Prof. George Karras

Consider again Mankiw, Romer, and Weil's (QJE, May 1992) neoclassical growth equation:

ln(y_T) - ln(y₀) = (1-e^-T)[(1-)^-1ln(s) - (1-)^-1ln(n) + (1-)^-1ln(h) - ln(y₀)], (0)

where y is GDP per adult, T is the terminal year, 0 the initial year, is the rate of convergence, s is the fraction of income invested in physical capital, n is the population growth rate (plus the depreciation rate and the rate of technological growth), h is human capital, and and are the output elasticities with respect to physical capital and human capital, respectively. Note that the equation is linear in the variables ln(y_T)-ln(y₀), ln(s), ln(n), ln(h), and ln(y₀), but highly nonlinear in the parameters , , and .

Because of the linearity of the equation in the variables, we have been able to estimate it in the linear form:

growth_i = ₀ + ₁*lny1960_i + ₂*lns_i + ₃*lnpop_i + ₄*lnschool_i + _i, (1)

where the variables are growthln(y₁₉₈₅)-ln(y₁₉₆₀), lny1960ln(y1960), lnsln(s), lnpopln(pop+0.05), and lnschoolln(school); the parameters are ₀, ₁-(1-e^-T), ₂(1-e^-T)/(1-), ₃-(1-e^-T)/(1-), and ₄(1-e^-T)/(1-); and is the error term. But note that this method does not directly estimate the structural parameters , , and , or obtain their standard errors.

Here we will estimate the structural parameters , , and directly by Non-Linear Least Squares (NLLS) and Maximum Likelihood (ML). A significant additional advantage of these nonlinear approaches is that they will give us direct estimates of standard errors for the structural parameters. To implement the nonlinear estimation, add a constant and an error term to (0) and write it as

growth_i = ₀ - (1-e^-T)*lny1960_i + (1-e^-T)(1-)^-1*lns_i

- (1-e^-T)(1-)^-1*lnpop_i + (1-e^-T)(1-)^-1*lnschool_i + _i, (2)

where the variables are defined as in equation (1), but now the parameters to be estimated are ₀, , , and .

NOTE: This example uses RATS for each of the estimation that follow.

allocate 98

*

* cross section data from Mankiw, Romer, Weil, QJE 1992 (Appendix)

data(unit=input,org=obs) / number y1960 y1985 growth pop iy school

1 2485 4371 4.8 2.6 24.1 4.5

2 1588 1171 0.8 2.1 5.8 1.8

... etc. ...

120 9523 12308 2.7 1.7 22.5 11.9

121 1781 2544 3.5 2.1 16.2 1.5

*

set y1960 = log(y1960)

set y1985 = log(y1985)

set growth = y1985 - y1960

set pop = log(pop/100.+0.05)

set iy = log(iy/100.)

set school = log(school/100.)

B1. Run OLS

linreg growth

# constant y1960 iy pop school

Dependent Variable GROWTH - Estimation by Least Squares

Usable Observations 98 Degrees of Freedom 93

Centered R**2 0.484306 R Bar **2 0.462126

Uncentered R**2 0.745475 T x R**2 73.057

Mean of Dependent Variable 0.4492766111

Std Error of Dependent Variable 0.4458053414

Standard Error of Estimate 0.3269532452

Sum of Squared Residuals 9.9415534807

Regression F(4,93) 21.8349

Significance Level of F 0.00000000

Durbin-Watson Statistic 2.134934

Variable Coeff Std Error T-Stat Signif

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

1. Constant 3.004870632 0.827868892 3.62965 0.00046387

2. Y1960 -0.286576570 0.061716598 -4.64343 0.00001123

3. IY 0.523736529 0.086847397 6.03054 0.00000003

4. POP -0.504594920 0.288579024 -1.74855 0.08366852

5. SCHOOL 0.229479466 0.059533600 3.85462 0.00021318

B2. Compute and display the initial values

compute ib0 = %beta(1)

compute isigmasq = %seesq

* note: %beta(2) = -[1-exp(-lambda*T)]

compute ilambda = - log(%beta(2)+1.)/(1985.-1960.)

* note: %beta(3) = - %beta(2)*alpha/(1-alpha)

compute ialpha = %beta(3)/(%beta(3)-%beta(2))

* note: %beta(5) = - %beta(2)*beta/(1-alpha)

compute ibeta = - %beta(5)*(1.-ialpha)/%beta(2)

display isigmasq ilambda ialpha ibeta

0.10690 0.01351 0.64634 0.28320

*

nlpar(subiterations=100)

C1. List the parameters and write the non-linear equation

nonlin b0 lambda alpha beta

*

frml equation = $

b0 - (1.-exp(-lambda*(1985.-1960.)))*y1960 $

+ (1.-exp(-lambda*(1985.-1960.)))*alpha/(1.-alpha)*iy $

- (1.-exp(-lambda*(1985.-1960.)))*alpha/(1.-alpha)*pop $

+ (1.-exp(-lambda*(1985.-1960.)))*beta/(1.-alpha)*school

C2. Set initial values equal to the values implied by the OLS estimation

compute b0 = ib0

compute lambda = ilambda

compute alpha = ialpha

compute beta = ibeta

C3. Estimate the equation by NLLS

nlls(frml=equation,iterations=100) growth

Dependent Variable GROWTH - Estimation by Nonlinear Least Squares

Iterations Taken 2

Usable Observations 98 Degrees of Freedom 94

Centered R**2 0.484285 R Bar **2 0.467826

Uncentered R**2 0.745465 T x R**2 73.056

Mean of Dependent Variable 0.4492766111

Std Error of Dependent Variable 0.4458053414

Standard Error of Estimate 0.3252161481

Sum of Squared Residuals 9.9419610413

Durbin-Watson Statistic 2.140439

Variable Coeff Std Error T-Stat Signif

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

1. B0 2.9678112818 0.5671820197 5.23256 0.00000101

2. LAMBDA 0.0135709153 0.0032908809 4.12379 0.00008039

3. ALPHA 0.6445455868 0.0485153349 13.28540 0.00000000

4. BETA 0.2847445520 0.0775753697 3.67055 0.00040160

Note that NLLS produces standard errors for each of the estimated parameters. In fact the entire variance-covariance matrix of the parameter vector is estimated. This allows us to proceed with inference as usual.

D1. List the parameters and write the log likelihood function (Note that the variance ² is now to be jointly estimated with he rest of the parameters)

nonlin sigmasq b0 lambda alpha beta

*

frml resid = growth - $

( b0 - (1.-exp(-lambda*(1985.-1960.)))*y1960 $

+ (1.-exp(-lambda*(1985.-1960.)))*alpha/(1.-alpha)*iy $

- (1.-exp(-lambda*(1985.-1960.)))*alpha/(1.-alpha)*pop $

+ (1.-exp(-lambda*(1985.-1960.)))*beta/(1.-alpha)*school )

*

frml logl = $

-.5*log(sigmasq) - .5*(1/sigmasq)*(resid**2)

D2. Set initial values equal to the values implied by the OLS estimation

compute sigmasq = isigmasq

compute b0 = ib0

compute lambda = ilambda

compute alpha = ialpha

compute beta = ibeta

D3. Estimate the equation by ML

maximize(method=bhhh,iterations=100) logl

Estimation by BHHH

Iterations Taken 10

Usable Observations 98 Degrees of Freedom 93

Function Value 63.12195118

Variable Coeff Std Error T-Stat Signif

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

1. SIGMASQ 0.1014280561 0.0136226552 7.44554 0.00000000

2. B0 2.9665559411 0.6541677030 4.53486 0.00000576

3. LAMBDA 0.0135658878 0.0041292501 3.28532 0.00101868

4. ALPHA 0.6446537726 0.0540997724 11.91602 0.00000000

5. BETA 0.2845499324 0.0694294595 4.09840 0.00004160

Once again, we have standard errors for each of the estimated parameters and the entire variance-covariance matrix of the parameter vector.

IMPORTANT NOTE: The specific instructions needed to implement NLLS and ML will differ from software to software, but the following three elements will always be there: (i) specification of the nonlinear equation (or likelihood function), (ii) setting of intitial values for the parameters to be estimated (otherwise 0 is assumed, which may be a very bad guess), and (iii) actual minimization (maximization) of the sum of squares (log likelihood).