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A Monte Carlo analysis of principal components and ridge regression with application to a fisheries evaluation model

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  • Essential to the testing of propositions in economic theory is the estimation of the parameters involved in relationships among economic variables. Probably the most widely used method of estimation is ordinary least squares (OLS). However, severe multicollinearity or near linear dependence of the explanatory variables can cause the OLS estimates to be unrealistic and imprecise due to large sampling variances. One common solution to the multicollinearity problem is to drop highly intercorrelated variables from the regression model. Unfortunately, variable deletion may result in serious specification bias and thus may be a poor method for alleviating multicollinearity. This paper investigates the use of two methods of biased linear estimation, principal components analysis and ridge regression, as alternatives to OLS estimation of the full model in the presence of multicollinearity. A biased linear estimator may be an attractive alternative if its mean square error (MSE) compares favorably with OLS. In this paper, three ridge estimators and two types of principal components estimators are compared to OLS in a series of Monte Carlo experiments and in an application to an empirical problem. The three ridge estimators are: (1) an estimator proposed by Lawless and Wang; (2) a fixed k-value estimator (k = 0.1) ; (3) RIDGM, an estimator proposed by Dempster, Wermuth, et al. The two types of principal components estimators are: (1) the traditional t-criteria for deleting principal components; (2) a proposed loss-function-related criterion for deleting principal components. In the Monte Carlo experiments, OLS and the biased estimators are applied to four data sets, each characterized by a different level of multicollinearity and various information-to-noise ratios. The Monte Carlo results indicate that all the biased estimators can be more effective than OLS (considering MSE) in estimating the parameters of the full model under conditions of high multicollinearity at low and moderate information-to-noise ratios. (The RIDGM estimator, however, produced lower MSE than OLS at all information-to-noise ratios in the data sets where multicollinearity was present.) For principal components analysis, the proposed loss-function-related criterion produced generally lower MSE than the traditional t-criteria. For ridge regression, the Lawless-Wang estimator, which is shown to minimize estimated MSE, produces generally lower MSE than the other ridge estimators in the Monte Carlo experiments. Also, the Lawless-Wang estimator was somewhat more effective overall than the proposed loss-function-related criterion for deleting components. Another comparison of the estimators is made in their application to an empirical problem, a recreation demand model specified by the travel cost method. The comparison of the estimators is based on estimated MSE and on prior information about the coefficients. In this particular case, the Lawless-Wang estimator appears to produce the best improvement over OLS. However, this empirical problem is merely an example of the application of the biased estimators rather than a crucial test of their effectiveness. The inability to judge the reliability of biased estimates, due to the unknown bias squared component of MSE, has been a serious limitation in the application of biased linear estimation to empirical problems. Brown, however, has proposed a method for estimating the MSE of ridge coefficients. His method is applied in the empirical example and in the Monte Carlo experiments to the ridge estimators, and in principle, to the proposed loss-function-related criterion for deleting principal components. For the Lawless-Wang ridge estimator and the proposed loss-function-related criterion, the suggested method for estimating MSE appears to produce good estimates of MSE under conditions of high multicollinearity at low and moderate information-to-noise ratios. In fact, at all but the highest information-to-noise ratio, the Monte Carlo results indicate that this estimate of MSE can be much more accurate than estimates of OLS variances.
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