### 经济代写|计量经济学作业代写Econometrics代考|Models and Data-Generating Processes

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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|Models and Data-Generating Processes

In economics, it is probably not often the case that a relationship like (2.01) actually represents the way in which a dependent variable is generated, as it might if $x_{t}(\beta)$ were a physical response function and $u_{t}$ merely represented errors in measuring $y_{t}$. Instead, it is usually a way of modeling how $y_{t}$ varies

with the values of certain variables. They may be the only variables about which we have information or the only ones that we are interested in for a particular purpose. If we had more information about potential explanatory variables, we might very well specify $x_{t}(\beta)$ differently so as to make use of that additional information.

It is sometimes desirable to make explicit the fact that $x_{t}(\beta)$ represents the conditional mean of $y_{t}$, that is, the mean of $y_{t}$ conditional on the values of a number of other variables. The set of variables on which $y_{t}$ is conditioned is often referred to as an information set. If $\Omega_{t}$ denotes the information set on which the expectation of $y_{t}$ is to be conditioned, one could define $x_{t}(\boldsymbol{\beta})$ formally as $E\left(y_{t} \mid \Omega_{t}\right)$. There may be more than one such information set. Thus we might well have both
$$x_{1 t}\left(\boldsymbol{\beta}{1}\right) \equiv E\left(y{t} \mid \Omega_{1 t}\right) \quad \text { and } \quad x_{2 t}\left(\boldsymbol{\beta}{2}\right) \equiv E\left(y{t} \mid \Omega_{2 t}\right)$$
where $\Omega_{1 t}$ and $\Omega_{2 t}$ denote two different information sets. The functions $x_{1 t}\left(\boldsymbol{\beta}{1}\right)$ and $x{2 t}\left(\boldsymbol{\beta}{2}\right)$ might well be quite different, and we might want to estimate both of them for different purposes. There are many circumstances in which we might not want to condition on all available information. For example, if the ultimate purpose of specifying a regression function is to use it for forecasting, there may be no point in conditioning on information that will not be available at the time the forecast is to be made. Even when we do want to take account of all available information, the fact that a certain variable belongs to $\Omega{t}$ does not imply that it will appear in $x_{t}(\boldsymbol{\beta})$, since its value may tell us nothing useful about the conditional mean of $y_{t}$, and including it may impair our ability to estimate how other variables affect that conditional mean.

For any given dependent variable $y_{t}$ and information set $\Omega_{t}$, one is always at liberty to consider the difference $y_{t}-E\left(y_{t} \mid \Omega_{t}\right)$ as the error term associated with the $t^{\text {th }}$ observation. But for a regression model to be applicable, these differences must generally have the i.i.d. property. Actually, it is possible, when the sample size is large, to deal with cases in which the error terms are independent, but identically distributed only as regards their means, and not necessarily as regards their variances. We will discuss techniques for dealing with such cases in Chapters 16 and 17 , in the latter of which we will also relax the independence assumption. As we will see in Chapter 3, however, conventional techniques for making inferences from regression models are unreliable when models lack the i.i.d. property, even when the regression function $x_{t}(\boldsymbol{\beta})$ is “correctly” specified. Thus we are in general not at liberty to choose an arbitrary information set and estimate a properly specified regression function based on it if we want to make inferences using conventional procedures.

## 经济代写|计量经济学作业代写Econometrics代考|Linear and Nonlinear Regression Functions

The general regression function $x_{t}(\beta)$ can be made specific in a very large number of ways. It is worthwhile to consider a number of special cases so as to get some idea of the variety of specific regression functions that are commonly used in practice.
The very simplest regression function is
$$x_{t}(\boldsymbol{\beta})=\beta_{1} \iota_{t}=\beta_{1},$$
where $\iota_{t}$ is the $t^{\text {th }}$ element of an $n$-vector $\iota$, each element of which is 1 . In this case, the model $(2.01)$ says that the conditional mean of $y_{t}$ is simply a constant. While this is a trivial example of a regression function, since $x_{t}(\boldsymbol{\beta})$ is the same for all $t$, it is nevertheless a good example to start with and to keep in mind. All regression functions are simply fancier versions of (2.10). And any regression function that cannot fit the data at least as well as (2.10) should be considered a highly unsatisfactory one.

The next-simplest regression function is the simple linear regression function
$$x_{t}(\beta)=\beta_{1}+\beta_{2} z_{t},$$
where $z_{t}$ is a single independent variable. Actually, an even simpler model would be one with a single independent variable and no constant term. However, in most applied problems it does not make sense to omit the constant term. Many linear regression functions are used as approximations to unknown conditional mean functions, and such approximations will rarely be accurate if they are constrained to pass through the origin. Equation (2.11) has two parameters, an intercept $\beta_{1}$ and a slope $\beta_{2}$. This function is linear in both variables ( $\iota_{t}$ and $z_{t}$, or just $z_{t}$ if one chooses not to call $\iota_{t}$ a variable) and parameters $\left(\beta_{1}\right.$ and $\beta_{2}$ ). Although this model is often too simple, it does have some advantages. Because it is very easy to graph $y_{t}$ against $z_{t}$, we can use such a graph to see what the regression function looks like, how well the model fits, and whether a linear relationship adequately describes the data. “Eyeballine” the data in this way is harder, and therefore much less often done, when a model involves more than one independent variable.

One ubvious generalization of (2.11) is the multiple linear regression function
$$x_{t}(\boldsymbol{\beta})=\beta_{1} z_{t 1}+\beta_{2} z_{t 2}+\beta_{3} z_{t 3}+\cdots+\beta_{k} z_{t k}$$

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

When we specify a regression model, we must specify two things: the regression function $x_{t}(\boldsymbol{\beta})$ and at least some of the properties of the error terms $u_{t}$. We have already seen how important the second of these can be. When we added errors with constant variance to the multiplicative regression function (2.13), we obtained a genuinely nonlinear regression model. But when we added errors that were proportional to the regression function, as in (2.15), and made use of the approximation $e^{w} \cong 1+w$, which is a very good one when $w$ is small, we obtained a loglinear regression model. It should be clear from this example that how we specify the error terms will have a major effect on the model which is actually estimated.

In (2.01) we specified that the error terms were independent with identical means of zero and variances $\sigma^{2}$, but we did not specify how they were actually distributed. Even these assumptions may often be too strong. They rule out any sort of dependence across observations and any type of variation over time or with the values of any of the independent variables. They also rule out distributions where the tails are so thick that the error terms do not have a finite variance. One such distribution is the Cauchy distribution. A random

variable that is distributed as Cauchy not only has no finite variance but no finite mean either. See Chapter 4 and Appendix B.

There are several meanings of the word independence in the literature on statistics and econometrics. Two random variables $z_{1}$ and $z_{2}$ are said to be stochastically independent if their joint probability distribution function $F\left(z_{1}, z_{2}\right)$ is equal to the product of their two marginal distribution functions $F\left(z_{1}, \infty\right)$ and $F\left(\infty, z_{2}\right)$. This is sometimes called independence in probability, but we will employ the former, more modern, terminology. Some authors say that two random variables $z_{1}$ and $z_{2}$ are linearly independent if $E\left(z_{1} z_{2}\right)=$ $E\left(z_{1}\right) E\left(z_{2}\right)$, a weaker condition, which is implied by stochastic independence but does not imply it. This terminology is unfortunate, because this meaning of “linearly independent” is not the same as its usual meaning in linear algebra, and we will therefore not use it. Instead, in this situation we will merely say that $z_{1}$ and $z_{2}$ are uncorrelated, or have zero covariance. If either $z_{1}$ or $z_{2}$ has mean zero and they are uncorrelated, $E\left(z_{1} z_{2}\right)=0$. There is a sense in which $z_{1}$ and $z_{2}$ are orthogonal in this situation, and we will sometimes use this terminology as well.

## 经济代写|计量经济学作业代写Econometrics代考|Models and Data-Generating Processes

X1吨(b1)≡和(是吨∣Ω1吨) 和 X2吨(b2)≡和(是吨∣Ω2吨)

## 经济代写|计量经济学作业代写Econometrics代考|Linear and Nonlinear Regression Functions

X吨(b)=b1我吨=b1,

X吨(b)=b1+b2和吨,

(2.11) 的一个明显概括是多元线性回归函数
X吨(b)=b1和吨1+b2和吨2+b3和吨3+⋯+bķ和吨ķ

## 广义线性模型代考

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