# European nominal and real convergences: joint process or rival dynamics?

Martine Carré

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Martine Carré. European nominal and real convergences: joint process or rival dynamics?. 1996. halshs-05106192

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Distributed under a Creative Commons Attribution 4.0 International License
CAHIERS ECO & MATHS

96.98

EUROPEAN NOMINAL AND REAL CONVERGENCES :
JOINT PROCESS OR RIVAL DYNAMICS ?

Martine CARRE

MAD
# European nominal and real convergences: joint process or rival dynamics?

Martine CARRE

MAD, Université de Paris I*†

First draft : February 1995

This draft : June 1996

## Abstract

Recent empirical studies reveal that the convergence speed of nominal and real variables are fairly different. In this paper, we study the temporal evolution of the mutual influence between the convergences of a nominal and a real variable. We first refer to σ-convergence analysis. In order to compare nominal and real paths, we connect the evolutions of the cross-section variances of the two variables. Second time, we study nominal and real convergence processes, as reciprocally conditioned within a same system. Nominal and real convergence are negatively correlated in the eighties, while the nineties are characterized by a double convergence movement.

**Key words:** Convergence, European integration

**JEL Citation Index:** C31,F15

## Résumé

Le débat induit par la ratification du traité de Maastricht a suscité différents travaux économétriques sur la convergence des économies européennes. Les résultats obtenus tendent à contraster les degrés de convergence respectifs des variables nominales et réelles. L'objet de cette contribution est d'étudier plus précisément l'évolution temporelle des différents effets croisés, qui peuvent s'exercer entre les dimensions réelles et nominales de la convergence. A cette fin, nous évaluons dans un premier temps, les mouvements suivis par les différents processus en terme de σ-convergence. Puis dans une dernière partie, nous estimons les deux types de convergence comme mutuellement conditionnés au sein d'un même système.

**Mots-Clefs :** Convergence, Intégration européenne

<footnote>*I thank P.Y. Hénin for helpful comments and suggestions. Moreover, I benefited from discussions with F. Collard, J.F. Jacques, and J.M. Tallon. I thank participants at the M.A.D. seminar, the A.F.S.E. Conference on European Integration (Nantes June 1995) and T2M Conference in Paris (January 1996). Errors and omissions remain my responsibility.</footnote>

<footnote>†Correspondence to M.A.D., Université Paris I, 90 rue de Tolbiac, 75634 Paris Cedex 13.</footnote>
# Introduction

Countries wishing to join the Economic and Monetary Union (E.M.U.) before the end of the 90s have to satisfy a certain number of convergence criteria specified in the Maastricht treaty. These criteria were elaborated in order for potential members to face a monetary union with similar structures, compatible with a common economic policy. Indeed, a common response to exogenous shocks could be inappropriate if the members are not close enough. In this view, convergence among European economies is a necessary stage before the final phase of a monetary union. The treaty implicitly underlines that exchange rates and price stability will favour growth and economic integration. Under this assumption, nominal convergence will lead to real convergence among European countries. The opposite, however, cannot be excluded *a priori*: nominal convergence could be a very costly process for the poorest European countries.

There have been a large number of empirical studies trying to assess whether European countries have converged. Some use the cointegration approach (Arde ni [1992]); others estimate models with variable parameters (Haldane and Hall [1991], Hall, Robertson and Wickens [1992]). These studies, whether using time series or cross section data, generally point to the existence of a movement of nominal con vergence together with real divergence during the 80s. A common feature of all these studies is that they analyze these two convergence processes independently. A con tribution of this paper is precisely to consider the relative evolution of nominal and real variables, in terms of conditional convergence. We propose a simultaneous exa mination of the convergence process for the nominal and real variables in a bivariate system. The study of a joint system thus enables us to empirically evaluate the link between nominal and real convergence.

The existence of such a link can have several theoretical justifications. As Blan chard and Muet [1993] argue, if and when European countries become part of a common currency area, “competitive disinflation” will be the only mechanism left to improve competitiveness. The convergence necessity expressed by the treaty seems to lead European countries to give up expansionnist and inflationary policies. But a country which fixes its exchange rate, starting with higher inflation than its tra ding partners, will initially further lose competitiveness and experience an increase in unemployment. Indeed, while inflation will decline, it will for some time remain higher than foreign inflation, thus leading to further real appreciation. Eventually however, inflation becomes less than foreign inflation, competitiveness will improve and demand will increase. Strategies of this type tend to be slow to give results. Furthemore, they seem, at this time, to be more successful at bringing down infla tion than at favoring economic growth. The one clearest gain from “competitiveness through disinflation” has been disinflation.

Barro [1995] argues that poor countries grow faster than rich countries and tend thereby to catch up, only if they have favorable settings for governement policies. He then shows empirically that although the adverse influence of inflation on growth looks small, the long term effects on standards of living are substantial<fnref n="1" />.

<footnote n="1">¹However, statistically significant results emerge only when high-inflation experiences are inclu-</footnote>
On the other hand, we might fear that the disinflationnist adjustment process be
painful when short-term unemployment becomes persistent. In the long run, one can
fear the appearance of high unemployment zones, in which long-term unemployed
would have a very low level of human capital. The long period of low activity might
depress investment, and slow down capital accumulation for a long time. This could
affect likewise the long-term growth trend of countries that were inflationnist at the
beginning. Cyclical G.N.P. depression could have persistent and structural effects.
Feldstein [1992] among others, notes that countries with the worst initial conditions
could well be driven to low growth paths with high unemployment. An increased
dispersion of unemployment rates will then, ceteris paribus, hinder the effect of
concerted policies, and could prevent countries with the highest rates to eventually
join the monetary union (Heylen, VanPoeck and VanGompel [1995]).

One can so wonder if the deflationary constraint imposed on country-members
does not constitute a threat to the process of real convergence during the transitory
phase. It is important to stress that, if nominal and real convergence are negatively
correlated, nominal constraints imposed by Maastricht could amplify real diver-
gences. In terms of economic policy, the issue is to choose between a policy that
would aim at nominal convergence, and a policy that would directly promote the
convergence of standards of living and activity levels, i.e. that would directly seek
real convergence.

The aim of this paper is not to test a particular theory, but rather to assess
the existence of a link between nominal and real convergence. We will focus on the
temporal evolution of the mutual influence between nominal and real dynamics. We
estimate a bivariate system including a real variable and a nominal one. We chose,
for the real variable, the logarithm of per capita G.N.P.. The nominal variable we
chose is the inflation rate<fnref n="2" />. The estimation of this system reveals the important
influence of the initial per capita G.N.P. level in the nominal convergence. In fact,
this effect has ceased to be significant when real divergence occured (in the beginning
of the eighties). Moreover, if we see again, in the nineties, a double convergence
movement, the two cross-effects between real and nominal convergences are not
significant anylonger.

In the first section, we define and comment the method used in this paper to
test nominal and real convergence. In the second section, we empirically evaluate
the paths followed by the different processes, using the σ-convergence methodology.
In order to analyze the link between nominal and real paths, we connect the cross-
section variances evolutions of the two variables in a same original figure. This
sets the scene for the last section, in which we estimate a system including both
nominal and real variables, using a two-equation model, which generalizes Barro's
equation for β-convergence. The two processes (real and nominal) are then seen as
joint processes in a unique bivariate-system. This method allows the estimation of
the mutual influences between the two types of movements. We then analyze their

ded in the sample.

<footnote n="2">²This is the nominal variable specified by the Maastricht treaty. However, in the long run, in a fixed exchange rate system, such as the E.M.U., convergence of inflation rates is not enough, and that of price levels is required.</footnote>
# 1 Interpreting convergence tests

In this section, we introduce and discuss the various concepts and techniques used in this paper. We then relate the different definitions of convergence to different empirical methods. This leads us to justify our choice to use cross-section empirical methods.

## 1.1 Defining and testing convergence

The notion of convergence can be given two different definitions (see Bernard and Dur- lauf [1996]). The first definition considers convergence as catching up, whereas the second definition sees it as the equality of long-run forecasts for countries at a fixed time.

In the sense of the first definition, countries *i* and *j* converge between dates *t* and *t* + *T* if disparity at date *t* of a variable - denoted *y*<sub>*i*,t*</sub> - is expected to decrease in value. In short, if *y*<sub>*i*,t*</sub> > *y*<sub>*j*,t*</sub>, then there is convergence between *t* and *t* + *T* if:

$$E(y_{i,t+T} - y_{j,t+T} | I_t) < y_{i,t} - y_{j,t}$$

where *I*<sub>*t*</sub> denotes all information available at *t*.

Countries *i* and *j* converge in the sense of the second definition if:

$$\lim_{k \to \infty} E(y_{i,t+k} - y_{j,t+k} | I_t) = 0$$

Cross-section methodologies are associated to the first definition. They consider the behavior of the difference of a given variable, say G.D.P., between two countries, over a fixed time interval and equates convergence with the tendency of this difference to narrow. In this line, the σ-convergence à la Barro [1991] is certainly the most intuitive approach. It requires the decrease over time of cross-dispersions for some given variables. So, if we let σ<sub>*t*</sub> be the empirical cross-variance of a variable<sup>3</sup>, we can observe a σ-convergence movement when the ratio σ<sub>*t*+*j*</sub>/σ<sub>*t*</sub> is less than to one, with *j* > 0. The second type of cross-section methodology considers the traditional β-convergence equation, coming from the neoclassical growth model. The focus is then the link between an economy's average growth and its initial income, through the β-convergence equation as Barro [1991] defined it:

$$T^{-1}(y_{j,T} - y_{j,0}) = a + by_{j,0} + u_{j,T} \quad (1)$$

$$\sigma_t = \left( N^{-1} \sum_{i=1}^{N} \left[ y_{i,t} - \left( N^{-1} \sum_{k=1}^{N} y_{k,t} \right) \right]^2 \right)^{1/2}$$

<footnote n="3">³σ<sub>*t*</sub> is the (N-sample-size) cross-section standard deviation, i.e.</footnote>
which tests the convergence of a variable, noted y, and where u is a white noise. β-convergence occurs when there exists a significant negative correlation between the initial level of a variable and its average rate of increase. Thus, the weaker the initial level of a variable is, the higher its average rate of increase will be.

The concepts of σ and β-convergence are clearly different. Quah [1993] show how it is possible for a negative relationship between initial income and growth to be compatible with a stable cross-section variance in output levels. We can note with Bernard and Durlauf [1996] that intuitively this can hold because shocks to country-specific growth rates can offset the effect of this negative coefficient. This can easily be proved, using the formulation of Lichtenberg [1994], by looking at the traditional β-convergence cross-regression :

$$y_{i,t+1} = (1 + \beta)y_{i,t} + u_i \quad (2)$$

where u is a white noise, and σ_u² its variance. Let σ_{t+1} (resp. σ_t) be the standard deviation of y_{t+1} (resp. y_t). One has : σ_{t+1}² = (1+β)²σ_t² + σ_u², and therefore : σ_t²/σ_{t+1}² = (1-(σ_u²/σ_{t+1}²)/(1+β)²)

The σ-convergence depends not only on the sign of β but also on the variance of the noise relatively to the variance of the estimated variable. Thus, β-convergence is meant to capture the idea of a long term convergence tendency, which can be disturbed momentarily by exogenous shocks. The σ-convergence analysis takes these shocks into account. This fact can be held against the σ-convergence concept, too sensitive to short-term shocks and thus hiding long-term links<fnref n="4" />.

Moreover, using a dynamic version of Galton's fallacy, Quah [1993] and Hart [1995] establish that a negative initial level coefficient in β-convergence regressions is consistent with an unchanging cross-section distribution. Starting from equation 2, where we let β' = 1 + β, we observe<fnref n="5" /> that :σ²_{t+1}/σ²_t = β'²/β², where β is the correlation coefficient between y_{i,t+1} and y_{i,t}.

The change in the variance of the variable can be then be decomposed into a regression effect - illustrated by the coefficient β' in the regression - and a size mobility effect - illustrated by the correlation coefficient ρ. It is possible for countries to diverge if β' > ρ, even if an estimate of β' is less than unity. A low value of ρ implies that there is little correlation between initial and final values and hence a large mobility through the distribution. We must then be careful when interpreting the results of the β-convergence regressions and use the information given the analysis of σ-convergence.

## 1.2 Which convergence notion?

Bernard and Durlauf [1996] show that cross-section tests used in this work - σ and β-methologies - may be interpreted with respect to the first definition of convergence. Nevertheless, they show that these tests fail to prove some evidence for the notion of

<footnote n="4">⁴Henin and LePen [1995] illustrate this discussion in an exercise, where they show how a simple diagrammatic representation may be associated to the distinction between σ and β-convergence.</footnote>

<footnote n="5">⁵Indeed $\beta' = \frac{\text{cov}(y_{i,t+1}, y_{i,t})}{\sigma_t^2}$ and $\rho = \frac{\text{cov}(y_{i,t+1}, y_{i,t})}{\sigma_{t+1}^2\sigma_t^2}$.</footnote>
convergence compatible with the second definition. Indeed, they tend to spuriously reject the null hypothesis of no convergence even when the variables of interest exhibit different steady states. This can occur if economies converging to low-output equilibria start far below their steady states, whereas economies converging to high-output equilibria start at values near their steady states.

This observation calls into question the usefulness of cross-section tests in assessing theoretical disputes in the growth literature. However, the first definition seems to be the most useful from the perspective of this empirical work. The Maastricht convergence conception is less restrictive than the second definition. The only relevant nominal criterion to judge if a country can join the U.E.M. is whether it has come closer to U.E.M. members over an horizon of a few years. Hence the criterion is a middle run criterion and does not consider the (infinite) long run.

The accession conditions require only that countries are on the way to converge, and not that they have already converged. Now, as Bernard and Durlauf [1996] argue, time series tests have poor properties when applied to data from economies in transition. In the time series tests, the convergence concept corresponds to the cointegration one and implies that convergence has already occured, while cross-section tests suppose that the data are in transition towards a limiting distribution. Cross-section method is then more useful for our study than time series tests.

We could finally remark, that in this exercise, it is essential to study if, on average, the processes of nominal and real convergences are significantly correlated in the European countries. More than the evolution of particular countries, we are interested in the link between these two aspects of the convergence. We chose then to use cross-section methodologies even if, as Quah [1996] argues, these methods deal only with the average behaviour and not the behaviour of an entire population. The cross-section tests cannot indeed identify groups of countries which are converging<fnref n="6" />.

## 2 Real and nominal σ-convergence

The data in this contribution come from different sources. The index of consumption prices comes from the I.M.F. data base. The real G.D.P. is extracted from the Summers and Heston [1991] data set. We consider all the members of the E.M.S. and E.F.T.A., except Greece. Taking Greece into account would greatly alter estimations for the nominal convergence without appreciably affecting the results for real convergence. The very high level of greek inflation rates and their important fluctuations introduce a bias in favour of the divergence hypothesis. Furthemore, all countries having the same weight in cross-section methods, Greece would have too important an effect on the results.

We start in a first subsection with the study of the path followed by the two processes using temporal graphical representations of univariate cross-section variances. We move then to the study of the link between nominal and real convergences.

<footnote n="6">⁶This class of tests cannot reflect the evolution of a part of the distribution compared with the evolution of another. The extreme case is a situation, where the variance is unchanged between two periods even though for example the richest countries have become the poorest and conversely. It justifies, for Quah, the necessity to study the evolution of the internal distribution.</footnote>
## 2.1 Study of univariate cross-section variances

We look first at the evolution of cross-countries variances for per capita G.N.P..

<figure><img src="image_2.png" /><figcaption>Figure 1: Evolution of cross-countries variances for per capita G.N.P.</figcaption></figure>

We note a global movement of convergence for the G.N.P per capita on the entire period (figure 1). We can however observe the increase of the dispersion in the early 80's (82-86), which seems contemporaneous with "restrictive" policies until the oil counter-shock. This temporary increase in the dispersion is mainly due to the behaviour of the countries in the periphery of the Union<fnref n="7" />. For the six first members of the Community - Germany, France, Benelux, and Italy -, the convergence has just slowed down during this period.

<figure><img src="image_3.png" /><figcaption>Figure 2: Evolution of cross-countries variances for inflation rate</figcaption></figure>

<footnote n="7">⁷We note especially the behaviour of Spain and Portugal.</footnote>
The dispersion of inflation rates cross-variances is very stable in the sixties (figure 2). Nevertheless, as we have seen before a constant variance can hide a crossed dynamic in the internal distribution. The variance stability can not be interpreted as reflecting inflation rates stability for the entire population. The oil shock brings along a very important increase of the dispersion. We must wait until the second half of the 80's to note a convergence movement despite the sudden increase of the variance in 1989-1990. This increase can be essentially imputed to the behaviour of Sweden, Germany and Portugal. The 90's are clearly marked by a large decrease of the dispersion.

If a first look at these figures suggest some first conclusions on the link between nominal and real convergences, it is difficult to compare them, to see their similarity or their differences. One contribution of this paper is therefore to connect in a same figure, nominal convergence with real convergence.

## 2.2 The link between real and nominal convergence

The $\sigma$-convergence hypothesis can be represented by the condition $\frac{\sigma_{y,t+8}}{\sigma_{y,t}} < 1$ where $\sigma_{y,t}$ is the standard error of a variable $y$ at date $t$.

The length of eight periods must be close to the length of a rolling filter, where we smooth the average increase rates. The convergence being a long-term process, the choice of this interval takes into account the length of the main economic cycles. This justifies that the length of the filter be long enough. On the other hand, the requirement to have a large number of points, in order to analyze the rolling evolution of the ratio, motivates a short enough length of the filter. The length of eight years seemed a good compromise.

The aim of this contribution justifies the choice of the rolling estimate over the recursive estimate. The convergence being a long-term process, the temporal dimension of samples plays an important part. The modifications occured over time could be hidden by the regular increase of the estimation period length$^8$. We need a constant period length to make comparisons easy. Moreover, the recursive method - which deals with the evolution of the ratio $\frac{\sigma_{y,t}}{\sigma_{y,t_0}}$ - leads to results that stongly depends on the base year. As a matter of fact, if the dispersion of the variables is extremely narrow during the reference year (this is the case for this exercise), the recursive results would be biased for the whole period in favour of the divergence hypothesis. A rolling measure can deal with this problem.

We therefore chose to use the rolling method, despite the fact that it makes difficult the identification of the date of possible breaking points. Indeed, as this method attaches the same weight to each observation, it is in fact impossible to know if the change in the evolution of the ratio could be imputed to the removal of the rolling sample's initial data or to the introduction of the final data.

<footnote>$^8$This problem would be even more serious in the next section when we estimate a $\beta$-convergence system. With a recursive analysis, the coefficient depends on the parameter $T$, which measures the length of the estimation period. Therefore, the convergence speed and the $R^2$ of the regressions are very sensitive to the changes of the period $T$.</footnote>
In order to make comparisons easier, we draw a figure cut out in four distinct zones, delimiting the areas in which nominal/real convergence occur. Thus, the four areas identified can be represented graphically, where the ratio $\sigma_{t+8}/\sigma_t$ is affected to the date $t+4$ (this amounts to center the data on each interval).

<figure><img src="image_4.png" /><figcaption>Figure 3: Convergence path in the inflation rates and per capita G.N.P.</figcaption></figure>

We can see important modifications in the link between nominal and real con-
vergence over time (figure 3). In the beginning of the sixties, there is a double
convergence movement. Conversely, we observe a period in the seventies, of real
convergence and nominal divergence. The beginning of the eighties (80-84) reveals
real divergence and nominal convergence. The end of the eighties is characterized
by the return of a double convergence movement.

One is then tented to conclude, that the nominal convergence observed during the eighties, has occurred at the price of a per capita G.N.P. divergence. This movement coincides with the creation of the E.M.S.. The deflationary policies seem to have insured for this period nominal convergence to the detriment of real convergence. Nevertheless, the analysis is more intricate and it is difficult to conclude only on the basis of these figures. We can hardly talk about links and correlations when the observed movements could well be entirely coincidental. It is therefore interesting to study the bivariate system - nominal and real -. Indeed, a weak point of the analysis so far is that we can not distinguish with a simple analysis of the σ-convergence the different dynamics of the processes, their respective stability, and especially their cross effects.

3 The estimate of a two dimensional system

We study a two-dimensional system, where each convergence equation is conditioned
by the initial level of both nominal and real variables. This system can be written
(without the noise) in the following way :
$$ \left( \begin{array}{c} \frac{x_T - x_0}{T} \\ \frac{y_T - y_0}{T} \end{array} \right) = \left( \begin{array}{c} a \\ b \end{array} \right) + \left( \begin{array}{cc} c & d \\ e & f \end{array} \right) \left( \begin{array}{c} x_0 \\ y_0 \end{array} \right) $$

where $x$ is a real variable and $y$ a nominal variable.

This joint dynamics can be analyzed as a representation of conditional convergence with a mutual conditioning. The model of conditional convergence predicts that countries can reach different stationary states: the convergence is conditioned by the determinants of the stationary state. Conditioning nominal (resp. real) convergence by the initial level of the real (resp. nominal) variable, we can then study the potential influences between the two types of convergence.

The case $e = d = 0$ corresponds to the unconditional convergence analysis. In that case, nominal and real convergence processes are disjoint. This situation will be called hereafter the “diagonal system”, or “disjoint model” as opposed to the “joint system”, where $e$ or $d$ is different from 0. The diagonal system is equivalent to the traditional $\beta$-convergence equation (see equation 1). The coefficient $b$ is then estimated by the O.L.S. method. Thus we can deduce from diagonal models the estimated convergence speeds for the two types of variables. One gets an estimator of $\lambda$:

$$ \hat{\lambda} = -\ln(1 + \hat{b}T)/T $$

where $\hat{b}$ is the estimated coefficient of the univariate regression.

The particular interest of the joint system characteristic equation is that it allows the study of different variables dynamics following the introduction of an initial gap between countries. The convergence will therefore be understood as a dynamic concept, i.e. the closing of initial gaps between countries. The basic system linking the average rate of increase, that is the average speed of a variable, with its initial level, can be rewritten so as to reveal the underlying dynamics. This simplified form without the noise can be expressed as in Cohen [1992]:

$$ \left\{ \begin{array}{l} \frac{d}{dt}x_t = a + cx_0 + dy_0 \\ \frac{d}{dt}y_t = b + ex_0 + fy_0 \end{array} \right. $$

The simple study of the signs of the coefficients in $\beta$-convergence models is not enough to conclude about the convergence of the variables. We must in the first place analyze the stability of the system. Indeed, if the system were not stable, we would estimate a conditional dynamics towards a moving target and not the convergence towards a stationary state. We will examine the stability of the system with the help of the usual criterium in linear dynamics : the analyze of the trace and the determinant. As in Cohen [1992]<fnref n="9" />, we test if the determinants are positive with Wald's tests.

The second section has made clear that important changes occured in the convergence processes. Thus, we will study these evolutions by comparing different periods.

<footnote n="9">⁹In order to know if we can reject the hypothesis $H_0$ that $\Theta = bc' - b'c > 0$, we perform the following test, presented by Cohen [1992] : Let be $u = (\frac{\partial \Theta}{\partial b}, \frac{\partial \Theta}{\partial c}, \frac{\partial \Theta}{\partial b'}, \frac{\partial \Theta}{\partial c'})$ so $u = (c', -b', -c, b)$.</footnote>

<footnote>We call $\hat{\Sigma}$ the covariance matrix of the estimated coefficients $(\hat{b}, \hat{c}, \hat{b}', \hat{c}')$, and note : $\hat{V}(\hat{\Theta}) = \hat{u}\hat{\Sigma}\hat{u}^T$. We calculate the student statistics of $\Theta : \epsilon = \frac{\hat{\Theta}}{\hat{V}(\hat{\Theta})^{1/2}}$.</footnote>
In a first time, we estimate a rolling evolution of the joint system. In a second time, we study more deeply some periods chosen on historical grounds and estimate the coefficients for four time periods : from 1964 to 1973, from 1972 to 1979, from 1979 to 1987, and lastly from 1987 to 1992. These sub-periods and the rolling periods are never exactly the same. Finding essentially the same results with rather different estimation periods increases the robustness of the conclusions.

## 3.1 Rolling estimations of joint system

We cross-estimate, for each rolling sub-period, the two-dimensional system, where convergence movements are reciprocally conditioned. We report the coefficients of these regressions in figures 4, 5, and 6.

<figure><img src="image_5.png" /><figcaption>Figure 4: Rolling estimates of the “direct-effect” coefficients in the nominal and real convergence equation.</figcaption></figure>

The coefficients called “direct” are the estimated coefficient of inflation initial level in the nominal convergence equation and the estimated coefficient of the initial per capita G.N.P. level in the real convergence equation. We see on the figure 4 the convergence or divergence tendencies of the two types of processes. A more careful study is however necessary to reach a conclusion, because we need to check the stability of the system and the influences of cross-effects. We nevertheless observe in a first analysis the existence of four distinct periods :

- From 1965 to 1971 : Nominal convergence-Real convergence

- From 1972 to 1978 : Nominal divergence-Real convergence

- From 1979 to 1984 : Nominal convergence-Real divergence

- From 1984 to 1988 : Nominal convergence-Real convergence
We must recall at this point that these dates are centered on rolling samples. Thus, the estimation set, for example, on 1965 is the estimation achieved on the interval 1961-1969. We did not report the confidence intervals on this figure: the coefficients are always significant, except around the two oil shocks for the nominal variable, and when divergence occured for the real variable.

Figure 5: Confidence interval of the rolling "cross-effect" coefficient in the nominal convergence equation. The interval is estimated at the 10% level.

<figure><img src="image_6.png" /></figure>

Figure 6: Confidence interval of the rolling "cross-effect" coefficient in the real convergence equation. The interval is estimated at the 10% level.

<figure><img src="image_7.png" /></figure>

The results concerning the cross-coefficients, reported in figures 5 and 6, show that the influence of the initial level of per capita G.N.P. on the growth rate of inflation is more significant than the impact of initial inflation level on the increase rate of the logarithm of per capita G.N.P.. We hence note an asymmetry in the
system. The aim of this paper is not to test any particular theory. Nevertheless,
we could maybe understand these results as indicating the presence of a significant
Phillips effect in the price equation and pointing out the absence of a competitivity
effect in the growth equation.

The coefficient of initial per capita G.N.P. level in the nominal convergence
equation is in fact significantly positive on a set of adjacent intervals, which define
the period 1973-1980, i.e. from 1969 to 1976 on figure 5. This period corresponds
to the beginning of the divergence movement for the nominal variable up to the
beginning of the divergence movement for the real variable. In this period, the weaker
the initial per capita G.N.P. of a country is, the higher the increase rate of per capita
G.N.P. is, and the weaker the inflation growth rate is.

In fact, this effect has ceased to be significant, when real divergence occured (in
the beginning of the eighties). In short, this evidence confirms the importance for
this period of wage and price control as a key component of the disinflation process.
In any case, it suggests that the policy was not simply relying on unemployment to
discipline costs, as noted by Blanchard and Muet [1993]. This sustained deflationary
effort is contemporaneous of real convergence slow-down.

## 3.2 Estimations on periods chosen on historical grounds

We consider sub-periods, chosen on historical grounds: the breaking points are taken
around the two oil shocks, and the oil counter-shock of 1986. An “independence”
test<fnref n="10" /> is made on each subperiod in order to identify situations, in which we can’t
reject the diagonal model in favour of the joint system. These situations reveal that
nominal and real processes are independent.

We report estimated coefficients of the diagonal models and of the joint system,
for the four time periods, in a single table, presented in an appendix. The χ² statistics
associated with the independence test is about 8,23 for the 1964-1973 time period,
4,47 for the 1972-1979 period, 0,737 for the 1979-1987 period, and 1,36 for the
last period. These results lead us to reject the diagonal model for two out of these
four periods. We will then focus on the study of the joint system coefficients for
all periods, except 1979-1987 and 1987-1992. For these periods, we accept the null
hypothesis and don't reject the diagonal model. It is then useless to study the joint
system and the non significant cross-effects. We must however remark the relative
importance of the coefficient linking the initial per capita G.N.P. level to the inflation
growth rate in the nominal convergence for the last subperiod. It is then nevertheless
interesting to study a triangular dynamics, where we constrain the real convergence
to be unconditional. We choose to take into account in the i.r.f. of the processes,
one of the cross-effects, because it is important in spite of its weak significativity.

<footnote n="10">¹⁰We recall here the test presented by Gourieroux and Monfort [1990] pp 604. Let be $Z_t = AW_t + u_t$ for $t = 1, ..T$ a linear multivariate regression model, where $Z_t$ is a s-vector, $W_t$ a k-vector, $A$ a ($s \times k$) matrix of unknown parameters, and $u_t$ a white noise. Calling $H_o$ the hypothesis on $A$, the likelihood ratio statistics is : $c_T^R = T \log \frac{det \hat{\Omega}_o}{det \hat{\Omega}} \sim \chi_{1-\alpha}^2(r)$ where $\hat{\Omega}_o$ et $\hat{\Omega}$ are the maximum likelihood constrained and unconstrained estimators of $\Omega$, $r$ the constraint's number, and $\alpha$ the hold level.</footnote>
Lastly, we can observe that the only subperiod (1979-1987), for which the real initial
level has really no more influence on nominal convergence, is the period, where real
convergence is no more realised.

The sign of the coefficients linking the average increase rates to their own initial
level is negative for the first period (1964-1973). This reveals a movement of nominal
and real convergences. Inflation rates converge about two times faster than real
per capita standards of living. Moreover, the determinant is, for this period, equal
to 0,00132 and the Wald's test statistics allow us to accept the assumption that
the determinant is positive at the 5% level. The trace is equal to -0,081 and is
significantly negative<fnref n="11" /> at the 1% level. Hence, the system is stable. All coefficients
of the system are significant and the cross-effects reveal a negative contribution to
the convergence, whose dynamics will be study with the help of figure 7 (reported
in appendix 2). The impact of an initial gap in standards of living on nominal
convergence is more important than the influence of the initial gap in inflation rates
on real convergence.

Conversely, we observe that in the 1972-1979 period, nominal convergence has ceased. The coefficients are both significantly positive in both univariate and bivariate analysis. The real convergence, however, continues. Its speed is nevertheless slower. We can note the importance of the initial per capita G.N.P. level in nominal convergence. If a country has a high initial standard of living, it will experience higher average increase rates of inflation. The determinant is then negative (-0,00541) and Wald's test allows us to reject the assumption that the determinant is positive. The trace being significantly positive at the 5% level<fnref n="12" />, the equilibrium is unstable, with possibility of a saddle point. The diverging influence of the nominal variable and the explosive character of the system explain the real diverging movement in the long run (figure 8 in appendix 2).

For the 1979-1987 period, the situation is reversed. Once again, nominal convergence occurs at the cost of real convergence. The only significant coefficient links the initial inflation rate level to its own average increase rate. We then observe a nominal unconditional convergence movement, that could be called dichotomic. The trace (-0,081) and the determinant (−6,94·10⁻⁵) are both negative. The equilibrium appears unstable with a possibility of a saddle point.

Lastly, we see again, for the 1987-1992 period, a double convergence movement : both direct coefficients are significantly negative. The convergence speed is very high. The study of the trace and the determinant reveal the stability of the joint system. The trace is significantly negative<fnref n="13" /> at the 1% level and the determinant is positive. Wald's test allows us to accept the assumption that the determinant is positive.

In opposite to 1964-1973, the convergence processes become for this last period relatively dichotomic. We observe that an initial gap in per capita G.N.P. involves a weaker nominal reaction. Moreover the closing of the gap is realized faster (figure 9, in appendix 2).

<footnote n="11">¹¹The student's statistics, that is associated with the trace is equal to -4,87</footnote>

<footnote n="12">¹²The trace is equal to 0,257 and its student statistics is equal to 2,26</footnote>

<footnote n="13">¹³The student statistics associated with the trace is equal to -6,62.</footnote>
Conclusion

The aim of the present contribution was the study of nominal and real convergence
processes, as reciprocally conditioned within a same system. The joint dynamics of
inflation and G.N.P. growth was consider as a framework to test for the interrelation
between nominal and real convergence among differents European countries.

A preliminary study of σ-convergence reveals that the nominal and real convergen-
ce are negatively correlated in the eighties. Contrary to what happened during the
seventies, nominal convergence occurs in the eighties at the cost of a real divergence.
This negative correlation disappears in the nineties. Thus, we observe at the end of
the sample period both a nominal and a real convergence movement.

The estimation of the bivariate system brings the mutual relationships between
these different movements forward. Initial per capita G.N.P. level influences the
nominal convergence. However, this effect has ceased to be significant, when real
convergence slowed down (in the beginning of the eighties). Moreover, if we see a-
gain, in the nineties, a double convergence movement, the two cross-effects are not
significant anylonger. It seems then difficult to say that the nominal convergence pro-
cess leads to real divergences. The convergence dynamics have become dichotomic,
and nominal convergence seems not to interfere with real convergence anymore.
# Appendix 1: Estimation of the diagonal and joint system

We can read, in the first line, the list of regressors, except the constant, and in the first column, the list of the two estimated variables. Student's statistics for the coefficients are reported between brackets. The $\lambda$ parameter is the convergence speed, calculated in the univariate system.

<table><caption>Table 1: Estimation of the diagonal and joint system</caption><thead><tr><th></th><th>1964-1973</th><th>Δp<sub>0</sub></th><th>ln y<sub>0</sub></th><th>R<sup>2</sup></th><th>λ</th></tr></thead><tbody><tr><td rowspan="2">Diagonal system</td><td>Δ<sup>2</sup>p</td><td>-0,052<br>(-2,57)</td><td></td><td>0,27</td><td>0,07</td></tr><tr><td>Δ ln y</td><td></td><td>-0,03<br>(-8,19)</td><td>0,81</td><td>0,035</td></tr><tr><td rowspan="2">Joint system</td><td>Δ<sup>2</sup>p</td><td>-0,051<br>(-3,12)</td><td>0,205<br>(2,25)</td><td>0,41</td><td></td></tr><tr><td>Δ ln y</td><td>0,001<br>(2,03)</td><td>-0,03<br>(-9,79)</td><td>0,84</td><td></td></tr><tr><td rowspan="2">1972-1979</td><td>Δp<sub>0</sub></td><td>0,29<br>(1,9)</td><td>0,001<br>(-0,47)</td><td>0,29</td><td></td></tr><tr><td>Δ ln y</td><td></td><td></td><td>0,35</td><td>0,025</td></tr><tr><td rowspan="2">Joint System</td><td>Δ<sup>2</sup>p</td><td>0,287<br>(2,27)</td><td>1,035<br>(2,17)</td><td>0,29</td><td></td></tr><tr><td>Δ ln y</td><td>-0,001<br>(-0,47)</td><td>-0,023<br>(-3,21)</td><td>0,3</td><td></td></tr><tr><td rowspan="2">1979-1987</td><td>Δp<sub>0</sub></td><td>0,083<br>(-6,81)</td><td></td><td>0,75</td><td>0,136</td></tr><tr><td>Δ ln y</td><td></td><td>0,003<br>(0,55)</td><td>-0,05</td><td>-0,0035</td></tr><tr><td rowspan="2">Joint system</td><td>Δ<sup>2</sup>p</td><td>-0,082<br>(-6,63)</td><td>-0,063<br>(-0,23)</td><td>0,73</td><td></td></tr><tr><td>Δ ln y</td><td>0,0002<br>(0,83)</td><td>0,001<br>(0,23)</td><td>-0,08</td><td></td></tr><tr><td rowspan="2">1987-1992</td><td>Δp<sub>0</sub></td><td>0,15<br>(-4,99)</td><td></td><td>0,61</td><td>0,277</td></tr><tr><td>Δ ln y</td><td></td><td>-0,041<br>(-3,42)</td><td>0,41</td><td>0,045</td></tr><tr><td rowspan="2">Joint system</td><td>Δ<sup>2</sup>p</td><td>-0,159<br>(-5,66)</td><td>0,33<br>(1,13)</td><td>0,61</td><td></td></tr><tr><td>Δ ln y</td><td>-0,0004<br>(-0,36)</td><td>-0,04<br>(-3,47)</td><td>0,37</td><td></td></tr></tbody></table>
Appendix 2: Impulse response functions to the introduction of an initial gap between countries

<figure><img src="image_8.png" /><figcaption>Figure 8: 1972-1979</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 9: 1987-1992</figcaption></figure>

<figure><img src="image_10.png" /></figure>
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