1. INTRODUCTION
A general instrumental variable (IV) estimation method, called the generalized method of instrumental variables (GMIV), is developed in this study for generalized linear models (GLMs), which include linear, probit, logistic, Poisson, and gamma regression models as special cases. The GMIV can potentially be used to solve the estimation problems in structural equation modeling (SEM) such as error-in-variable model, factor analysis, path analysis, and simultaneous equations model, when some of the structural equations are GLMs. Our GMIV estimator substantially generalizes the generalized method of moments (GMM) estimator proposed by Windmeijer and Santos Silva (1997) for a count regression model with the canonical link. A preliminary version of this general IV estimator has been implemented by Tsai, Shau, and Hu (2006), Shau, Tsai, and Hu (2006), and Lai and Hu (2006) for generalized simultaneous equation model (GSiEM), generalized path analysis (GPA), and generalized factor analysis (GFA) respectively.
A brief review of the usual IV estimation method is given in the following section, which is essential to the understanding of the proposed new estimator.
2. Review-- Instrumental Variables (IVs)
2.1 Definition
The method of instrumental variables (IVs) has been developed and used extensively in econometrics for estimating the structural parameters in the regression model where the regressor is correlated with the error term (Bowden and Turkington, 1984). For easy computation, let the “convergence in probability”(i.e., p ) be denoted by “plim”hereinafter.
2.2 The IV Estimators for a Linear Equation:
A linear equation is specified as
ε X Y
where Y is an n × 1 vector of response variables, X is an n × p matrix of covariates, is a p × 1 vector of parameters, and ε is an n × 1 vector of errors. Suppose that 0 ε X 1 T lim p n
so that the ordinary least squares (OLS), weighted least squares (WLS), and generalized least squares (GLS) estimators of would not be consistent. However, if we can find a set of q (qp) variables to form a matrix Z such that
0 ε Z 1 T lim p n
then we can use Z as an instrument for X and minimize the following quadratic form with respect to ) ( ) ( ) ( ) ( ) | ( T T T T T 1 W Z ε W Z εYX ZWZ YX S
to obtain a consistent IV estimator
Y P X X P X w w IV T 1 T ) ( ˆ
where W = [Var(ZTε)]-1is the weight matrix and Pw= ZWZT.
Next, a nonlinear equation can be specified as
ε X
Yf( ;)
One way to generalize the IV estimators from linear equations to nonlinear ones is to replace (Y - X) by (Y - f(X; )) in the minimization of the corresponding quadratic form. Thus, given an instrument Z for X, we can minimize the following quadratic form with respect to
)) ; ( ( )) ; ( ( ) ( ) ( ) | ( T T T T T 2 W Z ε W Z ε Y f X ZWZ Y f X S
to obtain a consistent IV estimator, where W = [Var(ZTε)]-1 is the weight matrix (Bowden and Turkington, 1984, Chap. 5, pp. 156-201; Amemiya, 1985, Chap. 8, pp. 245-266; Davidson and MacKinnon, 1993, pp. 661-667). This IV estimator ˆIV does not have a closed form, and thus it may have to be solved iteratively.
2.4Applications
As summarized by Bowden and Turkington (1984, pp. 3-10), the IV estimation method has four important applications: (1) the error-in-variable problem, (2) the self-selection problem, (3) the simultaneous equations model, and (4) the time series problem. We are particularly interested in the first three.
3.
Problem: IV for GLMs?
In practice, it is very difficult, if not impossible, to correctly specify a joint multivariate distribution for the observed correlated multiple responses of different types (e.g., continuous, binary, ordinal, and count), and thus the maximum likelihood estimation (MLE) approach is generally not feasible for correlated multivariate responses of various types. In contrast, the least squares (LS) minimization approach, including the IV and GMM estimation methods, is still a promising alternative. Thus, we generalize the current IV estimation method for GLMs to allow the multivariate response variables of such structural equations to be of various types (e.g., continuous, binary, ordinal, and count).
3.1 Problem
However, when the structural equation is of a GLM form, some additional difficulties in IV estimation arise. Inevitably, the link function of GLM makes the mean of the response variable “nonlinear”in the unknown structural parameters. Moreover, the variance function of the response variable in GLM usually depends on its mean. Thus, the usual IV estimator derived by minimizing the corresponding quadratic form for a nonlinear equation (see, e.g., Amemiya 1985, p. 246, Eq. (8.1.2)) would generally be inconsistent for structural GLMs.
3.2Solution
algorithm, to obtain consistent GMIV estimates of these regression parameters.
4.
The Iterative Reweighted Instrumental Variables (IRIV)
Algorithm
Model Specification
Tsai, Shau, and Hu (2006) consider the following two-equation partially recursive GSiEM/GPA model:
1 2 3 . 2 1 2 20 2 2 1 1 1 10 1 1 1 3 1 2 1 ) ( Y X X X X g y x x x x
The consistency and asymptotically normality of the proposed IV estimator for the regression coefficient of the second equation is proved, but the second equation is linear. Thus, we consider the following two-equation partially recursive GSiEM/GPA model: 1 2 3 . 2 1 2 20 2 2 2 1 1 1 10 1 1 1 3 1 2 1 ) ( ) ( Y X X g X X g y x x x x
where (X ,1 X ,2 X ) are the covariates (or exogenous variables),3 and1 are2 the means of the responses (or endogenous variables) Y and1 Y2 respectively,
ons distributi of family l Exponentia ~ and 2 1 Y Y
and g1 and g2 are the link function for and1 respectively.2
5. SIMULATIONS
Following the formulation of Hsiao (1986, Sec. 5.4, pp. 112-125), we generate the data for the two-equation partially recursive GSiEM/GPA model, Eqs. (1) and (2), from the following two equations
(4) ) ( (3) ) ( 1 2 3 . 2 1 2 20 2 2 2 1 1 1 10 1 1 1 3 1 2 1 Y X X g X X g y x x x x
5.1 A Partially Recursive Binomial- Binomial GSiEM/GPA Model
We specify the following partially recursive Binomial-Binomial GSiEM/GPA model, as an example of Eqs. (1) and (2), in the first simulation study
1 21 2 22 20 2 1 11 10 1 ) logit( ) logit( Y X X
where the two response variables are Y1~ Binomial(1, and1) ) , 1 ( Binomial 2 2 Y respectively.
5.2 A Partially Recursive Binomial- Poisson GSiEM/GPA Model
1 21 2 22 20 2 1 11 10 1 ) log( ) logit( Y X X
where the two response variables are Y1~ Binomial(1, ) and1 Y2 Poisson(2) respectively.
6. RESULTS
In the partially recursive Binomial- Binomial and Binomial-Poisson GSiEM/GPA Model, the naive estimator of for GLM is asymptotically biased, but the IV21 estimator of is asymptotically unbiased.21
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