# Robust Minimum Distance Inference in Structural Models ††thanks: Research
funded by Ministerio de Ciencia e Innovación, grant ECO2017-86675-P,
MCI/AEI/FEDER/UE, grant PGC 2018-096732-B-100, grant PID2021-127794NB-I00 and
Comunidad de Madrid, grants EPUC3M11 (VPRICIT) and H2019/HUM-589.

Joan Alegre
Universidad Carlos III de Madrid Juan Carlos Escanciano
Universidad Carlos III de Madrid

###### Abstract

This paper proposes minimum distance inference for a structural parameter of
interest, which is robust to the lack of identification of other structural
nuisance parameters. Some choices of the weighting matrix lead to asymptotic
chi-squared distributions with degrees of freedom that can be consistently
estimated from the data, even under partial identification. In any case,
knowledge of the level of under-identification is not required. We study the
power of our robust test. Several examples show the wide applicability of the
procedure and a Monte Carlo investigates its finite sample performance. Our
identification-robust inference method can be applied to make inferences on
both calibrated (fixed) parameters and any other structural parameter of
interest. We illustrate the method’s usefulness by applying it to a structural
model on the non-neutrality of monetary policy, as in Nakamura and Steinsson
(2018), where we empirically evaluate the validity of the calibrated
parameters and we carry out robust inference on the slope of the Phillips
curve and the information effect.

Keywords:

Minimum Distance Estimation; Partial Identification; Robust inference.

_JEL classification:_ C10, C12, C16

##  1 Introduction

Minimum distance inference is commonly used in structural econometric models,
with applications varying from dynamic and static games of imperfect
information (see, e.g., Pesendorfer and Schmidt-Dengler (2008)), dynamic
stochastic general equilibrium (DSGE) models (see, e.g., Rotemberg and
Woodford (1997), Amato and Laubach (2003), and Christiano et al. (2005)), to
any economic model estimated by simulated-based methods such as Simulated
Method of Moments (SMM) or Indirect Inference (II) (see, e.g., Nakamura and
Steinsson (2018)). Two key assumptions in the literature are that structural
parameters are point-identified and the corresponding Jacobian matrix is of
full column rank. In this paper, we relax these two conditions and propose
inference for a structural parameter of interest $\beta$ that is robust to
lack of under-identification of other nuisance structural parameter $\alpha$.
Moreover, it is not necessary to have prior knowledge of the degree of
identification of the vector $\alpha$. There is substantial theoretical and
empirical evidence that structural parameters in econometric models are
generally not identified, see the references below. However, the level of
underidentification is often unknown to the researcher. We propose an
inference method that is robust to such knowledge and also to singularities in
the optimal weighting matrix. Therefore, the proposed method provides a simple
robust alternative to the standard econometric practice. Notably, our approach
does not require the identification of the $\beta$ parameter either, as the
restricted model is used to compute the test statistic following a Lagrange
multiplier or score-type approach. As a result, this testing procedure is
particularly well-suited to make inferences on calibrated (fixed) parameters
in structural macroeconomic and microeconomic applications. We refer to this
application of our method as testing the validity of calibrated parameters.

In structural models estimated by minimum distance methods the degree of
identification of the structural parameters is often unknown. This is so
because of a lack of an analytic solution for the Jacobian (see, e.g.,
McFadden (1989) on SMM or Gourieroux et al. (1993) on II), or because the rank
of the Jacobian depends on an unknown population parameter (see, e.g,
Pesendorfer and Schmidt-Dengler (2008) on Dynamic Games, or Kalouptsidi et al.
(2021) on counterfactuals in Dynamic Discrete Choice models). In this paper,
we present an inference method for a parameter of interest that is robust to
the lack of identification of the nuisance parameter, both under the null and
under the alternative. Importantly, our method does not require any prior
knowledge about the unknown degree of identification of the model, which is
unknown but identified from data. Additionally, our proposed method has the
secondary benefit of providing an estimation of the degree of identification
of the model.

We describe our setting. Let $\theta\in\Theta\subset\mathbb{R}^{m}$ be a
reduced-form parameter which is identified from the data distribution $F_{0}$
as $\theta_{0}$. Let $\alpha\in\mathcal{A}\subset\mathbb{R}^{q}$ and
$\beta\in\mathcal{B}\subset\mathbb{R}^{p}$ be structural parameters.
Structural parameters are related to reduced-form parameters through the
mapping $g:\Theta\times\mathcal{A}\times\mathcal{B}\xrightarrow{}\Theta$ such
that

$\theta_{0}=g(\theta_{0},\alpha,\beta).$ (1)

This paper proposes a test for the hypothesis

$H_{0}:\beta=\beta_{0}\qquad vs\qquad H_{1}:\beta\neq\beta_{0},$ (2)

where $\beta_{0}$ is a known fixed value, and we allow for the nuisance
structural parameter $\alpha$ to be partially identified under the null, i.e.,
the set

$\mathcal{A}_{0}:=\left\\{\alpha\in\mathcal{A}\subset\mathbb{R}^{q}:\theta_{0}=g(\theta_{0},\alpha,\beta_{0})\right\\}$

is not necessarily a singleton. We assume that when the null hypothesis is
true there exists an $\alpha_{0}\in\mathcal{A}_{0}$, i.e, $\mathcal{A}_{0}$ is
non-empty. We also assume there exists an asymptotically normal estimator for
$\theta_{0},$ say $\widehat{\theta},$ satisfying

$\sqrt{n}\left(\widehat{\theta}-\theta_{0}\right)\rightarrow_{d}N\left(0,\Sigma\right),$

where $\Sigma$ might be a singular matrix. Consider a standard minimum
distance (MD) test statistic

$\widehat{F}(\beta_{0}):=\min_{\alpha\in\mathcal{A}}n\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)$
(3)

where $\widehat{W}$ is a consistent estimator for a positive semi-definite
symmetry non-stochastic matrix $W.$ Assume $g(\theta,\alpha,\beta_{0})$ is
twice continuously differentiable at $\theta_{0}\in\Theta\ $and
$\alpha_{0}\in\mathcal{A}_{0},$ and define the matrices
$\nabla_{\theta}g_{0}:=\partial
g(\theta_{0},\alpha_{0},\beta_{0})/\partial\theta^{\prime}$ and
$\nabla_{\alpha}g_{0}:=\partial
g(\theta_{0},\alpha_{0},\beta_{0})/\partial\alpha^{\prime}.$ Let $r_{\alpha}$
and $r_{\Sigma}$ denote the ranks of $\nabla_{\alpha}g_{0}$ and $\Sigma$,
respectively. We assume the rank of the variance and covariance matrix is
larger than the rank of the Jacobian, i.e., $r_{\Sigma}>r_{\alpha}$. Under the
regularity conditions below, $r_{\alpha}$ does not depend on the point
$\alpha_{0}\in\mathcal{A}_{0}$ where it is evaluated. Let $I_{m}$ be the
$m$-dimensional identity matrix. The main result of the paper shows that under
regularity conditions and $H_{0},$ if
$W=\left[(I_{m}-\nabla_{\theta}g_{0})\Sigma(I_{m}-\nabla_{\theta}g_{0})^{\prime}\right]^{\dagger}$,
where $A^{\dagger}$ denotes the Moore-Penrose generalized inverse of the
matrix $A$, then

$\widehat{F}(\beta_{0})\rightarrow_{d}\chi_{r_{\Sigma}-r_{\alpha}}^{2},$ (4)

where $\chi_{d}^{2}$ denotes a chi-squared distribution with $d$ degrees of
freedom. This result includes the case where $\alpha_{0}$ is point identified
and $r_{\alpha}=q$ (see Gourieroux and Monfort (1995) ). The more general case
discussed here allows for lack of identification of $\alpha_{0}$ and
$r_{\alpha}<q.$ The convergence in (4) suggests a feasible $\tau\%$ nominal
test with critical region
$\widehat{F}(\beta_{0})>\chi_{\widehat{r}_{\Sigma}-\widehat{r}_{\alpha},1-\tau}^{2},$
where $\widehat{r}_{\Sigma}-\widehat{r}_{\alpha}$ is a consistent estimator of
$r_{\Sigma}-r_{\alpha}$ and $\chi_{d,\tau}^{2}$ denotes the $\tau-$quantile of
$\chi_{d}^{2}.$ The proposed method applies general results on the second-
order differentiability properties of the mapping

$v(\theta):=\min_{\alpha\in\mathcal{A}}\left(\theta-g(\theta,\alpha,\beta_{0})\right)^{\prime}W\left(\theta-g(\theta,\alpha,\beta_{0})\right),$

at $\theta=\theta_{0},$ see Shapiro (1986). When $\alpha_{0}$ is a regular
point (see definition below) and other regularity conditions hold, $v(\theta)$
is twice continuously differentiable and it can be locally approximated by the
projection norm onto a tangent linear space of dimension $r_{\alpha}.$ That
is, the $q-$dimensional manifold

$\Theta_{0}:=\left\\{\theta\in\Theta\subset\mathbb{R}^{m}:\theta=g(\theta,\alpha,\beta_{0}),\hskip
7.11317pt\textit{for some}\hskip
7.11317pt\alpha\in\mathcal{A}\subset\mathbb{R}^{q}\right\\}$

has a tangent space of dimension $r_{\alpha}$ at $\theta_{0},$ which is
generated by the columns of $\nabla_{\alpha}g_{0}.$ This result does not
require identification of $\alpha_{0}$ or full column rank of
$\nabla_{\alpha}g_{0}.$ The asymptotic distribution of
$\widehat{F}(\beta_{0})$ then follows from a standard application of the
Delta-Method, after noticing that
$\widehat{F}(\beta_{0})=v(\widehat{\theta})+o_{P}(1)$.

One important novelty of our method is that it does not require any prior
knowledge about the degree of identification of the model, i.e, $r_{\alpha}$,
as long as $r_{\Sigma}>r_{\alpha}$. Such lack of knowledge about the degree of
identification arises whenever the Jacobian matrix depends on the population
parameter or there is no analytic solution for it. In SMM, for instance, no
analytical solutions of the Jacobian are available, though it can be
approximated. In structural models such as Dynamic Games or Dynamic Discrete
Choice models, a closed-form solution for $r_{\alpha}$ is available, but it
does depend on the population value of $\theta_{0}$, which is unknown. We
estimate the rank of $\Sigma$ and $\nabla_{\alpha}g_{0}$ using a hard
thresholding approach.

A prototypical application of our results is for testing the validity of
calibrated parameters in structural models. This refers to testing for
$H_{0}:\beta=\beta_{0}$ for calibrated values $\beta_{0}$ of a structural
parameter. “Validity” corresponds to a lack of rejection by our robust test.
Another application of our results could be for the construction of robust
confidence intervals for a structural parameter of interest by inverting our
robust test. We evaluate the finite sample performance of our tests (size and
power) in the context of two sets of Monte Carlo experiments, one for a static
Bayesian entry game and another for the New-Keynesian model in Nakamura and
Steinsson (2018). The Monte Carlo results confirm the robustness of our
method, relative to traditional non-robust tests based on asymptotic normality
under point-identification (t-tests), and show that estimation of the level of
under-identification does not have an impact on the performance of the robust
test.

We apply our robust method to study both the validity of the calibrated
parameters and the inference on the slope of the Phillips curve and the
information effect in a simplified version of Nakamura and Steinsson (2018).
We find their calibration to be valid according to our criteria, and obtain
robust confidence intervals for the slope of the Phillips curve and the
information effect, which are not only shorter than those obtained by the
classical t-test but also shorter than those suggested by the nonparametric
bootstrap. The identification-robust inference seems to be well motivated in
this application given that the Jacobian for nuisance structural parameters is
of reduced rank and the near singularity of the asymptotic variance matrix of
reduced form parameters.

The paper is organized as follows: after this introduction and a short
literature review, Section 3 discusses several examples of structural
econometric models where our results apply. Section 4 provides the asymptotic
null distribution and power theory for the proposed test. Section 5 discusses
the implementation. Section 6 investigates the finite sample performance of
the proposed methods, while Section 7 illustrates our method in the
application of Nakamura and Steinsson (2018). Finally, Section 8 concludes.
The main proofs of Section 4 are gathered into Appendix A.

##  2 Literature Review

This paper proposes an identification-robust inference method for parameters
of interest allowing for nuisance parameters to be unidentified both under the
null and under the alternative hypothesis, without prior knowledge of the
model’s identification degree.

Our problem can be placed in the literature of identification-robust inference
of a parameter of interest when the nuisance parameters are not identified
under the null. Some examples of this literature are Davies (1977), Engle
(1984), Andrews and Ploberger (1994), Hansen (1996), Stinchcombe and White
(1998), Conniffe (2001). Such literature indexes the test statistic by the
nuisance parameter, and then, finds the supremum of all possible test
statistics. The distribution of the supremum test statistic is not standard,
and they use Monte Carlo or resampling methods to find critical values for the
test. Our method leads to critical values that are known up to the degrees of
freedom, which can be estimated from data without resampling. Our environment
allows the model to be partially identified. Inference on partially identified
models has been treated by Horowitz and Manski (1998), Horowitz and Manski
(2000), Chernozhukov et al. (2002), Imbens and Manski (2004), Romano and
Shaikh (2008), Beresteanu and Molinari (2008), Chernozhukov et al. (2007),
Bugni (2010), Bugni (2016), and many others. As a common trait, such
literature has focused both on constructing confidence intervals that contain
the whole identified set and developing confidence intervals for the true
value. Most of the methods rely on subsampling or bootstrap to carry out
inference.

This paper also relates to the literature on testing with a singular
information matrix, such as Satorra (1989), Satorra (1992), Andrews (1987),
Ravikumar et al. (2000) or Dufour and Valéry (2016), among many others. Our
paper mainly differs from this literature in the assumption of point
identification of the nuisance parameter.

Inference under weak identification is also related to our problem. Some
papers in this literature are Stock and Wright (2000); Andrews and Cheng
(2012, 2013); Andrews and Mikusheva (2015, 2016a, 2016b); Han and McCloskey
(2019); Andrews and Guggenberger (2019); or Lee (2022). Kleibergen (2005)
assumes the full column rank of the Jacobian for the nuisance parameter.
Andrews and Mikusheva (2016b) gives finite sample bounds on the distribution
of minimum statistics used in minimum distance models. They assume the full
column rank of the Jacobian. Moreover, in their framework the model $g(\cdot)$
does not depend on the reduced-form parameter $\theta_{0}$. Andrews and
Guggenberger (2019) develop a singular weighting matrix robust test. They also
assume identification of the nuisance parameter. Andrews and Cheng (2012)
develops an estimation and inference method that is robust to lack of
identification (and thus weak identification) of nuisance parameters. There
are two main differences with our paper. First, to use their method, knowledge
about which parameters can be identified is required, while we are agnostic
about the identification problem. Second, the objective function does not
depend on the nuisance parameter when it is not identified, while we allow the
objective function to depend on the nuisance parameter even if it is not
identified. Han and McCloskey (2019) develops estimation and inference in a
GMM framework, with a Jacobian that is not full column rank. Han and McCloskey
(2019) works on the Andrews and Cheng (2012) environment, and therefore, prior
knowledge of the identification problem is required. Moreover, the objective
function cannot depend on unidentified nuisance parameters.

Antoine et al. (2022) develop an identification-robust inference of a
structural parameter of interest using a minimum distance estimation method.
The main difference with our paper is that we allow for under-identification
of the nuisance parameter. Antoine et al. (2022) also relies on bootstrap
methods to find the critical values of the test. We avoid using bootstrap,
making our procedure computationally less expensive for a large variety of
complex structural models.

##  3 Examples

###  3.1 Estimation of Games

The proposed method is motivated by the application of dynamic games by
Pesendorfer and Schmidt-Dengler (2008). These authors suggested the MD
estimator

$\widehat{\alpha}:=\arg\min_{\alpha\in\mathcal{A}}n\left(\widehat{\theta}-g(\widehat{\theta},\alpha)\right)^{\prime}\widehat{W}\left(\widehat{\theta}-g(\widehat{\theta},\alpha)\right),$
(5)

where $\widehat{\theta}$ is a consistent estimator for the ex-ante choice
probabilities $\theta_{0}$ and $g$ is the best response mapping, linking the
structural parameter $\alpha_{0}$ to the choice probability $\theta_{0}$. In
equilibrium $\theta_{0}=g(\theta_{0},\alpha_{0}).$ Much of the literature
assumes $\alpha_{0}$ is uniquely identified from the equilibrium conditions.
However, there is theoretical evidence that these models are not identified.
Non-identification results have been shown in the seminal paper by Rust (1994)
and Magnac and Thesmar (2002) for single-agent models and by Pesendorfer and
Schmidt-Dengler (2008) for multiple-agent models. Here, we present a
methodology that can be applied to partially identified models, and therefore,
has wider applicability than existing procedures. Pesendorfer and Schmidt-
Dengler (2008) show how many popular estimators fall under the class (5).
Specifically, they show that the moment estimator of Hotz and Miller (1993) or
the pseudo-maximum likelihood estimator of Aguirregabiria and Mira (2002) are
asymptotically equivalent to estimators in the class (5) for specific choices
of the plim of $\hat{W}$, $W.$ Here, we allow these choices as special cases,
while permitting the structural parameters to be under-identified.

###  3.2 Estimation of Dynamic Stochastic General Equilibrium Models

In this application $\theta_{0}$ is a vector of impulse response functions
obtained from a Structural Vector Autoregression model, and $g(\cdot)$ is the
link between structural parameters and reduced-form parameters. MD estimation
in this context has been proposed by Rotemberg and Woodford (1997), Amato and
Laubach (2003), and Christiano et al. (2005) , among many others. In this
application, $g$ does not depend on $\theta_{0}$.

###  3.3 Estimation using Simulated Based Methods

Simulated-based methods such as SMM and II also fit our environment, after an
appropriate modification that we give below in Section 3.3.1. Moreover, due to
the nature of their problem, the degree of identification of the model is
unknown. Furthermore, simulated-based methods are computationally demanding,
making bootstrap and subsampling inference not feasible computationally.
Therefore, our method addresses two main problems in this literature. It is
agnostic about the degree of identification, while it is also computationally
feasible.

####  3.3.1 Estimation with Simulated Method of Moments

This method is introduced in the seminal papers by McFadden (1989), Pakes and
Pollard (1989), Duffie and Singleton (1993), and Lee and Ingram (1991).

Assume we observe a sample of random variables $X_{i}$ of size $n$, where
$X_{i}$ follows a distribution $F(\cdot)$. Moreover, suppose we have a
structural economic model, indexed by parameters $(\alpha,\beta)$ that
generates a simulated random variable $\tilde{X}_{j}(\alpha,\beta)\sim
F_{\alpha,\beta}$. For a given measurable function:

$h:\mathcal{X}\xrightarrow{}\Theta,$ (6)

where $\mathcal{X}$ is the domain of $X_{i}$, let

$\theta_{0}=\int h(x_{i})dF(x_{i}).$ (7)

The reduced-form parameter $\theta_{0}$ is estimated using the sample analog
in (7). In SMM, the moment $g(\cdot)$ is defined by

$g(\alpha,\beta):=\int h(\tilde{x}_{i})dF_{\alpha,\beta}(\tilde{x}_{i}),$ (8)

which is not analytically known. We use Monte Carlo, and compute

$\hat{g}(\alpha,\beta)\equiv\frac{1}{B}\sum_{j=1}^{B}h(\tilde{x}_{j}(\alpha,\beta)),$
(9)

where $\\{x_{j}(\alpha,\beta)\\}$ are $B$ random draws from
$F_{\alpha,\beta}$. The SMM minimizes a quadratic objective function that
depends on the distance between the estimated reduced-form parameter (7) and
the simulated moment condition (9), i.e.,

$\tilde{F}(\beta_{0}):=\min_{\alpha\in\mathcal{A}}n\left(\widehat{\theta}-\hat{g}(\alpha,\beta_{0})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-\hat{g}(\alpha,\beta_{0})\right),$
(10)

where $\widehat{W}$ will be the estimated optimal weighting matrix of this
minimum distance problem. By choosing $B$ significantly large, one shows that
the numerical approximation of (9) has no impact on the asymptotic
distribution of (10),

$\hat{F}(\beta_{0})=\tilde{F}(\beta_{0})+o_{P}(1),$ (11)

and hence, SMM fits our framework with a $g(\cdot)$ given by (8) that does not
depend on $\theta$.

##  4 Minimum Distance Inference

This section investigates the asymptotic null distribution of
$\widehat{F}(\beta_{0}).$ The first condition requires the asymptotic
normality of the reduced-form estimator. The test statistic was defined in
equation (3) as

$\widehat{F}(\beta_{0}):=\min_{\alpha\in\mathcal{A}}n\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right).$

###### Assumption 1

The parameter $\theta_{0}$ belongs to the interior of
$\Theta\subset\mathbb{R}^{m}.$ Under $H_{0},$
$\sqrt{n}\left(\widehat{\theta}-\theta_{0}\right)\rightarrow_{d}N\left(0,\Sigma\right),$
where $\Sigma$ might be singular.

###### Assumption 2

The function $g(\theta,\alpha,\beta_{0})$ is twice continuously differentiable
at $\theta_{0}\in\Theta\ $and $\alpha_{0}\in\mathcal{A}$.

###### Assumption 3

The model is correctly specified,

$\theta_{0}=g(\theta_{0},\alpha,\beta)\hskip 8.5359pt\textit{for some}\hskip
2.84544pt\beta\in\mathcal{B}\hskip 2.84544pt\textit{and}\hskip
2.84544pt\alpha\in\mathcal{A},$ (12)

These assumptions are standard in the literature on MD inference. We say the
point $\alpha_{0}\in\mathcal{A}$ is locally regular if belongs to the interior
of $\mathcal{A}$ and the Jacobian matrix $\nabla_{\alpha}g(\alpha):=\partial
g(\theta_{0},\alpha,\beta_{0})/\partial\alpha^{\prime}$ has the same rank as
$\nabla_{\alpha}g_{0}\equiv\nabla_{\alpha}g(\alpha_{0})$, say $r_{\alpha},$
for every $\alpha$ in a neighborhood of $\alpha_{0}.$ We say the point
$\alpha_{0}\in\mathcal{A}$ is regular if it is locally regular and there exist
neighborhoods $\mathcal{U}$ and $\mathcal{V}$ of $\alpha_{0}$ and
$\theta_{0},$ respectively, such that
$\Theta_{0}\cap\mathcal{V}=g(\theta_{0},\mathcal{U},\beta_{0}),$ where
$\Theta_{0}:=\left\\{\theta\in\Theta\subset\mathbb{R}^{m}:\theta=g(\theta,\alpha,\beta_{0}),\hskip
7.11317pt\textit{for some}\hskip
7.11317pt\alpha\in\mathcal{A}\subset\mathbb{R}^{q}\right\\}$.

###### Assumption 4

(i) $\mathcal{A}$ is a compact subspace of $\mathbb{R}^{q}$; (ii) the
identified set $\mathcal{A}_{0}$ is connected; and (iii) all elements of
$\mathcal{A}_{0}$ are locally regular.

###### Assumption 5

$\widehat{W}=W+o_{P}(1)$, with $W$ a positive semi-definite and symmetric non-
stochastic matrix.

###### Assumption 6

$W$ is positive definite over the space spanned by the columns of
$\nabla_{\alpha}g_{0}$, i.e.,

$Rank(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0})=Rank(\nabla_{\alpha}g_{0}).$
(13)

###### Assumption 7

The matrix $I_{m}-\nabla_{\theta}g_{0}$ is non-singular.

Assumption 4$(i)$ is standard and it can be relaxed at the cost of introducing
further assumptions on $g.$ A subset $S$ of a topological space $\mathcal{X}$
is said to be connected whenever $S$ cannot be expressed as the disjoint union
of two non-empty open subsets of $\mathcal{X}$. When Conditions 4$(i)$,
4$(ii)$ and 4$(iii)$ do not hold, our test controls the size, although it
might become conservative, see Remark 4.2. Assumptions 5 and 6 are standard in
the literature. Assumption 7 is a local identification condition for reduced
form parameters.

###### Theorem 4.1

Let Assumptions 1 – 7 hold, and set
$W=\left((I_{m}-\nabla_{\theta}g_{0})\Sigma(I_{m}-\nabla_{\theta}g_{0})^{\prime}\right)^{\dagger}$.
Then,

$\widehat{F}(\beta_{0})\xrightarrow{d}\chi_{r_{\Sigma}-r_{\alpha}}^{2}$ (14)

###### Remark 4.1

If the gradient $\nabla_{\alpha}g$ is not available in closed form, one can
use finite-difference approximations, as justified in e.g. Hong et al. (2015).
For simplicity, we assume that a closed form of the derivatives is available.

###### Remark 4.2

If Assumptions 1 – 3, Assumption 5 – 7 hold, and
$W=\left(I_{m}-\nabla_{\theta}g_{0})\Sigma(I_{m}-\nabla_{\theta}g_{0})^{\prime}\right)^{\dagger}$,
then

$\limsup_{n\xrightarrow{}\infty}\Pr\\{\widehat{F}(\beta_{0})\geq c\\}\leq
Pr\\{\chi^{2}_{r_{\Sigma}-r_{\alpha}}\geq c\\}.$ (15)

See Theorem $3.3$ in Shapiro (1986). This result gives a conservative test for
cases where Assumption 4 does not hold. Violations of Assumption 4$(ii)$ might
occur, for example, when $\mathcal{A}_{0}$ is a discrete set. Violations of
Assumption 4$(iii)$ might occur whenever the model is reparametrized to avoid
boundary solutions. For example, let $g(\alpha)=\alpha$ and
$\mathcal{A}=\\{\alpha\in\mathbb{R}\hskip 2.84544pt|\hskip 2.84544pt\alpha\geq
0\\}$, we might reparametrize the model using the mapping
$\alpha\xrightarrow{}\alpha^{2}$. If $\alpha_{0}=0$, $\alpha_{0}$ is not a
regular point of $\mathcal{A}_{0}$.

If
$W=\left((I_{m}-\nabla_{\theta}g_{0})\Sigma(I_{m}-\nabla_{\theta}g_{0})^{\prime}\right)^{\dagger}$,
then the limiting distribution depends on the data-generating process only
through the rank of the matrix $\nabla_{\alpha}g_{0}.$ In this case, it is
natural to consider the sample analog

$\widehat{r}_{\alpha}=rank\left[\nabla_{\alpha}g(\widehat{\theta},\widehat{\alpha},\beta_{0})\right],$

where $\widehat{\alpha}$ solves the optimization problem

$\widehat{\alpha}=\arg\min_{\alpha\in\mathcal{A}}n\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)+\lambda_{n}\left|\alpha\right|^{2},$
(16)

and $\lambda_{n}$ is a penalization parameter such that $\lambda_{n}\downarrow
0$.

Furthermore, to satisfy Assumption 5 whenever
$W=\left((I_{m}-\nabla_{\theta}g_{0})\Sigma(I_{m}-\nabla_{\theta}g_{0})^{\prime}\right)^{\dagger}$,
an estimator based on the analog principle of $W$ might be inconsistent. This
is because the Moore-Penrose g-inverse is not continuous. Stewart (1969) gives
necessary and sufficient conditions to estimate consistently a Moore-Penrose
g-inverse. A detailed discussion on how to construct a consistent estimator of
$W$ is provided in Section 5.2.

###### Assumption 8

(i) $\widehat{\Sigma}=\Sigma+o_{P}(1);$ (ii)
$\alpha_{0}=\arg\min\left\\{\left|\alpha\right|:\alpha\in\mathcal{A}_{0}\right\\}$
is unique ; and (iii) $\lambda_{n}\rightarrow 0$ as $n\rightarrow\infty.$

Henceforth, to save notation we denote $d\equiv r_{\Sigma}-r_{\alpha}$ and
$\widehat{d}\equiv\widehat{r}_{\Sigma}-\widehat{r}_{\alpha}$. Consider the
oracle test
$\tilde{\phi}_{\tau}=1(\widehat{F}(\beta_{0})>\chi_{d,1-\tau}^{2}),$ the one
that uses the true degree’s of freedom $d$, where $1(A)$ denotes the indicator
function of the event $A$ (1 if $A$ holds, and 0 otherwise). Define the
asymptotic power function
$\pi_{\tau}(\beta)\equiv\lim_{n\rightarrow\infty}\mathbb{P}_{\beta}\left(\tilde{\phi}_{\tau}=1\right)$,
where $\mathbb{P}_{\beta}$ denotes the underlying probability when the true
parameter is $\beta.$ Call feasible test
$\hat{\phi}_{\tau}=1(\widehat{F}(\beta_{0})>\chi_{\widehat{d}}^{2})$, i.e, the
test using the estimated degree of freedom $\widehat{d}$.

###### Theorem 4.2

If $lim_{n\rightarrow\infty}\mathbb{P}_{\beta}(\widehat{d}=d)=1$ then, for all
$\beta$,

$\lim_{n\rightarrow\infty}\mathbb{P}_{\beta}\left(\hat{\phi}_{\tau}=1\right)=\pi_{\tau}(\beta).$
(17)

The next theorem justifies the proposed feasible test. Define the set

$\mathcal{B}_{1}:=\left\\{\beta_{1}\in\mathcal{B}\subset\mathbb{R}^{p}:\min_{\alpha\in\mathcal{A}}\left(g(\theta_{0},\alpha_{1},\beta_{1})-g(\theta_{0},\alpha,\beta_{0})\right)^{\prime}W\left(g(\theta_{0},\alpha_{1},\beta_{1})-g(\theta_{0},\alpha,\beta_{0})\right)>0\right\\}.$

where $\alpha_{1}\equiv\alpha_{1}(\beta_{1})$ is the minimum norm solution of
the problem

$\min_{\alpha\in\mathcal{A}}\left(\theta_{0}-g(\theta_{0},\alpha,\beta_{1})\right)^{\prime}W\left(\theta_{0}-g(\theta_{0},\alpha,\beta_{1})\right).$
(18)

###### Theorem 4.3

Let Assumptions 1 – 8 hold. Then, for all $\tau\in(0,1),$
$\pi_{\tau}(\beta_{0})=\tau,$ whereas $\pi_{\tau}(\beta)=1$ for all
$\beta\in\mathcal{B}_{1}$.

###### Remark 4.3

Theorem 4.3 shows that the feasible test controls size asymptotically, and it
is consistent for all $\beta\in\mathcal{B}_{1}$. If
$\beta=\beta_{1}\notin\mathcal{B}_{1}$, $\beta_{1}\neq\beta_{0}$, then there
is no information in the model to separate $(\alpha_{1},\beta_{1})$ from
$(\alpha_{0},\beta_{0})$, where $\alpha_{1}$ is such that
$g(\theta_{1},\alpha_{1},\beta_{1})=g(\theta_{0},\alpha_{0},\beta_{0})$. No
test has power against such observational equivalent cases.

Theorem 4.3 shows that the test have power if differences between
$g(\theta_{1},\alpha_{1},\beta_{1})$ and $g(\theta_{0},\alpha_{0},\beta_{0})$
can be detected when using $||x||_{W}=\left(x^{\prime}Wx\right)^{\frac{1}{2}}$
as seminorm, i.e,
$||g(\theta_{1},\alpha_{1},\beta_{1})-g(\theta_{0},\alpha_{0},\beta_{0})||_{W}>0$.

###### Remark 4.4

The test of Theorem 4.3 is straightforward to implement, and it can be used to
construct confidence regions for $\beta_{0}$ by the classical idea of
inverting the test statistic.

The next results give sufficient conditions to have local power under Pittman
alternatives. Let $\beta_{1n}=\beta_{0}+\frac{\delta}{\sqrt{n}}$, and with
some abuse of notation denote the asymptotic local power function by
$\pi_{\tau}(\delta)\equiv\lim_{n\rightarrow\infty}\mathbb{P}_{\beta_{1n}}\left(\tilde{\phi}_{\tau}=1\right)$.
We call $\delta$ the ”direction” of the alternative.

###### Assumption 9

$g(\theta,\alpha,\beta)$ is twice continuously differentiable at
$\theta_{0}\in\Theta\ $, $\alpha_{0}\in\mathcal{A}_{0}$, and
$\beta_{0}\in\mathcal{B}$.

Define the Jacobian of the parameter of interest as
$\nabla_{\beta}g_{0}:=\partial
g(\theta_{0},\alpha_{0},\beta_{0})/\partial\beta^{\prime}.$

###### Theorem 4.4

Under Assumptions 1 – 7$(i)$ and Assumption 9, the test will have non-trivial
local power whenever the direction $\delta$ is not in the null space of the
matrix $W\nabla_{\beta}g_{0}$, i.e,

$\pi_{\tau}(\delta)>\tau\iff\delta\notin Null(W\nabla_{\beta}g_{0}).$ (19)

###### Theorem 4.5

Under Assumptions 1 – 7$(i)$ and Assumption 9, the direction of maximum local
power, i.e. $\delta^{*}=arg\max_{||\delta||_{2}=1}\pi_{\tau}(\delta)$, is
given by

$\delta^{*}=arg\max_{||\delta||_{2}=1}\delta^{\prime}\nabla_{\beta}g_{0}^{\prime}W\nabla_{\beta}g_{0}\delta,$
(20)

that is, $\delta^{*}$ is the eigenvector associated to the maximum eigenvalue
of the matrix $\nabla_{\beta}g_{0}^{\prime}W\nabla_{\beta}g_{0}.$

###### Remark 4.5

Since $||\delta^{*}||_{2}=1$, elements of $\delta_{i}^{*}$ to the squared can
be interpreted as the relative importance of the element $\beta_{i}$ in the
direction of maximum local power. We call
$\sigma^{*}_{i}\equiv(\delta_{i}^{*})^{2}$ relative weights.

###### Remark 4.6

The value of $k^{*}$, obtained as the solution to the optimization problem

$k^{*}=\max_{||\delta||_{{2}=1}}\delta^{\prime}\nabla{\beta}g_{0}^{\prime}W\nabla_{\beta}g_{0}\delta,$
(21)

serves as the non-centrality parameter for the $\chi^{2}(d,k^{*})$ assymptotic
distribution associated with $\widehat{F}(\beta_{0})$ under the local
alternative. A larger $k^{*}$ value corresponds to a greater level of local
power. This value is the maximum eigenvalue of the matrix
$\nabla_{\beta}g_{0}^{\prime}W\nabla_{\beta}g_{0}$.

###### Remark 4.7

By the Rank-Nulity theorem and the result stated in Theorem 4.4, we can
determine the dimension of the subspace in $\mathbb{R}^{p}$ that consists of
all the directions our test cannot reject with non-trivial local power. This
subspace is denoted as $\mathcal{B}{\tau}=\\{\delta\in\mathbb{R}^{p}\hskip
2.84544pt:\hskip 2.84544ptW\nabla{\beta}g_{0}\delta=0\\}$. Consequently, the
dimension of $\mathcal{B}{\tau}$ is given by
$dim(\mathcal{B}{\tau})=p-Rank(W\nabla_{\beta}g_{0})$, and it can be estimated
accordingly.

##  5 Implementation

This section will illustrate the test algorithm, with additional details
provided in the following subsections.

  1. 1.

Fix $\beta=\beta_{0}$.

  2. 2.

Compute the reduced-form parameter estimator $\widehat{\theta}$.

  3. 3.

Compute the estimated asymptotic variance matrix of the estimated reduced-form
parameter $\hat{\Sigma}$.

  4. 4.

Solve problem (16) setting $\lambda_{n}$ by generalized cross-validation and
$W=\mathbf{I}_{m}$.

  5. 5.

Using $\widehat{\alpha}$ and $\widehat{\theta}$ compute
$I_{m}-\widehat{\nabla}_{\theta}g$.

  6. 6.

Estimate the rank $\widehat{r}_{\Sigma}$. Check Subsection 5.1 to follow a
detailed explanation.

  7. 7.

Estimate the optimal weighting matrix
$W=\left((I_{m}-\nabla_{\theta}g_{0})\Sigma(I_{m}-\nabla_{\theta}g_{0})^{\prime}\right)^{\dagger}$
as in Subsection 5.2.

  8. 8.

Solve problem (16) using $\hat{W}$ and update $\widehat{\alpha}$.

  9. 9.

Compute
$\widehat{F}(\beta_{0}):=n\left(\widehat{\theta}-g(\widehat{\theta},\widehat{\alpha},\beta_{0})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-g(\widehat{\theta},\widehat{\alpha},\beta_{0})\right)$.

  10. 10.

Estimate the rank $\widehat{r}_{\alpha}$ using the estimator of expression
(22). Check Subsection 5.1 to follow a detailed explanation. Compute
$\widehat{d}\equiv\widehat{r}_{\Sigma}-\widehat{r}_{\alpha}$.

  11. 11.

Reject null hypothesis at level $\tau$ when
$\hat{\phi}_{\tau}=1(\widehat{F}(\beta_{0})>\chi_{\widehat{d},1-\tau}^{2})$.

###  5.1 Rank estimation

In this subsection, we motivate the simple hard-thresholding method we use to
estimate the rank of a matrix. Take $\widehat{M}$ and $M$ as general positive
semi-definite symmetric matrices of dimension $m\times m$, with
$\widehat{M}\xrightarrow{p}M$, and $r_{M}=Rank(M)$. The estimator for the rank
of $M$ is:

$\widehat{r}_{M}=\\#\\{\widehat{\lambda}_{i}\geq\frac{1}{n^{b}},\hskip
5.69046pti\in 1,...,m\\},$ (22)

where $\widehat{\lambda}_{i}$ is the $i$th eigenvalue of $\widehat{M}$ and
$0<b<1$ is a tuning parameter. Previous literature such as Lütkepohl and Burda
(1997) or Dufour and Valéry (2016) has used $b=\frac{1}{3}$, but we instead
use $b=0.99$. Our choice is motivated by the better performance of our test in
the Monte Carlo exercise. We have tuned other values of $b>\frac{1}{2}$ and
obtained similar results. Future research will investigate cross validated
methods for the choice of $b$.

###### Assumption 10

Asymptotic normality of the estimated matrix $\widehat{M}$

$\sqrt{n}\left(Vec(\widehat{M})-Vec(M)\right)\xrightarrow{d}N_{m^{2}}(0,\Omega),$
(23)

with $s=Rank(\Omega)\leq m^{2}.$

###### Theorem 5.1

Under Assumption 10 $\widehat{r}_{M}\xrightarrow{p}r_{M}$.

Instead of using spectral cut-off methods, a common alternative is to use a
sequential testing procedure to estimate the rank. Statistical tests, such as
those proposed by Robin and Smith (2000) or Kleibergen (2005) can be employed
for this purpose. However, this approach requires a consistent estimator of
the asymptotic variance matrix, denoted as $\Omega$, of
$\sqrt{n}\left(Vec(\widehat{M})-Vec(M)\right)$. Obtaining such an estimator,
$\hat{\Omega}$, might be challenging in our context, particularly when a
closed-form solution for the matrix $M$ is unavailable. On the other hand,
spectral cut-off methods, such as those used by Lütkepohl and Burda (1997),
Dufour and Valéry (2016) or Lee (2022) among others, do not require estimating
$\Omega$. Our approach differs from theirs in that we exploit the fact that
the estimate $\widehat{\lambda}_{i}$ converges faster than $\sqrt{n}$ to the
population eigenvalue $\lambda_{i}$ whenever $\lambda_{i}=0$.
The following Theorem is taken from Robin and Smith (2000), see the Appendix
for a proof.

###### Theorem 5.2

Under Assumption 10, estimated eigenvalues whose population eigenvalue is 0
converge to the population parameter at a rate of $n^{-1}$. Assume population
eigenvalues are in descending order, then,

$\widehat{\lambda}_{i}=O_{P}\Big{(}\frac{1}{n}\Big{)}\hskip 28.45274pt\forall
i\in{r_{M}+1,\dots,m}.$ (24)

###### Remark 5.1

Since
$Rank(\nabla_{\alpha}g_{0})=Rank(\nabla_{\alpha}g_{0}\nabla_{\alpha}g_{0}^{\prime})$,
Theorem 5.1 can be used to estimate $\hat{r}_{\alpha}$, and noticing that
Assumption 10 holds whenever $\hat{\theta}$ is asymptotically normally
distributed.

To sum up, we illustrate the algorithm for the estimation of the rank of both
$\Sigma$ and $\nabla_{\alpha}g_{0}$.

  1. 1.

Estimate $\nabla_{\alpha}g_{0}\nabla_{\alpha}g_{0}^{\prime}$ and $\Sigma$.

  2. 2.

Do the singular value decomposition of both matrices and find the eigenvalues
of every matrix. In general terms,
$\widehat{M}=\widehat{Q}^{\prime}\widehat{\Lambda}\widehat{Q}$.

  3. 3.

Estimate the rank of $M$ using

$\widehat{r}_{M}=\\#\\{\widehat{\lambda}_{i}\geq\frac{1}{n^{b}},\hskip
5.69046pti\in 1,...,m\\},$

where $\widehat{\lambda}_{i}$ are the elements of the diagonal matrix
$\widehat{\Lambda}$ and $0<b<1$.

###  5.2 Estimation of Moore-Penrose g-inverse

In this subsection, we discuss how to implement a consistent estimate of the
optimal weighting matrix

$W=\left((I_{m}-\nabla_{\theta}g_{0})\widehat{\Sigma}(I_{m}-\nabla_{\theta}g_{0})^{\prime}\right)^{\dagger}.$
(25)

The main problem in consistently estimating $W$ is that Moore-Penrose
g-inverses are not continuous. An estimator based on the analog principle of
$W$ might be inconsistent. Define

$A\equiv(I_{m}-\nabla_{\theta}g_{0})\Sigma(I_{m}-\nabla_{\theta}g_{0})^{\prime},$
$\hat{A}\equiv(I_{m}-\hat{\nabla}_{\theta}g)\widehat{\Sigma}(I_{m}-\hat{\nabla}_{\theta}g)^{\prime}.$

Stewart (1969) gives necessary and sufficient conditions to estimate
consistently a Moore-Penrose g-inverse. Such conditions are: if
$\hat{A}\xrightarrow{p}A$, then

$\hat{A}^{\dagger}\xrightarrow{p}A^{\dagger}\iff
Pr\left(Rank(\hat{A})=Rank(A)\right)\xrightarrow{p}1.$

To ensure that such a condition holds, we truncate the eigenvalues of
$\widehat{A}$. First, use the spectral decomposition

$\hat{A}=\hat{R}^{\prime}\hat{P}\hat{R}.$ (26)

Let $\hat{p}_{j}$ be the diagonal elements of $\hat{P}$, i.e, the eigenvalues.
Then truncate the eigenvalues that are smaller than $\tfrac{1}{n^{b}}$,
$\tilde{p}_{j}=\hat{p}_{j}\mathbbm{1}{\\{\hat{p}_{j}>\frac{1}{n^{b}}\\}}$.
Define $\tilde{P}$ as the diagonal matrix composed by the $jth$ different
elements $\tilde{p}_{j}$, $\tilde{P}^{\dagger}$ as the diagonal matrix with
the $jth$ element given by $1/\tilde{p}_{j}$ when $\tilde{p}_{j}\neq 0$, and
zero otherwise, and compute

$\tilde{A}=\hat{R}\tilde{P}\hat{R}.$ (27)

Then, set

$\widehat{W}\equiv\tilde{A}^{\dagger}=\hat{R}\tilde{P}^{\dagger}\hat{R}.$ (28)

Following the results of Theorem 5.1
$Pr\left(Rank(\tilde{A})=Rank(A)\right)\xrightarrow{p}1$. Hence, the estimator
$\widehat{W}$ satisfies Stewart (1969) necessary and sufficient conditions for

$\widehat{W}\xrightarrow{p}W.$ (29)

  1. 1.

Compute the matrix
$\hat{A}\equiv(I_{m}-\hat{\nabla}_{\theta}g)\widehat{\Sigma}(I_{m}-\hat{\nabla}_{\theta}g)^{\prime}$.

  2. 2.

Get this spectral decomposition $\hat{A}=\hat{R}^{\prime}\hat{P}\hat{R}.$

  3. 3.

Compute $\tilde{P}$.

  4. 4.

Compute the matrix $\tilde{A}=\hat{R}\tilde{P}\hat{R}$.

  5. 5.

Set $\hat{W}=\tilde{A}^{\dagger}=\hat{R}\tilde{P}^{\dagger}\hat{R}$.

##  6 Monte Carlo

In the following subsections, we examine the finite sample properties of our
method using Monte Carlo in two different applications. First, in a Static
Bayesian Game proposed by Pesendorfer and Schmidt-Dengler (2008). Second, in a
setting that resembles the New-Keynesian model proposed by Nakamura and
Steinsson (2018).

###  6.1 Monte Carlo: Static Bayesian Game

In this section, we demonstrate our methodology using an example from
Pesendorfer and Schmidt-Dengler (2008). We examine the properties of our test
in finite samples, including size and power. The example we consider is a
static Bayesian Game involving two firms that must decide whether or not to
enter a market simultaneously. The payoffs for Firm 1 and Firm 2 are as
follows:

$\pi_{1}\left(a_{1},a_{2},s\right)=\begin{cases}\beta_{0}\cdot s&\text{ if
}a_{1}=\text{ active, }a_{2}=\text{ not active }\\\ \alpha_{1}\cdot s&\text{
if }a_{1}=\text{ active, }a_{2}=\text{ active }\\\ 0&\text{ otherwise.
}\end{cases}$

$\pi_{2}\left(a_{1},a_{2},s\right)=\begin{cases}\alpha_{2}\cdot s&\text{ if
}a_{2}=\text{ active, }a_{1}=\text{ not active }\\\ \alpha_{3}\cdot s&\text{
if }a_{2}=\text{ active, }a_{1}=\text{ active }\\\ 0&\text{ otherwise.
}\end{cases}$

The variable $s\in\\{s_{1},s_{2},s_{3}\\}$ is the state of the economy, with
$s_{1}<s_{2}<s_{3}$. The utility for a firm $i$ with state $s$ and other
firm’s action $j$, is defined by:

$u(a_{i},a_{j},s)=\pi_{i}(a_{i},a_{j},s)+\epsilon_{i},$

where $\epsilon_{i}$ is a random unobserved variable distributed as a standard
normal. Firm $i$th decides to be active whenever

$u(a_{1},a_{2},s)=\pi_{i}(a_{1},a_{2},s)+\epsilon_{i}>0.$

Firm $i$ has expectations about the probability of firm $j$ being active. In
equilibrium, such expectations will be equal to the actual probability of firm
$j$ choosing to be active, i.e

$\theta_{1,s}=g\left((1-\theta_{2,s})\cdot
s\cdot\beta_{0}+\theta_{2,s}\cdot\alpha_{1}\cdot s\right)\hskip
14.22636pt\forall s,$

and

$\theta_{2,s}=g\left((1-\theta_{1,s})\cdot
s\cdot\alpha_{2}+\theta_{1,s})\cdot\alpha_{3}\cdot s\right)\hskip
14.22636pt\forall s,$

where $\theta_{j,s}$ is the probability of choosing to be active for a firm
$j$ given state $s$. The mapping $g$ is the CDF of a standard normal
distribution. Our parameter of interest will be $\beta_{0}$, i.e, the
coefficient of firm 1’s payoff associated with being active when the other is
not. The remaining parameters are considered nuisance parameters.

####  6.1.1 Identification Problem

In this example, we have six moment conditions (three states for every single
firm) and four parameters, three of which are nuisance parameters. The
Jacobian of the nuisance parameter is

$\nabla_{\alpha}g(\theta,\alpha,\beta)\equiv\begin{pmatrix}\dot{g}_{1,1}\theta_{2,s_{1}}s_{1}&0&0\\\
\dot{g}_{1,2}\theta_{2,s_{2}}s_{2}&0&0\\\
\dot{g}_{1,3}\theta_{2,s_{3}}s_{3}&0&0\\\
0&\dot{g}_{2,1}(1-\theta_{1,s_{1}})s_{1}&\dot{g}_{2,1}\theta_{1,s_{2}}s_{1}\\\
0&\dot{g}_{2,2}(1-\theta_{1,s_{2}})s_{2}&\dot{g}_{2,2}\theta_{1,s_{2}}s_{2}\\\
0&\dot{g}_{2,3}(1-\theta_{1,s_{2}})s_{3}&\dot{g}_{2,3}\theta_{1,s_{3}}s_{3},\\\
\end{pmatrix}$ (30)

Where the density of the standard normal, denoted by $\dot{g}_{i,s}$, is
evaluated at $(1-\theta_{i,s})\cdot
s\cdot\beta+\theta_{i,s}\cdot\alpha_{1}\cdot s$ for
$s\in\\{s_{1},s_{2},s_{3}\\}$ and firm $i$. It is worth noting that when the
choice probabilities of firm one are one-half for all possible states of the
economy, i.e.,
$\theta_{1,s_{1}}=\theta_{1s_{2}}=\theta_{1,s_{3}}=\frac{1}{2}$, the
Jacobian’s rank becomes deficient.

$Rank(\nabla_{\alpha}g(\theta_{0},\alpha,\beta))=2<q.$ (31)

This simple example illustrates how the degree of identification might depend
on population values of the reduced form parameter, which are unknown.

####  6.1.2 Monte Carlo exercise

We test the hypothesis:

$H_{0}:\hskip 14.22636pt\beta=\beta_{0}\hskip 42.67912pt\textit{v.s}\hskip
42.67912ptH_{1}:\hskip 14.22636pt\beta\neq\beta_{0}.$ (32)

The value of $\beta_{0}$ is set to $\beta_{0}=1.5$ in the identified case
(case 1), and $\beta_{0}=0.3$ in the not identified case (case 2). We compare
three procedures: an Oracle test (Oracle) that assumes knowledge of the
degrees of freedom $d$, our robust test (Robust) that estimates $d$, and the
t-test (T-test) based on the asymptotic normality of the estimate of the
structural parameter $\beta_{0}$ under point-identification. A summary of the
performance of the different tests in terms of size for the identified case is
presented in Table 1. The findings indicate that rank estimation has a minimal
impact on the procedure’s overall level of uncertainty.

Table 1: Size of the test at 5%. Identified case. (Case 1): $Rank(\nabla_{\alpha}g)=3$ | $n=100$ | $n=250$ | $n=500$ | $n=1000$
---|---|---|---|---
Oracle | 0.067 | 0.053 | 0.050 | 0.045
Robust | 0.062 | 0.053 | 0.050 | 0.045
T-test | 0.055 | 0.047 | 0.046 | 0.049

Table 2 considers the unidentified case, the T-test presents large size
distortions with a larger size than $5\%$. In contrast, the Robust and Oracle
tests control the size. Again, the estimation of the rank does not affect the
performance of the Robust test.

Table 2: Size of the test at 5%. Not identified case. (Case 2): $Rank(\nabla_{\alpha}g)=2$ | $n=100$ | $n=250$ | $n=500$ | $n=1000$
---|---|---|---|---
Oracle | 0.058 | 0.051 | 0.052 | 0.050
Robust | 0.058 | 0.051 | 0.052 | 0.050
T-test | 0.128 | 0.12 | 0.113 | 0.114

In the following figures, we report the power of the different tests for the
Case 2 of not identification under the null. In this example, the parameter of
interest $\beta$ is identified under any possible alternative.

Figure 1:  Power of the test. (Case 2): Not identification under the null.

(a) Sample size $n=100$

(b) Sample size $n=1000$

As a consequence of $\beta$ being identified under the alternative, the t-test
has non-trivial power. Yet, the size is not controlled. The Robust and the
Oracle tests yield similar results in terms of power, suggesting that the
estimation of the degrees of freedom does not have a big impact in finite
samples.

###  6.2 Monte Carlo: Nakamura and Steinsson 2018

In this section, we carry out a Monte Carlo simulation based on Nakamura and
Steinsson (2018). Our analysis consists of two parts. Firstly, we assess the
finite sample properties by testing the validity of the calibrated (fixed)
parameters in terms of both the size and power of our method. Secondly, we
evaluate the finite sample properties of conducting inference on two
structural parameters of interest, namely the Phillips Curve slope and the
Information Effect. For these structural parameters, we compare the size and
power obtained from our robust inference method with those from standard
inference methods.

Nakamura and Steinsson (2018) provides compelling evidence of the non-
neutrality of monetary policy. They accomplish this by identifying the
information effect of Federal Reserve announcements on economic fundamental
beliefs. Specifically, Nakamura and Steinsson (2018) utilizes data on
unexpected changes in interest rates occurring within 30 minutes of a Fed
announcement. In addition, they construct and estimate a structural model
using the simulated method of moments. Their structural model involves the
estimation of $m=33$ reduced-form parameters and $q=5$ structural parameters.
The mapping function, $g(\cdot)$, has no closed-form solution, and it is
therefore estimated numerically as in Section 3.3.1.

Given that Nakamura and Steinsson (2018) uses two different databases to
estimate the reduced-form parameters, the matrix of variance and covariance
$\Sigma$ is not identified.

To illustrate our methodology, we employ a subset of moments and parameters,
estimating $m=8$ reduced-form parameters and $q=2$ structural parameters. The
latter corresponds to the slope of the Phillips Curve and the information
effect of monetary policy. On the other hand, Nakamura and Steinsson (2018)
calibrate eight parameters in their study, while we calibrate $k=11$
parameters. Notably, the reduced-form parameters are estimated from a single
database, enabling the identification of the variance and covariance matrix of
the estimated parameters.

As demonstrated in Section 7, the estimated rank of the Jacobian matrix is
deficient, and the asymptotic variance matrix of the reduced-form parameters,
$\widehat{\Sigma}$, is close to being singular. These limitations motivate the
use of robust inference techniques. It is worth noting that throughout the
Monte Carlo exercise, we employ the estimated $\hat{\Sigma}$ from Section 7 as
the true one. Table 3 presents the structural parameter values for different
data-generating processes.

Table 3:  Data Generating Process Structural Parameters.

Notation | Definition | Replication of Application
---|---|---
Calibration (fixed parameters)
$\rho$ | Subjective discount factor | 0.99
$\eta$ | Nominal rigidity | 0.75
$\omega$ | Elasticity of marginal cost to output | 2
$\gamma$ | Coeff on lagged inflation | 1
$\phi$ | Endogenous feedback in Taylor rule | 0.01
$\pi$ | Inflation target shock | 0
$\delta$ | Elasticity of substitution across varieties | 10
$\sigma$ | Intertemporal el of substitution | 0.50
$\rho_{1}$ | Autoregressive first root of monetary shock | 0.90
$\rho_{2}$ | Autoregressive second root of monetary shock | 0.79
$b$ | Consumption Habit | 0.95
Estimated parameters
$\kappa\zeta$ | Phillips Curve Slope | $1.0037\cdot 10^{-4}$
$\psi$ | Information Effect | $1.1617\cdot 10^{-6}$

  * •

Note. This DGP corresponds to the estimated values of the application in
Section 7.

####  6.2.1 Calibration Validity

We refer to calibration validity as not rejecting the calibrated value with
our robust test. In other words, we test if $\beta=\beta_{0}$, where
$\beta_{0}$ are the calibrated values.

In this subsection, we test the validity of the calibration of the modified
version of Nakamura and Steinsson (2018) explained above. The vector $\beta$
is composed by all calibrated (fixed) parameters

$\beta=\left(\rho,\eta,\omega,\gamma,\phi,\pi,\delta,\sigma,\rho_{1},\rho_{2},b\right)^{\prime}.$

The nuisance parameters are $\alpha=\left(\kappa\zeta,\psi\right)^{\prime}$.
There is no closed form solution for the mapping
$g:\mathcal{B}\times\mathcal{A}\xrightarrow{}\Theta$, with
$\mathcal{B}\subset\mathbb{R}^{11}$ and $\mathcal{A}\subset\mathbb{R}^{2}$. We
use numerical methods to estimate $g(\cdot)$. We test the null hypothesis:

$H_{0}:\beta=\beta_{0}\hskip 28.45274pt\textit{vs}\hskip
28.45274pt\beta\neq\beta_{0},$

where $\beta_{0}$ are the fixed parameter values shown in Table 3.

Table 4: Size of the Robust test at 5%. | $n=100$ | $n=250$ | $n=500$ | $n=1000$
---|---|---|---|---
Robust | 0.044 | 0.044 | 0.053 | 0.045
Oracle | 0.007 | 0.007 | 0.009 | 0.003

In the replication of the application, the model is identified, albeit weakly.
This is the reason why, in Table 4, it can be observed that using the oracle
test (which utilizes the true degree of freedom) yields a size different from
5%. The empirical size of the Robust test is satisfactory. To illustrate the
power of the test, we maintain all elements of $\beta$ constant except for the
shock to the inflation target, denoted as $\pi$. We focus on this alternative
hypothesis since $\pi$ contributes the most to the deviation with maximum
local power, as can be seen in Table 8 in Section 7.

Figure 2:  Power of the robust test. Calibration Validity.

####  6.2.2 Inference on structural Parameters: The Phillips Curve Slope
$\kappa\zeta$

In this subsection, we assume our previous calibration is valid. Moreover, the
vector of structural parameters of interest is $\beta=\kappa\zeta$, and the
nuisance parameter is $\alpha=\psi$. We test the hypothesis:

$H_{0}:\kappa\zeta=\kappa\zeta_{0}\hskip 28.45274ptvs\hskip
28.45274ptH_{1}:\kappa\zeta\neq\kappa\zeta_{0},$ (33)

where $\kappa\zeta_{0}$ are the structural values presented in Table 3 that
correspond to every different case of study.

Table 5: Size of the test at 5%. Inference on Phillips Curve Slope $\kappa\zeta$ Test | $n=100$ | $n=250$ | $n=500$ | $n=1000$
---|---|---|---|---
Replication of Application
Oracle | 0.016 | 0.009 | 0.015 | 0.011
Robust | 0.073 | 0.051 | 0.061 | 0.057
T-test | 0 | 0 | 0 | 0

Under the Null hypothesis, both the t-test and the Oracle test do not exhibit
a significance level of 5%. This deviation from the expected significance
level can be attributed to the weak identification of the model and the near
singularity of $\Sigma$ (covariance matrix). In contrast, the robust test has
a satisfactory size performance. Figure 3 reports the results on power. The
t-test has low power, while that of the Robust and Oracle tests is high for
deviations from the right of the null hypothesis. In contrast to the Static
Bayesian Game case, the model is unidentified under any alternative
hypothesis. Furthermore, the covariance matrix $\Sigma$ is nearly singular.
Consequently, the t-test fails to control the significance level. The Robust
test has non-trivial power for any alternative hypothesis. Nevertheless, the
power function is not symmetric around the null. Specifically, smaller
alternative hypotheses relative to the null hypothesis yield lower power
compared to larger alternatives. This would suggest that the derivative of the
mapping $g$ with respect to $\kappa\zeta$ is getting closer to 0 as
$\kappa\zeta$ goes to 0.

Figure 3:  Power of the test. Inference on Phillips Curve Slope estimate.

(a) Sample size $n=100$

(b) Sample size $n=1000$

####  6.2.3 Inference on Structural Parameters: The Information Effect $\psi$

In this subsection, the vector of structural parameters of interest is
$\beta=\psi$, and the nuisance parameter is $\alpha=\kappa\zeta$. Whereas the
rest of the structural parameters are assumed to be valid. We test the
hypothesis:

$H_{0}:\psi=\psi_{0}\hskip 28.45274ptvs\hskip
28.45274ptH_{1}:\psi\neq\psi_{0},$ (34)

where $\psi_{0}$ are the structural values presented in Table 3 that
correspond to every different case of study. In the following Table 6 and
Figure 4 we report the results for this case. A remarkable difference relative
to the previous section is that the Robust test has trivial power against any
alternative, due to the fact that there exist observational equivalent models
for any alternative $\beta_{1}\in\mathcal{B}$, i.e, $\forall$
$\beta_{1}\in\mathcal{B}$ $\beta_{1}\notin\mathcal{B}_{1}$, check Theorem 4.3
for further details.

Table 6: Size of the test at 5%. Inference on Information Effect $\psi$ Test | $n=100$ | $n=250$ | $n=500$ | $n=1000$
---|---|---|---|---
Replication of Application
Oracle | 0.011 | 0.004 | 0.014 | 0.006
Robust | 0.058 | 0.042 | 0.054 | 0.058
T-test | 0.214 | 0.118 | 0.055 | 0.119
Figure 4:  Power of the test. Inference on Information Effect $\psi$.

(a) Sample size $n=100$

(b) Sample size $n=1000$

In this particular application, it seems that the Information Effect is not
identified. This lack of identification could explain why there is trivial
power in the Robust Test.

##  7 Application: Inference in Nakamura and Steinsson 2018

In this section, we employ our robust inference method to re-analyze the study
by Nakamura and Steinsson (2018) and compare our results to those obtained
through standard methods. In particular, first, we check the validity of the
calibrated parameters in the main paper using our methodology. Second, we
compare the inferences obtained by inverting our test with those obtained by
standard inference methods.

To evaluate the robustness of their approach, we estimate the rank of the
Jacobian of the model evaluated at the estimated parameters, obtaining
$\hat{r}_{\alpha}=Rank(\nabla_{\alpha}g(\widehat{\alpha}))=4<q=5$, indicating
potential identification issues.

Unfortunately, in Nakamura and Steinsson (2018) the asymptotic variance matrix
of reduced form estimators is not identifiable due to the use of at least two
different databases. To circumvent this issue, we opted to employ a subset of
the estimated reduced form parameters, ensuring the identification of
$\Sigma$. Our estimation procedure involves computing $m=8$ reduced-form
parameters and $q=2$ structural parameters, the latter of which represent the
slope of the Phillips Curve and the information effect of monetary policy.
However, even though $\Sigma$ is identified in this simplified version of the
problem, it is nearly singular. Table 7 shows that the estimated rank of this
matrix is $Rank(\hat{\Sigma})=5$. As a result, we advocate for identification-
robust and singular-weighting-matrix-robust inference techniques in this
application. The null hypothesis is

$H_{0}:\beta=\beta_{0}\hskip 28.45274pt\textit{vs}\hskip
28.45274ptH_{1}:\beta\neq\beta_{0},$

where
$\beta_{0}=\left(\rho,\eta,\omega,\gamma,\phi,\pi,\delta,\sigma,\rho_{1},\rho_{2},b\right)^{\prime}$
are the values fixed in Table 3.

Table 7: Calibration validity | Statistic | $Rank(\hat{\Sigma})$ | $Rank(\nabla_{\alpha}\hat{g})$ | Critical value
---|---|---|---|---
Robust test | 3.18 | 5 | 1 | 9.487
(Minimum Eigenvalue) |  | ($2.6114\cdot 10^{-6}$) | ($3.2299\cdot 10^{-5}$) |

Table 7 shows that the null hypothesis is not rejected by our robust test.
Indicating that calibrated values are valid. Moreover, it is worth noting that
the model is not identified, as the estimated value of
$\widehat{r}_{\alpha}=Rank(\nabla_{\alpha}g(\widehat{\alpha}))$ is only one,
which falls short of the required value of $q=2$. We conducted an empirical
analysis of the local power and determined that the dimension of the local
nontrivial power space, denoted by $\mathcal{B}_{\tau}$, is eight. This means
that out of 11 parameters, only deviations in three parameters can lead to the
rejection of the test with local nontrivial power.

Table 8:  Contribution to Direction of Maximum Local Power. Notation | Definition | Relative Weights
---|---|---
Calibration (fixed parameters)
$\rho$ | Subjective discount factor | 0.006
$\eta$ | Nominal rigidity | 0.03
$\omega$ | Elasticity of marginal cost to output | 0
$\gamma$ | Coeff on lagged inflation | 0.006
$\phi$ | Endogenous feedback in Taylor rule | 0.148
$\pi$ | Inflation target shock | 0.674
$\delta$ | Elasticity of substitution across varieties | 0
$\sigma$ | Intertemporal el of substitution | 0
$\rho_{1}$ | Autoregressive first root of monetary shock | 0.13
$\rho_{2}$ | Autoregressive second root of monetary shock | 0.003
$b$ | Consumption Habit | 0.008

Table 8 illustrates the individual contributions of each calibrated parameter
towards the direction of maximum local power, as discussed in Remark 4.5. The
top three contributors to the maximum local power are the inflation target
shock, the endogenous feedback in the Taylor rule, and the first
autoregressive root of the monetary shock.

Table 9: Inference of structural parameters of interest $\beta$ | PC slope | Information effect
---|---|---
Estimator | $1.0037\cdot 10^{-4}$ | $1.1617\cdot 10^{-6}$
CI T-test | (-9.246, | 9.295) | (-0.5017 , | 0.5017)
CI Robust test | ($2.1983\cdot 10^{-5}$, | 0.0152) | (0, | 0.9805)
CI Bootstrap | ($4.087\cdot 10^{-7}$, | 0.0321) | ($1.7392\cdot 10^{-4}$, | 0.9999)

Table 9 presents a summary of the estimation and inference results for the
structural parameters. In comparison, the confidence intervals (CI) based on
the t-test are wider than those obtained using the bootstrap or the robust
test. Moreover, the CI using the robust test appears to be shorter. CI
bootstrap is the method that Nakamura and Steinsson (2018) uses, and it is
obtained using non-parametric bootstrap to estimate the finite sample
distribution of the estimates. Notably, the estimated information effect is
approximately zero, indicating that nothing of the real effects of monetary
policy stem from revisions in agents’ fundamental economic beliefs. However,
the estimation’s confidence intervals are wide, primarily due to the exclusive
use of inflation-related reduced-form parameters.

##  8 Conclusion

In this paper, we propose inference for structural parameters based on minimum
distance that is robust to identification problems of nuisance parameters.
Moreover, to implement this method is not necessary to have prior knowledge
about the identification degree of the model. Last but not least, this
inference method is computationally fast. Our robust inference methods can be
applied to test the validity of calibrated (fixed) parameters in structural
models under weak identification of nuisance parameters. They can be also
applied to obtain identification-robust confidence intervals by inverting our
robust test. Therefore, this method is suitable for a large class of
applications in structural models commonly employed in Macroeconomics and
Microeconomics.

##  Appendix Appendix A Main Proofs

Define

$f(\theta,\alpha):=\left(\theta-g(\theta,\alpha,\beta_{0})\right)^{\prime}W\left(\theta-g(\theta,\alpha,\beta_{0})\right),$
(A-35) $v(\theta):=\min_{\alpha\in\mathcal{A}}f(\theta,\alpha)$

and

$\mathcal{M}(\theta)=\arg\min_{\alpha\in\mathcal{A}}f(\theta,\alpha).$

We shall use Theorem 4.2 in Shapiro (1985), which for the sake of reference is
repeated here adapted to our case.

Assumption S1: There exists a constant $C\in\mathbb{R}$ such that
$v(\theta_{0})<C,$ and a compact set $\Pi$ such that
$\\{\alpha\in\mathcal{A}:f(\theta,\alpha)\leq C\\}\subset\Pi$ whenever
$\theta$ is in a neighborhood of $\theta_{0}.$

Assumption S2: For every
$\alpha_{0}\in\mathcal{M}(\theta_{0})\equiv\mathcal{A}_{0}$ there exists a
local diffeomorphism (a continuously differentiable mapping which is locally
one-to-one, and with inverse which is continuously differentiable)
$\alpha=h(\gamma)$ such that the function
$t(\theta,\gamma):=f(\theta,h(\gamma))$ has Hessian matrix non-singular at
$(\theta_{0},\gamma_{0}),$ where $\gamma_{0}:=h^{-1}(\alpha_{0}),$ with
respect to the variables $\gamma_{1},...,\gamma_{r},$ and $t$ is independent
of the variables $\gamma_{r+1},...,\gamma_{q}$ in a neighborhood of
$(\theta_{0},\gamma_{0}).$

###### Theorem A.1

(Shapiro (1985), Theorem 4.2): Suppose $f(\theta,\alpha)$ is twice
continuously differentiable, $\mathcal{M}(\theta_{0})$ is connected, and
Assumptions S1 and S2 hold. Then, $v(\theta)$ is twice continuously
differentiable in a neighborhood of $\theta_{0}$ and

$\nabla_{\theta}v(\theta_{0})=\nabla_{\theta}f(\theta_{0},\alpha_{0})$ (A-36)

$\nabla_{\theta\theta}^{2}v(\theta_{0})=\nabla_{\theta\theta}^{2}f(\theta_{0},\alpha_{0})-\nabla_{\theta\alpha}^{2}f(\theta_{0},\alpha_{0})\left(\nabla_{\alpha\alpha}^{2}f(\theta_{0},\alpha_{0})\right)^{\dagger}\nabla_{\theta\alpha}^{2}f(\theta_{0},\alpha_{0})^{\prime},$
(A-37)

where expressions in the right side of (A-36) and (A-37) give the same result
for all $\alpha_{0}\in M(\theta_{0}),$ and (A-37) is independent of a
particular choice of the generalized inverse.

Since $f$ is continuous and $\mathcal{A}$ is compact, Assumption S1 holds. The
following result follows from the well-known Rank Theorem (see e.g. Fisher,
1966).

###### Theorem A.2

All $\alpha_{0}\in\mathcal{A}_{0}$ are interior locally regular points of
$\mathcal{A}$. If $r<q,$ and Assumption 6 holds, i.e,
$Rank(\nabla_{\alpha}g_{0})=Rank(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0})$
then Assumption S2 holds.

Proof of Theorem 4.1 By Assumption 2, $\mathcal{M}(\theta_{0})$ is non-empty,
i.e. $\alpha_{0}\in\mathcal{M}(\theta_{0})$. We allow for partial
identification, i.e. $\mathcal{M}(\theta_{0})$ containing more than one point.
To prove the theorem we shall apply Theorem A.1 above. From the previous
discussion, the conditions of the theorem are satisfied under our conditions.
Simple calculus shows that for the particular case in (A-35)

$\displaystyle\nabla_{\theta}f(\theta_{0},\alpha_{0})$
$\displaystyle=2\left(\theta_{0}-g(\theta_{0},\alpha_{0},\beta_{0})\right)^{\prime}W(I_{m}-\nabla_{\theta}g_{0})$
$\displaystyle=0$

by the correct specification of the model, whereas

$\displaystyle\nabla_{\theta\theta}^{2}f(\theta_{0},\alpha_{0})$
$\displaystyle=2(I_{m}-\nabla_{\theta}g_{0})^{\prime}W(I_{m}-\nabla_{\theta}g_{0})$
$\displaystyle\nabla_{\theta\alpha}^{2}f(\theta_{0},\alpha_{0})$
$\displaystyle=-2(I_{m}-\nabla_{\theta}g_{0})^{\prime}W\nabla_{\alpha}g_{0}$
$\displaystyle\nabla_{\alpha\alpha}^{2}f(\theta_{0},\alpha_{0})$
$\displaystyle=2\nabla_{\alpha}g_{0}{}^{\prime}W\nabla_{\alpha}g_{0}.$

Write

$\displaystyle\widehat{F}(\beta_{0})$
$\displaystyle=\min_{\alpha\in\mathcal{A}}n\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)$
$\displaystyle=n\left(\widehat{\theta}-g(\widehat{\theta},\widehat{\alpha},\beta_{0})\right)^{\prime}W\left(\widehat{\theta}-g(\widehat{\theta},\widehat{\alpha},\beta_{0})\right)+$
$\displaystyle
n\left(\widehat{\theta}-g(\widehat{\theta},\widehat{\alpha},\beta_{0})\right)^{\prime}\left(\widehat{W}-W\right)\left(\widehat{\theta}-g(\widehat{\theta},\widehat{\alpha},\beta_{0})\right).$

By the consistency of $\widehat{W},$ the second term on the right-hand size is
bounded by

$\displaystyle
n\left|\left(\widehat{\theta}-g(\widehat{\theta},\widehat{\alpha},\beta_{0})\right)\right|^{2}\left|\widehat{W}-W\right|$
$\displaystyle=O_{P}(1)o_{P}(1)$ $\displaystyle=o_{P}(1),$

where the $O_{P}(1)$ term follows from applying the Delta-Method and the
arguments above to the mapping
$\theta\rightarrow\min_{\alpha\in\mathcal{A}}\left|\left(\theta-g(\theta,\alpha,\beta_{0})\right)\right|^{2}.$
Then,

$\widehat{F}(\beta_{0})=nv(\widehat{\theta})+o_{P}(1).$

By Theorem A.1 and the second order Delta-Method, we have

$\displaystyle\widehat{F}(\beta_{0})$
$\displaystyle=n\left(\widehat{\theta}-\theta_{0}\right)^{\prime}\frac{1}{2}\nabla_{\theta\theta}^{2}v(\theta_{0})\left(\widehat{\theta}-\theta_{0}\right)+o_{P}(1).$
$\displaystyle=Z^{\prime}UZ.$

Where
$U:=(I_{m}-\nabla_{\theta}g_{0})^{\prime}\left(W-W\nabla_{\alpha}g_{0}\left(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0}\right)^{\dagger}\nabla_{\alpha}g_{0}^{\prime}W\right)(I_{m}-\nabla_{\theta}g_{0}),$
and $Z\sim N_{m}(0,\Sigma)$.

Now we find the distribution of $Z^{\prime}UZ$. Rao et al. (1972) (P.171, Thm
9.2.1) gives necessary and sufficient conditions for quadratic functions of
correlated normal variables to be distributed as a chi-squared. In our
context, $Z\sim N_{m}(0,\Sigma)$. If the condition $\Sigma U\Sigma
U\Sigma=\Sigma U\Sigma$ holds, then $Z^{\prime}UZ\sim\chi^{2}_{tr(U\Sigma)}$.
We first define the following matrices:

$E=\left(I_{m}-\nabla_{\theta}g_{0}\right),$ (A-38)
$B=\left(I_{m}-\nabla_{\alpha}g_{0}(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0})\nabla_{\alpha}g_{0}^{\prime}W\right),$
(A-39)
$C=\left(I_{m}-W\nabla_{\alpha}g_{0}(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0})\nabla_{\alpha}g_{0}^{\prime}\right).$
(A-40)

Notice that,

$U=E^{\prime}WBE=E^{\prime}CWE.$ (A-41)

Using the two previous equalities in (A-41), and the fact that
$W=\left(E\Sigma E^{\prime}\right)^{\dagger}$ so that $WE\Sigma
E^{\prime}W=W$, we can obtain

$\Sigma U\Sigma U\Sigma=\Sigma E^{\prime}CWE\Sigma E^{\prime}WBE\Sigma=\Sigma
E^{\prime}CWBE\Sigma=\Sigma E^{\prime}WBBE\Sigma=\Sigma
E^{\prime}WBE\Sigma=\Sigma U\Sigma.$ (A-42)

Using Assumptions 6 and 7 and the definition of $B$

$tr(U\Sigma)=tr(E^{\prime}WE\Sigma)-tr(E^{\prime}W\nabla_{\alpha}g_{0}\left(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0}\right)^{\dagger}\nabla_{\alpha}g_{0}^{\prime}WE\Sigma)$
(A-43) $=tr(WE\Sigma
E^{\prime})-tr(\nabla_{\alpha}g_{0}\left(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0}\right)^{\dagger}\nabla_{\alpha}g_{0}^{\prime}W)$
(A-44) $=tr(WE\Sigma
E^{\prime})-tr(\left(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0}\right)^{\dagger}\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0})$
(A-45) $=Rank(E\Sigma
E^{\prime})-Rank(\nabla_{\alpha}g_{0}^{\prime}W\nabla_{\alpha}g_{0})=r_{\Sigma}-r_{\alpha}.$
(A-46)

_Q.E.D._

Proof of Theorem 4.2: Let the feasible test be

$\tilde{\phi}_{\tau}=1(\widehat{F}(\beta_{0})>\chi_{d}^{2})$ (A-47)

and the oracle test

$\hat{\phi}_{\tau}=1(\widehat{F}(\beta_{0})>\chi_{\widehat{d}}^{2}).$ (A-48)

The power function is defined as

$\pi_{\tau}(\beta)\equiv\lim_{n\rightarrow\infty}\mathbb{P}_{\beta}\left(\tilde{\phi}_{\tau}=1\right)$
(A-49)

$\mathbb{P}_{\beta}\left(\hat{\phi}_{\tau}=1\right)=\sum_{j=1}^{m}\mathbb{P}_{\beta}\left(\widehat{F}(\beta_{0})>\chi_{j}^{2}\right)\mathbb{P}_{\beta}(\widehat{d}=j)$
$=\mathbb{P}_{\beta}\left(\widehat{F}(\beta_{0})>\chi_{d}^{2}\right)\mathbb{P}_{\beta}(\widehat{d}=d)+\sum_{j=1,\hskip
2.84544ptj\neq
d}^{m}\mathbb{P}_{\beta}\left(\widehat{F}(\beta_{0})>\chi_{j}^{2}\right)\mathbb{P}_{\beta}(\widehat{d}=j),$
(A-50)

by consistency of the estimated degree of freedom $\widehat{d}$,
$lim_{n\rightarrow\infty}\mathbb{P}_{\beta}(\widehat{d}=d)=1$, equation (A-50)
equals

$=\mathbb{P}_{\beta}\left(\widehat{F}(\beta_{0})>\chi_{d}^{2}\right)\mathbb{P}_{\beta}(\widehat{d}=d)+o_{P}(1)=\mathbb{P}_{\beta}\left(\tilde{\phi}_{\tau}=1\right)+o_{P}(1).$
(A-51)

_Q.E.D._

Proof of Theorem 4.3: Let $\alpha_{1}$ and $\alpha_{0}$ be the plim of

$\widehat{\alpha}_{1}=argmin_{\alpha\in\mathcal{A}}\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{1})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-(g(\widehat{\theta},\alpha,\beta_{1})\right)+\lambda_{n}\left|\alpha\right|^{2},$
(A-52)

and

$\widehat{\alpha}_{0}=argmin_{\alpha\in\mathcal{A}}\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-(g(\widehat{\theta},\alpha,\beta_{0})\right)+\lambda_{n}\left|\alpha\right|^{2},$
(A-53)

respectively. Define
$g(\widehat{\theta},\widehat{\alpha}_{1},\beta_{1})\equiv\hat{g}_{1}$, and
$g(\widehat{\theta},\widehat{\alpha}_{0},\beta_{0})\equiv\hat{g}_{0}$. From
now on we assume we are under the alternative hypothesis $\beta_{1}$. Using
the definition of $\hat{F}(\beta_{0})$,

$\displaystyle\widehat{F}(\beta_{0})$
$\displaystyle=\min_{\alpha\in\mathcal{A}}n\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)^{\prime}\widehat{W}\left(\widehat{\theta}-g(\widehat{\theta},\alpha,\beta_{0})\right)$
$\displaystyle=n\left(\widehat{\theta}-\hat{g}_{0}\right)^{\prime}W\left(\widehat{\theta}-\hat{g}_{0}\right)\left(1+o_{p}(1)\right)$
$\displaystyle=n\left(\widehat{\theta}-\hat{g}_{1}+\hat{g}_{1}-\hat{g}_{0}\right)^{\prime}W\left(\widehat{\theta}-\hat{g}_{1}+\hat{g}_{1}-\hat{g}_{0}\right)\left(1+o_{p}(1)\right)$
$\displaystyle=n\left(\widehat{\theta}-\hat{g}_{1}+\hat{g}_{1}-\hat{g}_{0}\right)^{\prime}W\left(\widehat{\theta}-\hat{g}_{1}+\hat{g}_{1}-\hat{g}_{0}\right)\left(1+o_{p}(1)\right)$
$\displaystyle=[n\left(\widehat{\theta}-\hat{g}_{1}\right)^{\prime}W\left(\widehat{\theta}-\hat{g}_{1}\right)+2n\left(\hat{g}_{1}-\hat{g}_{0}\right)^{\prime}W\left(\widehat{\theta}-\hat{g}_{1}\right)+\left(\hat{g}_{1}-\hat{g}_{0}\right)^{\prime}W\left(\hat{g}_{1}-\hat{g}_{0}\right)]\left(1+o_{p}(1)\right)$
$\displaystyle=[O_{P}(1)+2O_{P}(\sqrt{n})+O_{P}(n)]\left(1+o_{P}(1)\right)$
$\displaystyle=O_{P}(n).$

Last equality is true whenever
$|g(\theta,\alpha_{1},\beta_{1})-g(\theta,\alpha_{0},\beta_{0})|>0$.
Therefore,

$min_{\alpha\in\mathcal{A}}|g(\theta_{0},\alpha,\beta_{1})-g(\theta_{0},\alpha_{0},\beta_{0})|>0$
(A-54)

is a sufficient condition for the third term to be $O_{P}(n)$, and hence, the
test to have power against $\beta=\beta_{1}$.

_Q.E.D._

Proof of Theorem  4.4. From proof of Theorem 4.3 we have that under Pittman
alternatives, and noticing that
$\sqrt{n}\left(\hat{g}_{1}-\hat{g}_{0}\right)=\widehat{\nabla}_{\beta}g_{0}\sqrt{n}(\beta_{1}-\beta_{0})=\widehat{\nabla}_{\beta}g_{0}\delta=\nabla_{\beta}g_{0}\delta+o_{P}(1)$.

$\widehat{F}(\beta_{0})=n\left(\widehat{\theta}-\hat{g}_{1}\right)^{\prime}W\left(\widehat{\theta}-\hat{g}_{1}\right)+2\delta^{\prime}\nabla_{\beta}g_{0}^{\prime}W\sqrt{n}\left(\widehat{\theta}-\hat{g}_{1}\right)+\delta^{\prime}\nabla_{\beta}g_{0}^{\prime}W\nabla_{\beta}g_{0}\delta+o_{P}(1).$
(A-55)

After noticing that the limit of this previous expression under the
alternative can be represented as:

$\lim_{n\xrightarrow{}\infty}\widehat{F}(\beta_{0})=Y^{\prime}WY+b^{\prime}WY+c,$
(A-56)

where $\sqrt{n}\left(\widehat{\theta}-\hat{g}_{1}\right)\xrightarrow{d}Y\sim
N_{m}(0,E\Sigma E^{\prime})$, $b=W\nabla_{\beta}g_{0}\delta$ and
$c=\delta^{\prime}\nabla_{\beta}g_{0}^{\prime}W\nabla_{\beta}g_{0}\delta$.
This quadratic form holds the necessary and sufficient conditions of Rao et
al. (1972) to be a non-central chi-squared distribution, $\chi^{2}(d,k)$,
where $d$ are the degrees of freedom and $k$ is the non-centrality parameter.
The value of $k$ can be found by

$k\equiv b^{\prime}E\Sigma E^{\prime}WE\Sigma E^{\prime}b$ (A-57)

After substituting the value of $b$ into previous equation, and rearranging
terms,

$k=\delta^{\prime}\nabla_{\beta}g_{0}^{\prime}W\nabla_{\beta}g_{0}\delta.$
(A-58)

A non-central chi-squared distribution $\chi^{2}(d,k^{\prime})$ stochastically
dominates $\chi^{2}(d,k)$, with $k^{\prime}>k$. Hence, If we are under the
Pittman alternative, and the direction $\delta$ is such that $k>0$, the power
function will satisfy $\pi_{\tau}(\beta_{1})>\pi_{\tau}(\beta_{0})=\tau$,
after noticing that the probability distribution of $\pi_{\tau}(\beta_{0})$
follows a chi-squared distribution with $k=0$.

_Q.E.D._

Proof of Theorem 5.1: Recall that $\widehat{M}$ and $M$ are of dimension
$m\times m$. Let $r\equiv Rank(M)$. Assume eigenvalues of $\widehat{M}$ are in
descending order. Define the following sequence of events:

$R_{n}\equiv\\{\widehat{r}.=r\\},$ (A-59)

Let $\hat{\lambda}_{i}$ and $\lambda_{i}$ be the $ith$ eigenvalue of
$\widehat{M}$ and $M$, respectively. Let

$E_{n}\equiv\\{\widehat{\lambda}_{r}>\frac{1}{n^{b}}\\},$ (A-60)

$F_{n}\equiv\\{\widehat{\lambda}_{r+1}<\frac{1}{n^{b}}\\},$ (A-61)

The idea of the proof is to show that $Pr(E_{n})\xrightarrow{}1$,
$Pr(F_{n})\xrightarrow{}1$ as $n$ grows to infinite, and hence,

$Pr(E_{n}\cap F_{n})=Pr(E_{n})+Pr(F_{n})-Pr(E_{n}\cup F_{n})\xrightarrow{}1.$
(A-62)

Since $E_{n}\cap F_{n}\subset R_{n}$, $Pr(R_{n})\geq Pr(E_{n}\cap
F_{n})\xrightarrow{}1$.
Recall assumption 10,

$\sqrt{n}\left(Vec(\widehat{A})-Vec(A)\right)\xrightarrow{d}N_{m^{2}}(0,\Omega).$
(A-63)

Define

$\widehat{B}\equiv\widehat{A}-A,$ (A-64)

the element of row $ith$ and column $jth$ of $\widehat{B}$ is defined by
$b_{ij}$. By (A-63) all the elements $b_{i,j}$ are
$O_{P}(\frac{1}{\sqrt{n}})$. Define $\hat{\mu}_{l}$ as the $lth$ eigenvalue of
the matrix $\widehat{B}$. By definition
$||\widehat{A}-A||_{2}=\max_{l\in\\{1,..,m\\}}\hat{\mu}_{l}$. Using the
Gershgorin circle theorem, the maximum eigenvalue of the matrix will be
bounded by the disc with the largest radius, i.e.,

$||\widehat{A}-A||_{2}\equiv\max_{l\in\\{1,..,m\\}}\hat{\mu}_{l}\leq\max_{i\in\\{1,..,m\\}}\sum_{j\neq
i}^{q}|b_{ij}|=O_{P}\left(\frac{1}{\sqrt{n}}\right).$ (A-65)

The last equality comes from the fact that all the elements of (A-64) are
bounded in probability at a rate $\sqrt{n}$. Equation (A-65) implies

$||\widehat{A}-A||_{2}=O_{P}\left(\frac{1}{\sqrt{n}}\right).$ (A-66)

Now we are going to use the fact that estimated eigenvalues whose population
eigenvalue is 0, converge to $0$ at a rate of $n^{-1}$. Under Assumption 10,

$\widehat{\lambda}_{l}=O_{P}\left(\frac{1}{n}\right)\hskip
14.22636pt\textit{for }r<l\leq m,$ (A-67)

as follows from Theorem 5.2.

Suppose the case of $r=0$, notice $\lambda_{l}=0$ for $l\in\\{1,..,m\\}$, and
$A=0$. Then by result (A-67)

$Pr(R_{n})=P(\max_{l\in\\{1,...,m\\}}\hat{\lambda}_{l}\leq\frac{1}{n^{b}})\xrightarrow{n\xrightarrow{}\infty}1,$
(A-68)

Suppose the case of $r>0$. Let $\bar{c}=\frac{\lambda_{r}}{2}>0$. By Weyl’s
perturbation theorem and (A-65):

$\max_{l\in\\{1,...,q\\}}|\hat{\lambda}_{l}-\lambda_{l}|\leq||\widehat{A}-A||_{2}=o_{P}(1).$
(A-69)

notice that for $n$ large enough we have $\frac{1}{n^{b}}<\bar{c}$ and using
(A-65),

$Pr(E_{n})=Pr(\hat{\lambda}_{r}\geq\frac{1}{n^{b}})\geq
Pr(\hat{\lambda}_{r}>\bar{c})=Pr(\hat{\lambda}_{r}-2\bar{c}>-\bar{c})=Pr((\hat{\lambda}_{r}-\lambda_{r})>-\bar{c})=$
$=Pr(-(\hat{\lambda}_{r}-\lambda_{r})<\bar{c})\geq
Pr(|(\hat{\lambda}_{r}-\lambda_{r})|<\bar{c})\geq
Pr(\max_{l\in\\{1,..,m\\}}|(\hat{\lambda}_{l}-\lambda_{l})|<\bar{c})\xrightarrow{n\xrightarrow{}\infty}1.$
(A-70)

Suppose $r=m$. Then, by (A-70) and noticing that $E_{n}\subset R_{n}$,
$Pr(R_{n})\xrightarrow{n\xrightarrow{}\infty}1$. Now, assume $r<m$, Recall
$F_{n}\\{\hat{\lambda}_{r+1}\leq\frac{1}{n^{b}}\\}$. By (A-67) and noticing
that $\lambda_{l}=0$ $\forall l>r$, we have:

$Pr(F_{n})=Pr(\hat{\lambda}_{r+1}\leq\frac{1}{n^{b}})\xrightarrow{n\xrightarrow{}\infty}1.$
(A-71)

Therefore $Pr(E_{n})\xrightarrow{n\xrightarrow{}\infty}1$ and
$Pr(F_{n})\xrightarrow{n\xrightarrow{}\infty}1$ which implies
$Pr(R_{n})\xrightarrow{n\xrightarrow{}\infty}1$. _Q.E.D._

Proof of Theorem 5.2 (Robin and Smith (2000), Theorem 3.1 and Theorem 3.2).
Let $K$ be a Gram matrix $K=MM^{\prime}$, where $M$ is an $m\times q$ matrix.
Perform the singular value decomposition of $M$, $M=C\Lambda D^{\prime}$,
where every column of $C$ and $D$ are eigenvectors of $MM^{\prime}$ and
$M^{\prime}M$ respectively. $\Lambda$ is a diagonal matrix of eigenvalues. We
order the eigenvalues in decreasing order. We partition
$C=[C_{r_{M}},C_{m-r_{M}}]$ and $D=[D_{r_{M}},D_{q-r_{M}}]$ where
$C_{m-r_{M}}$ and $D_{q-r_{M}}$ are the eigenvectors associated with
eigenvalues $\lambda_{i}=0$. The matrices $C$ and $D$ have the following
properties: $C^{\prime}C=I_{m}$, $D^{\prime}D=I_{q}$,
$C_{r_{M}}^{\prime}C_{m-r_{M}}=0$, $D_{r_{M}}^{\prime}D_{q-r_{M}}=0$.
Define the following $m-r_{M}\times m-r_{M}$ matrix $\hat{A}\equiv
C_{m-r_{M}}^{\prime}(\hat{M}-M)D_{q-r_{M}}$, The proof is divided into three
steps. First, we prove that the eigenvalues of the matrix
$\hat{A}\hat{A}^{\prime}$, denoted as $\tilde{\lambda}_{i}$ are
$O_{p}(n^{-1})$. Second, we prove that
$o_{p}(1)=|n\hat{A}\hat{A}^{\prime}-n\hat{\lambda}_{i}I_{m-r_{M}}|$ where
$|\cdot|$ is notation for the determinant, and $\hat{\lambda}_{i}$ are the
eigenvalues of $\hat{K}=\hat{M}\hat{M}^{\prime}$ for $i=r_{M}+1,...,m$. In
other words, we prove that the determinantal function when evaluated at
$\hat{\lambda}_{i}$ is $o_{p}(1)$. Third, we prove that
$o_{p}(1)=|n\hat{A}\hat{A}^{\prime}-n\hat{\lambda}_{i}I_{m-r_{M}}|$ implies
that $n\hat{\lambda}_{i}=n\tilde{\lambda}_{j}+o_{p}(1)$, for some
$i=r_{M}+1,...,m$ and $j=1,..,m-r_{M}$, and hence,
$\hat{\lambda}_{i}=O_{p}(n^{-1})$.

First part of the proof. By Assumption 10 and using the Delta method,

$\sqrt{n}Vec(\hat{A})\xrightarrow{d}N(0,(C_{m-r_{M}}^{\prime}\otimes
D_{q-r_{M}}^{\prime})\Omega(C_{m-r_{M}}\otimes D_{q-r_{M}})),$ (A-72)

therefore, by the continuous mapping theorem,
$nVec(\hat{A})^{\prime}Vec(\hat{A})=O_{p}(1)$. Moreover,

$nVec(\hat{A})^{\prime}Vec(\hat{A})=nTrace(\hat{A}\hat{A}^{\prime})=n\sum_{i=1}^{m-r_{M}}\tilde{\lambda}_{i}=O_{p}(1).$
(A-73)

Since $\tilde{\lambda}_{i}$ is non-negative, equation (A-73) implies
$n\tilde{\lambda}_{i}=O_{p}(1)$.

Second part of the proof. Represent $\hat{M}$ as $\hat{M}=M+(\hat{M}-M)$
notice that,

$C_{r_{M}}^{\prime}\hat{M}=C_{r_{M}}^{\prime}(C\Lambda
D^{\prime}+\hat{M}-M)=\Lambda_{r_{M}}D_{r_{M}}^{\prime}+C_{r_{M}}^{\prime}(\hat{M}-M)=\Lambda_{r_{M}}D_{r_{M}}^{\prime}+O_{p}(n^{-\nicefrac{{1}}{{2}}}),$
(A-74)

where $\Lambda_{r_{M}}$ is the eigenvalue matrix of $r$ non-zero eigenvalues
of $M$. The second equality in (A-74) comes from the orthogonality of
$C_{r_{M}}$ with $C_{m-r_{M}}$ and $C_{r_{M}}^{\prime}C_{r_{M}}=I_{r_{M}}$. On
the other hand,

$C_{m-r_{M}}^{\prime}\sqrt{n}\hat{M}=C_{m-r_{M}}^{\prime}\sqrt{n}(\hat{M}-M),$
(A-75)

after noticing that $C_{m-r_{M}}^{\prime}M=0$ by
$C_{m-r_{M}}^{\prime}C_{r_{M}}^{\prime}=0$, and that the last $m-r_{M}$
elements of $\Lambda$ being 0. By $\hat{\lambda}_{i}$ being an eigenvalue of
$\hat{M}\hat{M}^{\prime}$ we have that:

$\begin{split}0&=|\hat{M}\hat{M}^{\prime}-\hat{\lambda}_{i}I_{m}|=|(C_{r_{M}}^{\prime},C_{m-r_{M}}^{\prime}\sqrt{n})(\hat{M}\hat{M}^{\prime}-\hat{\lambda}_{i}I_{m})(C_{r_{M}},C_{m-r_{M}}\sqrt{n})|\\\
&=|\begin{pmatrix}C_{r_{M}}^{\prime}\hat{M}\hat{M}^{\prime}C_{r_{M}}&C_{r_{M}}^{\prime}\sqrt{n}\hat{M}\hat{M}^{\prime}C_{m-r_{M}}\\\
C_{m-r_{M}}^{\prime}\sqrt{n}\hat{M}\hat{M}^{\prime}C_{r_{M}}&C_{m-r_{M}}^{\prime}n\hat{M}\hat{M}^{\prime}C_{m-r_{M}}\end{pmatrix}|\\\
&=|\begin{pmatrix}\Lambda_{r_{m}}\Lambda_{r_{m}}^{\prime}+o_{p}(1)&\Lambda_{r_{m}}D_{r_{M}}^{\prime}\sqrt{n}(\hat{M}-M)^{\prime}C_{m-r_{M}}+o_{p}(1)\\\
C_{m-r_{M}}^{\prime}\sqrt{n}(\hat{M}-M)D_{r_{M}}\Lambda_{r_{m}}^{\prime}o_{p}(1)&C_{m-r_{M}}^{\prime}n(\hat{M}-M)(\hat{M}-M)^{\prime}C_{m-r_{M}}\end{pmatrix}-\hat{\lambda}_{i}\begin{pmatrix}I_{r_{M}}&0\\\
0&nI_{m-r_{M}}\end{pmatrix}|\end{split}$

where the last equality comes from (A-74) and (A-75). Using the continuity of
a determinant

$\begin{split}&=|\begin{pmatrix}\Lambda_{r_{m}}\Lambda_{r_{m}}^{\prime}&\Lambda_{r_{m}}D_{r_{M}}^{\prime}\sqrt{n}(\hat{M}-M)^{\prime}C_{m-r_{M}}\\\
C_{m-r_{M}}^{\prime}\sqrt{n}(\hat{M}-M)D_{r_{M}}\Lambda_{r_{m}}^{\prime}&C_{m-r_{M}}^{\prime}n(\hat{M}-M)(\hat{M}-M)^{\prime}C_{m-r_{M}}\end{pmatrix}-n\hat{\lambda}_{i}\begin{pmatrix}0&0\\\
0&I_{m-r_{M}}\end{pmatrix}|+o_{p}(1).\end{split}$

Now we use the fact that $|\begin{pmatrix}F&G\\\
H&J\end{pmatrix}|=|F||J-HF^{-1}G|$, for $F$ nonsingular, and
$D_{q-r_{M}}=I_{m-r_{M}}-D_{r_{M}}$

$\begin{split}0&=|\Lambda_{r_{m}}\Lambda_{r_{m}}^{\prime}||C_{m-r_{M}}^{\prime}n(\hat{M}-M)D_{q-r_{M}}D_{q-r_{M}}^{\prime}(\hat{M}-M)^{\prime}C_{m-r_{M}}-n\hat{\lambda}_{i}I_{m-r_{M}}|+o_{p}(1).\end{split}$

Therefore,

$|C_{m-r_{M}}^{\prime}n(\hat{M}-M)D_{q-r_{M}}D_{q-r_{M}}^{\prime}(\hat{M}-M)^{\prime}C_{m-r_{M}}-n\hat{\lambda}_{i}|=o_{p}(1).$
(A-76)

Third part of the proof. We use the representation of the determinant as the
roots of the characteristic polynomial evaluated at $\hat{\lambda}_{i}$:

$|C_{m-r_{M}}^{\prime}n(\hat{M}-M)(\hat{M}-M)^{\prime}C_{m-r_{M}}-n\hat{\lambda}_{i}|=(n\tilde{\lambda}_{1}-n\hat{\lambda}_{i})(n\tilde{\lambda}_{2}-n\hat{\lambda}_{i})\cdot...\cdot(n\tilde{\lambda}_{p-r_{M}}-n\hat{\lambda}_{i})=o_{p}(1).$
(A-77)

Equation $(\ref{characteristic pol})$ implies that
$(n\tilde{\lambda}_{j}-n\hat{\lambda}_{i})=o_{p}(1)$ for some
$j=1,...,m-r_{M}$. Hence, using the result obtained in the first step,
$n\tilde{\lambda}_{j}=O_{p}(1)$, and hence

$n\hat{\lambda}_{i}=O_{p}(1).$ (A-78)

_Q.E.D._

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