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

Once one leaves the context of ordinary (linear) least squares with fixed regressors and normally distributed errors, it is frequently impossible, or at least impractical, to obtain exact statistical results. It is therefore necessary to resort to asymptotic theory, that is, theory which applies to the case in which the sample size is infinitely large. Infinite samples are not available in this finite unverse, and only if they were would there be a context in which asymptotic theory was exact. Of course, since statistics itself would be quite unnecessary if samples were infinitely large, asymptotic theory would not be useful if it were exact. In practice, asymptotic theory is used as an approximation – sometimes a good one, sometimes not so good.

Most of the time, it is a pious hope rather than a firmly founded belief that asymptotic results have some relevance to the data with which one actually works. Unfortunately, more accurate approximations are available only in the simplest cases. At this time, it is probably fair to say that the principal means of getting evidence on these matters is to use Monte Carlo experiments, which we will discuss in the last chapter of this book. Since one cannot resort to a Monte Carlo experiment every time one obtains a test statistic or a set of estimates, a thorough knowledge of asymptotic theory is necessary in the present state of the art and science of econometrics. The purpose of this chapter is therefore to embark on the study of the asymptotic theory that will be used throughout the rest of the book. All of this theory is ultimately based on laws of large numbers and central limit theorems, and we will therefore spend considerable time discussing these fundamental results.

In this chapter, we discuss the basic ideas of, and mathematical prerequisites to, asymptotic theory in econometrics. We begin the next section by treating the fundamental notion of an infinite sequence, either of random or of nonrandom elements. Much of this material should be familiar to those who have studied calculus, but it is worth reviewing because it leads directly to the fundamental notions of limits and convergence, which allow us to state and prove a simple law of large numbers. In Section 4.3, we introduce the “big- $O$ ” “little-o” notation and show how the idea of a limit can be used to obtain more precise and detailed results than were obtained in Section 4.2. Data-generating processes capable of generating infinite sequences of data are introduced in Section 4.4, and this necessitates a little discussion of stochastic processes. Section $4.5$ then introduces the property of consistency of an estimator and shows how this property can often be established with the help of a law of large numbers. Asymptotic normality is the topic of Section $4.6$, and this property is obtained for some simple estimators by use of a central limit theorem. Then, in Section 4.7, we provide, mostly for the sake of later reference, a collection of definitions and theorems, the latter being laws of large numbers and central limit theorems much more sophisticated than those actually discussed in the text. In addition, we present in Section $4.7$ two sets of conditions, one centered on a law of large numbers, the other on a central limit theorem, which will be very useful subsequently as a summary of the regularity conditions needed for results proved in later chapters.

## 经济代写|计量经济学作业代写Econometrics代考|Sequences, Limits, and Convergence

The concept of infinity is one of unending fascination for mathematicians. One noted twentieth-century mathematician, Stanislaw Ulam, wrote that the continuing evolution of various notions of infinity is one of the chief driving forces behind research in mathematics (Ulam, 1976). However that may be, seemingly impractical and certainly unattainable infinities are at the heart of almost all valuable and useful applications of mathematics presently in use, among which we may count econometrics.

The reason for the widespread use of infinity is that it can provide workable approximations in circumstances in which exact results are difficult or impossible to obtain. The crucial mathematical operation which yields these approximations is that of passage to the limit, the limit being where the notion of infinity comes in. The limits of interest may be zero, finite, or infinite. Zero or finite limits usually provide the approximations that are sought: Things difficult to calculate in a realistic, finite, context are replaced by their limits as an approximation.

The first and most frequently encountered mathematical construct which may possess a limit is that of a sequence. A sequence is a countably infinite collection of things, such as numbers, vectors, matrices, or more general mathematical objects, and thus by its mere definition cannot be represented in the actual physical world. But some sequences are nevertheless very familiar. Consider the most famous sequence of all: the sequence
$${1,2,3, \ldots}$$
of the natural numbers. This is a simple-minded example perhaps, but one that exhibits some of the important properties which sequences may possess.

## 经济代写|计量经济学作业代写Econometrics代考|Rates of Convergence

We covered a lot of ground in the last section, so much so that we have by now, even if very briefly, touched on all the important purely mathematical topics to be discussed in this chapter. What remains is to flesh out the treatment of some matters and to begin to apply our theory to statistics and econometrics. The subject of this section is rates of convergence. In treating it we will introduce some very important notation, called the $\boldsymbol{O}, \boldsymbol{o}$ notation, which is read as “big-O, little- $o$ notation.” Here $O$ and $o$ stand for order and are often referred to as order symbols. Roughly speaking, when we say that some quantity is, say, $O(x)$, we mean that is of the same order, asymptotically, as the quantity $x$, while when we say that it is $o(x)$, we mean that it is of lower order than the quantity $x$. Just what this means will be made precise below.

In the last section, we discussed the random variable $b_{n}$ at some length and saw from (4.05) that its variance converged to zero, because it was proportional to $n^{-1}$. This implies that the sequence converges in probability to zero, and it can be seen that the higher moments of $b_{n}$, the third, fourth, and so on, must also tend to zero as $n \rightarrow \infty$. A somewhat tricky calculation, which interested readers are invited to try for themselves, reveals that the fourth moment of $b_{n}$ is
$$E\left(b_{n}^{4}\right)=\frac{3}{16} n^{-2}-\frac{1}{8} n^{-3}$$
that is, the sum of two terms, one proportional to $n^{-2}$ and the other to $n^{-3}$. The third moment of $b_{n}$, like the first, is zero, simply because the random variable is symmetric about zero, a fact which implies that all its odd-numbered moments vanish. Thus the second, third, and fourth moments of $b_{n}$ all converge to zero, but at different rates. Again, the two terms in the fourth moment (4.11) converge at different rates, and it is the term which is proportional to $n^{-2}$ that has the greatest importance asymptotically.

The word “asymptotically” has here been used in a slightly wider sense than we have used up to now. In Section 4.1, we said that asymptotic theory dealt with limits as some index, usually the sample size in econometrics, tends to infinity. Here we are concerned with rates of convergence rather than limits per se. Limits can be used to determine the rates of convergence of sequences as well as their limits: These rates of convergence can be defined as the limits of other sequences. For example, in the comparison of $n^{-2}$ and $n^{-3}$, the other sequence that interests us is the sequence of the ratio of $n^{-3}$ to $n^{-2}$, that is, the sequence $\left{n^{-1}\right}$. This last sequence has a limit of zero, and so, asymptotically, we can treat $n^{-3}$, or anything proportional to it, as zero in the presence of $n^{-2}$, or anything proportional to it. All of this can be expressed by the little-o notation, which expresses what is called the small-order relation: We write $n^{-3}=o\left(n^{-2}\right)$, meaning that $n^{-3}$ is of lower order than $n^{-2}$. In general, we have the following definition:
Definition 4.5.

1,2,3,…

## 经济代写|计量经济学作业代写Econometrics代考|Rates of Convergence

“渐近地”这个词在这里使用的含义比我们迄今为止使用的要广泛一些。在第 4.1 节中，我们说渐近理论将极限作为一些指标，通常是计量经济学中的样本量，趋于无穷大。在这里，我们关注的是收敛速度而不是限制本身。限制可用于确定序列的收敛速度及其限制：这些收敛速度可以定义为其他序列的限制。例如，在比较n−2和n−3，我们感兴趣的另一个序列是比率的序列n−3到n−2，即序列\左{n^{-1}\右}\左{n^{-1}\右}. 最后一个序列的极限为零，因此，我们可以渐近地对待n−3，或任何与它成比例的东西，在存在的情况下为零n−2，或任何与它成比例的东西。所有这些都可以用 little-o 表示法来表达，它表达了所谓的小阶关系：我们写n−3=这(n−2)， 意思是n−3低于n−2. 一般来说，我们有以下定义：

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