LIML

Output     Options     Examples     References

LIML computes the limited information maximum likelihood estimator for a single equation linear structural model. To estimate a single equation nonlinear model via LIML, use FIML with unrestricted reduced-form equations for the included endogenous variables (those which appear on the right hand side of the primary equation).

LIML (BEKKER, FEI, FEPRINT, FULLER=<scalar value>, INST=(<list of instruments>), SILENT, TERSE) <dependent variable> <list of rhs endogenous and exogenous variables> ;

Usage

LIML's form is identical to the 2SLS/INST command -- specify a list of instruments and the variables in the equation. LIML determines the list of endogenous variables, included exogenous variables, and excluded exogenous variables by comparing the instrument list with the variables in the equation. If there are no endogenous variables, OLSQ is used and a warning is printed. If the right hand side equation is exactly identified (number of endogenous variables equals number of excluded exogenous variables), LIML is equivalent to 2SLS, so 2SLS is used, and a warning is printed. If the equation is under-identified, an error message is printed (just like 2SLS).

Output

The output of LIML begins with an equation title, the name of the dependent variable and the lists of endogenous, included exogenous and excluded exogenous variables. The LIML eigenvalue and an F-test of the overidentifying restrictions are printed. The log likelihood at the solution is stored, but not printed.

If FULLER is used, the FULLER constant and the computed K-class value are also printed.

This is followed by various statistics on goodness-of-fit: the sum of squared residuals, the standard error of the regression, the R-squared, and the Durbin-Watson statistic for autocorrelation of the residuals.

The estimated concentration parameter (mu squared) and Cragg-Donald F-statistic (CDF) are also shown and stored. When there is a single right hand side endogenous variable, CDF is an F-statistic which tests if the excluded exogenous variables (called Z2) have zero coefficients in the reduced form; the concentration parameter is equal to CDF times the number of excluded exogenous variables. These statistics are also valid for multiple RHS endogenous variables. They can be used to assess whether the model has a "weak instruments" or "many instruments" problem. For a single RHS endogenous variable, the bias of 2SLS is proportional to rho/CDF (Nelson and Startz 1990) where rho is the correlation between the reduced form and structural equations), so low values of the CDF imply a high bias. For a single RHS endogenous variable, values of CDF lower than 10 are considered to be problematic (Staiger and Stock (1997), refined in Stock and Yogo (2004). Unrealistically low computed standard errors for 2SLS also occur in such a situation and the use of LIML(FULLER=1) or plain LIML with the Bekker SEs instead of 2SLS is generally advised because these estimators have much lower bias and properly sized standard errors, especially when the number of excluded instruments is large.

LIML(FULLER=1) and LIML can still suffer from a "weak instruments" problem. For a single RHS endogenous variable, values of mu squared lower than 10~15 suggest a problem (bias and/or standard errors that are too small - see Hansen, Hausman and Newey (2004). For larger values of mu squared, LIML(FULLER=1) and LIML have low bias and properly sized standard errors. LIML(FULLER=1) has a slightly higher median bias than plain LIML, but it is mean unbiased, and it has a smaller MSE than LIML (it has finite moments).

Following this is a table of right hand side variable names, estimated coefficients, standard errors and associated t-statistics. If the variance-covariance matrix has not been suppressed (see the SUPRES command), it is printed after this table. Finally, if the RESID and PLOTS options are on, a table and plot of the actual and fitted values of the dependent variable and the residuals is printed.

LIML also stores most of these results in data storage for later use. The table below lists the results available after a LIML command. The fitted values and residuals will only be stored if the RESID option is on (the default).

If the regression includes PDL variables, the following will also be stored:

Method

The LIML eigenvalue is the minimum eigenvalue of the following matrix:

H is the residual covariance matrix of the endogenous and dependent variables regressed on all of the instruments. H1 is the residual covariance matrix of the same variables regressed on just the included instruments. This eigenvalue is calculated by CACM algorithm 384. The FULLER constant term (if non-zero) is subtracted from this eigenvalue to yield K, which is then used in the standard K-class formula to compute the coefficients. The standard errors for the NOBEKKER option are computed from the K-class inverse matrix times the sum of squared residuals divided by (number of observations minus number of estimated coefficients).

The F-test for overidentifying restrictions is given by

FSTAT = (LAMBDA-1)*(T-NZ)*(K2-G1)

where LAMBDA is the LIML eigenvalue, T is the number of observations, NZ is the number of instruments, and K2-G1 is the number of overidentifying restrictions = number of excluded exogenous variables K2 minus the number of included endogenous variables G1. See Anderson et al (1986).

The reported log likelihood is the same as would be computed by FIML on the model plus additional unconstrained equations for the right hand side variables as functions of the instruments.  

Options

BEKKER/NOBEKKER specifies that Bekker standard errors are to be computed (see Hansen, Hausman, and Newey 2004). These standard errors are better for small samples and/or when there are large numbers of excluded instruments.

FEI/NOFEI  specifies whether a model with individual fixed effects is to be estimated. FREQ (PANEL) must be in effect.

FEPRINT/NOFEPRINT specifies whether the estimated effects and their standard errors are to be printed.

FULLER= value used to weight the eigenvalue towards zero. The formula used is

K = LAMBDA - FULLER/(T-NZ),

 where K is the K-class constant, LAMBDA is the LIML eigenvalue, T is the number of observations, and NZ is the number of instruments. FULLER=0 (default) is the standard LIML estimator, which is median-unbiased. FULLER=1 yields a mean-unbiased estimator. FULLER values between 0 and 8-16/(T-NZ-2) dominate LIML in small-sigma efficiency. The LIML estimator modified in this way has smaller tails than the standard LIML estimator, which gives it good small-sample properties (see the references for details)

INST= (list of instruments). This list should include all the exogenous variables in the equation being estimated as well as the other exogenous variables in the model. Do not forget to include a constant if there is one in the model. Weights are not supported at present.

SILENT/NOSILENT suppresses all output.

TERSE/NOTERSE prints minimal output (estimated coefficients and a summary statistic).

Examples

This example estimates the consumption function for the illustrative model in the TSP User's Manual, using the constant, trend, government expenditures (G), and the log of the money supply (LM) as instruments:

LIML (INST=(C,G,TIME,LM)) CONS C GNP ;

Other examples:

LIML (INST=(C,LOGR,LOGR(-1),LOGR(-2),LOGR(-3)) LOGP C LOGP(-1)LOGR ;

LIML (FULLER=1,INST=(C,LOGR,LOGR(-1),LOGR(-2),LOGR(-3)) LOGP C LOGP(-1) LOGR ;

References

Anderson, T. W., Kunitomo, Naoto, and Morimune, Kimio, "Comparing Single Equation Estimators in a Simultaneous Equation System," Econometric Theory 2 (1986), pp. 1-32.

Cragg, J. G., and S. G. Donald, "Testing Identifiability and Specification in Instrumental Variable Models," Econometric Theory 9 (1993), pp. 222-240.

Fuller, Wayne A., "Some Properties of a Modification of the Limited Information Estimator," Econometrica 45: 939-953.

Hansen, C., J. A. Hausman, and W. Newey, “Weak Instruments, Many Instruments, and Microeconometric Practice,” MIT, Cambridge, Mass: working paper, 2004.

Judge et al, The Theory and Practice of Econometrics, John Wiley & Sons, New York, 1981, pp. 531-533.

Maddala, G. S., Econometrics, McGraw-Hill Book Company, New York, 1977, Chapter 11, Appendix C.

Nelson, C. R., and R. Startz, "Some Further Results on the Exact Small Sample Properties of the Instrumental Variables Estimator," Econometrica 58 (1990), pp. 967-976.

Pindyck, Robert S., and Daniel L. Rubinfeld, Econometric Models and Economic Forecasts, McGraw- Hill Book Company, New York, 1976, Chapter 9, Appendix 9.4.

Rothenberg, T. J., "Approximating the Distributions of Econometric Estimators and Test Statistics," Ch. 15 in Z. Griliches and M. Intriligator (eds.), Handbook of Econometrics, Vol. II, Amsterdam: North Holland, pp. 881-935.

Staiger, D., and J. H. Stock, "Instrumental Variables Regression with Weak Instruments," Econometrica 65 (1997), pp. 557-586.

Stock, J. H., and M. Yogo, "Testing for Weak Instruments in Linear IV Regression," NBER Technical Working Paper No. 284, October 2002.

Stewart, G. E., Algorithm #384, Collected Algorithms from ACM Volume II, ACM, New York, N. Y.

Theil, Henri, Principles of Econometrics, John Wiley & Sons, New York, 1971, Chapter 10.