### 数学代写|计量经济学原理代写Principles of Econometrics代考|Estimating the Regression Parameters

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• Longitudinal Data Analysis 纵向数据分析
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## 数学代写|计量经济学原理代写Principles of Econometrics代考|Food Expenditure Model Data

We assume that the expenditure data in Table $2.1$ satisfy the assumptions SR1-SR5. That is, we assume that the regression model $y_{i}=\beta_{1}+\beta_{2} x_{i}+e_{i}$ describes a population relationship and that the random error has conditional expected value zero. This implies that the conditional expected value of household food expenditure is a linear function of income. The conditional variance of $y$, which is the same as that of the random error $e$, is assumed constant, implying that we are equally uncertain about the relationship between $y$ and $x$ for all observations. Given $\mathbf{x}$ the values of $y$ for different households are assumed uncorrelated with each other.

Given this theoretical model for explaining the sample observations on household food expenditure, the problem now is how to use the sample information in Table 2.1, specific values of $y_{i}$ and $x_{i}$, to estimate the unknown regression parameters $\beta_{1}$ and $\beta_{2}$. These parameters represent the unknown intercept and slope coefficients for the food expenditure-income relationship. If we represent the 40 data points as $\left(y_{i}, x_{i}\right), i=1, \ldots, N=40$, and plot them, we obtain the scatter diagram in Figure $2.6$.Our problem is to estimate the location of the mean expenditure line $E\left(y_{i} \mid \mathbf{x}\right)=\beta_{1}+\beta_{2} x_{i}$. We would expect this line to be somewhere in the middle of all the data points since it represents population mean, or average, behavior. To estimate $\beta_{1}$ and $\beta_{2}$, we could simply draw a freehand line through the middle of the data and then measure the slope and intercept with a ruler. The problem with this method is that different people would draw different lines, and the lack of a formal criterion makes it difficult to assess the accuracy of the method. Another method is to draw a line from the expenditure at the smallest income level, observation $i=1$, to the expenditure at the largest income level, $i=40$. This approach does provide a formal rule. However, it may not be a very good rule because it ignores information on the exact position of the remaining 38 observations. It would be better if we could devise a rule that uses all the information from all the data points.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|The Expected Values of b1 and b2

The OLS estimator $b_{2}$ is a random variable since its value is unknown until a sample is collected. What we will show is that if our model assumptions hold, then $E\left(b_{2} \mid \mathbf{x}\right)=\beta_{2}$; that is, given $\mathbf{x}$ the expected value of $b_{2}$ is equal to the true parameter $\beta_{2}$. When the expected value of any estimator of a parameter equals the true parameter value, then that estimator is unbiased. Since $E\left(b_{2} \mid \mathbf{x}\right)=\beta_{2}$, the least squares estimator $b_{2}$ given $x$ is an unbiased estimator of $\beta_{2}$. In $S$ ection $2.10$, we will show that the least squares estimator $b_{2}$ is unconditionally unbiased also, $E\left(b_{2}\right)=\beta_{2}$. The intuitive meaning of unbiasedness comes from the sampling interpretation of mathematical expectation. Recognize that one sample of size $N$ is just one of many samples that we could have been selected. If the formula for $b_{2}$ is used to estimate $\beta_{2}$ in each of those possible samples, then, if our assumptions are valid, the average value of the estimates $b_{2}$ obtained from all possible samples will be $\beta_{2}$.

We will show that this result is true so that we can illustrate the part played by the assumptions of the linear regression model. In (2.12), what parts are random? The parameter $\beta_{2}$ is not random. It is a population parameter we are trying to estimate. Conditional on $\mathbf{x}$ we can treat $x_{i}$ as if it is not random. Then, conditional on $\mathbf{x}, w_{i}$ is not random either, as it depends only on the values of $x_{i}$. The only random factors in (2.12) are the random error terms $e_{i}$. We can find the conditional expected value of $b_{2}$ using the fact that the expected value of a sum is the sum of the expected values:
\begin{aligned} E\left(b_{2} \mid \mathbf{x}\right) &=E\left(\beta_{2}+\sum w_{i} e_{i} \mid \mathbf{x}\right)=E\left(\beta_{2}+w_{1} e_{1}+w_{2} e_{2}+\cdots+w_{N} e_{N} \mid \mathbf{x}\right) \ &=E\left(\beta_{2}\right)+E\left(w_{1} e_{1} \mid \mathbf{x}\right)+E\left(w_{2} e_{2} \mid \mathbf{x}\right)+\cdots+E\left(w_{N} e_{N} \mid \mathbf{x}\right) \ &=\beta_{2}+\sum E\left(w_{i} e_{i} \mid \mathbf{x}\right) \ &=\beta_{2}+\sum w_{i} E\left(e_{i} \mid \mathbf{x}\right)=\beta_{2} \end{aligned}
The rules of expected values are fully discussed in the Probability Primer, Section P.5, and Appendix B. 1.1. In the last line of (2.13), we use two assumptions. First, $E\left(w_{i} e_{i} \mid \mathbf{x}\right)=w_{i} E\left(e_{i} \mid \mathbf{x}\right)$ because conditional on $\mathbf{x}$ the terms $w_{i}$ are not random, and constants can be factored out of expected values. Second, we have relied on the assumption that $E\left(e_{i} \mid \mathbf{x}\right)=0$. Actually, if $E\left(e_{i} \mid \mathbf{x}\right)=c$, where $c$ is any constant value, such as 3 , then $E\left(b_{2} \mid \mathbf{x}\right)=\beta_{2}$. Given $\mathbf{x}$, the OLS estimator $b_{2}$ is an unbiased estimator of the regression parameter $\beta_{2}$. On the other hand, if $E\left(e_{i} \mid \mathbf{x}\right) \neq 0$ and it depends on $\mathbf{x}$ in some way, then $b_{2}$ is a biased estimator of $\beta_{2}$. One leading case in which the assumption $E\left(e_{i} \mid \mathbf{x}\right)=0$ fails is due to omitted variables. Recall that $e_{i}$ contains everything else affecting $y_{i}$ other than $x_{i}$. If we have omitted anything that is important and that is correlated with $\mathbf{x}$ then we would expect that $E\left(e_{i} \mid \mathbf{x}\right) \neq 0$ and $E\left(b_{2} \mid \mathbf{x}\right) \neq \beta_{2}$. In Chapter 6 we discuss this omitted variables bias. Here we have shown that conditional on $\mathbf{x}$, and under SR1-SR5, the least squares estimator is linear and unbiased. In Section 2.10, we show that $E\left(b_{2}\right)=\beta_{2}$ without conditioning on $\mathbf{x}$.

The unbiasedness of the estimator $b_{2}$ is an important sampling property. On average, over all possible samples from the population, the least squares estimator is “correct,” on average, and this is one desirable property of an estimator. This statistical property by itself does not mean that $b_{2}$ is a good estimator of $\beta_{2}$, but it is part of the story. The unbiasedness property is related to what happens in all possible samples of data from the same population. The fact that $b_{2}$ is unbiased does not imply anything about what might happen in just one sample. An individual estimate (a number) $b_{2}$ may be near to, or far from, $\beta_{2}$. Since $\beta_{2}$ is never known we will never know, given

one sample, whether our estimate is “close” to $\beta_{2}$ or not. Thus, the estimate $b_{2}=10.21$ may be close to $\beta_{2}$ or not.

The least squares estimator $b_{1}$ of $\beta_{1}$ is also an unbiased estimator, and $E\left(b_{1} \mid \mathbf{x}\right)=\beta_{1}$ if the model assumptions hold.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Assessing the Least Squares Estimators

Using the food expenditure data, we have estimated the parameters of the regression model $y_{i}=\beta_{1}+\beta_{2} x_{i}+e_{i}$ using the least squares formulas in (2.7) and (2.8). We obtained the least squares estimates $b_{1}=83.42$ and $b_{2}=10.21$. It is natural, but, as we shall argue, misguided, to ask the question “How good are these estimates?” This question is not answerable. We will never know the true values of the population parameters $\beta_{1}$ or $\beta_{2}$, so we cannot say how close $b_{1}=83.42$ and $b_{2}=10.21$ are to the true values. The least squares estimates are numbers that may or may not be close to the true parameter values, and we will never know.

Rather than asking about the quality of the estimates we will take a step back and examine the quality of the least squares estimation procedure. The motivation for this approach is this: if we were to collect another sample of data, by choosing another set of 40 households to survey, we would have obtained different estimates $b_{1}$ and $b_{2}$, even if we had carefully selected households with the same incomes as in the initial sample. This sampling variation is unavoidable. Different samples will yield different estimates because household food expenditures, $y_{i}, i=1, \ldots, 40$, are random variables. Their values are not known until the sample is collected. Consequently, when viewed as an estimation procedure, $b_{1}$ and $b_{2}$ are also random variables, because their values depend on the random variable $y$. In this context, we call $b_{1}$ and $b_{2}$ the least squares estimators.
We can investigate the properties of the estimators $b_{1}$ and $b_{2}$, which are called their sampling properties, and deal with the following important questions:

1. If the least squares estimators $b_{1}$ and $b_{2}$ are random variables, then what are their expected values, variances, covariances, and probability distributions?
2. The least squares principle is only one way of using the data to obtain estimates of $\beta_{1}$ and $\beta_{2}$. How do the least squares estimators compare with other procedures that might be used, and how can we compare alternative estimators? For example, is there another estimator that has a higher probability of producing an estimate that is close to $\beta_{2}$ ?

We examine these questions in two steps to make things easier. In the first step, we investigate the properties of the least squares estimators conditional on the values of the explanatory variable in the sample. That is, conditional on $\mathbf{x}$. Making the analysis conditional on $\mathbf{x}$ is equivalent to saying that, when we consider all possible samples, the household income values in the sample stay the

same from one sample to the next; only the random errors and food expenditure values change. This assumption is clearly not realistic but it simplifies the analysis. By conditioning on $\mathbf{x}$, we are holding it constant, or fixed, meaning that we can treat the $x$-values as “not random.”

In the second step, considered in Section $2.10$, we return to the random sampling assumption and recognize that $\left(y_{i}, x_{i}\right)$ data pairs are random, and randomly selecting households from a population leads to food expenditures and incomes that are random. However, even in this case and treating $\mathbf{x}$ as random, we will discover that most of our conclusions that treated $\mathbf{x}$ as nonrandom remain the same.

In either case, whether we make the analysis conditional on $\mathbf{x}$ or make the analysis general by treating $\mathbf{x}$ as random, the answers to the questions above depend critically on whether the assumptions SR1-SR5 are satisfied. In later chapters, we will discuss how to check whether the assumptions we make hold in a specific application, and what we might do if one or more assumptions are shown not to hold.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|The Expected Values of b1 and b2

OLS 估计器b2是一个随机变量，因为在收集样本之前它的值是未知的。我们将展示的是，如果我们的模型假设成立，那么和(b2∣X)=b2; 也就是说，给定X的期望值b2等于 true 参数b2. 当参数的任何估计器的期望值等于真实参数值时，该估计器是无偏的。自从和(b2∣X)=b2, 最小二乘估计量b2给定X是一个无偏估计量b2. 在小号选举2.10，我们将证明最小二乘估计b2也是无条件无偏的，和(b2)=b2. 无偏性的直观含义来自数学期望的抽样解释。认识到一个大小的样本ñ只是我们可以选择的众多样本之一。如果公式为b2用于估计b2在每个可能的样本中，如果我们的假设是有效的，那么估计的平均值b2从所有可能的样本中获得b2.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Assessing the Least Squares Estimators

1. 如果最小二乘估计b1和b2是随机变量，那么它们的期望值、方差、协方差和概率分布是什么？
2. 最小二乘原理只是使用数据获得估计值的一种方法b1和b2. 最小二乘估计器如何与可能使用的其他程序进行比较，我们如何比较替代估计器？例如，是否有另一个估计器有更高的概率产生接近于b2 ?

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