# 经济代写|计量经济学代写Econometrics代考|Find2022

## 经济代写|计量经济学代写Econometrics代考|One-Step Efficient Estimation

It is sometimes easy to obtain consistent but inefficient estimates but relatively difficult to obtain NLS estimates. This may, for example, be the case when the nonlinear model to be estimated is really a linear model subject to nonlinear restrictions, as many rational expectations models are. In these circumstances, a useful result is that taking just one step from these initial consistent estimates, using the Gauss-Newton regression, yields estimates that are asymptotically equivalent to NLS estimates.

Let $\boldsymbol{\beta}$ denote the initial estimates, which are assumed to be root- $n$ consistent. The GNR is then
$$\boldsymbol{y}-\dot{\boldsymbol{x}}=\dot{\boldsymbol{X}} \boldsymbol{b}+\text { residuals, }$$
and the estimate of $\boldsymbol{b}$ from this regression is
$$\dot{\boldsymbol{b}}=\left(\dot{\boldsymbol{X}}^{\top} \dot{\boldsymbol{X}}\right)^{-1} \dot{\boldsymbol{X}}^{\top}(\boldsymbol{y}-\dot{\boldsymbol{x}}) .$$
Thus the one-step efficient estimator is
$$\dot{\boldsymbol{\beta}}=\dot{\boldsymbol{\beta}}+\dot{\boldsymbol{b}} .$$
Taylor expanding $\boldsymbol{x}(\boldsymbol{\beta})$ around $\boldsymbol{\beta}=\boldsymbol{\beta}_0$ yields
$$\dot{\boldsymbol{x}} \cong \boldsymbol{x}_0+\boldsymbol{X}_0\left(\boldsymbol{\beta}-\boldsymbol{\beta}_0\right),$$
where $\boldsymbol{x}_0 \equiv \boldsymbol{x}\left(\boldsymbol{\beta}_0\right)$ and $\boldsymbol{X}_0 \equiv \boldsymbol{X}\left(\boldsymbol{\beta}_0\right)$. Substituting this into (6.31), replacing $\boldsymbol{y}$ by its value under the DGP, $\boldsymbol{x}_0+\boldsymbol{u}$, and inserting appropriate powers of $n$ so that all quantities are $O(1)$, leads to the result that
\begin{aligned} n^{1 / 2} \dot{\boldsymbol{b}} & \cong n^{-1 / 2}\left(n^{-1} \dot{\boldsymbol{X}}^{\top} \dot{\boldsymbol{X}}\right)^{-1} \dot{\boldsymbol{X}}^{\top}\left(\boldsymbol{x}_0+\boldsymbol{u}-\boldsymbol{x}_0-\boldsymbol{X}_0\left(\boldsymbol{\boldsymbol { \beta }}-\boldsymbol{\beta}_0\right)\right) \ &=\left(n^{-1} \dot{\boldsymbol{X}}^{\top} \dot{\boldsymbol{X}}\right)^{-1}\left(n^{-1 / 2} \dot{\boldsymbol{X}}^{\top} \boldsymbol{u}-\left(n^{-1} \dot{\boldsymbol{X}}^{\top} \boldsymbol{X}_0\right) n^{1 / 2}\left(\boldsymbol{\boldsymbol { \beta }}-\boldsymbol{\beta}_0\right)\right) \end{aligned}
But notice that
$$n^{-1} \dot{\boldsymbol{X}}^{\top} \dot{\boldsymbol{X}} \stackrel{a}{=} n^{-1} \boldsymbol{X}_0^{\top} \boldsymbol{X}_0 \stackrel{a}{=} n^{-1} \dot{\boldsymbol{X}}^{\top} \boldsymbol{X}_0$$

## 经济代写|计量经济学代写Econometrics代考|Hypothesis Tests Using Any Consistent Estimates

The procedures for testing that we discussed in Sections $6.4$ and $6.5$ all involve evaluating the artificial regression at restricted NLS estimates and thus yield test statistics based on the LM principle. But when the restricted regression function is nonlinear, it is not always convenient to obtain NLS estimates. Luckily, one can perform tests by means of a GNR whenever any root- $n$ consistent estimates that satisfy the null hypothesis are available. We briefly discuss how to do so in this section.

Suppose we are dealing with the situation discussed in Section 6.4, in which the parameter vector $\boldsymbol{\beta}$ is partitioned as $\left[\boldsymbol{\beta}_1: \boldsymbol{\beta}_2\right]$, and the null hypothesis is that $\boldsymbol{\beta}_2=\mathbf{0}$. Assume that we have available a vector of root- $n$
$$y-\dot{x}=\dot{X}_1 b_1+\dot{X}_2 b_2+\text { residuals. }$$
The explained sum of squares from this regression is
$$(\boldsymbol{y}-\dot{\boldsymbol{x}})^{\top} \dot{\boldsymbol{P}}_1(\boldsymbol{y}-\dot{\boldsymbol{x}})+(\boldsymbol{y}-\dot{\boldsymbol{x}})^{\top} \dot{\boldsymbol{M}}_1 \dot{\boldsymbol{X}}_2\left(\dot{\boldsymbol{X}}_2^{\top} \dot{\boldsymbol{M}}_1 \dot{\boldsymbol{X}}_2\right)^{-1} \dot{\boldsymbol{X}}_2^{\top} \dot{\boldsymbol{M}}_1(\boldsymbol{y}-\dot{\boldsymbol{x}}) .$$
The first term here is the explained sum of squares from a regression of $\boldsymbol{y}-\dot{\boldsymbol{x}}$ on $\dot{\boldsymbol{X}}_1$ alone, and the second term is the increase in the explained sum of squares brought about by the inclusion of $\dot{\boldsymbol{X}}_2$. Note that the first term is in general not zero, because $\boldsymbol{\beta}_1$ will not in general satisfy the first-order conditions for NLS estimates of the restricted model.

## 经济代写|计量经济学代写Econometrics代考|One-Step Efficient Estimation

$$\boldsymbol{y}-\dot{\boldsymbol{x}}=\dot{\boldsymbol{X}} \boldsymbol{b}+\text { residuals, }$$

$$\dot{\boldsymbol{b}}=\left(\dot{\boldsymbol{X}}^{\top} \dot{\boldsymbol{X}}\right)^{-1} \dot{\boldsymbol{X}}^{\top}(\boldsymbol{y}-\dot{\boldsymbol{x}}) .$$

$$\dot{\boldsymbol{\beta}}=\dot{\boldsymbol{\beta}}+\dot{\boldsymbol{b}} .$$

$$\dot{\boldsymbol{x}} \cong \boldsymbol{x}_0+\boldsymbol{X}_0\left(\beta-\boldsymbol{\beta}_0\right),$$

$$n^{1 / 2} \dot{\boldsymbol{b}} \cong n^{-1 / 2}\left(n^{-1} \dot{\boldsymbol{X}}^{\top} \dot{\boldsymbol{X}}\right)^{-1} \dot{\boldsymbol{X}}^{\top}\left(\boldsymbol{x}_0+\boldsymbol{u}-\boldsymbol{x}_0-\boldsymbol{X}_0\left(\boldsymbol{\beta}-\boldsymbol{\beta}_0\right)\right) \quad=\left(n^{-1} \dot{\boldsymbol{X}}^{\top} \dot{\boldsymbol{X}}\right)^{-1}\left(n^{-1 / 2} \dot{\boldsymbol{X}}^{\top} \boldsymbol{u}\right.$$

$$n^{-1} \dot{\boldsymbol{X}}^{\top} \dot{\boldsymbol{X}} \stackrel{a}{=} n^{-1} \boldsymbol{X}_0^{\top} \boldsymbol{X}_0 \stackrel{a}{=} n^{-1} \dot{\boldsymbol{X}}^{\top} \boldsymbol{X}_0$$

## 经济代写|计量经济学代写Econometrics代考|Hypothesis Tests Using Any Consistent Estimates

$$y-\dot{x}=\dot{X}_1 b_1+\dot{X}_2 b_2+\text { residuals. }$$

$$(\boldsymbol{y}-\dot{\boldsymbol{x}})^{\top} \dot{\boldsymbol{P}}_1(\boldsymbol{y}-\dot{\boldsymbol{x}})+(\boldsymbol{y}-\dot{\boldsymbol{x}})^{\top} \dot{\boldsymbol{M}}_1 \dot{\boldsymbol{X}}_2\left(\dot{\boldsymbol{X}}_2^{\top} \dot{\boldsymbol{M}}_1 \dot{\boldsymbol{X}}_2\right)^{-1} \dot{\boldsymbol{X}}_2^{\top} \dot{\boldsymbol{M}}_1(\boldsymbol{y}-\dot{\boldsymbol{x}})$$

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