### 经济代写|计量经济学作业代写Econometrics代考|Restrictions and Pretest Estimators

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• Statistical Inference 统计推断
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• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|计量经济学作业代写Econometrics代考|Restrictions and Pretest Estimators

In the preceding three sections, we have discussed hypothesis testing at some length, but we have not said anything about one of the principal reasons for imposing and testing restrictions. In many cases, restrictions are not implied by any economic theory but are imposed by the investigator in the hope that a restricted model will be easier to estimate and will yield more efficient estimates than an unrestricted model. Tests of this sort of restriction include DWH tests (Chapter 7), tests for serial correlation (Chapter 10), common factor restriction tests (Chapter 10), tests for structural change (Chapter 11), and tests on the length of a distributed lag (Chapter 19). In these and many other cases, restrictions are tested in order to decide which model to use as a basis for inference about the parameters of interest and to weed out models that appear to be incompatible with the data. However, because estimation and testing are based on the same data, the properties of the final estimates may be very difficult to analyze. This is the problem of pretesting.

For simplicity, we will in this section consider only the case of linear regression models with fixed regressors, some coefficients of which are subject to zero restrictions. The restricted model will be (3.18), in which $\boldsymbol{y}$ is regressed on an $n \times(k-r)$ matrix $\boldsymbol{X}{1}$, and the unrestricted model will be (3.19), in which $\boldsymbol{y}$ is regressed on $\boldsymbol{X}{1}$ and an $n \times r$ matrix $\boldsymbol{X}{2}$. The OLS estimates of the parameters of the restricted model are $$\overline{\boldsymbol{\beta}}{1}=\left(\boldsymbol{X}{1}^{\top} \boldsymbol{X}{1}\right)^{-1} \boldsymbol{X}{1}^{\top} \boldsymbol{y}$$ The OLS estimates of these same parameters in the unrestricted model can easily he found hy using, the FWT. Thenrem. They are $$\hat{\boldsymbol{\beta}}{1}=\left(\boldsymbol{X}{1}^{\top} \boldsymbol{M}{2} \boldsymbol{X}{1}\right)^{-1} \boldsymbol{X}{1}^{\top} \boldsymbol{M}{2} \boldsymbol{y}$$ where $\boldsymbol{M}{2}$ denotes the matrix that projects orthogonally onto $\mathcal{S}^{\perp}\left(\boldsymbol{X}{2}\right)$. It is natural to ask how well the estimators $\overline{\boldsymbol{\beta}}{1}$ and $\hat{\boldsymbol{\beta}}{1}$ perform relative to each other. If the data are actually generated by the DGP (3.22), which is a special case of the restricted model, they are evidently both unbiased. However, as we will demonstrate in a moment, the restricted estimator $\overline{\boldsymbol{\beta}}{1}$ is more efficient than the unrestricted estimator $\hat{\beta}{1}$. One estimator is said to be more efficient than another if the covariance matrix of the inefficient estimator minus the covariance matrix of the efficient one is a positive semidefinite matrix; see Section 5.5. If $\overline{\boldsymbol{\beta}}{1}$ is more efficient than $\hat{\boldsymbol{\beta}}{1}$ in this sense, then any linear combination of the elements of $\bar{\beta}{1}$ must have variance no larger than the corresponding linear combination of the elements of $\hat{\beta}_{1}$.

The proof that $\overline{\boldsymbol{\beta}}{1}$ is more efficient than $\hat{\boldsymbol{\beta}}{1}$ under the DGP (3.22) is very simple. The difference between the covariance matrices of $\hat{\boldsymbol{\beta}}{1}$ and $\overline{\boldsymbol{\beta}}{1}$ is
$$\sigma_{0}^{2}\left(\boldsymbol{X}{1}^{\top} \boldsymbol{M}{2} \boldsymbol{X}{1}\right)^{-1}-\sigma{0}^{2}\left(\boldsymbol{X}{1}^{\top} \boldsymbol{X}{1}\right)^{-1}$$

## 经济代写|计量经济学作业代写Econometrics代考|pretest estimator

Pretest estimators are used all the time. Whenever we test some aspect of a model’s specification and then decide, on the basis of the test results, what version of the model to estimate or what estimation method to use, we are employing a pretest estimator. Unfortunately, the properties of pretest estimators are, in practice, very difficult to know. The problems can been from the example we have been studying. Suppose the restrictions hold. Then the estimator we would like to use is the restricted estimator, $\overline{\boldsymbol{\beta}}{1}$. But, $\alpha \%$ of the time, the $F$ test will incorrectly reject the null hypothesis and $\beta{1}$ will be equal to the unrestricted estimator $\hat{\boldsymbol{\beta}}{1}$ instead. Thus $\check{\boldsymbol{\beta}}{1}$ must be less efficient than $\overline{\boldsymbol{\beta}}{1}$ when the restrictions do in fact hold. Moreover, since the estimated covariance matrix reported by the regression package will not take the pretesting into account, inferences about $\boldsymbol{\beta}{1}$ may be misleading.

On the other hand, when the restrictions do not hold, we may or may not want to use the unrestricted estimator $\hat{\beta}{1}$. Depending on how much power the $F$ test has, $\check{\boldsymbol{\beta}}{1}$ will sometimes be equal to $\overline{\boldsymbol{\beta}}{1}$ and sometimes be equal to $\hat{\boldsymbol{\beta}}{1}$. It will certainly not be unbiased, because $\overline{\boldsymbol{\beta}}{1}$ is not unbiased, and it may be more or less efficient (in the sense of mean squared error) than the unrestricted estimator. Inferences about $\check{\boldsymbol{\beta}}{1}$ based on the usual estimated OLS covariance matrix for whichever of $\overline{\boldsymbol{\beta}}{1}$ and $\hat{\boldsymbol{\beta}}{1}$ it turns out to be equal to may be misleading, because they fail to take into account the pretesting that occurred previously.

In practice, there is often not very much that we can do about the problems caused by pretesting, except to recognize that pretesting adds an additional element of uncertainty to most problems of statistical inference. Since $\alpha$, the level of the preliminary test, will affect the properties of $\boldsymbol{\beta}_{1}$, it may be worthwhile to try using different values of $\alpha$. Conventional significance levels such as $.05$ are certainly not optimal in general, and there is a literature on how to choose better ones in specific cases; see, for example, Toyoda and Wallace (1976). However, real pretesting problems are much more complicated than the one we have discussed as an example or the ones that have been studied in the literature. Every time one subjects a model to any sort of test, the result of that test may affect the form of the final model, and the implied pretest estimator therefore becomes even more complicated. It is hard to see how this can be analyzed formally.

## 经济代写|计量经济学作业代写Econometrics代考|Conclusion

This chapter has provided an introduction to several important topics: estimation of covariance matrices for NLS estimates, the use of such covariance matrix estimates for constructing confidence intervals, basic ideas of hypothesis testing, the justification for testing linear restrictions on linear regression models by means of $t$ and $F$ tests, the three classical principles of hypothesis testing and their application to nonlinear regression models, and pretesting. At a number of points we were forced to be a little vague and to refer to results on the asymptotic properties of nonlinear least squares estimates that we have not yet proved. Proving those results will be the object of the next two chapters. Chapter 4 discusses the basic ideas of asymptotic analysis, including consistency, asymptotic normality, central limit theorems, laws of large numbers, and the use of “big- $O$ ” and “little-o” notation. Chapter 5 then uses these concepts to prove the consistency and asymptotic normality of nonlinear least squares estimates of univariate nonlinear regression models and to derive the asymptotic distributions of the test statistics discussed in this chapter. It also proves a number of related asymptotic results that will be useful later on.

## 经济代写|计量经济学作业代写Econometrics代考|Restrictions and Pretest Estimators

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