CHANKAROLYI.RPF

CHANKAROLYI.RPF demonstrates GMM on a system of non-linear equations. It is based upon the model for interest rates from Chan, Karolyi, Longstaff and Sanders(1992). The model generates a conditional mean and a conditional variance:

\begin{equation} r_{t + 1} - r_t = \alpha + \beta r_t + \varepsilon _{t + 1} \end{equation}

\begin{equation} E_t \left[ {\varepsilon _{t + 1} } \right] = 0,E_t \left[ {\varepsilon _{t + 1}^2 } \right] = \sigma ^2 r_t^{2\gamma } \end{equation}

Note how the VARIANCE formula uses the square of the EPS formula:

nonlin alpha beta gamma sigmasq

frml eps = y1{-1}-(1+beta)*y1-alpha

frml variance = eps(t)^2-sigmasq*y1^(2*gamma)

Different information sets (instruments) and different ways of computing the weight matrix will give different estimates. The example does four NLSYSTEMS, two of which show here are the just-identified and overidentified (2 conditions x 3 instruments vs 4 free parameters) estimated with the default NOZUDEP handling:

instruments constant y1
nlsystem(instruments) / eps variance
instruments constant y1{0 1}

nlsystem(instruments) / eps variance

Full Program

calendar(m) 1946:1

open data tbills.xls

data(format=xls,org=columns) 1946:01 2011:01 tb3ms

set y1 = tb3ms

nonlin alpha beta gamma sigmasq

frml eps = y1{-1}-(1+beta)*y1-alpha

frml variance = eps(t)^2-sigmasq*y1^(2*gamma)

* Guess values from a linear regression

linreg y1

# constant y1{1}

compute alpha=%beta(1),beta=%beta(2)-1

compute sigmasq=%sigmasq,gamma=0.0

* Just identified model

instruments constant y1

nlsystem(instruments) / eps variance

* Overidentified model

* With nozudep

instruments constant y1{0 1}

nlsystem(instruments) / eps variance

* Same thing, but correcting the covariance matrix and J-statistic

instruments constant y1{0 1}

nlsystem(instruments,robusterrors) / eps variance

* With ZUDEP

instruments constant y1{0 1}

nlsystem(instruments,zudep) / eps variance

Output

Linear Regression - Estimation by Least Squares

Dependent Variable Y1

Monthly Data From 1946:02 To 2011:01

Usable Observations 780

Degrees of Freedom 778

Centered R^2 0.9809291

R-Bar^2 0.9809046

Uncentered R^2 0.9940858

Mean of Dependent Variable 4.4570256410

Std Error of Dependent Variable 2.9901748609

Standard Error of Estimate 0.4132011179

Sum of Squared Residuals 132.83195749

Regression F(1,778) 40017.0673

Significance Level of F 0.0000000

Log Likelihood -416.3905

Durbin-Watson Statistic 1.3212

Variable Coeff Std Error T-Stat Signif

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

1. Constant 0.0418011052 0.0265713863 1.57316 0.11608778

2. Y1{1} 0.9905557662 0.0049517225 200.04266 0.00000000

GMM-Factored Weight Matrix

Convergence in 17 Iterations. Final criterion was 0.0000000 <= 0.0000100

Monthly Data From 1946:01 To 2011:01

Usable Observations 780

Skipped/Missing (from 781) 1

Function Value 4.70586311e-15

Variable Coeff Std Error T-Stat Signif

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

1. ALPHA 0.041801105 0.026537299 1.57518 0.11521418

2. BETA -0.009444234 0.004945370 -1.90971 0.05617027

3. GAMMA 1.595429690 0.378547682 4.21461 0.00002502

4. SIGMASQ 0.000477412 0.000879624 0.54275 0.58730500

GMM-Factored Weight Matrix

Convergence in 9 Iterations. Final criterion was 0.0000048 <= 0.0000100

Monthly Data From 1946:01 To 2011:01

Usable Observations 779

Skipped/Missing (from 781) 2

Function Value 96.31177513

J-Specification(2) 96.3118

Significance Level of J 0.0000000

Variable Coeff Std Error T-Stat Signif

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

1. ALPHA 0.042300267 0.026607082 1.58981 0.11187711

2. BETA -0.010951691 0.004909161 -2.23087 0.02568986

3. GAMMA 1.358737262 0.275738621 4.92763 0.00000083

4. SIGMASQ 0.001482979 0.001946244 0.76197 0.44607807

GMM-Factored Weight Matrix

Convergence in 7 Iterations. Final criterion was 0.0000079 <= 0.0000100

With Heteroscedasticity/Misspecification Adjusted Standard Errors

Monthly Data From 1946:01 To 2011:01

Usable Observations 779

Skipped/Missing (from 781) 2

Function Value 96.31179931

J-Specification(2) 96.3118

Significance Level of J 0.0000000

Variable Coeff Std Error T-Stat Signif

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

1. ALPHA 0.042300211 0.000031814 1329.62802 0.00000000

2. BETA -0.010951687 0.000007532 -1454.06978 0.00000000

3. GAMMA 1.358733495 0.001222156 1111.75164 0.00000000

4. SIGMASQ 0.001483006 0.000008438 175.74638 0.00000000

GMM-Continuously Updated Weight Matrix

Convergence in 7 Iterations. Final criterion was 0.0000048 <= 0.0000100

Monthly Data From 1946:01 To 2011:01

Usable Observations 779

Skipped/Missing (from 781) 2

Function Value 10.77230305

J-Specification(2) 10.7723

Significance Level of J 0.0045796

Variable Coeff Std Error T-Stat Signif

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

1. ALPHA 0.059769659 0.030763748 1.94286 0.05203306

2. BETA -0.013557828 0.008823849 -1.53650 0.12441624

3. GAMMA 1.309121407 0.194980317 6.71412 0.00000000

4. SIGMASQ 0.001206328 0.000876889 1.37569 0.16891741