# 经济代写|计量经济学代写Econometrics代考|EC5216

## 经济代写|计量经济学代写Econometrics代考|Transforming the Dependent Variable

When we introduced the concept of a regression function in Chapter 2, we defined it as the function that determines the mean of a dependent variable $y_t$ conditional on an information set $\Omega_t$. With this definition, we can always write
$$y_t=x_t(\boldsymbol{\beta})+u_t$$
and assert that $u_t$ has mean zero conditional on $\Omega_t$, provided that $x_t(\boldsymbol{\beta})$ has been specified correctly. But no matter how well $x_t(\boldsymbol{\beta})$ has been specified, we cannot assert that $u_t$ is i.i.d. or has any other desirable properties. In particular, there is no reason for it to be normally distributed, homoskedastic, or even symmetric. Yet we need $u_t$ to be homoskedastic if the NLS estimates $\hat{\boldsymbol{\beta}}$ are to be efficient and inferences based on the usual least squares covariance matrix estimator are to be valid. ${ }^1$ We also need $u_t$ to be symmetric (and preferably normally distributed or close to it) if asymptotic results are to provide a good guide to the properties of finite-sample estimators. Moreover, if we wish to predict $y_t$ conditional on $\Omega_t$ and construct any sort of forecast interval, we must know (or at least be able to estimate) the distribution of $u_t$.
If we can find the mean of $y_t$ conditional on $\Omega_t$, then we can presumably just as well find the conditional mean of any smooth monotonic function of $y_t$, say $\tau\left(y_t\right)$. For example, $\tau\left(y_t\right)$ might be $\log y_t, y_t^{1 / 2}$, or $y_t^2$. If we write
$$\tau\left(y_t\right)=E\left(\tau\left(y_t\right) \mid \Omega_t\right)+v_t$$
for some nonlinear $\tau(\cdot)$, then the error term $v_t$ cannot be normally and independently distributed, or n.i.d., if $u_t$ is n.i.d. in (14.01). Conversely, if $v_t$ is n.i.d. in (14.02), $u_t$ cannot be n.i.d. in (14.01).

## 经济代写|计量经济学代写Econometrics代考|The Box-Cox Transformation

The Box-Cox transformation is by far the most commonly used nonlinear transformation in statistics and econometrics. It is defined as
$$B(x, \lambda)= \begin{cases}\frac{x^\lambda-1}{\lambda} & \text { when } \lambda \neq 0 ; \ \log (x) & \text { when } \lambda=0,\end{cases}$$
where the argument $x$ must be positive. By l’Hôpital’s Rule, $\log x$ is the limit of $\left(x^\lambda-1\right) / \lambda$ as $\lambda \rightarrow 0$. Figure $14.1$ shows the Box-Cox transformation for various values of $\lambda$. In practice, $\lambda$ generally ranges from somewhat below 0 to somewhat above 1. It can be shown that $B\left(x, \lambda^{\prime}\right) \geq B\left(x, \lambda^{\prime \prime}\right)$ for $\lambda^{\prime} \geq \lambda^{\prime \prime}$, and this inequality is evident in the figure. Thus the amount of curvature induced by the Box-Cox transformation increases as $\lambda$ gets farther from 1 in either direction.

There are three varieties of Box-Cox model. We will refer to (14.04) and (14.05) with $\tau(\cdot)$ given by the Box-Cox transformation as the simple Box-Cox model and the transform-both-sides Box-Cox model, respectively. We will refer to (14.06) with this choice of $\tau(\cdot)$ as the conventional Box-Cox model, because it is by far the most commonly used in econometrics.

One reason for the popularity of the Box-Cox transformation is that it incorporates both the possibility of no transformation at all (when $\lambda=1$ ) and the possibility of a logarithmic transformation (when $\lambda=0$ ). Provided that the regressors include a constant term, subjecting the dependent variable to a Box-Cox transformation with $\lambda=1$ is equivalent to not transforming it at all.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Transforming the Dependent Variable

$$y_t=x_t(\boldsymbol{\beta})+u_t$$

$$\tau\left(y_t\right)=E\left(\tau\left(y_t\right) \mid \Omega_t\right)+v_t$$

## 经济代写|计量经济学代写Econometrics代考|The Box-Cox Transformation

Box-Cox 变换是迄今为止统计和计量经济学中最常用的非线性变换。它被定义为

$B\left(x, \lambda^{\prime}\right) \geq B\left(x, \lambda^{\prime \prime}\right)$ 为了 $\lambda^{\prime} \geq \lambda^{\prime \prime}$ ，并且这种不等式在图中很明显。因此，由 Box-Cox 变换引起的曲 率量增加为 $\lambda$ 在任一方向上都离 1 越来越远。

Box-Cox模型共有三种。我们将参考 (14.04) 和 (14.05) $\tau(\cdot)$ 由 Box-Cox 变换分别给出为简单 Box-Cox 模 型和变换两侧 Box-Cox 模型。我们将选择 (14.06) $\tau(\cdot)$ 与传统的 Box-Cox 模型一样，因为它是迄今为 止计量经济学中最常用的模型。

Box-Cox 变换流行的一个原因是它结合了完全没有变换的可能性 (当 $\lambda=1$ ) 和对数变换的可能性 (当 $\lambda=0$ ) 。假设回归量包括一个常数项，使因变量经受 Box-Cox 变换 $\lambda=1$ 相当于根本不改造。

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