systemfit: Linear Equation System Estimation

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/systemfit.R

Description

Fits a set of linear structural equations using Ordinary Least Squares (OLS), Weighted Least Squares (WLS), Seemingly Unrelated Regression (SUR), Two-Stage Least Squares (2SLS), Weighted Two-Stage Least Squares (W2SLS) or Three-Stage Least Squares (3SLS).

Usage

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systemfit( formula, method = "OLS",
           inst=NULL, data=list(),
           restrict.matrix = NULL, restrict.rhs = NULL, restrict.regMat = NULL,
           pooled = FALSE, control = systemfit.control( ... ), ... )

Arguments

formula

an object of class formula (for single-equation models) or (typically) a list of objects of class formula (for multiple-equation models); if argument data is of class pdata.frame (created with pdata.frame()), this argument must be a single object of class formula that represents the formula to be estimated for all individuals.

method

the estimation method, one of "OLS", "WLS", "SUR", "2SLS", "W2SLS", or "3SLS" (see details); iterated estimation methods can be specified by setting control parameter maxiter larger than 1 (e.g. 500).

inst

one-sided model formula specifying the instrumental variables (including exogenous explanatory variables) or a list of one-sided model formulas if different instruments should be used for the different equations (only needed for 2SLS, W2SLS, and 3SLS estimations).

data

an optional data frame containing the variables in the model. By default the variables are taken from the environment from which systemfit is called.

restrict.matrix

an optional j x k matrix to impose linear restrictions on the coefficients by restrict.matrix * b = restrict.rhs (j = number of restrictions, k = number of all coefficients, b = vector of all coefficients) or a character vector giving the restrictions in symbolic form (see documentation of linearHypothesis in package "car" for details). The number and the names of the coefficients can be obtained by estimating the system without restrictions and applying the coef method to the returned object.

restrict.rhs

an optional vector with j elements to impose linear restrictions (see restrict.matrix); default is a vector that contains j zeros.

restrict.regMat

an optional matrix to impose restrictions on the coefficients by post-multiplying the regressor matrix with this matrix (see details).

control

list of control parameters. The default is constructed by the function systemfit.control. See the documentation of systemfit.control for details.

pooled

logical, restrict coefficients to be equal in all equations (only for panel-like data).

...

arguments passed to systemfit.control.

Details

The estimation of systems of equations with unequal numbers of observations has not been thoroughly tested yet. Currently, systemfit calculates the residual covariance matrix only from the residuals/observations that are available in all equations.

If argument data is of class pdata.frame (created with pdata.frame() and thus, contains panel data in long format), argument formula must be a single equation that is applied to all individuals. In this case, argument pooled specifies whether the coefficients are restricted to be equal for all individuals.

If argument restrict.regMat is specified, the regressor matrix X is post-multiplied by this matrix: X^{*} = X \cdot restrict.regMat. Then, this modified regressor matrix X^{*} is used for the estimation of the coefficient vector b^{*}. This means that the coefficients of the original regressors (X), vector b, can be represented by b = restrict.regMat \cdot b^{*}. If restrict.regMat is a non-singular quadratic matrix, there are no restrictions on the coefficients imposed, but the coefficients b^{*} are linear combinations of the original coefficients b. If restrict.regMat has less columns than rows, linear restrictions are imposed on the coefficients b. However, imposing linear restrictions by the restrict.regMat matrix is less flexible than by providing the matrix restrict.matrix and the vector restrict.rhs. The advantage of imposing restrictions on the coefficients by the matrix restrict.regMat is that the matrix, which has to be inverted during the estimation, gets smaller by this procedure, while it gets larger if the restrictions are imposed by restrict.matrix and restrict.rhs.

In the context of multi-equation models, the term “weighted” in “weighted least squares” (WLS) and “weighted two-stage least squares” (W2SLS) means that the equations might have different weights and not that the observations have different weights.

It is important to realize the limitations on estimating the residuals covariance matrix imposed by the number of observations T in each equation. With g equations we estimate g*(g+1)/2 elements using T*g observations total. Beck and Katz (1995,1993) discuss the issue and the resulting overconfidence when the ratio of T/g is small (e.g. 3). Even for T/g=5 the estimate is unstable both numerically and statistically and the 95 approximately [0.5*s^2, 3*s^2], which is inadequate precision if the covariance matrix will be used for simulation of asset return paths either for investment or risk management decisions. For a starter on models with large cross-sections see Reichlin (2002). [This paragraph has been provided by Stephen C. Bond – Thanks!]

Value

systemfit returns a list of the class systemfit and contains all results that belong to the whole system. This list contains one special object: "eq". It is a list and contains one object for each estimated equation. These objects are of the class systemfit.equation and contain the results that belong only to the regarding equation.

The objects of the class systemfit and systemfit.equation have the following components (the elements of the latter are marked with an asterisk (*)):

call

the matched call.

method

estimation method.

rank

total number of linear independent coefficients = number of coefficients minus number of linear restrictions.

df.residual

degrees of freedom of the whole system.

iter

number of iteration steps.

coefficients

vector of all estimated coefficients.

coefCov

estimated covariance matrix of coefficients.

residCov

estimated residual covariance matrix.

residCovEst

residual covariance matrix used for estimation (only WLS, W2SLS, SUR and 3SLS).

restrict.matrix

the restriction matrix.

restrict.rhs

the restriction vector.

restrict.regMat

matrix used to impose restrictions on the coefficients by post-multiplying the regressor matrix with this matrix.

control

list of control parameters used for the estimation.

panelLike

logical. Was this an analysis with panel-like data?

## elements of the class systemfit.eq

eq

a list that contains the results that belong to the individual equations.

eqnLabel*

the label of this equation.

eqnNo*

the number of this equation.

terms*

the 'terms' object used for the ith equation.

inst*

instruments of the ith equation (only 2SLS, W2SLS, and 3SLS).

termsInst*

the 'terms' object of the instruments used for the ith equation (only 2SLS, W2SLS, and 3SLS).

rank*

number of linear independent coefficients in the ith equation (differs from the number of coefficients only if there are restrictions that are not cross-equation).

nCoef.sys*

total number of coefficients in all equations.

rank.sys*

total number of linear independent coefficients in all equations.

df.residual*

degrees of freedom of the ith equation.

df.residual.sys*

degrees of freedom of the whole system.

coefficients*

estimated coefficients of the ith equation.

covb*

estimated covariance matrix of coefficients.

model*

if requested (the default), the model frame of the ith equation.

modelInst*

if requested (the default), the model frame of the instrumental variables of the ith equation (only 2SLS, W2SLS, and 3SLS).

x*

if requested, the model matrix of the ith equation.

y*

if requested, the response of the ith equation.

z*

if requested, the matrix of instrumental variables of the ith equation (only 2SLS, W2SLS, and 3SLS).

fitted.values*

vector of fitted values of the ith equation.

residuals*

vector of residuals of the ith equation.

Author(s)

Arne Henningsen [email protected],
Jeff D. Hamann [email protected]

References

Beck, N.; J.N. Katz (1995) What to do (and not to do) with Time-Series Cross-Section Data, The American Political Science Review, 89, pp. 634-647.

Beck, N.; J.N. Katz; M.R. Alvarez; G. Garrett; P. Lange (1993) Government Partisanship, Labor Organization, and Macroeconomic Performance: a Corrigendum, American Political Science Review, 87, pp. 945-48.

Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Prentice Hall.

Judge, George G.; W. E. Griffiths; R. Carter Hill; Helmut Luetkepohl and Tsoung-Chao Lee (1985) The Theory and Practice of Econometrics, Second Edition, Wiley.

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

Reichlin, L. (2002) Factor models in large cross-sections of time series, Working Paper, ECARES and CEPR.

Schmidt, P. (1990) Three-Stage Least Squares with different Instruments for different equations, Journal of Econometrics 43, p. 389-394.

Theil, H. (1971) Principles of Econometrics, Wiley, New York.

See Also

lm and nlsystemfit

Examples

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data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## OLS estimation
fitols <- systemfit( system, data=Kmenta )
print( fitols )

## OLS estimation with 2 restrictions
Rrestr <- matrix(0,2,7)
Rrestr[1,3] <-  1
Rrestr[1,7] <- -1
Rrestr[2,2] <- -1
Rrestr[2,5] <-  1
qrestr <- c( 0, 0.5 )
fitols2 <- systemfit( system, data = Kmenta,
                      restrict.matrix = Rrestr, restrict.rhs = qrestr )
print( fitols2 )

## OLS estimation with the same 2 restrictions in symbolic form
restrict <- c( "demand_income - supply_trend = 0",
   "- demand_price + supply_price = 0.5" )
fitols2b <- systemfit( system, data = Kmenta, restrict.matrix = restrict )
print( fitols2b )

# test whether both restricted estimators are identical
all.equal( coef( fitols2 ), coef( fitols2b ) )

## OLS with restrictions on the coefficients by modifying the regressor matrix
## with argument restrict.regMat
modReg <- matrix( 0, 7, 6 )
colnames( modReg ) <- c( "demIntercept", "demPrice", "demIncome",
   "supIntercept", "supPrice2", "supTrend" )
modReg[ 1, "demIntercept" ] <- 1
modReg[ 2, "demPrice" ]     <- 1
modReg[ 3, "demIncome" ]    <- 1
modReg[ 4, "supIntercept" ] <- 1
modReg[ 5, "supPrice2" ]    <- 1
modReg[ 6, "supPrice2" ]    <- 1
modReg[ 7, "supTrend" ]     <- 1
fitols3 <- systemfit( system, data = Kmenta, restrict.regMat = modReg )
print( fitols3 )


## iterated SUR estimation
fitsur <- systemfit( system, "SUR", data = Kmenta, maxit = 100 )
print( fitsur )

## 2SLS estimation
inst <- ~ income + farmPrice + trend
fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta )
print( fit2sls )

## 2SLS estimation with different instruments in each equation
inst1 <- ~ income + farmPrice
inst2 <- ~ income + farmPrice + trend
instlist <- list( inst1, inst2 )
fit2sls2 <- systemfit( system, "2SLS", inst = instlist, data = Kmenta )
print( fit2sls2 )

## 3SLS estimation with GMM-3SLS formula
inst <- ~ income + farmPrice + trend
fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta,
   method3sls = "GMM" )
print( fit3sls )


## Examples how to use systemfit() with panel-like data
## Repeating the SUR estimations in Greene (2003, p. 351)
data( "GrunfeldGreene" )
library( "plm" )
GGPanel <- pdata.frame( GrunfeldGreene, c( "firm", "year" ) )
formulaGrunfeld <- invest ~ value + capital
# SUR
greeneSur <- systemfit( formulaGrunfeld, "SUR",
   data = GGPanel, methodResidCov = "noDfCor" )
summary( greeneSur )
# SUR Pooled
greeneSurPooled <- systemfit( formulaGrunfeld, "SUR",
   data = GGPanel, pooled = TRUE, methodResidCov = "noDfCor",
   residCovWeighted = TRUE )
summary( greeneSurPooled )

## Further examples are in the documentation to the data sets
## 'KleinI' and 'GrunfeldGreene'.

Example output

Loading required package: Matrix
Loading required package: car
Loading required package: lmtest
Loading required package: zoo

Attaching package: 'zoo'

The following objects are masked from 'package:base':

    as.Date, as.Date.numeric


Please cite the 'systemfit' package as:
Arne Henningsen and Jeff D. Hamann (2007). systemfit: A Package for Estimating Systems of Simultaneous Equations in R. Journal of Statistical Software 23(4), 1-40. http://www.jstatsoft.org/v23/i04/.

If you have questions, suggestions, or comments regarding the 'systemfit' package, please use a forum or 'tracker' at systemfit's R-Forge site:
https://r-forge.r-project.org/projects/systemfit/

systemfit results 
method: OLS 

Coefficients:
demand_(Intercept)       demand_price      demand_income supply_(Intercept) 
         99.895423          -0.316299           0.334636          58.275431 
      supply_price   supply_farmPrice       supply_trend 
          0.160367           0.248133           0.248302 

systemfit results 
method: OLS 

Coefficients:
demand_(Intercept)       demand_price      demand_income supply_(Intercept) 
        101.481708          -0.316799           0.318885          54.149420 
      supply_price   supply_farmPrice       supply_trend 
          0.183201           0.259528           0.318885 

systemfit results 
method: OLS 

Coefficients:
demand_(Intercept)       demand_price      demand_income supply_(Intercept) 
        101.481708          -0.316799           0.318885          54.149420 
      supply_price   supply_farmPrice       supply_trend 
          0.183201           0.259528           0.318885 
[1] TRUE

systemfit results 
method: OLS 

Coefficients:
demand_(Intercept)       demand_price      demand_income supply_(Intercept) 
         99.895423          -0.316299           0.334636          51.929646 
      supply_price   supply_farmPrice       supply_trend 
          0.236157           0.236157           0.240931 

systemfit results 
method: iterated SUR 

convergence achieved after 35 iterations

Coefficients:
demand_(Intercept)       demand_price      demand_income supply_(Intercept) 
         97.516307          -0.143687           0.182020          77.900537 
      supply_price   supply_farmPrice       supply_trend 
          0.105094           0.108410           0.191543 

systemfit results 
method: 2SLS 

Coefficients:
demand_(Intercept)       demand_price      demand_income supply_(Intercept) 
         94.633304          -0.243557           0.313992          49.532442 
      supply_price   supply_farmPrice       supply_trend 
          0.240076           0.255606           0.252924 

systemfit results 
method: 2SLS 

Coefficients:
demand_(Intercept)       demand_price      demand_income supply_(Intercept) 
        106.789358          -0.411599           0.361681          49.532442 
      supply_price   supply_farmPrice       supply_trend 
          0.240076           0.255606           0.252924 

systemfit results 
method: 3SLS 

Coefficients:
demand_(Intercept)       demand_price      demand_income supply_(Intercept) 
         94.633304          -0.243557           0.313992          52.197204 
      supply_price   supply_farmPrice       supply_trend 
          0.228589           0.228158           0.361138 
Loading required package: Formula

systemfit results 
method: SUR 

         N DF    SSR     detRCov   OLS-R2 McElroy-R2
system 100 85 347048 6.17788e+13 0.844042   0.868682

                  N DF       SSR       MSE     RMSE       R2   Adj R2
Chrysler         20 17   3056.98   179.823  13.4098 0.911862 0.901493
General.Electric 20 17  14009.12   824.066  28.7065 0.687636 0.650887
General.Motors   20 17 144320.88  8489.463  92.1383 0.920742 0.911417
US.Steel         20 17 183763.01 10809.589 103.9692 0.421959 0.353954
Westinghouse     20 17   1898.25   111.662  10.5670 0.726429 0.694244

The covariance matrix of the residuals used for estimation
                  Chrysler General.Electric General.Motors  US.Steel
Chrysler          149.8722         -21.3757       -282.756   418.079
General.Electric  -21.3757         660.8294        607.533   904.952
General.Motors   -282.7564         607.5331       7160.294 -2222.060
US.Steel          418.0786         904.9517      -2222.060  8896.416
Westinghouse       13.3070         176.4491        126.176   546.186
                 Westinghouse
Chrysler              13.3070
General.Electric     176.4491
General.Motors       126.1762
US.Steel             546.1856
Westinghouse          88.6617

The covariance matrix of the residuals
                   Chrysler General.Electric General.Motors  US.Steel
Chrysler          152.84923          2.04737       -313.704   455.089
General.Electric    2.04737        700.45575        605.336  1224.405
General.Motors   -313.70357        605.33650       7216.044 -2686.517
US.Steel          455.08946       1224.40545      -2686.517  9188.151
Westinghouse       16.66062        200.31627        129.887   652.716
                 Westinghouse
Chrysler              16.6606
General.Electric     200.3163
General.Motors       129.8866
US.Steel             652.7164
Westinghouse          94.9125

The correlations of the residuals
                    Chrysler General.Electric General.Motors  US.Steel
Chrysler          1.00000000       0.00625711      -0.298702  0.384018
General.Electric  0.00625711       1.00000000       0.269251  0.482637
General.Motors   -0.29870209       0.26925075       1.000000 -0.329933
US.Steel          0.38401758       0.48263726      -0.329933  1.000000
Westinghouse      0.13832413       0.77689848       0.156947  0.698954
                 Westinghouse
Chrysler             0.138324
General.Electric     0.776898
General.Motors       0.156947
US.Steel             0.698954
Westinghouse         1.000000


SUR estimates for 'Chrysler' (equation 1)
Model Formula: Chrysler_invest ~ Chrysler_value + Chrysler_capital
<environment: 0x8a93190>

              Estimate Std. Error  t value   Pr(>|t|)    
(Intercept)  0.5043036 11.5128290  0.04380 0.96557134    
value        0.0695456  0.0168975  4.11573 0.00072174 ***
capital      0.3085445  0.0258636 11.92971 1.1008e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 13.409796 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 3056.984521 MSE: 179.822619 Root MSE: 13.409796 
Multiple R-Squared: 0.911862 Adjusted R-Squared: 0.901493 


SUR estimates for 'General.Electric' (equation 2)
Model Formula: General.Electric_invest ~ General.Electric_value + General.Electric_capital
<environment: 0x8a93190>

               Estimate  Std. Error  t value   Pr(>|t|)    
(Intercept) -22.4389132  25.5185863 -0.87932  0.3914900    
value         0.0372914   0.0122631  3.04094  0.0073808 ** 
capital       0.1307830   0.0220497  5.93127 1.6424e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 28.706543 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 14009.115084 MSE: 824.065593 Root MSE: 28.706543 
Multiple R-Squared: 0.687636 Adjusted R-Squared: 0.650887 


SUR estimates for 'General.Motors' (equation 3)
Model Formula: General.Motors_invest ~ General.Motors_value + General.Motors_capital
<environment: 0x8a93190>

                Estimate   Std. Error  t value   Pr(>|t|)    
(Intercept) -162.3641052   89.4592324 -1.81495   0.087216 .  
value          0.1204930    0.0216291  5.57087 3.3806e-05 ***
capital        0.3827462    0.0327680 11.68047 1.5189e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 92.138284 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 144320.876426 MSE: 8489.463319 Root MSE: 92.138284 
Multiple R-Squared: 0.920742 Adjusted R-Squared: 0.911417 


SUR estimates for 'US.Steel' (equation 4)
Model Formula: US.Steel_invest ~ US.Steel_value + US.Steel_capital
<environment: 0x8a93190>

               Estimate  Std. Error t value  Pr(>|t|)   
(Intercept)  85.4232548 111.8774214 0.76354 0.4556062   
value         0.1014782   0.0547837 1.85234 0.0814214 . 
capital       0.3999914   0.1277946 3.12996 0.0060999 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 103.969173 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 183763.011429 MSE: 10809.588908 Root MSE: 103.969173 
Multiple R-Squared: 0.421959 Adjusted R-Squared: 0.353954 


SUR estimates for 'Westinghouse' (equation 5)
Model Formula: Westinghouse_invest ~ Westinghouse_value + Westinghouse_capital
<environment: 0x8a93190>

             Estimate Std. Error t value   Pr(>|t|)    
(Intercept) 1.0888770  6.2588045 0.17398 0.86393984    
value       0.0570091  0.0113623 5.01742 0.00010566 ***
capital     0.0415065  0.0412016 1.00740 0.32786736    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 10.56701 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 1898.249072 MSE: 111.66171 Root MSE: 10.56701 
Multiple R-Squared: 0.726429 Adjusted R-Squared: 0.694244 


systemfit results 
method: SUR 

         N DF     SSR     detRCov   OLS-R2 McElroy-R2
system 100 97 1604301 9.94613e+16 0.279053   0.843852

                  N DF      SSR       MSE     RMSE         R2     Adj R2
Chrysler         20 17   6112.2   359.541  18.9616   0.823775   0.803042
General.Electric 20 17 691132.1 40654.827 201.6304 -14.410331 -16.223311
General.Motors   20 17 201010.5 11824.147 108.7389   0.889609   0.876622
US.Steel         20 17 689379.5 40551.736 201.3746  -1.168498  -1.423616
Westinghouse     20 17  16667.1   980.421  31.3117  -1.402026  -1.684618

The covariance matrix of the residuals used for estimation
                  Chrysler General.Electric General.Motors  US.Steel
Chrysler           409.190         -2594.27       -196.755   2594.27
General.Electric -2594.272         36563.24      -3479.909 -28622.88
General.Motors    -196.755         -3479.91       8612.147    996.07
US.Steel          2594.272        -28622.88        996.070  32902.83
Westinghouse      -102.225          3797.41       -970.757  -2271.94
                 Westinghouse
Chrysler             -102.225
General.Electric     3797.408
General.Motors       -970.757
US.Steel            -2271.944
Westinghouse          777.975

The covariance matrix of the residuals
                    Chrysler General.Electric General.Motors  US.Steel
Chrysler           305.61001         -1966.65       -4.80523   2158.60
General.Electric -1966.64761         34556.60    -7160.66658 -28722.01
General.Motors      -4.80523         -7160.67    10050.52486   4439.99
US.Steel          2158.59516        -28722.01     4439.98867  34468.98
Westinghouse      -123.92048          4274.00    -1400.74696  -2893.73
                 Westinghouse
Chrysler             -123.920
General.Electric     4274.000
General.Motors      -1400.747
US.Steel            -2893.733
Westinghouse          833.357

The correlations of the residuals
                  Chrysler General.Electric General.Motors   US.Steel
Chrysler          1.000000         0.220150     -0.3447472  0.2007628
General.Electric  0.220150         1.000000     -0.2232503 -0.1586937
General.Motors   -0.344747        -0.223250      1.0000000 -0.0923957
US.Steel          0.200763        -0.158694     -0.0923957  1.0000000
Westinghouse      0.290701         0.897313     -0.3760456 -0.0757482
                 Westinghouse
Chrysler            0.2907014
General.Electric    0.8973130
General.Motors     -0.3760456
US.Steel           -0.0757482
Westinghouse        1.0000000


SUR estimates for 'Chrysler' (equation 1)
Model Formula: Chrysler_invest ~ Chrysler_value + Chrysler_capital
<environment: 0xa8ac720>

                Estimate   Std. Error t value   Pr(>|t|)    
(Intercept) -28.24669393   4.88823801 -5.7785 9.1246e-08 ***
value         0.08910091   0.00507226 17.5663 < 2.22e-16 ***
capital       0.33401503   0.01671254 19.9859 < 2.22e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 18.961571 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 6112.20028 MSE: 359.541193 Root MSE: 18.961571 
Multiple R-Squared: 0.823775 Adjusted R-Squared: 0.803042 


SUR estimates for 'General.Electric' (equation 2)
Model Formula: General.Electric_invest ~ General.Electric_value + General.Electric_capital
<environment: 0xa8ac720>

                Estimate   Std. Error t value   Pr(>|t|)    
(Intercept) -28.24669393   4.88823801 -5.7785 9.1246e-08 ***
value         0.08910091   0.00507226 17.5663 < 2.22e-16 ***
capital       0.33401503   0.01671254 19.9859 < 2.22e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 201.630421 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 691132.055958 MSE: 40654.826821 Root MSE: 201.630421 
Multiple R-Squared: -14.410331 Adjusted R-Squared: -16.223311 


SUR estimates for 'General.Motors' (equation 3)
Model Formula: General.Motors_invest ~ General.Motors_value + General.Motors_capital
<environment: 0xa8ac720>

                Estimate   Std. Error t value   Pr(>|t|)    
(Intercept) -28.24669393   4.88823801 -5.7785 9.1246e-08 ***
value         0.08910091   0.00507226 17.5663 < 2.22e-16 ***
capital       0.33401503   0.01671254 19.9859 < 2.22e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 108.738893 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 201010.497194 MSE: 11824.146894 Root MSE: 108.738893 
Multiple R-Squared: 0.889609 Adjusted R-Squared: 0.876622 


SUR estimates for 'US.Steel' (equation 4)
Model Formula: US.Steel_invest ~ US.Steel_value + US.Steel_capital
<environment: 0xa8ac720>

                Estimate   Std. Error t value   Pr(>|t|)    
(Intercept) -28.24669393   4.88823801 -5.7785 9.1246e-08 ***
value         0.08910091   0.00507226 17.5663 < 2.22e-16 ***
capital       0.33401503   0.01671254 19.9859 < 2.22e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 201.374617 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 689379.520435 MSE: 40551.736496 Root MSE: 201.374617 
Multiple R-Squared: -1.168498 Adjusted R-Squared: -1.423616 


SUR estimates for 'Westinghouse' (equation 5)
Model Formula: Westinghouse_invest ~ Westinghouse_value + Westinghouse_capital
<environment: 0xa8ac720>

                Estimate   Std. Error t value   Pr(>|t|)    
(Intercept) -28.24669393   4.88823801 -5.7785 9.1246e-08 ***
value         0.08910091   0.00507226 17.5663 < 2.22e-16 ***
capital       0.33401503   0.01671254 19.9859 < 2.22e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 31.311667 on 17 degrees of freedom
Number of observations: 20 Degrees of Freedom: 17 
SSR: 16667.148552 MSE: 980.420503 Root MSE: 31.311667 
Multiple R-Squared: -1.402026 Adjusted R-Squared: -1.684618 

systemfit documentation built on April 4, 2018, 5:03 p.m.