An Instrumental Variable Estimator for Mixed Indicators: Analytic Derivatives and Alternative Parameterizations

Abstract

Methodological development of the model-implied instrumental variable (MIIV) estimation framework has proved fruitful over the last three decades. Major milestones include Bollen’s (Psychometrika 61(1):109–121, 1996) original development of the MIIV estimator and its robustness properties for continuous endogenous variable SEMs, the extension of the MIIV estimator to ordered categorical endogenous variables (Bollen and Maydeu-Olivares in Psychometrika 72(3):309, 2007), and the introduction of a generalized method of moments estimator (Bollen et al., in Psychometrika 79(1):20–50, 2014). This paper furthers these developments by making several unique contributions not present in the prior literature: (1) we use matrix calculus to derive the analytic derivatives of the PIV estimator, (2) we extend the PIV estimator to apply to any mixture of binary, ordinal, and continuous variables, (3) we generalize the PIV model to include intercepts and means, (4) we devise a method to input known threshold values for ordinal observed variables, and (5) we enable a general parameterization that permits the estimation of means, variances, and covariances of the underlying variables to use as input into a SEM analysis with PIV. An empirical example illustrates a mixture of continuous variables and ordinal variables with fixed thresholds. We also include a simulation study to compare the performance of this novel estimator to WLSMV.

Appendix: Notation and Algebraic Results

1.1 Vec and Related Operators

Our derivations make use of the $\mathrm {vec}$ and related operators for transforming a matrix into a vector. Although these operators are commonly encountered in multivariate analysis there representations vary considerably by author. For this reason, we will provide definitions corresponding to our own usage. For a $p \times q$ matrix, $\mathbf {X}$, the $\mathrm {vec}$ operator is used to stack columnwise the q columns of $\mathbf {X}$ into a $pq \times 1$ vector without regard for any repeated or constant elements. Consider the matrix $\mathbf {A}_{p,q}$, where $\mathbf {a}_{1},\dots , \mathbf {a}_{q}$ are the columns of $\mathbf {A}_{p,q}$ taken in lexicon order then $\mathrm {vec}\,\mathbf {A}_{p,q} = [\mathbf {a}_{1}, \dots ,\mathbf {a}_{q} ]^{'}$.

The $\upsilon (\cdot )$ operator is also used heavily in these derivations. Here, $\upsilon (\cdot )$, can be understood as a generalization of the $\mathrm {vech}(\cdot )$ (vector-half) operator for symmetric matrices, or the $\mathrm {vecp}(\cdot )$ operator for strictly lower-triangular matrices, to any patterned matrix. A $p \times q$ matrix is labeled patterned if it contains $p^{*}=pq-s-v$ mathematically independent and variable elements, where s and v are the number of repeated, and constant elements, respectively. Covariance and correlation matrices are two examples of patterned matrices. For a $p \times p$ covariance matrix, $\mathbf {S}$, $p^{*} = 1/2p(p+1)$, $s = 1/2p(p+1)$, and $v=0$. Likewise, for the $p \times p$ correlation matrix, $\mathbf {P}$, $p^{*} = 1/2p(p-1)$, $s = 1/2p(p-1)$, and $v=p$. Consider the covariance matrix, $\mathbf {S}_{2,2}$, then $\upsilon (\mathbf {S})=[{s}_{1,1}, {s}_{2,1}, {s}_{2,2}]^{'}$. Similarly, for the correlation matrix, $\mathbf {P}_{2,2}$, $\upsilon (\mathbf {P})=[{s}_{2,1}]^{'}$. Generally, for any patterned matrix $\mathbf {X}_{p,q}$, $\upsilon (\mathbf {X})$ will be a $p^{*} \times 1$ vector.

1.2 The Kronecker Product

For the matrices $\mathbf {X}_{p,q}$ and $\mathbf {A}_{r,s}$, we define the Kronecker product as $\mathbf {X} \otimes \mathbf {B} = (x_{i,j}\mathbf {A})_{pr \times qs}$, for $i =1,\dots ,p$ and $j =1,\dots ,q$. A useful result linking the Kronecker product to the $\mathrm {vec}$ operator states $\mathrm {vec}(\mathbf {A}_{m,n}\mathbf {B}_{n,q}\mathbf {C}_{q,r}) = (\mathbf {C}^{'} \otimes \mathbf {A})\mathrm {vec}\mathbf {B}$. Other useful properties of the Kronecker product utilized in these derivations for simplifying resultant expressions are $(\mathbf {A} \otimes \mathbf {B}) (\mathbf {C} \otimes \mathbf {D}) = \mathbf {AC} \otimes \mathbf {BD}$ and $(\mathbf {A} \otimes \mathbf {B})^{'} = \mathbf {A}^{'} \otimes \mathbf {B}^{'}$.

1.3 The Commutation Matrix

Commutation (or vec-permutation) matrices can be used to translate between vectors $\mathrm {vec}\,\mathbf {X}$ and $\mathrm {vec}\,\mathbf {X}^{'}$. The vec-permutation operator, $\mathbf {K}_{p,q}$, is defined such that $\mathrm {vec}\,\mathbf {X}_{p,q} = \mathbf {K}_{p,q} \mathrm {vec}\,\mathbf {X}^{'}$. The commutation matrix plays a central role in the formulation of matrix derivatives using the vec operator, and the following derivative is used throughout. Consider the $m \times n$ matrix $\mathbf {X}$, then $\partial \mathrm {vec}(\mathbf {X}^{'}) / \partial \mathrm {vec}(\mathbf {X})^{'} = \mathbf {K}_{m,n}$. Note also that $\partial \mathrm {vec}(\mathbf {X}) / \partial \mathrm {vec}(\mathbf {X})^{'} = \mathbf {I}_{mn}$.

1.4 Matrix Derivatives and L-Structured Matrices

The results herein require taking partial derivatives with respect to lower-triangular, strictly lower-triangular, diagonal and arbitrarily patterned matrices. Furthermore, the solution matrices resulting from these matrix derivatives are themselves often known a priori to be symmetric or patterned. For these reasons, we rely on a number of results detailed by Magnus (1983) and Magnus and Neudecker (1986) for L-structured matrices. The use of L-structures allows us to derive our results in the most general way possible across the different parameterizations available. The following properties of L-Structured matrices are used throughout, $\mathrm {vec}(\mathbf {X})=\varvec{\Delta }\upsilon (\mathbf {X})$ and $\upsilon (\mathbf {X})=\varvec{\Delta }^{+}\mathrm {vec}(\mathbf {X})$.

It is useful to consider $\varvec{\Delta }$ as a generalized duplication matrix, and $\varvec{\Delta }^{+}$ as a generalized elimination matrix. If $\mathbf {X}$ is a symmetric $\varvec{\Delta }$ is $p^2 \times p(p+1)/2$, while $\varvec{\Delta }$ is $p^2 \times p(p-1)/2$ if $\mathbf {X}$ is strictly lower-triangular. The most interesting case occurs when $\mathbf {X}$ exhibits an arbitrary constellation of free, fixed, and repeating elements. In this case, a general method is needed for constructing $\varvec{\Delta }$ and $\varvec{\Delta }^{+}$ when the specific patterning of $\mathbf {X}$ is unknown (prior to the analysis). Fortunately, a result for this specific case was derived by Nel (1980, Definition 6.1.1). In the case of arbitrary patterning, $\varvec{\Delta }$ is $p^2 \times p^{*}$.

Extending these properties to the case of matrix derivatives, it can be shown that $\frac{\partial \mathrm {vec} (\mathbf {X})}{\partial \, \upsilon (\mathbf {X})^{'}} = \varvec{\Delta }$ and $\frac{\partial \,\upsilon (\mathbf {X})}{\partial \, \mathrm {vec} (\mathbf {X})^{'}} = \varvec{\Delta }^{+}$. It follows that if the matrix function $\mathbf {Z} = f(\mathbf {X})$,

$$\begin{aligned} \frac{\partial \mathrm {vec} (\mathbf {Z})}{\partial \upsilon (\mathbf {X})^{'}}= & {} \frac{\partial \mathrm {vec} (\mathbf {Z})}{\partial \mathrm {vec}(\mathbf {X})^{'}} \frac{\partial \mathrm {vec} (\mathbf {X})}{\partial \upsilon (\mathbf {X})^{'}} = \frac{\partial \mathrm {vec} (\mathbf {Z})}{\partial \mathrm {vec}(\mathbf {X})^{'}} \varvec{\Delta }, \end{aligned}$$

(38)

and if $\mathbf {Z}$ is also patterned,

$$\begin{aligned} \frac{\partial \upsilon (\mathbf {Z})}{\partial \, \upsilon (\mathbf {X})^{'}}= & {} \frac{\partial \upsilon (\mathbf {Z})}{\partial \mathrm {vec}(\mathbf {Z})^{'}} \frac{\partial \mathrm {vec} (\mathbf {Z})}{\partial \mathrm {vec}(\mathbf {X})^{'}} \frac{\partial \mathrm {vec} (\mathbf {X})}{\partial \upsilon (\mathbf {X})^{'}} = \varvec{\Delta }^{+} \frac{\partial \mathrm {vec} (\mathbf {Z})}{\partial \mathrm {vec}(\mathbf {X})^{'}} \varvec{\Delta }. \end{aligned}$$

(39)

1.5 Derivatives of Common Matrix Functions

The following derivatives are used throughout and will be restated here for clarity. For the following results, suppose $\mathbf {X}$ is $m \times n$, $\mathbf {U}$ is $p \times q$, and $\mathbf {V}$ is $q \times r$, where both $\mathbf {U}$ and $\mathbf {V}$ are matrix functions of $\mathbf {X}$, then

$$\begin{aligned} \frac{\partial \,\mathrm {vec}\, (\mathbf {U+V})}{\partial \, \mathrm {vec}\, (\mathbf {X})^{'}}= & {} \frac{\partial \,\mathrm {vec}\, (\mathbf {U})}{\partial \, \mathrm {vec}\, (\mathbf {X})^{'}} + \frac{\partial \,\mathrm {vec}\, (\mathbf {V})}{\partial \, \mathrm {vec}\, (\mathbf {X})^{'}}, \end{aligned}$$

(40)

and

$$\begin{aligned} \frac{\partial \,\mathrm {vec}\, (\mathbf {UV})}{\partial \, \mathrm {vec}\, (\mathbf {X})^{'}}= & {} (\mathbf {V} \otimes \mathbf {I}_{p})^{'} \frac{\partial \,\mathrm {vec}\, (\mathbf {U})}{\partial \, \mathrm {vec}\, (\mathbf {X})^{'}} + (\mathbf {I}_{r} \otimes \mathbf {U} ) \frac{\partial \,\mathrm {vec}\, (\mathbf {V})}{\partial \, \mathrm {vec}\, (\mathbf {X})^{'}}. \end{aligned}$$

(41)

In addition, we state a general rule for taking derivatives of matrix inverses, specifically if $\mathbf {Y} = \mathbf {X}^{-1}$, then

$$\begin{aligned} \frac{\partial \,\mathrm {vec} (\mathbf {Y})}{\partial \, \mathrm {vec} (\mathbf {X})^{'}}= & {} -(\mathbf {X}^{-1'} \otimes \mathbf {X}^{-1}). \end{aligned}$$

(42)

Now suppose $\mathbf {X}$ is a symmetric matrix, $\mathbf {A}$ is a matrix of constants, and $\mathbf {Y} = \mathbf {X}\mathbf {A}\mathbf {X}$ using successive applications of (41) we can show that,

$$\begin{aligned} \frac{\partial \,\mathrm {vec}(\mathbf {Y})}{\partial \, \mathrm {vec}(\mathbf {X})^{'}}= & {} (\mathbf {AX} \otimes \mathbf {I})^{'} + (\mathbf {I} \otimes \mathbf {XA} ). \end{aligned}$$

(43)

Suppose instead that $\mathbf {A}$ and $\mathbf {B}$ are matrices containing constant elements and $\mathbf {Y} = \mathbf {A}^{'}\mathbf {X}^{-1}\mathbf {B}$, then

$$\begin{aligned} \frac{\partial \,\mathrm {vec}(\mathbf {Y})}{\partial \, \mathrm {vec}(\mathbf {X})^{'}}= & {} -(\mathbf {B}^{'}\mathbf {X}^{-1'} \otimes \mathbf {A}^{'}\mathbf {X}^{-1}). \end{aligned}$$

(44)