Reduced forms and weak instrumentation

ABSTRACT

This paper develops exact finite sample and asymptotic distributions for a class of reduced form estimators and predictors, allowing for the presence of unidentified or weakly identified structural equations. Weak instrument asymptotic theory is developed directly from finite sample results, unifying earlier findings and showing the usefulness of structural information in making predictions from reduced form systems in applications. Asymptotic results are reported for predictions from models with many weak instruments. Of particular interest is the finding that, in unidentified and weakly identified structural models, partially restricted reduced form predictors have considerably smaller forecast mean square errors than unrestricted reduced forms. These results are related to the use of shrinkage methods in system-wide reduced form estimation.

KEYWORDS:

• Endogeneity
• exact distribution
• finite sample theory
• moment existence
• partial identification
• reduced form
• structural equation
• unidentified structure
• weak instrument

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• C23
• C32

Acknowledgements

A preliminary version of this paper was presented at the Emory University Conference honoring Esfandiar Maasoumi on 15 November 2014. The author thanks Aman Ullah and three referees for helpful comments and acknowledges support of the NSF (USA) under Grant SES 12-58258 and Grant NRF-2014S1A2A2027803 from the Korean Government.

Notes

¹By apparent identification, we mean that order conditions enumerating the number and form of the restrictions appear, prima facie, to indicate identification, but without the assurance of supporting rank conditions that confirm relevance, to use the terminology of Phillips (1989).

²The density of $\tilde{b},$ first derived in Bergstrom (1962) and used later in Nelson and Startz (1990), has the form

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{f_{\tilde{b}}\left(b\right)=\sqrt{\frac{{\lambda }_n}{2\pi {\sigma }^2}}\frac{1-\beta }{{\left(1-b\right)}^2}exp\left\{-\frac{n}{2{\sigma }^2}{\left(\frac{b-\beta }{1-b}\right)}^2\right\},}\end{array}}$

where ${\lambda }_n={\sum }_{t=1}^n{z}_t^2$ is the noncentrality parameter in this system. Clearly $\underset{b\to 1}{lim}{f}_{\tilde{b}}\left(b\right)=0.$ By contrast, the density of the least squares estimator of b is positive at b = 1, as shown in Phillips (2009). The density $f_{\tilde{b}}\left(b\right)$ is also well known to be bimodal (Phillips and Wickens, 1978; Phillips and Hajivassiliou, 1987; Nelson and Startz, 1990; Phillips, 2006; Fiorio et al., 2010). Bimodality is particularly evident in the presence of strong endogeneity (as in this example) when the instruments are weak (Phillips, 2006).

³We use the notation $V{\backsim }_d{N}_{n,m+1}\left(0,I_{n\left(m+1\right)}\right)$ to signify that the matrix V is normally distributed, i.e., the n(m+1) vector $vec\left(V\right){\backsim }_{d}N\left(0,I_{n\left(m+1\right)}\right).$

⁴Strictly speaking the density (16) is proportional to a multivariate t distribution. In particular, the distribution given by the density (16) is the distribution of $t_q∕q^{1∕2}$ with q = L+1 where t_q is multivariate t with q degrees of freedom. See Phillips (1989, Theorem 2.1).

⁵A matrix variate X has a (variance matrix) mixed normal distribution if the density of X is a compound distribution  of the form $p_X\left(x\right)={\int }_{A>0}\phi \left(x,A\right)p\left(A\right)\left(dA\right)$ where φ(x,A) is the matrix normal density with covariance matrix A and the integral is taken over the matrix space A>0 with respect to the invariant measure (dA) on the cone of positive definite matrices weighted by the probability density p(A) of A. See Muirhead (1982) for further details.

⁶The result follows by examining the expansion (29), letting K→∞, and noting that ${\left(\frac{K-m}{2}\right)}_j∕{\left(\frac{K}{2}\right)}_j\to 1$ as K→∞ for all j. Alternatively, we may use the large-parameter asymptotic expansion ${}_1{F}_1\left(a,c;x\right)=e^x\left[1+O\left({\left|c\right|}^{-1}\right)\right]$ which holds when c→∞ and c−a and x are bounded, as in the present case (see Erdélyi, 1953, p. 279).

⁷Recall from Footnote 4 that the exact density (16) of β_IV in the irrelevant instrument case is the scaled multivariate t- distribution $t_q∕q^{1∕2},$ with q = K−m+1 degrees of freedom, which collapses to the origin as K→∞.

⁸For a detailed analysis of the bias properties of IV estimation, but not prediction, readers are referred to Chao and Swanson (2007), who consider various cases that allow for different expansion rates of K and n passing to infinity in IV estimation under weak instrumentation.