### 经济代写|计量经济学作业代写Econometrics代考|Hypothesis Testing: Introduction

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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|Hypothesis Testing: Introduction

Economists frequently wish to test hypotheses about the regression models they estimate. Such hypotheses normally take the form of equality restrictions on some of the parameters. They might involve testing whether a single parameter takes on a certain value (say, $\beta_{2}=1$ ), whether two parameters are related in a specific way (say, $\beta_{3}=2 \beta_{4}$ ), whether a nonlinear restriction such as $\beta_{1} / \beta_{3}=\beta_{2} / \beta_{4}$ holds, or perhaps whether a whole set of linear and/or nonlinear restrictions holds. The hypothesis that the restriction or set of restrictions to be tested does in fact hold is called the null hypothesis and is often denoted $H_{0}$. The model in which the restrictions do not hold is usually called the alternative hypothesis, or sometimes the maintained hypothesis, and may be denoted $H_{1}$. The terminology “maintained hypothesis” reflects the fact that in a statistical test only the null hypothesis $H_{0}$ is under test. Rejecting $H_{0}$ does not in any way oblige us to accept $H_{1}$, since it is not $H_{1}$ that we are testing. Consider what would happen if the DGP were not a special case of $H_{1}$. Clearly both $H_{0}$ and $H_{1}$ would then be false, and it is quite possible that a test of $H_{0}$ would lead to its rejection. Other tests might well succeed in rejecting the false $H_{1}$, but only if it then played the role of the null hypothesis and some new maintained hypothesis were found.

All the hypothesis tests discussed in this book involve generating a test statistic. A test statistic, say $T$, is a random variable of which the probability distribution is known, either exactly or approximately, under the null hypothesis. We then see how likely the observed value of $T$ is to have occurred, according to that probability distribution. If $T$ is a number that could easily have occurred by chance, then we have no evidence against the null hypothesis $H_{0}$. However, if it is a number that would occur by chance only rarely, we do have evidence against the null, and we may well decide to reject it.

The classical way to perform a test is to divide the set of possible values of $T$ into two regions, the acceptance region and the rejection region (or critical rcgion). If $T$ falls into the acecptance region, the null hypothesis is accepted (or at any rate not rejected), while if it falls into the rejection region, it is rejected. ${ }^{5}$ For example, if $T$ were known to have a $\chi^{2}$ distribution, the acceptance rogion would consist of all valuca of $T$ cqual to or lcsis than a certain critical value, say $C$, and the rejection region would then consist of all values greater than $C .$ If instead $T$ were known to have a normal distribution, then for a two-tailed test the acceptance region would consist of all absolute values of $T$ less than or equal to $C$. Thus the rejection region would consist of

two parts, one part containing values greater than $C$ and one part containing values less than $-C$.

The size of a test is the probability that the test statistic will reject the null hypothesis when the latter is true. Let $\boldsymbol{\theta}$ denote the vector of parameters to be tested; $\Theta_{0}$, the set of values of $\theta$ that satisfy $H_{0}$; and $R$, the rejection region. Then the size of the test $T$ is
$$\alpha \equiv \operatorname{Pr}\left(T \in R \mid \boldsymbol{\theta} \in \Theta_{0}\right)$$

## 经济代写|计量经济学作业代写Econometrics代考|Hypothesis Testing in Linear Regression Models

All students of econometrics are familiar with $\boldsymbol{t}$ statistics for testing hypotheses about a single parameter and $\boldsymbol{F}$ statistics for testing hypotheses about several parameters jointly. If $\hat{\beta}{i}$ denotes the least squares estimate of the parameter $\beta{i}$, the $t$ statistic for testing the hypothesis that $\beta_{i}$ is equal to some specified value $\beta_{0 i}$ is simply expression (3.12), that is, $\hat{\beta}{i}-\beta{0 i}$ divided by the estimated standard error of $\hat{\beta}_{i}$. If $\hat{\beta}$ denotes a set of unrestricted least squares estimates and $\overline{\boldsymbol{\beta}}$ denotes a set of estimates subject to $r$ distinct restrictions, then the $F$ statistic for testing those restrictions may be calculated as
$$\frac{(\operatorname{SSR}(\overline{\boldsymbol{\beta}})-\operatorname{SSR}(\hat{\boldsymbol{\beta}})) / r}{\operatorname{SSR}(\hat{\boldsymbol{\beta}}) /(n-k)}=\frac{1}{r s^{2}}(\operatorname{SSR}(\overline{\boldsymbol{\beta}})-\operatorname{SSR}(\hat{\boldsymbol{\beta}}))$$
Tests based on $t$ and $F$ statistics may be either exact or approximate. In the very special case referred to at the end of the last section, in which the regression model and the restrictions are both linear in the parameters, the regressors are (or can be treated as) fixed in repeated samples, and the error terms are normally and independently distributed, ordinary $t$ and $F$ statistics

actually are distributed in finite samples under the null hypotheses according to their namesake distributions. Although this case is not encountered nearly as often as one might hope, these results are sufficiently important that they are worth a separate section. Moreover, it is useful to keep the linear case firmly in mind when considering the case of nonlinear regression models.
Consider the restricted model
$$\boldsymbol{y}=\boldsymbol{X}{1} \beta{1}+\boldsymbol{u}$$
and the unrestricted model
$$\boldsymbol{y}=\boldsymbol{X}{1} \boldsymbol{\beta}{1}+\boldsymbol{X}{2} \boldsymbol{\beta}{2}+\boldsymbol{u}$$

## 经济代写|计量经济学作业代写Econometrics代考|Hypothesis Testing in Nonlinear Regression Models

There are at least three different ways that we can derive test statistics for hypotheses about the parameters of nonlinear regression models. They are to utilize the Wald principle, the Lagrange multiplier principle, and the likelihood ratio principle. These yield what are often collectively referred to as the three “classical” test statistics. In this section, we introduce these three principles and show how they yield test statistics for hypotheses about $\boldsymbol{\beta}$ in nonlinear regression models (and implicitly in linear regression models as well, since linear models are simply a special case of nonlinear ones). The three principles are very widely applicable and will reappear in other contexts throughout the book. ${ }^{7}$ A formal treatment of these tests in the context of least squares will be provided in Chapter 5. They will be reintroduced in the context of maximum likelihood estimation in Chapter 8 , and a detailed treatment in that context will be provided in Chapter 13. Valuable referencer include Engle (1984) and Godfrey (1988), and an illuminating introductory discussion may be found in Buse (1982).

The Wald principle, which is due to Wald (1943), is to construct a test statistic based on unrestricted parameter estimates and an estimate of the unrestricted covariance matrix. If the hypothesis involves just one restriction, say that $\beta_{i}=\beta_{i}^{}$, then one can calculate the pseudo- $t$ statistic $$\frac{\hat{\beta}{i}-\beta{i}^{}}{\hat{S}\left(\hat{\beta}{i}\right)} .$$ We refer to this as a “pseudo-t” statistic because it will not actually have the Student’s $t$ distribution with $n-k$ degrees of freedom in finite samples when $x{t}(\boldsymbol{\beta})$ is nonlinear in the parameters, $x_{t}(\boldsymbol{\beta})$ depends on lagged values of $y_{t}$, or the errors $u_{t}$ are not normally distributed. However, it will be asymptotically distributed as $N(0,1)$ under quite weak conditions (see Chapter 5 ), and its finite-sample distribution is frequently approximated quite well by $t(n-k)$.
In the more general case in which there are $r$ restrictions rather than just one to be tested, Wald tests make use of the fact that if $v$ is a random $r$-vector which is normally distributed with mean vector zero and covariance matrix $\boldsymbol{\Lambda}$, then the quadratic form
$$\boldsymbol{v}^{\top} \boldsymbol{\Lambda}^{-1} \boldsymbol{v}$$
must be distributed as $\chi^{2}(r)$. This result is proved in Appendix $\mathrm{B}$, and we used it in Sections $3.3$ and $3.5$ above.

To construct an asymptotic Wald test, then, we simply have to find a vector of random variables that should under the null hypothesis be asymptotically normally distributed with mean vector zero and a covariance matrix which we can estimate. For example, suppose that $\beta$ is subject to the $r(\leq k)$ linearly independent restrictions
$$\boldsymbol{R} \boldsymbol{\beta}=\boldsymbol{r},$$

## 经济代写|计量经济学作业代写Econometrics代考|Hypothesis Testing in Linear Regression Models

(固态继电器⁡(b¯)−固态继电器⁡(b^))/r固态继电器⁡(b^)/(n−ķ)=1rs2(固态继电器⁡(b¯)−固态继电器⁡(b^))

## 经济代写|计量经济学作业代写Econometrics代考|Hypothesis Testing in Nonlinear Regression Models

Wald 原理源于 Wald (1943)，它是基于无限制参数估计和无限制协方差矩阵的估计构建检验统计量。如果假设只涉及一个限制，那么说b一世=b一世，那么可以计算出伪吨统计b^一世−b一世小号^(b^一世).我们将其称为“伪 t”统计量，因为它实际上没有学生的吨分布与n−ķ有限样本中的自由度X吨(b)参数是非线性的，X吨(b)取决于滞后值是吨，或错误在吨不是正态分布的。但是，它将渐近分布为ñ(0,1)在相当弱的条件下（见第 5 章），它的有限样本分布通常很好地近似为吨(n−ķ).

Rb=r,

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## MATLAB代写

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