Journal of Economic Structures

The Official Journal of the Pan-Pacific Association of Input-Output Studies (PAPAIOS)

Journal of Economic Structures Cover Image
Open Access

Restatement of the I-O Coefficient Stability Problem

Journal of Economic StructuresThe Official Journal of the Pan-Pacific Association of Input-Output Studies (PAPAIOS)20132:2

https://doi.org/10.1186/2193-2409-2-2

Received: 2 September 2012

Accepted: 23 January 2013

Published: 4 February 2013

Abstract

The capacity of input-output tables to reflect the structural peculiarities of an economy and to forecast, on this basis, its evolution, depends essentially on the characteristics of the matrix A—matrix of I-O (or technical) coefficients. However, the temporal behaviour of these coefficients is yet an open question. In most applications, the stability of matrix A is usually admitted. This is a reasonable assumption only for a short-medium term. In the case of longer intervals, the question is much more complicated.

We shall empirically discuss this problem by using Romanian input-output tables. Our statistical option was motivated inter alia by the existence of official annual data for two decades (1989–2009).

As an introduction, Sect. 1 characterises the general framework of paper. Section 2—The main characteristics of I-O coefficients as statistical time series—examines the variability of technical coefficients expressed in both volume and value terms. The analysis is convergent to other previous works, confirming that the evolution of these coefficients in real and nominal terms is roughly similar. The main finding of this section is that, on one hand, the I-O coefficients are volatile, but on the other, they are serially correlated.

Consequently, Sect. 3—Attractor hypothesis—examines a possible presence of attractors in corresponding statistical series. The paper describes a methodology to approximate these using new indicators obtained by summation—in columns and rows—of the technical coefficients (colsums sca j and rowsums sra i ). The RAS method is involved as a connecting technique between these indicators and sectoral data.

Section 4—Conclusions—presents the main conclusions of the research and outlines several possible future developments. The database and econometric analysis are presented in Statistical and Econometric Appendix.

JEL Classification: C12, C32, C43, C67.

Keywords

I-O coefficients Volatility Serial correlation Attractor RAS technique

1 Introduction

1. The capacity of input-output tables to reflect the structural peculiarities of the economy and to forecast, on this basis, its evolution, depends essentially on the characteristics of matrix A of I-O (or technical) coefficients. The so-called Leontief matrix [ ( I A ) 1 ] has proven to be a powerful analytical tool in the investigation of propagated effects induced by inter-industry production chains. Our paper utilises the methodological framework developed in [23, 24, 28, 41, 44].

The temporal behaviour of I-O coefficients is yet an open question. In most applications, the stability of matrix A is usually assumed. This comes from both classical and extended interpretations of the Cobb–Douglas production function. According to Sawyer (p. 327 in [38]), “Under the first of these alternative hypotheses, the a ij will be stable in volume terms. Under the second, the a ij will be stable in value terms”. Generally, the relative stability of the technical coefficients can be considered as a reasonable assumption for a short-medium term. In the case of longer intervals, the question is much more complicated.

2. We shall empirically discuss this problem by using Romanian input-output tables. Our statistical option was motivated inter alia by the existence of official annual data for two decades (1989–2009).

These tables are built on an extended classification comprising 105 branches [17]. To simplify computational operations, the present research relates to a more compact version of 10 sectors [11, 33], as described in Table 1.
Table 1

Sectoral structure of the Romanian input-output tables

Code

Definition

1

Agriculture, forestry, hunting, and fishing

2

Mining and quarrying

3

Production and distribution of electric and thermal power

4

Food, beverages, and tobacco

5

Textiles, leather, pulp and paper, furniture

6

Machinery and equipment, transport means, other metal products

7

Other manufacturing industries

8

Constructions

9

Transports, post, and telecommunications

10

Trade, business, and public services

The correspondence of this collapsed structure to the original extended nomenclature is detailed in [12]. As in any aggregation, the one proposed in Table 1 implies some losses of information.

Nevertheless, the chosen analysis classification remains sufficiently complex and relevant to involve in this discussion some conceptual anchors of chaos theory. Specifically, we investigate whether the I-O coefficients series could contain sets of attractor points. To answer this question, a methodology for their numerical estimation will be applied to the available data.

3. The robustness of structural changes analysis and of the sectoral dynamic general equilibrium models depends mainly on the temporal behaviour of I-O coefficients. These can be estimated:

• in volume terms (at constant prices), denoted as ca ij ; and

• in value terms (at current prices), usually denoted as a ij .

The first estimation concerns the real economy, while the second relates to the nominal one. These determinations are mediated by the relative prices ( reP ij ).

If cx ij represents the part of sector i’s production (at constant prices p 0 i ) used in sector j, and cX j —total output of the sector j (at constant prices p 0 j ), then:
ca ij = cx ij / cX j
(1)
and
a ij = x ij / X j
(2)

in which the same components of the above ratio are expressed in current prices ( p i and p j , respectively).

Introducing the indices P i = p i / p 0 i and P j = p j / p 0 j , we obtain
a ij = x ij / X j = cx ij P i / ( cX j P j ) = ( cx ij / cX j ) ( P i / P j ) = ca ij reP ij
(3)

where reP ij = P i / P j .

The I-O coefficients at constant prices were estimated using formula (3), which is equivalent to ca ij = a ij / reP ij .

Econometric estimations involve several aggregative indicators resulted from the technical coefficients in value terms, namely:

• Colsums ( sca j ), which summarises the I-O coefficients in columns,
sca j = j a ij with  j = fixed ; i = 1 , 2 , , n
(4)

These approximate the weight of intermediary consumption in the total output of every sector.

• Rowsums ( sra i ), which summarises the I-O coefficients in rows,
sra i = i a ij with  i = fixed ; j = 1 , 2 , , n
(5)

These approximate the contribution of each sector to the intermediary consumption of the entire economy.

2 The Main Characteristics of I-O Coefficients as Statistical Time Series

In the evaluation of the temporal features of I-O coefficients, three questions are relevant:

• Do some peculiarities exist in the co-movement of I-O coefficients real-nominal expression?

• Are I-O coefficients really stable?

• Are these coefficients serially correlated?

The following sections attempt to find answers to these problems.

1. Relating to the first question, in principle the dynamics of real and nominal I-O coefficients are interdependent. On the supply side, the modifications in production costs (reflected by ca ij ) influence the current prices of transactions. On the other hand, the changes in relative prices (reflected by a ij ) have an impact on the demand structure and, consequently, on the size of the output and the conditions (technology, human capital, etc.) in which this is achieved. Due to the complexity of economic life, in each historical period this interdependence has some specific features. This is the reason why statistical evaluation becomes important. Given these, the estimation of the synchronisation degree (SDa) of changes in a ij and ca ij can be conclusive.

1.1. Starting from some proposals advanced in the literature about economic structures and cycles, three concrete formulae are considered.

(a) The first could be referred to as the cosine synchronisation degree (SDa1) since it is estimated as a vectorial angle between time series of I-O coefficients in their double expressions:
SDa 1 = t ( a ij , t ca ij , t ) / [ ( t a ij , t 2 ) 1 / 2 ( t ca ij , t 2 ) 1 / 2 ]
(6)
(b) The well-known correlation coefficient is often applied in statistical comparisons of real-nominal economic time series (see, for instance, [1, 8, 9, 16, 20, 26, 35, 39]). This Galtung–Pearson synchronisation degree (SDa2) is calculated as a ratio of covariance of series a ij and ca ij to the product of their standard deviations, respectively:
SDa 2 = ( n t a ij , t ca ij , t t a ij , t tca ij , t ) / { [ ( n t a ij , t 2 ( t a ij , t ) 2 ) 1 / 2 ] [ ( n t ca ij , t 2 ( t ca ij , t ) 2 ) 1 / 2 ] }
(7)
(c) A third method used in the economic literature for such analysis is worth mentioning [6, 9, 16]. We shall refer to it as the binary synchronisation degree (SDa3), which measures the proportion in which the compared series evolve in the same direction. Technically, a dummy variable is used, its value being 1 when the respective I-O coefficient increases, and 0 when it decreases or stagnates. If such an alternative assignment is denoted as da ij for series a ij , and, correspondingly, as dca ij for series ca ij , then SDa3 is given as
Sda 3 = { t ( da ijt dca ijt ) + ( 1 da ijt ) ( 1 dca ijt ) } / n
(8)

n being the number of observations in the sample.

1.2. The above described SDa1, SDa2, and SDa3 do not raise special computational problems, and moreover, are easy to interpret. They have been applied in the series of all 100 technical coefficients, and the obtained results are synthesised in Fig. 1. Therefore, 95 % of SDa1 is positioned within 0.75–1 limits, and only 5 % do not exceed 0.75. At the same time, SDa2 is less than 0.5 in only one-fourth of cases; it is between 0.5–0.75 in 12 % of cases, and exceeds 0.75 in the rest (63 %). The last indicator is even more conclusive: SDa3 is within 0.75–1 in 87 % of cases, and less than 0.65 in none of the cases.
Fig. 1

Synchronisation degree (SDa) of changes in a ij and ca ij

Summarising, all calculated synchronisation degrees of changes in a ij and ca ij indicate that the I-O coefficients in both their expressions—in volume and value terms—evolve in a similar manner.

1.3. A more nuanced understanding of this interdependence could be obtained by determining the global variability degree of changes in all I-O coefficients, avca for ca ij and ava for a ij :
avca t = j ( wq it ( i ( ca ijt ca ijt 1 ) 2 ) 1 / 2 )
(9)
ava t = j ( wq it ( i ( a ijt a ijt 1 ) 2 ) 1 / 2 )
(10)

where wq i represents the weight of sector i in the total output of economy.

There were applied two unit root tests for ava and avca: ADF—Augmented Dickey–Fuller and PP—Phillips–Perron. All available options concerning the exogenous (no one, constant, constant plus linear trend) have been computed. The results are detailed in Table 2. Indulgently accepting the stationarity assumption, the pairwise Granger test statistically accredits a certain interconnection between the respective series only on a short run, with the causality direction from avca toward ava (probability of null hypothesis = 0.0881) for one lag, and converse, from ava toward avca (probability of null hypothesis = 0.0943) for two lags. More appropriate for non-stationary series, the test Toda–Yamamoto [43] indicates again on a short run (two lags) an influence of ava on avca (according to F-statistic, and Chi-square, the probability for null hypothesis “ava does not cause avca” represents 0.1107 and, respectively, 0.0869).
Table 2

Unit root tests for ava and avca

 

Exogenous

None

Constant

Constant, linear trend

Null hypothesis: ava has a unit root

Null hypothesis: avca has a unit root

Null hypothesis: ava has a unit root

Null hypothesis: avca has a unit root

Null hypothesis: ava has a unit root

Null hypothesis: avca has a unit root

Augmented Dickey–Fuller

 t-statistic

−0.893149

−1.306076

−2.758188

−2.402669

−3.582018

−2.83151

 Prob.

0.3163

0.1702

0.0831

0.1541

0.0589

0.205

Phillips–Perron

 Adj. t-statistic

−0.713021

−1.355274

−2.661355

−2.441546

−3.582018

−2.676491

 Prob.

0.3943

0.1567

0.0989

0.1445

0.0589

0.2553

Except for 4 years (1991, 2002–2003, and 2005), the ratio of ava to avca was <1 in all periods. This means that the changes in relative prices somehow attenuated the shifts in technical coefficients in volume terms.

2. The examination of the co-movement pattern of changes in the real and nominal expressions of I-O coefficients does not clarify if these are relatively stable (small annual changes) or significantly volatile. This is important for our analysis.

In the case of I-O coefficients, we shall adopt a larger interpretation of volatility as an integrating measure of the frequency and size of the changes registered in their evolution. A comprehensive analysis of volatility determinants exceeds the thematic perimeter of this paper. Briefly, we recall the following factors:

• the performance of preponderantly used technologies that redound to most aspects of costs (labour productivity, energy and raw material intensities, quality of goods and services, length of productive cycles, etc.);

• the dimension, and structure of domestic demand, which influence the scale efficiency and relative prices;

• the openness degree of the country, with its impact on firms’ access to external markets, on import substitution effects, and on productive factors migration;

• the institutional reforms that have a great role in both emerging and developed economies; and

• the operational consequences of macroeconomic policies that can facilitate or, on the contrary, hinder the fructification of comparative advantages for the respective economy.

Quantitatively, the volatility of a given indicator will be approximated by its variation coefficient calculated (for the entire available time series) as follows. If q t is the value of this indicator at moment t ( t = 1 , 2 , , s ) and ω q its level admitted as referential, then this coefficient ( C V ) is determined by
C V = [ ( t ( q t / ω q 1 ) 2 ) / s ] 1 / 2
(11)

In principle, ω q can differ depending on the objectives of analysis. As a first choice, we adopt the sample mean, accommodating expression (11) to the standard deviation formula largely used in modern statistics. Such an approach is suitable in forecasting the volatility for different interested horizons by simple extrapolation of its statistically registered level.

The proposed procedure consists of the following steps:

• For each interval two estimations of the respective indicator are determined: an upper and a lower level. The first is obtained by multiplying the mean of the previous series by ( 1 + C V ) , while the other results similarly but using ( 1 C V ) as a multiplier. We shall designate these values as Y for the upper level and y for the lower one.

• On this basis, two new means are also computed, mixing the corresponding previous series with Y and y: they will be represented by the symbols M, and m, respectively. The statistical volatility is applied again by multiplying the new M by ( 1 + C V ) and m by ( 1 C V ) . This procedure is continued as much as it is considered useful (the forecast period being denoted by τ = 1 , 2 , , n ).

• The difference ( Y y ) can be admitted as an error ( ef V ) attributable to the initially estimated volatility. The interpretation of results would be facilitated by equalising the starting sample mean to unity.

More formally, for the upper level, we have
Y τ 1 = ( 1 + C V ) M τ 2 , τ = 1 , 2 , , n
(12)
M τ 1 = ( ( s + τ 2 ) M τ 2 + Y τ 1 ) / ( s + τ 1 ) = ( ( s + τ 2 ) M τ 2 + ( 1 + C V ) M τ 2 ) / ( s + τ 1 ) = M τ 2 ( s + τ 2 + 1 + C V ) / ( s + τ 1 ) = M τ 2 ( s + τ 1 + C V ) / ( s + τ 1 ) = M τ 2 ( 1 + C V / ( s + τ 1 ) )
(13)
Y τ = ( 1 + C V ) M τ 1 = ( 1 + C V ) M τ 2 ( 1 + C V / ( s + τ 1 ) )
(14)
A simplification can be obtained by passing to indices ( IY τ = Y τ / Y τ 1 ):
IY τ = { ( 1 + C V ) M τ 2 ( 1 + C V / ( s + τ 1 ) ) } / [ ( 1 + C V ) M τ 2 ] = ( 1 + C V / ( s + τ 1 ) )
(15)
This relationship is valid for τ 2 since Y 0 = M 0 = 1 and Y 1 = ( 1 + C V ) M 0 = ( 1 + C V ) . Finally, we have
Y n = ( 1 + C V ) τ ( 1 + C V / ( s + τ 1 ) ) for  τ = 2 , , n
(16)
Symmetrically, the expression of y n is determined as
y n = ( 1 C V ) τ ( 1 C V / ( s + τ 1 ) ) , again for  τ = 2 , , n
(16a)
and
ef Vn = [ ( 1 + C V ) τ ( 1 + C V / ( s + τ 1 ) ) ] [ ( 1 C V ) τ ( 1 C V / ( s + τ 1 ) ) ]
(17)
Therefore, ef Vn is influenced mainly by C V , s, and τ. Figures 2(a) and 2(b) illustrate some indifference curves of the initial C V depending on s and m, estimated under the conditions given in Table 3.
Fig. 2

(a) Estimation of the initial C V depending on s and the final desirable ef V (Variant A). (b) Estimation of the initial C V depending on s and the final desirable ef V (Variant B)

Table 3

Estimation of the initial C V depending on s and the final desirable ef V

Variant

Forecasted interval

Final desirable ef V

C V 050 A

5

0.05

C V 075 A

5

0.075

C V 100 A

5

0.1

C V 125 A

5

0.125

C V 050 B

10

0.05

C V 075 B

10

0.075

C V 100 B

10

0.1

C V 125 B

10

0.125

The presented algorithm can be used in establishing a kind of taxonomy scale of I-O coefficients volatility. Toward this aim, it would be necessary to determine the desirable levels of ef V and the length of τ (that is, the value of n). A possible starting point in this sense can be the expectable financial risk induced by economic decisions linked to forecasted I-O coefficients. Addressing this question requires further research. A possible solution to this problem could be adequately extrapolated in other socio-economic fields.

Returning to the Romanian I-O tables, the variation coefficient, based on formula (11), was computed for all statistical series in 1989–2009 (100 ca ij and 100 corresponding a ij ). The results are summarised in Table 4, which shows that there is no I-O coefficient with C V < 0.05 and only one with C V < 0.1 ; instead, 85 % of ca ij and 73 % of a ij are characterised by C V > 0.3 . The hypothesis that the mean of all C V would be between 0.4–0.65 was tested for both series C V ca ij and C V a ij . The results are presented in Fig. 3.
Fig. 3

Probability for the mean of C V ca ij and C V a ij to be situated between 0.4–0.65 (tabulated on abscissa)

Table 4

Tabulation of statistical variation coefficients ( C V )

Limits of var. coeff.

C V ca ij

C V a ij

0.05–0.1

1

1

0.1–0.2

5

11

0.2–0.3

9

15

0.3–0.4

15

11

0.4–0.5

16

23

0.5–0.6

16

12

0.6–0.7

10

5

0.7–0.8

9

10

0.8–0.9

8

6

0.9–1

4

3

>1

7

3

Total

100

100

In many cases, the volatility is so high that the calculated ef V becomes abnormal even for very short intervals. As an example, the evolution of the error attributable to the initially estimated volatility ( ef V ) was determined for three cases: for C V = 0.1 (variant 1), C V = 0.2 (variant 2), and C V = 0.3 (variant 3), during τ = 1 , 2 , , 15 . The results of this exercise are denoted as ef V 1 , ef V 2 , and ef V 3 , and are summarised in Fig. 4. We recall that the computed data represent indices comparatively to the mean level of the statistical series (the mean equalised to 1). For C V = 0.3 , the difference between the forecasted limits of the respective indicator can reach 0.7 in five years and 0.8 in ten. Even for C V = 0.1 , the potential forecasting error is hardly acceptable. As we have already shown, the levels calculated for Romanian I-O tables are overall much higher than the simulated (in Fig. 4) values of C V .
Fig. 4

Simulated ef V for three variants of C V

3. Like other previous studies, the analysis of Romanian I-O tables confirms that the technical coefficients are volatile. What needs to be documented is the nature of this volatility, and the highly questionable factor is the presence of non-linearities in the respective statistical series. Such a possibility has been revealed in many economic indicators [3, 34]. In the case of Romanian I-O tables, we shall also examine whether the data regarding the technical coefficients are independent or, on the contrary, serially correlated.

It is widely accepted that: “The correlation sum in various embeddings can…be used as a measure of determinism in a time series” (p. 313 in [40]). The BDS test is sensitive to a large variety of possible deviations from independence in time series, including linear dependence, non-linear dependence, or chaos. Concerning this technique, our turns to the conceptual and applicative framework developed in [2, 6, 32]. Thus, the null hypothesis of independent and identically distributed (i.i.d.) data is checked against an unspecified alternative.

For the I-O tables examined in this paper, the BDS test was applied to both categories of coefficients—at constant ( ca ij ) and current prices ( a ij ). Concerning the embedding dimension, we sought to cover an extended range of possibilities. Due to the insufficient length of the statistical series, five such variants were adopted: 2, 3, 4, 5, and 6. As a principal guiding mark, the p-value for the tested null hypothesis was retained, computed for the sample data (normal probability) and for their random repetitions (bootstrap probability). Recent software provided both probabilities (normal and bootstrap) for three options related to the distance used for testing: the fraction of pairs, the standard deviations, and the fraction of range. Therefore, 30 p-values were computed for each technical coefficient, resulting in five dimensions, two tested series (original and bootstrap), and three distances.

The characterisation of the global distributions of the obtained p-values for all series of technical coefficients will be discussed. Two classifications are significant.

First, the p-values for all 3000 estimations are classified according to the following thresholds: under 0.05, 0.05–0.1, 0.1–0.25, and 0.25–1, presented in Fig. 5. This shows that in the case of ca ij , over 75 % of p-values (2252) are below 0.05; if the group 0.05–0.1 is added, the proportion reaches 80 %. The picture is similar for a ij : almost 72 % of tests are estimated with p-values of under 0.05, and approximately 76 % have p-values of less than 0.1. This means that, generally, the series of I-O coefficients (either at constant or current prices) are not independent.
Fig. 5

Distribution of BDS tests (whole sample) in terms of p-value

The second application sorts I-O coefficients depending on the number of registered BDS p-values under 0.05. Toward this aim, six classes are delimited: up to 5 times, 5–10, 10–15, 15–20, 20–25, and 25–30. Evidently, the sum of classes is equal to 100 (the totality of coefficients). Figure 6 synthesises this distribution, showing that in each of the 86 ca ij , at least 15 tests had p-values of under 0.05. The result is no different in the case of a ij coefficients: among 90 cases, at least 10 p-values were under 0.05. The similarity of the ca ij and a ij series suggests that the volatility of relative prices does not substantially influence the presence of serial correlation in the data.
Fig. 6

Classification of the technical coefficients depending on registered p-values under 0.05

Thus, in this section, we can conclude that, on one hand, the I-O coefficients are volatile, but on the other, they are serially correlated. Both statements have statistical support. More simply stated, we acknowledge a paradox because the high volatility indicates rather the presence of a quasi-disorder, while the serial correlation indicates a possible stable pattern in the analysed time series. The following section focuses on this exciting matter.

3 Attractor Hypothesis

The revealed contradictory combination of relatively high volatility of data and their consistent serial correlation generates a legitimate question: Is this contradiction a sign of a possible presence of an attractor in statistical series?

1. Generally, an attractor is considered a point or a closed subset of points (lines, surfaces, volumes), toward which a given system tends to evolve independently of its initial (starting) state [2931, 36, 37]. Three types are frequently mentioned:

• stable steady states,

• different types of cycles, and

• strange attractors.

The first type is relatively usual in Economics (“At best, the notion of equilibrium might, in practice, be identified with the notion of <attractor>”; p. 34 in [14]). The list of such examples is long, from the optimal rates of accumulation to the extended palette of Phillips curves.

Such points or lines need to be regarded rather as historical (that is, contextually determined) phenomena than as permanent, inflexible benchmarks. It is worth mentioning that some authors considered the “natural rate of unemployment” as a rather weak attractor (p. xiii in [4]).

Taking into account the numerous such applications in economics, the following systematisation of types of stable steady states would be useful:

• stable points,

• constant rates of movement (in different expressions, such as indices, elasticities, ratios, spreads, etc.), and

• bands of evolution.

All these are interesting perspectives in researching I-O tables. However, such a target would require many and sustained efforts. Our target is very narrow, namely, to attempt to identify in the studied statistical series some fixed points as possible attractors. This hypothesis will be used in two sub-variants: fixed points as such or slightly variable points with gradually decreasing influence of unknown factors (cumulated over a time parameter). Besides, the econometric analysis will concentrate on the dynamics of each I-O coefficient, considered separately and not in connection with other series.

Therefore, the evolution of I-O coefficients is conceived as an auto-regressive adaptive process, the differences between their actual and long-run levels being influenced by the past deviations. In the simplest form, such an application for Romanian input-output tables was developed in [10]. In a general notation, if y is the time series of interest, we would have the following relationship:
y t = y ˜ α ( y ( 1 ) y ˜ ) = y ˜ ( 1 + α ) α y ( 1 )
(18)
where y ˜ represents the long-run levels of y (or the attractor according to this paper’s terminology). It is assumed that 0 < | α | < 1 , which means that y tends asymptotically towards y ˜ . Correspondingly, the first-order difference operator d ( y ) is defined as
d ( y ) = y y ( 1 ) = y ˜ ( 1 + α ) α y ( 1 ) y ( 1 ) = y ˜ ( 1 + α ) ( 1 + α ) y ( 1 ) = a 0 a 1 y ( 1 )
(19)

The expression (19) contains the equivalencies a 0 = y ˜ ( 1 + α ) and a 1 = ( 1 + α ) .

To be more realistic, this determination will be relaxed by two amendments. On one hand, the last formula will be extended, with gradually diminishing influence of time. On the other, the auto-regressive process may involve lags of higher orders, not only of the first one, as in (19).

2. Even under such modifications, the approximation of possible attractor points requires the presence of at least one non-differentiated observation in the computational formula. Therefore, it would be preferable to use the statistical series stationary in levels ( I ( 0 ) ). Unfortunately, most of the available data do not observe such a restriction. From this point of view, two already mentioned unit root tests were applied: ADF—Augmented Dickey–Fuller and PP—Phillips–Perron test. Each was computed in three versions for the exogenous variables:

• none (denoted as 1),

• individual effects (denoted as 2), and

• individual effects and individual linear trends (denoted 3).

The p-values calculated for all 100 technical coefficients were grouped as follows: 0–0.05, 0.05–0.1, 0.01–0.25, and 0.25–1.

The corresponding distribution for the technical coefficients at constant prices ( ca ij ) is presented in Figs. 7 and 8. Both unit root tests (ADF and PP) show that in around 80 % of the cases, the p-values exceed 0.1. The same result is found for the technical coefficients at current prices (see Figs. 9 and 10).
Fig. 7

ADF tests for ca ij

Fig. 8

PP tests for ca ij

Fig. 9

ADF tests for a ij

Fig. 10

PP tests for a ij

At this point, we are confronted with a problem. The BDS test indicated the presence of temporal correlation in the data for technical coefficients (either at constant or at current prices). As previously mentioned, this finding would justify the identification of possible attractor points in their evolution. Since the series are not stationary in levels, in order to avoid the calculation of attractor points (as levels) by first- or second-order differentiation (a difficult computational task), an indirect way to approximate such points will be proposed.

The first step is to determine colsums ( sca j ) and rowsums ( sra i ) for the technical coefficients at current prices. The resulting series are given in Statistical and Econometric Appendix. With respect to these time series, PANEL analysis did not reveal compelling signs of common explicative parameters. For this reason, they were examined separately. Table 5 shows the p-values of the ADF and PP tests for the sca i series. In only three cases ( sca 2 , sca 3 , and sca 4 ) are the corresponding p-values situated in the proximity of 0.25. Consequently, the series sca i will be used as such in regressions.
Table 5

ADF and PP tests for sca i

Variable

Exogenous

ADF

PP

t-statistic

Prob.

t-statistic

Prob.

sca 1

Constant, linear trend

−4.54901

0.009

−4.52912

0.0094

sca 2

Constant

−2.02573

0.274

−2.00889

0.2809

sca 3

Constant

−3.98533

0.0073

−2.00269

0.2833

sca 4

Constant, linear trend

−4.79669

0.0072

−2.85646

0.1956

sca 5

Constant, linear trend

−6.12916

0.0005

−3.86767

0.0339

sca 6

Constant, linear trend

−5.45292

0.0026

−3.4261

0.0761

sca 7

Constant

−4.76606

0.0018

−2.99545

0.0525

sca 8

Constant

−5.00001

0.0008

−7.99152

0

sca 9

Constant

−4.47988

0.0028

−2.81411

0.0741

sca 10

Constant, linear trend

−4.43914

0.012

−7.71446

0

Table 6 presents the same indicators for sra i . The introduction of econometric estimations for series sra 5 , sra 8 , and sra 10 as such would clearly be too risky. Consequently, the first two were recalculated by the Hodrick–Prescott filter, obtaining for each the sub-series denoted as HP and HPd (difference between filter and primary data), respectively. The third series ( sra 10 ) was replaced with the corresponding logarithms. Table 7 shows the unit root test results, based on which the new series for sra 5 , sra 8 , and sra 10 were used in regressions.
Table 6

ADF and PP tests for sra i

Variable

Exogenous

ADF

PP

t-statistic

Prob.

t-statistic

Prob.

sra 1

Constant, linear trend

−3.06826

0.1399

−1.59124

0.1031

sra 2

Constant

−2.94275

0.0581

−2.91376

0.0614

sra 3

Constant

−3.51945

0.0183

−3.51945

0.0183

sra 4

Constant

−2.6057

0.1083

−2.6057

0.1083

sra 5

Constant, linear trend

−2.28894

0.4194

−2.54869

0.3041

sra 6

None

−2.36343

0.0209

−2.17192

0.0319

sra 7

Constant, linear trend

−4.96559

0.0044

−2.84798

0.1981

sra 8

Constant, linear trend

−2.34672

0.3929

−1.90162

0.6163

sra 9

Constant

−2.91805

0.0609

−2.91805

0.0609

sra 10

Constant

−1.22677

0.6415

−1.28041

0.6175

Table 7

ADF and PP tests for derived series sra 5 , sra 8 , and sra 10

Variable

Exogenous

ADF

PP

t-statistic

Prob.

t-statistic

Prob.

sra 5 HP

None

−2.48196

0.0168

−1.41255

0.1422

sra 5 HPd

None

−5.36025

0

−3.91121

0.0005

sra 8 HP

Constant

−3.84112

0.0116

−2.06376

0.5334

sra 8 HPd

None

−3.73356

0.0008

−3.89625

0.0005

sra 10 l

None

−4.16256

0.0003

−5.48654

0

The formula (19) with the mentioned amendments was investigated using different specifications. The proposed selection considered, beside the mentioned premises, the results of tests for omitted or redundant variables, and outliers, also. It has also tried to reduce the econometric compromises as much as possible. For the current paper, several types of relationships were retained according to the scheme given in Table 8. Sometimes dummy variables were introduced to decrease the influence of data outliers.
Table 8

Main econometric relationships

Variables (y)

Specification

sca 1 , sra 2 , sra 4 , sra 9 , log ( sra 10 )

d ( y ) = a 0 + a 1 y ( 1 ) , with possible a 1 y ( 3 ) or a 2 d ( y , 2 )

sca 8 , sca 10

d ( y ) = b 0 + b 1 y ( 1 ) + b 2 t / ( t + 1 ) , with possible b 0 = 0

sca 2 , sra 3 , sra 5 HPd

d ( y ) = c 0 + c 1 y ( 1 ) + c 2 d ( y ( 1 ) ) , with possible c 0 = 0 or c 1 y ( 2 )

sca 5 , sca 6 , sca 9

d ( y ) = d 0 + d 1 y ( 1 ) + d 2 d ( y ( 1 ) ) + d 3 d ( y ( 2 ) ) + d 4 t / ( t + 1 ) , with possible d 3 = 0

sra 8 HP , sra 8 HPd

d ( y ) = e 0 + e 1 y ( 1 ) + e 2 d ( y , 2 ) , with possible e 0 = e 1 = 0

sca 7 , sra 5 HP

d ( y ) = f 0 + f 1 y ( 1 ) + f 2 d ( y ( 1 ) ) + f 3 d ( y ( 2 ) ) + f 4 d ( y ( 3 ) ) + f 5 t / ( t + 1 ) with possible f 3 = f 4 = f 5 = 0

sra 1 , sra 6

d ( y ) = g 0 + g 1 y ( 1 ) + g 2 d ( y ( 1 ) ) + g 3 t 1 , with possible g 2 d ( y ( 2 ) )

sca 3

d ( y ) = h 0 + h 1 y ( 3 ) + h 2 t 1

sca 4 , sra 7

d ( y ) = i 0 + i 1 y ( 2 ) + i 2 d ( y , 2 ) + i 3 t / ( t + 1 ) or i 3 t 1

3. The OLS-solution of system SyS1scr (Statistical and Econometric Appendix) was submitted to econometric controls from four standpoints: (a) variance inflation factors, (b) Breusch–Pagan–Godfrey heteroskedasticity test, (c) correlogram squared residuals, and (d) stationarity of residuals.

Concerning the variance inflation factors (Table 9), it is conclusive that more than 77 % of the centred VIFs do not exceed 2, and approximately 15 % are situated between 2 and 3; even the rest do not surpass 5.3. Based on these results, we could accept that the specification of the system SyS1scr is not contaminated in an alarming manner by collinearity effects.
Table 9

Variance Inflation Factors—SyS1scr

Variable

Coefficient variance

Uncentred VIF

Centred VIF

Variable

Coefficient variance

Uncentred VIF

Centred VIF

c(1)

0.007439

181.7134

NA

c(39)

0.024642

1.450884

1.315947

c(2)

0.032656

182.2648

1.009286

c(40)

0.109405

22.57514

5.223014

c(501)

0.00087

1.062407

1.009286

c(510)

0.001322

1.149574

1.085709

c(3)

0.003984

74.7162

NA

c(41)

0.010673

94.24219

NA

c(4)

0.00863

74.77052

1.17408

c(42)

0.035631

93.71807

2.198215

c(5)

0.014339

1.296515

1.292649

c(43)

0.014986

2.1191

2.11897

c(6)

0.001936

128.8835

NA

c(511)

0.002527

1.174249

1.112446

c(7)

0.003913

153.3235

1.466782

c(44)

0.020214

88.31065

NA

c(8)

0.007181

6.302923

1.458253

c(45)

0.043413

87.61261

1.624957

c(9)

0.025715

4402.565

NA

c(46)

0.044123

1.654492

1.645571

c(10)

0.014377

1123.768

4.799226

c(512)

0.004665

1.072668

1.016212

c(11)

0.003776

1.17145

1.169737

c(47)

0.003327

95.02657

NA

c(12)

0.008835

1235.916

4.601097

c(48)

0.021225

94.48058

1.093589

c(505)

0.000188

1.696676

1.607377

c(513)

0.00079

1.128173

1.071764

c(13)

0.019685

1392.005

NA

c(514)

0.000754

1.077272

1.023409

c(14)

0.024542

623.8674

1.638509

c(49)

2.81E-06

426.7166

NA

c(15)

0.016727

1.35002

1.347123

c(50)

1.05E-05

395.778

2.751647

c(16)

0.00631

364.5846

1.357284

c(51)

0.000234

3.754934

2.751647

c(17)

0.034858

1965.638

NA

c(52)

0.025272

1.796378

 

c(18)

0.030144

650.8176

2.390064

c(53)

0.016926

1.920122

 

c(19)

0.023574

1.422988

1.404814

c(515)

0.000178

1.170656

 

c(20)

0.011187

515.4273

1.918844

c(516)

0.000184

1.211256

 

c(21)

0.081562

8973.245

NA

c(54)

0.003218

19.90196

NA

c(22)

0.092117

5429.118

3.120946

c(55)

0.007158

20.39908

1.264573

c(23)

0.042363

2.079116

2.060516

c(56)

0.017469

1.220558

1.218342

c(24)

0.031891

1.948496

1.866357

c(517)

0.003526

1.147892

1.087476

c(25)

0.033425

2.002502

1.928123

c(57)

0.014686

248.5112

NA

c(26)

0.012829

1189.042

1.690041

c(58)

0.00772

281.5666

1.506779

c(27)

0.002979

389.7016

NA

c(59)

0.003002

1.016724

1.00456

c(28)

0.009449

388.3649

1.064145

c(60)

0.015966

4.954369

1.509934

c(29)

0.005972

1.198711

1.182468

c(61)

6.02E-06

47.60604

NA

c(506)

0.000171

1.178583

1.116553

c(62)

1.66E-04

23.38802

1.507298

c(30)

0.005762

1013.246

NA

c(63)

1.57E + 00

11.50596

1.668578

c(31)

0.027859

860.7738

1.847807

c(518)

2.77E-06

1.152835

1.092159

c(32)

0.019306

2.0286

2.028597

c(519)

2.94E-06

1.223618

1.159217

c(33)

0.003382

494.3642

1.101297

c(64)

0.006251

1.235811

 

c(507)

0.000142

1.385103

1.308153

c(520)

0.000222

1.133899

 

c(34)

0.003091

324.9721

NA

c(521)

0.000216

1.101912

 

c(35)

0.00348

292.1113

2.021579

c(65)

0.003105

18.95904

NA

c(36)

0.005765

95.10627

1.474381

c(66)

0.017978

19.27486

1.020607

c(508)

0.000506

2.661904

2.528809

c(522)

0.003492

1.066213

1.012902

c(509)

0.000219

1.151599

1.094019

c(523)

0.003497

1.067792

1.014402

c(37)

0.011343

177.5241

NA

c(67)

0.000826

3.416318

NA

c(38)

0.042122

285.9858

4.932777

c(68)

0.000602

3.868874

1.182485

    

c(524)

0.00545

1.252043

1.182485

The test Breusch–Pagan–Godfrey (Table 10) indicates high enough probabilities for the rejection of heteroskedasticity hypothesis.
Table 10

SyS1scr: heteroskedasticity test Breusch–Pagan–Godfrey

Dependent variable: d ( sca 1 )

Dependent variable: d ( sra 2 )

 F-statistic

0.901062

Prob. F(2.17)

0.4247

 F-statistic

1.017491

Prob. F(4.14)

0.4318

 Obs*R-squared

1.916936

Prob. Chi-Square(2)

0.3835

 Obs*R-squared

4.279439

Prob. Chi-Square(4)

0.3695

 Scaled explained SS

0.928978

Prob. Chi-Square(2)

0.6285

 Scaled explained SS

0.96349

Prob. Chi-Square(4)

0.9153

Dependent variable: d ( sca 2 )

Dependent variable: d ( sra 3 )

 F-statistic

0.493489

Prob. F(4.14)

0.7408

 F-statistic

0.610519

Prob. F(3.15)

0.6185

 Obs*R-squared

2.347896

Prob. Chi-Square(4)

0.6721

 Obs*R-squared

2.067521

Prob. Chi-Square(3)

0.5585

 Scaled explained SS

1.07891

Prob. Chi-Square(4)

0.8976

 Scaled explained SS

0.52206

Prob. Chi-Square(3)

0.914

Dependent variable: d ( sca 3 )

Dependent variable: d ( sra 4 )

 F-statistic

0.880908

Prob. F(3.14)

0.4746

 F-statistic

0.329585

Prob. F(3.16)

0.804

 Obs*R-squared

2.858248

Prob. Chi-Square(3)

0.414

 Obs*R-squared

1.16401

Prob. Chi-Square(3)

0.7616

 Scaled explained SS

2.466576

Prob. Chi-Square(3)

0.4814

 Scaled explained SS

0.798201

Prob. Chi-Square(3)

0.8499

Dependent variable: d ( sca 4 )

Dependent variable: d ( sra 5 HP )

 F-statistic

1.613982

Prob. F(5.13)

0.2249

 F-statistic

0.335166

Prob. F(2.16)

0.7201

 Obs*R-squared

7.277122

Prob. Chi-Square(5)

0.2008

 Obs*R-squared

0.76401

Prob. Chi-Square(2)

0.6825

 Scaled explained SS

7.487449

Prob. Chi-Square(5)

0.1868

 Scaled explained SS

0.343187

Prob. Chi-Square(2)

0.8423

Dependent variable: d ( sca 5 )

Dependent variable: d ( sra 5 HPd )

 F-statistic

0.757351

Prob. F(3.15)

0.5352

 F-statistic

0.651693

Prob. F(4.14)

0.6351

 Obs*R-squared

2.499355

Prob. Chi-Square(3)

0.4754

 Obs*R-squared

2.982437

Prob. Chi-Square(4)

0.5608

 Scaled explained SS

3.524105

Prob. Chi-Square(3)

0.3176

 Scaled explained SS

1.916603

Prob. Chi-Square(4)

0.7511

Dependent variable: d ( sca 6 )

Dependent variable: d ( sra 6 )

 F-statistic

0.498536

Prob. F(3.15)

0.6889

 F-statistic

0.541944

Prob. F(4.14)

0.7077

 Obs*R-squared

1.722675

Prob. Chi-Square(3)

0.6319

 Obs*R-squared

2.547519

Prob. Chi-Square(4)

0.6361

 Scaled explained SS

2.27106

Prob. Chi-Square(3)

0.5181

 Scaled explained SS

2.828426

Prob. Chi-Square(4)

0.5869

Dependent variable: d ( sca 7 )

Dependent variable: d ( sra 7 )

 F-statistic

0.776423

Prob. F(5.11)

0.5866

 F-statistic

0.082417

Prob. F(4.14)

0.9865

 Obs*R-squared

4.434583

Prob. Chi-Square(5)

0.4887

 Obs*R-squared

0.437113

Prob. Chi-Square(4)

0.9793

 Scaled explained SS

1.311754

Prob. Chi-Square(5)

0.9337

 Scaled explained SS

0.426564

Prob. Chi-Square(4)

0.9802

Dependent variable: d ( sca 8 )

Dependent variable: d ( sra 8 HP )

 F-statistic

1.183406

Prob. F(4.14)

0.3604

 F-statistic

1.320582

Prob. F(5.13)

0.3151

 Obs*R-squared

4.800931

Prob. Chi-Square(4)

0.3083

 Obs*R-squared

6.399829

Prob. Chi-Square(5)

0.2692

 Scaled explained SS

4.819883

Prob. Chi-Square(4)

0.3063

 Scaled explained SS

2.752073

Prob. Chi-Square(5)

0.7381

Dependent variable: d ( sca 9 )

Dependent variable: d ( sra 8 HPd )

 F-statistic

0.63052

Prob. F(5.12)

0.6804

 F-statistic

0.724598

Prob. F(5.13)

0.617

 Obs*R-squared

3.745019

Prob. Chi-Square(5)

0.5867

 Obs*R-squared

4.141061

Prob. Chi-Square(5)

0.5293

 Scaled explained SS

1.619852

Prob. Chi-Square(5)

0.8988

 Scaled explained SS

1.882761

Prob. Chi-Square(5)

0.8651

Dependent variable: d ( sca 10 )

Dependent variable: d ( sra 9 )

 F-statistic

0.928894

Prob. F(4.15)

0.4733

 F-statistic

0.298999

Prob. F(3.16)

0.8256

 Obs*R-squared

3.970571

Prob. Chi-Square(4)

0.41

 Obs*R-squared

1.061723

Prob. Chi-Square(3)

0.7863

 Scaled explained SS

2.66595

Prob. Chi-Square(4)

0.6152

 Scaled explained SS

0.863016

Prob. Chi-Square(3)

0.8343

Dependent variable: d ( sra 1 )

Dependent variable: d ( sra 10 l )

 F-statistic

0.476573

Prob. F(5.12)

0.7871

 F-statistic

1.355643

Prob. F(2.15)

0.2876

 Obs*R-squared

2.982131

Prob. Chi-Square(5)

0.7027

 Obs*R-squared

2.755483

Prob. Chi-Square(2)

0.2521

 Scaled explained SS

0.508079

Prob. Chi-Square(5)

0.9918

 Scaled explained SS

1.404621

Prob. Chi-Square(2)

0.4954

The correlogram of squared residuals was computed for five lags (Table 11). In most cases, Q-statistics are associated with relatively large p-values, which attest a weak serial correlation in the residuals.
Table 11

Correlogram of residuals squared—SyS1scr

Lag

Dependent variable: d ( sca 1 )

Dependent variable: d ( sca 8 )

Dependent variable: d ( sra 5 HP )

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

1

−0.272

−0.272

1.7151

0.19

−0.257

−0.257

1.46

0.227

0.276

0.276

1.6847

0.194

2

−0.096

−0.184

1.9425

0.379

0.066

0

1.5608

0.458

−0.11

−0.201

1.9688

0.374

3

−0.035

−0.13

1.9746

0.578

−0.064

−0.051

1.6631

0.645

0.016

0.122

1.9751

0.578

4

0.164

0.107

2.7103

0.607

−0.091

−0.128

1.8818

0.757

−0.052

−0.134

2.046

0.727

5

−0.196

−0.148

3.8388

0.573

0.12

0.074

2.2944

0.807

−0.167

−0.102

2.8394

0.725

Lag

Dependent variable: d ( sca 2 )

Dependent variable: d ( sca 9 )

Dependent variable: d ( sra 5 HPd )

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

1

−0.092

−0.092

0.1874

0.665

−0.087

−0.087

0.1621

0.687

0.033

0.033

0.0247

0.875

2

0.174

0.167

0.9021

0.637

−0.144

−0.152

0.6266

0.731

−0.106

−0.107

0.2871

0.866

3

−0.118

−0.093

1.2507

0.741

−0.101

−0.133

0.871

0.832

0.117

0.126

0.6284

0.89

4

0.047

0.004

1.31

0.86

−0.169

−0.228

1.6068

0.808

−0.143

−0.17

1.1734

0.882

5

−0.056

−0.018

1.3989

0.924

0.086

−0.003

1.8114

0.875

−0.259

−0.226

3.0863

0.687

Lag

Dependent variable: d ( sca 3 )

Dependent variable: d ( sca 10 )

Dependent variable: d ( sra 6 )

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

1

−0.168

−0.168

0.5983

0.439

0.127

0.127

0.3738

0.541

−0.033

−0.033

0.0248

0.875

2

0.038

0.01

0.6304

0.73

0.004

−0.012

0.3743

0.829

0.286

0.285

1.9394

0.379

3

−0.044

−0.037

0.6771

0.879

0.171

0.175

1.1293

0.77

−0.238

−0.242

3.3536

0.34

4

−0.009

−0.023

0.6789

0.954

0.259

0.225

2.9779

0.562

−0.097

−0.202

3.6057

0.462

5

−0.198

−0.208

1.7687

0.88

−0.308

−0.394

5.763

0.33

0.004

0.173

3.6061

0.607

Lag

Dependent variable: d ( sca 4 )

Dependent variable: d ( sra 1 )

Dependent variable: d ( sra 7 )

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

1

−0.022

−0.022

0.0109

0.917

−0.034

−0.034

0.0249

0.875

−0.109

−0.109

0.2637

0.608

2

−0.036

−0.037

0.0421

0.979

−0.147

−0.148

0.5104

0.775

0.081

0.07

0.4167

0.812

3

0.267

0.266

1.821

0.61

−0.224

−0.241

1.7165

0.633

−0.043

−0.027

0.4618

0.927

4

−0.151

−0.154

2.4276

0.658

0.172

0.135

2.4809

0.648

−0.178

−0.194

1.3069

0.86

5

−0.11

−0.1

2.7737

0.735

−0.206

−0.286

3.6588

0.6

0.008

−0.026

1.3088

0.934

Lag

Dependent variable: d ( sca 5 )

Dependent variable: d ( sra 2 )

Dependent variable: d ( sra 8 HP )

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

1

0.083

0.083

0.1544

0.694

−0.05

−0.05

0.0559

0.813

−0.228

−0.228

1.1477

0.284

2

−0.177

−0.186

0.8918

0.64

−0.287

−0.29

1.9901

0.37

−0.085

−0.145

1.3186

0.517

3

−0.066

−0.035

1.0011

0.801

−0.03

−0.069

2.012

0.57

0.282

0.245

3.3084

0.346

4

−0.187

−0.22

1.9314

0.748

−0.192

−0.311

2.997

0.558

−0.224

−0.128

4.6405

0.326

5

0.222

0.262

3.3306

0.649

−0.182

−0.313

3.9447

0.557

0.061

0.039

4.7479

0.447

Lag

Dependent variable: d ( sca 6 )

Dependent variable: d ( sra 3 )

Dependent variable: d ( sra 8 HPd )

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

1

−0.141

−0.141

0.4389

0.508

−0.024

−0.024

0.0131

0.909

−0.224

−0.224

1.1139

0.291

2

−0.159

−0.182

1.0301

0.597

−0.188

−0.189

0.8438

0.656

−0.194

−0.257

1.9937

0.369

3

−0.103

−0.164

1.2972

0.73

−0.394

−0.419

4.709

0.194

0.145

0.037

2.5173

0.472

4

0.066

−0.012

1.4131

0.842

0.051

−0.05

4.7777

0.311

0.068

0.077

2.6402

0.62

5

0.149

0.122

2.0468

0.843

0.017

−0.178

4.7854

0.443

−0.23

−0.17

4.1494

0.528

Lag

Dependent variable: d ( sca 7 )

Dependent variable: d ( sra 4 )

Dependent variable: d ( sra 9 )

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

AC

PAC

Q-statistic

Prob.

1

0.072

0.072

0.1057

0.745

−0.001

−0.001

4.00E-05

0.995

−0.056

−0.056

0.0733

0.787

2

−0.102

−0.108

0.3299

0.848

0.119

0.119

0.3466

0.841

0.161

0.158

0.7045

0.703

3

−0.221

−0.208

1.4564

0.692

−0.005

−0.004

0.3472

0.951

−0.284

−0.276

2.7994

0.424

4

−0.312

−0.312

3.8731

0.423

−0.017

−0.032

0.3555

0.986

0.37

0.367

6.5593

0.161

5

−0.298

−0.381

6.2633

0.281

0.141

0.144

0.9389

0.967

−0.084

−0.038

6.7672

0.239

Lag

  

Dependent variable: d ( sra 10 l )

        

AC

PAC

Q-statistic

Prob.

1

        

−0.042

−0.042

0.0377

0.846

2

        

0.015

0.013

0.0426

0.979

3

        

−0.022

−0.021

0.0547

0.997

4

        

−0.268

−0.27

1.899

0.754

5

        

0.455

0.467

7.6334

0.178

Concerning the stationarity of residuals, both unit root tests ADF and PP were applied again, in all available options for exogenous (Table 12). There were thus generated 132 values of the probability the respective residual has a unit root. Out of these, 76.52 % are placed under 0.05, and 10.61 % between 0.05–0.1.
Table 12

ADF and PP unit root tests of residuals SyS1scr

 

Null hypothesis: ressca 1 has a unit root

Null hypothesis: ressca 2 has a unit root

Null hypothesis: ressca 3 has a unit root

Null hypothesis: ressca 4 has a unit root

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

ADF, exogenous: none

−1.986552

0.0475

−5.140927

0

−4.403522

0.0002

−3.786799

0.0008

ADF, exogenous: constant

−1.948802

0.3045

−4.981806

0.0012

−4.268814

0.0047

−3.671733

0.0145

ADF, exogenous: constant, linear trend

−3.465446

0.0724

−4.908056

0.0059

−4.258263

0.019

−3.603718

0.0583

PP, exogenous: none

−3.405567

0.0018

−13.3349

0.0001

−4.409299

0.0002

−3.786799

0.0008

PP, exogenous: constant

−3.315973

0.0286

−16.20088

0

−4.274438

0.0046

−3.671733

0.0145

PP, exogenous: constant, linear trend

−3.424937

0.0777

−15.56681

0.0001

−4.277874

0.0184

−3.606573

0.058

 

Null hypothesis: ressca 5 has a unit root

Null hypothesis: ressca 6 has a unit root

Null hypothesis: ressca 7 has a unit root

Null hypothesis: ressca 8 has a unit root

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

ADF, exogenous: none

−5.658349

0

−3.995118

0.0005

−5.895841

0

−3.597347

0.0012

ADF, exogenous: constant

−5.487764

0.0004

−3.86399

0.0099

−5.639502

0.0004

−3.488259

0.021

ADF, exogenous: constant, linear trend

−5.309006

0.0026

−3.774645

0.0431

−5.697499

0.0017

−3.470132

0.0735

PP, exogenous: none

−5.658844

0

−3.99659

0.0005

−5.802776

0

−3.570019

0.0013

PP, exogenous: constant

−5.488494

0.0004

−3.865979

0.0098

−5.559396

0.0005

−3.42576

0.0238

PP, exogenous: constant, linear trend

−5.30975

0.0026

−3.775747

0.043

−5.697499

0.0017

−3.584896

0.0602

 

Null hypothesis: ressca 9 has a unit root

Null hypothesis: ressca 10 has a unit root

Null hypothesis: ressra 1 has a unit root

Null hypothesis: ressra 2 has a unit root

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

ADF, exogenous: none

−3.789794

0.0008

−5.27384

0

−3.016457

0.0049

−4.043831

0.0004

ADF, exogenous: constant

−3.663534

0.0155

−5.162812

0.0008

−2.900826

0.066

−3.951321

0.0083

ADF, exogenous: constant, linear trend

−3.646379

0.0559

−5.140881

0.0039

−2.8125

0.2119

−3.97919

0.0298

PP, exogenous: none

−3.76958

0.0009

−7.353143

0

−2.908989

0.0063

−4.043831

0.0004

PP, exogenous: constant

−3.635529

0.0164

−7.09582

0

−2.790106

0.0805

−3.951321

0.0083

PP, exogenous: constant, linear trend

−3.557692

0.0649

−7.493081

0

−2.681021

0.2547

−3.973131

0.0301

 

Null hypothesis: ressra 3 has a unit root

Null hypothesis: ressra 4 has a unit root

Null hypothesis: ressra 5 HP has a unit root

Null hypothesis: ressra 5 HPd has a unit root

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

ADF, exogenous: none

−3.46127

0.0017

−5.532511

0

−3.222773

0.0031

−2.361507

0.0218

ADF, exogenous: constant

−3.361322

0.027

−5.373084

0.0004

−3.091733

0.0465

−2.123058

0.2389

ADF, exogenous: constant, linear trend

−3.142646

0.1267

−4.837124

0.0061

−2.932113

0.1776

−2.265232

0.4268

PP, exogenous: none

−3.46127

0.0017

−5.913703

0

−1.834051

0.0646

−5.664019

0

PP, exogenous: constant

−3.361322

0.027

−5.70976

0.0002

−1.356726

0.5795

−5.853202

0.0002

PP, exogenous: constant, linear trend

−3.142646

0.1267

−9.865782

0

−1.714644

0.7022

−9.964217

0

 

Null hypothesis: ressra 6 has a unit root

Null hypothesis: ressra 7 has a unit root

Null hypothesis: ressa 8 HP has a unit root

Null hypothesis: ressra 8 HPd has a unit root

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

t-statistic

Prob.

ADF, exogenous: none

−3.720831

0.0009

−4.171027

0.0003

−2.832449

0.0074

−5.387255

0

ADF, exogenous: constant

−3.612433

0.0164

−3.94738

0.0089

−3.102695

0.0481

−5.243256

0.0006

ADF, exogenous: constant, linear trend

−3.505032

0.0692

−3.777624

0.0445

−2.922023

0.1835

−5.189448

0.0032

PP, exogenous: none

−3.709671

0.0009

−3.557824

0.0013

−2.757837

0.0087

−6.014966

0

PP, exogenous: constant

−3.598596

0.0168

−3.321521

0.0292

−2.660895

0.0999

−6.301019

0.0001

PP, exogenous: constant, linear trend

−3.489806

0.071

−2.902353

0.1844

−2.489547

0.3283

−8.353103

0

 

Null hypothesis: ressra 9 has a unit root

Null hypothesis: ressra 10 l has a unit root

  

t-statistic

Prob.

t-statistic

Prob.

    

ADF, exogenous: none

−2.725678

0.0093

−6.80313

0

    

ADF, exogenous: constant

−2.640708

0.1026

−6.617948

0.0001

    

ADF, exogenous: constant, linear trend

−2.683911

0.2527

−6.352981

0.0005

    

PP, exogenous: none

−2.732364

0.0091

−6.767128

0

    

PP, exogenous: constant

−2.648713

0.1012

−6.586823

0.0001

    

PP, exogenous: constant, linear trend

−2.715594

0.2416

−6.352981

0.0005

    

The above presented tests (for collinearity, heteroskedasticity, serial correlation, and stationarity of residuals) show that OLS could be acceptable to estimate the system SyS1scr.

4. The system SyS1scr has been solved using other four techniques: Weighted Least Squares (WLS), Seemingly Unrelated Regression (SUR), Generalised linear models (GLM), and Generalised Method of Moments (GMM). The obtained results are detailed in Statistical and Econometric Appendix.

The solution induced by Weighted Least Squares slightly ameliorates the standard errors, maintaining, however, the parameters of equations practically at the same level as OLS. The differences between Seemingly Unrelated Regression and OLS regarding estimators and coefficients of determination are also insignificant. The same conclusion is valid for the Generalised Linear Models (applied with bootstrap).

The Generalised Method of Moments was involved in variant HAC for the time series (Bartlett and Variable Newey–West). Despite the large number enough of trials, the results were inconclusive. First, in order to obtain a plausible solution, it was necessary to break SyS1scr into three sub-systems—SyS1scaG, SyS1sraG, and SyS1sra8G—which have been separately computed. Secondly, the algorithm did not work with dummies, or these were not introduced casually, but according to the specification test about outliers.

Briefly, the comparative analysis of different techniques suggests as acceptable OLS method. Nevertheless, a problem persists. According to Statistical and Econometric Appendix (System Residual Cross-Correlations—OLS), the disturbances of some relationships represented in SyS1scr are correlated. They reflect, at great extent, the indubitable fact of inter-industry linkages. Obviously, there must be a consistent solution of the question hereby discussed. It could result from a re-specification of the entire system by explicit inclusion in the equations of the factors inducing cross-correlations among input-output technical coefficients, and subsequently applying computational methods that avoid simultaneity effects. But such an approach should need further interdisciplinary research. Until then, I am reluctant to involve techniques which somehow mechanically constrict the cross-correlations of I-O coefficients. Consequently, for the present OLS will keep being involved in the succeeding steps of our approach.

5. Based on the previous system, the fitted sca j f and sra i f can be obtained, but not a ij f as such. To approximate these, the RAS technique was applied. During its half-century existence [42], this method has registered extended applications, including in recent researches [7, 18, 19, 21, 22, 25, 27]. Usually, the starting matrix for every t is the statistical matrix A t 1 , which is adjusted by successive bi-proportional corrections in dependence on exogenously given sectoral outputs. The applicability of such a method for an emergent economy such as in Romania has already been documented [13].

The present paper slightly modifies this procedure, using sca j f and sra i f as column and row restrictions in a RAS algorithm. The resulting technical coefficients (denoted as ra ij ) are relevant from the present research perspective. Notably, ra ij are calculated using the fitted sca j f and sra i f . The formulae, however, are based on the hypothesis that the respective original statistical series contain attractor points. Consequently, the analysis of the differences resra ij = a ij ra ij can be informative. Given the independency of these differences, the assumption that sca j f and sra i f include attractor points and that the derived ra ij contain such compatible points becomes plausible since both sca j and sra i represent simple summations of the corresponding a ij .

Consequently, we return to the BDS test. As in the previous application, the test was applied to both probabilities (normal and bootstrap) in three options related to the distance (fraction of pairs, standard deviations, and fraction of range) and in five dimensions (2, 3, 4, 5, and 6). For each resra ij , 30 p-values were again computed (as before). The distribution of all 3000 p-values is described in Fig. 11. Only one fifth of the p-values do not exceed 0.05. This proportion falls to 8 % in the case of the bootstrap method, which is more relevant for relatively short series.
Fig. 11

Distribution of the BDS tests for resra ij

For this reason, as a general approximation, the serial independence of resra ij differences was assumed. Consequently, the probability of attractor points in the data for a ij cannot be neglected.

6. Further on, the attractor points will be estimated based on the following additional assumptions:

• It is admitted that in the proximity of an attractor point, the values of the respective technical coefficients are relatively stable. In other words, first- and higher-order differences tend to disappear.

• In terms of level, the value of the technical coefficient coincides or is close to that of the attractor point. The importance of the presence of observations in level (I(0) problem) in econometric formulae has already been outlined.

• The attractor points are conceived at long-run levels. For large values of t, it is admitted that t 1 0 and t / ( t + 1 ) 1 .

The scheme containing the main econometric relationships will be adapted to these assumptions, the result being the algebraical expressions of attractors in the 9 types of specifications (Table 13) included in SyS1scr. Their symbols are given the prefix a: asca j and asra i .
Table 13

Algebraical attractor definitions

Variables (y)

Approximating formula

sca 1 , sra 2 , sra 4 , sra 9 , log ( sra 10 )

ay = a 0 / a 1

sca 8 , sca 10

ay = ( b 0 + b 2 ) / b 1

sca 2 , sra 3

ay = c 0 / c 1

sca 5 , sca 6 , sca 9

ay = ( d 0 + d 4 ) / d 1

sra 8

ay = e 0 / e 1

sca 7 , sra 5

ay = ( f 0 + f 5 ) / f 1

sra 1 , sra 6

ay = g 0 / g 1

sca 3

ay = h 0 / h 1

sca 4 , sra 7

ay = ( i 0 + i 3 ) / i 1 or = i 0 / i 1

Table 14 presents the approximated attractors for colsums ( asca j ) and rowsums ( asra i ) of the I-O coefficients. These estimations were included as column–row restrictions in a new RAS application concerning all a ij . This algorithm was applied on a matrix compounded by the average levels of the respective statistical coefficients (for the entire interval 1989–2009). Table 15 presents the so-obtained attractor points ( aa ij ).
Table 14

Attractor-points for the colsums and rowsums of technical coefficients

Symbol

Estimation

Symbol

Estimation

asca 1

0.488059

asra 1

0.508254

asca 2

0.633969

asra 2

0.546414

asca 3

0.904387

asra 3

0.674086

asca 4

0.603476

asra 4

0.389116

asca 5

0.566348

asra 5

0.467036

asca 6

0.5619

asra 6

0.564482

asca 7

0.722865

asra 7

1.337777

asca 8

0.536487

asra 8

0.130711

asca 9

0.438797

asra 9

0.37335

asca 10

0.47579

asra 10

0.687186

Table 15

Attractor-points for individual technical coefficients ( aa ij )

 

j

1

2

3

4

5

6

7

8

9

10

aa 1 j

0.233951

0.001173

0.000136

0.232187

0.034418

0.000132

0.000947

0.000367

0.000277

0.004665

aa 2 j

0.001076

0.173019

0.270478

0.001162

0.00058

0.006661

0.073396

0.015125

0.002156

0.002762

aa 3 j

0.024686

0.090712

0.288858

0.022381

0.030654

0.04102

0.095837

0.023173

0.033597

0.023168

aa 4 j

0.053107

0.0026

0.001616

0.213718

0.009814

0.002945

0.007421

0.004082

0.005101

0.088714

aa 5 j

0.008545

0.011228

0.003287

0.017918

0.290144

0.019729

0.021376

0.025626

0.009881

0.059303

aa 6 j

0.017634

0.084122

0.043601

0.011139

0.022303

0.176451

0.037253

0.062144

0.078011

0.031824

aa 7 j

0.086517

0.086553

0.165247

0.028473

0.076022

0.194198

0.371607

0.173607

0.094814

0.060737

aa 8 j

0.00491

0.004865

0.014924

0.002182

0.002516

0.003279

0.00486

0.07097

0.007305

0.0149

aa 9 j

0.013468

0.067402

0.026501

0.014494

0.020011

0.02745

0.028172

0.02357

0.120064

0.032218

aa 10 j

0.023246

0.085211

0.051111

0.03394

0.055662

0.066024

0.05111

0.114896

0.068838

0.137147

4 Conclusions

The analysis of Romanian I-O tables (based on surveys for 21 consecutive years) reveals new evidence in favour of the statement that the technical coefficients are volatile (illustrated by the relatively high standard deviation of corresponding series). This affects both determinations of I-O coefficients, either in volume ( ca ij ) or in value terms ( a ij ); the first is referred to as real volatility and the second as nominal volatility. Their dynamic pattern is similar, as confirmed by three measures: (a) the vectorial angle between the series a ij and ca ij , (b) the Galtung–Pearson correlation (also a cosine of the vectorial angle but between their deviations against the mean) and (c) the binary synchronisation degree.

To verify whether or not the I-O coefficients are serially correlated, the BDS procedure was used as a test covering a large variety of possible deviations from independence in the time data. Again, both forms of technical coefficients were studied. Generally, the serial correlation could not be statistically rejected. It is important to mention that this conclusion resulted from a relatively extended database.

Due to these two circumstances—high volatility and serial correlation—the possible presence of attractors in the technical coefficients series was taken into consideration. Such points would be flexibly interpreted not as unchangeable levels but rather as historical (contextually determined) phenomena. This approach is similar to the manner in which other authors regarded the natural rate of unemployment, for instance, as a weak attractor. Consequently, the evolution of I-O coefficients was conceived as an auto-regressive adaptive process, the differences between the actual coefficients and their long-run levels being influenced by the precedent deviations. Since the available series for sectoral coefficients are, as a rule, non-stationary, more aggregate indicators were employed in econometric analysis (column and row sums of I-O coefficients). The RAS technique was used to transform these into sectoral estimations.

The paper’s approach can be considered as an attempt to conciliate the assumption of I-O coefficients’ stability with their undisputable volatility.

Further research could improve on the econometric estimations through structural specifications of the technical coefficients, including their stable co-movements. Thus, more complex econometric specifications must be cautiously adopted, but based on a solid economic motivation.

The possible presence of attractors in the series of I-O coefficients also opens a large research space. A deeper investigation of their determinants—technologies, inter-industry linkages, institutional factors—would be interesting from both the theoretical and the applicative perspective. In addition, it would be relevant to clarify the temporal stability of the attractors themselves.

Statistical and Econometric Appendix

Table 16

Column-sums of the technical coefficients at current prices

Year

sca 1

sca 2

sca 3

sca 4

sca 5

sca 6

sca 7

sca 8

sca 9

sca 10

1989

0.491558

0.569253

0.889023

0.76225

0.552646

0.650026

0.812164

0.712277

0.420045

0.496334

1990

0.387324

0.668055

0.956937

0.729538

0.585299

0.622568

0.799226

0.633274

0.466569

0.454735

1991

0.494798

0.663253

0.820943

0.750352

0.675304

0.700584

0.75585

0.622815

0.454685

0.378095

1992

0.498327

0.676749

0.779253

0.737931

0.668211

0.700656

0.742198

0.584504

0.404153

0.346181

1993

0.475942

0.633793

0.722954

0.676025

0.623013

0.659749

0.711613

0.569026

0.393578

0.343785

1994

0.447545

0.625294

0.656678

0.648452

0.561198

0.593076

0.693557

0.513575

0.362277

0.334192

1995

0.431615

0.736073

0.637111

0.657541

0.58757

0.587836

0.740255

0.568219

0.423969

0.283543

1996

0.448299

0.889495

0.705545

0.662567

0.620109

0.639375

0.745313

0.574133

0.434749

0.325155

1997

0.448678

0.885471

0.718332

0.718407

0.614082

0.643057

0.756277

0.568766

0.434086

0.383843

1998

0.500438

0.73034

0.711868

0.671709

0.589266

0.618621

0.750819

0.552626

0.412593

0.373144

1999

0.451623

0.649843

0.710166

0.689681

0.626521

0.645794

0.730459

0.521393

0.410677

0.373031

2000

0.471773

0.620211

0.728855

0.675673

0.578638

0.610767

0.712465

0.551344

0.410928

0.376375

2001

0.464331

0.557372

0.767713

0.626716

0.568639

0.589786

0.727921

0.562682

0.412289

0.420057

2002

0.483088

0.550141

0.765831

0.629274

0.569244

0.582466

0.711277

0.545703

0.412843

0.415685

2003

0.46985

0.636448

0.790705

0.651569

0.586742

0.612382

0.755768

0.558792

0.424581

0.423453

2004

0.470463

0.65786

0.793915

0.654237

0.590792

0.605497

0.748681

0.554347

0.434705

0.423932

2005

0.511133

0.660908

0.793131

0.621407

0.583903

0.581869

0.73793

0.544117

0.431047

0.416438

2006

0.505062

0.665331

0.793829

0.623597

0.585979

0.585482

0.735709

0.543761

0.433134

0.428825

2007

0.547584

0.664387

0.789971

0.623635

0.579907

0.57569

0.716971

0.53191

0.420637

0.425441

2008

0.534281

0.641298

0.796951

0.626553

0.582379

0.579771

0.721889

0.533723

0.425995

0.439046

2009

0.521289

0.624123

0.795017

0.624762

0.592213

0.57092

0.704817

0.541346

0.440151

0.445557

Table 17

Row-sums of the technical coefficients at current prices

Year

sra 1

sra 2

sra 3

sra 4

sra 5

sra 6

sra 7

sra 8

sra 9

sra 10

1989

0.879487

0.715536

0.460816

0.424076

0.458559

1.204335

1.512107

0.130762

0.464525

0.105374

1990

0.719968

0.681193

0.595892

0.420332

0.509075

1.137719

1.616523

0.126245

0.382477

0.114103

1991

0.799707

0.519268

0.740962

0.30728

0.55112

1.009835

1.719619

0.122188

0.430851

0.115852

1992

0.758225

0.559776

0.802719

0.323097

0.549017

0.850436

1.632821

0.053936

0.447255

0.160882

1993

0.828402

0.483196

0.668033

0.307312

0.476423

0.608609

1.417821

0.066092

0.700265

0.253325

1994

0.7714

0.513863

0.655478

0.365699

0.488487

0.548466

1.37154

0.071654

0.453217

0.19604

1995

0.720338

0.52918

0.629171

0.344625

0.538629

0.64714

1.444648

0.135043

0.388398

0.276562

1996

0.639786

0.589832

0.583761

0.443051

0.593274

0.77149

1.494947

0.121243

0.492754

0.3146

1997

0.650862

0.624606

0.690994

0.457173

0.549141

0.631484

1.607715

0.100695

0.416869

0.441459

1998

0.701866

0.470438

0.650553

0.375155

0.545836

0.701163

1.429193

0.120339

0.450406

0.466476

1999

0.625048

0.294633

0.851485

0.378145

0.535579

0.65026

1.283071

0.102068

0.546984

0.541914

2000

0.589872

0.455738

0.705737

0.421593

0.504906

0.606695

1.440411

0.099778

0.298352

0.613946

2001

0.560844

0.515316

0.696012

0.423083

0.510963

0.516993

1.462513

0.114338

0.280583

0.616862

2002

0.550552

0.472552

0.796213

0.424217

0.501515

0.471378

1.415838

0.123366

0.281333

0.628588

2003

0.604588

0.565419

0.779686

0.417632

0.475566

0.521081

1.341116

0.167739

0.326985

0.710478

2004

0.628928

0.584544

0.695776

0.419551

0.467175

0.534748

1.390866

0.161495

0.34003

0.711316

2005

0.594292

0.563607

0.625771

0.390167

0.44902

0.562245

1.447483

0.1435

0.355017

0.750782

2006

0.574519

0.63664

0.584961

0.392571

0.425354

0.574501

1.388618

0.196368

0.353318

0.773859

2007

0.542797

0.630615

0.579161

0.414989

0.408277

0.577103

1.402135

0.185372

0.369498

0.766186

2008

0.610861

0.557296

0.590074

0.399415

0.389158

0.569906

1.423817

0.192557

0.376025

0.772777

2009

0.603559

0.541821

0.735107

0.381838

0.384324

0.589331

1.234232

0.244792

0.419811

0.725381

Table 18

System SYS1scr: Specification

d ( sca 1 ) = c ( 1 ) + c ( 2 ) sca 1 ( 1 ) + c ( 501 ) d 90

d ( sca 2 ) = c ( 3 ) + c ( 4 ) sca 2 ( 1 ) + c ( 5 ) d ( sca 2 ( 1 ) ) + c ( 502 ) d 95 + c ( 503 ) d 96

d ( sca 3 ) = c ( 6 ) + c ( 7 ) sca 3 ( 3 ) + c ( 8 ) / t + c ( 504 ) d 96

d ( sca 4 ) = c ( 9 ) + c ( 10 ) sca 4 ( 2 ) + c ( 11 ) d ( sca 4 , 2 ) + c ( 12 ) t / ( t + 1 ) + c ( 505 ) d 99

d ( sca 5 ) = c ( 13 ) + c ( 14 ) sca 5 ( 1 ) + c ( 15 ) d ( sca 5 ( 1 ) ) + c ( 16 ) t / ( t + 1 )

d ( sca 6 ) = c ( 17 ) + c ( 18 ) sca 6 ( 1 ) + c ( 19 ) d ( sca 6 ( 1 ) ) + c ( 20 ) t / ( t + 1 )

d ( sca 7 ) = c ( 21 ) + c ( 22 ) sca 7 ( 1 ) + c ( 23 ) d ( sca 7 ( 1 ) ) + c ( 24 ) d ( sca 7 ( 2 ) ) + c ( 25 ) d ( sca 7 ( 3 ) ) + c ( 26 ) t / ( t + 1 )

d ( sca 8 ) = c ( 27 ) + c ( 28 ) sca 8 ( 1 ) + c ( 29 ) d ( sca 8 , 2 ) + c ( 506 ) d 96

d ( sca 9 ) = c ( 30 ) + c ( 31 ) sca 9 ( 1 ) + c ( 32 ) d ( sca 9 ( 2 ) ) + c ( 33 ) t / ( t + 1 ) + c ( 507 ) d 96

d ( sca 10 ) = c ( 34 ) + c ( 35 ) t / ( t + 1 ) + c ( 36 ) sca 10 ( 1 ) + c ( 508 ) d 90 + c ( 509 ) d 95

d ( sra 1 ) = c ( 37 ) + c ( 38 ) sra 1 ( 1 ) + c ( 39 ) d ( sra 1 ( 2 ) ) + c ( 40 ) / t + c ( 510 ) d 98

d ( sra 2 ) = c ( 41 ) + c ( 42 ) sra 2 ( 1 ) + c ( 43 ) d ( sra 2 , 2 ) + c ( 511 ) d 99

d ( sra 3 ) = c ( 44 ) + c ( 45 ) sra 3 ( 2 ) + c ( 46 ) d ( sra 3 ( 1 ) ) + c ( 512 ) d 99

d ( sra 4 ) = c ( 47 ) + c ( 48 ) sra 4 ( 1 ) + c ( 513 ) d 96 + c ( 514 ) d 91

d ( sra 5 HP ) = c ( 49 ) + c ( 50 ) sra 5 HP ( 1 ) + c ( 51 ) d ( sra 5 HP ( 1 ) )

d ( sra 5 HPd ) = c ( 52 ) sra 5 HPd ( 1 ) + c ( 53 ) d ( sra 5 HPd ( 1 ) ) + c ( 515 ) d 93 + c ( 516 ) d 96

d ( sra 6 ) = c ( 54 ) + c ( 55 ) sra 6 ( 1 ) + c ( 56 ) d ( sra 6 , 2 ) + c ( 517 ) d 93

d ( sra 7 ) = c ( 57 ) + c ( 58 ) sra 7 ( 2 ) + c ( 59 ) d ( sra 7 , 2 ) + c ( 60 ) / t

d ( sra 8 HP ) = c ( 61 ) + c ( 62 ) sra 8 HP ( 1 ) + c ( 63 ) d ( sra 8 HP , 2 ) + c ( 518 ) d 93 + c ( 519 ) d 94

d ( sra 8 HPd ) = c ( 64 ) d ( sra 8 HPd , 2 ) + c ( 520 ) d 92 + c ( 521 ) d 95

d ( sra 9 ) = c ( 65 ) + c ( 66 ) sra 9 ( 1 ) + c ( 522 ) d 93 + c ( 523 ) d 99

d ( sra 10 l ) = c ( 67 ) + c ( 68 ) sra 10 l ( 3 ) + c ( 524 ) d 94

Table 19

SYS1scr estimated by different methods—sample 1990–2009: OLS—ordinary least squares

 

Coefficient

Std. error

t-statistic

Prob.

 

Coefficient

Std. error

t-statistic

Prob.

c(1)

0.283078

0.08625

3.282047

0.001142962

c(38)

−0.81444

0.205237

−3.96828

8.92E-05

c(2)

−0.58001

0.18071

−3.20961

0.001462231

c(39)

0.33183

0.156979

2.113855

0.035291504

c(501)

−0.10221

0.029494

−3.46533

0.000600906

c(40)

1.076357

0.330764

3.254154

0.001257343

c(3)

0.278431

0.06312

4.411133

1.40E-05

c(510)

0.086243

0.036361

2.371851

0.018283202

c(4)

−0.43919

0.0929

−4.72754

3.39E-06

c(41)

0.23365

0.103312

2.261582

0.02438575

c(5)

0.408362

0.119747

3.410206

0.00073125

c(42)

−0.42761

0.188762

−2.26531

0.024153256

c(502)

0.11044

0.033033

3.343364

0.000924647

c(43)

0.285218

0.122418

2.329865

0.020427225

c(503)

0.153027

0.035167

4.351465

1.81E-05

c(511)

−0.20212

0.050267

−4.02093

7.22E-05

c(6)

0.125699

0.044001

2.85674

0.004556706

c(44)

0.521483

0.142176

3.667871

0.000285804

c(7)

−0.13899

0.062557

−2.22178

0.026988791

c(45)

−0.77361

0.208359

−3.71289

0.00024117

c(8)

−0.24993

0.084743

−2.9493

0.003416579

c(46)

−0.50663

0.210056

−2.41191

0.016424787

c(504)

0.074459

0.017457

4.265162

2.62E-05

c(512)

0.193523

0.068301

2.833381

0.004894547

c(9)

0.924228

0.160358

5.76353

1.92E-08

c(47)

0.13412

0.057677

2.325375

0.020669034

c(10)

−0.75929

0.119904

−6.33248

8.05E-10

c(48)

−0.34468

0.145689

−2.36584

0.018577283

c(11)

0.47183

0.061448

7.678555

1.92E-13

c(513)

0.083091

0.028105

2.956484

0.003340097

c(12)

−0.46602

0.093994

−4.95792

1.15E-06

c(514)

−0.10229

0.027463

−3.72469

0.000230612

c(505)

0.03589

0.013722

2.615495

0.009326674

c(49)

0.013136

0.001677

7.831778

6.93E-14

c(13)

1.064914

0.140304

7.590058

3.43E-13

c(50)

−0.02813

0.003245

−8.66814

2.14E-16

c(14)

−1.18973

0.15666

−7.59436

3.34E-13

c(51)

1.08189

0.015285

70.78026

3.87E-199

c(15)

0.454757

0.129331

3.516215

0.000500165

c(52)

−0.83309

0.158971

−5.24052

2.89E-07

c(16)

−0.39111

0.079436

−4.92362

1.36E-06

c(53)

0.320171

0.130101

2.460931

0.014378755

c(17)

1.216451

0.186703

6.515444

2.77E-10

c(515)

−0.05209

0.013328

−3.90808

0.000113298

c(18)

−1.08361

0.17362

−6.24131

1.36E-09

c(516)

0.042189

0.013558

3.111817

0.002024883

c(19)

0.465948

0.153538

3.034749

0.002602261

c(54)

0.135167

0.056727

2.382743

0.017760431

c(20)

−0.60757

0.105767

−5.74438

2.13E-08

c(55)

−0.23945

0.084606

−2.83021

0.004942134

c(21)

1.5781

0.285591

5.525739

6.75E-08

c(56)

0.284088

0.132169

2.149434

0.032340176

c(22)

−1.82179

0.303507

−6.00245

5.21E-09

c(517)

−0.14994

0.059384

−2.52487

0.012051054

c(23)

0.765847

0.205822

3.720919

0.000233936

c(57)

1.052576

0.121187

8.685539

1.89E-16

c(24)

0.756215

0.17858

4.234588

2.99E-05

c(58)

−0.78681

0.087866

−8.95465

2.72E-17

c(25)

0.559842

0.182826

3.062164

0.002381463

c(59)

0.522016

0.054793

9.527016

3.95E-19

c(26)

−0.26119

0.113264

−2.30606

0.021738264

c(60)

0.773993

0.126356

6.12549

2.62E-09

c(27)

0.20165

0.054576

3.69482

0.000258233

c(61)

−0.02272

0.002453

−9.26173

2.86E-18

c(28)

−0.37587

0.097205

−3.86677

0.000133287

c(62)

0.17378

0.012884

13.48847

3.24E-33

c(29)

0.414938

0.077282

5.369165

1.51E-07

c(63)

7.618111

1.251823

6.085614

3.27E-09

c(506)

0.038061

0.013083

2.909294

0.003872887

c(518)

−0.00564

0.001664

−3.38751

0.00079222

c(30)

0.226448

0.075907

2.983219

0.003068886

c(519)

−0.00521

0.001714

−3.04001

0.002558513

c(31)

−1.23202

0.16691

−7.38132

1.33E-12

c(64)

0.394639

0.079062

4.991487

9.80E-07

c(32)

0.557367

0.138947

4.011371

7.50E-05

c(520)

−0.03994

0.014916

−2.67737

0.007798128

c(33)

0.314156

0.058156

5.401991

1.28E-07

c(521)

0.039854

0.014704

2.710374

0.007078646

c(507)

0.044864

0.011907

3.767881

0.000195553

c(65)

0.274047

0.055718

4.918437

1.39E-06

c(34)

−0.15911

0.055598

−2.86172

0.004487515

c(66)

−0.73402

0.134082

−5.47443

8.81E-08

c(35)

0.371868

0.058992

6.303751

9.50E-10

c(522)

0.307258

0.059092

5.199688

3.54E-07

c(36)

−0.44718

0.075925

−5.88968

9.68E-09

c(523)

0.153138

0.059135

2.589621

0.010041828

c(508)

0.091543

0.022503

4.067977

5.96E-05

c(67)

−0.07714

0.028744

−2.68356

0.00765829

c(509)

−0.06748

0.014801

−4.55935

7.28E-06

c(68)

−0.20561

0.024533

−8.38109

1.62E-15

c(37)

0.413941

0.106503

3.886657

0.00012328

c(524)

−0.62241

0.073826

−8.43065

1.15E-15

Table 20

SYS1scr estimated by different methods—sample 1990–2009: WLS—weighted least squares

 

Coefficient

Std. error

t-statistic

Prob.

 

Coefficient

Std. error

t-statistic

Prob.

c(1)

0.283078

0.079519

3.55988

0.000426566

c(38)

−0.81444

0.174418

−4.66946

4.43E-06

c(2)

−0.58001

0.166606

−3.48131

0.000567383

c(39)

0.33183

0.133406

2.487366

0.013371939

c(501)

−0.10221

0.027192

−3.75867

0.000202576

c(40)

1.076357

0.281095

3.829153

0.000154358

c(3)

0.278431

0.054182

5.138814

4.79E-07

c(510)

0.086243

0.030901

2.79095

0.00556695

c(4)

−0.43919

0.079745

−5.50742

7.43E-08

c(41)

0.23365

0.091796

2.545326

0.011380659

c(5)

0.408362

0.10279

3.97277

8.76E-05

c(42)

−0.42761

0.167719

−2.54953

0.011247182

c(502)

0.11044

0.028355

3.894901

0.000119343

c(43)

0.285218

0.108771

2.622176

0.00914958

c(503)

0.153027

0.030187

5.069303

6.73E-07

c(511)

−0.20212

0.044664

−4.5254

8.47E-06

c(6)

0.125699

0.038805

3.239239

0.001322776

c(44)

0.521483

0.126327

4.128053

4.66E-05

c(7)

−0.13899

0.05517

−2.51926

0.012240942

c(45)

−0.77361

0.185132

−4.17872

3.77E-05

c(8)

−0.24993

0.074736

−3.3442

0.000921969

c(46)

−0.50663

0.186639

−2.71451

0.006992757

c(504)

0.074459

0.015396

4.836239

2.05E-06

c(512)

0.193523

0.060687

3.188865

0.001567833

c(9)

0.924228

0.13765

6.714309

8.48E-11

c(47)

0.13412

0.051588

2.599849

0.009753476

c(10)

−0.75929

0.102925

−7.37712

1.37E-12

c(48)

−0.34468

0.130308

−2.64509

0.00856481

c(11)

0.47183

0.052746

8.945246

2.91E-17

c(513)

0.083091

0.025138

3.30545

0.001054529

c(12)

−0.46602

0.080684

−5.7758

1.80E-08

c(514)

−0.10229

0.024564

−4.16433

4.01E-05

c(505)

0.03589

0.011779

3.04696

0.002501691

c(49)

0.013136

0.001539

8.534482

5.52E-16

c(13)

1.064914

0.124663

8.542329

5.22E-16

c(50)

−0.02813

0.002978

−9.44589

7.25E-19

c(14)

−1.18973

0.139196

−8.54717

5.05E-16

c(51)

1.08189

0.014027

77.131

1.49E-210

c(15)

0.454757

0.114914

3.957369

9.32E-05

c(52)

−0.83309

0.14125

−5.89801

9.25E-09

c(16)

−0.39111

0.070581

−5.54135

6.22E-08

c(53)

0.320171

0.115598

2.769686

0.005934489

c(17)

1.216451

0.16589

7.33289

1.82E-12

c(515)

−0.05209

0.011843

−4.3984

1.48E-05

c(18)

−1.08361

0.154265

−7.02436

1.27E-11

c(516)

0.042189

0.012046

3.502234

0.000526139

c(19)

0.465948

0.136422

3.415497

0.000717678

c(54)

0.135167

0.050404

2.681688

0.007700416

c(20)

−0.60757

0.093977

−6.46509

3.72E-10

c(55)

−0.23945

0.075174

−3.18529

0.001586701

c(21)

1.5781

0.229729

6.869395

3.31E-11

c(56)

0.284088

0.117435

2.419109

0.016109025

c(22)

−1.82179

0.244141

−7.46203

7.91E-13

c(517)

−0.14994

0.052764

−2.84165

0.004772473

c(23)

0.765847

0.165563

4.62571

5.40E-06

c(57)

1.052576

0.107678

9.775252

6.02E-20

c(24)

0.756215

0.14365

5.264284

2.57E-07

c(58)

−0.78681

0.078071

−10.0781

5.87E-21

c(25)

0.559842

0.147065

3.806769

0.000168351

c(59)

0.522016

0.048685

10.7223

3.70E-23

c(26)

−0.26119

0.091109

−2.86681

0.004417774

c(60)

0.773993

0.11227

6.894012

2.84E-11

c(27)

0.20165

0.048492

4.158383

4.11E-05

c(61)

−0.02272

0.002105

−10.7896

2.16E-23

c(28)

−0.37587

0.086369

−4.35191

1.81E-05

c(62)

0.17378

0.011059

15.71359

9.33E-42

c(29)

0.414938

0.068666

6.042796

4.16E-09

c(63)

7.618111

1.074559

7.089526

8.47E-12

c(506)

0.038061

0.011624

3.274303

0.001173709

c(518)

−0.00564

0.001428

−3.94633

9.73E-05

c(30)

0.226448

0.064509

3.510344

0.000510922

c(519)

−0.00521

0.001471

−3.5415

0.000456213

c(31)

−1.23202

0.141846

−8.68557

1.89E-16

c(64)

0.394639

0.072553

5.439346

1.06E-07

c(32)

0.557367

0.118082

4.720168

3.51E-06

c(520)

−0.03994

0.013688

−2.9176

0.003773847

c(33)

0.314156

0.049423

6.356506

7.00E-10

c(521)

0.039854

0.013493

2.953561

0.003371041

c(507)

0.044864

0.010119

4.433653

1.27E-05

c(65)

0.274047

0.049836

5.498979

7.76E-08

c(34)

−0.15911

0.048149

−3.30443

0.001058255

c(66)

−0.73402

0.119926

−6.12059

2.69E-09

c(35)

0.371868

0.051088

7.278945

2.56E-12

c(522)

0.307258

0.052853

5.813428

1.47E-08

c(36)

−0.44718

0.065753

−6.80082

5.02E-11

c(523)

0.153138

0.052892

2.895285

0.00404538

c(508)

0.091543

0.019489

4.697295

3.90E-06

c(67)

−0.07714

0.026239

−2.9397

0.003521415

c(509)

−0.06748

0.012818

−5.26469

2.56E-07

c(68)

−0.20561

0.022396

−9.18103

5.19E-18

c(37)

0.413941

0.09051

4.573417

6.84E-06

c(524)

−0.62241

0.067394

−9.23532

3.48E-18

Table 21

SYS1scr estimated by different methods—sample 1990–2009: SUR—seemingly unrelated regression

 

Coefficient

Std. error

t-statistic

Prob.

 

Coefficient

Std. error

t-statistic

Prob.

c(1)

0.234957

0.058079

4.045498

6.53E-05

c(38)

−0.84056

0.100483

−8.36515

1.81E-15

c(2)

−0.48089

0.121424

−3.9604

9.20E-05

c(39)

0.34597

0.079175

4.369712

1.68E-05

c(501)

−0.11051

0.01443

−7.65865

2.19E-13

c(40)

1.137276

0.173371

6.559794

2.13E-10

c(3)

0.272854

0.020134

13.55169

1.87E-33

c(510)

0.09166

0.016681

5.494938

7.92E-08

c(4)

−0.42622

0.028895

−14.751

4.96E-38

c(41)

0.249617

0.026854

9.295474

2.23E-18

c(5)

0.400249

0.039501

10.13262

3.85E-21

c(42)

−0.44738

0.046709

−9.57791

2.69E-19

c(502)

0.120347

0.01254

9.597298

2.32E-19

c(43)

0.270023

0.025781

10.4738

2.66E-22

c(503)

0.147105

0.013128

11.2054

7.52E-25

c(511)

−0.20361

0.012093

−16.8361

3.88E-46

c(6)

0.115054

0.014094

8.163217

7.33E-15

c(44)

0.548312

0.045754

11.98381

1.22E-27

c(7)

−0.12727

0.019723

−6.45284

4.00E-10

c(45)

−0.81821

0.065265

−12.5368

1.15E-29

c(8)

−0.24232

0.034329

−7.05859

1.03E-11

c(46)

−0.54969

0.065482

−8.39457

1.47E-15

c(504)

0.082137

0.006536

12.56645

8.97E-30

c(512)

0.194547

0.029549

6.583848

1.85E-10

c(9)

0.943513

0.043286

21.79729

1.75E-65

c(47)

0.15916

0.025087

6.344446

7.51E-10

c(10)

−0.75699

0.029589

−25.5831

9.35E-80

c(48)

−0.40294

0.062287

−6.46904

3.64E-10

c(11)

0.475233

0.013396

35.47534

1.29E-113

c(513)

0.090953

0.014434

6.301505

9.62E-10

c(12)

−0.48929

0.029803

−16.4175

1.68E-44

c(514)

−0.08508

0.012256

−6.94239

2.11E-11

c(505)

0.036983

0.002789

13.25975

2.35E-32

c(49)

0.01349

0.000625

21.57141

1.30E-64

c(13)

1.06234

0.053497

19.85786

5.74E-58

c(50)

−0.02875

0.001192

−24.1101

2.82E-74

c(14)

−1.21682

0.053302

−22.8289

1.97E-69

c(51)

1.085785

0.008216

132.1608

6.73E-284

c(15)

0.466725

0.044263

10.54438

1.52E-22

c(52)

−0.86156

0.060809

−14.1682

8.52E-36

c(16)

−0.37179

0.042773

−8.6921

1.80E-16

c(53)

0.328044

0.049865

6.578703

1.91E-10

c(17)

1.251569

0.07414

16.88106

2.59E-46

c(515)

−0.05817

0.005475

−10.624

8.09E-23

c(18)

−1.13099

0.066292

−17.0608

5.13E-47

c(516)

0.037513

0.005639

6.65296

1.22E-10

c(19)

0.461616

0.053557

8.619101

3.03E-16

c(54)

0.140982

0.026324

5.355615

1.62E-07

c(20)

−0.61549

0.050982

−12.0727

5.80E-28

c(55)

−0.24846

0.035189

−7.0609

1.01E-11

c(21)

1.591889

0.090008

17.68604

1.82E-49

c(56)

0.280768

0.038032

7.382508

1.32E-12

c(22)

−1.83248

0.092471

−19.8168

8.30E-58

c(517)

−0.17261

0.023242

−7.42684

9.93E-13

c(23)

0.774811

0.067097

11.54767

4.56E-26

c(57)

0.98605

0.044398

22.2092

4.59E-67

c(24)

0.739849

0.054287

13.62841

9.59E-34

c(58)

−0.74137

0.031644

−23.4283

1.05E-71

c(25)

0.558427

0.053313

10.47448

2.64E-22

c(59)

0.523095

0.017136

30.52647

2.46E-97

c(26)

−0.268

0.04471

−5.99429

5.45E-09

c(60)

0.773597

0.058144

13.30473

1.59E-32

c(27)

0.197595

0.020551

9.614966

2.03E-19

c(61)

−0.0229

0.001283

−17.8516

4.08E-50

c(28)

−0.36951

0.035719

−10.3451

7.32E-22

c(62)

0.175333

0.00685

25.59751

8.27E-80

c(29)

0.400592

0.018888

21.20883

3.25E-63

c(63)

7.671487

0.548712

13.98089

4.41E-35

c(506)

0.038445

0.004267

9.009607

1.82E-17

c(518)

−0.0055

0.000613

−8.96842

2.46E-17

c(30)

0.213569

0.029681

7.195569

4.35E-12

c(519)

−0.00513

0.000533

−9.62612

1.87E-19

c(31)

−1.18993

0.056699

−20.9867

2.35E-62

c(64)

0.430947

0.030779

14.0012

3.69E-35

c(32)

0.545298

0.040457

13.47857

3.53E-33

c(520)

−0.03993

0.006948

−5.7465

2.10E-08

c(33)

0.308919

0.028288

10.92042

7.55E-24

c(521)

0.037305

0.00608

6.135315

2.48E-09

c(507)

0.046084

0.004433

10.39612

4.90E-22

c(65)

0.244369

0.023271

10.50116

2.14E-22

c(34)

−0.13306

0.025963

−5.12513

5.12E-07

c(66)

−0.67082

0.050755

−13.2168

3.41E-32

c(35)

0.351948

0.02466

14.27219

3.41E-36

c(522)

0.304342

0.029036

10.48162

2.50E-22

c(36)

−0.46841

0.030641

−15.2871

4.22E-40

c(523)

0.148176

0.021366

6.9352

2.21E-11

c(508)

0.089716

0.007458

12.02994

8.30E-28

c(67)

−0.08253

0.016711

−4.93861

1.26E-06

c(509)

−0.06953

0.005683

−12.2339

1.50E-28

c(68)

−0.21785

0.011922

−18.2732

9.07E-52

c(37)

0.425266

0.052871

8.043437

1.66E-14

c(524)

−0.64568

0.026648

−24.2303

9.97E-75

Table 22

SYS1scr estimated by different methods—sample 1990–2009: GLM—generalized linear models with bootstrap

 

Coefficient

Std. error

z

Prob.

 

Coefficient

Std. error

z

Prob.

c(1)

0.283078

0.082689

3.42

0.001

c(38)

−0.81444

0.188264

−4.33

0

c(2)

−0.58001

0.170839

−3.4

0.001

c(39)

0.33183

0.159535

2.08

0.038

c(501)

−0.10221

0.004737

−21.58

0

c(40)

1.076356

0.332231

3.24

0.001

c(3)

0.278431

0.101581

2.74

0.006

c(510)

0.086243

0.011943

7.22

0

c(4)

−0.43919

0.153461

−2.86

0.004

c(41)

0.23365

0.088917

2.63

0.009

c(5)

0.408363

0.140692

2.9

0.004

c(42)

−0.42761

0.163947

−2.61

0.009

c(502)

0.11044

0.006218

17.76

0

c(43)

0.285218

0.091688

3.11

0.002

c(503)

0.153027

0.014458

10.58

0

c(511)

−0.20212

0.015286

−13.22

0

c(6)

0.125699

0.041312

3.04

0.002

c(44)

0.521483

0.108545

4.8

0

c(7)

−0.13899

0.050457

−2.75

0.006

c(45)

−0.77361

0.158828

−4.87

0

c(8)

−0.24993

0.085994

−2.91

0.004

c(46)

−0.50663

0.175199

−2.89

0.004

c(504)

0.074459

0.005317

14

0

c(512)

0.193523

0.011274

17.17

0

c(9)

0.924228

0.094083

9.82

0

c(47)

0.13412

0.047279

2.84

0.005

c(10)

−0.75929

0.067526

−11.24

0

c(48)

−0.34468

0.118808

−2.9

0.004

c(11)

0.47183

0.047361

9.96

0

c(513)

0.083091

0.007286

11.4

0

c(12)

−0.46602

0.056969

−8.18

0

c(514)

−0.10229

0.004784

−21.38

0

c(505)

0.03589

0.004272

8.4

0

c(49)

0.013136

0.001949

6.74

0

c(13)

1.064914

0.167703

6.35

0

c(50)

−0.02813

0.003768

−7.46

0

c(14)

−1.18973

0.193828

−6.14

0

c(51)

1.08189

0.01169

92.55

0

c(15)

0.454757

0.165187

2.75

0.006

c(52)

−0.83309

0.153003

−5.44

0

c(16)

−0.39111

0.087032

−4.49

0

c(53)

0.320171

0.137497

2.33

0.02

c(17)

1.216451

0.209223

5.81

0

c(515)

−0.05209

0.005145

−10.12

0

c(18)

−1.08361

0.17854

−6.07

0

c(516)

0.042189

0.005872

7.18

0

c(19)

0.465948

0.172046

2.71

0.007

c(54)

0.135167

0.065899

2.05

0.04

c(20)

−0.60757

0.12657

−4.8

0

c(55)

−0.23945

0.111845

−2.14

0.032

c(21)

1.5781

0.291682

5.41

0

c(56)

0.284088

0.142332

2

0.046

c(22)

−1.82179

0.352457

−5.17

0

c(517)

−0.14994

0.029437

−5.09

0

c(23)

0.765847

0.237994

3.22

0.001

c(57)

1.052575

0.100791

10.44

0

c(24)

0.756215

0.238152

3.18

0.001

c(58)

−0.78681

0.075003

−10.49

0

c(25)

0.559842

0.191824

2.92

0.004

c(59)

0.522016

0.060821

8.58

0

c(26)

−0.26119

0.109698

−2.38

0.017

c(60)

0.773993

0.15599

4.96

0

c(27)

0.20165

0.037095

5.44

0

c(61)

−0.02272

0.001404

−16.18

0

c(28)

−0.37587

0.06611

−5.69

0

c(62)

0.173781

0.007403

23.47

0

c(29)

0.414938

0.064205

6.46

0

c(63)

7.618141

0.782133

9.74

0

c(506)

0.038061

0.004809

7.91

0

c(518)

−0.00564

0.000399

−14.12

0

c(30)

0.226448

0.115659

1.96

0.05

c(519)

−0.00521

0.000485

−10.75

0

c(31)

−1.23202

0.246907

−4.99

0

c(64)

0.394639

0.049193

8.02

0

c(32)

0.557367

0.151548

3.68

0

c(520)

−0.03994

0.003189

−12.52

0

c(33)

0.314156

0.064189

4.89

0

c(521)

0.039854

0.002782

14.32

0

c(507)

0.044864

0.006176

7.26

0

c(65)

0.274047

0.036869

7.43

0

c(34)

−0.15911

0.032982

−4.82

0

c(66)

−0.73402

0.096623

−7.6

0

c(35)

0.371868

0.0525

7.08

0

c(522)

0.307258

0.009443

32.54

0

c(36)

−0.44718

0.072451

−6.17

0

c(523)

0.153138

0.009668

15.84

0

c(508)

0.091543

0.018952

4.83

0

c(67)

−0.07714

0.01713

−4.5

0

c(509)

−0.06748

0.004226

−15.97

0

c(68)

−0.20561

0.019253

−10.68

0

c(37)

0.413941

0.095647

4.33

0

c(524)

−0.62241

0.030516

−20.4

0

Table 23

Comparative estimation output OLS–SUR

Equation: d ( sca 1 ) = c ( 1 ) + c ( 2 ) sca 1 ( 1 ) + c ( 501 ) d 90

OLS

SUR

R-squared

0.592041

Mean dependent var.

0.001486537

R-squared

0.582578

Mean dependent var.

0.001486537

Adjusted R-squared

0.544045

S.D. dependent var.

0.04237619

Adjusted R-squared

0.533469

S.D. dependent var.

0.04237619

S.E. of regression

0.028614

Sum squared resid.

0.013919204

S.E. of regression

0.028944

Sum squared resid.

0.014242074

Durbin–Watson stat.

1.538095

  

Durbin–Watson stat.

1.721237

  

Equation: d ( sca 2 ) = c ( 3 ) + c ( 4 ) sca 2 ( 1 ) + c ( 5 ) d ( sca 2 ( 1 ) ) + c ( 502 ) d 95 + c ( 503 ) d 96

OLS

SUR

R-squared

0.827101

Mean dependent var.

−0.00231224

R-squared

0.822768

Mean dependent var.

−0.00231224

Adjusted R-squared

0.777702

S.D. dependent var.

0.067510188

Adjusted R-squared

0.77213

S.D. dependent var.

0.067510188

S.E. of regression

0.03183

Sum squared resid.

0.014184143

S.E. of regression

0.032227

Sum squared resid.

0.014539665

Durbin–Watson stat.

2.754131

  

Durbin–Watson stat.

2.693471

  

Equation: d ( sca 3 ) = c ( 6 ) + c ( 7 ) sca 3 ( 3 ) + c ( 8 ) / t + c ( 504 ) d 96

OLS

SUR

R-squared

0.778591

Mean dependent var.

−0.00144036

R-squared

0.773913

Mean dependent var.

−0.00144036

Adjusted R-squared

0.731147

S.D. dependent var.

0.031713173

Adjusted R-squared

0.725466

S.D. dependent var.

0.031713173

S.E. of regression

0.016444

Sum squared resid.

0.003785497

S.E. of regression

0.016616

Sum squared resid.

0.003865477

Durbin–Watson stat.

2.002707

  

Durbin–Watson stat.

1.987606

  

Equation: d ( sca 4 ) = c ( 9 ) + c ( 10 ) sca 4 ( 2 ) + c ( 11 ) d ( sca 4 , 2 ) + c ( 12 ) t / ( t + 1 ) + c ( 505 ) d 96

OLS

SUR

R-squared

0.893072

Mean dependent var.

−0.00551455

R-squared

0.890453

Mean dependent var.

−0.00551455

Adjusted R-squared

0.862522

S.D. dependent var.

0.028411694

Adjusted R-squared

0.859154

S.D. dependent var.

0.028411694

S.E. of regression

0.010535

Sum squared resid.

0.001553663

S.E. of regression

0.010663

Sum squared resid.

0.00159172

Durbin–Watson stat.

1.833085

  

Durbin–Watson stat.

1.813972

  

Equation: d ( sca 5 ) = c ( 13 ) + c ( 14 ) sca 5 ( 1 ) + c ( 15 ) d ( sca 5 ( 1 ) ) + c ( 16 ) t / ( t + 1 )

OLS

SUR

R-squared

0.805431

Mean dependent var.

0.000363901

R-squared

0.801683

Mean dependent var.

0.000363901

Adjusted R-squared

0.766517

S.D. dependent var.

0.033923317

Adjusted R-squared

0.76202

S.D. dependent var.

0.033923317

S.E. of regression

0.016392

Sum squared resid.

0.004030354

S.E. of regression

0.016549

Sum squared resid.

0.004107978

Durbin–Watson stat.

2.597269

  

Durbin–Watson stat.

2.536249

  

Equation: d ( sca 6 ) = c ( 17 ) + c ( 18 ) sca 6 ( 1 ) + c ( 19 ) d ( sca 6 ( 1 ) ) + c ( 20 ) t / ( t + 1 )

OLS

SUR

R-squared

0.741118

Mean dependent var.

−0.00271833

R-squared

0.737032

Mean dependent var.

−0.00271833

Adjusted R-squared

0.689341

S.D. dependent var.

0.03293319

Adjusted R-squared

0.684438

S.D. dependent var.

0.03293319

S.E. of regression

0.018356

Sum squared resid.

0.005054087

S.E. of regression

0.0185

Sum squared resid.

0.005133851

Durbin–Watson stat.

1.930535

  

Durbin–Watson stat.

1.811626

  

Equation: d ( sca 7 ) = c ( 21 ) + c ( 22 ) sca 7 ( 1 ) + c ( 23 ) d ( sca 7 ( 1 ) ) + c ( 24 ) d ( sca 7 ( 2 ) ) + c ( 25 ) d ( sca 7 ( 3 ) ) + c ( 26 ) t / ( t + 1 )

OLS

SUR

R-squared

0.776545

Mean dependent var.

−0.00219888

R-squared

0.775659

Mean dependent var.

−0.00219888

Adjusted R-squared

0.674974

S.D. dependent var.

0.021803938

Adjusted R-squared

0.673686

S.D. dependent var.

0.021803938

S.E. of regression

0.012431

Sum squared resid.

0.001699731

S.E. of regression

0.012455

Sum squared resid.

0.001706466

Durbin–Watson stat.

2.45839

  

Durbin–Watson stat.

2.449675

  

Equation: d ( sca 8 ) = c ( 27 ) + c ( 28 ) sca 8 ( 1 ) + c ( 29 ) d ( sca 8 , 2 ) + c ( 506 ) d 96

OLS

SUR

R-squared

0.79341

Mean dependent var.

−0.00483833

R-squared

0.792092

Mean dependent var.

−0.00483833

Adjusted R-squared

0.752092

S.D. dependent var.

0.024203015

Adjusted R-squared

0.75051

S.D. dependent var.

0.024203015

S.E. of regression

0.012051

Sum squared resid.

0.002178319

S.E. of regression

0.012089

Sum squared resid.

0.002192212

Durbin–Watson stat.

1.725176

  

Durbin–Watson stat.

1.776138

  

Equation: d ( sca 9 ) = c ( 30 ) + c ( 31 ) sca 9 ( 1 ) + c ( 32 ) d ( sca 9 ( 2 ) ) + c ( 33 ) t / ( t + 1 ) + c ( 507 ) d 96

OLS

SUR

R-squared

0.846468

Mean dependent var.

−0.00080741

R-squared

0.845128

Mean dependent var.

−0.00080741

Adjusted R-squared

0.799227

S.D. dependent var.

0.022579243

Adjusted R-squared

0.797475

S.D. dependent var.

0.022579243

S.E. of regression

0.010117

Sum squared resid.

0.001330662

S.E. of regression

0.010161

Sum squared resid.

0.001342271

Durbin–Watson stat.

1.832536

  

Durbin–Watson stat.

1.928251

  

Equation: d ( sca 10 ) = c ( 34 ) + c ( 35 ) t / ( t + 1 ) + c ( 36 ) sca 10 ( 1 ) + c ( 508 ) d 90 + c ( 509 ) d 95

OLS

SUR

R-squared

0.848972

Mean dependent var.

−0.00253882

R-squared

0.846468

Mean dependent var.

−0.00253882

Adjusted R-squared

0.808698

S.D. dependent var.

0.031535026

Adjusted R-squared

0.805526

S.D. dependent var.

0.031535026

S.E. of regression

0.013793

Sum squared resid.

0.002853624

S.E. of regression

0.013907

Sum squared resid.

0.002900946

Durbin–Watson stat.

2.4873

  

Durbin–Watson stat.

2.432552

  

Equation: d ( sra 1 ) = c ( 37 ) + c ( 38 ) sra 1 ( 1 ) + c ( 39 ) d ( sra 1 ( 2 ) ) + c ( 40 ) / t + c ( 510 ) d 98

OLS

SUR

R-squared

0.611242

Mean dependent var.

−0.01089713

R-squared

0.609433

Mean dependent var.

−0.01089713

Adjusted R-squared

0.491625

S.D. dependent var.

0.047563903

Adjusted R-squared

0.489258

S.D. dependent var.

0.047563903

S.E. of regression

0.033913

Sum squared resid.

0.014951435

S.E. of regression

0.033992

Sum squared resid.

0.015021033

Durbin–Watson stat.

1.439341

  

Durbin–Watson stat.

1.431766

  

Equation: d ( sra 2 ) = c ( 41 ) + c ( 42 ) sra 2 ( 1 ) + c ( 43 ) d ( sra 2 , 2 ) + c ( 511 ) d 99

OLS

SUR

R-squared

0.777682

Mean dependent var.

−0.00733534

R-squared

0.773894

Mean dependent var.

−0.00733534

Adjusted R-squared

0.733218

S.D. dependent var.

0.089810856

Adjusted R-squared

0.728673

S.D. dependent var.

0.089810856

S.E. of regression

0.046388

Sum squared resid.

0.032277867

S.E. of regression

0.046782

Sum squared resid.

0.032827865

Durbin–Watson stat.

1.66777

  

Durbin–Watson stat.

1.64911

  

Equation: d ( sra 3 ) = c ( 44 ) + c ( 45 ) sra 3 ( 2 ) + c ( 46 ) d ( sra 3 ( 1 ) ) + c ( 512 ) d 99

OLS

SUR

R-squared

0.604261

Mean dependent var.

0.007327141

R-squared

0.601432

Mean dependent var.

0.007327141

Adjusted R-squared

0.525113

S.D. dependent var.

0.095697545

Adjusted R-squared

0.521718

S.D. dependent var.

0.095697545

S.E. of regression

0.065947

Sum squared resid.

0.065235376

S.E. of regression

0.066182

Sum squared resid.

0.065701731

Durbin–Watson stat.

1.611126

  

Durbin–Watson stat.

1.521897

  

Equation: d ( sra 4 ) = c ( 47 ) + c ( 48 ) sra 4 ( 1 ) + c ( 513 ) d 96 + c ( 514 ) d 91

OLS

SUR

R-squared

0.701614

Mean dependent var.

−0.00211187

R-squared

0.683711

Mean dependent var.

−0.00211187

Adjusted R-squared

0.645667

S.D. dependent var.

0.044451546

Adjusted R-squared

0.624407

S.D. dependent var.

0.044451546

S.E. of regression

0.02646

Sum squared resid.

0.011202251

S.E. of regression

0.027242

Sum squared resid.

0.011874394

Durbin–Watson stat.

2.497302

  

Durbin–Watson stat.

2.320523

  

Equation: d ( sra 5 HP ) = c ( 49 ) + c ( 50 ) sra 5 HP ( 1 ) + c ( 51 ) d ( sra 5 HP ( 1 ) )

OLS

SUR

R-squared

0.998586

Mean dependent var.

−0.00657374

R-squared

0.998573

Mean dependent var.

−0.00657374

Adjusted R-squared

0.998409

S.D. dependent var.

0.00887352

Adjusted R-squared

0.998394

S.D. dependent var.

0.00887352

S.E. of regression

0.000354

Sum squared resid.

2.00E-06

S.E. of regression

0.000356

Sum squared resid.

2.02E-06

Durbin–Watson stat.

0.584091

  

Durbin–Watson stat.

0.585266

  

Equation: d ( sra 5 HPd ) = c ( 52 ) sra 5 HPd ( 1 ) + c ( 53 ) d ( sra 5 HPd ( 1 ) ) + c ( 515 ) d 93 + c ( 516 ) d 96

OLS

SUR

R-squared

0.852908

Mean dependent var.

7.88E-06

R-squared

0.848174

Mean dependent var.

7.88E-06

Adjusted R-squared

0.82349

S.D. dependent var.

0.029321112

Adjusted R-squared

0.817809

S.D. dependent var.

0.029321112

S.E. of regression

0.012319

Sum squared resid.

0.00227626

S.E. of regression

0.012515

Sum squared resid.

0.002349518

Durbin–Watson stat.

1.956297

  

Durbin–Watson stat.

1.905226

  

Equation: d ( sra 6 ) = c ( 54 ) + c ( 55 ) sra 6 ( 1 ) + c ( 56 ) d ( sra 6 , 2 ) + c ( 517 ) d 93

OLS

SUR

R-squared

0.705175

Mean dependent var.

−0.02886253

R-squared

0.701184

Mean dependent var.

−0.02886253

Adjusted R-squared

0.64621

S.D. dependent var.

0.093185537

Adjusted R-squared

0.64142

S.D. dependent var.

0.093185537

S.E. of regression

0.055427

Sum squared resid.

0.046082199

S.E. of regression

0.055801

Sum squared resid.

0.046706118

Durbin–Watson stat.

1.764954

  

Durbin–Watson stat.

1.907709

  

Equation: d ( sra 7 ) = c ( 57 ) + c ( 58 ) sra 7 ( 2 ) + c ( 59 ) d ( sra 7 , 2 ) + c ( 60 ) / t

OLS

SUR

R-squared

0.920327

Mean dependent var.

−0.02012059

R-squared

0.918205

Mean dependent var.

−0.02012059

Adjusted R-squared

0.904393

S.D. dependent var.

0.108371317

Adjusted R-squared

0.901847

S.D. dependent var.

0.108371317

S.E. of regression

0.033509

Sum squared resid.

0.016842702

S.E. of regression

0.033952

Sum squared resid.

0.017291216

Durbin–Watson stat.

1.736261

  

Durbin–Watson stat.

1.701985

  

Equation: d ( sra 8 HP ) = c ( 61 ) + c ( 62 ) sra 8 HP ( 1 ) + c ( 63 ) d ( sra 8 HP , 2 ) + c ( 518 ) d 93 + c ( 519 ) d 94

OLS

SUR

R-squared

0.941453

Mean dependent var.

0.005926559

R-squared

0.941163

Mean dependent var.

0.005926559

Adjusted R-squared

0.924725

S.D. dependent var.

0.005647312

Adjusted R-squared

0.924352

S.D. dependent var.

0.005647312

S.E. of regression

0.001549

Sum squared resid.

3.36E-05

S.E. of regression

0.001553

Sum squared resid.

3.38E-05

Durbin–Watson stat.

1.30744

  

Durbin–Watson stat.

1.274754

  

Equation: d ( sra 8 HPd ) = c ( 64 ) d ( sra 8 HPd , 2 ) + c ( 520 ) d 92 + c ( 521 ) d 95

OLS

SUR

R-squared

0.806027

Mean dependent var.

0.000312728

R-squared

0.803115

Mean dependent var.

0.000312728

Adjusted R-squared

0.78178

S.D. dependent var.

0.029986054

Adjusted R-squared

0.778504

S.D. dependent var.

0.029986054

S.E. of regression

0.014008

Sum squared resid.

0.00313945

S.E. of regression

0.014112

Sum squared resid.

0.003186578

Durbin–Watson stat.

2.438696

  

Durbin–Watson stat.

2.477959

  

Equation: d ( sra 9 ) = c ( 65 ) + c ( 66 ) sra 9 ( 1 ) + c ( 522 ) d 93 + c ( 523 ) d 99

OLS

SUR

R-squared

0.774555

Mean dependent var.

−0.00223569

R-squared

0.769816

Mean dependent var.

−0.00223569

Adjusted R-squared

0.732284

S.D. dependent var.

0.110603027

Adjusted R-squared

0.726657

S.D. dependent var.

0.110603027

S.E. of regression

0.057227

Sum squared resid.

0.052399624

S.E. of regression

0.057826

Sum squared resid.

0.053501025

Durbin–Watson stat.

1.192039

  

Durbin–Watson stat.

1.327546

  

Equation: d ( sra 10 l ) = c ( 67 ) + c ( 68 ) sra 10 l ( 3 ) + c ( 524 ) d 94

OLS

SUR

R-squared

0.871203

Mean dependent var.

0.10191042

R-squared

0.867635

Mean dependent var.

0.10191042

Adjusted R-squared

0.85403

S.D. dependent var.

0.172691047

Adjusted R-squared

0.849986

S.D. dependent var.

0.172691047

S.E. of regression

0.065978

Sum squared resid.

0.065297402

S.E. of regression

0.066886

Sum squared resid.

0.067106068

Durbin–Watson stat.

2.849506

  

Durbin–Watson stat.

2.662251

  
Table 24

Generalized method of moments—time series (HAC): Kernel: Bartlett, bandwidth: Variable Newey–West (5), no prewhitening

SYS1scaG

d ( sca 1 ) = c ( 1 ) + c ( 2 ) sca 1 ( 1 ) @ sca 1 ( 1 )

d ( sca 2 ) = c ( 3 ) + c ( 4 ) sca 2 ( 1 ) + c ( 5 ) d ( sca 2 ( 1 ) ) @ sca 2 ( 1 ) d ( sca 2 ( 1 ) )

d ( sca 3 ) = c ( 6 ) + c ( 7 ) sca 3 ( 3 ) + c ( 8 ) / t @ sca 10 ( 3 ) 1 / t

d ( sca 4 ) = c ( 9 ) + c ( 10 ) sca 4 ( 2 ) + c ( 11 ) d ( sca 4 , 2 ) + c ( 12 ) t / ( t + 1 ) @ sca 6 ( 2 ) d ( sca 4 , 2 ) t / ( t + 1 )

d ( sca 5 ) = c ( 13 ) + c ( 14 ) sca 5 ( 1 ) + c ( 15 ) d ( sca 5 ( 1 ) ) + c ( 16 ) t / ( t + 1 ) @ sca 6 ( 1 ) d ( sca 6 ( 1 ) ) t / ( t + 1 )

d ( sca 6 ) = c ( 17 ) + c ( 18 ) sca 6 ( 1 ) + c ( 19 ) d ( sca 6 ( 1 ) ) + c ( 20 ) t / ( t + 1 ) @ sca 4 ( 1 ) d ( sca 5 ( 1 ) ) t / ( t + 1 )

d ( sca 7 ) = c ( 21 ) + c ( 22 ) sca 7 ( 1 ) + c ( 23 ) d ( sca 7 ( 1 ) ) + c ( 24 ) d ( sca 7 ( 2 ) ) + c ( 25 ) d ( sca 7 ( 3 ) ) + c ( 26 ) t / ( t + 1 ) @ sca 8 ( 1 ) d ( sca 7 ( 1 ) ) d ( sca 7 ( 2 ) ) d ( sca 7 ( 3 ) ) t / ( t + 1 )

d ( sca 8 ) = c ( 27 ) + c ( 28 ) sca 8 ( 1 ) + c ( 29 ) d ( sca 8 , 2 ) @ sca 7 ( 1 ) d ( sca 8 , 2 )

d ( sca 9 ) = c ( 30 ) + c ( 31 ) sca 9 ( 1 ) + c ( 32 ) d ( sca 9 ( 2 ) ) + c ( 33 ) t / ( t + 1 ) @ sca 9 ( 1 ) d ( sca 9 ( 2 ) ) t / ( t + 1 )

d ( sca 10 ) = c ( 34 ) + c ( 35 ) t / ( t + 1 ) + c ( 36 ) sca 10 ( 1 ) @ t / ( t + 1 ) sca 3 ( 1 )

SYS1sraG

d ( sra 1 ) = c ( 37 ) + c ( 38 ) sra 1 ( 1 ) + c ( 39 ) d ( sra 1 ( 2 ) ) + c ( 40 ) / t @ sra 10 ( 1 ) d ( sra 1 ( 2 ) ) 1 / t

d ( sra 2 ) = c ( 41 ) + c ( 42 ) sra 2 ( 1 ) + c ( 43 ) d ( sra 2 , 2 ) @ sra 3 ( 1 ) d ( sra 2 , 2 )

d ( sra 3 ) = c ( 44 ) + c ( 45 ) sra 3 ( 2 ) + c ( 46 ) d ( sra 3 ( 1 ) ) @ sra 2 ( 2 ) d ( sra 3 ( 1 ) )

d ( sra 4 ) = c ( 47 ) + c ( 48 ) sra 4 ( 1 ) @ sra 4 ( 1 )

d ( sra 5 HP ) = c ( 49 ) + c ( 50 ) sra 5 HP ( 1 ) + c ( 51 ) d ( sra 5 HP ( 1 ) ) @ sra 8 HP ( 1 ) d ( sra 8 HP ( 1 ) )

d ( sra 5 HPd ) = c ( 52 ) sra 5 HPd ( 1 ) + c ( 53 ) d ( sra 5 HPd ( 1 ) ) @ sra 5 HPd ( 1 ) d ( sra 5 ( 1 ) )

d ( sra 6 ) = c ( 54 ) + c ( 55 ) sra 6 ( 1 ) + c ( 56 ) d ( sra 6 , 2 ) @ sra 10 l ( 1 ) d ( sra 6 , 2 )

d ( sra 7 ) = c ( 57 ) + c ( 58 ) sra 7 ( 2 ) + c ( 59 ) d ( sra 7 , 2 ) + c ( 60 ) / t @ sra 7 ( 2 ) d ( sra 7 , 2 ) 1 / t

d ( sra 9 ) = c ( 65 ) + c ( 66 ) sra 9 ( 1 ) @ sra 9 ( 1 )

d ( sra 10 l ) = c ( 67 ) + c ( 68 ) sra 10 l ( 3 ) @ sra 10 ( 3 )

SYS1sra8G

d ( sra 8 HP ) = c ( 61 ) + c ( 62 ) sra 8 HP ( 1 ) + c ( 63 ) d ( sra 8 HP , 2 ) @ sca 1 ( 1 ) d ( sra 1 )

d ( sra 8 HPd ) = c ( 64 ) d ( sra 8 HPd , 2 ) @ d ( sra 8 )

 

Estimation

Coefficient

Std. error

t-statistic

Prob.

c(1)

0.306601

0.125989

2.43355

0.0161112

c(2)

−0.64008

0.280023

−2.2858

0.0236481

c(3)

0.306749

0.040223

7.626242

2.45E-12

c(4)

−0.4616

0.050503

−9.13994

3.75E-16

c(5)

0.582774

0.115473

5.046853

1.27E-06

c(6)

0.13037

0.028106

4.638562

7.51E-06

c(7)

−0.14457

0.042163

−3.42883

0.0007803

c(8)

−0.2129

0.056051

−3.79838

0.0002101

c(9)

0.940451

0.141303

6.65555

4.82E-10

c(10)

−0.78012

0.122989

−6.34302

2.45E-09

c(11)

0.540342

0.030012

18.00435

1.60E-39

c(12)

−0.46646

0.070064

−6.65759

4.77E-10

c(13)

1.086867

0.046148

23.55169

1.38E-52

c(14)

−1.22309

0.051193

−23.8916

2.48E-53

c(15)

0.492234

0.07655

6.430267

1.56E-09

c(16)

−0.39337

0.0287

−13.7063

2.39E-28

c(17)

1.01984

0.101599

10.03791

1.66E-18

c(18)

−0.8892

0.095385

−9.32216

1.26E-16

c(19)

0.457512

0.122083

3.747537

0.0002531

c(20)

−0.52278

0.058669

−8.91068

1.46E-15

c(21)

1.512662

0.228653

6.61553

5.95E-10

c(22)

−1.74819

0.253013

−6.90948

1.25E-10

c(23)

0.730626

0.163282

4.474613

1.49E-05

c(24)

0.729524

0.077765

9.381098

8.85E-17

c(25)

0.532158

0.139174

3.823693

0.0001914

c(26)

−0.24892

0.065536

−3.7982

0.0002102

c(27)

0.206372

0.052335

3.943296

0.0001223

c(28)

−0.38015

0.0913

−4.16373

5.23E-05

c(29)

0.343602

0.049807

6.898688

1.33E-10

c(30)

0.178478

0.045995

3.880364

0.000155

c(31)

−1.00173

0.120293

−8.32743

4.48E-14

c(32)

0.312532

0.117606

2.65744

0.0087153

c(33)

0.263697

0.032537

8.10445

1.62E-13

c(34)

−0.11243

0.05481

−2.0513

0.041954

c(35)

0.223479

0.040543

5.512165

1.48E-07

c(36)

−0.22669

0.066706

−3.39839

0.0008656

c(37)

0.410337

0.078538

5.224706

5.30E-07

c(38)

−0.79341

0.152393

−5.20636

5.77E-07

c(39)

0.246874

0.109319

2.258289

0.0252624

c(40)

1.014128

0.223294

4.54167

1.08E-05

c(41)

0.158335

0.081972

1.931566

0.0551585

c(42)

−0.3044

0.14473

−2.10325

0.0369891

c(42)

−0.3044

0.14473

−2.10325

0.0369891

c(43)

0.353333

0.085985

4.109225

6.29E-05

c(44)

0.607509

0.186953

3.249529

0.0014058

c(45)

−0.88958

0.272764

−3.26134

0.001352

c(46)

−0.59086

0.180956

−3.2652

0.0013348

c(47)

0.200071

0.023126

8.651512

4.78E-15

c(48)

−0.51659

0.051403

−10.0498

9.01E-19

c(49)

0.012438

0.001323

9.400312

5.06E-17

c(50)

−0.02672

0.002535

−10.5384

4.19E-20

c(51)

1.079729

0.012712

84.93893

2.99E-136

c(52)

−1.19794

0.103191

−11.6089

4.67E-23

c(53)

0.660945

0.10392

6.360122

1.97E-09

c(54)

0.175152

0.019104

9.168604

2.09E-16

c(55)

−0.31296

0.031561

−9.91593

2.08E-18

c(56)

0.292367

0.030034

9.734402

6.42E-18

c(57)

1.046782

0.020545

50.9516

1.23E-101

c(58)

−0.78373

0.015172