CESIS Electronic Working Paper Series

CESIS

Electronic Working Paper Series

Paper No. 51

On the specification of regression models with spatial dependence

¹

- an application of the accessibility concept

Martin Andersson (JIBS) & Urban Gråsjö (HTU)

Dec 2005

The Royal Institute of technology Centre of Excellence for studies in Science and Innovation http://www.infra.kth.se/cesis/cesis/publications/working_papers/index.htm Corresponding author:martin.andersson@ihh.hj.se

1 Status of the paper: will be presented at the 45^th annual meeting of the Western Regional Science Association, Santa Fe, New Mexico, US

On the specification of regression models with spatial dependence

- an application of the accessibility concept

^ψ

Martin Andersson^a & Urban Gråsjö^b

(martin.andersson@ihh.hj.se / urban.grasjo@htu.se)

Abstract

Using the taxonomy by Anselin (2003), this paper investigates how the inclusion of spatially discounted variables on the ‘right-hand-side’ (RHS) in empirical spatial models affects the extent of spatial autocorrelation. The basic proposition is that the inclusion of inputs external to the spatial observation in question as a separate variable reveals spatial dependence via the parameter estimate. One of the advantages of this method is that it allows for a direct interpretation. The paper also tests to what extent significance of the estimated parameters of the spatially discounted explanatory variables can be interpreted as evidence of spatial dependence. Additionally, the paper advocates the use of the accessibility concept for spatial weights. Accessibility is related to spatial interaction theory and can be motivated theoretically by adhering to the preference structure in random choice theory. Monte Carlo Simulations show that the coefficient estimates of the accessibility variables are significantly different from zero in the case of modelled effects. The rejection frequency of the three typical tests (Moran’s I, LM-lag and LM-err) is significantly reduced when these additional variables are included in the model.

When the coefficient estimates of the accessibility variables are statistically significant, it suggests that problems of spatial autocorrelation are significantly reduced. Significance of the accessibility variables can be interpreted as spatial dependence

JEL Classifications: R15, C31, C51

Keywords: accessibility, spatial dependence, spatial econometrics, Monte Carlo Simulations, spatial spillovers

1. Introduction

The so-called ‘1^st law of geography’ (Tobler, 1970) states that everything in space is related but the relatedness of things decreases with distance. In any research that acknowledges such a law,

ψ This paper is part of Urban Grasjö’s dissertation.

a Jönköping International Business School (JIBS) and Centre for Science and Innovation Studies (CESIS).

JIBS, P.O Box 1026, SE-551 11, Jönköping Sweden

bCorresponding Author: University of Trollhättan/Uddevalla (HTU) and Centre for Science and Innovation Studies (CESIS). HTU, P.O. Box 795, SE-451 26 Uddevalla, Sweden

spatial dependence among spatial units should be perceived as a generic occurrence that is subject to distance-related friction phenomena. Spatial dependence implies e.g. that activities in one region have an effect on the activities in another region. Distance enters in the sense that the strength of such inter-regional effects decreases with the distance between the regions. The axiom in regional science, for instance, is that ‘interaction decreases with distance’, (Beckmann, 2000).

Several theoretical approaches in regional and urban economics explicitly presume the existence of spatial dependence². The theoretical underpinnings in the literature on the relationship between geography and innovation, for instance, focus extensively on various forms of spatial externalities, (Audretsch, 2003). Spatial externalities refer to externalities that are distance sensitive and whose spatial range is limited. Of course, such externalities can be either pecuniary (mediated by the market) or technological (pure externalities), (cf. Breschi & Lissoni, 2001a,b & Johansson, 2004). Spatial externalities essentially imply that the ‘inputs’ in a given location has an effect on the ‘outputs’ in other locations and that this effect in turn diminishes with distance. In this rough sense, the existence of spatial externalities implies the existence of spatial dependence.

This paper starts from the proposition that any empirical spatial model that is formulated with reference to a theory which assumes spatial externalities – and thus spatial dependence – should be specified such that it treats space as being continuous. This essentially means that the model should include the ‘inputs’ in an area larger than the spatial observations themselves. The paper also proposes that by including inputs external to the spatial observation in question as a separate variable, spatial dependence is revealed by the parameter estimate, which allows for direct interpretation. The purpose of the paper is: (i) to examine if the inclusion of spatially discounted variables on the ‘right-hand-side’ (RHS) in empirical spatial models removes (or reduces) the problem of spatial autocorrelation among the residuals, (ii) to test if significance of the estimated parameters of the spatially discounted variables implies that estimation of a model without such variables leads to spatial autocorrelation. In order to achieve this, the paper utilizes Swedish data on R&D and patent applications to the European Patent Office (EPO) to estimate a Knowledge Production Function (KPF). Using Monte-Carlo simulations, the paper also presents some tests of how the inclusion of spatially discounted variables on the RHS affect the test statistics of the most common tests for spatial dependence. The ideas in the present paper are related to the work by McMillen (2003), who stresses the importance of model specification in spatial econometric models and maintains that (spatial) autocorrelation is often produced spuriously by model misspecification. McMillen’s (2003) results suggest that (p.215) “… omitted explanatory variables that are correlated over space and misspecified spatial effects will produce spatially correlated residuals, even when the true model errors are independent”.

A fundamental problem in applied spatial econometrics concerns the specification of the spatial interaction structure, i.e. the structure of the spatial weight matrix, (Florax & Rey, 1995).

Moreover, Fingleton (2003) remarks that the spatial weight matrix applied in many empirical studies is not underwritten by a strong theory and that the assumptions behind the chosen weight matrix are often not tested. In the context of the present paper, the inputs in other spatial units

2 Spatial dependence has implicitly been emphasized for a long time in urban and regional economics. The classic work by Harris (1954), for instance, illustrated how the location of production is affected by the market potential. In Harris’ study the market potential of a location was defined as the location’s internal demand along with the distance-weighted sum of the demand at other locations. Such effects relate to the traditional Central-Place-System (CPS) framework, in which consumers travel to the central location to consume high-order goods, (c.f. Dicken &

Lloyd, 1990).

should optimally be spatially discounted in a way that reflects the distance sensitiveness of the effects (or externalities) involved. With respect to the spatial discounting procedure, this paper advocates the use of accessibility as a measure of potential opportunities, (c.f. Weibull, 1980).

Throughout the paper, the spatial weight matrix is based on the concept of accessibility as a measure of potential of opportunities.

The remainder of the paper is organised as follows: in Section 2 an overview of standard spatial dependence models and some standard tests are presented. Section 3 discusses the accessibility concept and shows how it can be used to incorporate spatially discounted variables on the RHS in empirical spatial models. Section 4 presents an empirical application of the method described in Section 3 by estimating a KPF across Swedish municipalities. Section 5 presents Monte Carlo simulations of the method applied in Section 4 to investigate to what extent the results in Section 4 can be generalized. Conclusions are given in Section 6.

2. Spatial dependence in Empirical Models

Potential statistical problems associated with dependence among observations in cross-sectional data are extensively treated in spatial econometrics literature (e.g. Anselin, 1988a; and Anselin &

Florax, 1995). Anselin (1988a) refers to two types of spatial dependence: substantive spatial dependence and nuisance dependence (see also Anselin & Florax, 1995 and Florax & van der Vlist, 2003). The first deals with the spatial interaction of the variable of interest, e.g. the dependent variable of the regression model. The second is about the spatial dependence between the ignored variables in the model, which reflects the error terms. While substantive spatial dependence necessitates the development of spatially explicit models, nuisance dependence involves adjustments of existing specifications, for example to express neighbourhood effects in the model (Dubin, 1992).

2.1 Specifications accounting for spatial dependence

A presence of any kind of spatial dependence can invalidate regression results. In the case of spatial error autocorrelation, OLS parameter estimates are inefficient and in presence of spatial lag dependence parameters become biased and inconsistent (Anselin, 1988a). The general expression for the spatial lag model is:

u x Wy

y=ρ + β+ , (2.1)

where y is the dependent variable, W is a spatial weight matrix, Wy is a vector of lagged dependent observations ρ is a spatial autoregressive parameter, x is a matrix of independent variables, β is a vector of regression parameters and u is a vector of independent disturbance terms, u ~ N(0,σ²).

The standard spatial model with autoregressive disturbances represents an alternative form of spatial dependence. Spatial error autocorrelation is modelled as follows:

ε β +

= x

y _(2.2a)

u W +

=λ ε

ε (2.2b)

where

ε

is the spatially autoregressive error term, λ is the parameter of the spatially autoregressive errors Wε. The reduced form of (2.2a) then becomes:

u W I x

y= β+( −λ )⁻¹ (2.3)

The researcher’s job is to determine which model (spatial lag or spatial error) that best fits the data. Thus, the job is to determine whether ρ = 0 or λ = 0. It could be the case that both differ from zero and the question is then which model to choose³.

The standard taxonomy of spatial lag and error models is further developed in Anselin (2003) (see also Anselin & Bera, 1998; and Anselin, 2001). He distinguishes between a global and a local range of dependence, and the way in which this affects a regression specification of spatially lagged dependent variables (Wy), spatially lagged explanatory variables (Wx) and spatially lagged error terms (Wu). The developed taxonomy has two dimensions. The primary dimension is whether the spatial correlation in the reduced form pertains only to unmodelled effects (error terms), to modelled effects (included explanatory variables), or to both.

Specification tests and theoretical arguments should suggest the nature of the externalities and dictate the proper alternative. The second dimension in the taxonomy is the distinction between global and local spillovers. In the reduced form this boils down to the inclusion of a spatial multiplier effect of the form (I – λW)^-1 versus a simple spatial lag term using spatial weights W. ⁴ The taxonomy is presented in Table 2.1. Note that Wy only appears on the RHS for models that incorporate global spillovers.

Table 2.1: Taxonomy of Structural Forms

Local externalities Global externalities Unmodelled

effects, u y=xβ+u+γWu y=λWy+xβ−λWxβ+u Modelled effects,

x y=xβ+Wxρ+u y=ρWy+xβ+u−ρWu Wu

u Wx x

y= β+ ρ+ +γ y=(ρ+λ)Wy−ρλW²y+xβ−λWxβ+u−ρWu Both u and x

Wu u Wx x

y= β+ ρ+ +ρ y=ρWy+xβ+u Source: Anselin (2003)

If considering modelled effects and global spatial spillovers the specification is:

u x W I

y=( −ρ )⁻¹ β+ (2.4)

and after multiplying with (I –ρW):

3If ρ = 0 and λ = 0 then OLS is applicable (if other necessary conditions are met). Anselin & Rey (1991) showed in a Monte Carlo study that if tests for spatial lag and spatial error are both significant, the larger of the two statistics probably indicates the correct model.

4 The definition of geometric series gives (I−ρW)⁻¹=I+ρW+ρ²W²+..., and the interpretation is that every location is correlated with every other location in the system, but closer locations more so (since in most cases

<1 ρ ).

Wu u

x Wy

y=ρ + β + −ρ (2.5)

This model contains both a spatially lagged dependent variable as well as a spatial moving average (SMA) error.⁵ When there are local spillovers in the explanatory variables the specification is instead:

u Wx x

y= β+ ρ+ ^(2.6)

where ρ is not a scalar as in (2.4) but a column vector matching the column dimension of Wx. If W is a 1^st order contiguity matrix, this model would be appropriate when the proper spatial range of the explanatory variables is the location and its immediate neighbours (and not neighbours’

neighbours).⁶

Section 3 demonstrates how the inclusion of accessibilities on the RHS can account for global spillovers without estimating an equation like (2.5), which requires Maximum Likelihood (ML).

The general idea is to use an expression similar to the one for local spillovers (2.6) but with a weight matrix, that incorporates all locations (not only the neighbours).

2.2 Tests for spatial dependence

There are several test developed to detect spatial dependence. The most widely applied test statistic is Moran’s I. It was originally adopted only on single variables but Cliff and Ord (1972) and Hordijk (1974) applied the principle for spatial autocorrelation to the residuals of regression models for cross-sectional data. Computation of Moran’s I is achieved by division of the spatial co-variation by the total variation. Resultant values are in the range from -1 to 1. The general formula for computing Moran’s I is:

⎟⎠

⎜ ⎞

⎝

= ⎛

e e

We e S I N

'

' (2.7)

where

N = number observations, S = sum of all elements in W, W = spatial weight matrix and e = vector of residuals (OLS).

The test on the null hypothesis that there is no spatial autocorrelation between observed values over the N observations can be conducted based on the standardised statistic:

5 ,

)0

( ) ) (

( V I

I E I I

Z = − (2.8)

where

5 Equation (2.5) is a special case of the spatial autoregressive moving average (SARMA) model, see e.g. Huang 1984; Anselin, 1988; Anselin & Florax, 1995; Anselin & Bera 1998.

6 A first order contiguity matrix is a matrix, which has non-zero elements for locations with common boundaries.

2 2

2 2

)) ( ) (

2 )(

(

)) (

) ( ) ' ) (

(

) ) (

(

I K E

N K N

trMW MW

tr MWMW tr

S I N V

K N

MW tr S I N E

⎟⎟−

⎠

⎜⎜ ⎞

⎝

⎛

+

−

−

+

⋅ +

⎟⎠

⎜ ⎞

⎝

=⎛

⋅ −

=

(2.9)

where

tr = trace operator (sum of the diagonal elements), M =I−x(x'x)⁻¹x' (Projection matrix) and K = number of explanatory variables.

In Anselin & Rey (1991), an extensive set of simulations experiments was carried out, comparing Moran’s I to two Lagrange multiplier tests for a wide range models and error distributions, and for both error and lag forms of spatial dependence. Their conclusion confirmed the theoretical findings on the power of Moran’s I (King 1981), but also indicated a tendency for this test to have power against several types of alternatives, including non-normality, heteroscedasticity and different forms of spatial dependence. The weakness of Moran’s I is that it does not have a direct correspondence with a particular alternative hypothesis, i.e. if it is spatial error dependence or spatial lag dependence that is detected.

Tests for spatial error versus spatial lag can be conducted by using the Lagrange Multiplier (LM) principle (see e.g Burridge, 1980; Anselin, 1988b; Anselin & Florax, 1995). The LM-err statistic is defined as:

2 2

2

' ' ) '

( ⎟

⎠

⎜ ⎞

⎝

⎛

= +

− ee

We e W W W tr err N

LM (2.10)

The LM-lag statistic is as follows:

2

2 '

' ) '

' (

) ( )'

( ⎟

⎠

⎜ ⎞

⎝

⎛ + +

=

− ée

Wy e W W W e tr

e Wx M Wx lag N

LM β β ^(2.11)

with β for the vector with OLS coefficient estimates and the other notations are as before. Both the LM-err and the LM lag statistic are distributed as χ² with one degree of freedom.

3. Accessibility in Spatial Econometric Models

3.1 Spatial discounting and the accessibility concept

As stated in the introduction, empirical models formulated with reference to a theory which assumes spatial externalities – and thus spatial dependence – should be specified such that it treats space as being continuous. Since the effects of inputs from other locations are expected to diminish with distance, the question is how these inputs should be spatially discounted.

Few would disagree on that the extent of spatial externalities (or spatial dependence in general) between localities depends on the frequency of various types of interaction between those localities⁷. This means that a spatial discounting procedure should optimally relate to concepts from spatial interaction theory. Accessibility is precisely such a concept.

The accessibility concept has a long history in both regional science and transport economics.

According to Martellato, Nijkamp & Reggiani (1998, p163), Hansen (1959) provided one of the first foundations for the use of accessibility ‘theory‘ and defined accessibility as potential of opportunities for interaction⁸. In surveying the literature, Weibull (1980, p.54) remarks that interpretations of accessibility usually relate to (see also Pirie, 1979; Jones, 1981):

i. Nearness ii. Proximity

iii. Ease of spatial interaction

iv. Potential of opportunities for interaction

v. Potentiality of contacts with activities and supplies

The most popular interpretation of accessibility relate to (iii) and (iv) above, which emphasize the link between accessibility and interaction. Thus, high accessibility between two locations translates into a high potential for interaction and, hence, a high potential for spatial externalities between the locations⁹. In its most general form, the total accessibility of location i to an arbitrary opportunity x, A_i^Xcan be written as:

) ( )

( ...

) ( ...

)

( 1 1

1 ij

n

j j

in n ii

i X i

i x f c x f c x f c x f c

A = + + + + =

∑

= (3.1)

where f(c) denotes non-increasing function of distance. This function is often referred to as the distance-deterrence (or distance-decay) function. Note that (3.1) also includes location i’s internal accessibility to opportunity x.

Starting from the general expression in (3.1), there are several alternative ways in which numerical values of a location’s accessibility can be calculated. However, by using an axiomatic approach to the measurement of accessibility, Weibull (1976) narrowed down the measurement of accessibility to those measures that satisfy certain axioms.

7 For instance, spatial externalities that are mediated by the labour market – i.e. pecuniary externalities – depend on the interaction on the labour market, which is a market in which mobility is highly limited by distance.

8Baradaran & Ramjerdi (2001) note that this way of defining accessibility is closely associated with gravity models based on the interaction of masses.

9 In view of this, Karlsson & Manduchi (2001) have maintained that accessibility makes the general concept of geographical proximity, which is often emphasized in the literature on knowledge spillovers, operational.

Weibull (1976) starts from the point of view that accessibility is related to choice contexts for spatial interaction and that a choice context is represented by a configuration of opportunities for spatial interaction. Let d denote the distance from a point of reference and let a denote the attractiveness (e.g. size of an opportunity) at a location in question. A configuration c is then defined as an n-tuple of opportunities:

n i i i n

n a d a

d a

d a d

c =〈( ₁, ₁);( ₂, ₂);...;( , )〉=〈( , )〉₌₁ (3.2) where n is a finite positive integer, n∈ N =

{

1,2,....

}

¹⁰. The author then defines an accessibility measure as a function that to any configuration c attributes a finite and non-negative real number f

( )

c , f :C→ R₊, where C is the class of all configurations c . There are then six axioms that a meaningful and consistent accessibility measure should fulfil. For the sake of clarity, these axioms are reproduced below (see the original work for details)¹¹:

Axiom 1: For any configuration c:

(

^〈(d1^,a1^);...;(di^,ai^);...;(dj^,aj^);...;(dn^,an^)〉

)

⁼

f

(

^〈(d1^,a1^);...;(dj^,aj^);...;(di^,ai^);...;(dn^,an^)〉

)

f

, which states that the ordering of opportunities in any c should not affect the accessibility value.

Axiom 2: (a)

( ) (

i i ⁿi

)

n i i i i

i d f d a f d a

d ≤ ^* ⇒ 〈( , )〉₌₁ ≥ 〈( ^*, )〉₌₁ , for any attraction.

(a)

( ) (

i i ⁿi

)

n i i i i

i a f d a f d a

a ≤ ^* ⇒ 〈( , )〉₌₁ ≤ 〈( , ^*)〉₌₁ , for any distance.

, which states that the accessibility value should not increase with increasing distances and not decrease with increasing attractions.

Axiom 3: f₀(a)= f

(

〈(0,a)〉

)

(a) f₀ is continuous.

(b) f₀ is strictly increasing, i.e. x₁ <x₂ ⇒ f₀(x₁)< f₀(x₂)

, which is a sharpening of Axiom 2 in the case of a single opportunity at zero distance.

Axiom 4: f

(

〈(d₁,a₁);(d₂,a₂)〉

)

<lim_a→+∞ f

(

〈(0,a)〉

)

for every pair of opportunities

) ,

(d₁ a₁ and (d₂,a₂).

10 In e.g a country, the spatial distribution of an opportunity across locations and the infrastructure connecting these locations results in a given spatial configuration upon which accessibility calculations are made.

11 The presentation here builds on Weibull (1976, pp.359-362).

, which states that a single opportunity with infinite attraction situated at zero distance is better than any pair of opportunities with finite attractions.

Axiom 5: (a) f

(

〈(d₁,0)〉∪c

)

)= f

( )

c for any distance and configuration.

(b) f

(

〈 d⁽ 1^,0)〉

)

=⁰ for any distance.

, which states that opportunities with zero attraction should not contribute to the accessibility value.

Axiom 6: ^f

( ) ( )

^{c* = f c** ⇒ f}

(

^c*∪c

) (

^{= f c** ∪c}

)

, which means preferential independence. If two configurations are equivalent and a new set of opportunities, c , is superimposed on the two cases, then the two new configurations should be equivalent.

An accessibility measure that satisfies these axioms fulfils requirements of consistency and meaningfulness.

A very common and popular way of calculating an accessibility value when the accessibility is interpreted as potential of opportunity is to use an exponential distance-decay function in (3.1), (see e.g. Martellato, Nijkamp & Reggiani, 1998). In this case we have:

{

ij

}

ij t

c

f( )= exp −β (3.3)

where tij is the time distance between location i and j, and β is a time distance-friction (or sensitivity) parameter. Using (3.3) the total accessibility of location i to an arbitrary opportunity x, A_i^X, becomes:

{

ij

}

n

j j

X

i x t

A =

∑

=₁ exp−β (3.4)

Weibull (1976) shows that this type of accessibility measure satisfies axiom (1)-(6) above. Hence, the accessibility measure in (3.4) satisfies criteria of meaningfulness and consistency. For the interpretation of accessibility, it should be noted that the accessibility value in (3.4) may improve in two ways, either by an increase in the size of the opportunity, x_j, or by a reduction in the time distance between location i and j.

The accessibility measure in (3.4) can also be motivated theoretically by relating it to the preference structure in random choice theory. This procedure starts from a stochastic specification of the utility an individual (or firm/organization) in location i derives from accessing an opportunity x in location j, Uij. A simple form of such a specification is shown below:

ij ij

ij V

U = +ε

(3.5)

ij ij j

ij x c t

V = ln −φ −α

where Vij is the deterministically known utility and εij denotes the random influence from non- observed factors. xj is the opportunity in location j, cij denotes the cost of travelling from i to j, t _ij the time distance between the locations and εij random influence from non-observed factors.

Assuming that ε_ij is IID and extreme value (Gumbel) distributed, the probability that an individual in location i will choose to interact with location j (in the sense of accessing opportunity x in location j) is given by¹²:

{ } _∑

_∈

{ }

= _{ij j M ij}

ij V V

P exp exp (3.6)

where the set M ={1,…,i,…,j,…,n} contains all the possible locations to access opportunity x. In (3.6), the numerator represents the preference value accessing opportunity x in location j whereas the denominator is the sum of all such preference values. This means that, ceteris paribus, the probability of interaction between location i and j with respect to the given individual increases with the size of the attraction factor x in location j and decreases with the cost of accessing xj from location i.

Assuming that the cost of traveling between the locations, cij, is proportional to the time distance, such that c_ij =α_ct_ij, (c.f. Johansson Klaesson & Olsson, 2002), the denominator in (3.6) can be expressed as:

{ }

∑

∈ −

= _{j M j ij}

x

i x t

A exp β (3.7)

whereβ =(φα_c +α). The expression in (3.7) is exactly the one in (3.4). Obviously, if all individuals have a similar utility as in (3.5), a location with a high value of (3.7), i.e. high accessibility, is likely to experience a high level of interaction with other locations. Both the size of the attractor (x) and time distances in (3.7) are arguments in the preference function in (3.5).

3.2 Incorporating accessibility in spatial models

Having motivated the accessibility measure in (3.4) (and 3.7) we consider an example of model specification. Suppose we are interested in the relationship between y (output) and x (input).

Suppose also that the theory suggests that inputs in locations other than i are likely to have an effect on the output in that location, but that such inter-locational effects diminishes with distance. This could, for instance, be due to spatial externalities. Then, a natural way of specifying the empirical model is:

i i

i Wx u

y = β+ (3.8)

where Wi x is A . This specification is highly related to the specification in (2.4). In fact, (3.8) _i^X shows how the inclusion of accessibility on the RHS can account for global spillovers without

12 This condition is derived in several texts, see inter alia Train (1986) and Maddala (1986).

estimating an equation like (2.5), which requires ML-estimation. Equation (3.8) can in principle be estimated by means of OLS.

However, a drawback with the specification in (3.8) is that the inclusion of a single variable that measures the total accessibility does not provide any information about the spatial range of (potential) externalities. This problem can be solved by making a distinction between different spatial scales. Suppose for instance that the locations under study are municipalities. A typical municipality belongs to a local labour market (LLM) region. An LLM-region consists of a number of municipalities that together constitute an integrated labour market. LLM-regions are connected to other LLM-regions by means of economic and infrastructure networks. The same prevails for the different municipalities within a LLM region. Moreover, each municipality can also be looked upon as a number of nodes connected by the same type of networks. The borders between LLM-regions are in general characterized by a decline in the intensity of economic interaction including commuting compared to the intraregional interaction.

With reference to such a structure, it is possible to define three different spatial levels with different characteristics in terms of mobility and interaction opportunities. Because of this, it is also possible to construct three different categories of accessibility. Specifically, the total accessibility of a location (municipality) to a specific opportunity can be decomposed into local, intra-regional and inter-regional:

X iOR X iR X iL X

i A A A

A = + + (3.9)

where

{

_{L ii}

}

i X

iL x t

A = exp −β , local accessibility to opportunity x for location i.

{ }

∑

∈ ≠ −

= _{r R r i r R ir}

X

iR x t

A _, exp β , intra-regional accessibility to opportunity x for location i.

{ }

∑

∉ −

= _{k R k}exp _{OR ik}

X

iOR x t

A β , inter-regional accessibility to opportunity x for location i.

In the equations above, r defines locations within the own region R, and k defines locations in other regions. It is also evident that the value of β depends on whether the interaction is local (within location i), intra-regional (between locations in a region), or inter-regional (location i and j in different regions). Rewriting (3.8) to include the decomposition in (3.9) gives:

i i

i i

i W x W x W x u

y = ₁ β₁+ ₂ β₂+ ₃ β₃+ (3.10)

where W_i1x=A_iL^X, W_i2x=A_iR^X and W_i3x= A_iOR^X . W1 is a weight matrix with wii ≠ 0 and wij = 0, W2

is weight matrix with wii = 0 and wij ≠ 0 if location i and j in the same region and W3 a weight matrix with wii = 0 and wik ≠ 0 if location i and k in different regions. Observe that (3.10) is still a specification with modelled effects and global spillovers. However, the inclusion of each the three components separately means that the effect from each component can be revealed and compared with each other. Suppose for instance thatβˆ₁ and βˆ₂ turn out to be positive and significantly different from zero at the pre-specified significance level, while βˆ₃ turns out to be insignificant. Then, this is a direct indication of that the externalities tend to be bounded within the LLM region. Moreover, if a specification without W_i2x and W_i3x results in spatial

autocorrelation and the specification in (3.10) solves the problem, then (3.10) successfully models the effects involved.

The method described above have been used in a series of papers, (see e.g. Gråsjö, 2004 Andersson & Ejermo, 2004a,b; Andersson & Karlsson, 2005). These papers have employed the general accessibility measure described above as well as the decomposition in (3.9). In Gråsjö (2004) the number of patent applications in a municipality is regressed against the accessibility to university and company R&D on three different geographical levels (local, intra-regional and inter-regional). Gråsjö (2005) finds similar qualitative results when regressing the number of high-value export goods and export value across municipalities in Sweden on the accessibility to human capital and R&D. Spatial dependence in the residuals is tested by using the standardised statistic Moran’s I. The result indicates no evidence for spatial autocorrelation in the error terms.

Moreover, in Anderson & Karlsson (2005), the growth in value-added per employee across Swedish municipalities is regressed on the accessibility to knowledge resources. As in Gråsjö (2004, 2005), the Moran’s I showed no sign of spatial autocorrelation among the residuals when the model specification included the accessibility variables. However, it is necessary to conduct further studies in order to establish that the inclusion of spatially weighted explanatory variables, by using the accessibility concept, is a solution to potential spatial dependence.

4. An Empirical Application

4.1 Presentation of model

This section employs the procedure described in Section 3 by estimating a knowledge production function across Swedish municipalities¹³. The number of patent applications in municipality i is used as the dependent variable. Local, intra-regional and inter-regional accessibility to university and company R&D are explanatory variables. The time sensitivity parameter value βL is set to 0.02, βR to 0.1 and βOR to 0.05. Johansson, Klaesson & Olsson (2003) estimated these values by using data on commuting flows within and between Swedish municipalities in 1990 and 1998.

The data consists of a large number of zeroes, so Cobb-Douglas production is not applicable.¹⁴ Therefore an additive linear model is used. The model to be estimated takes the following form:

i D cR iOR D

cR iR D

cR iL D

uR iOR D

uR iR D

uR iL

i b b A b A b A b A b A b A u

Pat = ₁+ ₂ ^& + ₃ ^& + ₄ ^& + ₅ ^& + ₆ ^& + ₇ ^& + (4.1)

where uR&D denoted university R&D and cR&D denotes company R&D. Both types of R&D are measured in man-years. In order to check if the spatial weights defined in Equation (3.10) perform better than simple one-zero weights, OLS estimations of (4.1) are compared with OLS estimations of:

i D cR iOR D

cR iR D

cR iL D

uR iXR D

uR iR D

uR iL

i b b x b x b x b x b x b x u

Pat = ₁+ ₂ ^& + ₃ ^& + ₄ ^& + ₅ ^& + ₆ ^& + ₇ ^& + (4.2)

13 See Ejermo (2004,2005) for an extensive consideration of this kind of models with applications on Swedish data R&D and patent data.

14 Moreover, it could be argued that university R&D and company R&D are more like substitutes to each other, which implies a linear production function.

In (4.2) xiL, xiR and xiOR denotes R&D efforts locally, within own LLM region, and in other LLM regions. The error terms are investigated whether or not they are spatially autocorrelated. The estimations and the test results are presented in Section 4.3.

4.2 Weight matrices

The dimension of a spatial weight matrix W is given by the number of observations. A matrix with a weight wij reflects the spatial relation between observation i and j. Depending on the expected structure of spatial dependence, wij can represent either contiguity relations or it can model the role of distance in dependence. If two observations are contiguous, i.e. they share a common border (first order contiguity) or are located within a given distance band, the value of wij is larger than zero, and zero otherwise. In a simple binary matrix the weight wij = 1 if i and j are contiguous and if the elements are row-standardised, i.e. every element is divided by the respective row-sum, then 0 < wij < 1. If spatial dependence is expected to be determined by distance relations, a matrix element is based on the distance between i and j, i.e. their inverse or the square of the inverse. Distance is often measured by the physical distance, but a better way to measure it is to use the time it takes to travel between different locations (Beckman, 2000). Time distances are also crucial when it comes to attend to business meetings and also to spatial borders of labour markets (see e.g. Johansson & Klaesson, 2002 and Hugosson & Johansson, 2001 for the Swedish case).

Regressions on the patent data are conducted with two types of weight matrices; one with accessibility weights according to Equation (3.3) and (3.10), which enters Equation (4.1) and the other with binary weights, which enters Equation (4.2). When checking for spatial dependence in the error structure and spatial autocorrelation in the variables (i.e. doing the tests) the results from using a row standardised binary matrix is compared with the results from an inverse time distance matrix. In the binary matrix, the weight wij is greater than zero if municipality i and j are in the same LLM-region, and zero otherwise. The inverse distance matrix has a weight greater than zero if i and j are within certain time distance bands from each other. The distance bands used are 30, 60, 90 and 120 minutes for the variable tests and additionally 180, 240 and 300 minutes for the regression tests.

4.3 Estimation and test results Spatial autocorrelation in variables

Number of patents is a yearly average during the period 1994-1999 for the municipalities in Sweden. Two types of patent variables are used: (i) patents registered by inventor and (ii) patents registered by proprietor/applicant. The last is often the company where the inventor is employed.

If all companies are located in municipalities where the inventors live, then the two variables are identical, but this, however, is rarely the case. Accessibility to university and company R&D are computed using university R&D measured in man years (full-time equivalents) during the period 1993/94-1999 for Swedish municipalities.

Table 4.1: Moran’s I values of variables for different weight matrices

Binary t<30 min t<60 min t<90min t<120min Number of patents, inventor 0.082

(0.99)

0.551 (3.07)

0.290 (2.37)

0.206 (2.02)

0.157 (1.75) Number of patents, proprietor/applicant 0.021

(0.28)

0.149 (0.84)

0.077 (0.65)

0.058 (0.59)

0.043 (0.51) Accessibility to university R&D, local -0.015

(-0.13)

0.014 (0.10)

0.025 (0.23)

0.012 (0.15)

0.007 (0.11) University R&D, local -0.016

(-0.14)

0.007 (0.06)

0.020 (0.19)

0.009 (0.12)

0.004 (0.08) Accessibility to company R&D, local 0.012

(0.18)

0.140 (0.79)

0.067 (0.57)

0.048 (0.50)

0.035 (0.42) Company R&D, local 0.012

(0.18)

0.142 (0.81)

0.068 (0.58)

0.049 (0.50)

0.035 (0.42) Z(I) values in parenthesis

As can be seen from Table 4.1, the number of patents in a municipality registered by inventor is the only variable that is spatially autocorrelated. For all weight matrices, except the binary and the one with a time distance band less than 120 minutes, the Moran’s I is statistically significant.

The binary matrix has especially difficult to pick up potential spatial autocorrelation.

Spatial dependence in error terms

In order to establish if the inclusion of spatially weighted explanatory variables by means of the accessibility concept can reduce spatial dependence, the regressions are conducted with patents registered by inventor as the dependent variable. Then we have a spatially dependent variable on the LHS, whose spatial effects we are trying to model. Two questions are in focus.

1. Does the inclusion of accessibility have any affect on spatial dependence in the residuals?

2. Does the variable separation on different geographical levels (local, intra- regional and inter-regional) result in error terms that are spatially independent?

Test statistics for Moran’s I and for LM-err and LM-lag is computed to assess the questions above. The first tests are conducted on a simple specification, with only university and company R&D as explanatory variables. If the tests indicate spatial dependence the model must be changed in order to capture the spatial effects. Following the procedure discussed in Section 3, the strategy here is to add variables (first intra-regional and then if necessary inter-regional variables) on the RHS to account for the spatial effects. Regression and test results are presented in Table 4.2. In this table, all three tests, i.e. Moran’s I, LM-err and LM-lag, are carried out based on a binary W- matrix. However, the table also reports the Moran’s I test statistic when using different cut-off values for the inverse time-distance matrix. As can be seen from the table, the spatial reach of spatial dependence among the residuals is reduced when we add accessibility variables on the RHS. A comparison of R1 and R2 in Table 4.2 answers the first question above. The result indicates that the accessibility concept has a minor advantage, since R2 has higher p values for the test statistics compared to R1. This is true for all weight matrices. But the error terms in both R1 and R2 are still spatially dependent (except with the inverse time distance matrix, t < 300 minutes in R2). In R3 and R4 the intra-regional variables are included and then the tests are not

able to pick up any spatial dependence. Thus, specifications like R3 and R4 do model the spatial structure of patent registered by the inventor’s home address (i.e. the municipality where he/she lives).

Table 4.2: Regression results and Moran’s I , LM-err and LM-lag values for different weight matrices

R1 R2 R3 R4 University R&D,

local 0.029

(3.68)

0.029 (3.79) University R&D,

intra-regional 0.00005

(0.04) Company R&D,

local

0.376 (6.12)

0.367 (5.83) Company R&D,

intra-regional

0.013 (2.94) Access to university

R&D, local 0.037

(3.70)

0.031 (3.62) Access to university

R&D, intra-regional -0.018

(-1.23) Access to company

R&D, local

0.421 (5.97)

0.440 (6.23) Access to company

R&D, intra-regional

0.200 (2.54)

Adjusted R² 0.892 0.889 0.911 0.918

LM-err, binary 12.27 (0.0005)

11.32 (0.0008)

0.013

(0.908) 0.036 (0.848) LM-lag, binary 0.413

(0.521)

0.385 (0.535)

0.0002 (0.988)

0.020 (0.887) Moran’s I, binary 0.176

(0.020)

0.169 (0.024)

-0.010 (0.499)

-0.010 (0.486) Moran’s I, t < 30 min 0.872

(6.0E-7)

0.798 (4.6E-6)

0.133 (0.415)

0.115 (0.474) Moran’s I, t < 60 min 0.417

(0.0003)

0.379 (0.001)

-0.011

(0.499) 0.019 (0.410) Moran’s I, t < 90 min 0.304

(0.001)

0.276 (0.003)

-0.001 (0.474)

0.010 (0.433) Moran’s I, t < 120 min 0.242

(0.004)

0.220 (0.007)

0.0004 (0.467)

0.013 (0.418) Moran’s I, t < 180 min 0.170

(0.014)

0.155 (0.022)

-0.0006 (0.472)

0.008 (0.434) Moran’s I, t < 240 min 0.134

(0.028)

0.121 (0.041)

-0.002 (0.0.479)

0.004 (0.447) Moran’s I, t < 300 min 0.118

(0.038)

0.107 (0.053)

-0.002 (0.478)

0.003 (0.454)

*) White’s (1980) robust standard errors are used for t values. t values in parenthesis for parameter estimates

**) The intra-regional variables in R3 and R4 are collinear, but this does not harm the residuals

***) p values in parenthesis for Moran’s I , LM-err and LM-lag

****) R1 and R3 are modifications of Equation (4.2) and R2 and R4 are modifications of Equation (4.1)

*****) In binary matrix: weight wij > 0 if municipality i and j belong to the same LLM-region, 0 otherwise.

******) Non-binary matrix: wij = 1/(tij)

5. Monte Carlo Simulations

Using a regional KPF on Swedish data, the preceding section illustrated how the inclusion of spatially discounted explanatory variables on the RHS reduced spatial autocorrelation.

The purpose of this section is to test whether the same result emerge in a more general setting.

Using the taxonomy by Anselin (2003) in the Monte Carlo simulations, we test how the inclusion of spatially discounted variables on the RHS affects the extent of spatial autocorrelation. Data for the dependent variable and/or the error term is generated such that it is spatially dependent. Then, similar to Table 4.2, a comparison is made between models with and without spatially discounted variables. Rejection frequencies of some common test for spatial autocorrelation (Moran’s I, LM- lag and LM-err)are then presented for each model. Parameter accuracy is assessed by checking bias and Root Mean Square Error (RMSE) for the estimated parameters of each model.

Moreover, to what extent significance of the estimated parameters of the spatially discounted explanatory variables can be interpreted as evidence of spatial dependence is also assessed.

5.1 Design of the simulation framework

Locations are normally generated randomly in Monte Carlo simulations. However, we use the locations of the Swedish municipalities. There are several reasons for this:

1) a uniform distribution of locations is often used, but is not very realistic. It is more probable that the locations are clustered.

2) the municipalities in Sweden are divided into LLM-regions, which makes it possible to test intra-regional and inter-regional effects separately.

3) the distances between the municipalities are real travelling time, which is also more realistic than the often used Euclidean distance.

Regarding the data generating process, independent variables consist of an intercept and an x- variable drawn from a uniform distribution with range 0-5. This data remains the same for all repetitions. The ‘true’ y-values and the error structures are generated according to the structural equations of Anselin’s (2003) new taxonomy (see Table 5.1).

Table 5.1: Structural models in the data generating process Global spillovers True values, y Error, e 1 Unmodelled effects xβ (I−λW)⁻¹u

2 Modelled effects (I−ρW)⁻¹xβ u 3 Both effects, global (I−ρW)⁻¹xβ (I−λW)⁻¹u

Local spillovers

4 Unmodelled effects xβ (I+λW)u

5 Modelled effects xβ +Wxρ u 6 Both effects xβ +Wxρ (I+λW)u

*) Note that ρ is a scalar in 2 and 3 and a column vector matching the column dimension of Wx in 5 and 6

The ‘true’ values of the regression coefficients are β1 = 0.5 and β2 = 1.0. W is a weight matrix with inversed distances. Three different weight matrices are used in the data generating process:

• W60, with wij ≠ 0 if tij < 60 minutes, zero otherwise

• W₁₂₀, with wij ≠ 0 if tij < 120 min, zero otherwise

• W₄₅, with wij ≠ 0 if tij < 45 min, zero otherwise

W60 and W120 are used for global spillovers and W45 for local spillovers. The reason for using different weight matrices is to check the importance of inter-regional accessibility. The average number of joins for a location is then 7 for W60, 24 for W120 and 4 for W45. The error term u is drawn from a normal distribution with mean zero and variance one. The parameters ρ and λ determine the strength of the spatial dependence. Our aim is to generate data such that estimation of a model likey_i =b₁+b₂x_i +ε_i results in spatial autocorrelation. For this reason, the following parameter values have been used:

• ρ = 0.4, λ = 0.75 for W₆₀

• ρ = 0.3, λ = 0.65 for W120

• ρ = 0.6, λ = 2.0 for W45

which fulfils our purpose. Altogether nine experiments are conducted. Each experiment is repeated 1000 times. Every repetition contains three different OLS regressions: 1) without accessibilities, yi(xi), 2) with intra-regional accessibility, yi(xi,A ) and 3) with intra- and inter-_iR^X regional accessibility, yi(xi,A ,_iR^X A ), with x matrices according to_iOR^{X 15}:

OLS 1:

⎟⎟

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎜⎜

⎝

⎛

=

288 2 1

1 . .

. . 1 1

x x x

x , OLS 2:

⎟⎟

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎜⎜

⎝

⎛

=

X R X

R X

R

A x

A x

A x

x

288 288

2 2

1 1

1

. . .

. . . 1 1

, OLS 3:

⎟⎟

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎜⎜

⎝

⎛

=

X OR X

OR X OR

X R X

R X R

A A A

A x

A x

A x

x

288 2 1

288 288

2 2

1 1

. .

1

. . .

. . . 1 1

whereA and _iR^X A are defined as in (3.9). Thus, the same principle is applied here as in section _iOR^X 5, i.e. adding the accessibility variables one by one. Note that the values of β3 and β4 are set to zero in OLS 2 and 3.

For each experiment spatial dependence is tested for by using the Moran’s I statistic, and the Lagrange Multiplier test statistics LM-err and LM-lag. We report how the rejection frequency of each of these tests is affected by the inclusion of accessibility variables on the RHS. Our main focus is on how the inclusion of spatially weighted explanatory variables, by means of the accessibility concept, affects spatial dependence. How the tests react to different weight matrices

15 As can be seen from the notation, local accessibility is not used in the simulations. Hence, for a given location (observation) the local inputs, i.e. the inputs ‘inside’ this location, are not spatially discounted whereas all inputs in other locations are spatially discounted. This is because local accessibilities do not have any significant effect on spatial autocorrelation among the residuals, (see Table 4.2).