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经济代写|计量经济学代写Introduction to Econometrics代考|ECO400 Hypothesis testing

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经济代写|计量经济学代写Introduction to Econometrics代考|Hypothesis testing

To test a set of linear restrictions on the model coefficients, the most common approach is to use a Wald test. We write the set of $q$ linearly independent restrictions as
$$H_0: R \beta=r$$
with the alternative that at least one restriction is violated. Here, $R$ denotes a $q \times K$ matrix of constants, and $r$ a $q$-dimensional vector of constants. For example, if we wish to test that $\beta_2=0$ and $\beta_3=0$, we have
$$R=\left(\begin{array}{llll} 0 & 1 & 0 & \ldots \ 0 & 0 & 1 & \ldots \end{array}\right)$$
and $r=(0,0)^{\prime}$. The Wald test statistic is given by a quadratic form in $R \hat{\beta}-r$, weighted by the inverse of the corresponding estimated covariance matrix. That is,
$$\xi_W=(R \hat{\beta}-r)^{\prime}\left[R \hat{V}(\hat{\beta}) R^{\prime}\right]^{-1}(R \hat{\beta}-r)$$

经济代写|计量经济学代写Introduction to Econometrics代考|p-values and p-hacking

Most modern software provides $p$-values with any test that is done. A $p$-value denotes the probability, under the null hypothesis, to find the reported value of the test statistic or a more extreme one. If the $p$-value is smaller than the significance level (e.g., $5 \%$ ), the null hypothesis is rejected. Checking $p$-values allows researchers to draw their conclusions without consulting the appropriate critical values, making them a convenient piece of information. It also shows the sensitivity of the decision to reject the null hypothesis with respect to the choice of significance level. However, $p$-values are often misinterpreted or misused, as stressed by a recent statement of the American Statistical Association (Wasserstein and Lazar, 2016). For example, it is inappropriate (though a common mistake) to interpret a $p$-value as giving the probability that the null hypothesis is true.

Unfortunately, in empirical work some researchers are overly obsessed with obtaining “significant” results and finding $p$-values smaller than $0.05$ (and this also extends to journal editors). If publication decisions depend on the statistical significance of research findings, the literature as a whole will overstate the size of the true effect. This is referred to as publication bias (or “file drawer” bias). For example, investigating more than 50,000 tests published in three leading economic journals, Brodeur et al. (2016) conclude that the distribution of $p$-values indicates both selection by journals as well as a tendency of researchers to inflate the value of almost-rejected tests by choosing slightly more “significant” specifications. Their analysis is extended in Brodeur et al. (2020), with a focus on inference methods used in causal analysis.
The problem of publication bias relates to the broader problem of $p$-hacking. Even if the null hypothesis is correct, there is always a small probability of rejecting it (corresponding to the size of the test). Such type I errors are rather likely to happen if we use a sequence of many tests to select the regressors to include in the model. This process is referred to as data snooping, data mining or $p$-hacking (see Leamer, 1978; Lovell, 1983). As a result, an extensive specification search may pick up accidental patterns in the data and deliver a seemingly “significant” result with no genuine interpretation or meaning. This problem is potentially a serious issue in empirical finance, where many scholars are using the same databases (such as the Center for Research in Security Prices (CRSP) and Compustat). For example, Lo and MacKinlay (1990) analyse data snooping biases in tests of financial asset pricing models, while Sullivan et al. (2001) analyse the extent to which the presence of calendar effects in stock returns can be attributed to data snooping. Harvey et al. (2016) provide a critical account of the literature on factor models explaining the cross-section of asset returns. To accommodate for the inherent data mining, they suggest that a new factor needs to clear a much higher hurdle, with a $t$-statistic greater than 3.0. However, as argued by Harvey (2017), simply raising the threshold for significance is insufficient, and may unintendedly increase the amount of data mining and, in turn, publication bias. Recently, Mitton (2021) documents large variation in empirical methodology in corporate finance regressions in top finance journals, enabling selective reporting that results from $p$-hacking and publication bias.

经济代写|计量经济学代写Introduction to Econometrics代考|Hypothesis testing

$$H_0: R \beta=r$$

$$R=\left(\begin{array}{llllllll} 0 & 1 & 0 & \ldots & 0 & 0 & 1 & \ldots \end{array}\right)$$

$$\xi_W=(R \hat{\beta}-r)^{\prime}\left[R \hat{V}(\hat{\beta}) R^{\prime}\right]^{-1}(R \hat{\beta}-r)$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。