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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Nonlinear Least Squares

In the preceding chapter, we introduced some of the fundamental ideas of asymptotic analysis and stated some essential results from probability theory. In this chapter, we use those ideas and results to prove a number of important properties of the nonlinear least squares estimator.

In the next section, we discuss the concept of asymptotic identifiability of parametrized models and, in particular, of models to be estimated by NLS. In Section 5.3, we move on to treat the consistency of the NLS estimator for asymptotically identified models. In Section $5.4$, we discuss its asymptotic normality and also derive the asymptotic covariance matrix of the NLS estimator. This leads, in Section $5.5$, to the asymptotic efficiency of NLS, which we prove by extending the well-known Gauss-Markov Theorem for linear regression models to the nonlinear case. In Section $5.6$, we deal with various useful properties of NLS residuals. Finally, in Section 5.7, we consider the asymptotic distributions of the test statistics introduced in Section $3.6$ for testing restrictions on model parameters.

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

When we speak in econometrics of models to be estimated or tested, we refer to sets of DGPs. When we indulge in asymptotic theory, the DGPs in question must be stochastic processes, for the reasons laid out in Chapter 4. Without further ado then, let us denote a model that is to be estimated, tested, or both, as $M$ and a typical DGP belonging to $M$ as $\mu$. Precisely what we mean by this notation should become clear shortly.

The simplest model in econometrics is the linear regression model, but even for it there are several different ways in which it can be specified. One possibility is to write
$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}, \quad \boldsymbol{u} \sim N\left(\mathbf{0}, \sigma^{2} \mathbf{I}_{n}\right)$$

where $\boldsymbol{y}$ and $\boldsymbol{u}$ are $n$-vectors and $\boldsymbol{X}$ is a nonrandom $n \times k$ matrix. Then the (possibly implicit) assumptions are made that $\boldsymbol{X}$ can be defined by some rule (see Section 4.2) for all positive integers $n$ larger than some suitable value and that, for all such $n, \boldsymbol{y}$ follows the $N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^{2} \mathbf{I}{n}\right)$ distribution. This distribution is unique if the parameters $\beta$ and $\sigma^{2}$ are specified. We may therefore say that the DGP is completely characterized by the model parameters. In other words, knowledge of the model parameters $\beta$ and $\sigma^{2}$ uniquely identify an element $\mu$ of $M$. On the other hand, the linear regression model can also be written as $$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{\Pi ID}\left(\mathbf{0}, \sigma^{2} \mathbf{I}{n}\right)$$
with no assumption of normality. Many aspects of the theory of linear regressions are just as applicable to $(5.02)$ as to $(5.01)$; for instance, the OLS estimator is unbiased, and its covariance matrix is $\sigma^{2}\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1}$. But the distribution of the vector $\boldsymbol{u}$, and hence also that of $\boldsymbol{y}$, is now only partially characterized even when $\beta$ and $\sigma^{2}$ are known. For example, the errors $u_{t}$ could be skewed to the left or to the right, could have fourth moments larger or smaller than $3 \sigma^{4}$, or might even possess no moments of order higher than, say, the sixth. DGPs with all sorts of properties, some of them very strange, are special cases of the linear regression model if it is defined by (5.02) rather than $(5.01)$.

We may call the sets of DGPs associated with (5.01) and (5.02) $M_{1}$ and $M_{2}$, respectively. These sets of DGPs are different, $M_{1}$ being in fact a proper subset of $M_{2}$. Although for any DGP $\mu \in M_{2}$ there is a $\beta$ and a $\sigma^{2}$ that correspond to, and partially characterize, $\mu$, the inverse relation does not exist. For a given $\beta$ and $\sigma^{2}$ there is an infinite number of DGPs in $\mathbb{M}{2}$ (only one of which is in $M{1}$ ) that all correspond to the same $\beta$ and $\sigma^{2}$. Thus we must for our present purposes consider $(5.01)$ and $(5.02)$ as different models even though the parameters used in them are the same.

The vast majority of statistical and econometric procedures for estimating models make use, as does the linear regression model, of model parameters. Typically, it is these parameters that we will be interested in estimating. As with the linear regression model, the parameters may or may not fully characterize a DGP in the model. In either case, it must be possible to associate a parameter vector in a unique way to any DGP $\mu$ in the model MI, even if the same parameter vector is associated with many DGPs.

## 经济代写|计量经济学作业代写Econometrics代考|Consistency of the NLS Estimator

A univariate “nonlinear regression model” has up to now been expressed in the form
$$\boldsymbol{y}=\boldsymbol{x}(\boldsymbol{\beta})+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{\Pi D}\left(\mathbf{0}, \sigma^{2} \mathbf{I}{\mathrm{n}}\right)$$ where $\boldsymbol{y}, \boldsymbol{x}(\boldsymbol{\beta})$, and $\boldsymbol{u}$ are $n$-vectors for some sample size $n$. The model parameters are therefore $\boldsymbol{\beta}$ and either $\sigma$ or $\sigma^{2}$. The regression function $x{t}(\boldsymbol{\beta})$, which is the $t^{\text {th }}$ element of $\boldsymbol{x}(\boldsymbol{\beta})$, will in general depend on a row vector of variables $Z_{t}$. The specification of the vector of error terms $\boldsymbol{u}$ is not complete,

since the distribution of the $u_{t}$ ‘s has not been specified. Thus, for a sample of size $n$, the model M described by $(5.08)$ is the set of all DGPs generating samples $\boldsymbol{y}$ of size $n$ such that the expectation of $y_{t}$ conditional on some information set $\Omega_{t}$ that includes $Z_{t}$ is $x_{t}(\boldsymbol{\beta})$ for some parameter vector $\boldsymbol{\beta} \in \mathbb{R}^{k}$, and such that the differences $y_{t}-x_{t}(\beta)$ are independently distributed error terms with common variance $\sigma^{2}$, usually unknown.

It will be convenient to generalize this specification of the DGPs in M a little, in order to be able to treat dynamic models, that is, models in which there are lagged dependent variables. Therefore, we explicitly recognize the possibility that the regression function $x_{t}(\boldsymbol{\beta})$ may include among its (until now implicit) dependences an arbitrary but bounded number of lags of the dependent variable itself. Thus $x_{t}$ may depend on $y_{t-1}, y_{t-2}, \ldots, y_{t-l}$, where $l$ is a fixed positive integer that does not depend on the sample size. When the model uses time-series data, we will therefore take $x_{t}(\boldsymbol{\beta})$ to mean the expectation of $y_{t}$ conditional on an information set that includes the entire past of the dependent variable, which we can denote by $\left{y_{s}\right}_{s=1}^{t-1}$, and also the entire history of the exogenous variables up to and including the period $t$, that is, $\left{\boldsymbol{Z}{s}\right}{s=1}^{t}$. The requirements on the disturbance vector $\boldsymbol{u}$ are unchanged.
For asymptotic theory to be applicable, we must next provide a rule for extending (5.08) to samples of arbitrarily large size. For models which are not dynamic (including models estimated with cross-section data, of course), so that there are no time trends or lagged dependent variables in the regression functions $x_{t}$, there is nothing to prevent the simple use of the fixcd-inrepeated-samples notion that we discussed in Section 4.4. Specifically, we consider only sample sizes that are integer multiples of the actual sample size $m$ and then assume that $x_{N m+t}(\boldsymbol{\beta})=x_{t}(\boldsymbol{\beta})$ for $N>1$. This assumption makes the asymptotics of nondynamic models very simple compared with those for dynamic models. 3

Some econometricians would argue that the above solution is too simpleminded when one is working with time-series data and would prefer a rule like the following. The variables $Z_{t}$ appearing in the regression functions will usually themselves display regularities as time series and may be susceptible to modeling as one of the standard stochastic processes used in time-series analysis; we will discuss these standard processes at somewhat greater length in Chapter 10. In order to extend the DGP (5.08), the out-of-sample values for the $Z_{t}$ ‘s should themselves be regarded as random, being generated by appropriate processes. The introduction of this additional randomness complicates the asymptotic analysis a little, but not really a lot, since one would always assume that the stochastic processes generating the $Z_{t}$ ‘s were independent of the stochastic process generating the disturbance vector $\boldsymbol{u}$.

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