# Correlation between Predictions from Equation i and j

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### Description

correlation returns a vector of the correlations between the predictions of two equations in a set of equations. The correlation between the predictions is defined as,

r_{ijk} = \frac{x'_{ik}C_{ij}x_{jk}}{√{(x'_{ik}C_{ii}x_{ik})(x'_{jk}C_{jj}x_{jk})}}

where r_{ijk} is the correlation between the predicted values of equation i and j and C_{ij} is the cross-equation variance-covariance matrix between equations i and j.

### Usage

 1 correlation.systemfit( results, eqni, eqnj ) 

### Arguments

 results an object of type systemfit. eqni index for equation i eqnj index for equation j

### Value

correlation returns a vector of the correlations between the predicted values in equation i and equation j.

### Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com

### References

Greene, W. H. (1993) Econometric Analysis, Second Edition, Macmillan.

Hasenauer, H; Monserud, R and T. Gregoire. (1998) Using Simultansous Regression Techniques with Individual-Tree Growth Models. Forest Science. 44(1):87-95

Kmenta, J. (1997) Elements of Econometrics, Second Edition, University of Michigan Publishing

systemfit
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 data( "Kmenta" ) eqDemand <- consump ~ price + income eqSupply <- consump ~ price + farmPrice + trend inst <- ~ income + farmPrice + trend system <- list( demand = eqDemand, supply = eqSupply ) ## perform 2SLS on each of the equations in the system fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta ) print( fit2sls ) print( fit2sls$rcov ) ## perform the 3SLS fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta ) print( fit3sls ) print( "covariance of residuals used for estimation (from 2sls)" ) print( fit3sls$rcovest ) print( "covariance of residuals" ) print( fit3sls$rcov ) ## examine the correlation between the predicted values ## of suppy and demand by plotting the correlation over ## the value of q r12 <- correlation.systemfit( fit3sls, 1, 2 ) plot( Kmenta$consump, r12, main="correlation between predictions from supply and demand" )