See also separate section on regression.

Key Terms and Concepts

• Two stages in 2SLS refer to (1) a stage in which new dependent or endogenous variables are created to substitute for the original ones, and (2) a stage in which the regression is computed in OLS fashion, but using the newly created variables. The purpose of the first stage is to create new dependent variables which do not violate OLS regression's recursivity assumption.

• A problematic causal variable is an endogenous variable whose disturbance term is posited to be correlated with the disturbance term of another endogenous variable on which it has a direct effect. Problematic causal variables are replaced by substitutes in the first stage of 2SLS.

• Instruments are the variables used in the first stage of 2SLS to create the new variables (called instrumental variables) which replace the problematic causal variables. This is accomplished using OLS regression, with the problematic causal variable as the dependent and instrumental variables as the independents. The instruments are the exogenous variables with direct or indirect causal paths (straight arrows in the diagram) to the problematic causal variable but which have no direct causal path to the endogenous variable whose disturbance term is correlated with that of the problematic causal variable. The predicted values of this regression equation are the values of the new causal variable which replaces the problematic causal variable.

• Stage two. The new variable will be uncorrelated with the disturbance term of the endogenous variable. When similar replacements are done for all problematic causal variables, the recursivity assumption will no longer be violated. The researcher then proceeds to stage 2 of 2SLS, which is simply an ordinary least squares (OLS) regression, but using the newly created instrumental variables. In a path model with multiple endogenous variables, there will be one regression for each endogenous variable. In each case the dependent will be a given endogenous variable and the independents will be those variables with direct effects on it (straight arrows going into it), but using the substitute new variables wherever the original ones were problematic causal variables.

Assumptions

• The recursivity assumption in OLS regression is that the model should not involve feedback loops. Not only should there be no circular direct effects (the straight arrows in a path diagram leading, for example, from A to B to C to A), but one must also assume the disturbance terms for the endogenous variable(s) (that is, variables with incoming arrows) are uncorrelated with its or their respective causal variables (that is, those variables with outgoing arrows leading to the endogenous variable or variables). Thus, for instance, the model should not contain a situation such as one where endogenous variable Y1 is partly caused by endogenous variable Y2, yet there is a correlation between the disturbance term for Y1 and the disturbance term for Y2.

• Regression model assumptions. As with other regression models, the model must be correctly specified, relationships must be homoscedastic (error variance is similar for the range of the response variables), and error terms must be normally distributed. The other assumptions of 2SLS are mostly the same as for regression.

• Testing assumptions. Failure to filter the data to assure assumptions are met may result in critical errors of interpretation. Econometricans have developed tests for specification error, heteroscedasticity, error normality, and more, all of which are can be employed for 2SLS models. See Pagan (1984), and Pesaran and Taylor (1999), and for an example, Eoczkowski and Farrell (1998).

Frequently Asked Questions

• Will 2SLS parameters be much different from OLS coefficients for the same data?
• What computer software supports 2SLS?
• Why is MLE generally preferred to 2SLS in estimating path parameters?
• In SEM, is there any reason to use 2SLS instead of MLE?
• How is 2SLS used to test for selection bias?
• How is the intercept interpreted in 2SLS?
• May one apply 2SLS to cointegrated time series?

• Will 2SLS parameters be much different from OLS coefficients for the same data?
  A comparison of methods by de Leeuw and Kreft (1986) reported that 2SLS and OLS (as well as weighted least squares GLS) gave similar parameter estimates for the same data.

• What computer software supports 2SLS?
  SPSS has a 2SLS module. LISREL, the leading tructural equation modeling package from Scientific Software International, started implementing 2SLS estimation with version 8.30.

• Why is MLE generally preferred to 2SLS in estimating path parameters?
  Maximum likelihood estimation (MLE) is more common than 2SLS for estimating path parameters in non-recursive models because the MLE estimates take the entire model into account, whereas 2SLS estimates are computed based on one portion of the model at a time. That is, MLE is a "full informational" whereas 2SLS is a "partial informational" technique. For these reasons, some researchers prefer MLE over 2SLS.

• In SEM, is there any reason to use 2SLS instead of MLE?
  It is true that path estimation in structural equation modeling (SEM) typically uses maximum likelihood estimation (MLE) for non-recursive models. However, two-stage least squares (2SLS), not being an iterative strategy like MLE, is faster computationally and requires less computer memory. It also does not require the computer or researcher to posit starting points for the estimates, mistakes in which may (rarely) lead to lack of convergence in MLE. The 2SLS approach also does not require distributional assumptions be made about the exogenous variables. As Bollen (1996) shows, 2SLS can be used to estimate both the measurement and structural models in SEM.

  A strong reason for using 2SLS, noted by Eddie Eoczkowski, "is that 2SLS estimators of SEMs can make use of all the advances made by econometricans in diagnostic testing of models." Oscztrowski writes, "My concern with many published SEM results is that insufficient effort is devoted to checking the assumptions which underpin estimation. Only if underpinning assumptions are valid will the resulting estimates be consistent, efficient and permit valid testing. In other words, too many fragile SEM estimates get through without rigorous checking. Effectively by subjecting the final estimated model to a battery of checks only good strong robust models will get through and hence may be of some use to practitioners." For an example, see Eoczkowski and Farrell (1998), who use non-nested tests and Bollen's (1996) 2SLS estimator for SEMs to compare non-nested SEMs to facilitate the choice of alternative measurement scales for the same latent construct. (Eoczkowski, private communication, 8/18/2000).

  As of Amos version 16, Amos does not support 2SLS estimation. LISREL, however, does. A description of the LISREL procedure is found in Schmacker & Lomax (2004: 386-388).

• How is 2SLS used to test for selection bias?
  Selection bias is a form of correlated error which violates an assumption of regression analysis and many other procedures. It may occur when using treatment vs. comparison group membership as an independent variable, because measured factors such as gender, race, education, etc., may cause differential selection into the two groups and these factors may also be causes of the dependent variable, causing correlation of error of an independent (group membership) with the dependent. The 2SLS technique can be used to test for selection bias provided all important factors accounting for selection are included in the set of measured independent variables. When selection bias causes correlated error due to unmeasured variables, 2SLS cannot be used as a test for that bias.

  The general strategy of using 2SLS to test for selection bias is (1) in the first stage, predict participation/non-participation in the treatment group, based on gender, race, education, and other measured independents thought to be relevant to selection; then (2) use the estimated probability of selection computed in stage 1 as an independent variable in stage 2. If its OLS regression coefficient is not significant, it is concluded that selection bias is not a problem. An illustration is found in Heinrich (1998).

  Stage 1. Using OLS regression, the dependent variable is participation in the treatment group = 1, non-participation = 0. The independents are gender, race, education, and other measured independents thought to be relevant to selection into the treatment group. Naturally, the researcher should have anticipated the need for this test of selection bias and should have measured all factors which enter into the staff decision to admit a subject into the treatment group. The higher the R² for this equation, the more complete the set of selection-related independents, but since selection may also be at random, a low or moderate R² is not in itself a reason to reject the appropriateness of the 2SLS procedure. However, failure to include an independent important to selection can radically affect the coefficients in stage 1 and the resulting estimates of the probability-of-selection variable (see below).

  Stage 2. Based on stage 1, a probability-of-selection variable is calculated for each case in the sample. This is simply the regression estimate of the dependent in stage 1. Then an OLS regression is run with the actual dependent variable of interest as dependent. The independents include the probability-of-selection variable and any other variable which is an independent in the model for that dependent variable (these do not necessarily include stage 1 independents thought to influence selection). If the b coefficient for the probability-of-selection variable is not significant, the researcher concludes selection bias is not a problem. The researcher may also run this second-stage regression with and without the probability-of-selection variable. If the regression coefficients are similar for the other independent variables in the two regressions, this is further indication that selection bias is not a problem.

  There are significant limitations on this method of testing for selection bias. First, 2SLS does not test for selection bias due to unmeasured variables. Second, it does not test for nonlinear and interaction effects among measured variables unless explicitly modeled in stage 1. Third, if in Stage 2 the probability-of-selection variable and the grouping variable (treatment/comparison) are both included in the equation as independents, when there is multicollinearity between these two variables, the significance of the b coefficient for the probability-of-selection variable and the grouping variable may be biased, leading to erroneous conclusions about the presence or absence of selection bias.

• How is the intercept interpreted in 2SLS?
  The intercept is the expected value of the dependent when all the explanatory variables are set to zero. Thus the 2SLS intercept is interpreted the same as for OLS, MLE, WLS, and other forms of estimation. In all forms of estimation, setting all explanatory variables to zero may be nonsensical in substantive terms, rendering interpretation of the intercept meaningless. Generally, the intercept is not interpreted in studies employing 2SLS.

• May one apply 2SLS to cointegrated time series?
  A time series is cointegrated if some linear combination of the series is stationary. For instance, consumption tends to be a roughly constant proportion of income over the long term, so (ln income) minus (ln consumption) approaches stationarity. Some researchers have held econometric time series analysis is preferable to structural equation modeling with 2SLS estimation if time series are cointegrated. However, Hsiao (1997) demonstrated that 2SLS estimation and Wald-type test statistics used in SEM remain good approximations even when data are cointegrated.

Bibliography

• Berry, W. D. (1984). Nonrecursive causal models. Beverly Hills, CA: Sage Publications, Ch. 5. Covers 2SLS with emphasis on the identification problem.

• Bollen, K.A (1996) 'An Alternative Two Stage Least Squares (2SLS) Estimator for Latent Variable Equations.' Psychometrika, 61: 109-121.

• de Leeuw, J. and I. G. G. Kreft (1986). Random coefficient models for multilevel analysis. Journal of Educational Statistics 11: 57-86.

• Hsiao, Cheng (1997). Statistical properties of the two-stage least squares estimator under cointegration. Review of Economic Studies 64: 385-398. Finds 2SLS estimation is robust in the face of cointegrated time series.

• Kline, Rex B. (1998). Principles and practice of structural equation modeling. NY: Guilford Press, pp.175-180. One section contains a brief, readable introduction to the subject.

• Pagan, A. (1984) 'Model Evaluation by Variable Addition.' ch 5, pp103-134 in D.F. Hendry and K.F. Wallis (eds) Econometrics and Quantitative Economics, Blackwell: Oxford.

• Pesaran, M and L. Taylor (1999) 'Diagnostics for IV regressions.' Oxford Bulletin of Economics and Statistics, 61(2), 255-281.

  Examples

• Heinrich, Carolyn J. (1998). Returns to education and training for the highly disadvantaged: What does it take to make an impact? Evaluation Review 22(5): 637-667. See especially the appendix for discussion of use of 2SLS in a public administration context to test for selection bias.

• Eoczkowski, E. and M. Farrell (1998) 'Discriminating between Measurement Scales using Non-Nested Tests and Two Stage Least Squares Estimators: The Case of Market Orientation.' International Journal of Research in Marketing, 15: 349-366.