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An Empirical Analysis for Austrian Data

Gerhard Thury,MichaelWfiger

Zusammenfassung

Realisationen yon Zeitreihen werden von Sonderfaktoren oft stark beeinflu~t. Beobachtun-

gen yon bkonomischen Grbl3en sind davon nicht ausgenommen. Die Ank(Jndigung und das

Inkraftsetzen neuer Verordnungen, grbi%ere Anderungen in der Wirtschaftspolitik oder der

Steuergesetzgebung und ~hnliche Ereignisse kbnnen Anlal~ f0r betr~chtliche Verzerrungen

in bkonomischen Zeitreihen sein. Das Auftreten solcher Ausreil~er fi3hrt zu falsch identifi-

zierten Modellen und schlecht gesch~tzten Modellparametern, was sich letztlich in unge-

nauen Prognosen und falschen Empfehlungen f0r die Wirtschaftspolitik niederschl~gt. Bis-

her wurde die Existenz einer Ausreil3erproblematik bestenfalls in einer Fu~note angemerkt,

ansonsten aber ignoriert, well das notwendige Instrumentarium f0r eine erfolgversprechen-

de Behandlung des Problems fehlte. In der Zwischenzeit wurden Verfahren for die Aus-

schaltung von Ausreil~ern entwickelt, die in der vorliegenden Arbeit auf ihre praktische

Brauchbarkeit getestet werden. Wir wenden diese Verfahren auf drei ~konomische Zeitrei-

hen, namlich Einzelhandelsumsatze, Kauf yon dauerhaften Konsumg0tern und Autok~iufe

an. Diese Zeitreihen werden deshalb gewahlt, well sie eine Anzahl yon Ausreil3ern aufwei-

sen, deren Ursache, Datierung und Grbr~enordnung bekannt ist. Es I~l~t sich somit 0ber-

pr0fen, ob die getesteten Verfahren in der Lage sind, diese Ausreil3er aufzusp0ren und zu

eliminieren.

Abstract

Time series are often subject to the influence of non-repetitive events. Economic variables

make here no exception. For example, the announcement and implementation of new

regulations, major changes in economic policy or in the tax legislation, and similar events

may cause substantial disturbances in economic time series. The presence of outliers may

lead to wrongly identified models and inappropriately estimated model parameters giving

rise to poor forecasts and erroneous conclusions. In the past, these problems had mostly

to be ignored, because simple yet efficient techniques for the treatment of outliers did not

exist. The situation improved slightly when B o x - T i a o (1975) proposed intervention analy-

sis. However, the fact that a detailed knowledge of the structure of the series to be ana-

lysed is required for a successful application of this technique, is a severe restriction for its

use in practical work. But, in the meantime, there exist already techniques which solve the

outlier problem more or less automatically. For a detailed discussion of these techniques

Empirica - AUSTRIAN ECONOMIC PAPERS

and their computer implementation see Chen - Uu - H u d a k (1990). It is the aim of this pa-

per to gain information on the reliability of these methods in practical situations. For this

purpose, we apply them in the analysis of three Austrian economic time series, namely re-

tail sales, purchases of durables, and car purchases. We believe that these series are well

suited for our objective. They are strongly contaminated by outliers and, additionally, there

already exist sophisticated intervention models which can serve as benchmarks in the

comparison.

proposed by B o x - Jenkins (1976) is often used in applied work to describe the behaviour

of seasonal time series. Here, Yt is an observed time series, which is not contaminated by

the presence of outliers, and a t is a white noise series, a ( B ) , O(B), q)(B), O(BS), (k(B'),

a* (B') are polynomials in the backshift operator B. All roots of 0 (B), r (B), O (B s), r (BS)

lie outside the unit circle, and all roots of a (B) and a * (B s) are on the unit circle.

It became clear very soon, however, that this model class is too restrictive to adequately

describe most economic time series since these series are heavily contaminated by the

presence of calendar effects and outliers. To assume that an observed series Z t follows the

more general model

(2) Zt = Yt + f ( t )

might help to overcome these problems. Here, Yt follows the above mentioned ARIMA

model, a n d f ( t ) is a parametric function capturing calendar and outlier effects.

In the present paper, we concentrate primarily on outlier effects and, therefore, the form of

f ( t ) is determined by the type of outlier present. We consider four types of outliers. These

are additive outliers (AO), innovational outliers frO), level shifts (LS), and temporary

changes (TC). To simplify presentation, we use in the theoretical part of this paper a simple

model of the form

The effect of each type of outlier can often be seen more clearly in a simulated series.

Therefore, n = 65 observations are simulated from the model

(1-0.6B)Y t = at

Outlier Detection and Adjustment

Figure 1

6.0

5.0

4-.0

,3.0

2.0

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Figure 2

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7 15 19 25 31 57 45 49 55 61

Empirica - AUSTRIAN ECONOMIC PAPERS

with

o"a = 1.0.

An additive outlier (AO) is an event that affects a series for one time period only. A record-

ing error is an example of an AO. For an AO, the f u n c t i o n f ( t ) becomes wA I t (T), and the

observed series is given by the model

(4) Zt = Yt+w41t(T),

where I t (T) is 1 at t = T and 0 otherwise, and w.4 measures the size of the error. To illus-

trate the effect of an AO on an observed series, we add an AO at time T = 30 with wA = 5.0

to our simulated AR (1) process. The result is shown in Figure 2. We see that all observa-

tions are unchanged, except the value at t = 30 is now 5.318 (instead of 0.318 previously).

The effect of an I 0 on a time series is more intricate than that of other types of outliers.

The model for the observed series is given by

(5) Zt = 0 (B)

~b(B){a,+ w i I t (T)}

= Yt+wzlr

As indicated by its name, an lO affects the innovation of a process, as the noise series a t is

called sometimes in the literature. From equation (5) we see that, when an I 0 occurs at

t = T, the effect of this outlier on Zr+k, for k _> 0, is equal to wx ~rk where w1 is the initial effect

and ~k is the k-th coefficient of

o@

9z(B) = (1 + ~ I B + ~1B2+...) = r

For a stationary series, an I 0 will produce a temporary effect since the ~ decay to zero ex-

ponentially. A graph of our simulated AR (1) process with an I 0 at t = 30 and w] = 5.0 is

shown in Figure 3. We observe that the values for t = 30, 31, and 32 are all above their

corresponding values of Figure 1. Closer analysis reveals that the effect can be observed

through t = 43.

Outlier Detection and Adjustment

Figure 3

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Figure 4

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Empirica - AUSTRIAN ECONOMIC PAPERS

A level shift produces an abrupt and permanent step change. The model can be written as

1

(6) Zt = Yt + ' - ~ - ~ w z It(T)

= Yt+wLSt(T),

where St (T) is a step function such that Se (T) has the value 0 for t < T and the value 1 for

t>__ T. The effect of an LS on our simulated AR (1) process is shown in Figure 4. We note

that the level shift causes a jump in the mean level of the series from t = 30 onwards. The

deviations from these mean levels remain unchanged.

A temporary change is an event having an initial impact that decays exponentially accord-

ing to some dampening factor 8. We can represent the observed series as

1

(7) Zt = Yt+-~-~wclt(T), 0 < 8 < 1.

Figure 5

Temporary. change at t = 30 in a simulated A R (1) process

6.0

4.8

3.6

2.4

1.2

0.0

-1.2

-2.4

7 1,3 19 25 ..31 37 4,3 49 55 61

We see that AO and LS are two boundary cases of a TC, where 8 = 0 and 8 = 1, respec-

tively. For illustrative purposes, we include a TC at time t = 30 with Wc= 5.0 and 8 = 0.8

Outlier Detection and Adjustment

into our simulated AR (1) process. The result is displayed in Figure 5. The graph looks

similar to that of ] 0 . Since ,~ is relatively close to 1, the effect of the outlier is discernible for

a number of periods (here through to t = 59).

Our above illustrative graphs of the effects of different outlier types on observed series

might nourish the erroneous belief that it is possible to spot outliers by a close inspection

of graphs of the contaminated series. For a simple AR (1) process useful information will

be obtained. But, for more complicated processes as they are encountered in practice,

there is not much to be gained from such an inspection. The residuals of a fitted model,

however, might provide useful information for detecting outliers. As we mentioned earlier,

outliers in time series can give rise to wrongly identified models and poor estimates of the

model parameters. Therefore, it is unclear how useful these residuals really are for outlier

detection. To decide this question, we first derive the impact of outliers on an "ideal" resid-

ual series when the underlying A R M A model as well as all of its parameters are known. We

can then contrast this situation with the information we can obtain from the actual residuals

of a fitted model

(8) et = = ( B ) Z~,

A R M A model. When this model and its parameters are known, these weights can be calcu-

lated by equating coefficients in

If we observe a single outlier at time t = T, we can rewrite et according to the type of outlier.

We have

WL

6' t = l_B=(B) lt(T)+at for an L%

~4~C

et - 1 - J~B ~(B) I t (T) + a for a TC.

Empirica - AUSTRIAN ECONOMIC PAPERS

Wc, i = 1,

wz, i=2,

(11) et = wi xi t + at, where w~ =

wL , i=3,

Wc, i = 4.

where x i t = 0 for all i and t < T, and x~t = 1 for all i and t = T. For T + k (k = 1, 2 . . . . . n-T) the

values of x~t are

(12) i = 1 (AO)

i = 2 (10) O,

k

i = 3 (LS) 1-~,,~.,

j=l

k-1

i = 4 (TC) (~k _ T. 6 k-j 5 - =k.

]=1

Additional insight may be gained by calculating these different e t series for our simulated

AR (1) process. Here, we have the advantage that, since we are dealing with a simulated

series, we also know the noise series at. This allows us to isolate the various outlier ef-

fects.

Figure 6 shows the residual series for an I 0 at t = 3 0 We indeed observe an impact at that

point of time. Whether other values are also affected by this outlier is hardly to tell from

this graph. However, since we are operating in an artificial world, we can subtract the noise

a t from et. The resultant series is shown in Figure 7. Now, it is perfectly clear that the only

value affected by the l O is at time t = 30. It is surprising to detect that an outlier which af-

fects a large number of values in the observed series, affects only one value of the residual

series.

Next, we analyse the case of an additive outlier. Since an AO affects the observed series

for one period only, it could be suspected that this is also the case with the residual series.

From equations ( 1 1 ) and (12), however, we see immediately that this is not the case. We

observe an impact in the period in which the A O occurs, which then decays according to

the :~ weights of the underlying ARMA model. We illustrate this phenomenon by graphing

the residual series of the simulated AR (1) process containing an AO. Inspection of this

graph reveals that two observations might be affected, namely the values for t = 30 and

Outlier Detection and Adjustment

Figure 6

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Figure 7

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Empirica - AUSTRIAN ECONOMIC PAPERS

Figure 8

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Figure 9

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-4.0 I I I I I 1 I I i I I I I I t I I I I I I I ', ', ', ', ', ', ~ t ', ', t ', ', ', ', I ', ', ', ', + t ', ', ', t ', ', ', ~ ', '+ ', '+ [ ', ', '+ '+ I I I

7 13 19 25 31 ,37 43 49 55 61

Outlier Detection and Adjustment

Figure 10

6.0

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-3.0 IIIIIIIIIiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

7 t3 19 25 31 37 43 49 55 61

Figure 11

6.0

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-,,3.0

-4-.0 llllllllllllllllllllllllllllllllllllllllllllllllllllllllll.I-,-l.-.H~b

7 15 19 25 31 57 43 49 55 61

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Figure 12

Residual series e for the simulated All (1) process with a TC

6.0

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-3.0 i ', [ ', ', ~ ', ~ [ [ [ [ [ [ [ [ [ [ ~ I [ [ ~ t [ ~ [ t I [ [ I t ~ ', I ~ ', [ ', ', ', ', ', [ ', t ', [ I ', ', [ [ ', ', ', ', ', ', [ [ [ [

7 13 19 25 31 37 43 49 55 61

Figure 13

Residual series of Figure 12 with the underlying noise removed

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t = 31. But again, the information provided by the graph of the residual series is not very

exact. Subtracting the noise from et produces a clear-cut picture of the effect of an AO on a

residual series. The resultant graph can be found in Figure 9. Only two values are different

Outlier Detection and Adjustment

from zero, those at t = 30 and 31. This result follows from the structure of our simulated

series. For an AR (1) process only 7q is non-zero. For a more complex process, several of

the ~r weights would be non-zero and, consequently, more values of the residual series

would be affected.

Finally, we come to the effects of level shift and temporary change. Again, the pattern of

the impacts is related to the ~rweights of the underlying ARMA model. From e q u a t i o n s (11)

and (12), we see that in the case of an LS, this impact is related to the cumulative sum of

the Jrweights. Figure 10 illustrates the effect of an LS at t = 30. We observe a large initial

impact at t = 30 followed by a step increase from t = 31 onwards. Subtracting the noise

from et produces exactly this reaction pattern more clearly, as we can see from Figure 11.

Unlike a level shift, a temporary change does not produce a permanent shift, neither in the

observations nor in the related residual series. Instead, we observe an initial impact at the

time of the occurrence of the outlier, which is down-weighted in the following periods, and

vanishes eventually. The graph of the residual series et for the simulated AR (1) process

with a TC at t = 30 is shown in Figure 12. We note that this graph is hardly discernible from

the effects of an I 0 . Consequently, it will be difficult to differentiate between these two

types of outliers in practical work. Only when the noise is removed from et, we can see that

there exist certain differences in the impacts of these two outliers. The corresponding se-

ries is shown in Figure 13.

In practice, the ARMA parameters and the residual error (ra are unknown, but estimates

can be obtained from the data. The residual series of the fitted model may then be used to

search for outliers. C h a n g - Tiao - C h e n (1988) propose a procedure for detecting outliers

by analysing the residual series. Since type and date of occurrence of a possible outlier are

unknown, the procedure works sequentially through time and calculates four test statistics

(one for each type of outlier) for each point of time. The largest test statistic (in absolute

value) for each outlier type and its corresponding time index are retained. From this set,

again the largest test statistic is chosen and is compared with some pre-specified critical

value. If it is larger than this critical value, the procedure has detected an outlier and has

also identified its type. The residual series is then adjusted for this outlier, and a new esti-

mate of ~ra is computed. The process of detecting outliers and adjusting the residual series

is repeated until no additional outliers can be found. Finally, an adjusted model is esti-

mated, in which intervention variables are included in order to remove the effects of de-

tected outliers. Unfortunately, this procedure has two serious drawbacks. First, there is no

guarantee that the iterative search for outliers is efficient. Second, and more importantly,

the presence of outliers may give rise to wrongly identified and poorly estimated ARMA

models. As a consequence, spurious outliers may be detected.

Empirica - AUSTRIAN ECONOMIC PAPERS

Table 1

Estimation results for an AR (1) fit of the simulated AIR (1) process

AO at t = 30 0.367 0.t16 1A85

I0 att = 30 0.504 0.108 1.150

LS at t = 30 0.944 0.047 1.262

TC at t =30 0.567 0.103 1.153

Table 2

Outlier detection for the fitted simulated AR (1) process with various outliers added

^ ^

t w~ Outlier type found cra

Adjusted Unadjusted

AO at t = 30 30 5.53 AO 0 888 1.185

(6.44)

/(3 at t = 30 30 5.68 10 O.884 1.150

(6.38)

L,.g' at t = 30 30 5.99 I0 0 848 1.262

(5.90)

20 - 3.43 1(3

(- 3.76)

6 2.67 I0

(3.13)

W e illustrate this problem using our simulated AR (1) process. The estimates of @, its

standard error, and the estimate of cra for each of the five simulated series are given in Ta-

ble 1. Apart from the LS case, the estimates of ~ and era are rather close to the true values.

Consequently, the additive and the innovational outlier are correctly detected, as can be

seen from Table 2. But, as expected, the level shift causes problems. Not only the type is

wrong ( I 0 instead o f / S ) , but also two spurious outliers are inappropriately identified.

In order to avoid these problems, C h e n - Liu (1990) propose an iterative procedure for the

joint estimation of model parameters and outlier effects. This procedure differs from that

described above in several respects. First, the outlier detection is performed iteratively

based on the adjusted residuals as well as the adjusted observations. That is, when an

outlier is detected its effect is removed both from the observed series and from the residual

series of the fitted model. By adjusting observations, the procedure avoids the need to for-

mulate and estimate an intervention model. The three steps 1. outlier detection, 2. outlier

adjustment, and 3. parameter estimation based on the adjusted series are iterated until no

outliers are found. Secondly, the outlier detection is based on robust estimates of the

Outlier Detection and Adjustment

model parameters. Finally, outlier effects are jointly estimated using multiple regression.

As a result, this procedure should produce better estimates of model parameters and

should reduce spurious outliers and masking effects in outlier detection. Most of the em-

pirical results in this paper are derived by applying this procedure, which is part of the

newest release of the SCA statistical package.

We now use the outlier detection and adjustment procedure of C h e n - l_iu (1990) in order

to analyse three Austrian economic time series: retail sales, car purchases, and purchases

of durables. The rationale behind this choice is rather pragmatic. Since we want to test the

reliability of this outlier detection procedure in practical situations, we choose series which

are, as we know for certain, heavily contaminated by outliers. For the chosen series, we

not only know that they contain big outliers, but we also have exact information about their

date of occurrence. Consequently, these series can be regarded as an ideal test for an out-

lier detection procedure which is not based on prior knowledge, but utilises only the infor-

mation contained in the observed series. A large number of the outliers in our series is

caused by fiscal policy measures. These measures are listed in Table 3.

Table 3

Fiscal policy measures affecting consumer expenditure

1968, September 1 Introduction of surtax on purchases of new cars

1970, December 31 Abrogation of surtax

1973, January 1 Introduction of V. A. T,

1975, January 1 Reduction in tariffs vis-a-vis the Common Market

1978, January 1 Introduction of special V. A. T. rate on "luxury goods"

1984, January 1 Increase in V. A. T.

1985, October 1 Catalytic converter regulation

For retail sales we have monthly information, but only quarterly data exist for car pur-

chases and expenditures on durable goods. Because of the heavy computational burden

we were forced to restrict the period to be analysed to the time span 1970-1990 in the case

of retail sales. Furthermore the sample A C F of the differenced retail sales series

[(l-B) (1-B 12) In (]1i)] showed a confused pattern which is most likely due to the influence

of the trading day effects in the data (see Liu, 1980, Tsay, 1984). Analysing the spectrum

we really found a marked peak at 0.696 ~r. This is evidence that there are calendar effects

in this time series ( C l e v e l a n d - Devlin, 1982, KohlmOIler, 1987). We therefore included cal-

endar variables in the model explaining retail sales.

The outcome of model parameter estimation and outlier detection and adjustment are

shown in Table 4. The first model given is an ARIMA model without any adjustment. In the

second model, the procedure proposed by C h e n - Liu (1990) is applied to detect and re-

Empirica - AUSTRIAN ECONOMIC PAPERS

move outliers. In the SCA terminology, it is the OESTIM paragraph which is used to esti-

mate this model. The third model, finally, is a re-estimated version of an intervention model

which was originally published in Thury (1986) and Thury - WOger (1989). For ease of

presentation, only the estimates of the moving average parameters of this model are given

here. We see that the parameter estimates and the standard errors for the unadjusted

model diverge substantially from those of the other models. Ignoring outlier effects seems

to have indeed very serious negative consequences for the resulting parameter estimates

and the goodness of fit. The effects of outliers on forecasts are analysed in Ledolter (1989,

1990).

Table 4

A R I M A models for retail sales

ARIMA model Interventionmodel

Without outlieradjustment With outlier adjustment

01 0.5863 0.7008 0.7090

(10.95) (14.89) (14.36)

012 0.5152 0.1846 0.1836

(9.22) (2.84) (2.73)

SE 0.0265 0.0185 0.0166

Outliers

% Type

1973, January - 0.056 AO

(- 4.28)

1974, December - 0.076 AO

(- 5.76)

1977, October 0.059 LS

(5.44)

1978, January - 0.146 LS

(- 13.22)

1978, April - 0.044 AO

(- 3.32)

1983, September 0.046 TC

(3.83)

1983, December 0.090 AO

(6.78)

1984, January - 0.062 AO

(- 4.7O)

The lower part of Table 4 contains the detected outliers. Most of the effects of the men-

tioned fiscal policy measures are correctly detected, as far as their time of occurrence and

their type is concerned. The introduction of the V. A. T. rate in 1973 caused a postpone-

Outlier Detection and Adjustment

ment of purchases and so did the announcement of the reduction in tariffs vis-a-vis the

Common Market at the end of 1974. The applied procedure of estimating and adjusting for

outliers delivers AO (i. e., a negative outlier) in those cases. The announcement of the in-

troduction of a special V. A. T. rate on "luxury goods" triggered in advance purchases over

a period from October to December 1977 (i. e., LS in the outlier terminology ), which were

followed by a drop in purchases in 1978 (i. e., LS in January and AO in April). The general

increase in the V. A. T. rate in 1984 also caused outliers. We observe increased purchases

starting ifi September 1983, which at first level off (i. e., a TC in the outlier terminology) and

reach a new peak in December. The following purchasing restraint caused an AO (i. e.,

negative outlier) in January 1984. The abrogation of surtax on new cars and the catalytic

converter regulation did not cause any outlier in the series of retail trade probably because

expenditures on cars only take a small part of total sales.

In most cases the size of the detected outliers is in close accordance with the magnitude of

the estimated intervention parameters. Marked differences can only be observed at the

beginning of 1978 (see Figure 14).

Figure 14

Comparison of outlier- and intervention-adjusted retail sales

400 9

l --, Outlieredjusted

350 InterventionGdj.

500

250

.i

it

i!

J

200

150

100

70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

Up to this point the two procedures (intervention model or outlier procedure) yielded the

same results. Thereafter the two series differ only in the levels and exhibit parallel move-

ments. The hypothesis that in a regression of one series on the other the slope coefficient

is 1 cannot be rejected. In this case, therefore, our procedure for detecting and adjusting

outliers seems to make the formulation and estimation of an intervention model unneces-

sary.

Empirica - AUSTRIAN ECONOMIC PAPERS

Table 5

A R I M A models for car purchases

ARIMA model Intervention model

Without outlier adjustment With outlier adjustment

Constant 0.0670 0.0764 0.0794

(5.47) (4.51) (4.45)

01 0.5533 0.3057 0.2363

(6.57) (3.44) (2.44)

O4 0.8110 0.4853 0.4492

(15.20) (6.50) (5.63)

0.8382 0.7929 0.7909

(22.63) (23.30) (19.63)

EE 0.2317 0.1054 0.0993

Outliers

wi Type

1968, 3rd quarter 0.550 I0

(5.21)

1988, 4th quarter - 1.668 TC

(- 19.01)

1969, 1st quarter 0.223 AO

(2.74)

1970, 1st quarter - 0.271 TC

(- 3.08)

1972, 4th quarter 0.579 AO

(7.13)

1974, 1st quarter - 0.257 LE

(- 3.06)

1977, 4th quarter 0.672 1(9

(7.49)

1978, 1st quarter - 1.248 TC

(- 13.92)

1984, 2nd quarter - 0.300 TC

(- 3.43)

1985, 4th quarter - 0.418 [0

(- 3.97)

We turn next to an analysis of car purchases, a series which, definitely, is affected most by

the fiscal policy measures mentioned above. We dispose of rather noisy, quarterly obser-

vations covering the period 1954 to 1990. The outcome of model parameter estimation and

outlier detection and adjustment are shown in Table 5. The third model is a re-estimated

version of an intervention model w h i c h was originally published in Thury (1988). The esti-

mates of the autoregressive and moving average parameters and standard errors for the

Outlier Detection and Adjustment

unadjusted model diverge even more substantially than in the case of the retail sales from

those of the other models, since car purchases are most affected by external effects.

Again most of the effects of the fiscal policy measures are correctly detected, as far as

their time of occurrence and their type is concerned. Both the introduction of the surtax in

1968 and the introduction of a special V. A. T. rate on "luxury goods" gave rise to a similar

reaction pattern. We observe increased purchases (i. e., an innovational outlier) immedi-

ately before the introduction of the respective policy measure, and a corresponding tempo-

rary restraint (i. e., a TC in the outlier terminology) in the acquisition of cars after the intro-

duction. For both policy measures, the size of the detected outliers is in close accordance

with the magnitude of the estimated intervention parameters. In addition to the outliers due

to the known fiscal policy measures, other outliers are detected as we can see from Ta-

ble 5.

One should not rush into classifying them as spurious without further reflection. The tem-

porary change at the beginning of 1970 and the level shift at the beginning of 1974 might

well be explained by the economic situation prevailing at that time. In the beginning of 1970

a discussion about abolishing the surtax on purchases of new cars started, a surtax which

was indeed repealed at the end of this year. Thus, it was only rational for a prospective

buyer to postpone the purchase of a new car in this situation. An observed temporary drop

in car sales was the consequence. Similarly, good reasons for the existence of a negative

level shift in 1974 can be given. At that time, petrol prices were increasing tremendously,

and it was even unclear whether enough petrol would be available in the future. Conse-

quently, some restraint in acquiring new cars should not be a surprise. Only for the additive

outlier in 1969, we have no economic interpretation at hand. It may either be a trivial re-

cording error, or its cause has simply been forgotten during the twenty years.

The fact, that we find outliers besides the well known fiscal policy effects, is an interesting

piece of information. It illustrates a serious limitation of intervention analysis, namely the

requirement to have some prior information about the occurrence of possible outliers. Oth-

erwise, it may be difficult to discover them by a purely visual inspection of the data. Auto-

matic outlier detection offers the possibility of avoiding this problem at low costs.

As a third example, we analyse the purchases of durables. The various ARIMA models are

given in Table 6. Again, we observe significant differences in the parameter estimates of

the pure ARIMA model without outlier adjustment and those of the other two models. But,

in contrast to the case of car purchases and retail sales, we observe here also a marked

difference in the estimate of the residual standard error between the ARIMA model with

outlier adjustment and the intervention model. This estimate is significantly lower for the

intervention model. The automatic procedure removes fewer outiiers than are considered in

the intervention model. Some of the effects of the fiscal policy measures are much smaller

here than in the case of car purchases above and, therefore, can be hardly detected by an

Empirica - AUSTRIAN ECONOMIC PAPERS

Table 6

ARIMA models for purchases of durables: Step I

ARIMA model Interventionmodel

Without outlieradjustment With outlieradjustment

Constant 0.0515 0.0615 0.0577

(4.39) (6.99) (3.70)

0.4740 0.3185 0.3179

(5.95) (3.54) (3.33)

04 0.6760 0.4699 0.4433

(10.20) (6.16) (5.45)

# 0.8951 0.842o 0.84o5

(26.43) (23.00) (18.89)

SE 0.0774 0.0646 0.0483

Outliers

% Type

1968, 4th quarter - 0.232 TC

(- 4.39)

1972, 4th quarter 0.162 AO

(3.35)

1978, 1st quarter - 0.323 AO

(- 6.68)

1984, lstquarter - 0.173 LS

(- 3.30)

automatic procedure. They are masked by the presence of a relatively large number of

similarly sized values in the residual series. Removing all these values by reducing the

critical value of the outlier test statistic will disturb the parameter estimates of the underly-

ing ARIMA model. As a consequence, spurious outliers may occur. For series with such a

structure, the estimation of intervention models might be advantageous provided, however,

that the necessary prior information is available. It would then be possible to capture the

effects of fiscal policy measures without disturbing the parameter estimates of the underly-

ing ARIMA model, as was successfully done in the intervention model of Table 6. In cases

where this prior information is not available an appropriate application of the automatic

outlier detection procedure might still be possible by proceeding in a stepwise fashion. In a

first step, the procedure is applied to the original series, detected outliers are then re-

moved, and an adjusted series is generated. In a second step, the procedure is applied to

this adjusted series.

Outlier Detection and Adjustment

Table 6~Continued

ARIMA models for purchases of durables: Step 2

ARIMA model Interventionmodel

Without outlier adjustment With outlieradjustment

Constant 0.0610

(7.01)

0r o.3229

(3.60)

o4 o.4817

(6.38)

# 0.8441

(23.47)

SE 0.0452

Outliers

wi Type

1963, 1st quarter - 0.125 AO

(- 3.67)

1968, 3rd quarter 0.105 TC

(2.62)

1971, 1st quarter 0.156 I0

(3.44)

1977, 4th quarter 0.253 AO

(7.3O)

1978, 2rid quarter - 0.162 LS

(- 4.85)

1988, 1st quarter 0.166 10

(3.66)

The results of such a stepwise outlier detection are given in the second part of Table 6. The

outcome is more than satisfactory. The parameter estimates for the underlying model

hardly change in step 2, the estimate of the residual error, however, has dropped signifi-

cantly. Additionally, we detect now an acceptable number of economically interpretable

outliers: Proceeding in this stepwise fashion, we obtain results that are fully congruent with

the findings for car purchases.

5. Concluding remarks

The procedure proposed by Chert - Liu (1990) to detect and remove outliers enables us to

derive more adequate models from the time series considered. The improvement over the

usual ARIMA model is enormous, the differences to a sophisticated intervention model are

small. This procedure at least yields helpful information in the formulation of an adequate

Empirica - AUSTRIAN ECONOMIC PAPERS

the formulation and estimation of an intervention model by applying ~n outlier adjustment

procedure. In some cases we obtained more information by automatic outlier detection.

We found some outliers in addition to the well known policy effects. On the other hand, if

the automatic procedure removes fewer outliers than are known, because they are masked

by the presence of a relative large number of similarly sized values in the residual series a

stepwise application of the automatic outlier detection procedure might help. In a first step,

the procedure is applied to the original series, detected outliers are then removed and an

adjusted series is generated. In a second step, the procedure is applied to this adjusted

series. Our empirical results also showed that when reducing the critical value of the outlier

test statistic an increased number of spurious outliers will be found not only because of the

increased risk for type I error but also as a consequence of disturbed parameter estimates

of the underlying ARIMA model.

6. References

Box, G. E. P., Jenkins, J. M., Time Series Analysis: Forecasting and Control, 2rid Edition, Holden-Day, San

Francisco, 1976.

Box, G. E. P., Tiao, G. C., "Intervention Analysis with Applications to ~Economic and Environmental Prob-

lems", Journal of the American Statistical Association, 1975, 70(349), pp. 70-79.

Chang, I., Tiao, G.C., Chen, C., "Estimation of Time Series Parameters in the Presence of Outliers",

Technometrics, 1988, 30(2), pp. 193-204.

Chen, C., Liu, L.-M., "Joint Estimation of Model Parameters and Outlier Effects in Time Series", Scientific

Computing Associates, Lisle, Working Paper, 1990, (116).

Chen, C., Liu, L.-M., Hudak, G. B., Outlier Detection and Adjustment in Time Series Modeling and Forecast-

ing, Scientific Computing Associates, Lisle, 1990.

Cleveland, W. S., DevUn, S.J., "Calendar Effects in Monthly Times Series: Modelling and Adjustment",

Journal of the American Statistical Association, 1982, 77(379), pp. 520-528.

KohlmQIler, G., "Analyse yon Zeitreihen mit Kalenderunregelmal3igkeiten",Quantitative Okonomie, 1987, (2).

Ledolter, J., "The Effects of Additive Outliers on the Forecasts from ARIMA Models", International Journal of

Forecasting, 1989, 5, pp. 231-240.

Ledolter, J., "Outlier Diagnostics in Time Series Analysis", Journal of Time Series Analysis, 1990, 11,

pp. 317-324.

Liu, L.-M., "Analysis of Time Series with Calendar Effects", Management Science, 1980, 26, pp. 106-112.

Thury, G., "The Consequences of Trading Day Variation and Calendar Effects for ARIMA Model Building

and Seasonal Adjustment", Empirica, 1986, 13(1), pp. 3-26.

Thury, G., "Intervention Analysis of Consumer Expenditure in Austria", Empirica, 1988, 15(2), pp. 295-325.

Outlier Detection and Adjustment

Thury, G., W0ger, M., "Des Weihnachtsgeschaft im Einzelhander', WlFO-Monatsberichte, 1989, 62(12),

pp. 700-705.

Tsay, R. S., "Regression Models with Time Series Errors", Journal of the American Statistical Association,

1984, 79(385), pp. 118-124.

Correspondence:

Gerhard Thury, Michael W0ger

C)sterreichisches Institut fur Wirtschaftsforschung

Postfach 91

A-1103 Wien