Data Base and Model Calibration

Abstract

In Chaps. 3, 4 and 5 we have developed a simple but increasingly complex general equilibrium model. Starting with the standard textbook version of a private, closed economy we showed how the different pieces of the model interact with each other to give rise to a system of equations that capture and describe market equilibrium. The addition of the government and the external sector, as well as the modification of the labor market to allow for unemployment, provided a more realistic picture of an actual economy and laid the grounds for the study of various policy issues. In each of the examples used, however, the specification of the model parameters was arbitrary, except for a choice of units that yielded convenient solution values for prices and output levels.

Appendices

Appendix 1

Table 6.12 An empirical social accounting matrix

Appendix 2: GAMS Code for the Updating of a Social Accounting Matrix

$TITLE SAM UPDATING: CHAPTER 6 Tables 6.5,6.6,6.7 OPTION NLP=CONOPT; SET I accounts in SAM /1*6/; ALIAS (J,I); TABLE A0(I,J) prior SAM 1 2 3 4 5 6 1 20 50 0 0 15 15 2 30 25 0 0 35 10 3 40 10 0 0 0 0 4 10 15 0 0 0 0 5 0 0 30 20 0 0 6 0 0 20 5 0 0; PARAMETER Z1(I) new marginals /1 104.99 2 105.23 3 62.00 4 32.61 5 65.29 6 29.32 /; VARIABLES D Minimand X(I,J) Estimated SAM cells SICOS(I) Cosine similarity; D.L = 0; X.L(I,J) = A0(I,J); X.LO(I,J) = 0; SICOS.L(I) = 0.5; SICOS.LO(J)=0; SICOS.UP(J) = 1; EQUATIONS GOAL_E Objective function: Entropy GOAL_L Objective function: Least squares GOAL_C Objective function: Cosine similarity ROWSUM(I) Row sum restrictions COLSUM(I) Column sum restrictions ZERO(I,J) Zero structure SIMCOS(I) Column similarity; $ONTEXT Three possible objective functions according to three closeness philosophies: entropy, least squares and similarity. The user can select which of the corresponding three models will be solved. Solution data is saved in a MS Excel sheet $OFFTEXT GOAL_E.. D =E= SUM((I,J)$A0(I,J), X(I,J)*LOG(X(I,J)/A0(I,J))); GOAL_L.. D =E= SUM((I,J)$A0(I,J), SQR(X(I,J)-A0(I,J))) ; GOAL_C.. D =E= -SUM(J, SICOS(J)); ROWSUM(I).. SUM(J, X(I,J)) =E= Z1(I); COLSUM(I).. SUM(J, X(J,I)) =E= Z1(I); ZERO(I,J).. X(I,J)$(A0(I,J) EQ 0) =E= 0; SIMCOS(I).. SICOS(I) =E= SUM(J$A0(J,I),X(J,I)*A0(J,I)) / SQRT(SUM(J$A0(J,I),SQR(X(J,I)))*SUM(J$A0(J,I),SQR(A0(J,I)))); MODEL ENTROPY /GOAL_E, ROWSUM, COLSUM, ZERO/; MODEL LEASTSQUARES /GOAL_L, ROWSUM, COLSUM, ZERO/; MODEL COSINE /GOAL_C, ROWSUM, COLSUM, ZERO, SIMCOS/; PARAMETER SAM(I,J) to upload updating result; SOLVE ENTROPY MINIMIZING D USING NLP; SAM(I,J) = X.L(I,J); $LIBINCLUDE XLDUMP SAM UPDATES.XLS ENTROPY SOLVE LEASTSQUARES MINIMIZING D USING NLP; SAM(I,J) = X.L(I,J); $LIBINCLUDE XLDUMP SAM UPDATES.XLS LEASTSQUARES SOLVE COSINE MINIMIZING D USING NLP; SAM(I,J) = X.L(I,J); $LIBINCLUDE XLDUMP SAM UPDATES.XLS SIMILARITY

Appendix 3: Proximity Indicators

Consider two square matrices ${\mathbf{A}}^1=\left({a}_{ij}^1\right)$ and ${\widehat{\mathbf{A}}}^1=\left({\widehat{a}}_{ij}^1\right)$ such that for all j = 1 , 2 ,  …  , n we have:

$$ \sum_{i=1}^n{a}_{ij}^1=\sum_{i=1}^n{a}_{ji}^1=\sum_{i=1}^n{\widehat{a}}_{ij}^1=\sum_{i=1}^n{\widehat{a}}_{ji}^1={z}_j $$

Define:

$$ {b}_{ij}^1={a}_{ij}^1/{z}_j,{\widehat{b}}_{ij}^1={\widehat{a}}_{ij}^1/{z}_j,{\omega}_j={z}_j/{\sum}_i{z}_i,\overline{a}=\left(1/{n}^2\right)\cdot {\sum}_{i,j}{a}_{ij}^1\ \mathrm{and}\ \widehat{\overline{a}}=\left(1/{n}^2\right)\cdot {\sum}_{i,j}{\widehat{a}}_{ij}^1. $$

By the way, notice that in our example we have in fact that $\overline{a}=\widehat{\overline{a}}$. Then:

Le Masné (1990) proximity index with LM ∈ [0, 1]:

$$ LM=\sum_{j=1}^n{\omega}_j\cdot \left(1-0.5\cdot \sum_{i=1}^n\left|{b}_{ij}^1-{\widehat{b}}_{ij}^1\right|\right) $$

Chenery and Watanabe (1958) proximity index with CW ∈ [0, 1]:

$$ CW=\sum_{j=1}^n{\omega}_j\cdot \frac{\sum_{i=1}^n\left|{a}_{ij}^1-{\widehat{a}}_{ij}^1\right|}{0.5\cdot \sum_{i=1}^n\left({a}_{ij}^1+{\widehat{a}}_{ij}^1\right)} $$

Pearson correlation with R ² ∈ [0, 1]:

$$ R=\frac{\sum_{i=1}^n\sum_{j=1}^n\left({a}_{ij}^1-\overline{a}\right)\cdot \left({\widehat{a}}_{ij}^1-\widehat{\overline{a}}\right)}{\sqrt{\sum_{i=1}^n\sum_{j=1}^n{\left({a}_{ij}^1-\overline{a}\right)}^2\cdot \sum_{i=1}^n\sum_{j=1}^n{\left({\widehat{a}}_{ij}^1-\widehat{\overline{a}}\right)}^2}} $$