Finite Sample Limited Information Inference Methods for Structural Equations and Models With Generated Regressors

Abstract

We propose exact tests and confidence sets for various structural models typically estimated by IV methods, such as models with unobserved regressors, which remain valid despite the presence of identification problems or weak instruments. Two approaches are considered: (1) an instrument substitution method, which generalizes the Anderson-Rubin procedure, and (2) a samplesplit method, that allows the use of "generated regressors." Projection techniques are also proposed for inference on general parameter transformations. The asymptotic theory of the tests under weaker assumptions is discussed. Simulation results are presented. The suggested techniques are applied to a model of Tobin's q and to a model of academic performance.

Further results on projection-based inference in IV regressions with weak, collinear or missing instruments

“…Finally, we think that the analytical results presented here on quadric confidence sets can be useful in other contexts involving, for example, errors-in-variables models (see Dufour and Jasiak, 2001), non-linear models, and dynamic models. Such extensions would go beyond the scope of the present paper.…”

“…In the special case where ðb 0 ; g 0 1 Þ 0 reduces to a single parameter (i.e., G ¼ 1 and k 11 ¼ 0), the set C ðb;g 1 Þ ðaÞ has a closed-form solution involving a quadratic inequality: C b ðaÞ ¼ fb 0 : ab 2 0 þ bb 0 þ cp0g, where a, b and c are simple functions of the data and thecritical value. The set C b ðaÞ can be viewed as an extension of the quadratic forms described in Dufour and Jasiak (2001) and Zivot et al (1998).…”

“…2 See, for example, Nelson and Startz (1990a, b), Buse (1992), Hall et al (1996), Dufour (1997), Staiger and Stock (1997), Wang and Zivot (1998), Zivot et al (1998), Startz et al (1999), Chao and Swanson (2000), Stock and Wright (2000), Dufour and Jasiak (2001), Hahn and Hausman (2002a, b), Kleibergen (2002Kleibergen ( , 2004Kleibergen ( , 2005, Moreira (2003a, b), Moreira and Poi (2001), Yogo (2002, 2003), , Perron (2003), Wright (2003Wright ( , 2002, Bekker and Kleibergen (2003), Hall and Peixe (2003), Forchini and Hillier (2003), Andrews et al (2004), Dufour and Taamouti (2004), and the reviews of and Dufour (2003). 3 We borrow the terminology ''robust to weak instruments'' from Stock et al (2002, p. 518).…”

“…To produce inference on transformations of model parameters, we consider the projection technique described in Dufour (1990Dufour ( , 1997, Wang and Zivot (1998), Dufour and Jasiak (2001) and Dufour and Taamouti (2005). This technique produces exact confidence sets in the sense that the probability of covering the true parameter value is at least as large as the stated level (in accordance with the standard definition of Lehmann 1986, Sections 3.1 and 3.5).…”

On the precision of Calvo parameter estimates in structural NKPC models

“…4 For comprehensive surveys on dealing with some of these issues in the presence of identification problems, see Stock, Wright, and Yogo (2002) and Dufour (2003). See also Dufour (1997), Staiger and Stock (1997), Wang and Zivot (1998), Zivot, Startz, and Nelson (1998), Dufour and Jasiak (2001), Kleibergen (2002), Kleibergen (2005), Dufour and Taamouti (2005), Dufour and Taamouti (2007), Andrews, Moreira, and Stock (2006), Hoogerheide, Kaashoek, and van Dijk (2007), Joseph and Kiviet (2005), Kiviet and Niemczyk (2007), Bolduc, Khalaf, and Moyneur (2008), Beaulieu, Dufour, and Khalaf (2008) and Chaudhuri and Zivot to weak-instruments problems have been gaining credibility in macroeconomics largely due to their focus on the NKPC. 5 The findings from the latter suggest that data may be weakly informative on key NKPC parameters, which may cast doubt on many commonly held paradigms in this literature.…”