Market Access and Regional Wage Structure : Estimating the Helpman-Hanson Model for Sweden

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Market Access and Regional

Wage Structure

E s t i m a t i n g t h e H e l p m a n - H a n s o n M o d e l f o r S w e d e n

Master thesis in Economics

Author:

Lars Lundström, 880210-2411

Tutors:

Prof. Börje Johansson

Ph.D. Lina Bjerke

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Table of Contents

1

Introduction ... 1

2

Theoretical and empirical concern about the

Helpman-Hanson model ... 2

3

Data and Econometric Issues ... 4

4

Empirical Analysis ... 9

4.1

Harris Model ... 9

4.2

Helpman-Hanson Model ... 11

5

Conclusion ... 15

6

References ... 18

Appendix A ... 20

Derivation of the estimation equation ... 20

Appendix B ... 24

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2

Theoretical and empirical concern about the

Helpman-Hanson model

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Figure 1

Income per km

Figure 2

Housing per km

Figure 3

Wage levels

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4

Empirical Analysis

4.1 Harris Model

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4.2 Helpman-Hanson Model

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Appendix A

Derivation of the estimation equation

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Appendix B

Estimation Code

// Include header files and import the maximization package #include <oxstd.h>

#include <oxfloat.h> #import <maximize>

// Declare a few variables

static decl s_vY, s_mX, s_iEval = 0, vEps;

// The initialize function initializes variable-matrices based on an excel table and also sets // the initial starting values for the parameters

Initialize(const mDat, const avY, const amX, const avP) {

// In this case you would want to have the wage data in the first column (column [0]) // and the other variables in the next three columns

avY[0] = mDat[][0]; amX[0] = mDat[][0:3];

// Set initial starting values for the parameters based on previous research avP[0] = < 0 ; 5 ; 0.9 ; 0.1 >;

// Print a message stating the initial parameter values

println("\rThe starting values for the parameters are: \r", "%r", {"beta", "sigma", "delta", “tao”}, avP[0]);

}

// The AvgLnLikRegr function is essentially the core equation that is run multiple times // in order to optimize the parameter values

AvgLnLiklRegr(const vP, const adLnPdf, const avScore, const amHess) {

// Declare the variables used and store each of the parameters into separate // variable names

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// Initialize the variables dC = 0; dKLoop = 0; iN = 290; vw = s_vY; vY = s_mX[][1]; vW = s_mX[][0]; vH = s_mX[][2]; vD = s_mX[][3]; vEps = constant(.NaN,iN,1);

// The outer loop that runs through all the iN regions and stores the difference // between the calculated wage and the actual wage for the region in the vector // vEps

for(j = 0; j<iN; j++) {

dKLoop = 0;

// The inner loop that aggregates all the regions weighted by distance // for each of the regions in the outer loop. This is the summation in // the original equation.

for(k = 0; k < iN; k++) {

dKLoop = dKLoop + (vY[k].^(sigma+(1-sigma)/delta) .* vW[k].^(sigma-1)/delta) .* vH[k].^((1-delta).*(sigma-vW[k].^(sigma-1)/delta) .* exp(vD[dC] .*(tao.*(1-sigma)) );

dC = dC + 1; }

// Stores the difference between the actual and the calculated value vEps[j] = log(vw[j]) - (beta + (1/sigma)*log(dKLoop));

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// Store the negative of the squared sum of the error terms in the adLnPdf variable. // The reason for using the negative value is of course that we want to maximize // the result later.

adLnPdf[0] = -vEps'*vEps;

// Increment the number of times we have run the estimation and store it in a variable s_iEval = s_iEval + 1;

// If the SSE is a number and is not missing we return it if(adLnPdf[0] != .NaN) {

return !ismissing(adLnPdf[0]); }

}

// The Estimate function maximizes the returned value from the AvgLnLikRegr function by // changing the parameter values using the MaxBFGS function.

Estimate(const avP, const adLnPdf) {

decl ir;

// set the maximum number of estimations to 10000 MaxControl(10000, -1);

ir = MaxBFGS(AvgLnLiklRegr, avP, adLnPdf, 0, 1); return ir;

}

// Output the number of estimations, the convergence, the optimal parameter values together // with their standard errors, the epsilon vector and a Hess matrix.

Output(const ir, const vP, const dLnPdf) {

decl mHess, mS2, dSigma, dStdError;

println("\rThe number of calls of the log likelihood function = ", s_iEval); println("MaxBFGS returns '", ir, "' meaning '", MaxConvergenceMsg(ir),"'."); Num2Derivative(AvgLnLiklRegr, vP, &mHess);

mS2 = invert(mHess*mHess')/(rows(vEps)); dSigma = (1/rows(vEps))*vEps'*vEps; mS2 = dSigma*mS2;

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println("\rOptimal parameter values are: \r", "%r", {"beta", "sigma", "delta"}, "%c", {"Estimates", "Std Error"}, vP~dStdError');

println("The epsilon vector equals in this case: \r", vEps); println(mHess);

}

// The main function that, at execution of the program, loads the excel table and then // initializes the entire estimation procedure.

main() {

decl mDat, vP, ir, dLnPdf, avScore; mDat = loadmat("PATH to the excel-file"); Initialize(mDat, &s_vY, &s_mX, &vP); ir = Estimate(&vP, &dLnPdf);

Output(ir,vP,dLnPdf); }