China: To what extent is provincial growth determined by growth in contiguous provinces?

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China: To what extent is provincial growth

determined by growth in contiguous provinces?

Master Thesis (‘Doctoraal Scriptie’)

Faculty of Economics

University of Groningen

by

Gelijn Werner *

Groningen

31 August, 2007

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Preface,

a word of thanks in Dutch

‘Geen tijd meer voor een voorwoord, je moet het zo inleveren’ zit ik te denken. Afijn, dan toch kort. Al is het alleen al om een aantal mensen te bedanken voor hun steun aan deze afrondende scriptie van mijn doctoraal studie Algemene Economie aan de Rijksuniversiteit te Groningen. Allereerst natuurlijk de heer Elhorst. Bij mijn eerste bezoek aan hem in december 2006 kwam hij binnen vijf minuten met een onderwerp dat aansloot bij mijn ideeën –macro-economie, economische groei en China. Vervolgens heeft hij mij op geduldige en intellectuele wijze geïntroduceerd in de voor deze scriptie relevante ins en outs van het geroemde

wiskundige computerprogramma MATLAB (ook dank aan de heer Pim Heijnen voor zijn MATLAB snelcursus); hij heeft me geïntroduceerd in een, voor mij, nieuw vakgebied, ruimtelijke econometrie; en hij heeft me uiteraard voorzien van waardevolle kritiek, waar ik de nodige lering uit kon trekken.

Daarnaast wil ik graag mijn ouders, Hans en Liesbeth Werner, bedanken voor hun morele steun, net als mijn lieve vriendin, Anouk Dechesne.

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Abstract

This paper investigates whether space is a determinant of China’s provincial economic growth in China: is the relative location of China’s provinces a determinant of economic growth? This proposition is investigated by means of a spatial econometric specification known as a first-order spatial autoregressive lag model. This specification links the growth rates of provinces to the initial income levels and some local characteristics in its home province as well as in contiguous provinces and besides to the growth rates in contiguous provinces. Empirical analysis is done using provincial data from China over the period 1980-2002. This analysis learns us that space is a determinant of China’s provincial economic growth and that the relative location of China’s provinces is a determinant of economic growth. The model specification itself, however, can be considered as a specification that is lacking strength.

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1 Introduction

Between 1978 and 2002 China has achieved a real annual GDP growth rate of 9.5 percent on average.1_{This average is remarkably high compared to the world’s average during this}

period, 2.3 percent.2 This high growth rate has made people to regard China as a major player in the world economy and its economic reforms as a huge success. Under the banner of “allowing some to get rich first” Deng Xiaoping and his policy makers placed priority on efficiency over equality (Kwan, 2005). This, among other factors, had the side effect of large uneven regional development. Figure 1 gives GDP per capita per province in 2003, whereby the provinces are subdivided into three categories: coastal provinces, central provinces and western provinces. Figure 1 shows that, on overall, the coastal provinces are richer than other provinces. This strokes with the fact that rapid economic growth mainly characterized the coastal provinces, leaving inland provinces behind in terms of economic growth. In 1980, real GDP per capita in coastal provinces was 1.80 times larger than that in western provinces. Via 1.80 in 1990, it became 2.63 times larger in 2002. This not only points to economic

disparities, but also to widening disparities. In other words, divergence took place. This increasing inequality is an interesting feature of the Chinese development, already studied by many researchers. Some studies focus on the determinants of these widening disparities (Fujita and Hu, 2001; Buckley, 2002; Zhang and Felmingham, 2002; Zhang and Zhang, 2003; Jones and Li, 2003; Zhao and Tong, 2000). They consider for example geographical factors, business cycles and investments. Other studies focus on the regional differences in wages and its determinants (Tsui, 1998; Gustafsson and Shi, 2002; Meng, 2004).3 The last group of studies focus on the impact of economic reform policies since 1978 (Demurger et al., 2002; Zhang, Elhorst and Van Witteloostuijn, 2006).

As one can see, many studies have focused on economic differences and divergence between regions within China. However, regional economies in China not only differ, they also interact. Economic interaction has been studied widely. This brings me to the spatial

economics literature that has investigated the determinants of economic growth empirically.

1_{Source: China Statistical Yearbook 2004.} 2_{Source: World Development Indicators.}

3_{Since provinces can be seen as regions, studies regarding to regional economies concern provincial economies}

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Economic growth can be better understood by an appropriate appreciation considering economic interaction between regions, because regions’ income levels are interdependent. Among others, Fingleton (2003) provides the example of the savings rate. Since capital can move freely across regional borders, regional savings do not have to equalize regional investments. This makes one region’s economy dependent on another region’s economy (through savings) and this applies for Chinese regional economies as well. Different kind of interdependencies have been theoretically explained. These theoretical explanations

correspond to empirically found causal factors of interdependencies, such as interregional migration of production factors (Ertur and LeGallo, 2003) or general spillovers (LeGallo et al., 2003). Examples of papers dealing with interregional migration in China are Suzuki and Suzuki (2007), Wei (1997) and Day (1994). A paper dealing with spillovers is provided by Groenewold, Lee and Chen (2005). It analyses whether inter-regional spillovers take place in China, using a six-region vectorautoregressive model and extensive sensitivity analysis. The first to model data with space and time dynamics within one model was Badinger et al. (2004). A weakness of this model, however, is that space and time are not modeled

simultaneously. They first accounted for regional dependency through space, and then for time dynamics. It is intuitively appealing however that time and space are interdependent and should be modeled simultaneously. Short-term and long-term changes in any region may impact other regions at different moments in time. Elhorst, Piras and Arbia (2006) have presented a test whether the relative location of region is a determinant of economic growth by simultaneously modeling time and space. They do this by setting up a first-order

autoregressive lag model, as developed by Elhorst (2001). This model consists of a growth equation in which regional growth depends not only on factors in its own region, but also on factors in neighbouring regions, taking time dynamics into account. This model is used to provide a more appropriate description of regional economic interaction.

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1977-2002.4_{This does not mean that the same outcomes can be found when using different}

samples of regions.5 Combining China and the spatial econometrics approach brings me to the main question in this paper. The present paper tries to answer the question whether space is a determinant of economic growth in China: is the relative location of China’s provinces a determinant of economic growth? We deal with this problem by means of the empirical model that is called a first-order spatial autoregressive lag model. This is to provide a more

satisfactory picture than previous studies did, of how Chinese provincial economies interact with one another.

Virtually, the methodology used is the same as in Elhorst, Piras and Arbia’s study “Growth and convergence in a multi-regional model with space-time dynamics” (2006). This implies that a growth equation is set up in which the annual growth rate of per capita GDP in one region depends on its initial income level, and the rates of saving, population growth, technological progress and depreciation in the province itself. In addition, it depends on the income levels and rates in neighbouring provinces. This approach is preceded by a non-spatial cross-section approach and time-series cross-section approach to gain better understanding of the results. The equation is set up by making use of regional data from China over the period 1980-2002. The data consists of twenty-nine provinces. Besides providing an answer to the main question stated earlier, the present paper is an empirical application of the rather new first-order spatial autoregressive lag model.

In what follows, Section 2 presents the theoretical background. The benchmark Solow-Swan model is discussed, before going into the spatial extension to this model, as provided by Ertur and Koch (2005). Section 3 briefly discusses the spatial econometric literature. Next, Section 4 provides the hypotheses in the present paper. Section 5 discusses the data and Section 5 gives the results of the empirical analysis, before section 6 comes with a conclusion.

4_{Regions were situated in 15 countries: Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland,}

Italy, Luxembourg, the Netherlands, Portugal, Spain, Sweden and the United Kingdom.

5_{Also, changing the time-span can influence the outcomes drastically. However, in the present study we modify}

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2. Theoretical background

This section provides the theory behind the model we use to answer the main question in this paper. The first subsection gives the standard Solow-Swan model. This model provides the economic foundation for the spatially extended model, as it was developed by Ertur and Koch (2005). The second part presents this spatially extended model.

2.1 The Solow-Swan model

The basic Solow-Swan model (Solow, 1956; Swan, 1956) is an economic growth model based on the Harrod-Domar model (Harrod 1939; Domar, 1946). The Harrod-Domar model is used in development economics to explain an economy's growth rate in terms of the level of saving and productivity of capital. It suggests that there is no natural reason for an economy to have balanced growth. The Solow-Swan model extended the Harrod-Domar model by including a new term, productivity growth. The difference between the two models is that the Harrod-Domar sees capital accumulation as the engine of economic growth, whereas the Solow-Swan model sees technological progress as the engine. One outdated feature of the model is that technology –and thereby economic growth- is exogenously given. However, even today the model continues to be of great theoretical and empirical importance.

The model assumes a production function which includes only two production factors, namely labour and capital, producing a homogenous good in one single sector in a closed economy. Furthermore, it assumes efficient use of resources, constant returns to scale and labour-augmenting technological progress. The basis production function is as follows:

Y(t) = K(t) [A(t)L(t)]1-, 0 < 1. ( i )6

Y is defined as output, K as capital, L as labour and A as technology. By assumption, A and L

grow at the given rates g and n, respectively. This gives:

L(t) = L(0) ent , (ii)

A(t) = A(0) egt . (iii)

6_{We are aware of the unusualness of using Roman figures here. However, I choose to do so in remembrance of}

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Investments equal savings by assumption, where savings are exogenously given. The savings rate is given as s. In order to get to the steady state in this model, an equation for k is set up, where k is defined as the stock of capital per effective unit of labour (k = K/AL), y as the level of output per effective unit of labour (y = Y/AL) and as the depreciation rate. The equation is as follows:

k = sy(t) – (n+g+ ) k(t) = sk(t) – (n+g+ ) k(t) (iv)

Optimizing equation (iv) by setting k equal to zero gives the steady state value to which k converges:

k* = ( s/[n+g+ ]) 1/(1- ). (v)

where k* is the steady-state value of k. Figure 2 gives a graphical representation of the model equilibrium. At point A the lines of (n+d)k and sy intersect, marking the equilibrium.7 When this equilibrium is reached, the steady state capital-labor ratio is constant and growth equals the sum of exogenously given population growth and technological progress (Heijdra and Van der Ploeg, 2002). The steady-state per capita income, q, is given by:

ln q(t) = ln A(0) + gt + [ /(1- )] ln [s/(n+g+ )]. (vi)

Using a Tayor expansion as in Mankiw et al. (1992) an easy-to-work-with growth equation can be constructed:

ln(qt/qt-T) / T = 0 + 1 ln(qt-T) + 2 ln s/(n+g+ )/ T + , (vii)

where T gives the time span under consideration in the growth period studied here, is the annual speed of convergence ( = - log (1+ 1T)/T) and is basically defined as an

autonomous, normally distributed error term. Furthermore, 0 = (1-e- T) ln A(0) + g[t-e- T

(t-T)], 1 = -(1-e- T)/T and 2 = [ /(1- ) (1-e- T). Now one can estimate equation (vii) by

Ordinary Least Squares (OLS) regression. This method of regression was already developed

7_{Note that the depreciation rate is denoted as d in the graph and that technology is not included yet. However,}

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in 1795 by C.F. Gauss (see Stigler, 1986). The steady-state per capita income, qt, can be

recovered from equation (vii):

ln(qt) = - ( 0/ 1) – ( 2/ 1)ln [s/(n+g+ )], (viii)

where the negative signs are explained by the negative sign of 1. A negative sign of 1 is a

necessary condition for convergence. Equations (vii) and (viii) provide us with, among others, two possible tests. Testing whether 2 is significantly different from zero in equation (vii)

answers the question whether s, n, g and affect the growth rate of an economy. Testing whether –( 2/ 1) is significantly different from zero in equation (viii) answers the question

whether s, n, g and affect the steady-state position of an economy.

A fundamental outcome of the Solow-Swan model using OLS regression is that in the end, leaving s, n, g and constant, economies tend towards the same equilibrium growth path for capital and output per capita. This implies that poor economies grow faster than rich

economies. On the long term each economy will follow its own steady state growth rate and all economies become equally rich. It is important to note that if one of the independent variables (s, n, g, or the initial income level) differs among economies, the equilibrium levels differ too. If this is so, convergence is not called absolute anymore, but conditional. Poor economies might still grow faster than rich countries, however conditional upon the values of s, n, g, and the initial income level.

As stated earlier, the Solow-Swan model has been passed by more advanced models that possess more realistic characteristics. They deal with the main weakness of the Solow-Swan model, namely that technology is being determined exogenously. Heijdra and Van der Ploeg (2002) give an overview of endogenous growth models. The book distinguishes three major streams: “capital-fundamentalist” models that generate perpetual growth by abandoning some key element from the basic Solow-Swan model; models that focus on the accumulation of the “growth engine” human capital; and models that focus on R&D as the engine of growth. Nevertheless, the exogenous Solow-Swan growth model seems suitable for adoption here, just as was done by Elhorst, Piras and Arbia (2006). Reason for this is that it estimates the precise law of motion generated by the Solow-Swan growth model. Next to that, frequently agrees better with the physical capital’s share. This capital share of output is supposed to be

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estimating the value of , the Solow-Swan specification of the model can be tested as an additional means. The estimator for is found by:

= 2 / ( 2- 1). (ix)

2.2 Spatial extension to the Solow-Swan model

The basic Solow-Swan model as presented in the previous paragraph does not include spatial interaction effects. Since we are interested in spatial relationships between neighbouring provinces, we need a model that includes spatial effects. Therefore, the model needs to be extended. This subsection shows a Solow-Swan model extended with spatial interaction effects. This model has been set up in a spatial econometric setting by Ertur and Koch (2005). They model technological interdependence between economies by including physical capital externalities as suggested by the Frankel-Arrow-Romer model (Arrow, 1962; Frankel, 1962; Romer, 1986). Besides, spatial externalities in knowledge are included:

Ai(t) = (t)ki (t), (x)

where i (=1,2,..,N) refers to a specific economy. To see that it is just an extension of the basic Solow-Swan model, one can set ϕ and ρ equal to 0. Equation (x) consists of three terms. The first term, (t), is identical to A(t) in equation (iii) which belonged to the basic Solow-Swan model, only stated as (t) instead of A(t):

(t) = (0) egt_. _(xi)

The assumption is that technological development is identical to all economies and is determined exogenously. The second term refers to physical capital generating knowledge spillovers. The state of technology in region/country i is assumed to be related to the level of physical capital per worker [=ki(t)] in that specific region/country. The parameter ϕ, with

0<ϕ<1, gives the relative size of that physical capital per worker. The third term then refers

to spatial externalities in knowledge. A distance decay function, ρwij, describes how economy

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common use in the spatial econometrics literature to use one out of two possible measures, an inverse-distance measure or a binary contiguity measure. The first one measures the real physical distance or travel time between economy i and economy j (dij). By setting wij = 1/dij,

a pattern of spatial relations between economies is used. The second measure is a discrete binary measure that describes spatial relations between economies. This measure uses wij=1

to indicate that economies are contiguous and wij=0 to indicate that economies are not

contiguous. It is intuitively appealing that ρ>0.

A spatial counterpart of equation (vi) gives the steady-state for y, including spatial interdependence:

ln q(t) = ρ (1-α)/(1-α-ϕ) W lnq(t) + 1/(1-ϕ) ln Ω(0) + gt +

(α+ϕ)/(1-α-ϕ) ln [s/(n+g+δ)] + ρ[α/(1-α-ϕ)] W ln [s/(n+g+δ)], (xii)

where t, ln q(t), ln Ω(0) and ln [s/(n+g+δ)] denote N x 1 vectors and [W = wij] is a N x N

matrix, providing the spatial interdependence weights. In the same fashion as done in section 2.1, using a Taylor expansion produces an easy-to-work-with growth equation:

ln(qt/qt-T) / T = ρ~ W [ln(qt/qt-T) / T] + 0 + 1 ln(qt-T) +

2 ln s/(n+g+ )/ T + β3W(ln(qt-T) + β4W ln [s/(n+g+ )/ T] + , (xiii)

where 0 is a complicated function of the parameters (see Koch and Ertur, 2005),

1 = -(1-e- T)/T,

2 = [ /(1- ) (1-e- T),

β3 = ρ (1-α)/(1-α-ϕ) (1-e- T) / T,

β4 = -ρα/(1-α-ϕ) (1-e- T), and

ρ~ = ρ(1-α)/(1-α-ϕ).

Estimates of imply the unknown values of , and :

= 4 / ( 4 – 3), (xiv)

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= ( 4 – 3) / ( 1 – 2). (xvi)

Equations (xiv), (xv) and (xvi) are restricted by 3 + ρ~ 1= 0, otherwise , and are

overdetermined. Just as applies for , the annual convergence speed, , the capital share in income, and , the physical capital externalities, are the same as in the basic Solow-Swan model as described in section 2.1. The parameter , measuring the impact of the spatial arrangement, can not be found in the basic model.

The spatial extension to the basic Solow-Swan model brings extra complications with it, making it more difficult to derive an expression for the steady-state per capita income, qt. The

first step is to multiply equation (xiii) by T and then to rewrite it as:

B ln q(t) = A ln q(t-T) + x (xvii)

where x = T 0 + T 2ln[s/(n+g+ )] +T 4 W ln[s/(n+g+ )],

A = (1+T 1)I + T 3W – ρ~ W, and

B = I - ρ~ W.

Elhorst (2001) has found that the eigenvalues of the matrix AB-1 should lie inside the unit circle, in order for the model to converge. Row-normalizing W assures that the largest eigenvalue is equal to 1. This simplifies to ( 1 + 3) < 0. Following Elhorst, a steady-state

value of qt can be obtained by :

ln q(t) = (B-A)-1 x . (xviii)

This implies that:

ln q(t) = (-T 1 – T 3W)-1 [T 0 +

T 2ln[s/(n+g+ )] +T 4 W ln[s/(n+g+ )], (xix)

Which can be rewritten as:

ln q(t) = [I + ( 3/ 1)W]-1 [-( 0/ 1) –

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Equation (xx) is the spatially extended Solow-Swan model counterpart of equation (viii). Just as in the basic Solow-Swan model, it can be tested whether s, n, g and in a particular economy affect the growth rate in that economy by looking at 2. In the spatially extended

model, this follows from equation (xii). If this coefficient is significantly different from zero, the test provides evidence in favor of the hypothesis that s, n, g and in one particular economy affect the economic growth rate in that economy. Similarly, it can be tested whether

s, n, g and in a particular economy affect the steady-state position of that economy by

looking at –( 2 / 1). In the spatially extended model, this follows from equation (xx). The

new thing is that space is also involved. By looking at 2 in equation (xii) and –( 2 / 1) in

equation (xx) one can also test whether s, n, g and in a particular economy affect the growth rate in neighbouring economies and the steady-state position of neighbouring economies, respectively.

This brings me to the central idea that characterizes the spatially extended Solow-Swan model. This central idea is that growth also depends on growth and growth related factors in neighbouring economies. To put it more precize, the per capita GDP growth rate now also depends on the per capita GDP growth rates in neighbouring economies and on s, n, g, and the initial income level in neighbouring economies.

3 Spatial econometric background

Now that the previous section has provided the setup of the model specification used, it is time to lay it against the spatial econometric background.

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assumption that explanatory variables are fixed in repeated sampling, whereas the fact that heterogeneity occurs in the modeled relationships violates the assumption that a single linear relationship exists across the sample data observations.

The first fruitful developments within the field of spatial econometrics took place during the 1970s. The seminal work of Cliff and Ord (1975) popularized the field of spatial statistics and spatial econometrics. The first one to provide a complete treatment of many elements of spatial econometrics was Anselin (1988). Ever since, the field of spatial econometrics has developed quickly.

Today, it is common use to apply spatial econometric techniques to estimate models, usually making use of a spatial weight matrix that accounts for the relative location of different economies (Arbia, 2006). The research field has done so in two directions (Abreu et al., 2005; Anselin and Bera, 1998): The spatial lag model and the spatial error model. Ord (1975) already provided Maximum Likelihood methods for estimating the spatial lag and spatial error SAR models in 1975. Currently, the spatial lag and the spatial error model are the two main economic growth models focused on spatial interaction used in spatial econometrics literature yet. The spatial lag model is defined as follows:

ln(qt/qt-T) / T = ρ~ W [ln(qt/qt-T) / T] + 0 +

1 ln(qt-T) + 2 ln s/(n+g+ )/ T + . (xxi)

The spatial lag model as described in equation (xxi) states that the growth rate of one province depends on the growth rates in neighbouring provinces and on a set of specific local

characteristics that are observed. It includes a spatially lagged variable. The spatial error model is defined as follows :

ln(qt/qt-T) / T = 0 + 1 ln(qt-T) + 2 ln s/(n+g+ )/ T + , (xxii)

where = W + , E( ) = 0, E( `) = 2 IN. The model includes a spatial autocorrelation

coefficient, called . Following Anselin and Bera (1998) equation (xxii) can be rewritten as follows:

ln(qt/qt-T) / T = W ln(qt/qt-T) / T + 0 + 1 ln(qt-T) +

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The spatial autocorrelation error model as presented in equations (xxii) and (xxiii) includes some local characteristics, just as in the spatial lag model. Its remaining structure is identical. Furthermore, it has a well-behaved error-term too. New is that the spatial error model includes two more explanatory variables. This makes the model more general than the spatial lag model, making it more interesting for applications in spatial econometrics.

The model we use is very similar to the above-mentioned models. Reason to use this specification is that it might provide a more satisfactory description of economic interaction than previous models (see the explanation in Elhorst, Piras, Arbia , 2006). It has the same structure as the spatial lag model. Nevertheless, the spatial lag model lacks the coefficients 3

and 4 in the equation, as a result of what it lacks two spatial explanatory variables. The

distributed lag model as formulated by Elhorst (2001) is similar to the spatial error model as presented in equation (xxiii), the difference being that the spatial error model imposes two nonlinear factor constraints; 3 = - 1 and 4 = - 1. Note that the first constraint is also

mentioned in the previous subsection as a necessary condition to avoid overdetermination of the parameters of the autoregressive distributed lag model. The first-order spatial

autoregressive lag model is more general than the spatial error model. Reason to test for the spatial error model is that this model includes less parameters and therefore gains in terms of statistical efficiency. If the restrictions do not hold statistically, the spatial error model must be rejected in favor of the first-order spatial autoregressive lag model.

4 Hypotheses

The main question in the present paper is, as introduced in Section 1, whether space is a determinant of economic growth in China: is the relative location of China’s provinces a determinant of economic growth? This problem is set within a specific spatial econometric framework that is called the first-order spatial autoregressive lag model. To be able to answer the main question quantitatively, we duplicate the four hypotheses drafted by Elhorst, Piras and Arbia (2006) and test these hypotheses using data of Chinese provinces:

Hypothesis 1:

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We test this basic hypothesis by verifying whether ρ~ in equation (xiii) is significantly

different from zero, where ρ~ is the coefficient of the variable W* ln(qt/qt-T)/T, which stands

for the contiguous GDP growth rates per capita. Hypothesis 2:

The rate of growth of a particular Chinese region is affected by s, n, g , in its neighbouring Chinese regions.

We test this hypothesis by verifying whether 4 in equation (xiii) is significantly different

from zero, where 4 is the coefficient of the variable W* ln s/(n+g+ )/ T, which stands for

contiguous local characteristics. Hypothesis 3:

The steady-state position of a particular Chinese province and s, n, g and in its neighbouring Chinese provinces have a direct relationship.

This can be tested by verifying whether –( 4/ 1) in equation (xx) is significantly different

from zero. Hypothesis 4:

The steady-state position of a particular Chinese provinces is related to s, n, g and in its neighbouring Chinese provinces due to indirect and induced effects.

This can be tested by verifying whether –( 3/ 1) in equation (xx) is significantly different

from zero.

Some remarks on the model. As shown in Section 2.2, a necessary condition for convergence is that the coefficients of the initial income-level, both of the home province and of

neighbouring provinces, sum up to a value below zero: 1+ 3 < 0. When discussing the

results in Section 6, it is investigated whether this restriction is satisfied.

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different points of time. The inclusion of these fixed effects have great impact on the speed of convergence. To see why, we quote Elhorst, Piras and Arbia (2006; pp. 13):

‘… we can say that the growth model with fixed effects measures the time required before shocks caused by changes in s, n, g and are settled conditional upon these fixed effects, while the growth model without fixed effects measures the time required before these relatively persistent differences disappear as well.’ Following this argumentation, it is

reasonable to believe that the speed of convergence is supposed to be higher using the fixed-effects model.

A note on the way the model is estimated. Since the growth rates in different provinces are simultaneously determined, the model has been estimated by maximum likelyhood. We do this following Anselin and Bera (1998) and Arbia (2006), who treat spatial dependence in linear regression models rigorously. Just as in Elhorst, Piras and Arbia (2006) we make use of a MATLAB routine to estimate the model. This routine for cross-sectional was developed by LeSage (2005) and extended for spatial panels with and without fixed effects by Elhorst (2003).8

Besides the main objective, we investigate some additional objectives. The model outcomes give an idea on the value of the first-order spatial autoregressive lag model. Furthermore, we consider two tests ; one with respect to the spatial lag model, one with respect to the spatial error model. The first test is based on the significance level of the coefficients of the variables

W* ln(qt-T) and W* ln [s/(n+g+δ)]/T: 3 and 1. If these two coefficients are significantly

different from zero, the spatial lag model must be rejected. The second test is based on the model restrictions: 3 = - 1 and 4 = - 1. These restrictions are tested by means of a

Waldtest. The model outcomes provide values for 2(1) as well as attached probabilities. If these probalities are low enough, the restrictions must be rejected.

8_{Matlab routines as developed by LeSage are freely available at}_{http://www.spatial-econometrics.com}_{. Matlab}

routines as extended by Elhorst are freely available at

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5 Data issues

The data we use have already been collected by Zhang, Elhorst and Van Witteloostuijn (2006). The dataset they constructed contains all information I need for answering my main question. It consists of numerical time-series data for 29 Chinese provinces, over the period 1980-2002. The provinces are: Beijing, Tianjin, Hebei, Shanxi, Neimenggu, Liaoning, Jinlin, Helongjiang, Shanghai, Jiangsu, Zhejiang, Anhui, Fujian, Jiangxi, Shandong, Henan, Hubei, Hunan, Guangdong, Guangxi, Hainan, Sichuan, Guizhou, Xinjiang, Shaanxi, Gansu, Qinghai, Ningxia, Yunnan.

The variables we pick out of the data are GDP per capita in each province, domestic

investment per capita per province and population per province. Out of the data on GDP per capita we extract the initial GDP per capita per province in each time-span we consider and the growth rate of GDP per capita per province in each time-span we consider. Out of the data on domestic investment per capita we abstract the savings rate s. We do so by dividing

domestic investments by GDP for each province in each time-span. Out of the data on population we extract the population growth rate n in each province in each used time-span. Exogenous technological development, g, and depreciation, , are assumed to sum up to 0,05, as is commonly done.9 This number is assumed to be universal and not province- and time-specific. Therefore, the number of 0.05 is assumed for every province in every time-span. Since we are interested in growth dynamics between provinces and growth is a long-term process, the time-span is 23 years. In order to set up W as a spatial matrix, indicating which provinces are contiguous, we checked the map of China. Figure 3 provides the adopted map. It shows China’s provincial division. Table 11 provides an overview of which province is neighbouring which province. Some remarks on this point: firstly, Tibet and Taiwan are left out. Secondly, regional border were changing during the time span under study. This implies that the arrangement of provinces in 1980 is different from that in 2002. However, to be able to study data succesfully, a coherent set of data is needed. This means that the province Chongqing, which was part of Sichuan up till 14 march 1997 and is a distinct province now, is not studied separately.

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A note on the length of the considered time-span. As we use data from 1980 up to 2002, yearly observations are available for each provincial variable for 23 one-year time spans. However, besides stock variables we also consider flow variables. Since we do not know the growth rate in 1980, we can only start in 1981. This reduces the number of yearly

observations available for each provincial variable back to 22.

Since data are available for time-spans of one year, it seems intuitively appealing to consider the model using yearly observations. Economic growth in a particular province in a specific year is determined by initial levels of GDP per capita that year and by s, n, g and in that year.10_{This is an example of time-series cross-section (TSCS) data. However, there is one}

big but about this. Convergence is a longer term phenomenon and the model outcomes may be influenced by short run shocks too much in the case of TSCS based on yearly numbers. To avoid these short run shocks one might look at data in one single cross-section. This implies that data are averaged over the period 1980-2002. To put it more precize, s, n, g and are averaged per province for the whole period and only the initial levels of GDP per capita in 1980 are used instead of yearly differing initial levels of GDP per capita. Single cross-section modelling has some obvious disadvantages as well. It assumes that, besides g and , n and s are constant over the sample period. This does not correspond with the real-life fact that these variables change over time. It does not take this change into account. Furthermore, single cross-section modelling reads past the fact that convergence might be reached through different growth paths. Finally, single cross-section modelling does not make use of all available information. It makes use only of data at the beginning of the sample period and at the end. This makes the analysis less valuable.

Because of the disadvantages of these two extremes, an intermediate solution can be found in doing a time-series cross-section regression for a period somewhere in between. This brings a complication with it, since 22 as a figure can not easily be divided. We decided to split up the data up into two periods of five years and two periods of six years: 1981-1985, 1986-1990, 1991-1996, 1997-2002. This narrows the shortcomings of the above-mentioned single cross-section regression and TSCS regression based on yearly data.

In the pooled regression we are executing, the number of observations in the single cross-section equals the number of provinces : 29. The number of observations in the time-series cross-section regression using yearly data equals the number of provinces times the number of

10_{The initial GDP per capita can be seen as the end value of GDP per capita in the year the regarding time-span}

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years in the sample period : 29 * 22 = 638. For the time-series cross-section regression using five- and six-yearly data the number of observations becomes : 29 * 4 = 116.

6 Empirical analysis

This section provides the results of the empirical analysis. We perform this analysis by first looking at the results from the basic Solow-Swan model. Secondly, by looking at the key outcomes as given by the spatially extended regressions. Thirdly, by testing the hypotheses. Fourthly and finally, by answering to co-questions. All estimation results can be found in the Tables. Tables 1-5 exhibit the basic Solow-Swan results, whereas Tables 6-10 show the spatial results.11

6.1 Basic results

Tables 1 –3 provide the basic Solow-Swan OLS estimation results without fixed effects. These results include values for the parameters and . Table 1 gives the results modelling the data in one single cross-section over a sample period of 22 years. These results show that the sign of the initial income-level variable is negative, just asthe sign of the variable ln

[s/(n+g+δ)]/T. Both variables have an insignificant effect. The convergence speed, given by

, is 0.49% per annum. The parameter α, giving the capital share of output, is 1.38. This value is outside the interval on which it is defined: 0 < < 1. Tables 2 and 3 respectively give the results in the TSCS approaches over a one-year time-span and over a five-year time-span. Moving to these Tables one might notice different results. According to both TSCS

approaches the sign for the initial income-level variable is negative, just as according to the single cross-section approach. Furthermore, the constant takes quite the same value. This time, however, in both TSCS cases the variable has a significant effect. Looking at the sign for the variable ln [s/(n+g+δ)]/T, one can see that both in the case of yearly observations as

well as in the case of of five- and six-yearly observations the sign is positive. Nevertheless, only in the case of yearly observations the variable ln [s/(n+g+δ)]/T has a significant effect.

The speed of convergence according to the TSCS approaches is much higher than according to single cross-section approach. When pooling time-series cross-sectional data over one-year

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time spans, it is 1.98% per year, pooling TSCS data over five- and six-year time spans, it is 1.54%. The parameter α for yearly observations is calculated to be 0.77, whereas the share for five- and six-yearly observations is calculated to be 0.67.

When comparing the different basic Solow-Swan model regressions without fixed effects, large differences exist between the TSCS results and the single-cross section results. The single-cross section Table includes some remarkable results. Firstly, the variable ln

[s/(n+g+δ)]/T shows a negative sign, indicating that the GDP growth rate per capita depends

negatively on its own region’s savings rate. Secondly, the two relevant variables, the initial income-level and ln [s/(n+g+δ)]/T, are not significant. Thirdly, the speed of convergence per

year, 0.49 is remarkably low. Finally, α takes a value outside the interval on which it is defined. This drives me towards the TSCS approaches. The TSCS approaches show more plausible results in that they stroke better with common sense, they are better defended by earlier works and they are better supported by theoretical predictions. The variable ln

[s/(n+g+δ)]/T shows a positive sign, the α is within the interval on which it is defined, the

speed of convergence is more realistic and, including the constant, five out of the six

explanatory variables are significant. Only the variable ln [s/(n+g+δ)] for the OLS case over

five- and six-yearly observations is not significant.

Tables 4 and 5 provide the basic Solow-Swan OLS estimation results with fixed effects. These results only apply for the time-series cross-section approaches, respectively for one-year time-spans and for five- and six-one-year time-spans. The results are less diverse as between the TSCS regressions without fixed effects. Both variables, the initial income-level and ln

[s/(n+g+δ)]/T, show a significant positive effect.12_{The speed of convergence is, respectively}

for one-year and for five- and six-year time-spans, 9.25% and 7.00% per annum. These figures might seem striking high, also compared to the results without fixed effects. However, the fact that the convergence speed is higher than in the case without fixed effects strokes well with previous studies (Islam, 1995; Caselli et al., 1996; Elhorst, Piras and Arbia, 2006). The values found for the parameter α are a lot lower than in the model without fixed effects. In the case of one-year time-spans the value for α is 0.20. When modelling over five- and six-year time-spans the value for α is 0.37. These values for are much more in line with expectations

than in the case without fixed effects. We build these expectations on the work of Bai et al.

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(2006). This work estimates that the real rate of return to capital in China was around 25 percent during 1978-93, fell during 1993-98, and fluctuated around 20 percent since 1998.

6.2 Spatial results

The estimation results for the SAR model are reported in Tables 6-10. Before looking at the bare estimation results, we take a glance at some other aspects. Recall from Section 2.2 and Section 3.1 that convergence only takes place if the sum of the coefficients of the variables that account for the initial income level at home and across the regional borders is negative. Looking at the sum of 1 + 3 in the five regressions, We notice that the sum is negative four

times. Only in the single cross-section regression without fixed effects, this condition for convergence to take place is not satisfied. This makes the the single cross-section over 22 years without fixed effects suspicious. Another aspect of interest is the explanatory power. The explanatory power of the model is indicated by R2 in every regression. The results exhibit

a strong pattern for the value of R2. Overall, the value of R2 increases when extending the time-span, when including fixed effects and when including spatial effects. For example, the spatial results for one-year time-spans with fixed effects show a higher explanatory power than spatial results for one-year time-spans without fixed effects. Or spatial results without without fixed effects for five- and six-year time-spans exhibit a higher value than the non-spatial results without fixed effects for the same time-spans. The only exception is the non-spatial single cross-section model. This case has already been put under suspicion, now it is even more.

In all SAR model estimates, spatial variables are added concerning the GDP per capita growth rate, the initial income-level and ln [s/(n+g+δ)]/T. In addition to the coefficients and their

degree of significance, we also look at the parameters , and .

Tables 6 – 8 provide the spatially extended Solow-Swan results without fixed effects. To start with similarities between the results, the three Tables show that in the case without fixed effects the three variables denoted as ln [s/(n+g+δ)]/T, contiguous GDP per capita growth

rate and contiguous initial income-level show a positive sign over every time-span. The two variables initial income-level and contiguous ln [s/(n+g+δ)]/T are negatively signed, again

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regression over five- and six-year time-spans shows the most significant effects. The largest values are found for the variable that accounts for the contiguous GDP growth rate per capita, especially when following the TSCS approaches.

The negative sign of the variable W*ln [s/(n+g+δ)]/T seems to contradict the common idea

that the growth rate of income per capita in one region is positively related to the saving rates of neighbouring regions. However, this coefficient only measures the direct effect. In order to find the overall effect, one should look at both the variables ln [s/(n+g+δ)]/T and W*ln

[s/(n+g+δ)]/T. If the sum of the coefficients of these two variables is positive, that is if

indirect and induced effects are included as well, one obtains the overall effect. The sum of the coefficients, 2 + 4, is positive in four of the five regressions. Only in the single

cross-section regression (without fixed effects) it is not. Since this regression is very much disputable we assume the overall effect of the neighbouring saving rates to be positive. Next, we look at the parameters. The speed of convergence, , over five- and six-year time-spans is 1.70% per annum. This is higher than over yearly time-time-spans (1.50%) and over a cross-section time span of 22 years (1.58%). The values for according to the TSCS

approaches satisfy the condition for convergence mentioned in section 2.2 and section 5.2.2. On the other hand, the value for according to the single cross-section approach does not. The capital share per output, , shows values that are inside the interval on which it is defined. Using the SAR estimator based on yearly observations, the value for is 0.07. Based on five- and six-yearly observations, the value becomes 0.58, and based on a single cross-section of 22 years it is 0.69. These values are odd, considering the expectations we have on the capital share of output. Following the theoretical foundation set up in section 2, the value for , indicating physical capital externalities, must be inside the interval (0,1). In case of yearly observations, the parameter takes the value of 0.51. However, in the other two cases the value takes a negative sign, which is obviously outside the interval on which it is defined.13

13_{For the article “Growth and Convergence in a Multi-Regional Model with Space-Time Dynamics” by Elhorst,}

Piras and Arbia (2006) Elhorst has done some Matlab programming that provides corrected values for and . These values are corrected as following:

α~= | | .[( + )/(| |+| |)], ϕ~= | | . [( + )/(| |+| |)],

where α~denotes the corrected value of and ϕ~ the corrected value of .

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As the single-cross section regression without fixed effects exhibits such disputable results, we will ignore this regression from now on.

Tables 9 and 10 give the SAR model estimates for the model with fixed effects. Again, as in the case without spatial variables, results only apply for the time-series cross-section

approaches, Tables 9 and 10 respectively for one-year time-spans and for five- and six-year time-spans. Both regressions show a positive sign for all explanatory variables, except for the variable home initial income-level, which shows a negative sign. Furhermore, apart from the variable contiguous ln [s/(n+g+δ)], all variables show a significant effect in both regressions.

In both cases the variable accounting for contiguous GDP growth per capita shows the largest coefficient estimate.

The convergence speed with fixed effects for one-year time-spans and for five- and six-year time-spans are 10.10% per year and 8.03 % per year, respectively. In both cases, the value for

satisfies the condition for convergence. The parameter takes values of 0.42 and 0.36 for one-year time-spans and five- and six-year time-spans, respectively. In both cases the parameter has a negative sign.14

6.3 Testing the hypotheses

Going through the spatial results, we discuss the hypotheses tested in the present paper. To start with the first hypothesis, we look at the parameter ρ~ . In all four accepted spatial regressions this parameter is significantly different from zero. In all four cases the

(asymptotic) t-value is very high. More generally speaking, the largest coefficient values are, as stated earlier, found for the variable that accounts for the contiguous GDP growth rate per capita, ρ~ . All together we have found strong evidence in favour of the first hypothesis, which states that the growth rate of a particular Chinese province is related to that of its contiguous Chinese provinces.

The second hypothesis, whether the rate of growth of a particular Chinese provinces is affected by s, n, g , in its contiguous Chinese provinces, is tested by looking at 4 in

equation (xiii), the coefficient of the variable W* ln [s/(n+g+δ)]/T. As noted before, this

variable has a negative sign in the regressions without fixed effects, and a positive sign in the regressions with fixed effects. Only in the model which seems to be less fitted for use here –

14_{The corrected value for and for in the fixed-effects TSCS spatially extended regressions respectively are}

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the single cross-section sar model- the coefficient is significant. Since in all cases 4 is not

significantly different from zero, the second hypothesis must be rejected.

In order to mention anything about the third hypothesis, we look at –( 4/ 1) in equation (xx).

Tables 6 – 10 also give results for this set of coefficients, which are derived from equation (xiii). The coefficient of the variable W* ln [s/(n+g+δ)] is divided by the coefficient of

ln(qt-T) to look for the direct relationship between the steady-state position of a particular

Chinese province and s, n, g and in its neighbouring Chinese provinces. Looking for ( 4/ 1)

in the tables, one sees that -( 4/ 1) has a negative value when studied without fixed effects,

and a positive value when studied with fixed effects.15 In neither case ( 4/ 1) is significantly

different from zero, as a result of which the third hypothesis must be rejected.

The fourth and last hypothesis states that the steady-state position of a particular Chinese province is related to s, n, g and in its contiguous Chinese provinces due to indirect and induced effects. Studying -( 3/ 1) is to provide the answer. The value for ( 3/ 1) in equation

(xx), distracted from the values of 3 and 1 in equation (xiii), is negative in all regressions.

This implies that the value for -( 3/ 1) is positive in all cases. Furthermore, ( 3/ 1), and

therefore -( 3/ 1), is significantly different from zero. Therefore we conclude that we have

strong evidence to support the hypothesis that the steady-state position of one Chinese province is related to s, n, g and in its neighbouring Chinese provinces due to indirect and induced effects.

6.4 Model testing

The model as developed theoretically by Ertur and Koch (2005) and applied empirically by Elhorst, Piras and Arbia (2006), obviously, includes spatial variables. These variables are added for their supposed relevance. If such relevant variables are omitted from the equation, the estimated coefficients are biased and inconsistent. Elhorst, Piras and Arbia (2006) already checked for biasness, by comparing the the speed of convergence in the OLS regressions with that in the SAR model with spatial variables. They concluded that “…we can say that the

objection against previous studies on economic growth, that they suffer from an omitted regressor bias if the spatially lagged dependent variable is ignored, is correct…”.

We did the same check on biasness. In all accepted cases, the inclusion of spatial variables increases the speed of convergence. Based on the Chinese data from 1980 till 2002, this

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makes me to agree, cautiously, with their conclusion that extra explanatory spatial variables should be included.

Should this be done by the first-order spatial autoregressive lag model? To get to the answer, we first test the spatial lag model and the spatial error model. In Section 3 we mentioned the spatial lag model and the spatial error model as being the two main spatial econometric models. The first-order spatial autoregressive lag model as applied in the present paper is (slightly) different from both models. Section 4 provided a way to test whether the spatial lag model and the spatial error model are acceptable on the data. Given the estimation results, one can easily see that the spatial lag model must be rejected. This is because 3 or 4 is

significantly different from zero in every regression. This implies that the spatial lag model is not acceptable when describing growth using Chinese provincial data over the period 1980-2002. It seems to be suffering from biasness and from too much simplicity. To test the spatial error model restrictions, 3 = - 1 and 4 = - 1, Wald-tests are performed. Tables 6-10

report values for this test. Except for the suspicious single-cross section model without fixed, the values show that the probabilities are too high to reject the restrictions. In other words, the restrictions can not be rejected and the spatial error model seems acceptable on the data. The fact that the spatial error model restrictions can not be rejected is a negative result for the value of the first-order spatial autoregressive lag model. Another harmful finding is that the parameter values are outside the interval on which they are defined in many instances. These empirical findings do not stroke with the theoretical background as developed by Ertur and Koch (2005).

6 Conclusions

We have tried to find an answer whether the growth rate in one Chinese province is related to that of its contiguous provinces. Although this has been done before, we have used a new specification that is called the first-order spatial autoregressive lag model, as it was done by Elhorst, Piras and Arbia (2006) for European regions, to answer this question. Together with the answer to this question, some other conclusion can be made as well.

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notion that the convergence speeds are underestimated in the empirical case without spatial effects. This suggests that studies that do not include spatial variables suffer from an omitted regressor bias. The point made here is strengthened by the fact that the explanatory power, indicated by R2, increases strongly when adding spatial effects.

In order to study spatial dependencies between the Chinese provinces we have used the first-order spatial autoregressive lag model. We have formulated four hypotheses that we have tested. The first, main, hypothesis, whether the growth rate in one Chinese province is related to that of its neighbouring provinces’growth rate, is accepted. This implies that the main question can be answered with a ‘yes’: Provincial growth rates are determined by contiguous growth rates to a large extent. The second and third hypotheses, whether the growth rate and the steady-state position of a particular Chinese province is directly affected by the local characteristics s, n, g and in neighbouring provinces respectively, must be rejected. No significant evidence is found which supports these suppositions. The fourth hypothesis is accepted, implying that the steady-state position of a particular Chinese province is related to the local characteristics s, n, g and in its contiguous Chinese provinces, due to indirect and induced effects. One can conclude that Chinese provinces’ growth rates as a matter of fact do depend on each other, although no significant direct link is found between growth rates and contiguous local characteristics.

Besides looking for answers regarding to China, the present paper also provides a means of testing the empirical model we use, the first-order spatial autoregressive lag model.

Summarized, the model has not passed the test very well. No evidence has been found to reject the restrictions that are part of the spatial error model. This implies that we can not use the surplus of restrictions as an arguments to use the autoregressive lag model instead of the spatial error model.

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A thing to note about the present paper is that it finds convergence to take place in all accepted regressions. In section 1 we stated that divergence took place between China’s provinces over the period 1980-2002. Evidently, these two conclusions do not match.16 A possible reason is the pattern of convergence and divergence. Following Bing (2003) we know that China’s provincial growth rates converged in the 1980s and diverged in the 1990s. Therefore, it might be interesting for further researchers to test whether the outcomes differ when splitting up the sample period.

16_{Considering the fact that we expect to find divergence to take place puts the rejected spatial single}

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Tables and figures

Table 1, non-spatial solow-swan model, cross-section – 22 years period, without fixed effects

Dependent Variable = ln(qt/qt-T) / T R-squared = 0.1167 Rbar-squared = 0.0487 sigma^2 = 0.0002 Durbin-Watson = 2.1177 Nobs, Nvars = 29, 3 *************************************************************** Variable Coefficient t-statistic t-probability

constant 0.188082 6.222275 0.000001 ln(qt-T) -0.004674 -0.841330 0.407835 ln [s/(n+g+δ)]/T -0.016942 -0.916447 0.367855 *************************************************************** Speed of convergence = 0.4933 Implied = 1.3810

Table 2, , non-spatial solow-swan model, TSCS - yearly observations, without effects

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Table 3, non-spatial solow-swan model, TSCS – five- and six-yearly observations, without fixed effects Dependent Variable = ln(qt/qt-T) / T R-squared = 0.0666 Rbar-squared = 0.0501 sigma^2 = 0.0022 Durbin-Watson = 0.3061 Nobs, Nvars = 116, 3 *************************************************************** Variable Coefficient t-statistic t-probability

constant 0.195483 6.587572 0.000000 ln(qt-T) -0.014788 -2.700203 0.007995 ln [s/(n+g+δ)]/T 0.029829 1.284594 0.201561 *************************************************************** Speed of convergence = 1.5425 Implied = 0.6686

Table 4, , non-spatial solow-swan model, TSCS – yearly observations, with regional and time-period fixed effects

ln(qt-T) -0.088364 -7.197284 0.000000

ln [s/(n+g+δ)]/T 0.021460 2.612728 0.009195

***************************************************************

Speed of convergence = 9.2514

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Table 5, , non-spatial solow-swan model, TSCS – five- and six-yearly observations, with regional and time-period fixed effects

ln(qt-T) -0.058133 -5.371137 0.000000

ln [s/(n+g+δ)]/T 0.033570 2.924626 0.004162

***************************************************************

Speed of convergence = 7.0049

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Table 6, sar estimator, cross-section – 22 years period, without fixed effects Dependent Variable = ln(qt/qt-T) / T R-squared = 0.4111 Rbar-squared = 0.3130 sigma^2 = 0.0001 Nobs, Nvars = 29, 5 log-likelihood = 101.10229 # of iterations = 15

min and max rho = -1.0000, 1.0000

total time in secs = 1.1720

time for lndet = 0.2970

time for t-stats = 0.0630

No lndet approximation used

*************************************************************** Variable Coefficient Asymptot t-stat z-probability

constant 0.089617 1.524986 0.127263 ln(qt-T) -0.013350 -2.500486 0.012402 ln [s/(n+g+δ)]/T 0.009373 0.494669 0.620834 W* ln(qt-T) 0.034458 3.266988 0.001087 W* ln [s/(n+g+δ)]/T -0.078429 -2.370064 0.017785 W* ln(qt/qt-T) / T 0.164961 0.728127 0.466536 *************************************************************** Speed of convergence = 1.5806 Implied + implied = 0.4125 Implied = 0.6948 Implied = -0.2823 Implied + implied = 0.4125 Implied (corrected) = 0.2933 Implied (corrected) = 0.1192 Implied = 4.9680 2.1458 11.7675 6.9567 Stationarity = 0.0211 ρ~ / 1 (+ t-value) = -0.7021 (-0.5265) 3 / 1 (+ t-value) = -2.5811 (-2.6814) 4 / 1 (+ t-value) = 5.8747 (1.7220)

Waldtest restr. 1 (+ prob.) = 9.1100 (0.0025)

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Table 7: sar estimator, TSCS – yearly observations, without fixed effects Dependent Variable = ln(qt/qt-T) / T R-squared = 0.6892 Rbar-squared = 0.6867 sigma^2 = 0.0016 Nobs,Nvar,TNvar = 638, 5, 6 log-likelihood = 1085.5464 # of iterations = 15

min and max rho = -1.0000, 1.0000

time for lndet = 0.0620 time for t-stats = 0.0320 No lndet approximation used

constant 0.044255 3.424302 0.000616 ln(qt-T) -0.014857 -3.399855 0.000674 ln [s/(n+g+δ)]/T 0.020172 2.553651 0.010660 W* ln(qt-T) 0.009004 1.879985 0.060110 W* ln [s/(n+g+δ)]/T -0.000634 -0.056621 0.954847 W* ln(qt/qt-T) / T 0.768962 32.058886 0.000000 *************************************************************** Speed of convergence = 1.4968 Implied = 0.0658 Implied = 0.5101 Implied + implied = 0.5759 Implied (corrected) = 0.0658 Implied (corrected) = 0.5101 Implied = 0.2752 0.2752 0.2752 0.2752 Stationarity = -0.0059 ρ~ / 1 (+ t-value) = -1.3578 (-2.2976) 3 / 1 (+ t-value) = -0.6061 (-3.2738) 4 / 1 (+ t-value) = 0.0427 (0.0567)

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Table 8: sar estimator, TSCS – five- and six-yearly observations, without fixed effects Dependent Variable = ln(qt/qt-T) / T R-squared = 0.8193 Rbar-squared = 0.8111 sigma^2 = 0.0004 Nobs,Nvar,TNvar = 116, 5, 6 log-likelihood = 271.28368 # of iterations = 21

min and max rho = -1.0000, 1.0000

time for t-stats = 0.0150

*************************************************************** Variable Coefficient Asymptot t-stat z-probability constant 0.029510 1.815879 0.069389 ln(qt-T) -0.016258 -3.154761 0.001606 ln [s/(n+g+δ)]/T 0.021355 1.725835 0.084377 W* ln(qt-T) 0.014325 2.427117 0.015219 W* ln [s/(n+g+δ)]/T -0.019452 -0.944051 0.345143 W* * ln(qt/qt-T)/T 0.856979 21.590000 0.000000 *************************************************************** Speed of convergence = 1.7031 Implied = 0.5759 Implied = -0.0081 Implied + implied = 0.5678 Implied (corrected) = 0.5599 Implied (corrected) = 0.0079 Implied = 0.8980 0.8653 0.9237 0.8945 Stationarity = -0.0019 ρ~ / 1 (+ t-value) = -1.3135 (-1.7359) 3 / 1 (+ t-value) = -0.8811 (-4.1951) 4 / 1 (+ t-value) = 1.1964 (0.9208)

Waldtest restr.1+2 (+ prob.) = 0.0156 (0.9922)

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Table 9: sar estimator, TSCS – yearly observations, with fixed effects Dependent Variable = ln(qt/qt-T) / T R-squared = 0.7763 Rbar-squared = 0.7555 sigma^2 = 0.0012 Nobs,Nvar,TNvar = 638, 4, 55 log-likelihood = 1246.6826 # of iterations = 15

min and max rho = -1.0000, 1.0000

ln(qt-T) -0.096072 -7.454640 0.000000 ln [s/(n+g+δ)]/T 0.017539 2.133853 0.032855 W* ln(qt-T) 0.044874 2.272843 0.023036 W* ln [s/(n+g+δ)]/T 0.009383 0.577448 0.563637 W* ln(qt/qt-T)/T 0.286987 6.051164 0.000000 *************************************************************** Speed of convergence = 10.1006 Implied = -0.2644 Implied = 0.4188 Implied + implied = 0.1544 Implied (corrected) = 0.0597 Implied (corrected) = 0.0946 Implied = 0.3124 0.4201 -1.3824 -0.4811 Stationarity = -0.0512 ρ~ / 1 (+ t-value) = -0.1826 (-2.0231) 3 / 1 (+ t-value) = -0.4671 (-2.4608) 4 / 1 (+ t-value) = -0.0977 (-0.5772)

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Table 10: sar estimator, TSCS – five- and six-yearly observations, with fixed effects Dependent Variable = ln(qt/qt-T) / T R-squared = 0.9326 Rbar-squared = 0.9018 sigma^2 = 0.0002 Nobs,Nvar,TNvar = 116, 4, 37 log-likelihood = 340.47603 # of iterations = 19

min and max rho = -1.0000, 1.0000

ln(qt-T) -0.064904 -6.350333 0.000000 ln [s/(n+g+δ)]/T 0.024477 2.257455 0.023980 W* ln(qt-T) 0.045606 2.775967 0.005504 W* ln [s/(n+g+δ)]/T 0.003433 0.166168 0.868024 W* ln(qt/qt-T) / T 0.491978 5.304798 0.000000 *************************************************************** Speed of convergence = 8.0284 Implied = -0.0814 Implied = 0.3553 Implied + implied = 0.2739 Implied (corrected) = 0.0511 Implied (corrected) = 0.2228 Implied = 0.4718 0.5377 -0.7524 -0.1073 Stationarity = -0.0193 ρ~ / 1 (+ t-value) = -0.3771 (-2.0937) 3 / 1 (+ t-value) = -0.7027 (-3.0214) 4 / 1 (+ t-value) = -0.0529 (-0.1661)

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Figure 2, Solow-Swan equilibrium

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Figure 3, provincial map of China, 2007