### 经济代写|计量经济学作业代写Econometrics代考| The Probability Approach to Econometrics

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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

## 经济代写|计量经济学作业代写Econometrics代考|What is Econometrics

The term “econometrics” is believed to have been crafted by Ragnar Frisch (1895-1973) of Norway, one of the three principal founders of the Econometric Society, first editor of the journal Econometrica, and co-winner of the first Nobel Memorial Prize in Economic Sciences in 1969 . It is therefore fitting that we turn to Frisch’s own words in the introduction to the first issue of Econometrica to describe the discipline.
A word of explanation regarding the term econometrics may be in order. Its definition is implied in the statement of the scope of the [Econometric] Society, in Section I of the Constitution, which reads: “The Econometric Society is an international society for the advancement of economic theory in its relation to statistics and mathematics…. Its main object shall he tn promnte studies that aim at a unificatinn af the thenretical=quantitative and the empirical-quantitative approach to economic problems…”

But there are several aspects of the quantitative approach to economics, and no single one of these aspects, taken by itself, should be confounded with econometrics. Thus, econometrics is by no means the same as economic statistics. Nor is it identical with what we call general economic theory, although a considerable portion of this theory has a defininitely quantitative character. Nor should econometrics be taken as synonomous with the application of mathematics to economics. Experience has shown that each of these three view points, that of statistics, economic theory, and mathematics, is a necessary, but not by itself a sufficient, condition for a real understanding of the quantitative relations in modern economic life. It is the unification of all three that is powerful. And it is this unification that constitutes econometrics.
Ragnar Frisch, Econometrica, (1933), 1, pp. 1-2.
This definition remains valid today, although some terms have evolved somewhat in their usage. Today, we would say that econometrics is the unified study of economic models, mathematical statistics, and economic data.

Within the field of econometrics there are sub-divisions and specializations. Econometric theory concerns the development of tools and methods, and the study of the properties of econometric methods. Applied econometrics is a term describing the development of quantitative economic models and the application of econometric methods to these models using economic data.

## 经济代写|计量经济学作业代写Econometrics代考|The Probability Approach to Econometrics

The unifying methodology of modern econometrics was articulated by Trygve Haavelmo (1911-1999) of Norway, winner of the 1989 Nobel Memorial Prize in Economic Sciences, in his seminal paper “The probability approach in econometrics” (1944). Haavelmo argued that quantitative economic models must necessarily be probability models (by which today we would mean stochastic). Deterministic models are blatently inconsistent with observed economic quantities, and it is incoherent to apply deterministic models to non-deterministic data. Economic models should be explicitly designed to incorporate randomness; stochastic errors should not be simply added to deterministic models to make them random. Once we acknowledge that an economic model is a probability model, it follows naturally that an appropriate tool way to quantify, estimate, and conduct inferences about the economy is through the powerful theory of mathematical statistics. The appropriate method for a quantitative economic analysis follows from the probabilistic construction of the economic model.

Haavelmo’s probability approach was quickly embraced by the economics profession. Today no quantitative work in economics shuns its fundamental vision.

While all economists embrace the probability approach, there has been some evolution in its implementation.

The structural approach is the closest to Haavelmo’s original idea. A probabilistic economic model is specified, and the quantitative analysis performed under the assumption that the economic model is correctly specified. Researchers often describe this as “taking their model seriously” The structural approach typically leads to likelihood-based analysis, including maximum likelihood and Bayesian estimation.

A criticism of the structural approach is that it is misleading to treat an economic model as correctly specified. Rather, it is more accurate to view a model as a useful abstraction or approximation. In this case, how should we interpret structural econometric analysis? The quasi-structural approach to inference views a structural economic model as an approximation rather than the truth. This theory has led to the concepts of the pseudo-true value (the parameter value defined by the estimation problem), the quasi-likelihood function, quasi-MLE, and quasi-likelihood inference.

Closely related is the semiparametric approach. A probabilistic economic model is partially specified but some features are left unspecified. This approach typically leads to estimation methods such as least-squares and the Generalized Method of Moments. The semiparametric approach dominates contemporary econometrics, and is the main focus of this textbook.

Another branch of quantitative structural economics is the calibration approach. Similar to the quasi-structural approach, the calibration approach interprets structural models as approximations and hence inherently false. The difference is that the calibrationist literature rejects mathematical statistics (deeming classical theory as inappropriate for approximate models) and instead selects parameters by matching model and data moments using non-statistical ad hoc ${ }^{1}$ methods.

## 经济代写|计量经济学作业代写Econometrics代考|Econometric Terms and Notation

In a typical application, an econometrician has a set of repeated measurements on a set of variables. For example, in a labor application the variables could include weekly earnings, educational attainment, age, and other descriptive characteristics. We call this information the data, dataset, or sample.

We use the term observations to refer to the distinct repeated measurements on the variables. An individual observation often corresponds to a specific economic unit, such as a person, household, corporation, firm, organization, country, state, city or other geographical region. An individual observation could also be a measurement at a point in time, such as quarterly GDP or a daily interest rate.

Economists typically denote variables by the italicized roman characters $y, x$, and/or $z$. The convention in econometrics is to use the character $y$ to denote the variable to be explained, while the characters $x$ and $z$ are used to denote the conditioning (explaining) variables.

Following mathematical convention, real numbers (elements of the real line $\mathbb{R}$, also called scalars) are written using lower case italics such as $x$, and vectors (elements of $R^{k}$ ) by lower case bold italics such as $\boldsymbol{x}$, e.g.
$$\boldsymbol{x}=\left(\begin{array}{c} x_{1} \ x_{2} \ \vdots \ x_{k} \end{array}\right)$$
Upper case bold italics such as $\boldsymbol{X}$ are used for matrices.
We denote the number of observations by the natural number $n$, and subscript the variables by the index $i$ to denote the individual observation, e.g. $y_{i}, \boldsymbol{x}{i}$ and $z{i}$. In some contexts we use indices other than $i$, such as in time series applications where the index $t$ is common. In panel studies we typically use the double index it to refer to individual $i$ at a time period $t$.
The $i^{\text {th }}$ observation is the set $\left(y_{i}, x_{i}, z_{i}\right)$.
The sample is the set $\left{\left(y_{i}, x_{i}, z_{i}\right): i=1, \ldots, n\right}$.
It is proper mathematical practice to use upper case $X$ for random variables and lower case $x$ for realizations or specific values. Since we use upper case to denote matrices, the distinction between random variables and their realizations is not rigorously followed in econometric notation. Thus the notation $y_{i}$ will in some places refer to a random variable, and in other places a specific realization. This is undesirable but there is little to be done about it without terrifically complicating the notation. Hopefully there will be no confusion as the use should be evident from the context.

We typically use Greek letters such as $\beta, \theta$ and $\sigma^{2}$ to denote unknown parameters of an econometric model, and use boldface, e.g. $\boldsymbol{\beta}$ or $\boldsymbol{\theta}$, when these are vector-valued. Estimators are typically denoted by putting a hat ” $\wedge$ “, tilde ” $\sim$ ” or bar “-” over the corresponding letter, e.g. $\widehat{\beta}$ and $\tilde{\beta}$ are estimators of $\beta$.
The covariance matrix of an econometric estimator will typically be written using the capital boldface $\boldsymbol{V}$, often with a subscript to denote the estimator, e.g. $\boldsymbol{V}{\widehat{\boldsymbol{\beta}}}=\operatorname{var}[\widehat{\boldsymbol{\beta}}]$ as the covariance matrix for $\widehat{\boldsymbol{\beta}}$. Hopefully without causing confusion, we will use the notation $V{\boldsymbol{\beta}}=\operatorname{avar}[\widehat{\boldsymbol{\beta}}]$ to denote the asymptotic covariance matrix of $\sqrt{n}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})$ (the variance of the asymptotic distribution). Estimators will be denoted by appending hats or tildes, e.g. $\hat{V}{\beta}$ is an estimator of $V{\beta}$.

## 经济代写|计量经济学作业代写Econometrics代考|What is Econometrics

“计量经济学”一词被认为是由挪威的 Ragnar Frisch (1895-1973) 创造的，他是计量经济学学会的三位主要创始人之一，《计量经济学》杂志的第一任编辑，也是第一届诺贝尔纪念奖的共同获得者1969 年获得经济科学博士学位。因此，我们在《计量经济学》第一期的导言中用弗里施自己的话来描述这门学科是恰当的。

Ragnar Frisch, Econometrica, (1933), 1, pp. 1-2。

## 经济代写|计量经济学作业代写Econometrics代考|The Probability Approach to Econometrics

Haavelmo 的概率方法很快被经济学界所接受。今天，经济学中的任何量化工作都不会回避其基本愿景。

X=(X1 X2 ⋮ Xķ)

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