This paper analyses the influence of 13 stylized facts of the German economy on the West German business cycles from 1955 to 1994. The method used in this investigation is Statistical Experimental Design with orthogonal factors. We are looking for all existing Plackett-Burman designs realizable by coded observations of these data. The plans are then analysed by regression with forward selection and various classification methods to extract the relevant variables for separating upswing and downswing of the cycles. The results are compared with already existing studies on this topic. --

Preusser, Andrea

Pumplün, Constanze

Weihs, Claus

English

Research Papers in Economics

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Pumplün, Constanze; Weihs, Claus; Preusser, Andrea
Working Paper
Experimental Design for Variable
Selection in data bases
Technical Report / Universität Dortmund, SFB 475 Komplexitätsreduktion in Multivariaten
Datenstrukturen, No. 2004,72
Provided in cooperation with:
Technische Universität Dortmund
Suggested citation: Pumplün, Constanze; Weihs, Claus; Preusser, Andrea (2004) :
Experimental Design for Variable Selection in data bases, Technical Report / Universität
Dortmund, SFB 475 Komplexitätsreduktion in Multivariaten Datenstrukturen, No. 2004,72, http://
hdl.handle.net/10419/22585Experimental Design for Variable Selection in
data bases
Constanze Pumpl¨ un, Claus Weihs, and Andrea Preusser
University of Dortmund
⋆
Department of Statistics
44221 Dortmund, Germany
Abstract. This paper analyses the inﬂuence of 13 stylized facts of the German
economy on the West German business cycles from 1955 to 1994. The method
used in this investigation is Statistical Experimental Design with orthogonal fac-
tors. We are looking for all existing Plackett-Burman designs realizable by coded
observations of these data. The plans are then analysed by regression with forward
selection and various classiﬁcation methods to extract the relevant variables for
separating upswing and downswing of the cycles. The results are compared with
already existing studies on this topic.
1 Introduction
In the following, existing data are analysed using the method of statistical
experimental design. The aim of experimental design is to estimate factor
eﬀects with the highest accuracy possible. Usually, an experimental design
with ﬁxed factor levels is taken and the response of the experiment is used
to ﬁnd factors of high inﬂuence with as few experiments as possible. Thus
the optimal factors determining the response are found faster and with less
expense than by carrying out all experiments with all possible factor level
combinations. In order to detect the variables which do inﬂuence the up- and
downswing phases of the economy, we use a special type of screening plans,
namely Plackett-Burman plans. Contrary to the method of full factorial de-
signs, which investigate main eﬀects and all possible interactions, these plans
are employed to ﬁnd only the main eﬀects in the model.
The original data used here are highly correlated. In order to eliminate
these correlations, the data are coded by -1 and +1 only and then special ob-
servations are selected building Plackett-Burman plans. The main advantage
of this method is that it selects the most important factors not disturbed by
correlations in the data. By this procedure, on the one hand, the data are
reduced by the discrete coding by -1 and +1 and on the other hand by choos-
ing special observations only. In order to at least partially compensate this,
we are analysing all existing Plackett-Burman plans with respect to the data
⋆ This work has been supported by the Deutsche Forschungsgemeinschaft, Sonder-
forschungsbereich 475. We also thank Uwe Ligges and Karsten Luebke for their
support.2 Constanze Pumpl¨ un, Claus Weihs, and Andrea Preusser
and ﬁnally choose those variables which are, what we call, uniquely correlated
to the up- and down phases of the economy. The following investigations are
based on 13 stylized facts of the West German economy (cf. Heilemann and
M¨ unch (1999)) which have been selected by Heilemann and M¨ unch to ex-
plain the German business cycle. There exists already a number of papers
which analyse and interpret these data based on, e.g. classiﬁcation methods
like linear discriminant analysis and time series analysis (cp. Heilemann and
M¨ unch (1999), Weihs, R¨ ohl and Theis (1999), Weihs and Garczarek (2002)).
In this paper in the ﬁrst step, we code the data to -1 and +1 and in the
second step we look for all Plackett-Burman plans in the coded data. All
these plans are analysed by stepwise regression with forward selection, by
unpruned classiﬁcation trees, by trees consisting only of the tree stump and
by stepwise linear discriminant analysis (cp. R¨ over (2003)). All this is based
on an a priori classiﬁcation of the response in the phases ‘up’, and ‘down’
in the years under investigation, based on Heilemann and M¨ unch (1999).
Finally, the variables which have turned out to be important are compared
with the results of existing studies.
2 Data
The predictor data set consists of 13 variables which have been measured
quarterly (157 quarters) in the years 1955/4 to 1994/4 (price index base
is 1991) (cf. Heilemann and M¨ unch (1999)). The variables (and their ab-
breviations) are real-gross-national product-gr (BSP91JW), real-private-con-
sumption-gr (CP91JW), government-deﬁcit-rate (DEFRATE), wage-and
salary-earners-gr (EWAJW), net-export-rate (EXIMRATE), money-supply-
M1-gr (GM1JW), real-investment-in-equipment-gr (IAU91JW), real-
investment-in-construction-gr (IB91JW), unit-labour-cost-gr (LSTKJW),
GNP-price-deﬂator-gr (PBSPJW), consumer-price-index-gr (PCPJW),
nominal-short-term-interest-rate (ZINSK), real-long-term-interest-rate
(ZINSLR). The letters ‘gr’ are an abbreviation of ‘growth rates relative to
last years corresponding quarter’.
3 Plackett-Burman designs
Heilemann and M¨ unch (1999) distinguish 4 phases of the business cycle:
‘upswing’, ‘upper turning point’, ‘downswing’ and ‘lower turning point’. Each
quarter has been assigned one of these phases which we assume to be the
correct one. Here only the phases ‘up-’, and ‘downswing’ are considered.
Therefore, the phases ‘upper turning point’ and ‘lower turning point’ are
split in the middle, i.e. if, e.g., the ‘upper turning point’ phase lasts for k
quarters, k ∈ N,[k/2] quarters will be added to the ‘upswing’ phase and
k −[k/2] quarters will be added to the succeeding ‘downswing’ phase, where
[x] denotes the so called Gauß brackets, i.e. the largest integer less or equalExperimental Design for Variable Selection 3
to x, x ∈ N. An analogous convention holds for the ‘lower turning point’
phase. These two phases ‘upswing’ and ‘downswing’ are coded by 0 and 1,
respectively. Note that two phase consideration is standard in business cycle
analysis. Thus, it is the natural starting point for our studies. Extensions to
4 classes are planned.
Plackett-Burman plans only exist if the number of experiments n is a
multiple of four and the number of variables is n-1 (cf. Plackett and Burman
(1946), Weihs and Jessenberger (1999)). The Plackett-Burman plan for n = 8
is shown in Table 1.
x1 x2 x3 x4 x5 x6 x7
1 - - - - - - -
2 - - + - + + +
3 + - - + - + +
4 + + - - + - +
5 + + + - - + -
6 - + + + - - +
7 + - + + + - -
8 - + - + + + -
Table 1. Plackett-Burman plan with 8 experiments.
The second row is called generating row, as it generates the rows 3–8 of
the matrix by being shifted one position to the right at each step. Plackett-
Burman plans are orthogonal arrays in the sense of (Hedayat et al. (1999)),
they are of the form OA(4λ,4λ − 1,2,2), λ ∈ N, (λ = 2 in Table 1), i.e.
each factor has only two levels -1 and +1, the sum of each column is 0 and
columns are pairwise orthogonal. If an 8th column consisting only of +1’s
is added to the matrix, one gets a unique Hadamard matrix of order 8 (cp.
Hedayat et al. (1999)). Therefore it is necessary to code the existing data in
+1 and -1, in order to look for Plackett-Burman designs. For each variable,
all values less than its median are taken as -1 and all values greater than or
equal to its median are taken as +1. As there are 13 variables, one looks for
Plackett-Burman plans with n = 8 or n = 12 in the coded data. 113 diﬀerent
plans were found for n = 8 and none for n = 12.
The algorithm for ﬁnding these plans is ﬁrst to look for all rows which
contain at least seven times the number -1. The corresponding columns are
then searched for the generating row. After this has been found, the search
continues for the generating row shifted one position to the right, etc. This
process has to be carried out for all possible permutations of the original seven
columns. A much faster algorithm has been suggested by S.Haustein (private
communication), where one looks for the base row u0 = (− − − − − − −)
and then searches for a row v in the corresponding columns with Hamming
distance 4 to u0. After this has been found, one looks for a row v1 with
Hamming distance 4 to u0 and v. This process is continued until eight rows
have been found which are equidistant with Hamming distance 4. These eight4 Constanze Pumpl¨ un, Claus Weihs, and Andrea Preusser
rows form a Plackett-Burman plan for n = 8, because the Plackett-Burman
plan for n = 8 is an orthogonal array of the form OA(8,8 − 1,2,2) and
this class has only one isomorphism class. Here two arrays are said to be
isomorphic (cf. Hedayat et al. (1999)), if one can be obtained from the other
by permutations of rows, columns or factor levels.
In the following investigations, a linear screening model is used, y = Xβ+
ǫ, where X = (1,A) is an (n × n) matrix with 1 = (1,1,1,...)t and A the
Plackett-Burman matrix. β is the vector of unknown coeﬃcients, y the result
vector with the coded business cycle phases and ε the error vector.
4 Results
4.1 Stepwise regression by forward selection
113 diﬀerent Plackett-Burmanplans were found by the method described in 3.
When evaluating these plans by stepwise regression with forward selection
with respect to y (cp. Weihs, Jessenberger (1999)), we used the F-test at
level 0.2. Figures 1, and 2 show the absolute and the relative frequency of the
selected variables (dark bars). The light bars show how often each variable
appears in all 113 Plackett-Burman plans. Figure 1 thus shows that each
variable is at least once in a plan (light bars). The variables which turn
out to be most important by this method are ‘DEFRATE’, ‘EXIMRATE’,
‘LSTKJW’, ‘IAU91JW’ and ‘ZINSK’(cp. Figure 2). If one uses the F-test
with level 0.05 one gets the same variables except ‘EXIMRATE’. It is also
interesting that in almost half of all cases none of the variables turns out
to be important. Furthermore it strikes that for all variables the dark bars
are rather small, compared to the light ones. That means that although a
variable appears often in the plans it is chosen only a few times as important
concerning the up- and down of the economy.
4.2 Classiﬁcation methods
In the 113 plans, variables are selected also by diﬀerent classiﬁcation meth-
ods, i.e.unpruned classiﬁcation trees (TreeAllNodes), classiﬁcation trees with
only the tree stump (TreeStump) and stepwise linear discriminant analysis
(cp. R¨ over (2003)). Figures 3, and 4 again show the absolute and the rel-
ative frequency of selected variables by the diﬀerent methods. The number
in brackets following the variable name indicates how often the variable ap-
pears in a Plackett-Burman plan. Classiﬁcation by unpruned trees yields as
important variables ‘BSP91JW’, ‘CP91JW’, ‘DEFRATE’ and ‘EXIMRATE’.
Using only the tree stump yields the same variables without ‘CP91JW’ as
important. This is the same result one gets by stepwise linear discriminant
analysis. On the whole, these three classiﬁcation methods yield similar results
but on diﬀerent levels.Experimental Design for Variable Selection 5
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regression selection
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Fig.1. Absolute frequency of variable selected by stepwise regression with forward
selection.
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8 regression selection
PB−plan
Fig.2. Relative frequency of variable selected by stepwise regression with forward
selection.
For all used classiﬁcation methods as well as for stepwise regression with
forward selection it is important to know how the rows which build the
Plackett-Burman plans are distributed. This is illustrated in Figure 5 which
shows how often each row is contained in a plan. Note that the outstanding
row number 72 refers to the 4th quarter of 1972 and row number 145 to the
1st quarter of 1991. These years are special years from an economic point of
view, as in 1972 the German economy suﬀered from the oil price shock. The
German uniﬁcation inﬂuences the post 1990 data, an eﬀect shown in the ﬁrst
quarter of 1991.
4.3 Variable assessment
If one wants to decide which of the above variables plays a dominant role
with respect to the business cycle, it is important to assess their correlation6 Constanze Pumpl¨ un, Claus Weihs, and Andrea Preusser
B
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StepLDA
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Fig.3. Absolute frequency of variables selected in Plackett-Burman design.
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Fig.4. Relative frequency of variables selected in Plackett-Burman design.
in all those Plackett-Burman plans where the corresponding variable was
included. It turns out (see Table 2) that unit labour costs(‘LSTKJW’) is
clearly positively correlated to y (84% of all cases) and the government deﬁcit
(‘DEFRATE’) can still be considered as positive correlated (78% of all cases),
taking into account a possible error margin. No variable is clearly negatively
correlated to y. Hence, one may ﬁnally consider those variables as important
which on the one hand are chosen most often, both by regression and by
classiﬁcation, and which on the other hand possess a distinct positive or
negative correlation to y. Using this decision criterion, one gets ‘unit labour
costs’ (‘LSTKJW’) and ‘government deﬁcit’ (‘DEFRATE’) as variables which
clearly determine the West German business cycles.Experimental Design for Variable Selection 7
rownumber
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Fig.5. Absolute frequency of rows in all Plackett-Burman plans.
In previous studies of this topic (cp. e.g. Weihs and Garczarek (2002),
Weihs et al. (1999)) the variables most inﬂuential for the West German busi-
ness cycle in the 4 phase case were ‘wage and salary earners’ (‘EWAJW’)
and ‘unit labour costs’ (‘LSTKJW’). Moreover, if one compares the above
method to stepwise regression by forward selection on the whole data set,
Positive Negative No Cor. % positive
BSP91JW 18 41 17 24
CP91JW 32 24 26 39
DEFRATE 62 3 14 78
EWAJW 13 2 5 65
EXIMRATE 19 5 4 68
GM1JW 8 53 19 10
IAU91JW 5 40 16 8
IB91JW 21 51 29 21
LSTKJW 36 1 6 84
PBSPJW 29 6 15 58
PCPJW 24 6 9 49
ZINSK 50 14 13 65
ZINSLR 25 18 12 45
Table 2. Correlation with respect to y.8 Constanze Pumpl¨ un, Claus Weihs, and Andrea Preusser
again taking level 0.2 in the F-test, the model ‘LSTKJW’ + ‘IAU91JW’ +
‘DEFRATE’ + ‘ZINSK’ + ‘CP91JW’ + ‘BSP91JW’ is chosen. This strongly
indicates the importance of ‘LSTKJW’ and ‘DEFRATE’. Also stepwise linear
discriminant analysis, classiﬁcation by unpruned trees and classiﬁcation trees
using only the tree stump were applied on the whole data set. The application
of unpruned classiﬁcation trees shows ‘IAU91JW’ to be the most important
variable, as does classiﬁcation trees using only the tree stump. Stepwise linear
discriminant analysis shows that besides ‘IAU91JW’, also two other variables
are important, ‘LSTKJW’ and ‘PCPJW’.
5 Conclusion
‘Unit labour costs’ (‘LSTKJW’) has been detected as an important variable
by this method as well as by previous methods (cp. 4.3). This strongly indi-
cates that this variable has a great inﬂuence on the West German business
cycle. The question why the ‘government deﬁcit’ (‘DEFRATE’) turns out to
be important here, but does not so in previous studies, requires a thorough
analysis of the inﬂuence of the methods applied here on the results. The ad-
vantage of using Plackett-Burman plans lies in the clean and easy selection of
variables in determining the important variables. This is only a ﬁrst step in
this direction. Right now, we are investigating only the correlations of those
variables with the business cycle, which have turned out to be important in
the above described investigations. A next step could be to investigate a sim-
ilar procedure with full factorial designs or fractional factorial designs. These
plans also respect orthogonality, but in addition permit interactions between
the factors.
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Experimental Design for Variable Selection in data bases

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