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Gradients Uses of gradients in artificial regressions Checking convergence of NLS estimates (gallant1.prg) Omitted variable LM test (gallant2.prg) Omitted variable LM test robust to heteroskedasticity (gallant3.prg) Checking convergence of NLS estimates  

The following program illustrates how to use an artificial regression of the residuals on the gradients to check for convergence of the non-linear least squares estimates. 'uses of gradients 'checking NLS estimates by an artificial regression 'see Davidson-MacKinnon 1993, chapter 6 'last revised 3/25/2004 'change path to program path %path = @runpath cd %path 'create workfile wfcreate gallant1 u 1 30 'read data from plain text file read(b2) amstat.dat t y x1 x2 x3 'starting values c(1) = -0.04866 c(2) = 1.03884 c(3) = -0.73792 c(4) = -0.51362 'replicate Gallant (1987) fig 5a (p.35) equation eq1.ls(c=1e-9,m=1000,showopts,deriv=aa) y = c(1)*x1 + c(2)*x2 + c(4)*exp(c(3)*x3) show eq1.output 'get residuals eq1.makeresid res 'get derivative series eq1.makederivs(n=grd1) 'run Gauss-Newton regression 'all coefs, t-stats, and R2 should be within convergence 'tolerance if NLS converged equation eq2.ls res grd1 show eq2.output ^Top

Omitted variable LM test The following program computes a Lagrange multiplier test statistic for an omitted variable (@trend) in a nonlinear regression model. See Davidson and MacKinnon (1993), Estimation and Inference in Econometrics, Chapter 6 for theoretical background. 'uses of gradients 'LM test for omitted variable in NLS 'see Davidson-MacKinnon 1993, chapter 6 'last revised 3/25/2004 'change path to program path %path = @runpath cd %path 'create workfile wfcreate gallant1 u 1 30 'read data from plain text file read(b2) amstat.dat t y x1 x2 x3 'starting values c(1) = -0.04866 c(2) = 1.03884 c(3) = -0.73792 c(4) = -0.51362 'estimate restricted model 'replicate Gallant (1987) fig 5a (p.35) equation eq1.ls(c=1e-9,m=1000,showopts,deriv=aa) y = c(1)*x1 + c(2)*x2 + c(4)*exp(c(3)*x3) show eq1.output 'get residuals eq1.makeresid res 'get derivative series eq1.makederivs(n=grd1) 'run Gauss-Newton regression (6.17) equation eq2.ls res @trend grd1 show eq2.output 'compute test statistic as n*R^2_u scalar lmstat = eq2.@regobs * (1 - eq2.@ssr/@sumsq(res)) 'show results in table table out out(1,1) = "Omitted variable test based on Gauss-Newton regression:" out(2,1) = "n*R^2_u = " out(2,2) = lmstat out(2,3) = "p-value = " out(2,4) = 1 - @cchisq(lmstat,1) 't-statistic in test regression !tstat = eq2.@tstats(1) 'degrees of freedom !df = eq2.@regobs - eq2.@ncoef out(3,1) = "t-stat = " out(3,2) = !tstat out(3,3) = "p-value = " out(3,4) = 2*(1-@ctdist(@abs(!tstat),!df)) show out ^Top

Omitted variable LM test robust to heteroskedasticity The following program computes a Lagrange multiplier test statistic for an omitted variable (@trend) in a nonlinear regression model. The test statistic is designed to be robust to heteroskedasticity of unknown form. See Davidson and MacKinnon (1993), Estimation and Inference in Econometrics, Chapter 6 for theoretical background. 'uses of gradients 'hetero-robust Gauss-Newton regression 'see Davidson-MacKinnon 1993, chapter 11.6 'last revised 3/25/2004 'change path to program path %path = @runpath cd %path 'create workfile wfcreate gallant1 u 1 30 'read data from plain text file read(b2) amstat.dat t y x1 x2 x3 'starting values c(1) = -0.04866 c(2) = 1.03884 c(3) = -0.73792 c(4) = -0.51362 'estimate restricted model 'replicate Gallant (1987) fig 5a (p.35) equation eq1.ls(c=1e-9,m=1000,showopts,deriv=aa) y = c(1)*x1 + c(2)*x2 + c(4)*exp(c(3)*x3) show eq1.output 'get residuals eq1.makeresid res 'get derivative series eq1.makederivs(n=grd1) 'step 1: regress test variable on derivs and get resids equation eq2.ls @trend grd1 eq2.makeresids zres 'step 2: weight step 1 resids by restricted resids series zu = zres*res 'step 3: run robust Gauss-Newton regression (11.69) eq2.ls 1 zu 'step 4: compute test statistic n*R^2_u scalar lmstat2 = eq2.@regobs - eq2.@ssr 'show results in table table out out(1,1) = "Omitted variable test based on heteroskedasticity robust Gauss-Newton regression:" out(2,1) = "n*R^2_u = " out(2,2) = lmstat2 out(2,3) = "p-value = " out(2,4) = 1 - @cchisq(lmstat2,1) 'one-variable case can be computed by "regular" Gauss-Newton regression with White standard errors equation eq2.ls(h) res @trend grd1 show eq2.output 'robust t-statistic in test regression !tstat = eq2.@tstats(1) 'degrees of freedom !df = eq2.@regobs - eq2.@ncoef out(3,1) = "robust t-stat = " out(3,2) = !tstat out(3,3) = "p-value = " out(3,4) = 2*(1-@ctdist(@abs(!tstat),!df)) show out ^Top