On the performance of block-bootstrap continuously updated GMM for a class of non-linear conditional moment models

Abstract

In the context of the continuously updated generalized-methods-of-moments (GMM), this study evaluates the finite sample properties of Wald- and criterion-based bootstrap inference for a class of models defined by non-linear conditional moment functions. This work provides simulation evidence that validates the moving block-bootstrap (MBB) as an alternative to asymptotic approximation for robust finite sample GMM inference. The study considers data generating processes with highly non-linear conditional moment functions, weak instruments, and near failure of the identification condition. In the absence of a consensus on best practice when identification is weak, Monte Carlo results of this study are encouraging to the empirical researchers. For criterion-based tests, the MBB performs fairly well in reducing the error in the rejection frequency that occurs when first-order asymptotic critical values are used. In particular, it is possible to improve finite sample inference by inverting bootstrap Wald-type statistics which are commonly used in practice The bootstrap percentile-$t$ confidence intervals performed better than the asymptotic confidence intervals but only marginally in weakly identified specifications with high non-linear moment functions.

Appendices

Appendix 1: Martingale differences

This proof is a special case of the general theory and proof developed in Andrews (2002).

Let $\{g^{*}_{t}\}$, where $g^{*}_{t}=g^{*}(X_{t},\widehat{\theta })$, be a martingale first difference sequence satisfying

$$\begin{aligned} \mathbb{E }\,(g^{*}_{t}g^{*'}_{s})=0,\quad \text{ for } t \ne s. \end{aligned}$$

This implies that the matrix $\widetilde{W}_{n}$ for the non-overlapping blocks reduces to

$$\begin{aligned} \widetilde{W}_{n}&= \frac{1}{n}\sum _{We=0}^{\phi -1}\sum _{j=1}^{\omega }g^{*}_{i\omega +j}g^{*\prime }_{i\omega +j}\\&= \frac{1}{n}\sum _{t=1}^{n}g^{*}_{t}g^{*\prime }_{t}=\varOmega ^{*}_{n}(\widehat{\theta }). \end{aligned}$$

Plugging $\widetilde{W}_{n}$ in $V_{n}$ and using $W_{n}=\varOmega ^{*}(\widehat{\theta })^{-1}$ leads to

$$\begin{aligned} V_{n} =\widehat{M}_{n}\widehat{M}_{n}. \end{aligned}$$

(27)

Let $\widehat{M}_{n}=I_{m\times q}-\widehat{P}_{n}$ where$\widehat{P}_{n}=\widehat{\varOmega }_{n}^{-\frac{1}{2}}\widehat{G}_{n} [\widehat{G}_{n}^{\prime }\widehat{\varOmega }_{n}^{-1}\widehat{G}_{n}]^{-1} \widehat{G}_{n}^{\prime }\widehat{\varOmega }_{n}^{-\frac{1}{2}}. \widehat{P}_{n}$ is an idempotent matrix, that is $\widehat{P}_{n}=\widehat{P}_{n}'$ and $\widehat{P}_{n}=\widehat{P}_{n}\widehat{P}_{n}$.Therefore,

$$\begin{aligned} V_{n}&= \widehat{M}_{n}\widehat{M}_{n} \\&= I_{m\times q}-\widehat{P}_{n}+\widehat{P}_{n}\widehat{P}_{n}=I_{m\times q} \end{aligned}$$

Appendix 2: Quadrature approximation

The following is a description of the quadrature approximation as outlined in Tauchen (1986a), Tauchen and Hussey (1991) and Kocherlakota (1990). See these references for further details about these derivations.

The Euler equation for the calibrated economy can be written in terms of prices and dividends,

$$\begin{aligned} E_{\theta }\left[ \beta c_{t+1}^{-\gamma }(1+v_{t+1})d_{t+1}\bigm | \mathfrak I _{t}\right] =v_{t} \end{aligned}$$

(28)

where we denote by $d_{t}=\frac{D_{t}}{D_{t-1}}$ the dividend growth, $v_{t}=\frac{P_{t}}{D_{t}}$ the price-dividend ratio and $c_{t}=\frac{C_{t+1}}{C_{t}}$ the consumption growth.

Our economy is similar to the one described by Kocherlakota (1990) with three assets: the Risk free $R_{f}$ which pays one unit of consumption, the market portfolio $MP$ which pays $C_{t}$ in period $t$ and the stock market $SM$ with dividend pay-offs $D_{t}$ in period $t$. Equation (28) implies a conditional moment restriction for each asset $j\in \{MP,SM,R_{f}\}$.

The only driving random processes in the model are $c_{t}$ and $d_{t}$, and so, conditional on the past, $P_{t}$, or equivalently, $v_{t}$ is a deterministic function of $c_{t}$ and $d_{t}$ implicitly given by the Euler equation (28).

The deterministic function that gives $v_{t}$ as a solution to (28) cannot be found in closed analytic form. The quadrature approximation uses a finite state Markov process to approximate the bivariate vector autoregression, and enables the approximate solution to be obtained by matrix inversion.

The approximation involves fitting a 8 state Markov chain to log consumption growth and log dividend growth calibrated so as to approximate the first order VAR in (25).

Let $\widetilde{c}(l)$ and $\widetilde{d}(l),\,l=1,\ldots ,16,$ denote the abscissa for the $8-point$ quadrature rule. Each combination of abscissa $\{(k,k'), k=1,\ldots ,8; k'=1,\ldots ,8\}$ defines a state $s_{j}$, for $j=1,\ldots ,\widetilde{N}$, where $\widetilde{N}=8^{2}$. Let $\widetilde{c}_{s_{j}}$ and $\widetilde{d}_{s_{j}}$ denote the values of $c$ and $d$ in state $s_{j}$ and let $\widetilde{Y}_{s_{j}}=(\widetilde{c}_{s_{j}},\widetilde{d}_{s_{j}})$.

The transition matrix $Pi$ for the Markov process defined by $Pi_{k,j}=P\left( Y_{t+1}=\widetilde{y}_{s_{j}}|\right.$ $\left. Y_{t}=\widetilde{y}_{s_{k}}\right)$ where,

$$\begin{aligned} Pi_{k,j}=\frac{p(\widetilde{y}_{s_{j}}|\widetilde{y}_{s_{k}})}{S(\widetilde{y}_{s_{k}})p(\widetilde{y}_{s_{j}})}w_{j} \end{aligned}$$

(29)

where $S(x)=\sum _{l=1}^{\widetilde{N}}\frac{p(\widetilde{y}_{s_{l}}|x)}{p(\widetilde{y}_{s_{l}})}w_{l}$. The Gaussian rule defines $p(x|\widetilde{y}_{s_{k}})$ and $p(x)$ as density functions for the bi-variate normals $N(\widetilde{y}_{s_{k}},\Sigma )$ and $N({\varvec{\mu }},\Sigma _{Y})$, respectively, where $\Sigma _{Y}$ solves $\Sigma _{Y}=\varPhi \Sigma _{Y}\varPhi '+\Sigma$. The weights $w_{j}$ are computed using a Hermite Gauss rule.

The solution to the integral in the Euler equation (28) is characterized by the solution to the system of $\widetilde{N}$ linear equations given by the discrete approximation in

$$\begin{aligned} \beta \sum _{j=1}^{\widetilde{N}} \Pi _{k,j}(\widetilde{c}_{s_{j}})^{-\gamma }(1+\widetilde{v}_{s_{j}}) \widetilde{d}_{s_{j}}=\widetilde{v}_{s_{k}},&\quad k=1,\ldots ,\widetilde{N} \end{aligned}$$

(30)

The solution exists if all the eigenvalues of the $\widetilde{N}\times \widetilde{N}$ matrix ${{\fancyscript{S}}}$ defined by the elements ${{\fancyscript{S}}}_{k,j}=\beta \Pi _{k,j}(\widetilde{c}_{s_{j}})^{-\gamma } \widetilde{d}_{s_{j}}$ lie within the unit circle. The solution is characterized by,

$$\begin{aligned} \widetilde{v}={(\mathbf{I}_{\widetilde{\varvec{N}}}-{\fancyscript{S}})^{-1}{\fancyscript{S}} {\iota }_{\widetilde{N}}} \end{aligned}$$

(31)

where ${\iota }_{\widetilde{N}}$ is a $\widetilde{N}\times 1$ column vector of ones. For the market portfolio, the dividend ratio is equal to $\widetilde{d}_{MP,s_{j}}=\widetilde{c}_{s_{j}}$ and therefore ${{\fancyscript{S}}}_{k,j}=\beta \Pi _{k,j}(\widetilde{c}_{s_{j}})^{1-\gamma }$. From the series of the equilibrium price-dividend ratios $\widetilde{v}_{i}, i\in \{F,MP,SM\}$, the corresponding returns are computed as:

$$\begin{aligned} \widetilde{r}_{MP,s_{k}s_{j}}&= \widetilde{c}_{s_{j}} \frac{1+\widetilde{v}_{MP,s_{j}}}{\widetilde{v}_{MP,s_{k}}}\\ \widetilde{r}_{SM,s_{k}s_{j}}&= \widetilde{d}_{s_{j}} \frac{1+\widetilde{v}_{SM,s_{j}}}{\widetilde{v}_{SM,s_{k}}}\\ \widetilde{r}_{F,s_{k}}&= 1/\left( \sum _{j=1}^{\widetilde{T}} \beta \Pi _{k,j}(\widetilde{c}_{s_{j}})^{-\gamma }\right) . \end{aligned}$$