### 经济代写|计量经济学作业代写Econometrics代考| Consistency and Laws of Large Numbers

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• 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|Consistency and Laws of Large Numbers

We begin this scetion by introducing the notion of consistency, one of the most basic ideas of asymptotic theory. When one is interested in estimating parameters from data, it is desirable that the parameter estimates should have certain properties. In Chapters 2 and 3 , we saw that, under certain regularity

conditions, the OLS estimator is unbiased and follows a normal distribution with a covariance matrix that is known up to a factor of the error variance, which factor can itself be estimated in an unbiased manner. We were not able in those chapters to prove any corresponding results for the NLS estimator, and it was remarked that asymptotic theory would be necessary in order to do so. Consistency is the first of the desirable asymptotic properties that an estimator may possess. In Chapter 5 we will provide conditions under which the NLS estimator is consistent. Here we will content ourselves with introducing the notion itself and illustrating the close link that exists between laws of large numbers and proofs of consistency.

An estimator $\hat{\beta}$ of a vector of parameters $\beta$ is said to be consistent if it converges to its true value as the sample size tends to infinity. That statement is not false or even seriously misleading, but it implicitly makes a number of assumptions and uses undefined terms. Let us try to rectify this and, in so doing, gain a better understanding of what consistency means.

First, how can an estimator converge? It can do so if we convert it to a sequence. To this end, we write $\hat{\beta}^{n}$ for the estimator that results from a sample of size $n$ and then define the estimator $\hat{\beta}$ itself as the sequence $\left{\hat{\beta}^{n}\right}_{n=m \text {. }}$. The lower limit $m$ of the sequence will usually be assumed to be the smallest sample size that allows $\hat{\beta}^{n}$ to be computed. For example, if we denote the regressand and regressor matrix for a linear regression done on a sample of size $n$ by $\boldsymbol{y}^{n}$ and $\boldsymbol{X}^{n}$, respectively, and if $\boldsymbol{X}^{n}$ is an $n \times k$ matrix, then $m$ cannot be any smaller than $k$, the number of regressors. For $n>k$ we have as usual that $\hat{\beta}^{n}=\left(\left(\boldsymbol{X}^{n}\right)^{\top} \boldsymbol{X}^{n}\right)^{-1}\left(\boldsymbol{X}^{n}\right)^{\top} \boldsymbol{y}^{n}$, and this formula embodies the rule which generates the sequence $\hat{\beta}$.

An element of a sequence $\hat{\boldsymbol{\beta}}$ is a random variable. If it is to converge to a true value, we must say what kind of convergence we have in mind. since we have seen that more than one kind is available. If we use almost sure convergence, we will say that we have strong consistency or that the estimator is strongly consistent. Sometimes such a claim is possible. More frequently we use convergence in probability and so obtain only weak consistency. Here “strong” and “weak” are used in the same sense as in the definitions of strong and weak laws of large numbers.

Next, what is meant by the “true value”? We answer this question in detail in the next chapter, but here we must at least note that convergence of a sequence of random variables to any kind of limit depends on the rule, or DGP, which generated the sequence. For example, if the rule ensures that, for any sample size $n$, the regressand and regressor matrix of a linear regression are in fact related by the equation
$$\boldsymbol{y}^{n}=\boldsymbol{X}^{n} \beta_{0}+\boldsymbol{u}^{n}$$
for some fixed vector $\beta_{0}$, with $\boldsymbol{u}^{n}$ an $n$-vector of white noise errors, then the true value for this DGP will be $\beta_{0}$. The estimator $\hat{\beta}$, to be consistent, should

converge, under the $\operatorname{DGP}(4.19)$, to $\beta_{0}$ whatever the fixed value $\beta_{0}$ happens to be. However, if the DGP is such that (4.19) does not hold for any $\boldsymbol{\beta}_{0}$ at all, then we cannot give any meaning to the term “consistency” as we are using it at present.

After this preamble, we can finally investigate consistency in a particular case. We could take as an example the linear regression (4.19), but that would lead us into consideration of too many side issues that will be dealt with in the next chapter. Instead, we will consider the very instructive example that is afforded by the Fundamental Theorem of Statistics, a simple version of which we will now prove. This theorem, which is indeed fundamental to all statistical inference, states that if we sample randomly with replacement from a population, the empirical distribution function is consistent for the population distribution function.

Let us formalize this statement and then prove it. The term population is used in its statistical sense of a set, finite or infinite, from which independent random draws can be made. Each such draw is a member of the population. By random sampling with replacement is meant a procedure which ensures that in each draw the probability that any given member of the population is drawn is unchanging. A random sample will be a finite set of draws. Formally, the population is represented by a c.d.f. $F(x)$ for a scalar random variable $x$. The draws from the population are identified with different, independent, realizations of $x$.

## 经济代写|计量经济学作业代写Econometrics代考|6 Asymptotic Normality and Central Limit Theorems

There is the same sort of close connection between the property of asymptotic normality and central limit theorems as there is between consistency and laws of large numbers. The easiest way to demonstrate this close connection is by means of an example. Suppose that samples are generated by random drawings from distributions with an unknown mean $\mu$ and unknown and variable variances. For example, it might be that the variance of the distribution from which the $t^{\text {th }}$ observation is drawn is
$$\sigma_{t}^{2} \equiv \omega^{2}\left(1+\frac{1}{2}(t(\bmod 3))\right) .$$
Then $\sigma_{t}^{2}$ will take on the values $\omega^{2}, 1.5 \omega^{2}$, and $2 \omega^{2}$ with equal probability. Thus $\sigma_{t}^{2}$ varies systematically with $t$ but always remains within certain limits, in this case $\omega^{2}$ and $2 \omega^{2}$.

We will suppose that the investigator does not know the exact relation (4.26) and is prepared to assume only that the variances $\sigma_{t}^{2}$ vary between two positive bounds and average out asymptotically to some value $\sigma_{0}^{2}$, which may or not be known, defined as
$$\sigma_{0}^{2} \equiv \lim {n \rightarrow \infty}\left(\frac{1}{n} \sum{t=1}^{n} \sigma_{t}^{2}\right)$$
The sample mean may still be used as an estimator of the population mean, since our law of large numbers, Theorem 4.1, is applicable. The investigator is also prepared to assume that the distributions from which the observations are drawn have absolute third moments that are bounded, and so we too will assume that this is so. The investigator wishes to perform asymptotic statistical inference on the estimate derived from a realized sample and is therefore

interested in the nondegenerate asymptotic distribution of the sample mean as an estimator. We saw in Section $4.3$ that for this purpose we should look at the distribution of $n^{1 / 2}\left(m_{1}-\mu\right)$, where $m_{1}$ is the sample mean. Specifically, we wish to study
$$n^{1 / 2}\left(m_{1}-\mu\right)=n^{-1 / 2} \sum_{t=1}^{n}\left(y_{t}-\mu\right),$$
where $y_{t}-\mu$ has variance $\sigma_{t}^{2}$.
We begin by stating the following simple central limit theorem.
Theorem 4.2. Simple Central Limit Theorem. (Lyapunov)
Let $\left{y_{t}\right}$ be a sequence of independent, centered random variables with variances $\sigma_{t}^{2}$ such that $\sigma^{2} \leq \sigma_{t}^{2} \leq \bar{\sigma}^{2}$ for two finite positive constants, $\sigma^{2}$ and $\bar{\sigma}^{2}$, and absolute third moments $\mu_{3}$ such that $\mu_{3} \leq \bar{\mu}{3}$ for a finite constant $\bar{\mu}{3}$. Further, let
$$\sigma_{0}^{2} \equiv \lim {n \rightarrow \infty}\left(\frac{1}{n} \sum{t=1}^{n} \sigma_{t}^{2}\right)$$
exist. Then the sequence
$$\left{n^{-1 / 2} \sum_{t=1}^{n} y_{t}\right}$$
tends in distribution to a limit characterized by the normal distribution with mean zero and variance $\sigma_{0}^{2}$.

## 经济代写|计量经济学作业代写Econometrics代考|Some Useful Results

This section is intended to serve as a reference for much of the rest of the book. We will essentially make a list (with occasional commentary but without proofs) of useful definitions and theorems. At the end of this we will present two sets of regularity conditions that will each have a set of desirable implications. Later, we will be able to make assumptions by which one or other of these whole sets of regularity conditions is satisfied and thereby be able to draw without further ado a wide variety of useful conclusions.

To begin with, we will concentrate on laws of large numbers and the properties that allow them to be satisfied. In all of these theorems, we consider a sequence of sums $\left{S_{n}\right}$ where
$$S_{n}=\frac{1}{n} \sum_{t=1}^{n} y_{t}$$
The random variables $y_{t}$ will be referred to as the (random) summands. First, we present a theorem with very little in the way of moment restrictions on the random summands but very strong restrictions on their homogeneity.
Theorem 4.3. (Khinchin)
If the random variables $y_{t}$ of the sequence $\left{y_{t}\right}$ are mutually independent and all distributed according to the same distribution, which possesses a mean of $\mu$, then
$$\operatorname{Pr}\left(\lim {n \rightarrow \infty} S{n}=\mu\right)=1$$
Only the existence of the first moment is required, but all the summands must be identically distributed. Notice that the identical mean of the summands means that we need not bother to center the variables $y_{t}$.

Next, we present a theorem due to Kolmogorov, which still requires independence of the summands, and now existence of their second moments, but very little else in the way of homogeneity.

## 经济代写|计量经济学作业代写Econometrics代考|6 Asymptotic Normality and Central Limit Theorems

σ吨2≡ω2(1+12(吨(反对3))).

σ02≡林n→∞(1n∑吨=1nσ吨2)

n1/2(米1−μ)=n−1/2∑吨=1n(是吨−μ),

σ02≡林n→∞(1n∑吨=1nσ吨2)

\left{n^{-1 / 2} \sum_{t=1}^{n} y_{t}\right}\left{n^{-1 / 2} \sum_{t=1}^{n} y_{t}\right}

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