# Inventories dynamics and sectoral propagation mechanisms an empirical study with French and American data

Mélika Ben Salem, Jean-François Jacques

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Mélika Ben Salem, Jean-François Jacques. Inventories dynamics and sectoral propagation mechanisms an empirical study with French and American data. 2025. halshs-04986999

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CAHIERS ECO & MATHS

96.34

INVENTORIES DYNAMICS AND
SECTORAL PROPAGATION MECHANISMS
AN EMPIRICAL STUDY WITH FRENCH AND AMERICAN DATA

Mélika BEN SALEM

Jean-François JACQUES

MAD
# Inventories dynamics and sectoral propagation mechanisms

An empirical study with French and American data*

Mélika Ben Salem†

MAD (Université Paris I) and Université d'Evry-Val d'Essonne

<figure><img src="image_2.png" /></figure>

Jean-François Jacques‡

MAD (Université Paris I) and Université de Paris IX

First draft February 1995 this draft March 1996

## Abstract

The aim of this paper is to reveal empirically two kinds of mechanisms which operate on inventories: one is demand related and the other is supply related. Accordingly, we first test the presence of cointegration relationships in the dynamic bivariate inventories-sales model then we perform Granger-causality tests. It allows for a full specification of the dynamic properties of these series.

Our results suggest that inventories movements stem from demand factors. Moreover, inventories and sales are linked in the long run, especially in the trade sectors. In France the dataset used prevents us from concluding in this way.

**Keywords : inventories, business cycle, causality**

## Résumé

Cet article se propose d'étudier empiriquement l'existence de deux types de mécanismes inhérents aux stocks, l'un dépendant de la demande et l'autre de l'offre. Dans cette perspective, nous testons d'abord la présence de relations de cointégration dans le modèle bivarié stocks-ventes, ensuite nous appliquons les tests de causalité au sens de Granger. Cette démarche permet de spécifier complètement les propriétés dynamiques de ces variables.

Nos résultats montrent que les fluctuations de stocks sont engendrées par les déterminants de la demande. De plus, les stocks et les ventes sont dépendantes à long terme, notamment dans les secteurs de la distribution. Seulement, la base de données françaises ne nous permet pas d'établir clairement ce résultat.

**Mots-clés : stocks, cycle, causalité**

JEL: E22 E32

<footnote>*We thank P. Fève, S. Grégoir and F. Postel-Vinay for useful comments. A previous version of this paper has also benefited from discussions during presentations at the MAD workshop and at the 7th World Congress of the Econometric Society, Tokyo, August 1995. All remaining mistakes are of our owns.</footnote>

<footnote>†M.A.D., unité associée CNRS-URA 0926, Université de Paris I, 90 rue de Tolbiac, 75634 Paris Cedex 13, France, email: bensalem@univ-paris1.fr.</footnote>

<footnote>‡J.F. Jacques thanks the CREST for financial support.</footnote>
Introduction

It is well known since Abramovitz [1950] that inventory movements are dominant fea-
tures of business cycles. It is often accepted (Maccini [1991]) that inventories transmit
shocks to the economy and amplify them without being at the origin of the business
cycles. For example, Blinder and Fisher [1981] argue that inventories play a direct role
in the propagation of monetary shocks. Not many empirical studies directly inquire
about inventories as a variable transmitting aggregate shocks to production. The main
concern of these contributions is to test the implications of the linear quadratic model.
For two reasons, we think that the relevant framework for this research is sectoral: it
allows for identification of precise nature of the transmission mechanism, and of the
sectors which play a major role in that mechanism. In this paper we focus on the for-
mer problem. We especially want to show that production or sales influence inventory
movements. In other words we ask the origin of the inventories fluctuations empirically.

As it has been said previously, there exist two mechanisms: one is demand related and the other is supply related.

The first is analysed in the production smoothing model. When the technology presents decreasing returns to scale and demand is a random variable, the firm tries to keep production constant, inventories serving as a buffer. In that case, the demand deplets inventories, which are then built up by production.

However, authors (such as Blinder [1986]) have shown that the empirical variance of production is higher than the one of sales. Hence they introduce supply shocks (a cost shock) in the model to reproduce this fact. The mechanism is the following: when a favorable supply shock occurs the producer accumulates inventories in order to benefit from these better conditions of production. The producer sells his production afterwards and the level of inventories decreases. Hence production could vary more than sales.

Without claiming to test any model, the aim of our empirical study is to detect one
of the sequences presented above. We use the Granger causality test to achieve that.
This definition of causality is related to the improvement of prediction. Consequently
the demand mechanism relies on forecast of production and inventories investment,
taking the past of sales into account. On the contrary the supply mechanism includes
the past of production, which is relevant to predict at best inventories investment and
sales. In both cases inventories are the link between sales and production, and could
amplify the fluctuations of production.

Our causality tests are performed using the Vector AutoRegression framework. A preliminary step consists of determining at best the joint dynamics of the variables. Indeed, Toda and Phillips [1991] have extended causality test to the case of the cointegrated variables. The cointegration results robustness allows first for checking the evidence of multicointegration as Granger and Lee [1990] show it on American data and in second place for testing this property on french data.

The first and second sections are devoted respectively to the univariate and biva-
riate specifications of the time series. We interpret the implications of such dynamics
representation in terms of causality. The last section carries out some economic inter-
# 1 Univariate analysis of inventories and sales dynamics

## 1.1 The data

French data are extracted from the “Comptes Nationaux Trimestriels” (Quaterly National Accounts) of INSEE. Inventories and sales are classified by branches, sixteen of them (see appendix 1 for a review of branches). French inventories cannot be distinguished according to the nature of the holder, whether producer or trader, contrarily to U.S. data. French data are quarterly, seasonally adjusted, measured in 1980 french francs and cover the period 1970:1 through 1993:2. The sample size is 93. Sales series result from subtracting inventories variation series from production series. The level of inventories is evaluated at the end of each quarter. Classification by branches is similar to a classification “by product”, regardless of who owns it.

U.S. data, final sales and inventories, are taken from the CITIBANK data base. Figures are monthly for the period 1967:1 to 1991:8 for final sales and level of inventory at the end of each month in 1982 constant dollars and seasonally adjusted. Production figures are obtained by adding inventories variation series to sales series. The sample size is 307 observations. The data are described in more detail further in the appendix 1.

Contrarily to the classification by branches, classification by sectors allows for determining the holder's identity: a sector is a set of production (or trade) units which main activity is to produce (or to sell) a particular good.

## 1.2 Unit root tests

Since the paper of Nelson and Plosser [1982], unit root tests are usually applied to univariate time series before beginning the study of their joint dynamics. We use a sequential strategy including the Dickey-Fuller tests (ADF, LM) and the Schmidt-Phillips one, called D.F./S.P. in the tables 1, 2. The lag length has been selected using BIC and Hannan criteria.

The presence of a shift in the deterministic component has been also tested according to Andrews and Zivot [1992]'s strategy. The aim of applying such a test is to check the robustness of the unit root in these series. Contrarily to Perron [1989]'s test, the date of the break is estimated (see appendix 2 for the test's methodology.).

### 1.2.1 France

The values of table 1 are consistent with sales and inventories being I(1). Seeing that inventories seem to be a random walk, production is I(1) because it is the sum of a stationary variable (variations of inventories) and a non-stationary one (sales).

We find pervasive and robust evidence of unit roots in the French series. However, sales of household durable goods and farm inventories seem to be stationary because of the presence of a break in their deterministic trend.

### 1.2.2 U.S.A.

The results shown in table 2 are quite similar to the French case, seeing that inventories dynamics are characterized by a random walk, plus a deterministic trend. In most
<table><caption>Table 1: Univariate Dynamics of French data</caption><thead><tr><th colspan="4">Sales</th><th colspan="3">Inventories</th></tr><tr><th>Sectors</th><th>Lags</th><th>D.F./S.P.</th><th>Zivot-Andrews</th><th>Lags</th><th>D.F./S.P.</th><th>Zivot-Andrews</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>2</td><td>0</td><td>I(1)+T</td><td>I(1)</td><td>5</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>3</td><td>2</td><td>I(1)</td><td>I(1)</td><td>1</td><td>I(1)</td><td>I(1)</td></tr><tr><td>4</td><td>7</td><td>I(1)</td><td>I(0)+T<sub>86:1</sub></td><td>1</td><td>I(1)</td><td>I(1)</td></tr><tr><td>5</td><td>1</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>6</td><td>0</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>7</td><td>1</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)</td><td>I(1)</td></tr><tr><td>8</td><td>0</td><td>I(1)+T</td><td>I(1)</td><td>0</td><td>I(1)</td><td>I(1)</td></tr><tr><td>9</td><td>0</td><td>I(1)</td><td>I(1)</td><td>3</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>10</td><td>0</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(0)+C<sub>73:1</sub>+T<sub>73:1</sub></td></tr><tr><td>11</td><td>5</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>12</td><td>1</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>13</td><td>0</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>14</td><td>1</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr></tbody></table>

<table><caption>Table 2: Univariate Dynamics of U.S. data</caption><thead><tr><th colspan="4">Sales</th><th colspan="3">Inventories</th></tr><tr><th>Sectors</th><th>Lags</th><th>D.F./S.P.</th><th>Zivot-Andrews</th><th>Lags</th><th>D.F./S.P.</th><th>Zivot-Andrews</th></tr></thead><tbody><tr><td>1</td><td>2</td><td>I(1)+T</td><td>I(1)</td><td>3</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>2</td><td>3</td><td>I(1)+T</td><td>I(1)</td><td>3</td><td>I(1)</td><td>I(1)</td></tr><tr><td>3</td><td>5</td><td>I(1)</td><td>I(1)</td><td>3</td><td>I(1)</td><td>I(1)</td></tr><tr><td>4</td><td>0</td><td>I(1)</td><td>I(1)</td><td>1</td><td>I(1)</td><td>I(1)</td></tr><tr><td>5</td><td>1</td><td>I(1)</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>6</td><td>3</td><td>I(1)+T</td><td>I(1)</td><td>3</td><td>I(1)</td><td>I(0)+C<sub>80:2</sub>+T<sub>80:2</sub></td></tr><tr><td>7</td><td>4</td><td>I(1)+T</td><td>I(1)</td><td>3</td><td>I(1)</td><td>I(1)</td></tr><tr><td>8</td><td>5</td><td>I(1)</td><td>I(0)+C<sub>79:4</sub>+T<sub>79:4</sub></td><td>8</td><td>I(1)</td><td>I(1)</td></tr><tr><td>9</td><td>5</td><td>I(1)</td><td>I(0)+C<sub>78:8</sub>+T<sub>78:8</sub> *</td><td>13</td><td>I(1)</td><td>I(0)+T<sub>79:8</sub></td></tr><tr><td>10</td><td>3</td><td>I(1)</td><td>I(1)</td><td>4</td><td>I(1)</td><td>I(1)</td></tr><tr><td>11</td><td>9</td><td>I(1)</td><td>I(1)</td><td>2</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>12</td><td>1</td><td>I(1)+T</td><td>I(1)</td><td>4</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>13</td><td>2</td><td>I(1)+T</td><td>I(0)+T<sub>86:6</sub> *</td><td>0</td><td>I(1)</td><td>I(0)+C<sub>74:6</sub>+T<sub>74:6</sub></td></tr><tr><td>14</td><td>0</td><td>I(1)+T</td><td>I(1)</td><td>3</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>15</td><td>3</td><td>I(0)+T+C</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(0)+C<sub>73:2</sub>+T<sub>73:2</sub></td></tr><tr><td>16</td><td>1</td><td>I(1)+T</td><td>I(1)</td><td>3</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>17</td><td>9</td><td>I(1)</td><td>I(0)+T<sub>79:6</sub></td><td>1</td><td>I(1)</td><td>I(0)+C<sub>79:1</sub>+T<sub>79:1</sub></td></tr><tr><td>18</td><td>0</td><td>I(1)</td><td>I(1)</td><td>0</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>19</td><td>9</td><td>I(0)+T+C</td><td>I(1)</td><td>1</td><td>I(1)</td><td>I(0)+C<sub>75:11</sub>+T<sub>75:11</sub></td></tr><tr><td>20</td><td>5</td><td>I(1)+T</td><td>I(1)</td><td>6</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>21</td><td>5</td><td>I(0)+T+C</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(0)+T<sub>81:6</sub></td></tr><tr><td>22</td><td>1</td><td>I(0)+T+C</td><td>I(1)</td><td>14</td><td>I(1)</td><td>I(1)</td></tr><tr><td>23</td><td>1</td><td>I(1)+T</td><td>I(0)+C<sub>85:3</sub>+T<sub>85:3</sub></td><td>3</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>24</td><td>1</td><td>I(1)+T</td><td>I(1)</td><td>0</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>25</td><td>5</td><td>I(1)+T</td><td>I(1)</td><td>11</td><td>I(1)</td><td>I(1)</td></tr><tr><td>26</td><td>2</td><td>I(1)+T</td><td>I(1)</td><td>1</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>27</td><td>2</td><td>I(1)</td><td>I(1)</td><td>1</td><td>I(1)</td><td>I(1)</td></tr><tr><td>28</td><td>7</td><td>I(1)+T</td><td>I(1)</td><td>7</td><td>I(1)+T</td><td>I(1)</td></tr><tr><td>29</td><td>2</td><td>I(1)+T</td><td>I(1)</td><td>0</td><td>I(1)+T</td><td>I(1)</td></tr></tbody></table>

* : significant level of test of 10%
sectors, sales and inventories remain integrated of order one. For inventories, the break in the deterministic trend has happened at dates known to be at the start of a recession. Indeed inventories seem to be reduced dramatically during times of recession as if the inventory holders had benefited from it. Nevertheless, unit root tests performed on sub periods lead to accept the unit root hypothesis. This is why the I(1) hypothesis is accepted on the whole period.

## 2 Bivariate analysis of inventories-sales dynamics

Accounting for identity between inventories variations, sales and production, corresponds in the time series framework to a first cointegration relationship. Moreover, we can test a deeper level of cointegration between inventories levels and sales, which is called multicointegration.

To take this phenomenon into account, Granger and Lee [1989] have proved another representation theorem for the production-sales dynamics than the Engle and Granger [1987]'s one. Gregoir and Laroque [1990] broaden the field of its application and show that it is easier to study the bivariate inventory-sales dynamics.

### 2.1 Cointegration

The residual-based test of Engle and Granger [1987] for cointegration does not implement serial correlation corrections which are used for ruling out nuisance parameters under the null of no cointegration. For this reason, we apply two statistics proposed by Phillips and Ouiaris [1990], whose limiting distributions are free of nuisance parameters (see appendix 4).

#### 2.1.1 France

<table><caption>Table 3: Cointegration (Demeaned) on French data</caption><thead><tr><th rowspan="2">Sectors</th><th colspan="2">Sales on Inventories</th><th colspan="2">Inventories on sales</th></tr><tr><th>З̂<sub>α</sub></th><th>З̂<sub>t</sub></th><th>З̂<sub>α</sub></th><th>З̂<sub>t</sub></th></tr></thead><tbody><tr><td>1</td><td>-11.24</td><td>-2.39</td><td>-10.73</td><td>-2.31</td></tr><tr><td>2</td><td>-2.95</td><td>-0.74</td><td>-2.26</td><td>-0.54</td></tr><tr><td>3</td><td>-3.93</td><td>-1.69</td><td>-6.25</td><td>-1.77</td></tr><tr><td>4</td><td>-4.97</td><td>-1.14</td><td>-10.25</td><td>-2.04</td></tr><tr><td>5</td><td>-3.37</td><td>-1.32</td><td>-4.25</td><td>-1.72</td></tr><tr><td>6</td><td>-6.53</td><td>-1.56</td><td>-7.55</td><td>-1.82</td></tr><tr><td>7</td><td>-8.08</td><td>-2.06</td><td>-8.56</td><td>-2.18</td></tr><tr><td>8</td><td>-7.81</td><td>-1.85</td><td>-10.18</td><td>-2.25</td></tr><tr><td>9</td><td>-12.97</td><td>-2.72</td><td>-17.02</td><td>-3.05</td></tr><tr><td>10</td><td>-12.13</td><td>-2.66</td><td>-10.10</td><td>-2.21</td></tr><tr><td>11</td><td>-1.66</td><td>-0.67</td><td>-2.19</td><td>-0.88</td></tr><tr><td>12</td><td>-7.07</td><td>-1.86</td><td>-7.90</td><td>-2.05</td></tr><tr><td>13</td><td>-6.12</td><td>-1.80</td><td>-6.87</td><td>-2.04</td></tr><tr><td>14</td><td>-11.90</td><td>-2.51</td><td>-11.39</td><td>-2.38</td></tr></tbody></table>

The bold-faced statistics mean cointegration

* : significant level of test of 10%

The hypothesis of no cointegration cannot be rejected at the aggregate level. There is no multicointegration in French data.
<table><caption>Table 4: Cointegration (Demeaned)</caption><thead><tr><th rowspan="2">Sectors</th><th colspan="2">Sales on Inventories</th><th colspan="2">Inventories on sales</th></tr><tr><th>Ẑ<sub>α</sub></th><th>Ẑ<sub>t</sub></th><th>Ẑ<sub>α</sub></th><th>Ẑ<sub>t</sub></th></tr></thead><tbody><tr><td>1</td><td>-7.73</td><td>-1.88</td><td>-8.33</td><td>-2.06</td></tr><tr><td>2</td><td>-1.58</td><td>-0.58</td><td>-3.89</td><td>-1.49</td></tr><tr><td>3</td><td>-2.59</td><td>-0.85</td><td>-3.81</td><td>-1.57</td></tr><tr><td>4</td><td>-15.95</td><td>-2.89</td><td>-5.95</td><td>-1.72</td></tr><tr><td>5</td><td>-12.06</td><td>-2.30</td><td>-4.38</td><td>-1.68</td></tr><tr><td>6</td><td>1.16</td><td>0.75</td><td>-2.20</td><td>-1.45</td></tr><tr><td>7</td><td>-3.69</td><td>-0.89</td><td>-4.96</td><td>-1.21</td></tr><tr><td>8</td><td>-34.94</td><td>-4.30</td><td>-21.80</td><td>-3.39</td></tr><tr><td>9</td><td>-19.12*</td><td>-3.12*</td><td>-9.12</td><td>-2.15</td></tr><tr><td>10</td><td>-67.44</td><td>-6.04</td><td>-50.85</td><td>-5.31</td></tr><tr><td>11</td><td>-3.77</td><td>-1.08</td><td>-4.69</td><td>-1.41</td></tr><tr><td>12</td><td>-3.48</td><td>-1.09</td><td>-5.91</td><td>-1.79</td></tr><tr><td>13</td><td>-2.27</td><td>-0.76</td><td>-7.02</td><td>-2.05</td></tr><tr><td>14</td><td>-6.76</td><td>-1.70</td><td>-7.53</td><td>-2.01</td></tr><tr><td>15</td><td>-29.48</td><td>-3.89</td><td>-27.08</td><td>-3.69</td></tr><tr><td>16</td><td>-19.24*</td><td>-3.16</td><td>-18.86</td><td>-3.21</td></tr><tr><td>17</td><td>-12.86</td><td>-2.50</td><td>-12.26</td><td>-2.51</td></tr><tr><td>18</td><td>-5.95</td><td>-1.66</td><td>-6.46</td><td>-1.95</td></tr><tr><td>19</td><td>-1.61</td><td>-0.58</td><td>-5.40</td><td>-1.83</td></tr><tr><td>20</td><td>-29.46</td><td>-3.96</td><td>-29.17</td><td>-3.95</td></tr><tr><td>21</td><td>-11.02</td><td>-2.37</td><td>-10.87</td><td>-2.38</td></tr><tr><td>22</td><td>-24.22</td><td>-3.58</td><td>-23.45</td><td>-3.49</td></tr><tr><td>23</td><td>-27.49</td><td>-3.81</td><td>-26.85</td><td>-3.71</td></tr><tr><td>24</td><td>-38.63</td><td>-4.59</td><td>-37.58</td><td>-4.51</td></tr><tr><td>25</td><td>-51.67</td><td>-5.36</td><td>-50.97</td><td>-5.31</td></tr><tr><td>26</td><td>-59.20</td><td>-5.73</td><td>-57.60</td><td>-5.66</td></tr><tr><td>27</td><td>-71.06</td><td>-6.32</td><td>-65.28</td><td>-6.03</td></tr><tr><td>28</td><td>-9.91</td><td>-2.37</td><td>-8.95</td><td>-2.13</td></tr><tr><td>29</td><td>-15.47</td><td>-2.79</td><td>-11.85</td><td>-2.21</td></tr></tbody></table>

The bold-faced statistics mean cointegration

* : significant level of test of 10%

The results available in table 4 contrast with those obtained in the French case: intermediate and trade sectors present inventories cointegrated with their sales. Manufacturing sectors as machines (6 and 7), and other durable goods (11), are not characterized by a stable relation between inventories and sales. The difference with French data can be explained by the fact that we cannot distinguish between producers and sellers who hold the inventories. Therefore we cannot find any evidence of cointegration in French data if the behavior of manufacturers and that of retailers are in fact different.

This cointegration analysis shows two kinds of behavior: one for the trade sectors and intermediate goods industries, and another one particular to manufacturing sectors. Inventories of the former sectors can be explained by supplying guests first. Accordingly levels of inventories and sales should be linked together to avoid stockout.

Our result can be compared with those of Granger and Lee [1989] inasmuch as we used the same American data set. They find evidence of multicointegration at the aggregate level rather than at the disaggregate one. Inventories and sales (or inventories
and production) are cointegrated according to Granger and Lee [1989] in the sectors (2), (20) and (25), although the former two sectors are a little bit surprising with cointegration residuals being not I(0)<fnref n="1" />. They point out the lack of relevance of such results: cointegration in the aggregate sector implies cointegration at a less aggregate level. Moreover, they accept multicointegration in five sectors only, (1), (3), (7), (16), (29), whereas our analysis leads to find cointegration for other sectors ((8), (9), (10), (15), (16), (20), (22), (23), (24), (25), (26), (27)). This difference is perhaps due to the corrected-bias term added to the ADF statistics, chosen optimally by applying a Bartlett window to whiten the autocorrelated residuals.

## 2.2 Causality

In the multivariate framework described below, Granger causality tests are a relevant way to reveal intra-sectoral propagation mechanisms. Two mechanisms can be distinguished:

(A) a production-smoothing mechanism stemming from inventories if the following causal pattern is identified:

$$Sales \rightarrow Inventories \rightarrow Production$$

(B) a *contrario* a sales-smoothing mechanism stemming from inventories in the following case:

$$Production \rightarrow Inventories \rightarrow Sales$$

The first mechanism refers to the inventory production-smoothing model (see Blinder [1986]). Production smoothing arises from variable demand and rising marginal costs. This model results in depleting inventories, when a demand shock occurs, to build them up steadily afterwards. The second mechanism can indicate a lack of production-smoothing. The origin of this lack is either fixed costs in the technology or favorable supply shocks.

At this stage, we must draw attention to two great matters that clearly limit our interpretation. The first is of economic relevance, the second is of statistic.

**Other mechanisms** First of all, another interpretation of the mechanism (B) has been advocated by Blinder [1986]. Suppose a firm expects futures sales. In that case it wants to build inventories. So past productions help predicting future demand. This mechanism is not production related as we argue, but rather demand related.

Other causal patterns are possible. For example, for case (A), no causality running from inventories to production means that production-smoothing mechanism is efficient because production evolves independently from demand variations. Beason [1993] implements a test of the production smoothing mechanism based on this idea.

In case (B), no causality between sales and inventories is possibly interpreted as efficient demand smoothing inasmuch as demand is independent from productivity shocks.

Howether, we show in the next paragraph that it is not relevant to split up pieces from the whole causality chain (A) or (B). Efficient smoothing is then a false interpretation of the operating mechanism not based on the whole causality pattern. Indeed, pieces of the causality chains are held in by both mechanisms which makes what we call *induced causality* arise.

<footnote n="1">¹See their ADF statistics of table I for columns $w_{2,t}$ et $u_{2,t}$, p. S150.</footnote>
**Induced Causality** Because variables are linked by an accounting identity, *induced causality* can occur.

The causality tests are implemented in the bivariate error correction model production-inventories series<fnref n="2" />, with one lag. It has the following form<fnref n="3" />:

$$ \Delta P_t = \mu_{11}(\alpha P_{t-1} + \beta I_{t-1}) + \mu_{12}\Delta P_{t-1} + \mu_{13}\Delta I_{t-1} + u_{1t} $$

$$ \Delta I_t = \mu_{21}(\alpha P_{t-1} + \beta I_{t-1}) + \mu_{22}\Delta P_{t-1} + \mu_{23}\Delta I_{t-1} + u_{2t} $$

The VECM in terms of sales and inventories is easily inferred from that defined below through the identity $P = S + \Delta I$:

$$ \Delta S_t = (\mu_{11} - \mu_{21})(\alpha S_{t-1} + \beta I_{t-1}) + (\mu_{12} - \mu_{22})\Delta S_{t-1} + (\mu_{13} + \mu_{12} + 1 - \mu_{23} - \mu_{22} + \alpha(\mu_{11} - \mu_{21}))\Delta I_{t-1} + (\mu_{22} - \mu_{12})\Delta I_{t-2} + u_{1t} $$

$$ \Delta I_t = \mu_{21}(\alpha S_{t-1} + \beta I_{t-1}) + \mu_{22}\Delta S_{t-1} + (\mu_{23} + \mu_{22} + \alpha\mu_{21})\Delta I_{t-1} + \mu_{22}\Delta I_{t-2} + u_{2t} $$

If we assume either $\mu_{21} \neq \mu_{22}$, or $\mu_{12} + 1 - \mu_{23} - \mu_{22} - \alpha\mu_{21} \neq 0$ or $\mu_{21} \neq 0$, we obtain the following results:

* We accept no causality from inventories to production if and only if we reject no causality from inventories to sales.

* We reject no causality from production to inventories if and only if we reject no causality from sales to inventories.

Accordingly we accept the “(A) mechanism”, estimating the bivariate system production-inventories, if and only if we reject the hypothesis of no causality from production to inventories and from inventories to production; we accept “mechanism (B)” if and only if we reject the hypothesis of no causality from production to inventories and accept the hypothesis of no causality from inventories to production.

We perform this testing strategy over both systems (production-inventories) and (sales-inventories) to check the robustness of our result. We do not conclude in the event of non similar results (which happens probably when the restrictions over parameters are not met).

In spite of these drawbacks, the analysis of causality brings out some results that deserve economic interpretation. Indeed we insist on our empirical approach: we do not use the Granger causality test to establish that the configuration of estimated parameters is implied by a specific model such as the linear quadratic one, but rather to identify the variables which improve forecast of variables.

The causality tests are derived from the strategy of Toda and Phillips [1991], described in appendix 5.

### 2.2.1 France

The results in table 5 show rather a causality running from sales to inventories. We identify this result as the (A) case prevailing in the following sectors : automobile

<footnote n="2">²This representation is equivalent to that of sales and inventories.</footnote>

<footnote n="3">³We have shown that this result can be easily extended with a VECM including more lags.</footnote>
<table><caption>Table 5: Causality (without cointegration) on French data</caption><thead><tr><th>Sectors</th><th>Lags</th><th>Prod. ⇒ Inv.</th><th>Inv. ⇒ Prod.</th><th>Mechanism</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>2</td><td>73</td><td>B</td></tr><tr><td>2</td><td>1</td><td>5.8*</td><td>0</td><td>A</td></tr><tr><td>3</td><td>1</td><td>27</td><td>0</td><td></td></tr><tr><td>4</td><td>1</td><td>11</td><td>27</td><td></td></tr><tr><td>5</td><td>1</td><td>0</td><td>5.8*</td><td>A</td></tr><tr><td>6</td><td>1</td><td>16</td><td>81</td><td></td></tr><tr><td>7</td><td>1</td><td>2</td><td>27</td><td>B</td></tr><tr><td>8</td><td>1</td><td>10*</td><td>0</td><td>A</td></tr><tr><td>9</td><td>1</td><td>4</td><td>6*</td><td>A</td></tr><tr><td>10</td><td>1</td><td>40</td><td>0</td><td></td></tr><tr><td>11</td><td>1</td><td>0</td><td>17</td><td>B</td></tr><tr><td>12</td><td>1</td><td>76</td><td>7*</td><td></td></tr><tr><td>13</td><td>1</td><td>12</td><td>0</td><td></td></tr><tr><td>14</td><td>1</td><td>1</td><td>8*</td><td>A</td></tr></tbody></table>

The bold-faced statistics mean causality

* : significant level test of 10%

(2), durable goods (5), intermediate goods (8), energy (9). The pattern (B) can be identified in the following sectors: industry and trade (1), goods and services (7) and farm products (11).

In conclusion, it is possible to advocate in favor of a propagation mechanism of aggregate shocks, conveying through the inventory variable, especially for the manufacturing and intermediate goods sectors. At the aggregate level ((1) and (7)), inventories dynamics is lead by production.

### 2.2.2 U.S.A.

<table><caption>Table 6: Causality (without cointegration) on U.S. data</caption><thead><tr><th>Sectors</th><th>Lags</th><th>Prod. ⇒ Inv.</th><th>Inv. ⇒ Prod.</th><th>Mechanism</th></tr></thead><tbody><tr><td>1</td><td>3</td><td>0</td><td>0</td><td>A</td></tr><tr><td>2</td><td>3</td><td>0</td><td>0</td><td></td></tr><tr><td>3</td><td>3</td><td>0</td><td>0</td><td>A</td></tr><tr><td>4</td><td>1</td><td>58</td><td>20</td><td></td></tr><tr><td>5</td><td>1</td><td>15</td><td>0</td><td></td></tr><tr><td>6</td><td>3</td><td>58</td><td>4</td><td></td></tr><tr><td>7</td><td>3</td><td>43</td><td>2</td><td></td></tr><tr><td>11</td><td>1</td><td>4</td><td>0</td><td></td></tr><tr><td>12</td><td>1</td><td>22</td><td>0</td><td></td></tr><tr><td>13</td><td>1</td><td>54</td><td>13</td><td></td></tr><tr><td>18</td><td>1</td><td>4</td><td>0</td><td>A</td></tr><tr><td>29</td><td>1</td><td>55</td><td>0</td><td></td></tr></tbody></table>

The bold-faced statistics mean causality

* : significant level test of 10%

Causality tests, which provide the results presented in tables 6 and 7, are more difficult to implement, due to possible multicointegration in different sectors. We implement the two-steps strategy of Toda and Phillips [1991] to consider the effects of such a phenomenon. Accordingly, in the error correction model, the joint significance of the past regressors expressed in differentiated terms is tested by a standard Fisher test in
<table><caption>Table 7: Causality (with Demeaned Cointegration) on U.S. data</caption><thead><tr><th rowspan="3">Sectors</th><th rowspan="3">Lags</th><th colspan="8">Causality between Inventories and Production</th><th rowspan="3">Mechanism</th></tr><tr><th colspan="4">Production ⇒ Inventories</th><th colspan="4">Inventories ⇒ Production</th></tr><tr><th colspan="2">Res1</th><th colspan="2">Res2</th><th colspan="2">Res1</th><th colspan="2">Res2</th></tr><tr><th></th><th></th><th></th><th></th><th></th><th>SR</th><th>LR</th><th>SR</th><th>LR</th><th>SR</th><th>LR</th><th></th></tr></thead><tbody><tr><td>8</td><td>1</td><td>56</td><td>0</td><td>35</td><td>0</td><td>0</td><td>41</td><td>0</td><td>19</td><td></td></tr><tr><td>9</td><td>1</td><td>83</td><td>0</td><td>77</td><td>0</td><td>0</td><td>6.5*</td><td>0</td><td>19</td><td>A</td></tr><tr><td>10</td><td>2</td><td>36</td><td>0</td><td>39</td><td>0</td><td>0</td><td>66</td><td>0</td><td>55</td><td>A</td></tr><tr><td>15</td><td>2</td><td>20</td><td>0</td><td>19</td><td>0</td><td>0</td><td>10*</td><td>0</td><td>22</td><td>A</td></tr><tr><td>16</td><td>1</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>26</td><td>0</td><td>61</td><td></td></tr><tr><td>20</td><td>1</td><td>46</td><td>0</td><td>47</td><td>0</td><td>0</td><td>90</td><td>0</td><td>96</td><td></td></tr><tr><td>22</td><td>1</td><td>36</td><td>9*</td><td>36</td><td>10*</td><td>0</td><td>19</td><td>0</td><td>17</td><td>A</td></tr><tr><td>23</td><td>1</td><td>65</td><td>2.5</td><td>68</td><td>3.5</td><td>0</td><td>46</td><td>0</td><td>29</td><td></td></tr><tr><td>24</td><td>1</td><td>31</td><td>4</td><td>31</td><td>3.5</td><td>9*</td><td>14</td><td>10*</td><td>12</td><td>A</td></tr><tr><td>25</td><td>1</td><td>47</td><td>0</td><td>47</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>A</td></tr><tr><td>26</td><td>1</td><td>84</td><td>0</td><td>83</td><td>0</td><td>0</td><td>4</td><td>0</td><td>7*</td><td></td></tr><tr><td>27</td><td>1</td><td>69</td><td>0</td><td>70</td><td>0</td><td>0</td><td>4</td><td>0</td><td>14</td><td></td></tr></tbody></table>

The bold-faced statistics mean causality

* : significant level test of 10%

each equations. There is no causality if both statistics cannot allow to reject the null hypothesis.

Given the arbitrary choice of the cointegration regression, we have done both regressions: production on inventories and inventories on production, which respectively result in residuals res1 and res2. To test causality, we have estimated the bivariate error correction model in production and inventories, including the possible error correction term (res1 or res2).

Sectors without cointegration are those where there is a production smoothing pattern, like other durable goods (11), rubber (18), and aggregate sectors (1), (2) and (3). For sectors with cointegration on the whole sample, mechanism (A) prevails.

This analysis of causality could help investigating the channels of propagation of aggregate shocks in various sectors of the economy. As a matter of fact, causality runs from trade sectors to manufacturing ones, and consequently from the latter to intermediate goods sectors. Indeed it is of some importance to point out the case of retailers, whose inventories are lead by production.

## 3 Conclusion

Inventories play a major role in economic fluctuations. Many authors argue that inventories investment is the source of instability in the cycle (for example Blinder and Maccini [1991]). This paper is an attempt to identify what kind of mechanisms plays on inventories. To address this question, we implement Granger causality test in the vector autoregressive framework.

The first step consists in looking for cointegration relationships between inventories and production. Contrary to Granger and Lee [1990] we find long run relationships in trade sectors rather than in aggregate sectors.

The second step is related to the shocks propagation mechanism through sectors. It seems that past sales help predicting future productions through inventories fluctuations in most American sectors. In France the results are more contrasted: this mechanism prevails only in automobile, durable goods, intermediate goods and energy sectors.
The other mechanism (supply mechanism) arises in the following sectors: industry and trade, goods and services and farm products, mostly aggregate sectors.

Notwithstanding the obvious limitations of our analysis, the causality results suggest that the inventories are likely to be the link between sectors.
# Appendix 1: the data

## 3.1 French branchs

1) Manufacturing and trade

2) Automobile and transport

3) Building (civil and farm infrastructure)

4) Household durable goods

5) Industrial durable goods

6) Consumer goods

7) Trading goods and service

8) Intermediate goods

9) Energy

10) Farm products

11) Other farm products

12) Industrial product

13) Manufacturing product

14) Non-Industrial product

## 3.2 U.S. sectors

1) Manufacturing and trade (2 + 20 + 25)

2) Manufacturing (3 + 12)

3) Durable goods manufacturing (4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)

4) Primary metal

5) Fabricated metal

6) Machinery (Except electrical)

7) Electrical machinery

8) Total transportation equipment (9 + 10)

9) Motors

10) Other equipment

11) Other durable good manufacturing

12) Non-durable goods manufacturing (13 + 14)

13) Foods and kindred products
14) Non food (15 + 16 + 17 + 18 + 19)

15) Paper and allied products

16) Chemicals and allied products

17) Petroleum and coal products

18) Rubber and plastic products

19) Other non-durable goods manufacturing

20) Merchant wholesalers (21 + 22)

21) Durable good wholesalers

22) Non-durable goods wholesalers (23 + 24)

23) Groceries and farm products wholesalers

24) Other non-durable good wholesalers

25) Retail trade (26 + 27)

26) Durable goods retailers (27 ⊂ 26)

27) Auto dealers

28) Non-durable goods retailers (29 ⊂ 28)

29) Food stores
## Appendix 2: Critical values for unit roots tests

The critical values are extracted from Dickey and Fuller [1979] and Dickey and Fuller [1981].

<table><thead><tr><th colspan="9">France (sample size 100)</th></tr><tr><th>Significant level</th><th>Eq. (1)</th><th>t<sub>β</sub></th><th>t<sub>β̂</sub></th><th>t<sub>δ</sub></th><th>t<sub>δ̂</sub></th><th>Φ<sub>1</sub></th><th>Φ<sub>2</sub></th><th>Φ<sub>3</sub></th><th>Schmidt-Phillips</th></tr></thead><tbody><tr><td>5%</td><td>-3.45</td><td>-2.89</td><td>-1.95</td><td>1.96</td><td>1.96</td><td>4.71</td><td>4.88</td><td>6.49</td><td>-3.04</td></tr><tr><td>10%</td><td>-3.15</td><td>-2.58</td><td>-1.61</td><td>1.60</td><td>1.60</td><td>3.86</td><td>4.16</td><td>5.47</td><td>-2.76</td></tr></tbody><tbody><tr><th colspan="9">U.S.A. (sample size 250)</th></tr><tr><td>5%</td><td>-3.42</td><td>-2.87</td><td>-1.95</td><td>1.96</td><td>1.96</td><td>4.61</td><td>4.71</td><td>6.30</td><td>-3.06</td></tr><tr><td>10%</td><td>-3.13</td><td>-2.57</td><td>-1.62</td><td>1.60</td><td>1.60</td><td>3.79</td><td>4.05</td><td>5.36</td><td>-2.77</td></tr></tbody></table>

## Appendix 3: Unit root tests with a breaking trend alternative

Andrews and Zivot [1992] investigate the possibility raised by Perron [1989] that aggregate economic time series can be characterized as being stationary around broken trend lines. Unlike this author, they treat the break as unknown *a priori*. The three models considered by Andrews and Zivot [1992] are the “Crash model” (here model (A)) which provides information about whether there has been a jump in the trend under the alternative hypothesis, the changing growth model (model (B)) which reveals whether there has been a change in the slope of the trend, and the third model (C) including the two kinds of breaks. Under the null, the series $x_t$ is characterized by a random walk whereas under the alternative, it is defined in the three following ways :

$$ (A) \quad x_t = \mu_0 + \mu_1 t + \mu_2 C_t(\lambda) \qquad + \alpha x_{t-1} + \beta(L) \Delta x_{t-1} + \varepsilon_t \quad t = 1, \dots, T $$

$$ (B) \quad x_t = \mu_0 + \mu_1 t + \mu_3 T_t(\lambda) \qquad + \alpha x_{t-1} + \beta(L) \Delta x_{t-1} + \varepsilon_t \quad t = 1, \dots, T $$

$$ (C) \quad x_t = \mu_0 + \mu_1 t + \mu_2 C_t(\lambda) + \mu_3 T_t(\lambda) \qquad + \alpha x_{t-1} + \beta(L) \Delta x_{t-1} + \varepsilon_t \quad t = 1, \dots, T $$

where $\beta(L)$ is a lag polynomial of order p. These models allow for a single or at most two shifts in the deterministic trend at an unknown but identical date. The deterministic regressor $C_t(\lambda)$ captures a shift in mean at period $T\lambda$ :

$$ C_t(\lambda) = \begin{cases} 1 & \text{if } t > T\lambda \\ 0 & \text{else} \end{cases} $$

whereas $T_t(\lambda)$ defines a shift in trend (a change in slope) :

$$ T_t(\lambda) = \begin{cases} t - T\lambda & \text{if } t > T\lambda \\ 0 & \text{else} \end{cases} $$

Each model characterizes $x_t$ in the following manner :

<table><tr><td>Model (A)</td><td>I(0)+C<sub>T<sub>B</sub></sub></td></tr><tr><td>Model (B)</td><td>I(0)+T<sub>T<sub>B</sub></sub></td></tr><tr><td>Model (C)</td><td>I(0)+T<sub>T<sub>B</sub></sub>+C<sub>T<sub>B</sub></sub></td></tr></table>
The choice of the break date is a two-steps procedure. First compute the Student statistic of $\alpha$, $t_{\alpha}(\lambda)$, conditionnal to $\lambda$, in the test $H_0: \alpha = 1$ against $H_1: \alpha < 1$, for each $\lambda = \frac{1}{T}, \frac{2}{T}, \dots, 1$. The estimation of the break date relies on the minimisation of $t_{\alpha}(\lambda)$ over $\lambda$. The criteria is then the following :

$$t_{\alpha}[\lambda_{\inf}] = \inf_{\lambda \in \Lambda} t_{\alpha}(\lambda)$$

for each $\lambda$ in $[0,1]$

The critical values tabulated by Andrews and Zivot [1992] are higher than those computed conditionnally to a known break, and are independent of the choice of the shift date in the deterministic trend.

<table><caption>Table 8: Critical Values of Trend-Break Tests</caption><thead><tr><th>Sign. level</th><th>(A)</th><th>(B)</th><th>(C)</th></tr></thead><tbody><tr><td>5 %</td><td>-4.80</td><td>-4.42</td><td>-5.08</td></tr><tr><td>10 %</td><td>-4.58</td><td>-4.11</td><td>-4.82</td></tr></tbody></table>

## Appendix 4: Cointegration test

The empirical analysis of long run relationships between inventories levels and either sales or production is achieved in the residual-based tests framework by Engle and Granger [1987]. The test statistics, $Z_t$ and $Z_{\alpha}$ including serial correlation correction, are proposed by Phillips [1987]. Consider the following cointegration regressions :

$$I_t = a_1 + b_1 P_t + res1_t$$

$$P_t = a_2 + b_2 I_t + res2_t$$

$$I_t = a_3 + b_3 S_t + res3_t$$

$$S_t = a_4 + b_4 I_t + res4_t$$

where $I$ being inventories levels, $P$ production, and $S$ sales. The null hypothesis of no cointegration is tested by implementing unit root tests on the demeaned residuals $res1$, $res2$, $res3$ and $res4$. These residuals can be detrended in introducing a deterministic trend in each regression.

Once residuals estimated, the AR(1) : $res_t = \alpha res_{t-1} + k_t$, can be performed in substituting $res$ by $res1$, $res2$, $res3$ and $res4$. Then the two following statistics can be computed :

$$Z_{\alpha} = T(\alpha - 1) - \frac{1}{2}(s_{Tl}^2 - s_k^2) \left( \frac{\sum_{t=1}^{T} res_{t-1}^2}{T^2} \right)^{-1}$$ $$Z_t = \left( \sum_{t=1}^{T} res_{t-1}^2 \right)^{\frac{1}{2}} \frac{\alpha - 1}{s_{Tl}} - \frac{1}{2}(s_{Tl}^2 - s_k^2) \left[ s_{Tl} \left( \frac{\sum_{t=1}^{T} res_{t-1}^2}{T^2} \right)^{\frac{1}{2}} \right]^{-1}$$

where $s_k^2 = \frac{\sum_{t=1}^{T} k_{t-1}^2}{T}$

and $s_{Tl}^2 = \frac{\sum_{t=1}^{T} k_{t-1}^2}{T} + 2 \frac{\sum_{s=1}^{l} w_{sl} \sum_{t=s+1}^{T} k_t k_{t-s}}{T}$
<table><caption>Table 9: Critical Values of cointegration tests</caption><thead><tr><th rowspan="2">Significant level</th><th colspan="2">Z<sub>α</sub></th><th colspan="2">Z<sub>t</sub></th></tr><tr><th>Demeaned</th><th>Demeaned and Detrended</th><th>Demeaned</th><th>Demeaned and Detrended</th></tr></thead><tbody><tr><td>5%</td><td>-20.49</td><td>-32.22</td><td>-3.36</td><td>-4.15</td></tr><tr><td>10%</td><td>-17.03</td><td>-27.78</td><td>-3.06</td><td>-3.84</td></tr></tbody></table>

The weight for each autocovariance<fnref n="4" /> depends upon *l*, the Bartlett's window<fnref n="5" />, proposed by Andrews [1991]. This optimal lag is chosen according to the value of the auto-regressive coefficient in the following regression: $Δres_t = ρΔres_{t-1} + u_t$, where $res_t$ is any cointegration residual. The parameter $a(1)$ is computed as $\frac{4ρ^2}{(1-ρ)^2(1+ρ)^2}$.

Phillips and Ouiaris [1990] tabulate a full set of critical values. They allow for demeaned and detrended data and cointegration regressions with up to five variables.

<footnote n="4">⁴$w_{sl} = 1 - \frac{s}{l+1}$</footnote>

<footnote n="5">⁵$l = 1.1447(a(1)T)^{\frac{1}{3}} - 1$</footnote>
# Appendix 5: Causality tests

The causality tests rely on the statistical definition proposed by Granger, and are applied according to the strategy of Toda and Phillips [1991].

**Definition** : a variable Y causes a variable X if, given the past of X, the knowledge of the past of Y allows to improve the forecast ex ante of the current value of X.

$$E(X_t|X_{t-1}, Y_{t-1}) \neq E(X_t|X_{t-1})$$

$$E(X_t|X_{t-1}, Y_{t-1}) \neq E(X_t|X_{t-1}, Y_t)$$

## Two-steps strategy of tests

1. Regress $\Delta X_t$ on its past, the past of $\Delta Y_t$ and the possible cointegration relationship ($\alpha X_{t-1} + \beta Y_{t-1}$):

$$\Delta X_t = \mu_1 + \mu_2(\alpha X_{t-1} + \beta Y_{t-1}) + \sum_{i=1}^{p} \mu_{1i} \Delta X_{t-i} + \sum_{i=0}^{p} \mu_{2i} \Delta Y_{t-i} + u_{1t}$$

2. Regress $\Delta Y_t$ on its past, the past of $\Delta X_t$ and the possible cointegration relationship ($\alpha X_{t-1} + \beta Y_{t-1}$):

$$\Delta Y_t = \mu_3 + \mu_4(\alpha X_{t-1} + \beta Y_{t-1}) + \sum_{i=1}^{p} \mu_{3i} \Delta X_{t-i} + \sum_{i=0}^{p} \mu_{4i} \Delta Y_{t-i} + u_{2t}$$

Then compute the T-statistics in the following cases:

1. $H_0: \mu_{2i} = 0$. If we reject $H_0$, we reject no causality from Y to X. If not, you must do the second step.

2. $H_0: \mu_2 = 0$. If we reject $H_0$, we reject no causality from Y to X.

This strategy is valid under the hypothesis of a well-specified cointegration relationship. This condition is satisfied by the exclusion tests achieved in the Johansen and Juselius [1990]'s procedure.
# Appendix 6

<table><caption>Table 10: Cointegration (Demeaned and Detrended) on American data</caption><thead><tr><th rowspan="2">Sectors</th><th colspan="2">Sales on Inventories</th><th colspan="2">Inventories on sales</th></tr><tr><th>Ẑ<sub>α</sub></th><th>Ẑ<sub>t</sub></th><th>Ẑ<sub>α</sub></th><th>Ẑ<sub>t</sub></th></tr></thead><tbody><tr><td>1</td><td>-8.13</td><td>-2.02</td><td>-7.07</td><td>-1.61</td></tr><tr><td>2</td><td>-10.20</td><td>-2.27</td><td>-4.76</td><td>-1.44</td></tr><tr><td>3</td><td>-8.80</td><td>-2.12</td><td>-5.80</td><td>-1.51</td></tr><tr><td>4</td><td>-24.18</td><td>-3.56</td><td>-4.44</td><td>-1.57</td></tr><tr><td>5</td><td>-21.18</td><td>-3.29</td><td>-3.53</td><td>-1.23</td></tr><tr><td>6</td><td>-6.54</td><td>-1.86</td><td>-2.09</td><td>-0.78</td></tr><tr><td>7</td><td>-9.63</td><td>-2.18</td><td>1.76</td><td>0.68</td></tr><tr><td>8</td><td>-38.41</td><td>-4.52</td><td>-8.61</td><td>-2.10</td></tr><tr><td>9</td><td>-55.32</td><td>-5.49</td><td>-21.15</td><td>-3.36</td></tr><tr><td>10</td><td>-67.48</td><td>-6.04</td><td>-34.38</td><td>-4.29</td></tr><tr><td>11</td><td>-11.10</td><td>-2.40</td><td>-3.26</td><td>0.93</td></tr><tr><td>12</td><td>-15.96</td><td>-2.85</td><td>-4.82</td><td>-1.65</td></tr><tr><td>13</td><td>-42.26</td><td>-4.77</td><td>-9.94</td><td>-2.35</td></tr><tr><td>14</td><td>-11.88</td><td>-2.48</td><td>-7.19</td><td>-1.95</td></tr><tr><td>15</td><td>-38.57</td><td>-4.42</td><td>-10.04</td><td>-2.19</td></tr><tr><td>16</td><td>-17.55</td><td>-3.02</td><td>-11.96</td><td>-2.64</td></tr><tr><td>17</td><td>-14.71</td><td>-2.83</td><td>-11.74</td><td>-2.39</td></tr><tr><td>18</td><td>-12.48</td><td>-2.53</td><td>-3.02</td><td>-1.30</td></tr><tr><td>19</td><td>-34.43</td><td>-4.25</td><td>-4.87</td><td>-1.57</td></tr><tr><td>20</td><td>-28.58*</td><td>-3.91*</td><td>-22.64</td><td>-3.38</td></tr><tr><td>21</td><td>-14.06</td><td>-2.68</td><td>-6.91</td><td>-1.69</td></tr><tr><td>22</td><td>-52.01</td><td>-5.28</td><td>-13.80</td><td>-2.67</td></tr><tr><td>23</td><td>-26.36</td><td>-3.63</td><td>-22.55</td><td>-3.42</td></tr><tr><td>24</td><td>-71.43</td><td>-6.33</td><td>-31.60*</td><td>-4.13*</td></tr><tr><td>25</td><td>-61.81</td><td>-5.88</td><td>-48.14</td><td>-5.17</td></tr><tr><td>26</td><td>-57.70</td><td>-5.66</td><td>-30.14*</td><td>-4.06*</td></tr><tr><td>27</td><td>-69.76</td><td>-6.34</td><td>-27.32</td><td>-6.87</td></tr><tr><td>28</td><td>-19.21</td><td>-3.17</td><td>-7.49</td><td>-1.96</td></tr><tr><td>29</td><td>-24.99</td><td>-3.58</td><td>-6.25</td><td>-1.69</td></tr></tbody></table>
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