. First, the original dataset has been divided by 100 to prevent numerical overflows of

the maximization procedure. The presented results have transformed back by multiplication with 100

again. Then, within the models M1 to M4, a log and a logit transformation has been applied to the

series, namely ln(ˆy_{t}) and ln(

ˆy_{t}

100−ˆy_{t}

). The logit transformation is developed for proportions, such as the

victimization series. For log and logit transformations M3 and M4 have to be adjusted in the following

way. To get the difference ˆy_{dif,t }= ˆy_{ISM,t }− ˆy_{SM}_{IV }_{,t }in the intervention variable (4.11), it has to be put

as a ratio : ˆy_{dif,t }=

ˆy_{ISM,t}

. Taking the log of this ratio when implementing it in the model, results in

ˆy_{SMIV ,t}

the difference ln(ˆy_{dif,t}) = ln(ˆy_{ISM,t}) − ln(ˆy_{SM}_{IV }_{,t}).

As discussed in section 3.1, the amount of oversampling by local authorities in the ISM is fluctuating

sharply regionally. Also, CAPI was left out as a data collection method. In this case, the assumption that

variance estimates are inversely proportional with sample size might be invalid. Therefore, for models

M1, M3, M4 and M5, next to the original file for sample sizes an adapted file for sample size has been

used where the sample sizes of the ISM are adjusted to 19000 for all years, marked by ⋆. This is the

sample size of the part of the ISM conducted by Statistics Netherlands, as discussed in section 3.1. It is

35

expected that this adjustment might lead to more plausible results.

5.2 Model improvements

5.2.1 Likelihood ratio tests

This section discusses the applied Likelihood Ratio Tests as discussed in section 4.2.5. Recall that in

M1 - M5 every variance of an irregular term is considered separately. That is, for every breakdown k,

both the the variances of the irregular terms of the transition equation and the measurement equation

are estimated independent from each other. The number of hyperparameters to be estimated in such

becomes 12 for models M1, M2 and M3, 15 for model M4 and 10 for M5. To lower the number of

hyperparameters to be estimated, it has been considered to assume that the variances of the irregular

terms of either the measurement equation or the transition equation are equal among the k breakdowns.

These constraints are represented in (4.16). They can be easily implemented in the multivariate state

space model described in section 4 by setting the hyperparameters equal to each other in the estimation

process. To test for this assumption, the classical Likelihood Ratio Test (LRT) can be used. Here, the

variances of the irregular terms are set equal to each other under the null, and unrestricted under the

alternative. The ratio

( _{)}

likelihood(null)

LR = −2ln _{likelihood}_{(}_{alternative}_{)}

(

= −2 ln^{(}likelihood(null)^{) }− ln^{(}likelihood(alternative)^{))}

is then chi-square distributed, with 5 degrees of freedom M1-M4 and 4 degrees of freedom for M5. Note

that for M4, the variances of the police series have not been restricted, such that also for M4 there are

5 degrees of freedom. The results are provided in table 5.

Table 5 – Results Likelihood Ratio Tests

Model LR measurement equation^{∗ }LR transition equation^{@}

M1 29.60 0.46

M1 ⋆ 43.60 1.10

M2 36.66 2.56

M3 54.14 9.98

M3 ⋆ 51.56 Out of memory

M4 50.44 5.48

M4 ⋆ 50.96 5.48

M5 Out of memory 0.00

M5 ⋆ Out of memory 0.00

∗ - the variances of the irregular terms of the measurement equation are set equal under the null.

@ - the variances of the irregular terms of the transition equation are set equal under the null.

Bold value - The null that the variances of the irregular terms are equal can be rejected.

The results in table 5 point out that the null hypothesis of setting the variances of the measurement

equation equal can be rejected. Hence, setting these variances equal would worsen the fit of the model.

36

Note that since M5 and M5 ⋆ ran out of memory with the applied restriction, the LRT could not be

performed for these models. The null hypothesis of setting the variances of the transition equation equal

could not be rejected for no model. The LRT could not be performed on M3 ⋆ since the program ended

with an out of memory message. Following the results of Likelihood Ratio Tests, it has been decided to

set the variances of the irregular terms of the transition equation equal to each other in Q_{t}.

5.2.2 Adjustment of variance structure

The models M1-M5 as described in section 5.1 have been implemented with the restriction on the

variances of the transition equation as discussed in section 5.2.1. The resulting diagnostic tests showed

severe signs of heteroskedasticity for the Property series for almost all models. When looking at figure 1,

it seems that a change in the variance of Property in 2005 could cause this heteroskedasticity. Therefore

it has been decided to model the variance of the measurement equation for Property separately in the

years 2005-2010, and check whether this adjustment solves the heteroskedasticity problem. The adjusted

variance of the measurement equation in H_{t }is implemented as

⎧

⎪_{⎨}

V ar(ϕ_{t,P roperty}) = _{⎪⎩}

1

n_{t }^{(}^{σ}^{2}_{ϕ,P }) if 1992 ≤ t ≤ 2004

roperty_{1}

1

n_{t }^{(}^{σ}^{2}_{ϕ,P }) if 2005 ≤ t ≤ 2010

roperty_{2}

, (5.1)

for M1, M2, M3 and M4. For the series on national level of M5 it holds that

⎧

⎪_{⎨}

V ar(ϕ_{t,P roperty}) = _{⎪⎩}

1

n_{t }^{(}^{σ}^{2}_{ϕ,P }) if 1992 ≤ t ≤ 2004

roperty_{1}

1

n_{t }^{(}^{σ}^{2}_{ϕ,P }) if 2005 ≤ t ≤ 2009

roperty_{2}

, (5.2)

and for the male and female subpopulations of M5 it holds that

V ar(ϕ_{t,P roperty,}_{h}) =

⎧

⎪_{⎨}

⎪_{⎩}

1

n

(σ

t,h

2_{ϕ,P }) if 1992 ≤ t ≤ 2004

roperty_{1}

1

n

(σ

t,h

2_{ϕ,P }) if 2005 ≤ t ≤ 2009

roperty_{2}

. (5.3)

The results for the the heteroskedasticity tests under both the original and adjusted variance of the

irregular term of the measurement equation for the Property series are given in table 6. Note that the

confidence interval for the heteroskedasticity test as defined in section 4.3.6 is equal to

CIHSK = [F ^{7}

7( ^{0}^{.}^{05}

2

)^{, F}

7

7(1− ^{0}^{.}^{05}

2

_{)}] = [0.20, 5.00].

Following the results in table 6, it can be concluded that there are no signs for heteroskedasticity in

the Property series with the adjusted variance model. Therefore, it has been decided to implement this

adjustment for all models to estimate the discontinuities as described in section 4.3. Note that with

this assumption there are no ”out of memory” messages present. The heteroskedasticity test and the

37

remaining diagnostic tests will be discussed in depth in section 5.4.

Model

Table 6 – Results for the heteroskedasticity tests for M1-M5

Heterosked. Breakdowns

test Violence Property Vandalism No stop* Other ^{Total}^{„}

M1 No adjustment 1.16 10.17 5.12 0.30 0.71 4.46

M1 ⋆ No adjustment 0.71 8.35 2.80 0.30 0.50 2.77

M2 No adjustment 0.66 5.57 1.31 0.28 0.40 2.07

M3 No adjustment 0.77 8.36 4.67 0.29 0.46 3.00

M3 ⋆ No adjustment Out of memory.

M4 No adjustment 0.77 8.35 4.66 0.29 0.45 3.08(0.29)

M4 ⋆ No adjustment 0.66 7.96 2.98 0.29 0.46 2.65(0.29)

M5 No adjustment

National 1.32 12.39 4.19 0.59 - 4.25

Men 2.97 8.70 2.49 0.76 - 5.41

Women 0.69 3.60 3.63 0.79 - 2.62

M5 ⋆ No adjustment

National 0.91 11.15 3.36 0.59 - 3.42

Men 1.98 7.13 1.28 0.76 - 3.41

Women 0.65 3.40 3.42 0.79 - 2.44

M1 Adjusted var.^{@ }1.16 2.73 5.12 0.30 0.71 4.46

M1 ⋆ Adjusted var.^{@ }0.71 2.82 2.80 0.30 0.50 2.77

M2 Adjusted var.^{@ }0.66 1.31 1.31 0.28 0.40 2.07

M3 Adjusted var.^{@ }0.77 3.14 4.67 0.29 0.46 3.00

M3 ⋆ Adjusted var.^{@ }0.66 2.95 2.99 0.30 0.46 2.58

M4 Adjusted var.^{@ }0.77 3.34 4.67 0.29 0.46 3.08(0.29)

M4 ⋆ Adjusted var.^{@ }0.66 2.94 2.98 0.30 0.45 2.64(0.29)

M5 Adjusted var.

National^{… }1.32 4.46 4.19 0.59 - 4.25

Men^{$ }2.97 2.22 2.49 0.76 - 5.41

Women^{$ }0.69 1.87 3.63 0.79 - 2.62

M5 ⋆ Adjusted var.

National^{… }0.91 4.13 3.36 0.59 - 3.42

Men^{$ }1.98 2.25 1.28 0.76 - 3.41

Women^{$ }0.65 1.68 3.42 0.79 - 2.44

* - ”Failure to stop after an accident”, ⋆ - Sample size of ISM is adjusted to 19.000,

„ - Police series in brackets, @ - The variance of the Property series is adjusted to (5.1),

… - The variance of the Property series is adjusted to (5.2).

$ - The variance of the Property series is adjusted to (5.3).

Bold value - the outcome is outside the confidence interval for the statistic.

The results of the Likelihood Ratio Tests for the models with the adjusted variance of the measurement

equation of the Property series are presented in table 7. These results also show that the null of setting

the variances of the transition equation equal to each other could not be rejected for no model. Therefore,

also in the models with the adjusted variance for the Property series, the variances of the irregular terms

of the transition equation are set equal in the estimation process described in section 4.3.5.

38

Model

Table 7 – Results Likelihood Ratio Tests with adjusted variance for Property

LR transition equation^{@}

M1 1.45

M1 ⋆ 0.44

M2 0.88

M3 9.84

M3 ⋆ 3.92

M4 10.51

M4 ⋆ 5.86

M5 0.00

M5 ⋆ 0.00

@ - the variances of the irregular terms of the transition equation are set equal under the null.

Bold value - The null that the variances of the irregular terms are equal can be rejected.

5.3 Estimated discontinuities

The estimated discontinuities ^{ˆ}

β_{k }are presented in tables 8, 9 and 10. Within the tables, the discontinuities

which are significantly different from zero are made bold.

Transformation None refers to the untransformed series. The log and the logit models are reported

in the log and logit scale, respectively. Initially it was planned that the logit transformation would

be applied to all models. However, after the results for M1 and M2 pointed out that these results

were almost identical to the results with the log transformation, it has been decided to disregard the

logit transformation for M3 and M4. Furthermore, for all models discussed in this paper the parameter

estimation converged ”strong”, meaning that the model was a good fit for the data.

5.3.1 Estimated discontinuities M1-M4

The estimated discontinuities for the models M1 - M4 are provided in tables 8 and 9. For those models

it is directly seen that for the breakdowns Violence, Failure to stop after an accident and Other there

were no significant discontinuities. Hence, it appears that the series Violence, Failure to stop after an

accident and Other are less sensitive for survey redesigns. These are exactly the three series with the

lowest percentages in the dataset. This might explain why the coefficients of the intervention variables

for these series are not significant, since series with lower percentages are also expected to have lower

discontinuities such that they are less likely to be significant.

The breakdowns Property, Vandalism and Total victimization, on the contrary, appear to be more

sensitive to survey redesigns. The first two series are the series with the highest rates, and the latter

one is the total victimization percentage. For Intervention 1, from LPSS to PSLC, only the Vandalism

series is significant for some models. Recall from section 3 that the main difference between these surveys

was that the LPSS focused on victimization, whereas the PSLC was a more general survey. However,

most questions from the LPSS remained the same in the PSLC. This might explain why this redesign

39