Workshop Lecture 5: Spatial Econometric Modeling of Origin-Destination flows

Transcription

1 Workshop Lecture 5: Spatial Econometric Modeling of Origin-Destination flows James P. LeSage Fields Endowed Chair of Urban and Regional Economics McCoy College of Business Administration Department of Finance and Economics Texas State University - San Marcos San Marcos, TX March 2014

2 The gravity/spatial interaction model The terms gravity models and spatial interaction models has been used in the literature to label models that focus on least-squares regression-based models that attempt to explain variation in flows between origins and destinations, Sen and Smith (1995). Table 1: A flow matrix (population migration ) Destination/Origin Origin 1 Origin 2... Origin n Alabama Arizona Wyoming Destination 1 (Alabama) 1,392,507 2, Destination 2 (Arizona) 3,452 1,669,415 4,458. Destination n (Wyoming) 517 2, ,232 1

3 A typical least-squares gravity model y = αι N + X d β d + X o β o + gγ + ε (1) where: y is the OD flow matrix vectorized g is a vector derived from the matrix of O-D distances vectorized (N = n 2 x1) ι N is an n 2 x1 intercept vector ε N(0,σ 2 I n 2) α, β o,β d,γ are parameters to be estimated X d = X ι n, we repeat the matrix X n times to produce an N by k matrix representing destination characteristics. X o = ι n X, a second matrix that represents origin characteristics, again repeats the same matrix X. 2

4 Spatial dependence of flows The notion that use of distance functions in conventional spatial interaction models effectively capture spatial dependence in the interregional flows being analyzed has been challenged in recent work by Porojon (2001) for the case of international trade flows, Ming and Pace (2004) for retail sales. The residuals from conventional models were found to exhibit spatial dependence. No one has used spatial lags typically found in spatial econometric methods in these models. Griffith and Jones (1980) made a similar observation and developed some of the ideas set forth here, but could not implement these methods computationally. Map demonstration of spatial dependence. 3

5 Interregional flows in a spatial regression context In contrast to the traditional regression-based gravity model, a spatial econometric model of the variation in origin-destination flows would be characterized by: 1) reliance on spatial lags of the dependent variable vector, which we refer to as a spatial autoregressive model (SAR): y = ρw y + Xβ + ε, or 2) of the disturbance terms, which we label a spatial error model (SEM): y = Xβ + u, u = ρw u + ε, or 3) perhaps spatial lags of both kinds, which we denote as the general spatial model (SAC): y = ρw y + Xβ + u u = ρw u + ε. 4

6 Spatial weight matrices for n 2 OD flows A key issue is how to construct a meaningful spatial weight matrix in the case where the n 2 by 1 vector of observations reflect flows from all origins to all destinations. Staring with a typical n by n first-order contiguity weight matrix W we can repeat this using: W o = I n W = W W W (2) W o y, represents an average of flows from neighbors to each origin state to each of the destinations, which we label origin-based spatial dependence. 5

7 An example of origin-based spatial dependence As an example, consider a single row i of the spatial lag vector W o y that represents flows from the origin state of Florida to the destination state of Washington. First-order contiguous neighbors to the origin Florida are Alabama and Georgia, and neighbors to the destination Washington are Oregon and Idaho. The spatial lag W o y would represent an average of the flows from Alabama and Georgia (neighbors to the origin) to the destination state Washington. 6

8 Destination-based spatial dependence A second type of spatial dependence that could arise in the gravity model would be destination-based dependence. Intuitively, it seems plausible that forces leading to flows from an origin state to a destination state may create similar flows to nearby or neighboring destinations. W d = W I n (3) W d y represents an average of flows from an origin state to neighbors of the destination state. Using our example of flows from the origin state of Florida to the destination state of Washington, the ith row of the spatial lag vector W d y represent an average of flows from Florida to Idaho and Oregon, states that neighbor the destination state of Washington. 7

9 Origin-to-destination-based dependence A third type of dependence to consider is reflected in the product W o W d =(I n W ) (W I n )=W W. (W o W d )y = W w y reflects an average of flows from neighbors to the origin state to neighbors of the destination state. As a motivation for this, consider spatial autoregressive successive filtering. (I N ρ 1 W o )(I N ρ 2 W d )y = αι+x d β d +X o β o +gγ+ε (4) y = ρ 1 W o y+ρ 2 W d y ρ 1 ρ 2 W w y+αι+x d β d +X o β o +gγ+ε (5) 8

10 A family of SAR O-D models A family of nine different model specifications is proposed. For the SAR model: y = ρ 1 W o y + ρ 2 W d y + ρ 3 W w y + αι N + X d β d + X o β o + γg + ε Nine different models that can be derived by placing various restrictions on the parameters ρ i,i =1,...,3. Since the statistical theory for testing parameter restrictions is welldeveloped, this seems desirable from an applied specification search viewpoint. 1. The restriction: ρ 1 = ρ 2 = ρ 3 = 0, produces the least-squares model 2. The restriction: ρ 2 = ρ 3 =0, results in a model based on a single weight matrix W o 3. The restriction: ρ 1 = ρ 3 =0, results in a model based on a single weight matrix W d 4. The restriction: ρ 1 = ρ 2 =0, results in a model based on a single weight matrix W w 5. The restriction: ρ 1 = ρ 2,ρ 3 =0, results in a model based on a single weight matrix constructed using W o + W d. 9

11 6. The restriction: ρ 1 = ρ 2,ρ 3 = ρ 2 1 = ρ2 2, produces another single weight matrix model based on W o +W d + W w. 7. The restriction: ρ 3 = 0, leads to a model with separable origin and destination autoregressive dependence embodied in the two weight matrices W o and W d, 8. The restriction: ρ 3 = ρ 1 ρ 2 results in a successive filtering or model involving both origin W o, and destination W d dependence as well as product separable interaction W w,constrained 9. The unrestricted model involving the three matrices W o, W d,andw w yields the ninth member of the family of models. 10

12 Estimation issues For the case of migration flows between n = 3, 110 US counties, we have N = n 2 = 9.61 million, so computation becomes difficult. Some elementary manipulations of the moment matrix can illuminate a computationally efficient approach to estimation of the least-squares model based on: the n by k matrix X, the n by n matrix Y of O-D flows, and traces of the n by n matrix G of O-D distances. Z Z = Similarly, Z y N nx X 0 X Gι n 0 0 nx X X Gι n 0 ι n G X ι n G X tr ( G 2) (6) Z y =[ι n Yι n X Yι n X Y ι n tr (GY )] (7) ˆβ =(Z Z) 1 Z y 11

13 Models 1) to 6) involving a single weight matrix W j,j = o, d We note that the concentrated log-likelihood function for model specifications 1) to 6) based on a single spatial weight matrix, which we denote W j, concentrated with respect to the parameters β and σ will take the form: LnL(ρ) ln I N ρw j N 2 ln(e(ρ) e(ρ)) (8) Where e(ρ) e(ρ) represents the sum of squared errors expressed as a function of the scalar parameter ρ alone after concentrating out the parameters β, σ, and C denotes a constant not depending on ρ. These models can be estimated using conventional maximum likelihood or Bayesian algorithms. But, we can simplify calculation of the log-determinant term considerably. 12

14 The log-determinant in Models 1) to 6) ln I N ρw j = n t=1 ρ t tr(w t j ) = n ln I n ρw For the case of a single spatial weight matrix W j, j = o, d, w, we can rely on algorithms for computing the logged determinant of an n by n matrix ln I n ρw, when working with a vector of N = n 2 origin-destination flows. For the earlier example of n =3, 100 US counties and N = 9, 610, 000, we should be able to solve these estimation problems in a matter of seconds on desktop computers when using computationally efficient sparse algorithms for the n by n log-determinant portion of the problem (see LeSage and Pace, 2004). t 13

15 Models 7) to 9) involving W o,w d,w w The concentrated log-likelihood for this model is: LogL = C + log (I N ρ 1 W o )(I N ρ 2 W d ) (N/2)log(e(ρ 1,ρ 2 ) e(ρ 1,ρ 2 )) Using the mixed-product rule for Kronecker products ln (I N ρ 1 W o )(I N ρ 2 W d ) = n(ln I n ρ 1 W +ln I n ρ 2 W ) We provide a computationally efficient way to compute this log-determinant as well as I N ρ 1 W o ρ 2 W d ρ 3 W w in the paper. 14

16 Maximizing the likelihood for Models 7) to 9) LogL = C + log I N ρ 1 W o ρ 2 W d ρ 3 W w (N/2)log(e(ρ 1,ρ 2 ) e(ρ 1,ρ 2 )) The log-determinant already taken care of. e(ρ 1,ρ 2,ρ 3 ) e(ρ 1,ρ 2,ρ 3 ) can be constructed in a computationally efficient way using some tricks. Y ρ 1 WY ρ 2 YW ρ 3 WYW (9) = αι n ι n + Xβ dι n + ι nβ o X + γg + E eventually,the sum-of-squared residuals component of the log-likelihood for the family of O-D models can be expressed as a quadratic form involving the parameters placed in a vector: τ =[1 ρ 1 ρ 2 ρ 3 ] and a small matrix Q that need not be recomputed as values of ρ 1,ρ 2,ρ 3 change during optimization. e(ρ 1,ρ 2,ρ 3 ) e(ρ 1,ρ 2,ρ 3 )=τ Qτ Q = Ê Ê = tr(ê iêj) = Ê i Ê j τ =[1 ρ 1 ρ 2 ρ 3 ] (10) 15

17 An applied illustration Using population migration flows over the periods and for the 48 contiguous US states plus the District of Columbia, n =49, n 2 = 2401, asmall example. Dependent variable y is the growth rate in interstate migration flows from the earlier to later period, intrastate flows are set to zero. Explanatory variables are origin and destination characteristics for the states in 1990 taken from the 1990 US Census. All variables are log transformed. Table 2: Explanatory variables used in the model Variable name Description near retired log (pop aged 60-64/ pop in 1990) retired log (pop aged 65-74/ pop in 1990) sales log (proportion of work force in sales in 1990) diffstate log (proportion of pop living in a different state in 1985) foreign born log (proportion of foreign born pop in 1990) grade9 log (< 9th grade as highest degree in 1990/pop > age 25) associate log (associate degree in 1990/pop > age 25) college log (college as highest degree in 1990/pop > age 25) grad prof log (graduate or professional degree in 1990/pop > age 25) rents log (median rent in 1990) unemp ratio of 1995 unemployment rate to 1990 unemployment rate pc income log (per capita income in 1990) area log (1990 state area in square miles) 16

18 Model comparisons Table 3: Log Likelihoods for alternative models Model Log Likelihood LR test versus Critical Value Model 9 (α =0.05) 9: ρ 1,ρ 2,ρ 3 unrestricted : ρ 3 = χ 2 (1) = : ρ 3 = ρ 1 ρ χ 2 (1) = : ρ 1 = ρ 2,ρ 3 = χ 2 (2) = : ρ 1 = ρ 2,ρ 3 = ρ χ 2 (2) = : ρ 2 = ρ 3 = χ 2 (2) = : ρ 1 =0,ρ 3 = χ 2 (2) = : ρ 1 =0,ρ 2 = χ 2 (2) = : ρ 1 =0,ρ 2 =0,ρ 3 = χ 2 (3) = 7.82 Table 4: ρ 1,ρ 2,ρ 3 estimates for various models Model ρ 1,ρ 2,ρ 3 OLS 0 W J = W d only W J = Wo only W j = Wo + W d + W od W J = Wo + W d (I N ρ 1 Wo ρ 2 W d + ρ 1 ρ 2 WoW d ) , (I N ρ 1 Wo ρ 2 W d ) , (I N ρ 1 Wo ρ 2 W d + ρ 3 WoW d ) , ,

19 Table 5: Maximum Likelihood Estimates Variable Coefficient t-statistic(plevel) constant (0.4869) D nearretirement (0.6891) D retired (0.4909) D sales (0.1207) D diffstate (0.0016) D foreignborn (0.2606) D grade (0.0000) D associate (0.0023) D college (0.2999) D gradprof (0.0908) D rents (0.0001) D area (0.2246) D unemp (0.0071) D pcincome (0.0775) O nearretirement (0.9153) O retired (0.1065) O sales (0.3533) O diffstate (0.0000) O foreignborn (0.0000) O grade (0.9977) O associate (0.0178) O college (0.0020) O gradprof (0.0000) O rents (0.0000) O area (0.0000) O unemp (0.0013) O pcincome (0.0000) log(distance) (0.8077) ρ (0.0000) ρ (0.0000) ρ (0.8658) 18

20 Other estimation issues (i) The presence of large flows on the diagonal of the flow matrix relative to much smaller flows for offdiagonal elements, reflecting a large degree of intraregional connectivity. (ii) the presence of numerous zeros for large flow matrices as in the case of migration flows between US counties, reflecting a lack of interaction between numerous regions in the sample. For example, a sample of 3,109 US county-level migration flows for 1990 contain zeros in over 90 percent of the 9.6 million flow matrix elements, invalidating simple transformations to achieve normality. Migration is a relatively rare event, so Poisson models are more appropriate estimation methods for this type of sample data. This is taken up in Section 5 of the paper. (iii) the fat-tailed nature of the distribution of the vectorized flow matrix, even after transformation to growth rates over time (as illustrated by Figure 1 for the state-level migration flows example). (iv) we need to correctly interpret these estimates in terms of partial derivatives. 19

21 Figure 1: Distribution of Annualized Migration Growth Rates 450 Histogram Normal plot Migration flow growth rates to Frequency 20

22 Bayesian Robust estimates For such cases, LeSage (1997) sets forth Bayesian Markov Chain Monte Carlo (MCMC) estimation procedures for conventional spatial regression models. These models can accommodate fat-tailed disturbance distributions following an approach introduced by Geweke (1993). A set of latent variance scalars is introduced to produce different variance estimates for each observation, that is, we replace ε Normal(0,σ 2 I N ),with: y = ρ 1 W o y + ρ 2 W d y + ρ 3 W w y + αι N + X d β d + X o β o + γg + ε ε Normal[0,σ 2 V ] (11) V = diag(v ij ) i =1,...,n, j =1,...,n Formally, our priors can be expressed as: δ constant (12) r/v ij iid χ 2 (r) (13) 1/σ 2 Γ(ν 1,ν 2 ) (14) ρ i U( 1, 1), i =1,...,3 (15) 21

23 Table 6: Origin-Destination pairs for variance scalar estimates greater than 10 Origin State Destination State Posterior mean v i RI ND DE DC VT SD VT NE ND RI DE ND VT WY RI WV RI WY MT DE WY RI DE SD ND CT SD CT NH SD VT MN ID DE IA VT

24 Conclusions regarding estimation It is relatively easy to incorporate spatial lags into conventional least-squares gravity models The concentrated log-likelihood takes a form where we have a log-determinant and sum-of-squared errors term. The log-determinant takes a particularly convenient form, making it easy to handle models with n 2 in the millions The sum-of-squared errors can be easily computed for variousvaluesofρ 1,ρ 2,ρ 3 : e(ρ 1,ρ 2,ρ 3 ) e(ρ 1,ρ 2,ρ 3 )=τ Qτ (16) 23

25 Interpreting estimates from OD flow models Spatial spillovers in these models can take the form of spillovers to both regions/observations neighboring the origin or destination in the dyadic relationships that characterize origin-destination flows as well as network effects that impact all other regions in the network. Making one region, say i =1more/less attractive: (Potentially) increases/decreases inflows from all j = 2,...,n other regions (destination effect) (Potentially) decreases/increases outflows to all j = 2,...,n other regions (origin effect) (Potentially) increases/decreases inflows to regions k neighboring region i (network effects) (Potentially) decreases/increases outflows to regions k neighboring region j (network effects) (Potentially) increases/decreases flows within region i (intraregional effect) We set forth scalar summary expressions for the various types of effects that average over all regions similar to regression estimates, and associated (simulated) measures of dispersion that can be used to produce inferences about the size and significance of these effects. 24

26 Interpreting the non-spatial model Typical regression model, re-stated using matrix algebra rather than i, j dyad notation. y = αι n 2 + X o β o + X d β d + γg + ε (17) y is an n 2 1 vector of (logged) flows constructed by stacking the columns of the n n flow matrix Y, an n n matrix of (logged) distances D(i, j) between the n destination and origin regions is converted to a vector by stacking columns to form the n 2 1 vector g, and associated parameter γ. LeSage and Pace (2008) show that X o = ι n X, where X is an n k matrix of characteristics for the n regions, represents a Kronecker product, and ι n is an n 1 vector of ones. (This assumes a destination-centric flow matrix where the i, jth element represents a flow from region i to j.) The Kronecker product repeats the same values of the n regions in a strategic fashion to create a vector (or matrix) of sizes associated with each origin region, hence use of the notation X o to represent these explanatory variables reflecting origin characteristics. The matrix/vector X d = X ι n, arranges the n regional characteristics to match the vector y, producing explanatory variables associated with each destination region. 25

27 The vectors β o and β d are k 1 parameter vectors associated with the origin and destination region characteristics, respectively. 26

28 The impact of a change in region 3, (X 3 )on flows ΔY ΔX 3 = (18) Where β d =1,β o =0.5, The role of the independence assumption is clear, where we see from row 3 that the change of outflows from region 3 to all other regions equals 0.5, which is the value of the coefficient β o in our example. Similarly, column 3 exhibits identical changes in inflows to region 3, taking the value 1 of the coefficient β d in our example. The diagonal (3,3) element reflects a response equal to β o + β d, the sum of the changes in flows in- and outof region 3, reflecting the change in intraregional flows arising from the change in X 3. We have only 2n non-zero changes in flows by virtue of the independence assumption. All changes involving flows in- and out-of regions other than region 3 are zero. 27

29 Scalar summary measures of a change in X i,i=1,...,n on flows A sequence of n, n n matrices Y i, one associated with each ΔX i. Origin effects: n i=1 ( k i ( Y i,ik X i )/n)/n Destination effects: n i=1 ( k i ( Y i,ki X i )/n)/n Intraregional effects: n i=1 ( k=i ( Y i,kk X i )/n)/n ΔX i Origin effects Destination effects Intraregional effects Network effects Total effects Table 7: Scalar summary measures of effects for the non-spatial model N.B. these are not equal to ˆβ d, ˆβ o, so people are misinterpreting regression estimates from these models. A key point: you cannot change X d while holding X o constant (the definition of a partial derivative). So, β o,β d do not reflect traditional regression-based partial derivatives. A change in X i implies changes in both X o and X d, which is taken into account here. 28

30 The spatial model impact of a change in region 3, (X 3 )onflows y = ρ o W o y + ρ d W d y + ρ w W w y + X d β d + X o β o + γg + ε W o = I n W, W d = W I n,w w = W W ΔY ΔX 3 = (19) Elements not in row and column 3 are network effects. We cumulate these for i =1,...,n, then average over the n changes in X i,i =1,...,n to produce a scalar summary measure of network effects. Note also that elements in the third row and column are not equal, but decay with distance from region 3. 29

31 Scalar summary effects for the spatial model Origin effects: n i=1 ( k i ( Y i,ik X i )/n)/n Destination effects: n i=1 ( k i ( Y i,ki X i )/n)/n Intraregional effects: n i=1 ( k=i ( Y i,kk X i )/n)/n Network effects: n i=1 ( k i ( j i ( Y i,kj X )/n)/n)/n i vec(y ) X r i J i=2 = = B 1 (ι n J i β r o + J i ι n β r d ) B = (I n 2 ρ d W d ρ o W o ρ w W w ) We have n sets of vec(y )(n 2 1) outcomes,onefor each change in X r i,i = 1,...,n which we denote Y i,. above, and reference individual elements using Y i,jk. 30

32 ΔX r Spatial ΔX r Non-Spatial Origin effects Destination effects Intraregional effects Network effects Total effects Table 8: Scalar summary measures of effects for the spatial and non-spatial gravity models Total effects: n i=1 ( n j=1 ( n k=1 ( Y i,jk X i )/n)/n)/n 31

33 An applied example Commuting-to-work flows for 60 regions (Quartiers Tlse) in Toulouse, France Flows were constructed from the declared home and work addresses of the active population Only two explanatory variables (plus distance): Workers residing in each district and Employment (jobs) located in each district Typical log-log specification, 15% of the flows are zero values so log(1+flows) was used The data set comes from the INSEE census in 1999 and the location variables from IGN maps. The work location variable is known at a fine scale (ilot). Workers in the defense sector and workers moving to variable sites or working at home are excluded. 52% of workers in the region come from outside the region and 19% of residents in the region work outside. These were excluded to form a closed system of flows between the 60 districts. 32

34 OLS estimates and scalar summary effects estimates Ordinary Least-squares Estimates R-squared = **************************************************** Variable Coefficient t-stat t-prob constant D_residents D_jobs O_residents O_jobs log(distance) Variables orig mean median lower 0.05 upper 0.95 residents jobs Variables dest mean median lower 0.05 upper 0.95 residents jobs Variables intra mean median lower 0.05 upper 0.95 residents jobs Variables total mean median lower 0.05 upper 0.95 residents jobs

35 What do these effects mean? A 1 percent increase in workers residing in a district leads to a 0.87 percent increase in outflows of commuters to work (residents origin-effects) A 1 percent increase in jobs located in a district leads to a 1.05 percent increase in commuters flowing into the district (jobs destination effects) A 1 percent increase in residents (or jobs) located in a district leads to 0.01 percent increase in commutingto-work flows within that district (intraregional effects). Suggests most residents commute outside the district where they live and most jobs employ residents from outside the district. A 1 percent increase in residents living in a destination (place of work) district has a (negative) (significant) effect on commuting-to-work inflows to the district (residents destination effects). A 1 percent increase in jobs located in an origin (place of its workers residence) district would have a negative (significant) effect on commuting-to-work outflows from the district (jobs origin effects). N.B. contrast with OLS Network effects are zero by construction meaning that more/less flows from one district do not imply more/less flows from neighboring districts. 34

36 The total effect of increasing jobs located in a typical district by 1 percent is an increase in (cumulative) commuting-to-work flows by 1.03 percent (jobs total effect) The total effect of increasing (working) residents located in a typical district by 1 percent is an increase in (cumulative) commuting-to-work flows by 0.74 percent (residents total effect) This contrast (1.03 vs. 0.74) may suggest that residential districts typically have more places of employment than commercial districts have places of residence. (Which seems incorrect.) 35

37 Spatial scalar summary effects estimates log flows coefficients t-stat lower 0.05 upper 0.95 constant D_residents D_jobs O_residents O_jobs log(distance) rho_o rho_d rho_w Variables orig mean median lower 0.05 upper 0.95 residents jobs Variables dest mean median lower 0.05 upper 0.95 residents jobs Variables netw mean median lower 0.05 upper 0.95 residents jobs Variables intra mean median lower 0.05 upper 0.95 residents jobs Variables total mean median lower 0.05 upper 0.95 residents jobs

38 What do these effects mean? A 1 percent increase in residents in a district leads to a 1.43 percent increase in outflows of commuters to work (residents origin-effects) A 1 percent increase in jobs located in a district leads to a 1.23 percent increase in commuters flowing into the district (jobs destination effects) A 1 percent increase in residents (or jobs) located in a district leads to 0.02 percent increase in commuting-towork flows within that district (intraregional effects). Suggests most workers commute outside the district where they live and most jobs employ people from outside the district. A 1 percent increase in residents living in a destination (place of work) district has no (significant) effect on commuting-to-work in/outflows from the district (workers destination effects). N.B. contrast with OLS A 1 percent increase in jobs located in an origin (place of its workers residence) district would have no (significant) effect on commuting-to-work in/outflows from the district (jobs origin effects). N.B. contrast with OLS A 1 percent increase in residents living in a district would lead to a 16.6 percent (cumulative) increase in commuting-to-work flows throughout the 60 district region (residents network effects). Implies strong clustering of residential districts. 37

39 A 1 percent increase in jobs located in a district would lead to a 13.8 percent (cumulative) increase in commuting-to-work flows throughout the 60 district region (jobs network effects). Implies strong clustering of employment districts. The total effect of increasing residents in the typical district by 1 percent is an increase in (cumulative) commuting-to-work flows by 18.2 percent. The total effect of increasing jobs located in a typical district by 1 percent is an increase in (cumulative) commuting-to-work flows by 15.3 percent. This contrast (18.2 vs. 15.3) suggests that residential districts typically have slightly less places of employment than commercial districts have places of residence. (Which seems correct). 38

40 Figure 2: Employers/jobs Moran Scatterplot x x Figure 3: Employers/jobs Moran Map

41 Figure 4: Residents/workers Moran Scatterplot x x Figure 5: Residents/workers Moran Map

42 Conclusions regarding interpretation The most important aspect of commuting-to-work flows (or almost any flows) are the network effects. These arise due to residential and employment clustering. These effects are assumed to be zero by conventional OLS gravity models due to the independence assumption. OLS estimates are biased and inconsistent. Practitioners have mis-interpreted the coefficients from OLS gravity models for years. Interpreting coefficient estimates from OLS models as if they were partial derivatives is wrong. Interpreting coefficient estimates from spatial models as if they were partial derivatives is wrong. The true partial derivatives are more complicated to calculate, especially the measures of dispersion associated with these. MCMC estimation and draws were used here to calculate measures of dispersion. 41

43 Alternative (simpler) models y = Xβ + Δ d φ + Δ o ψ + ε (20) φ = Z d α d + ρ d Wφ + u (21) ψ = Z o α o + ρ o Wψ + v (22) A family of models φ = Z d α d + ρ d Wφ + u (23) ψ = Z o α o + ρ o Wψ + v (24) φ = Z d α d + u (25) ψ = Z o α o + v (26) φ = ρ d Wφ + u (27) ψ = ρ o Wψ + v (28) 42

44 Simultaneous dependence To see the simultaneous nature of the prior dependence structure, consider how a change in trade openness associated with a single country impacts the vector of effects parameters in the model. Working with the outward effects we can write: φ 1. φn = K k=1 S k (W ) S k (W ) 1n.... S k (W ) n1... S k (W )nn z k,1. z k,n + V(W)ιnα d,0 + V(W)u S k (W) = V(W)α d,k V(W) = (In ρ d W) 1 = In + ρ d W + ρ 2 d W We use α d,0 to represent the parameter from the vector α d associated with the intercept and α d,k to denote the parameter associated with the kth variable from the matrix Z d. In addition, we use S k (W ) ij to indicate the i, jth element of the matrix S k (W). Using V (W ) i to represent the ith row of V(W), wecanwritethepartial derivative expression in (29). φ i = K [S k (W ) i1 z k,1 + S k (W ) i2 z k, S k (W ) in z k,n ] k=1 + V (W ) i ιnα d,0 + V (W ) i u φ i z k,j = S k (W ) ij (29) 43

45 The implication of a non-zero cross-partial in (29) is that a change in trade openness (more generally, the kth variable in the matrix Z d ) in a single country j represented by elements of the matrix Z d (which we denote z k,j ) will give rise to changes in effects parameters of countries i j. Since we can change openness in all countries i = 1,...,n, this leads to an n n matrix of partial derivative responses shown in (30). φ Z d,k = (I n ρ d W) 1 α d,k (30) Diagonal elements of the matrix represent own-partials and off-diagonal elements are the cross-partial derivatives. If we are interested in the impact of changes in variables in the matrix Z d on trade flows, we need to take the matrix Δ d into account. We note that: y/ Z d,k = y φ φ Z d,k φ = Δ d ( Z ) d,k = (ιn In)(In ρ d W) 1 α d,k = (ιn In)(Inα d,k + ρ d Wα d,k + ρ 2 d W2 α d,k +...) 44