Exploring Brexit with dynamic spatial panel models : some possible outcomes for employment across the EU regions

Starting with a reduced form derived from standard urban economics theory, this paper estimates the possible job-shortfall across UK and EU regions using a time-space dynamic panel data model with a Spatial Moving Average Random E⁄ects (SMA-RE) structure of the disturbances. The paper provides a logical rational for the presence of spatial and temporal dependencies involving the endogenous variable, leading to estimates based on a dynamic spatial Generalized Moments (GM) estimator proposed by Baltagi, Fingleton and Pirotte (2018). Given state-of-the art interregional trade estimates, the simulations are based on a linear predictor which utilizes di⁄erent regional interdependency matrices accord-ing to assumptions about interregional trade post-Brexit.


Introduction
What will be the impact across the EU regions 1 of a shock to the UK and EU economy following Brexit? What I want to do is show possible, speculative, predictions of the impact of Brexit on employment across the EU. The Brexit e¤ect will not be con…ned to the UK, but how far will it spread, and how long will it last? I am using a state of the art modelling approach, but the predictions, like all predictions, can only be very imprecise. Nevertheless, we can look at the outcome qualitatively. The main outcome of my e¤ort is that spatial and temporal spillovers are a signi…cant feature. In other words, the impact of Brexit will extend beyond the shores of the UK and persist into the future.
I emphasize that these are speculative predictions regarding the impact of Brexit across the EU and UK regions. The predictions are presented with a high level of caution, so the numbers SHOULD NOT be taken too literally. They are naturally dependent on the assumptions underpinning the model. Di¤erent assumptions regarding model structure, relevant data and estimation techniques can all potentially a¤ect the predictions of the Brexit impact or of what would otherwise happen had Brexit not been the result of the UK referendum. Nevertheless the predicted % impacts in terms of jobshortfall are robustly predictable given what is assumed regarding the overall % reduction in trade between UK and EU regions. Also it is reasonably insensitive to what is assumed about future paths of drivers of employment. This suggests that the Brexit e¤ect is perhaps somewhat immune to how we predict it, within certain reasonable parameters naturally. If the Brexit e¤ect is in a sense neutral to the modelling and prediction e¤ect, then we might be able to take the predictions a bit more seriously than would otherwise be the case. Of course, as always, extreme caution should be exercised.
The simulations in the paper are the outcomes of assumed changes in across-EU region connectivity due to trade barriers erected on exit from the EU. Using state-of-the-art trade interregional ‡ow estimates, the level of trade between post-Brexit EU regions and UK regions is reduced compared with the trade ‡ows that would otherwise exist. The simulated impact of Brexit is an estimate of the % job-shortfall that would occur were there to be no additional stimulus, in the form of changed levels of output and capital in response to new trade deals as mooted by the pro-Brexit lobby. In reality it might be the case that the UK and EU's employment shortfall, as predicted by this analysis, will be o¤set by the introduction of new trade deals with the rest of the World, resulting in a boost to output and capital investment and consequently employment. Moreover, a boost to the UK economy could increase demand across Europe and lead to an increase in employment, contrary to what is predicted here which is based on a reduction in trade. However this is very much speculation and there is much uncertainty as to the real outcome as a result of Brexit. What is attempted here is a simulation of outcomes under strict assumptions, which may or may not hold.
More speci…cally, in order to estimate of the Brexit e¤ect, I estimate employment levels across N = 255 EU regions both with and without Brexit. The explicit drivers of employment are output and capital, which are approximated by Gross Value Added (GVA) and a function of Gross Fixed Capital Formation (GFCF) respectively. Estimation of a model with employment related to output and capital is based on a viable data series over the period 2001 to 2010 2 . Di¤erent assumptions can be made about post 2011 paths for GVA and GFCF, given that accessible data with the same geography are not available over the more recent period. One assumption might be that post 2011 paths are driven by each region's average growth over the period 1991 to 2011. A second assumption might be to hold GVA and GFCF at the 2011 levels through into the future. Interestingly di¤erent assumptions have relatively little e¤ect on outcomes.

The Model
The reduced form used as a basis to simulate the Brexit e¤ect assumes that employment partly depends on level of output, as measured by GVA (Gross Value Added), denoted by q t , and (a proxy for) the level of capital within the region, based on GFCF (Gross Fixed Capital Formation), which is denoted by k t . To show this we start with the theoretical model given as equation (1), which is based on equation (30) given in the Appendix. The N by 1 vector e d t is the density of employment per unit area, and a t is the level of e¢ciency of labour at time t; so that the product e d t a t is the number of labour e¢ciency units. This is related to e q t ; which is a measure of output in the competitive …nal goods and services sector in each region at time t;via the constant parameters and e ;thus In order to obtain total output q t ; it is assumed that e q t = q t , in which is an N by 1 vector giving the share of total output in each region that is competitive …nal goods and services output. For simplicity of estimation it is assumed that is constant over time. Also the employment levels are e t = he d t in which h is the area of land in each region. Taking logs gives ln + ln q t = ln + e ln e t + e ln a t e ln h Rearranging (2) gives ln e t = 1 e (ln + ln q t ln ) ln a t + ln h To obtain (3) I assume that labour e¢ciency a t = qt e kt , with more e¢cient labour having a higher level of output per unit of capital e k t . As shown below in equation (13), an approximation to the log level of capital is ln e k t = ln e a + e b ln k t ; hence ln a t = ln q t + ln e a e b ln k t ; and from this ln e t = 1 e (ln + ln q t ln ) + ln h ln q t ln e a + e b ln k t Collecting together constants as c and reorganising gives in which the error term " t captures the time-invariant regional heterogeneity in land h and in shares , which are unobservable, as given in equation (11).
In the dynamic context, it is reasonable to assume that disparities in employment levels across locations will persist as an equilibrium outcome to unchanging and fundamental causes. We therefore assume that (log) employment levels across regions, denoted by the N by 1 vector ln e t at time t will persist at dynamically stable levels so that ln e t = ln e t 1 unless there 4 are changes in the levels of q t or k t , or changes in interregional trade, or changes in unobservable e¤ects. If such a disturbance occurs at time t and is ephemeral, then ln e t 6 = ln e t 1 but over a subsequent period of quiescence t ! T then once again we expect employment levels to converge on a new equilibrium at which ln e T = ln e T 1 : Assume data are observed where ln e t 6 = ln e t 1 but tending to converge, so that ln e t = f (ln e t 1 ), and an autoregressive process is assumed, hence in which & is an N by 1 vector and is a scalar parameter. In the long-run with abs( ) < 1; and with no subsequent disturbances, the process converges to ln e T = & (1 ) . Consider next connectivity between regions in the form of a matrix W N ;which is is a time-invariant N by N matrix where N is the number of regions. For purposes of interpreting parameter estimates we normalize by dividing W N by the maximum eigenvalue of W N to give 3 W N . Using this normalization, the maximum eigenvalue of W N is 1, and the continuous range for which in which 1 is a scalar spatial autoregressive parameter: Given (5), logic dictates that Subtracting (6) from (5) leads to another logically consistent expression in which the spatial dependence implied by (6) can be seen in (7) as an explicit cause of variation in ln e t :Thus The matrix W N retains (scaled) absolute levels rather than shares as the basis of interregional connectivity, and we make the standard assumptions for a weights matrix, that it comprises …xed (non-stochastic) non-negative values with zeros on the leading diagonal and its row and column sums are uniformly bounded in absolute value, and maintain the same assumption for B 1 N = (I N 1 W N ) 1 (Elhorst, 2014, p. 99 in which B N = (I N 1 W N ) ; C N = ( I N + W N ) and I N is an identity matrix of order N . In order to solve equation (7), given appropriate parameter restrictions, equation (7) converges to ln e T = (B N C N ) 1 B N &: Introducing additional covariates by writing B N & = (c + x ) ; in which c is a constant N by 1 vector, x is an N by k matrix and is a k by 1 vector, gives ln e t = B 1 N [C N ln e t 1 + c + x ] In order to maintain dynamically stable simulations, following Elhorst (2001Elhorst ( ,2014, Parent andLeSage (2011, p. 478, 2012, p. 731) and Debarsy, Ertur and LeSage (2012, p. 162), requires the largest characteristic root (e max ) of B 1 N C N to be less than 1. This restriction ensures that employment converges to equilibrium levels ln e T = (B N C N ) 1 (c + x ).
Additional realism is introduced in three ways. First, the restriction that = 1 is removed since 1 and are unknown, so that is free to vary. However we anticipate that b b 1 b : Second, taking account of the variables in equation (4), the time invariant matrix x is replaced by timevarying matrix 4 x t : Third, spatially dependent unobservables are represented by the error term " t : Although the system may, depending on B 1 N C N , still tend towards equilibrium, equilibrium will be continuously disturbed and new equilibrium levels established as t varies. For simplicity of estimation, inter-regional connectivity is assumed to remain constant over the estimation period. These considerations lead to the model given in equations (8,9,10,11), which is a time-space dynamic panel data model, thus x 2t ] and = [ 1 2 ] T , equation (8) can be stated more explicitly as ln e t = c + ln e t 1 + 1 W N ln e t + 1 ln q t + ::: i = 1; :::; N; t = 1; :::; T The disturbances " t capture the e¤ects of the spatially dependent unobserved variables, with a compound structure (11) comprising time-invariant unobserved interregional heterogeneity represented by i with i = 1; :::; N and unobserved idiosyncratic shocks represented by it ; i = 1; :::; N; t = 1; :::; T . These are assumed to be independent of each other and are collectively represented by u it . It is important to recognize that the i s represent unobserved factors creating interregional heterogeneity perhaps as a result of di¤erences in industrial structure and physical and sociocultural environment, which in the short run can be treated as time-invariant. In the longer run, one might introduce a factor structure in which it = T i F t where i is a (r x 1) vector of factor loadings and F t is a (r x 1) vector of common factors, such as shocks to industrial structure and environment, with heterogenous regional e¤ects i ;in which case it = i1 F 1t + ::: + ir F rt : One might assume that in the short run the common factors do not vary over time, so that F t = F . Hence i = it = i1 F 1 + ::: + ir F r : Most usually the assumption is that spatial dependence is an autoregressive (SAR-RE) process, such that " t = 2 M N " t +u t :However in this paper the assumption for the error process is a spatial moving average process (SMA-RE) as in equation (10), This means that the error process is such that a shock in a region a¤ects only neighbouring regions as de…ned by a row standardized interregional contiguity matrix 5 M N . In contrast, an SAR-RE process would entail shocks a¤ecting all regions.There are two reasons for this. First, Assuming SMA-RE rather than SAR-RE errors improves the predictive performance of the 5 The matrix M N has e e e 1 max = 1;where e e e is the vector of purely real characteristic roots of M N : We assume that M N has the same properties as W N ; and with the restriction that e e e 1 min < < e e e 1 max = 1 one guarantees the invertibility of G N as in equation (21): estimator, as described in Section 6. Secondly, SMA-RE errors might proxy for omitted spillovers, which otherwise might be captured by the spatial lags W N x t : This is pertinent since the presence of W N x t on the right hand side of (8) could adversely a¤ect estimation. As explained by Fingleton, Le Gallo and Pirotte (2017) and Baltagi, Fingleton and Pirotte(2018), an SMA-RE error speci…cation 'mitigates against the problem for instrumental variable estimation identi…ed by Pace et al. (2012)'. In two-stage least squares (2SLS) estimation, the instrument set should comprise the 'exogenous' variables (x t ) and their spatial lags (W N x t ), and kept to a low order to avoid linear dependence and retain full column rank for the matrix of instruments Prucha 1998, 1999) . The performance of the estimation procedure could be suboptimal, as explained by Pace et al. (2012), by including W N x t among the set of explanatory variables. This is because with spatial lags (W N x t ) among the set of regressors, then spatial lags of the spatial lags . .) feature among the instruments, and this could lead to a weak instrument problem. To avoid this, SMA-RE errors are adopted as an alternative way to capture local spillovers.

Data
In estimating equation (9), data for employment (e t ), output as measured by Gross Value Added (GVA, q t ) and capital as proxied by a function of Gross Fixed Capital Formation (GFCF, k t ), both denominated in e2005m, are taken from the Cambridge Econometrics European Regional Economic database, with observations over the 10 year period 2001 to 2010 used to estimate the model. Data are also available for 2011 2012, but are held back to allow out-of-sample prediction tests of the model and some rivals. k t is used to re ‡ect capital stock e k t , for which data are unavailable, on the basis of a simple relationship which is assumed to exist between the two variables. k t measures gross net investment (acquisitions minus disposals of produced …xed assets) in …xed capital assets and so provides an indicator of changes to the stock of capital. The assumption is that k t is a non-linear function of a constant fraction e a of e k t so that As a test of the viability of this approximation, assume a standard model for the evolution of capital stock which is depreciating at a constant rate e d so that in which T is a large number. One problem with (14) is that it requires the initial capital stock at time t = 1;i.e. e k 1 : However given arbitrary values for e k 1 and e d, values for e a and e b can be found whereby (13) provides a reasonable approximation to the outcome of iterations (14). A more realistic test is provided by the existence of both (albeit experimental estimates of) capital stock 6 (Derbyshire et al, 2010) and of well-founded GFCF data. Using the latest available data for both k t and e k t , which is for the year, t = 2008, and taking logs of (13), leads to a loglinear regression of ln e k t on ln k t which gives OLS estimates of the constant ln e a 1 = 2:4546 (t ratio = 13:5628) and slope e b = 1:0195(t ratio = 50:8118);with R 2 = 0:8888:The plot of ln k t against ln e k t shows a signi…cant linear relationship and no evidence of outliers or of heteroscedasticity It thus appears that the model given as equation (12) provides a good approximation. The estimated e a = 0:0859 suggests the approximate proportion of the capital stock that is invested, and, by comparison, P k t / P e k t = 0:0686: The matrix W N is based on estimated bilateral trade ‡ows between EU NUTS2 regions. The data come from the PBL (the Netherlands Environmental Assessment Agency) 7 who developed a new methodology which is close to that of Simini et al (2012). Details of the methodology are given in Thiessen et al. (2013Thiessen et al. ( ,2013aThiessen et al. ( ,2013b, see also Gianelle(2014). The method follows a top-down approach and therefore is consistent with the national accounts of the di¤erent countries. Given the total international exports and imports on the country level, interregional trade ‡ows are derived using data on business travel (services) and on freight transport (goods). Additionally, exports that went to EU destination countries' …nal demand 8 were also included. Trade ‡ows involving regions of non-EU countries Switzerland and Norway were obtained on the basis of interregional trade ‡ows estimated by the best linear disaggregation method of Chow and Lin(1971), which was initially used to break down annual time series into quarterly series (see Abeysinghe andLee, 1998, Doran and. In this, commencing with aggregate trade values 9 between 21 EU counties, these were allocated to the NUTS2 regions. A parallel approach has been used by Polasek, Verduras and Sellner (2010), Vidoli and Mazziotta (2010), and Fingleton, Garretsen and Martin (2015). More detail of the method is provided in the Appendix. Finally, OLS regression of the log PBL trade ‡ows on log Chow-Lin trade ‡ows produced parameters used to predict the missing PBL regional trade ‡ows for Switzerland and Norway using the values for these regions obtained via the Chow-Lin approach. For estimation, the start-of-period trade ‡ows for the year 2000 is used. This year is chosen because it is the earliest available, so it is treated as exogenous to e t ,q t and k t , for t = 2001 to 2010. Prediction is based on the 2010 trade ‡ows supplemented in the same way by Chow-Lin data. Estimates are also given in Appendix Table A3 based on a W N matrix constructed entirely from the Chow-Lin trade ‡ows. These simply use great circle distances and year 2000 GVA levels, and so are also be assumed to be exogenous. The comparative predictive performance of each set of estimates is discussed in Section 6.

Estimator for the time-space dynamic panel data model
Comprehensive overviews of spatial panel econometrics are given by Pesaran(2015, Chapters 29 and 30) and Baltagi(2013, Chapter 13) which highlight its growing importance for the applied econometrician. The estimator used in this paper, introduced by Baltagi et al (2018), adds to the available methodology by allowing a wider range of spatial interaction e¤ects which include the spatial lag of the temporal lag of the dependent variable W N ln e t 1 , 2.Final consumption expenditure by government 3.Net capital formation 4.Inventory adjustment 9 They are downloadable from http://cid.econ.ucdavis.edu/data/undata/undata.html, see also Feenstra et al. (2005). thus avoiding bias due to constraints necessary for dynamic stability and stationarity, and also by allowing spatial moving average compound error dependence rather than the usual autoregressive compound error process found in the majority of spatial econometric models. The estimator, which is applied to equation (8), is based on the earlier paper by Baltagi et al. (2014), which extends the approach of Arellano and Bond (1991) by the introduction of extra moments in line with the presence and availability of spatial lags (see also Bouayad-Agha and Védrine, 2010). Since the estimator is described elsewhere, a simple outline sketch is provided here focussing on the treatment of regressors as predetermined rather than exogenous 10 . Hence in equation (9), ln q t and ln k t are considered to be predetermined alongside endogenous left hand side variables ln e t 1 ; W N ln e t 1 and W N ln e t : Focussing on the endogenous dependent variable ln e t , the instruments include ln e t lagged by two periods, and its spatial lag W N ln e t also lagged by two periods, so that the moments equations (15,16) hold assuming it is serially uncorrelated and E( it ; it 2 ) = 0:Thus following Baltagi et al (2007) in which E denotes the expectation. Also, if we were to assume exogenous rather than predetermined regressors (x 1 ; x 2 ) this leads to (17) Z t = ln e 1 ; :::; ln e t 2; W N ln e 1 ; :::; W N ln e t 2 ; x 11 ; :::; x 1T ; x 21 ; :::; x 2T ; W N x 11 ; :::; W N x 1T ; W N x 21 ; :::; for t = 3; :::; T:Given that in (17) the regressors (x 1 ; x 2 ) are exogenous, the moments equations are satis…ed including the entire set x 11 ; :::; x 1T ; x 21 ; :::; x 2T ; ; W N x 11 ; :::; W N x 1T and W N x 21 ; :::; W N x 2T regardless of time t. As explained in Baltagi et. al.(2018), additional instruments can be generated via the matrix W 2 N ;but for simplicity these are omitted from the estimators used in the current paper.
Strict exogeneity rules out any feedback from past shocks to current values of the variable, and the need to accommodate feedback leads to the preferred estimator based on predetermined regressors (see Bond, 2002, Pesaran, 2015. Predetermined regressors are contemporaneously uncorrelated, so that corr(x t , t ) = 0; but do depend on earlier shocks so that, for example, corr(x t , t 1 ) 6 = 0:This means that an adjustment to ln e;which embodies ; at time t does not have an instantaneous e¤ect on output and capital investment time t but takes e¤ect at t + 1 and later. This allows an extension to the set of instruments (compared with assuming endogeneity, where all endogenous variables are lagged by two periods), by the inclusion of x 1t 1 ; x 2t 1 ; W N x 1t 1 and W N x 2t 1 so that ln e 1 ; :::; ln e t 2; W N ln e 1 ; :::; W N ln e t 2 ; x 11 ; :::x 1t 2 ; x 1t 1 ; x 21 ; :::x 2t 2 ; x 2t 1 ; W N x 11 ; :::W N x 1t 2 ; W N x 1t 1 ; W N x 21 ; ::: Given the set of instruments as in equation (18), these are used to obtain initial estimates of ; 1 ; , 1 and 2 ; having …rst di¤erenced the data to eliminate the time invariant individual e¤ects which are correlated with the lagged dependent variable. The resulting estimates are then used to give estimated errors which lead to estimates of the parameters of the spatial moving average error process, namely 2 ; 2 and 2 using moments equations given in Fingleton(2008). Given these, preliminary one-stage consistent spatial GM estimates are obtained, followed by the two-stage Spatial GM estimates of ; 2 ; and based on a robust version of the variance-covariance matrix: 12  (8) Table 2 shows that the estimate for the spatial lag of the temporal lag ( W N ln e t 1 ) is not dissimilar to b b 1 , in line with expectation stemming from an equilibrium process. Also the Table 2 estimates are stationary and dynamically stable, as shown by the largest characteristic root of B 1 N C N which is equal to 0:6757, and the stationary bounds for 2 are e e 1 min = 1:1239 < 2 < e e 1 max = 1: Observe that the negative values of b 2 imply positive spatial dependence among the errors. Appendix Table A give the estimates of some rival estimators, including one with SMA-RE errors but assuming exogenous regressors (Table A1), and with SAR-RE errors assuming predetermined regressors (Table A2). As noted in Section 6, the predictive ability of these rivals is not as good as obtained via the preferred estimates summarised in Table 2.

Prediction
In order to support the preferred model summarised by Table 2, a crossvalidation strategy is employed to assess the performance of competing estimators 'by comparing their predictive ability on data which have not been used in model estimation' (Anselin, 1988). Out-of-sample predictions of the level of employment across regions are obtained for the years 2011 and 2012 using 2011 and 2012 data combined with the parameter estimates obtained for data over the estimation period 2001 to 2010.
Following Chamberlain (1984), Sevestre and Trognon (1996), and Baltagi et. al.(2014), the linear predictor is in which E [:] denotes the expectation, so this can be seen to be identical to equation (8) but with expectations. With regard to the estimate of the time-invariant component of the error term , assuming a spatial moving average error process gives equation (8) rewritten thus In order to obtain estimates b (t) estimates b  (22) for T + 1 = 2011; in which x 1T +1 = ln q T +1 and x 2T +1 = ln k T +1 ; t = 1; :::; T: For two step ahead 11 , x 1T +2 = ln q T +2 and x 2T +2 = ln k T +2 : Figure 1 shows a close correlation between predicted log employment ln b e T +1 and observed log employment, suggesting that the preferred estimator giving the Table 2 estimates would be a good basis for simulating the impact on employment following Brexit. The preference for the Table 2 estimates is based on the mean of the (ln e i;T +s ln b e i;T +s ) 2 =N for s = 1; 2;denoted by RM SE. In the case of Table 2, RM SE = 0:0721:Rival estimators (Appendix Table A), give less accurate one-and two-step ahead predictions. In the case of assuming SMA-RE errors and exogenous regressors, RM SE = 0:1791. Assuming SAR-RE errors with predetermined regressors gives RM SE = 0:2890. Note that in the case of SAR-RE errors, b G N = (I N b 2 M N ) 1 in equations (19,20,22). Table A also gives estimates relating to SMA-RE errors and predetermined regressors, but are based on W N derived using the Chow-Lin approach. In this case RM SE = 0:2529;providing support for the choice of W N based on the PBL trade data. Table A also give estimates based on SMA-RE errors and predetermined (and exogenous) regressors, but with the additional variables W N x 1 ;the spatial lag of ln q, with parameter 3 ,and W N x 2 ;the spatial lag of lnk, with parameter 4 , with W N given by the PBL trade data. This is thus a form of spatial Durbin speci…cation, but with regressors x t = (x 1t ; x 2t ; W N x 1t ; W N x 2t ) the additional covariates evidently cause a problem of weak instruments, giving dynamically unstable nonstationary estimates, as re ‡ected by the largest characteristic root of B 1 N C N equal to 1:0663 (1:9041) and, with x in equation (21) and (22) RM SE = 7:4403 (3:0918): The same spatial Durbin speci…cation again assuming predetermined regressors but with 2 restricted to zero gives a largest characteristic root equal to 1:1127 and RM SE = 3:3017. The same spatial Durbin speci…cation assuming exogenous regressors and with a spatial autoregressive (SAR) error process gives a largest characteristic root equal to 2:489 and RM SE = 20:9333.These results point to the viability of the Table  2 estimates for prediction purposes.

Simulating the Brexit e¤ect
The approach adopted is to use the parameters estimates in Table 2 to predict the impact on employment of presumably reduced trade between the UK and the remaining EU regions in the year 2020 and beyond. Attention is focussed on 2020 and later, given that the UK's formal exit from the EU is scheduled for the …rst half of 2019, so 2020 will the …rst full year outside the EU. Given a lack of appropriate and accessible data, for instance with the same geography as up to 2011, beyond 2011 employment could be predicted on the basis of alternative assumptions about the level of q and k in 2020. One assumption might be that q and k in 2020 are at the same level as observed in each region in 2011. An alternative assumption could be that from 2011 onwards they grow at their historical rates, taken over the period 1991 to 2011 in each region. On this basis on average the level of q and k in 2025 is approximately 25% more than the 2011 levels. However it is easy to show by simulation, using alternative levels of x T +s in equation (23) that the assumption made about the future path of q and k only has a minor e¤ect on the equilibrium % job shortfall across regions. For example, a 100% increase in the 2011 levels of q and k only increases the job shortfall in London from 1:829% to 1:927%, a factor of approximately 1:054, assuming a 2% reduction in trade. Similarly assuming a 16% reduction in trade in increases London's job shortfall from 14:58% to 15:36%. Table 3 summarizes the outcomes for various trade reductions. Commensurate e¤ects occur across all regions, as exempli…ed by the outcomes for Paris in Table 4. The conclusion is that irrespective of the % reduction in trade, doubling the 2011 levels of q and k only increases the % job shortfall by a factor of approximately 1:05.
Regardless of the assumptions regarding future levels of q and k, from = 2020 onwards there are two scenarios, on based on the trade ‡ows assuming no Brexit e¤ect, and the other assuming a Brexit e¤ect on trade ‡ows, and the di¤erence between them is taken as the Brexit e¤ect. Regarding the no-Brexit e¤ect scenario, this applies matrix W N ,which is based on the latest available trade ‡ows pertaining to the year 2010. The prediction is then given by the solution to equation (24) : Also x is an (N by 2) matrix containing the forward projections ln q and ln k ; thus The second scenario is to assume that bilateral trade between the UK regions and the (remaining) EU regions is, for example, 2% lower than it would otherwise be. Thus of the N = 255 UK plus EU regions, there are N 2 N = 64; 770 bilateral trade ‡ows in any one year involving the regions. With 37 UK regions and 218 EU regions, (2 x 37 x 218) = 16; 132 interregional trade ‡ows are assumed to be 2% smaller than under an assumption of no Brexit e¤ect. This Brexit-a¤ected trade ‡ow matrix is denoted by f W N which leads to e Thus the % job shortfall is ln e e ln b e : The assumption of a 2% reduction in trade is of course an arbitrary one, and in ‡uenced at the time of writing by negotiations, debates and proposals relating to the possibility of a transition period in which the pattern of trade might not be too dissimilar to the typical pattern assumed for the pre-Brexit period. Any % change could be assumed, and indeed in an ideal world one might wish to make changes to trade on an individual region by region and sector by sector basis rather than assume that trade reduces by the same amount across all regions and all sectors. However this is very much the unknown, although some sectorally speci…c estimates are given below which also have geographically nuanced outcomes. Initially the simulation adopts the simple assumption that all trade between EU and UK regions is reduced by 2%, although the resulting geographical pattern is robust to the % reduction assumed. This is exempli…ed by Tables 3 and 4. These show that doubling the reduction in trade approximately doubles the % job shortfall. Also these ratios hold regardless of the di¤erent assumptions one makes about the future path of q and k. For example, assuming 2011 levels, the equilibrium % job shortfall in London goes from 7:304% assuming an 8% trade reduction, to 14:578% assuming a 16% trade reduction, and from 7:697% to 15:362% if one doubles the 2011 levels of q and k. A similar pattern is shown for Paris in Table 4, although the % job shortfalls are about 50% of those for London.  The total impact of Brexit can be broken down into the impact of restricted trade in manufactured goods 12 , the impact of restricted trade in services 13 , and the impact of restricted trade in other sectors 14 . These sectoral trade patterns have di¤erent geographies and therefore the impacts have di¤erent geographical distributions to the outcomes assuming a global reduction across all sectors. In order to simulate the impact of restricted trade in manufactured goods, the employment levels assuming no disruption to trade across all sectors is compared to the levels of employment in which overall trade is reduced solely as a result of the reduction by 2% in manufactured goods trade between EU regions and UK regions. A similar comparison is made for employment based on no trade reduction versus employment based on reduced overall trade as a consequence of trade between EU and UK regions involving services being reduced by 2%.

Results
The initial outcomes relate to a reduction in EU-UK trade of 2% across all sectors. I predict % change in employment across the EU and UK regions assuming the 2011 levels for q;and k. On this basis Figure 2 shows the dynamic paths for each region to 2050, with convergence to steady state occurring after 2030. From this it is evident that the maximum equilibrium job shortfall is 1:829%;in the case of Inner London;with most other regions falling below 1%. Figure 3 shows the geographical pattern of the Brexit impact equal to ln e e ln b e for = 2025, indicating a maximum shortfall by 2025 of 1:72% (Inner London). The picture which emerges from the simulation is that the negative Brexit impact is bilateral, with both UK regions and EU regions likely to see an employment shortfall. Figure 3 shows larger negative impacts in regions with strong trading links to the UK, most notably in the Paris region ( 0:87%), the Southern and Eastern region of Ireland ( 1:05%), and the Oberbayern region centred on Munich( 0:75%). Figure 4 gives the frequency distribution of the Figure 3 data, highlighting the fact that despite some large impacts, for about 160 of the 255 regions, Brexit is likely to have close to zero e¤ect on employment. Figure 5 shows that within the UK, Inner ( 1:72%) and Outer London ( 0:99%) are expected to have the biggest % shortfall by 2025, with impacts generally higher along the Thames valley in Berkshire, Bucks and Oxfordshire ( 0:83) towards Gloucestershire, Wiltshire and North Somerset ( 0:9%). Generally %s are higher around the Greater South East and in some of the large conurbations (Birmingham 0:6%, Manchester 0:57%, West Yorkshire 0:51%) than in more rural and peripheral regions. Figure 6 gives the frequency distribution of the Figure 5 data, emphasizing the Inner London outlier, with the majority of regions having a job shortfall of less than 0:5%. As noted above, if one were to assume di¤erent reductions in trade other than 2%, the outcomes for employment would be di¤erent, but proportional to the 2% impact, and the geographical pattern would be essentially identical.  Figure 7 shows that the geography of the impact due to 2% less industry trade is very similar to the overall pattern shown in Figure 3. However a comparison of Figures 4 and 8 emphasizes the di¤erences in the levels of impact, with the maximum level, equal to 0:86% for Inner London. Figures 9 and 10 show the % job shortfall in the UK and Ireland. Again the impact in the South and East of Britain and especially London is clear, and the e¤ect on Ireland remains pronounced. Compared with industry, the impact of reduced trade in services is much less symmetrical, with the bulk of the job shortfall occurring in Britain and Ireland. This is clear from Figures  11 and 12. Inner London clearly stands out as having the most signi…cant impact compared with all other EU regions, and apart from the South and Eastern region of Ireland, almost all non-zero employment shortfalls occur in Great Britain. Figures 13 and 14 emphasize the polarized e¤ect of service trade reduction, with Inner London standing out with a job shortfall of 0:62%. Outer London is predicted to see a job shortfall of 0:29% by 2025, but Southern and Eastern Ireland see a comparable e¤ect, equal to 0:3%. All other regions have predictions of 0:2% or less.

Conclusion
In the paper the impact of Brexit is measured in terms of the job-shortfall, which is the reduction in the number of jobs in each region due to Brexit assuming no alternative sources of employment are put in place. This of course might be a false assumption, as the pro-Brexit lobby has consistently emphasized the potential stimulus of new trade deals with other non-EU countries. Therefore our impact as re ‡ected in the maps of job-shortfall indicate those regions which will be in greatest need of alternative employment sources to compensate for the job-shortfall likely as a result of Brexit. Thus the paper is not predicting a job-loss per se, simply a potential job loss without successful alternative trade arrangements post-Brexit. The paper shows negative impacts on employment which a¤ect not only the UK regions but also employment levels in EU regions, especially those which are close trading partners. This pan-European interregional interdependency is captured in the model by spatial and temporal interactions based on state-of-the-art trade ‡ow estimates which determine the strength of interdependence. This means that employment within a region depends on the levels of output and capital within the region, but also on demand coming from other regions which are trading partners. The approach adopted has been to assume a reduction in trade between EU and UK regions which gives a corresponding reduction in demand. While the impact levels are proportional to the trade reduction assumed, the model implies that the geography of Brexit impact is robust to the assumptions made. In summary, within the UK the predicted job-shortfall by 2025 tends to be concentrated in the greater South East in and around London, plus some major conurbations. Within the EU regions, while the impact is spread more widely and for most regions is quite limited, Southern and Eastern Ireland, Paris and Oberbayern, Stuggart and Dusseldorf stand out as regions likely to see signi…cant job shortfalls. It would seem that not only is it important for the UK to seek new trade agreements, but also the EU. Breaking down the impact sectorally, it appears that the major part of the overall job shortfall is attributable to a loss of manufacturing trade. The impact of reduced trade in services appears to be less, but interestingly is much less symmetrically distributed across EU and UK regions than for manufacturing, with the bulk of the job-shortfall focussed on UK regions, especially London, but also on Ireland.
These are speculative simulations, conditional on important assumptions, so the numbers SHOULD NOT be taken too literally. Great caution is needed in interpreting the validity and value of any 'prediction' e¤ort. It is worth recalling the words of George Box : "All models are wrong but some are useful". David Spiegelhalter, Professor of the Public Understanding of Risk at the University of Cambridge, refers to Donald Rumsfeld as the patron saint of Risk Analysis. He will be remembered for famously saying that "but there are also unknown unknowns. There are things we do not know we don't know". We should therefore put forward predictions with all due humility, but clearly and without fear, because we don't want to come across as 'dithering scientists'. In defence of the approach adopted, there is support from the words of Pesaran(1990), who points out that 'Econometric models are important tools for forecasting and policy analysis, and it is unlikely that they will be discarded in the future. The challenge is to recognise their limitations and to work towards turning them into more reliable and e¤ective tools. There seem to be no viable alternatives'.

Theoretical background
The theoretical background to the model speci…cation commencing with equation (1) is derived from standard urban economics as given by Abdel-Rahman and Fujita(1990), Ciccone and Hall(1996) and Fujita and Thisse (2002) Assume that the equilibrium output of each service …rm is i e and there are g …rms, depending on the total services e¤ective labour. We obtain g by dividing the total services e¤ective labour by the e¤ective labour per …rm, thus g = (1 e )e e ai e + s In (26), (1 e ) equals the services labour share of total e¤ective labour e e under competitive equilibrium in the labour market:For each services …rm, we have internal returns to scale, with s denoting the …xed labour requirement and a the marginal labour requirement, so that ai e + s is the e¤ective labour per …rm at equilibrium. From the CES production function we obtain where e is a measure of monopoly power in the monopolistically competitive services sector, and e e 1 is the constant elasticity of substitution. Substituting i into equation (25) gives which simpli…es to e q = e e (1+(1 e )(e 1)) = e e e This shows that with = 1 there are increasing returns (e > 1) if service …rms are relevant to output in the competitive sector e < 1 and also possess monopoly power (e > 1) : However < 1 indicates that land is also a relevant factor, and depending on the value of any tendency to increasing returns could be o¤set by the congestion e¤ect caused by the restriction of production to a unit of land, leading to diminishing returns.  N ) matrix T e N , in which the subscript e N means national). There are e N = 21 unobserved intra-country trade ‡ows, thus giving p = 420 observations for the year 2000. The e N 2 = 441 'observations' are the dependent variable in a Weighted Least Squares regression model. All observed trade ‡ows are given the weight 1, and those unobserved weighted zero. In terms of parameter estimates, an entirely equivalent procedure is to estimate the 420 observed trade ‡ows by OLS. In the regression, the explanatory regressors are great circle distances between country pairs (G e N ), and the product of each pair of country's national GVA level (e q e N ) in the year 2000, so that given the ( e N 1) vector e q e N , Q e N = e q e N e q > e N is an ( e N e N ) matrix. Subsequently Q e N ; G e N and T e N are reshaped as ( e N 2 1) vectors q e N , g e N and t e N , and together with c e N which is an ( e N 2 1) vector of ones, these variables provide the input for the regression denoted by equation (31), thus giving the estimates b e N . Interregional trade estimates are based on the (national level) regression parameter estimates b e N and on the estimated regression residuals b e e N . Thus, trade ‡ows between regions, denoted by t e R , are obtained by applying the national level estimates b e N to the regressors measured at the regional level, denoted by g e R and q e R . Also, an equal share of the national level residuals b e e N is added to regions corresponding to country pairs. This process is summarised by the two equations:

Other estimates
ln t e N = e N ;1 c e N + e N ;2 ln g e N + e N ;3 ln q e N + e e N with log bilateral interregional trade ‡ows ln t e R then being obtained using ln t e R = e N ;1 c e R + e N ;2 ln g e R + e N ;3 ln q e R + b e e R .
To obtain the ( e R 2 1) vector b e e R , we calculate the ( e R e R) matrix b V e R is an ( e R e R) matrix containing the inter-country residuals b e e N allocated equally to all regions corresponding to country pairs. Also there are block diagonal zeros as a result of the unobserved intra-country trade ‡ows, and zeros across 4 rows representing 4 regions equal to the countries Estonia, Lithuania, Latvia and Luxembourg which were excluded from the initial regression. Finally, b V e R is reshaped as the ( e R 2 1) vector b e e R of equation (32) to give ln t e R . The resulting e R e R matrix of interregional trade ‡ows is W N = T e R .