SIMULEST.RPF

SIMULEST.RPF is an example of estimation of a simultaneous equations model using all of the main techniques (2SLS, 3SLS, LIML, FIML).

The base model is taken from Pindyck and Rubinfeld (1998), p. 390. Their model is

$\begin{array}{l} {C_t} = {\alpha _C} + {\beta _C}{\kern 1pt} {Y_t} + {\gamma _C}{C_{t - 1}} + {u_{{\kern 1pt} Ct}}{\kern 1pt} \\ {I_t} = {\alpha _I} + {\beta _I}\left( {{Y_{t - 1}} - {Y_{t - 2}}} \right) + {\gamma _I}{Y_t} + {\delta _I}{R_{t - 4}} + {u_{It}}{\kern 1pt} \\ {R_t} = {\alpha _R} + {\beta _R}{Y_t} + {\gamma _R}\left( {{Y_t} - {Y_{t - 1}}} \right) + {\delta _R}\left( {{M_t} - {M_{t - 1}}} \right) + {\varepsilon _R}\left( {{R_{t - 1}} + {R_{t - 2}}} \right) + {u_{Rt}} \\ {Y_t} \equiv {C_t} + {I_t} + {G_t} \\ \end{array}$

The data set consists of the following variables, quarterly from 1947:1 to 1988:1. However, the estimates are done using a common estimation range of 1950:1 through 1985:4. If RATS were left to determine the range, estimation would start in 1948:1 as the start of the range is determined by lag 4 on RATE in the instrument set.

CONS (C)	= real personal consumption
INVEST (I)	= real gross domestic investment
GNP (Y)	= real GNP net of exports and imports
GOVT (G)	= real government purchases of goods and services
MONEY (M)	= M1
RATE (R)	= 90 day Treasury bill rate

The variables $Y_t-Y_{t-1}$ (YDIFF), $R_t+R_{t-1}$ (RSUM) and $M_t-M_{t-1}$ (MDIFF) and are created from these:

set ydiff = gnp-gnp{1}

set rsum = rate+rate{1}

set mdiff = m-m{1}

Though we will start with the original set of equations, we will end up using a slightly different form for the investment equation, adding an $I_{t-1}$ term to the second equation above. This is done because an estimate of the autocorrelation coefficient on the residuals in the original equation comes in very close to one.

The instrument set for all equations will be the full list of exogenous (GOVT and MDIFF) and pre-determined variables (lags and functions of lags of the four endogenous variables) that appear in the model. Note that you need to include the CONSTANT.

instruments constant cons{1} ydiff{1} gnp{1} govt $

mdiff rsum{1} rate{4}

Two-Stage Least Squares

You do two-stage least squares by setting the instrument set (already done) and using LINREG with the INST option.

linreg(inst) cons 1950:1 1985:4

# constant gnp cons{1}

linreg(inst) invest 1950:1 1985:4

# constant ydiff{1} gnp rate{4}

linreg(inst) rate 1950:1 1985:4

# constant gnp ydiff mdiff rsum{1}

Autocorrelated Errors

To correct for first-order autocorrelated errors, use AR1 with the INST option. If you choose METHOD=CORC, RATS uses Fair’s (1970) procedure. This requires, for consistency, that you include as instruments the lags of the variables involved in the regression. The alternative is METHOD=HILU, which does not have this requirement.

With the speed of current computers, there is no practical advantage to using the “Cochrane–Orcutt” style iterative procedures. Particularly in simultaneous equations, it is quite possible to have the function being optimized show multiple peaks when graphed against the autoregressive parameter. The search method of HILU will make it much more likely that you will find the best estimates. In fact, in the example model, we found a difference between the two techniques, and altered the model as a result.

A common criticism of simultaneous equations models where many of the equations have to be estimated with autocorrelation corrections is that much of the explanatory power of the “model” actually comes not from the structural relationship, but from the autoregressive errors. If your original equations are showing very low Durbin-Watsons, it would probably be a good idea to rethink them as dynamic equations.

The following expands the instrument set as required by Fair’s procedure and re-estimates the investment equation, using both HILU and CORC.

instruments constant cons{1 2} ydiff{1 2} gnp{1} govt{0 1} $

mdiff{0 1} rate{1 to 5}

ar1(method=hilu,inst) invest 1950:1 1985:4

# constant ydiff{1} gnp rate{4}

ar1(method=corc,inst) invest 1950:1 1985:4

# constant ydiff{1} gnp rate{4}

The two AR1 instructions produce dramatically different results even though they are (theoretically) minimizing the same function. Because of this, all further estimates use an investment equation which includes lagged investment as an explanatory variable. This incorporates the dynamics into the equation rather than tacking it on to the error term. The change in the model requires a change in the instrument set, since INVEST{1} is now in the model:

instruments constant cons{1} ydiff{1} gnp{1} invest{1} $

govt mdiff rsum{1} rate{4}

Three Stage Least Squares

You can estimate the system by three stage least squares (3SLS) using the INST option with either SUR or NLSYSTEM.

To use SUR, you must do the following:

1.Define the equations in the system using a set of EQUATION instructions.

2.Set up the instruments list using INSTRUMENTS.

3.Estimate the model using SUR(INST).

To use NLSYSTEM, you need to

1.Set the list of parameters to be estimated using the instruction NONLIN. If you might be doing a good deal of experimenting with your equations, it would be a good idea to set up a separate PARMSET for each equation, and then combine them when you estimate.

2.Create formulas for the equations with FRML instructions.

3.Set up the instruments list using INSTRUMENTS.

4.Estimate the model with NLSYSTEM(INST).

NLSYSTEM and SUR should produce almost identical results for any linear model which has no restrictions on the parameters. This uses SUR:

equation consleq cons

# constant gnp cons{1}

equation investleq invest

# constant invest{1} ydiff{1} gnp rate{4}

equation rateleq rate

# constant gnp ydiff mdiff rsum{1}

group prmodel consleq investleq rateleq

sur(inst,model=prmodel,iterations=10) * 1950:1 1985:4

and this uses NLSYSTEM:

nonlin(parmset=structural) c0 c1 c2 i0 i1 i2 i3 i4 r0 r1 r2 r3 r4

frml consnl cons = c0+c1*gnp+c2*cons{1}

frml investnl invest = i0+i1*invest{1}+i2*ydiff{1}+$

i3*gnp+i4*rate{4}

frml ratenl rate = r0+r1*gnp+r2*ydiff+r3*mdiff+r4*rsum{1}

nlsystem(inst,parmset=structural,cvout=v) 1950:1 1985:4 $

consnl investnl ratenl

Limited Information Maximum Likelihood (LIML)

LIML actually was used before the now much more popular 2SLS as a single equation (limited information) estimator for simultaneous equation models. For a description of it, see, for instance, Greene (2012). While it has some theoretical advantages over 2SLS, it doesn’t appear to provide superior estimates. It can be done with RATS using the procedure @LIML. Just as with LINREG(INST) for 2SLS, set the instrument list in advance using INSTRUMENTS. The syntax is then familiar:

@liml depvar start end

# list of explanatory variables

@liml cons 1950:1 1985:4

# constant gnp cons{1}

@liml invest 1950:1 1985:4

# constant invest{1} ydiff{1} gnp rate{4}

@liml rate 1950:1 1985:4

# constant gnp ydiff mdiff rsum{1}

Full Information Maximum Likelihood (FIML)

RATS has no specific instruction to do FIML for simultaneous equations. However, for small systems, you can implement FIML using the instruction NLSYSTEM. This assumes that you have Normal residuals with an unrestricted covariance matrix. You would set everything up as you would for 3SLS, except that, instead of using the INSTRUMENTS option, you include the option JACOBIAN=FRML for the Jacobian. For the linear model $\bf{Y}_t \Gamma = {\bf{X}}_t {\rm B} + {\bf{u}}_t $, this will provide a formula which computes $|G|$. For the example model, the Jacobian reduces to 1–C1–I3, but we write out the full expression.

frml jacobian = %det(||1.0 ,0.0 ,0.0 ,-c1|$

0.0 ,1.0 ,0.0 ,-i3|$

0.0 ,0.0 ,1.0 ,0.0|$

-1.0,-1.0 ,0.0 ,1.0||)

nlsystem(parmset=structural,cvout=v,jacobian=jacobian,iters=200,$
title="FIML Estimates") 1950:1 1985:4 consnl investnl ratenl

Full Program

open data prsmall.xls

cal(q) 1947

data(format=xls,org=obs) 1947:1 1988:1

set ydiff = gnp-gnp{1}

set rsum = rate+rate{1}

set mdiff = m-m{1}

instruments constant cons{1} ydiff{1} gnp{1} govt $

mdiff rsum{1} rate{4}

* First, we estimate the equations by 2SLS.

linreg(inst) cons 1950:1 1985:4

# constant gnp cons{1}

linreg(inst) invest 1950:1 1985:4

# constant ydiff{1} gnp rate{4}

linreg(inst) rate 1950:1 1985:4

# constant gnp ydiff mdiff rsum{1}

* Pindyck and Rubinfeld note the low Durbin-Watson on the investment

* equation. This is re-estimated using AR1. The Hildreth-Lu procedure

* gives a very different result from Cochrane-Orcutt (actually Fair's

* procedure), with a rho effectively of 1.0. We therefore replace P&R's

* investment equation with one redone to provide better dynamic behavior.

instruments constant cons{1 2} ydiff{1 2} gnp{1} govt{0 1} $

mdiff{0 1} rate{1 to 5}

ar1(method=hilu,inst) invest 1950:1 1985:4

# constant ydiff{1} gnp rate{4}

ar1(method=corc,inst) invest 1950:1 1985:4

# constant ydiff{1} gnp rate{4}

* The investment equation is altered by adding lagged investment to the

* explanatory variables. The complete model is then re-estimated (using

* 2SLS again) with an updated set of instruments

instruments constant cons{1} ydiff{1} gnp{1} invest{1} $

govt mdiff rsum{1} rate{4}

linreg(inst,frml=conseq) cons 1950:1 1985:4

# constant gnp cons{1}

linreg(inst,frml=investeq) invest 1950:1 1985:4

# constant invest{1} ydiff{1} gnp rate{4}

linreg(inst,frml=rateeq) rate 1950:1 1985:4

# constant gnp ydiff mdiff rsum{1}

* We now estimate the model using 3SLS with the instruction SUR.

equation consleq cons

# constant gnp cons{1}

equation investleq invest

# constant invest{1} ydiff{1} gnp rate{4}

equation rateleq rate

# constant gnp ydiff mdiff rsum{1}

group prmodel consleq investleq rateleq

sur(inst,model=prmodel,iterations=100) * 1950:1 1985:4

* We next estimate the model using 3SLS with the instruction NLSYSTEM.

* (Not necessary here but would be if there were some non-linearities).

nonlin(parmset=structural,zeros) c0 c1 c2 i0 i1 i2 i3 i4 r0 r1 r2 r3 r4

frml consnl cons = c0+c1*gnp+c2*cons{1}

frml investnl invest = i0+i1*invest{1}+i2*ydiff{1}+$

i3*gnp+i4*rate{4}

frml ratenl rate = r0+r1*gnp+r2*ydiff+r3*mdiff+r4*rsum{1}

nlsystem(inst,parmset=structural,cvout=v) 1950:1 1985:4 $

consnl investnl ratenl

* LIML estimation uses the LIML procedure

@liml cons 1950:1 1985:4

# constant gnp cons{1}

@liml invest 1950:1 1985:4

# constant invest{1} ydiff{1} gnp rate{4}

@liml rate 1950:1 1985:4

# constant gnp ydiff mdiff rsum{1}

* Finally, we estimate the model by FIML. This adds to the standard

* likelihood for a multivariate Normal model, the log det of the

* Jacobian of the transformation from residuals to endogenous variables.

* This is done using the Jacobian option, which gives the determinant.

* In this case, this actually would reduce to 1-c1-i3, but we write out

* the full expression.

frml jacobian = %det(||1.0 ,0.0 ,0.0 ,-c1|$

0.0 ,1.0 ,0.0 ,-i3|$

0.0 ,0.0 ,1.0 ,0.0|$

-1.0,-1.0 ,0.0 ,1.0||)

nlsystem(parmset=structural,cvout=v,jacobian=jacobian,iters=200,$

title="FIML Estimates") 1950:1 1985:4 consnl investnl ratenl

Output

Linear Regression - Estimation by Instrumental Variables

Dependent Variable CONS

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 141

Mean of Dependent Variable 1411.1625000

Std Error of Dependent Variable 486.7321052

Standard Error of Estimate 11.3243791

Sum of Squared Residuals 18082.060169

J-Specification(5) 61.0820

Significance Level of J 0.0000000

Durbin-Watson Statistic 1.6280

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -3.743302427 4.966793544 -0.75367 0.45230661

2. GNP 0.028215936 0.018274085 1.54404 0.12481985

3. CONS{1} 0.964528206 0.027301301 35.32902 0.00000000

Linear Regression - Estimation by Instrumental Variables

Dependent Variable INVEST

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 140

Mean of Dependent Variable 383.83541667

Std Error of Dependent Variable 131.09401018

Standard Error of Estimate 23.87349707

Sum of Squared Residuals 79792.140737

J-Specification(4) 92.9231

Significance Level of J 0.0000000

Durbin-Watson Statistic 0.5451

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -63.82538832 8.16893092 -7.81319 0.00000000

2. YDIFF{1} 0.17929434 0.07713636 2.32438 0.02154365

3. GNP 0.21621773 0.00567178 38.12167 0.00000000

4. RATE{4} -10.90217143 1.22579661 -8.89395 0.00000000

Linear Regression - Estimation by Instrumental Variables

Dependent Variable RATE

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 139

Mean of Dependent Variable 5.1534027778

Std Error of Dependent Variable 3.2689143326

Standard Error of Estimate 0.9010717880

Sum of Squared Residuals 112.85832103

J-Specification(3) 9.9890

Significance Level of J 0.0186601

Durbin-Watson Statistic 1.2413

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -0.769457067 0.334490446 -2.30039 0.02291495

2. GNP 0.000874065 0.000291413 2.99940 0.00320641

3. YDIFF 0.001085252 0.006484148 0.16737 0.86732217

4. MDIFF -0.066131484 0.017867890 -3.70114 0.00030848

5. RSUM{1} 0.386253361 0.032303605 11.95697 0.00000000

Regression with AR1 - Estimation by Instrumental Variables

Dependent Variable INVEST

Quarterly Data From 1950:02 To 1985:04

Usable Observations 143

Degrees of Freedom 138

Mean of Dependent Variable 385.10769231

Std Error of Dependent Variable 130.65960726

Standard Error of Estimate 11.49120841

Sum of Squared Residuals 18222.606170

Durbin-Watson Statistic 1.9859

Q(35-1) 25.7318

Significance Level of Q 0.8449820

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -79194.55526 13284.47342 -5.96144 0.00000002

2. YDIFF{1} 0.02770 0.03215 0.86177 0.39031026

3. GNP 0.58813 0.04778 12.30957 0.00000000

4. RATE{4} -0.49612 1.24855 -0.39736 0.69171816

5. RHO 0.99990 0.00337 296.28215 0.00000000

Regression with AR1 - Estimation by Instrumental Variables

Dependent Variable INVEST

Quarterly Data From 1950:02 To 1985:04

Usable Observations 143

Degrees of Freedom 138

Mean of Dependent Variable 385.10769231

Std Error of Dependent Variable 130.65960726

Standard Error of Estimate 15.97848755

Sum of Squared Residuals 35233.064905

Durbin-Watson Statistic 1.8752

Q(35-1) 42.0270

Significance Level of Q 0.1622199

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -61.40753139 22.24527145 -2.76048 0.00655668

2. YDIFF{1} 0.10246895 0.04770245 2.14809 0.03345317

3. GNP 0.20480112 0.01027980 19.92267 0.00000000

4. RATE{4} -6.26124578 1.56015119 -4.01323 0.00009779

5. RHO 0.78905420 0.05297324 14.89534 0.00000000

Linear Regression - Estimation by Instrumental Variables

Dependent Variable CONS

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 141

Mean of Dependent Variable 1411.1625000

Std Error of Dependent Variable 486.7321052

Standard Error of Estimate 11.3267610

Sum of Squared Residuals 18089.667678

J-Specification(6) 61.1065

Significance Level of J 0.0000000

Durbin-Watson Statistic 1.6284

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -3.670622295 4.957241122 -0.74046 0.46025417

2. GNP 0.027886587 0.018218752 1.53065 0.12809568

3. CONS{1} 0.965018982 0.027219090 35.45376 0.00000000

Linear Regression - Estimation by Instrumental Variables

Dependent Variable INVEST

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 139

Mean of Dependent Variable 383.83541667

Std Error of Dependent Variable 131.09401018

Standard Error of Estimate 16.78975787

Sum of Squared Residuals 39183.539726

J-Specification(4) 15.3558

Significance Level of J 0.0040175

Durbin-Watson Statistic 2.0147

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -25.01460746 6.44309646 -3.88239 0.00015926

2. INVEST{1} 0.64206142 0.04808629 13.35228 0.00000000

3. YDIFF{1} 0.21567242 0.05431865 3.97050 0.00011457

4. GNP 0.07953995 0.01099385 7.23495 0.00000000

5. RATE{4} -4.59031825 0.98390949 -4.66539 0.00000716

Linear Regression - Estimation by Instrumental Variables

Dependent Variable RATE

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 139

Mean of Dependent Variable 5.1534027778

Std Error of Dependent Variable 3.2689143326

Standard Error of Estimate 0.9355393107

Sum of Squared Residuals 121.65749845

J-Specification(4) 12.6151

Significance Level of J 0.0133178

Durbin-Watson Statistic 1.2208

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -0.839398549 0.345175636 -2.43180 0.01629650

2. GNP 0.000992823 0.000295518 3.35960 0.00100771

3. YDIFF -0.002971527 0.006356691 -0.46746 0.64089971

4. MDIFF -0.059860827 0.018232136 -3.28326 0.00129814

5. RSUM{1} 0.373238376 0.032776485 11.38738 0.00000000

Linear Systems - Estimation by GMM-Factored Weight Matrix (3SLS)

Iterations Taken 22

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

J-Specification(14) 93.9316

Significance Level of J 0.0000000

Dependent Variable CONS

Mean of Dependent Variable 1411.1625000

Std Error of Dependent Variable 486.7321052

Standard Error of Estimate 11.3061877

Sum of Squared Residuals 18407.502834

Durbin-Watson Statistic 1.6369

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -1.310489539 4.844085691 -0.27053 0.78674952

2. GNP 0.016637966 0.017537130 0.94873 0.34275898

3. CONS{1} 0.981868303 0.026195595 37.48219 0.00000000

Dependent Variable INVEST

Mean of Dependent Variable 383.83541667

Std Error of Dependent Variable 131.09401018

Standard Error of Estimate 16.69511093

Sum of Squared Residuals 40136.648959

Durbin-Watson Statistic 1.7262

Variable Coeff Std Error T-Stat Signif

************************************************************************************

4. Constant -26.77413334 6.03182829 -4.43881 0.00000905

5. INVEST{1} 0.62900972 0.04069816 15.45548 0.00000000

6. YDIFF{1} 0.10089273 0.04471907 2.25615 0.02406151

7. GNP 0.08441125 0.00937220 9.00655 0.00000000

8. RATE{4} -5.07290797 0.84845596 -5.97899 0.00000000

Dependent Variable RATE

Mean of Dependent Variable 5.1534027778

Std Error of Dependent Variable 3.2689143326

Standard Error of Estimate 1.0702067790

Sum of Squared Residuals 164.92932716

Durbin-Watson Statistic 1.2639

Variable Coeff Std Error T-Stat Signif

************************************************************************************

9. Constant -0.713970944 0.369638934 -1.93154 0.05341679

10. GNP 0.000999767 0.000289942 3.44817 0.00056440

11. YDIFF -0.013923726 0.005972398 -2.33135 0.01973513

12. MDIFF -0.061750288 0.015646163 -3.94667 0.00007924

13. RSUM{1} 0.379550597 0.031360902 12.10267 0.00000000

Covariance\Correlation Matrix of Residuals

CONS INVEST RATE

CONS 127.82988079 0.05492486 0.30609482

INVEST 10.36751028 278.72672888 0.62238984

RATE 3.70373475 11.12037683 1.14534255

GMM-Factored Weight Matrix

Convergence in 22 Iterations. Final criterion was 0.0000039 <= 0.0000100

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Function Value 93.93159806

J-Specification(14) 93.9316

Significance Level of J 0.0000000

Dependent Variable CONS

Mean of Dependent Variable 1411.1625000

Std Error of Dependent Variable 486.7321052

Standard Error of Estimate 11.3061875

Sum of Squared Residuals 18407.502180

Durbin-Watson Statistic 1.6369

Dependent Variable INVEST

Mean of Dependent Variable 383.83541667

Std Error of Dependent Variable 131.09401018

Standard Error of Estimate 16.69511096

Sum of Squared Residuals 40136.649092

Durbin-Watson Statistic 1.7262

Dependent Variable RATE

Mean of Dependent Variable 5.1534027778

Std Error of Dependent Variable 3.2689143326

Standard Error of Estimate 1.0702067986

Sum of Squared Residuals 164.92933322

Durbin-Watson Statistic 1.2639

Variable Coeff Std Error T-Stat Signif

***************************************************************************************

1. C0 -1.31049410 4.84408536 -0.27053 0.78674878

2. C1 0.01663799 0.01753713 0.94873 0.34275838

3. C2 0.98186827 0.02619559 37.48219 0.00000000

4. I0 -26.77413281 6.03182846 -4.43881 0.00000905

5. I1 0.62900972 0.04069816 15.45548 0.00000000

6. I2 0.10089273 0.04471907 2.25615 0.02406151

7. I3 0.08441125 0.00937220 9.00655 0.00000000

8. I4 -5.07290792 0.84845595 -5.97899 0.00000000

9. R0 -0.71397099 0.36963894 -1.93154 0.05341678

10. R1 0.00099977 0.00028994 3.44817 0.00056440

11. R2 -0.01392373 0.00597240 -2.33135 0.01973512

12. R3 -0.06175029 0.01564616 -3.94667 0.00007924

13. R4 0.37955059 0.03136090 12.10267 0.00000000

Linear Regression - Estimation by LIML

Dependent Variable CONS

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 141

Centered R^2 0.9994405

R-Bar^2 0.9994326

Uncentered R^2 0.9999409

Mean of Dependent Variable 1411.1625000

Std Error of Dependent Variable 486.7321052

Standard Error of Estimate 11.5939059

Sum of Squared Residuals 18953.030316

Regression F(2,141) 125945.7304

Significance Level of F 0.0000000

Log Likelihood -555.6804

Durbin-Watson Statistic 1.6350

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 1.8710360083 5.2754295101 0.35467 0.72336682

2. GNP 0.0027746082 0.0197620627 0.14040 0.88854374

3. CONS{1} 1.0024393238 0.0295164389 33.96207 0.00000000

LIML Specification Test

Chi-Squared(6)= 107.089139 with Significance Level 0.00000000

Linear Regression - Estimation by LIML

Dependent Variable INVEST

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 139

Centered R^2 0.9839850

R-Bar^2 0.9835241

Uncentered R^2 0.9983374

Mean of Dependent Variable 383.83541667

Std Error of Dependent Variable 131.09401018

Standard Error of Estimate 16.82701273

Sum of Squared Residuals 39357.621703

Regression F(4,139) 2135.0899

Significance Level of F 0.0000000

Log Likelihood -608.2926

Durbin-Watson Statistic 2.0314

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -24.15069393 6.46227176 -3.73718 0.00027096

2. INVEST{1} 0.65057260 0.04825642 13.48158 0.00000000

3. YDIFF{1} 0.21775768 0.05444255 3.99977 0.00010258

4. GNP 0.07745179 0.01103494 7.01877 0.00000000

5. RATE{4} -4.45470801 0.98687987 -4.51393 0.00001345

LIML Specification Test

Chi-Squared(4)= 17.841923 with Significance Level 0.00132504

Linear Regression - Estimation by LIML

Dependent Variable RATE

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Degrees of Freedom 139

Centered R^2 0.9069518

R-Bar^2 0.9042741

Uncentered R^2 0.9734352

Mean of Dependent Variable 5.1534027778

Std Error of Dependent Variable 3.2689143326

Standard Error of Estimate 1.0113888399

Sum of Squared Residuals 142.18412657

Regression F(4,139) 338.7123

Significance Level of F 0.0000000

Log Likelihood -203.4134

Durbin-Watson Statistic 1.2077

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -0.957697916 0.378707653 -2.52886 0.01256033

2. GNP 0.001193690 0.000337771 3.53402 0.00055621

3. YDIFF -0.009833182 0.007826592 -1.25638 0.21108524

4. MDIFF -0.049254607 0.020543092 -2.39762 0.01783026

5. RSUM{1} 0.351224765 0.037416125 9.38699 0.00000000

LIML Specification Test

Chi-Squared(4)= 13.348844 with Significance Level 0.00969128

FIML Estimates

Convergence in 73 Iterations. Final criterion was 0.0000077 <= 0.0000100

Quarterly Data From 1950:01 To 1985:04

Usable Observations 144

Log Likelihood -1335.2871

Dependent Variable CONS

Mean of Dependent Variable 1411.1625000

Std Error of Dependent Variable 486.7321052

Standard Error of Estimate 11.5817648

Sum of Squared Residuals 19315.767782

Durbin-Watson Statistic 1.6291

Dependent Variable INVEST

Mean of Dependent Variable 383.83541667

Std Error of Dependent Variable 131.09401018

Standard Error of Estimate 16.67944487

Sum of Squared Residuals 40061.358872

Durbin-Watson Statistic 1.7629

Dependent Variable RATE

Mean of Dependent Variable 5.1534027778

Std Error of Dependent Variable 3.2689143326

Standard Error of Estimate 1.0253194085

Sum of Squared Residuals 151.38430408

Durbin-Watson Statistic 1.2672

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. C0 3.42547966 4.75020661 0.72112 0.47083431

2. C1 -0.00465438 0.01672408 -0.27830 0.78077911

3. C2 1.01357026 0.02498662 40.56452 0.00000000

4. I0 -25.79744169 6.05768407 -4.25863 0.00002057

5. I1 0.63679041 0.04089645 15.57080 0.00000000

6. I2 0.11251043 0.04395753 2.55953 0.01048150

7. I3 0.08225606 0.00947648 8.68002 0.00000000

8. I4 -4.90819918 0.86118764 -5.69934 0.00000001

9. R0 -0.63893440 0.34540833 -1.84979 0.06434319

10. R1 0.00089217 0.00023999 3.71759 0.00020113

11. R2 -0.01053334 0.00295132 -3.56902 0.00035832

12. R3 -0.06561464 0.01389994 -4.72050 0.00000235

13. R4 0.39059767 0.02597049 15.04006 0.00000000