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Table of Contents
INTRODUCING AUTOREGRESSIVE ELEMENTS
IN SYSTEM DYNAMICS MODELS
Pablo ALVAREZ-DE-TOLEDO
Adolfo CRESPO"
Fernando NUNEZ
Department of Industrial Management, School of Engineering
University of Seville
Camino de los Descubrimientos s/n.
41092 Sevilla, SPAIN
Telephone: +34 954 487217/5/3, FAX: +34 954 486112
e-mail: pablo@pluto.us.es , adolfo.crespo@esi.us.es , fnunez@esi.us.es
Carlos USABIAGA
Department of Economics and Business
University Pablo de Olavide
Ctra. Utrera, km. | - 41013 Sevilla, SPAIN
Telephone: +34 954 349358, FAX: +34 954 349339
e-mail: cusaiba@dee.upo.es
Abstract
Autoregressive, vector autoregressive and structural vector autoregressive models may be
described, in general, as those models that explain, at least partially, the values of a variable or set
of variables, based on the past values of this variable or set of variables. During the last decades
these models have increased their presence and importance within the field of economic and
econometric analysis. It has been found that this kind of simple models, with a small number of
variables and parameters, can seriously compete in terms of their forecasting capabilities with the
large macroeconomic models, with hundreds of variables and parameters, developed during the
fifties and sixties.
This paper explains how System Dynamics models built using Vensim simulation
environment may easily incorporate the main elements of autoregressive models. In order to do that
we have developed a structural autoregressive model using stock and flow diagrams built with
Vensim software and provided the code for the mathematical formulation in a way that this tool
can be later used in System Dynamics models. This tool provides short term forecasting capabilities
to System Dynamics models built using Vensim. As an illustration, we present an application to the
study of the Spanish labor market.
Keywords: Autoregressive Models, System Dynamics Models, Impulse-Response Functions,
Forecasting, Labor Market.
* Author in charge of correspondence.
1. Introduction
In this paper we present an approximation to autoregressive models —autoregressive (AR),
vector autoregressive (VAR) and structural vector autoregressive (SVAR)- from the point of view
of the usefulness that they can provide to the System Dynamics (SD) modelers. The purpose of the
paper is to use the SVAR methodology to elaborate an stock and flow diagram and the
corresponding formulation and code written in Vensim, in a way that this new tool (macro) can be
used to simulate and forecast the behavior of a variable in the short term when the past information
of this variable and other related variables is known. Moreover, the proposed model allows to build
the map of contemporaneous relations among the considered variables. Therefore, the
aforementioned model does not try to be a SD model in itself, but a tool to be used with Vensim
when building a wider SD model. This tool will endow the wider model with the required
endogenous structure to increase its short term forecasting capabilities.
VAR models allow us to analyze the dynamic relations among a set of variables and offer
bigger possibilities to study and contrast theoretical models. Sims (1980) also mentioned that an
additional interest of estimating VAR models is the type of information derived from the estimated
set of equations. For example, it is possible to analyze the sign, the intensity, the timing and the
persistence that each one of the stochastic innovations have on the variables of the model, by means
of the impulse-response functions. Another basic element of the VAR analysis is the variance
decomposition of the forecasting error, from which it is possible to study the relative weight of
every disturbance in the variability of the model endogenous variables.
The SVAR models appear as a response to the criticism received by VAR models regarding
their absence of theoretical background. In this way, a VAR model turns out to be a reduced form of
a dynamic structural model -theoretical-, which can be estimated from its reduced form and from a
set of restrictions on the model parameters.
The rest of the paper is organized as follows. The second section describes briefly the
autoregressive models methodology, providing the appropriate references for a more detailed study.
The third section describes how to implement an autoregressive stock and flow model using the
Vensim simulation environment. This model is composed by two sub-models that articulate the
SVAR structure. The fourth section develops an example, based on a previous work which applied
the SVAR methodology to the Spanish labor market. Finally, the fifth section concludes.
2. Introduction to AR, VAR and SVAR models
AR models may be described as those in which a variable is explained, at least partially,
depending on its past values. VAR models can be understood as a vector generalization of AR
models’. During the last decades these models have increased their presence and importance within
the field of economic and econometric analysis. It has been found that this kind of simple models,
with a small number of variables and parameters, can seriously compete in terms of their
forecasting capabilities with the large macroeconomic models, with hundreds of variables and
parameters, developed during the fifties and sixties.
The VAR models relate several variables in a form such that the value that each of them takes
in a period of time is related to the values that the same variable and all other variables take in
previous periods. A VAR model can be formulated as follows:
Ye= D1 Yer + De yer +...+ Dp Yep + e+ & []
! Autoregressive models methodology (AR, VAR and SVAR) can be found in Hamilton (1994).
where yt, Yt1,.--, Yep are vectors (n x 1) containing the values of the variables in periods t, t-1,...,
t-p; B;, @2,..., B, are matrices (n x n) containing the model parameters that can be estimated; ¢ is a
vector (n x 1) containing the model constants, that can equally be estimated; and & is a vector of
random perturbation terms, also denominated innovations, as this is the only new information that
enters in period ¢, with respect to what it is already available from previous periods. In this model
Q=E(é &’) is the variance-covariance matrix (n x n) of the innovations. This matrix can be a non-
diagonal matrix reflecting the fact that innovations can be contemporaneously correlated among
them. The model in [1] is denoted VAR(p), where the order p is the number of time lags of the
model. Equation [1] can be written:
(In - BL - DpL? -...- DpL?) ye = WL) ye = + & (2]
where ®(L) is a matrix polynomial in the lag operator” with (n x n) matrices @; , and I, represents
the identity matrix of order n.
Under suitable conditions, VAR models can be transformed in moving average infinite-order
vector models (reduced moving average form):
Ye = €ct Weer +P 2e.2t... +H = (nt PLL’ +...) + P= VL) et [3]
where &, &-1, &-2,... are the innovations vectors in t, t-1,... ; Wis a vector of constants and ‘W(L) is a
matrix polynomial in the lag operator with infinite (n x n) matrices ;. Comparing [2] and [3],
W(L)=@(L)" should apply.
Notice that, as I) is the identity matrix, in the VAR model in equation [2] each element of the
vector yt (endogenous variables determined within the system) is expressed as a function of lagged
values of all the elements in the same vector (variables predetermined in previous periods).
However, do not appear contemporaneous relations among the variables, in other words, each
variable is not related to the values of the others in that same period. Thus, equation [2] can be
viewed as the reduced autoregressive form that could be obtained from a SVAR model in which
there would be a relation among endogenous variables for the current time period:
B(L) y:= (Bo - BLL - Bol? -...- Bp”) y:=k + [4]
where k is a vector (n x 1) with constants, u, a vector (m x 1) of perturbations, which in this
structural model are called structural shocks, and B(Z) is a matrix polynomial in the lag operator
with (n x n) matrices Bj.
Notice how Bo denotes the contemporaneous relations among the endogenous variables’ y,.
Moreover, it is common to suppose that u; are standardized structural shocks, not
contemporaneously correlated to each other, so that their variance-covariance matrix is the identity
(E (uu) =D.
Equation [4] is the structural autoregressive form of the model. If we pre-multiply both
members of the equation by Bo | we would obtain [2] -reduced autoregressive form- and, vice versa,
known the matrix Bo, the structural autoregressive form of the model can be obtained pre-
multiplying by Bo in both members of [2]. However, it can be demonstrated that the information
° The lag operator L is defined as follows: L x, = x)1/ L x, = Xp2/ 65 L? X= Xrp-
~ The model in its structural form [4] cannot be directly estimated by OLS in a consistent way, as there are endogenous
variables among the system regressors. It is therefore required to implement the estimation in its reduced form [2]. This
problem in the OLS estimation of the models in its structural form is discussed in Greene (2003).
contained in y; is not enough to identify the matrix Bo, and some additional restrictions are required.
These additional restrictions can be obtained from the implications that theoretical models have on
the expected behavior of the variables y,. In this sense, it can be affirmed that whereas in the VAR
model of equation [2] the theoretical requirements are minimum (the set of variables whose
interaction is going to be analyzed and the number of time lags to be included), in the SVAR model
a greater theoretical content can be found, given by the model from which the above mentioned
additional restrictions are obtained previously.
Like VAR models in reduced form [2], VAR models in its structural form [4], under suitable
conditions, can be transformed into the structural moving average form:
ye= C(L) ur + h= (Cot Cit Col" +... +C,0'+...) uth [5]
where h is a vector (” x 1) of constants, and C(L) = B(L) ‘
Equation [5] represents the model [1] in moving average form with orthogonal shocks. Each
element ij of the matrix C, of the polynomial C(L) -c*j- identifies the effect of an orthogonal unitary
shock in yj (j=1,..,n) at date t -uj- on the variable y; (i=1,..,.n) at date tts, under the assumption that
there is not other kind of shock at date t or earlier. Therefore, the elements of matrix C, describe the
temporary effect of a unitary shock —impulse- on the model variables —response-. The matrix of
dynamic multipliers C; is known as the orthogonalized impulse-response function. To obtain the
matrix polynomial C(L) it is required to calculate previously the matrix Bo; and once we know Bo,
we can obtain u; from ¢,, and C(L) from ¥(L) according to the equation C(L) = ¥(L) By.
Another basic element in VAR analysis, besides the impulse-response functions, is the
variance decomposition of the forecasting error, from which it is possible to study the relative
weight of each shock in the variability of the model endogenous variables. Thus, the weight of a
shock in yj at date t (uj) in the variability of the variable y; at date tts (yit+s) will be given by:
(cj)
(ch)?
ja
{6]
3. Design of a structural autoregressive stock and flow model in Vensim
We will now show how it is possible to implement a stock and flow SVAR model using
Vensim (version 4.0"). We will explain how we have implemented this model by constructing two
basic stock-flow sub-models (sub-model 1 and sub-model 2), each of which corresponds to different
phases of the analysis process, as it will be showed in detail later. We want to remark here that it is
not the purpose of this paper to design a SD model, but to provide a sort of “macro” that will facilitate
the SD modelers the incorporation of SVAR methodology elements.
The core of both sub-models is the forecast of the variables in every period from their values
in the previous periods, according to the reduced autoregressive form of equation [2]. The main
difference between both sub-models is that sub-model | uses the real data of the variables in the
previous periods, whereas sub-model 2 uses the forecasts for the previous periods given by the sub-
model. For this reason, we can say that the forecast horizon is one period in the first sub-model, and
multi-period in the second one.
4 We have used Vensim 4.0, which is a Trade Mark of Ventana System Inc.
Sub-model 1 is used to estimate the parameters of the model in the reduced autoregressive
form [2] from a series of real values of the variables. This sub-model analyzes the fit between the
forecasts of the estimated model [2] and the series of real values of the variables, computing the
differences between them (residuals or estimated values of innovations ¢) and their variance-
covariance matrix.
With the values of the estimated parameters, sub-model 2 is used to calculate the Bo matrix,
with the additional restrictions that the theoretical model imposes on the dynamics of the variables.
After that, we can obtain the structural autoregressive form from the reduced autoregressive form.
Sub-model 2 also allows simulating the response of the variables to different impulses, in innovations
€ or in structural shocks u, -impulse-response functions-. These functions are also related to the
moving average forms of the model. In order to obtain all these results sub-model 2 is used in three
versions, differing only in the magnitude of the innovations.
In the next two sections we present in detail the analysis process that both sub-models follow.
In appendix A the corresponding code for both sub-models is presented, in an application to the
study of the Spanish labor market.
3.1. Detailed explanation of sub-model 1
Sub-model | stock and flow diagram can be observed in figure 1 (the names of the variables
will be indicated within quotation marks as they are explained).
First, the sub-model reads the data imported from an external file obtaining "variables in t". In
order to obtain the lags structure of the model, the level variables "variables in t-i" are generated
(where i means the order of the lag) and updated at the end of each period. This is done with an
inflow named “incr var” that is used to store data of vector y;, in that period and in the p-1 past
periods, as lagged data for the following period, and an outflow named “decr var” which eliminates
previously stored data in the level variable.
As we said before, the core of sub-model | is the forecast of the variables for every period
("variables forecast in t") from their values in the previous periods, according to the model in reduced
autoregressive form [2]. The model parameters to estimate are "variable in t-i coefficients" and the
"constants". The variable "Time" is used to control the periods in which the model is initialized,
introducing the real values of the variables as first lags.
The estimation of the model parameters, starting from the series of the variables real values, is
done using a modified Powell Method? included in the “calibration” option of Vensim. By doing so,
the values obtained for the parameters minimize the sum of the squared residuals (real values of the
variables minus forecasts of the model) for all the periods that compose the estimation interval. A
joint estimation is done for all the equations that compose [2], corresponding to each variable in the
vector y;, giving the same weight to the sums of the squares of every equation residuals in the global
payoff function to minimize.
* Among the numerical optimization techniques, the direct-search method that does not evaluate the gradient is most
suitable for the analysis of dynamics of complex nonlinear control systems. The Powell method (Powell, 1964) is well
known in order to have an ultimate fast convergence among direct-search methods. The basic idea behind the Powell
method is to break the V dimensional minimization down into N separate one-dimensional (1D) minimization problems.
Then, for each 1D problem a binary search is implemented to find the local minimum within a given range.
Furthermore, on subsequent iterations, an estimate is made of the best directions to use for the 1D searches.
Some problems, however, are not always assured of optimal solutions because the direction vectors are not
always linearly independent. To overcome this difficulty, the method was revised (Powell,1968) by introducing new
criteria for the formation of linearly independent direction vectors. This revised method, which is the one used in this
paper, is called “The Modified Powell Method”.
decr var
: z variables in t-i
variables in :
i la coefficients
variables in t zl variables
forecast in t
iner var ha
constants
<Time>
innovations
CSP previous cov PINAL
re P TIME>
Nw,
cov
Figure 1. Stock and flow diagram of sub-model 1
With the estimated values of the parameters we obtain the estimated “innovations”, and from
those values we obtain the estimate of the variance—covariance matrix "cov". The flow variable "inc"
increases in every period the accumulated level "previous cov" of the sum of the residuals products
for all the previous periods. Finally, the variable "FINAL TIME” provides the number of periods that
it is necessary to take into account in this process.
3.2. Detailed explanation of sub-model 2
In this section, we present in detail the analysis process that sub-model 2 follows, divided in four
steps. In order to obtain the results of this sub-model, three versions of the same model are developed
differing only in the magnitude of the innovations. The first version corresponds to step 1), the second
to step 2) and the third to steps 3) and 4). The sub-model 2 stock and flow diagram can be observed in
figure 2.
1) Obtaining the polynomial matrix YL) corresponding to the reduced moving average form,
and the impulse-response functions (non orthogonalized)
The non orthogonalized impulse-response functions are obtained as a result of the simulation®
of the response of the vector of variables y; to impulses in the innovations &. The specification "non
orthogonalized" refers to the fact that innovations appear contemporaneously correlated among
them. The response obtained as a result of the simulation is "variables forecast in t", which
corresponds to the vector y;, obtained by means of the model in reduced autoregressive form of
equation [2], with "variables in t-i coefficients" estimated in the sub-model 1. The "variables in t-i",
in this sub-model 2, are generated from the forecast in t’, updating them at the end of every period
with an input flow, "incr var", that stores as lagged data for the following period the forecast in this
® See Hamilton (1994).
7 Initially, their values are made equal to 0.
period and the variables in the p-1 previous periods, and an output flow, "decr var", which
eliminates the information previously stored.
vera in vdfe
decr var | coefficients
variables in Na 4 i
i pe - duration,
variables all
pe) forecast nt [*Y innovations
. NY
iner var magnitude
a cov
payoff eH
a” s covl
<FINAL
TIME Time>
Figure 2. Stock and flow diagram of sub-model 2
The impulse are the "innovations" ¢, that are made equal to | in the initial period for the
corresponding variable of the vector y;, whereas they are made equal to zero for the remaining
variables in this period and for all the variables in the following periods. Therefore, n simulations will
be required. Each simulation will depend on which variable of the vector y experiments the initial
unitary impulse. Nevertheless, it is possible to use subscripts in order to carry out all the simulations
at the same time. The innovations are obtained as the product of "duration", which establishes the
time that the innovation lasts (in this case, an initial impulse that disappears later), by the "magnitude"
of the same innovation (in this case, the first version of sub-model 2, the "magnitude" is 1 for the
variable that experiences the impulse and 0 for the others).
From the impulse-response functions we can obtain the matrix ‘W(L) corresponding to the
moving average reduced form. It is sufficient to note that in (L) the term corresponding to the lag s
is composed by the elements of the impulse-response functions corresponding to the period s of
simulation.
2) Obtaining the matrix S = Bo', the structural autoregressive form, the structural moving
average form, and the structural shocks
As it was previously exposed in section 2, pre-multiplying both members of the reduced
autoregressive form [2] by Bo, the structural autoregressive form [4] can be obtained and, vice
versa, known the matrix S = Bo’, it is possible to obtain [2] from [4], pre-multiplying both members
of this equation by S. Given that £ (u; u;’) = I and & = S uw, then:
E(e@e)=Q=SS’ [7]
where Q is the variance-covariance matrix of the innovations ¢, estimated in the sub-model 1. As Q
is a n x n symmetrical matrix, the equation [7] provides (n’+n)/2 conditions to identify the n°
elements of S. The other (n°-n)/2 conditions, as it was exposed in section 2, are obtained as
implications of theoretical models.
Since the sub-model 2 corresponds to the reduced autoregressive form, we must consider that,
according to the equation ¢, = S ui, an unitary value of one of the shocks u; is equivalent to a vector
of innovations ¢ of magnitude equal to the respective column of the matrix S. Therefore, the matrix
S that we look for will be composed by the values for the variable "magnitude" in the second
version of the sub-model 2.
The numerical optimization is guided by the fulfillment of the aforementioned conditions, by
means of the maximization of a vector of n° variables "payoff", giving the same weight to all these
variables. The first (n?-n)/2 variables capture the theoretical restrictions required for the identification
of the SVAR model, while the remaining (n°+n)/2 guarantee the fulfillment of equation [7].
Regarding theoretical restrictions, sometimes they directly influence the matrix $ = By’, while in other
cases, as it happens in our application, they affect dynamic forecasts of the model acting indirectly on
S. On the other hand, the additional (n°+n)/2 restrictions are the square of the differences among all
non identical elements of the symmetrical matrices SS' and Q, with negative sign. Initially, in the
second version of the sub-model 2, we impose initial unitary values to all the elements of
"magnitude", and therefore to all the elements of S. After that, the process of optimization continues
until the values of the above mentioned elements that approximate the payoff sufficiently to its
maximum possible value are found. In this maximum value, the last (n°+n)/2 variables of the payoff
function will have a value equal to zero and the theoretical restrictions will be fulfilled. In the
variables "cov" and "cov" (with the corresponding subscripts) are respectively the elements of the
matrix © estimated in the sub-model 1, and the elements of the product SS’, obtained from the values
of "magnitude" forming the matrix S. The variables “Time” and “FINAL TIME" are used to control
that the payoff is calculated in the period corresponding to the theoretical restrictions used.
Once the matrix § = By’ has been obtained, pre-multiplying both members of the reduced
autoregressive form [2] by Bo, the structural autoregressive form [4] and the structural shocks u, = Bo &
are obtained. The structural moving average form [5] can also be obtained from the moving average
reduced form [3], obtained in step 1), multiplying ‘W(L) by S, since, as ¢,=S uy, we get:
y= ¥(L) at... =WL)SS'e+...=CL) ut... [8]
where C(L) = Y(L)S is the polynomial matrix corresponding to the structural moving average
form.
3) Obtaining the orthogonalized impulse-response functions
In the third version of sub-model 2 the elements of "magnitude" are made equal to the values
obtained for S in the previous step. Each of the parallel simulations thus carried out with the reduced
autoregressive form corresponds to unitary values in the initial period of each one of the structural
shocks. Therefore, the values obtained in the simulations of the variables in “variables forecast in t’”,
represent the orthogonalized impulse-response functions for these variables. The specification
"orthogonalized" refers to the fact that the structural shocks are not contemporaneously correlated
among each other.
4) Variance decomposition of the forecasting error
From the values, in each period, of the variables forecast in the orthogonalized impulse-response
functions of the previous step, we can obtain the decomposition of the variance of the forecasting error
8
"vdfe" in the same period. This forecasting error is originated by the responses to each one of the n
structural shocks. So, for each variable, the percentage that supposes the square of its value in each
simulation is calculated in relation to the sum of all the squares.
4. An application to the Spanish labor market
We present now an application of the model explained in the previous section, implementing
in Vensim 4.0 a SVAR model referred to the Spanish labor market®, developed by Dolado and
Gomez (1997), following Blanchard and Diamond (1989).
Dolado and Gomez (1997) SVAR model focuses on the quarterly series of three variables:
unemployment (U), vacancies (V), and labor force (L). As we will see, in this model the vector y; is
obtained from a few previous transformations, and it is composed by the variables v1=A(v-u),
v2=Au and v3=Al, where v, u and | are the logarithms of V, U and L, and where A indicates the
first difference of the corresponding variable. These three transformed variables correspond
respectively to the rates of growth of the vacancies/unemployment ratio, unemployment and labor
force.
Relating each of these three transformed variables with the lagged values (up to 4 quarters) of
all of them, the reduced autoregressive form [2] is derived, including also a vector of dummy quarterly
variables d; with its coefficients matrix D, to control for the seasonal effects:
OL) y=e+Ddat & [9]
As it was mentioned in section 2, in this reduced autoregressive form, contemporaneous relations
do not appear among the variables, that is, each variable is not related to the values of the others in the
same period. These contemporaneous relations do appear in the structural autoregressive form [4]. The
matrix Bo, within the polynomial in the lag operator B(L), reflects the contemporaneous relations
among the variables. As it was also exposed in section 2, the information contained in the time series y;
is not sufficient to identify the elements of Bo, and therefore it is necessary to add restrictions. These
restrictions can be obtained from the implications that theoretical models may have on the expected
behavior of the variables y,.
Dolado and Gomez (1997) use a theoretical model, following a flow approach’? to labor
market, made up of four blocks: the flows of job creation and job destruction, the hiring process
through a matching function between vacancies and unemployment, the wage determination as a
function of the excess demand in the labor market, and the labor supply or labor force as a function of
wages and unemployment. All this is used to obtain a relation among the transformed variables that
compose the vector y; in the structural autoregressive form [4]. At the same time, the structural
shocks u;, are identified using three types of disturbances in the economy: aggregate activity shocks,
due to disturbances in the different components of aggregate demand, reallocation shocks, due to
disturbances affecting the efficiency in the matching process between vacancies and unemployed
(skill mismatch, geographical mismatch ...) and /abor force shocks, due to disturbances that affect
directly this variable (women participation in the labor market ...). The additional restrictions for
the identification of Bo, obtained as implications of this theoretical model, are that a labor force
5 Pioneering works in the application of the SVAR analysis to the labor market are Blanchard y Quah (1989), Bean
(1992) and Gali (1992).
° Other work in this line, for the Andalusian labor market, is Usabiaga et al. (2001).
1 A detailed analysis of the labor market, following the flow approach, can be found in Blanchard and Diamond (1992)
and Pissarides (2000).
shock does not have permanent effects on unemployment and vacancies and that a reallocation
shock does not have permanent effects on the vacancies/unemployment ratio.
In order to develop this application with Vensim we are going to follow the steps described in
the previous section with both sub-models. Since these steps correspond faithfully to those followed
in the process of econometric estimation of a SVAR model, the numerical results obtained are
practically identical to those of Dolado and Gomez". In this application, the sub-models 1 and 2 are
renamed " labor 1 " and" labor 2 " respectively, and their code is shown in appendix A.
4.1. Sub-model labor 1
The stock and flow diagram corresponding to the sub-model labor | is presented in figure 3.
—— re ies dummies
coefficients
variables in t-i
deer var coefficients
variables
forecast in t}+—
variables in
ti
Variables in|
t
‘Time
1 eS previous cov
innovations
“a _ FINAL
IE.
TIME
Figure 3. Stock and flow diagram of labor 1
First, the model reads the "data" imported from an external file: quarterly series’? of values
for vacancies (V), unemployment (U) and labor force (L), besides the quarterly dummy variables.
Later, several data initial transformations are made, obtaining "variables in t" and the
"dummies". The variables obtained are v1=A(v-u), v2=Au and v3=Al, composing the vector y;. In
order to calculate these differences, the level variables "data in t-1" need to be calculated first, and
are updated at the end of every period with an inflow "incr dat" that stores the data of this period as
lagged information for the following period, and an outflow "decr dat" that eliminates the data
stored previously. Moreover, the four seasonal dummies (d1, d2, d3, d4) are reduced to three (t1, t2,
13), in order to avoid the problem of perfect collinearity among them, defining:
" We have replicated the analysis developed in Dolado and Gomez (1997) using the econometric software Eviews 3.1.
”? Tn appendix B we show the data that we have used in our analysis.
10
tl=d1-d4
t2=d2-d4 [10]
t3=d3-d4
As we said previously, the core of labor 1 is the forecast of the variables for every period
"variables forecast in t" from their values in the previous periods. The model parameters to estimate
are "variable coefficients in t-i", "dummies coefficients" and the "constants". Using the “calibration”
option of Vensim, with model 1, the following values are obtained for the parameters:
vit" -0,0227 0,1266 -1,3112 7,5704 vi "t-1"
v2 "te = 0,0154 + -0,0174 0,5078 -0,6107 v2 "t-1" +
v3 "t" 0,0023 -0,0026 -0,0215 0,1264 v3 "t-1"
-0,2134 0,3097 0,9093 vi "t-2"
+ -0,0073 0,1930 -2,1762 v2 "t-2" +
0,0002 -0,0439 -0,0313 v3 "t-2"
0,2090 -0,1916 -6,1599 vi "t-3"
+ -0,0103 -0,1398 1,1385 v2 "t-3" +
0,0007 0,0039 0,1545 v3 "t-3"
0,0646 0,9359 7,8355 vi "t-4"
+ 0,0368 0.1115 -2,4865 v2 "t-4" +
0,0009 0,0313 0,0049 v3 "t-4"
-0,0562 0,3256 -0,2187 tl
+ 0,0095 -0,0570 0,0275 2
-0,0008 -0,0005 0,0048 13
With the estimated values of the parameters, we obtain the estimated “innovations” and the
estimate of the variance—covariance matrix "cov":
vil v2 v3
vl 0,020200 -0,000163 — 0,000091
v2 -0,000163 0,000386 0,000017
v3 0,000091 0,000017 0,000010
Sub-model labor 2
The stock and flow diagram corresponding to the sub-model labor 2 is presented in figure 4.
1) Obtaining the polynomial matrix ‘¥(L) corresponding to the reduced moving average form,
and _ the impulse-response functions (non orthogonalized)
In the case of our application to the labor market, the terms of ‘W(Z) are 3x3 matrices and their
elements correspond to the response of each one of the three variables (v1, v2, v3) to each one of the
three simulated unitary impulses’*.
2) Obtaining the matrix S = Bo', the structural autoregressive form, the structural moving
average form, and the structural shocks
8 The numerical results obtained at this step are not relevant since impulse-response functions are non orthogonalized.
ll
As in our example Q is a 3 x 3 symmetrical matrix, the equation [7] provides six conditions in
order to identify the nine elements of S. The other three conditions are obtained as implications of
the theoretical model. These conditions are that a labor force shock does not have permanent effects
on unemployment and vacancies, and that a reallocation shock does not have permanent effects on
the vacancies/unemployment ratio.
variables in t-i
coefficients
“Ne
duration
variables i
go forecast in t/“*~ innovations
. IY
incr var i
~ ra magnitude
accumulated| le —| accumaiated tS;
covl payoff |t—] forecast forecast_| inc previous
accumulated
A \ forecast
FINAL
TIME <Time: vdfe
accumulated} |
forecast v
Figure 4. Stock and flow diagram of labor 2
The numerical optimization is guided by the fulfillment of the aforementioned conditions, by
means of the maximization of a vector of nine variables "payoff", giving the same weight to all these
variables. The first three ([pol], [po2], [po3]) are the square of the forecast in the final period (long
term) of u and v, responding to an unitary labor force shock, and the square of the forecast in the final
period of v-u responding to an unitary reallocation shock, all of them with negative sign. The other six
([po4], [po5], [po6], [po7], [po8], [po9]) are the square of the differences among all six non identical
elements of the symmetrical matrices S S ' and Q, also with negative sign. That value of S is reached
when Q = § S’, and the forecast of u and v in the final period responding to an unitary labor force
shock, and the forecast of v-u responding to an unitary reallocation shock in that final period are all of
them zero. Remember that the matrix S that we look for will be composed by the values that the
variable "magnitude" takes in our model (of each innovation in each of the three simulations). The
result obtained after the simulation is as follows:
0,1161 0,0745 = -0,0343
Seo -0,0054 — 0,0127 0,014
0,0018 -0,0006 — 0,0025
The properties of the theoretical model refer to the values of u, v and v-u in the long term. As
the variable "variables forecast in t” corresponds to the first differences A(v-u), Au y Al, the model
recovers v-u, u and | accumulating the forecast in the variable "accumulated forecast”. The level
12
“previous accumulated forecast” is updated at the end of every period with the inflow “inc previous
accumulated forecast”, that is equal to the forecast of the variables obtained in this period, and
"accumulated forecast" is obtained by adding this forecast to the one accumulated previously
(notice that the updating of the level variables at the end of every period is not registered in the
output of the model until the following period). "Accumulated forecast v" is then obtained adding
the levels v-u and u.
3) Obtaining the orthogonalized impulse-response functions
In the third version of labor 2, the elements of "magnitude" are made equal to the values
obtained for S in the previous step. Therefore, the values obtained in the simulations of the variables u
and | in "accumulated forecast", and of v in "accumulated forecast v", represent the orthogonalized
impulse-response functions for these variables, recovered from their first differences. The
specification "orthogonalized" refers to the fact that the structural shocks are not contemporaneously
correlated among each other.
As an example, we represent in figure 5 the impulse-response functions obtained for the
variable unemployment [u = In(U)]. A positive shock of aggregate activity (j1) permanently reduces
unemployment, whereas a positive reallocation shock (j2) provokes a permanent increase of the
same variable. Finally, a positive labor force shock increases unemployment in the short term;
however, in the long term, as job creation and destruction adjust to the decrease of the real wages
associated with the increase of unemployment, the effects on unemployment will tend to disappear.
Figure 5. Orthogonalized impulse-response functions.
Unemployment
{u, j2] > 7 33)
fu. JJ
SS eee
1 1 rrr cht
0.05 Pag pisi 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
-0,1 -
Time (quarter)
4) Variance decomposition of the forecasting error
From the values, in each period, of u and | in "accumulated forecast" and v in "accumulated
forecast v" in the orthogonalized impulse-response functions of the previous step, we can obtain the
variance decomposition of the forecasting error "vdfe" in the same period for u, | and v. This
forecasting error is originated by the responses to each one of the three structural shocks.
In figure 6 we represent the variance decomposition of the forecasting error for
unemployment [u = In(U)]. In the very short term the structural shocks with more weight in
unemployment variability are labor force (j3) and reallocation (j2). However, in the medium and
long term the shocks of aggregate activity (jl) and reallocation (j2) explain completely the
unemployment variability".
Figure 6. Variance decomposition of the forecasting error.
Unemployment
{u, j2] =. i3) |
—leid
0% a T T TI rt T
1 3 5 7 9 111315 1719 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Time (Quarter)
5. Conclusion
Our main goal has been the creation of a tool or “macro” to be used with Vensim simulation
environment which, applied on a given SD model, provides autoregressive endogenous structure
and short term forecasting capabilities. The main effort has been done searching for the
correspondence, in stock-flow modeling, of the main concepts and procedures that appear in the
SVAR model.
In order to develop this stock-flow version of the SVAR model, we have built two sub-models
using the Vensim simulation environment. Each of these sub-models corresponds to different phases
of the process of the SVAR analysis. The lagged variables, essential in the SVAR analysis, are now
treated as level variables. The calculation procedures have been similar to those of the original
econometric SVAR analysis, although the analytical resolution of some of the steps of the problem
has been done through simulation within the stock-flow sub-models.
The core of the stock-flow sub-models built is the reduced autoregressive form of the SVAR
analysis. The responses to the structural shocks have been obtained transforming them into non
orthogonalized innovations, by means of the corresponding matrix, which also has been estimated
with the second one of these sub-models.
As an illustration, we present an application to the study of the Spanish labor market. The
results obtained (estimates of the parameters, impulse-response functions, decomposition of the
variance of the forecasting error) with the stock-flow sub-models reproduce faithfully those of the
original application of the SVAR analysis.
's An important feature of the SVAR methodology is that the results obtained from the analysis of the impulse-response
functions and the variance decomposition of the forecasting error are related to the identification restrictions adopted.
14
References
— Usabiaga, C. (Dir.), Alvarez-de-Toledo, P., Crespo, A. and Nujiez, F. 2001. Comparacion entre
las Técnicas de Analisis Shift-Share y de Economias Virtuales, Vector Autorregresivo y
Dinamica de Sistemas: Aplicaciones al Mercado de Trabajo Andaluz. Research Project financed
by the Andalusian Government (Andalusian Statistics Institute).
— Bean, C. 1992. Identifying the Causes of British Unemployment. Center for Economic
Performance (London School of Economics), Working Paper n° 276.
— Blanchard, O.J. and Diamond, P. 1989.The Beveridge Curve. Brookings Papers on Economic
Activity, 1, pp. 1-60.
— Blanchard, O.J. and Diamond, P. 1992. The Flow Approach to Labor Markets. American
Economic Review, 82(2), pp. 354-359.
— Blanchard, O. and Quah, D.T. 1989. The Dynamic Effects of Aggregate Demand and Supply
Disturbances. American Economic Review, 79(4), pp. 655-673.
— Dolado, J.J. and Gomez, R. 1997. La Relacién entre Desempleo y Vacantes en Espaiia:
Perturbaciones Agregadas y de Reasignacion. Jnvestigaciones Economicas, 21(3), pp. 441-472.
— Gali, J. 1992. How Well Does the IS-LM Model Fit Postwar US Data. Quarterly Journal of
Economics, 107(2), pp. 709-738.
— Greene, W.H. 2003. Econometric Analysis (fifth edition), Prentice Hall, New York.
— Hamilton, J.D. 1994. Time Series Analysis, Princeton University Press, Princeton.
— Pissarides, C.A. 2000. Equilibrium Unemployment Theory, MIT Press, Cambridge (Mass.).
— Powell, M.J.D. 1964. An Efficient Method for Finding the Minimum of a Function of Several
Variables Without Calculating Derivatives. Computer Journal, 7(2), pp. 155-62.
— Powell, M.J.D. 1968. On the Calculation of Orthogonal Vectors. Computer Journal, 11(2), pp.
302-304.
— Schweikhardt, R.G. 1973. Labor Market Dynamics I, MIT Press, Cambridge (Mass.).
— Sims, C.A. 1980. Macroeconomics and Reality. Econometrica, 48(1), pp. 1-48.
APPENDIX A
SUB-MODEL LABOR 1 VENSIM CODE:
Data input:
data[x]
x: u,v,pa,d1,d2,d3,d4
Obtaining the data lagged one period:
"data in t-1"[x]= INTEG (incr dat[x]-decr dat[x], 1)
incr dat[x]= data[x]
decr dat[x]= "data in t-1"[x]
Obtaining the model variables:
Variables in t[v1]= LN(data[v])-LN("data in t-1"[v])-LN(data[u])+LN("data in t-1" [u])
Variables in t[v2]= LN(data[u])-LN("data in t-1"[u])
Variables in t[v3]= LN(data[pa])-LN("data in t-1"[pa])
variables: v1,v2,v3
Autoregressive Structure:
"variables in t-i"[variables,lag]= INTEG (incr var[variables,lag]-decr var [variables,lag], 0)
incr var[variables,"t-1"]= Variables in t[variables]
incr var[variables,"t-2"]= "variables in t-i"[variables,"t-1"]
incr var[variables,"t-3"]= "variables in t-i"[variables,"t-2"]
incr var[variables,"t-4"]= "variables in t-i"[variables,"t-3"]
decr var[variables,lag]= "variables in t-i"[ variables, lag]
lag: "t-1","t-2","-3","-4"
Dummies:
dummies[t1]= data[d1]-data[d4]
dummies[t2]= data[d2]-data[d4]
dummies[t3]= data[d3]-data[d4]
quarters: t1, 2, t3
Variables forecast:
variables forecast in t[variables]= IF THEN ELSE (Time>=6, SUM("variables in t-i"
[variables! lag!]*"variables in t-i coefficients" [variables, variables! ,lag!]) + constants[variables]+
SUM(dummies[quarter!]*Dummies coefficients[variables,quarter!]),
Variables in t[variables])
Coefficients to estimate:
"variables in t-i coefficients" [variables, variables, lag]= 0
Dummies coefficients[variables,quarter]= 0
constants[variables]= 0
Obtaining the innovations and the variance-covariance matrix:
innovations[variables]= Variables in t[variables]-variables forecast in t[ variables]
inc[v1v1]= innovations[v1 ]*innovations[v1]
inc[v2v2]= innovations[v2]*innovations[v2]
inc[v3v3]= innovations[v3]*innovations[v3]
inc[vlv2]= innovations[v1 ]*innovations[v2]
inc[v1v3]= innovations[v1 ]*innovations[v3]
inc[v2v3]= innovations[v2]*innovations[v3]
previous cov[cova]= INTEG (inc[cova]/(FINAL TIME-5-16),0)
cov[cova]= previous cov[cova]+inc[cova]/(FINAL TIME-5-16)
cova: vl v1,v2v2,v3v3,vlv2,v1v3,v2v3
Simulation control parameters:
FINAL TIME = 72
~ Quarter
~ The final time for the simulation.
INITIAL TIME =1
~ Quarter
~ The initial time for the simulation.
SAVEPER = |
~ Quarter
~ The frequency with which output is stored.
TIME STEP = 1
~ Quarter
~ The time step for the simulation.
SUB-MODEL LABOR 2 VENSIM CODE:
Autoregressive Structure:
"variables in t-i"[variables,lag,j]= INTEG (incr var[variables,lag,j]-decr var [variables,lag,j], 0)
incr var[variables,"t-1",j]= variables forecast in t[variables,j]
incr var[variables,"t-2",j]= "variables in t-i"[variables,"t-1",j]
incr var[variables,"t-3",j]= "variables in t-i"[variables,"t-2",j]
incr var[variables,"t-4",j]= "variables in t-i"[variables,"t-3",j]
decr var[variables,lag,j]= "variables in t-i"[variables,lag,j]
lag: "t-1","t-2","t-3","t-4"
variables : v1, v2, v3
Variables forecast:
variables forecast in t[variables,j]= SUM("variables in t-i"[variables! ,lag!,j] * "variables in t-i
coefficients" [variables, variables!,lag!]) + innovations[variables,j]
J: J 1253
Estimated Coefficients:
"variables in t-i coefficients"[v1,v1,lag] = 0.12661, -0.213419, 0.209034, 0.0646055
"variables in t-i coefficients"[v1,v2,lag] = - 1.31123,0.30969,-0.19163,0.935878
"variables in t-i coefficients"[v1,v3,lag] = 7.57044,0.909342,- 6.15989,7.8355
"variables in t-i coefficients"[v2,v1,lag] = - 0.0174353,- 0.00726969,- 0.0102628,0.0367592
"variables in t-i coefficients"[v2,v2,lag] = 0.507797,0.193024,- 0.13983,0.11152
"variables in t-i coefficients"[v2,v3,lag] = - 0.610707,- 2.17617,1.13846,- 2.48651
"variables in t-i coefficients"[v3,v1,lag] = - 0.00261 106,0.000206853,0.000728788,0.0009468
"variables in t-i coefficients"[v3,v2,lag] = - 0.0214941,- 0.0438737,0.00386141,0.0312963
"variables in t-i coefficients"[v3,v3,lag] = 0.12642,- 0.0313129,0.154495,0.00487324
Obtaining the variables in levels:
accumulated forecast v[j]= accumulated forecast[v 1 ,j]+accumulated forecast[v2,j]
accumulated forecast[variables,j]= previous accumulated forecast[variables,j] + variables forecast
in t[variables,j]
inc previous accumulated forecast[variables,j]= variables forecast in t[variables,j]
previous accumulated forecast[variables,j]=INTEG(inc previous accumulated forecast[variables,j],
0)
Obtaining the matrix S (only in version 2):
payoff[pol]= IF THEN ELSE(Time = FINAL TIME, -accumulated forecast[v1,j2]/2 , 0 )
payoff[po2]= IF THEN ELSE(Time = FINAL TIME, -accumulated forecast[v1,j3]/2 , 0 )
payoff[po3]= IF THEN ELSE(Time = FINAL TIME, -accumulated forecast[v2,j3]/2 , 0 )
payoff[po4]= IF THEN ELSE(Time = FINAL TIME, -(1-(cov1[v1,j1]/cov[v1,j1]))*2 , 0)
payoff[po5]= IF THEN ELSE(Time = FINAL TIME, -(1-(cov1[v1,j2]/cov[v1,j2]))2 , 0)
payoff[po6]= IF THEN ELSE(Time = FINAL TIME, -(1-(cov1[v1,j3]/cov[v1,j3]))2 , 0)
payoff[po7]= IF THEN ELSE(Time = FINAL TIME, -(1-(cov1[v2,j2]/cov[v2,j2]))2 , 0)
payoff[po8]= IF THEN ELSE(Time = FINAL TIME, -(1-(cov1[v2,j3]/cov[v2,j3]))2 , 0)
payoff[po9]= IF THEN ELSE(Time = FINAL TIME, -(1-(cov1[v3,j3]/cov[v3,j3]))2 , 0)
po: pol,po2,po3,po4,po5,po06,po7,po8,po9
cov1[v1,j1] = magnitude[pol ]‘2+magnitude[po4]*2+magnitude[po5]}*2
covl[v1,j2] =
magnitude[pol ]*magnitude[po7]+magnitude[po4]*magnitude[po2]+magnitude[po5]*
magnitude[po6]
covl[v1,j3] =
magnitude[pol ]*magnitude[po8]+magnitude[po4]*magnitude[po9]+magnitude[po5]*
magnitude[po3]
cov 1[v2,j2] = magnitude[po7]2+magnitude[po2]*2+magnitude[po6]}2
covl[v2,j3] =
magnitude[po7]*magnitude[po8]+magnitude[po2]*magnitude[po9]+magnitude[po6]*
magnitude[po3]
cov1[v3,j3] = magnitude[po8]\2+magnitude[po9]2+magnitude[po3]*2
covl[v2,j1] =
magnitude[pol ]*magnitude[po7]+magnitude[po4]*magnitude[po2]+magnitude[po5]*
magnitude[po6]
covl[v3,j1] =
magnitude[po 1 ]*magnitude[po8]+magnitude[po4]*magnitude[po9]+magnitude[po5]*
magnitude[po3]
cov1[v3,j2] =
magnitude[po7]*magnitude[po8]+magnitude[po2]*magnitude[po9]+magnitude[po6]*
magnitude[po3]
cov[v1,j] = 0.0202,-0.000162531,9.0526e-005
cov[v2,j] = -0.000162531,0.000385541,1.74239e-005
cov[v3,j] = 9.0526e-005,1.74239e-005,1.02585e-005
Innovations:
Version 1:
innovations[variables,j]= duration*magnitude[variables,j]
magnitude[v1.j]= 1,0,0
magnitude[v2,j]= 0,1,0
magnitude[v3,j]= 0,0,1
duration = PULSE(1,1)
Version 2:
innovations[v1,j1] = duration*magnitude[po1]
innovations[v1 ,j2] = duration*magnitude[po4]
innovations[v1,j3] = duration*magnitude[po5]
innovations[v2,j1] = duration*magnitude[po7]
innovations[v2,j2] = duration*magnitude[po2]
innovations[v2,j3] = duration*magnitude[po6]
innovations[v3,j1] = duration*magnitude[po8]
innovations[v3,j2] = duration*magnitude[po9]
innovations[v3,j3] = duration*magnitude[po3]
magnitude[pol]= 1
magnitude[po2]= 1
magnitude[po3]= 1
magnitude[po4]= 1
magnitude[po5]= 1
magnitude[po6]= 1
magnitude[po7]= (cov[v1 ,j2]-magnitude[po4]*magnitude[po2]-magnitude[po5] *magnitude[po6]) /
magnitude[pol]
magnitude[po8]= (((cov[v1,j3]-magnitude[po5]*magnitude[po3])*magnitude[po2])-(cov[v2,j3]-
magnitude[po6] *magnitude[po3])*magnitude[po4])/(magnitude[po2]*magnitude[pol ]-
magnitude[po7] * magnitude[po4])
magnitude[po9]= (cov[v1 ,j3]-magnitude[po5 ]*magnitude[po3]-magnitude[po1 ]*magnitude[po8}]) /
magnitude[po4]
duration = PULSE(1,1)
Version 3:
innovations[variables,j] = duration*magnitude[variables,j]
magnitude[v1,j] = 0.116078,0.0745076,-0.0342579
magnitude[v2,j] = -0.0054,0.0127004,0.0139544
magnitude[v3,j] = 0.0018,-0.000581161,0.00251562
duration = PULSE(1,1)
Obtaining the variance decomposition of the forecasting error “vdfe” (only in version 3):
vdfe[v1,j]= accumulated forecast[v2,j]/2 / (accumulated forecast[v2,j1]*2t+accumulated
forecast[v2,j2]/2+accumulated forecast[v2,j3]/2)
vdfe[v2,j]= accumulated forecast[v3,j]/2 / (accumulated forecast[v3,j1]2t+accumulated
forecast[v3,j2]/2+accumulated forecast[v3,j3]/2)
vdfe[v3,j]= accumulated forecast v[j]‘2 / (accumulated forecast v[j1]2+accumulated forecast
v[j2]’2+accumulated forecast v[j3]*2)
Simulation control parameters:
FINAL TIME = 100
~ Quarter
~ The final time for the simulation.
INITIAL TIME =1
~ Quarter
~ The initial time for the simulation.
SAVEPER = 1
~ Quarter
~ The frequency with which output is stored.
TIME STEP = 1
~ Quarter
~ The time step for the simulation.
APPENDIX B
QUARTERLY SERIES OF VALUES FOR VACANCIES, UNEMPLOYMENT, LABOR
FORCE AND DUMMIES
Source: Labor Force Survey (published by INE), Employment Statistics (published by INEM)
Quarters Labor
1977:01- | Vacancies | Unemployment | Force | 1: 1% quarter | 1: 2“ quarter | 1: 3° quarter | 1: 4" quarter
1994:04 (Thnds) (Thnds) (Thnds) 0: rest 0: rest 0: rest 0: rest
Time Data[V] Data[U] Data[L] | Datafd1] Data[d2] Data[d3] Data[d4]
1 22,8 644 13265 1 0 0 0
2 28,7 634 13285 0 1 0 0
3 23,3 704 13337 0 0 1 0
4 22,3 750 13380 0 0 0 1
5 22,4 846 13377 1 0 0 0
6 30,8 864 13287 0 1 0 0
7 28 934 13302 0 0 1 0
8 25,4 996 13306 0 0 0 1
9 28,8 1061 13294 1 0 0 0
10 33,6 1061 13257 0 1 0 0
"1 26,8 1137 13327 0 0 1 0
12 271 1241 13337 0 0 0 1
13 30,3 1384 13340 1 0 0 0
14 25,2 1449 13261 0 1 0 0
15 18,2 1504 13279 0 0 1 0
16 174 1631 13273 0 0 0 1
7 16.4 1755 13300 1 0 0 0
18 16,9 1798 13251 0 1 0 0
19 21,2 1891 13354 0 0 1 0
20 26,6 2002 13375 0 0 0 1
21 29 2077 13423 1 0 0 0
22 33,9 2052 13419 0 1 0 0
23 30,6 2148 13493 0 0 1 0
24 33,7 2248 13572 0 0 0 1
25 42,4 2336 13550 1 0 0 0
26 60,2 2275 13561 0 1 0 0
27 57.3 2352 13648 0 0 1 0
28 47.9 2453 13703 0 0 0 1
29 515 2670 13679 1 0 0 0
30 64,9 2681 13623 0 1 0 0
31 53,9 2745 13691 0 0 1 0
32 50,7 2907 13719 0 0 0 1
33 59,7 2963 13739 1 0 0 0
34 98,3 2934 13705 0 1 0 0
35 82,8 2931 13800 0 0 1 0
36 80,5 2981 13853 oO 0 0 1
20
Quarters Labor
1977:01- Vacancies | Unemployment | Force | 1: 1“ quarter | 1: 2" quarter | 1: 3° quarter | 1: 4" quarter
1994:04 (Thnds) (Thnds) (Thnds) 0: rest 0: rest 0: rest 0: rest
Time Data[V] Data[U] Data[L] | Datafd1] Data[d2] Data[d3] Data[d4]
37 103.6 3017 13916 1 0 0 0
38 126,7 2932 13983 0 1 0 0
39 82,6 2894 14042 0 0 1 0
40 75.4 2925 14145 0 0 0 1
mu 96,4 2992 14232 1 0 0 0
42 113,8 2936 14266 0 1 0 0
43 101 2918 14440 0 0 1 0
44 97.5 2904 14498 0 0 0 1
45 114.6 2941 14553 1 0 0 0
46 136,7 2899 14608 0 1 0 0
47 120,3 2850 14701 0 0 1 0
48 105.9 2701 14621 0 0 0 1
49 1113, 2698 14702 1 0 0 0
50 151.6 2555 14750 0 1 0 0
51 138,7 2468 14895 0 0 1 0
52 1245, 2522 14930 0 0 0 1
53 128.5, 2511 14993 1 0 0 0
54 183.6 2438 14996 0 1 0 0
55 129 2392 15048 0 0 1 0
56 125.8 2424 15045 0 0 0 1
57 114,4 2421 15000 1 0 0 0
58 140.4 2388 15011 0 1 0 0
59 119,2 2480 15157 0 0 1 0
60 105.5, 2566 15126 0 0 0 1
61 100,4 2632 15082 1 0 0 0
62 109.8 2686 15144 0 1 0 0
63 107 2789 15202 0 0 1 0
64 109 3047 15193 0 0 0 1
65 79,1 3300 15181 1 0 0 0
66 101,9 3397 15265 0 1 0 0
67 81,7 3546 15423 0 0 1 0
68 724 3682 15406 0 0 0 1
69 65,4 3793 15428 1 0 0 0
70 90,5 3763 15491 0 1 0 0
ra 86,9 3698 15486 0 0 1 0
72 61,9 3698 15468 0 o ) 1
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