# 经济代写|计量经济学代写Econometrics代考|BEA472

## 经济代写|计量经济学代写Econometrics代考|Estimating Regression Models with AR

Suppose that we want to estimate a nonlinear regression model with error terms that follow an $\mathrm{AR}(1)$ process:
$$y_t=x_t(\boldsymbol{\beta})+u_t, \quad u_t=\rho u_{t-1}+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right) .$$
Because $u_{t-1}=y_{t-1}-x_{t-1}(\boldsymbol{\beta})$, this model can be rewritten as
$$y_t=x_t(\boldsymbol{\beta})+\rho\left(y_{t-1}-x_{t-1}(\boldsymbol{\beta})\right)+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right),$$
which is also a nonlinear regression model, but one with error terms that are (by assumption) serially uncorrelated. Since (10.12) is a nonlinear regression model with well-behaved error terms, it seems natural to estimate it by nonlinear least squares and to make inferences about it by using the Gauss-Newton regression. The regression function is simply
$$x_t^{\prime}(\boldsymbol{\beta}, \rho)=x_t(\boldsymbol{\beta})+\rho\left(y_{t-1}-x_{t-1}(\boldsymbol{\beta})\right),$$
which depends on $\rho$ as well as on $\boldsymbol{\beta}$.
There are two potential problems with (10.12). First of all, the regression function $x_t^{\prime}(\boldsymbol{\beta}, \rho)$ necessarily depends on $y_{t-1}$, whether or not $x_t(\boldsymbol{\beta})$ depends on any lagged values of the dependent variable. As we saw in Chapter 5 , this dependence does not prevent nonlinear least squares from having desirable asymptotic properties provided that certain regularity conditions are satisfied. It can be shown that, as long as $x_t(\boldsymbol{\beta})$ satisfies the regularity conditions of Theorems $5.1$ and $5.2$ and the stationarity condition that $|\rho|<1$ holds, this will indeed be the case for (10.12). However, if the stationarity condition did not hold, standard results about nonlinear least squares, in particular Theorem 5.2, the asymptotic normality theorem, would no longer apply to (10.12).

The second problem with (10.12) is what to do about the first observation. Presumably we do not have data for $y_0$ and for all the exogenous and predetermined variables needed to evaluate $x_0(\boldsymbol{\beta})$, since if we did the sample would not have started with the observation corresponding to $t=1$. Thus we cannot evaluate $x_1^{\prime}(\boldsymbol{\beta}, \rho)$, which depends on $y_0$ and $x_0(\boldsymbol{\beta})$. The easiest solution to this problem is simply to drop the first observation, requiring that (10.12) hold only for observations 2 through $n$. Dropping one observation makes no difference asymptotically, and so we can safely do so whenever the sample size is reasonably large.

## 经济代写|计量经济学代写Econometrics代考|Higher-Order AR Processes

Although the $\mathrm{AR}(1)$ process (10.01) is by far the most popular one in applied econometric work, there are many other stochastic processes that could reasonably be used to describe the evolution of error terms over time. Anything resembling a complete treatment of this topic would lead us far afield, into the vast literature on time-series methods. This literature, which evolved quite independently of econometrics and has influenced it substantially in recent years, deals with many aspects of the modeling of time series but especially with models in which variables depend only (or at least primarily) on their own past values. Such models are obviously appropriate for describing the evolution of many physical systems and may be appropriate for some economic systems as well. However, much of the use of time-series methods in econometrics has been to model the evolution of the error terms that adhere to more conventional regression models, and we will treat only that aspect of time-series methods here. A classic reference on times-series techniques is Box and Jenkins (1976), some books that may be more accessible to economists are Harvey (1981, 1989) and Granger and Newbold (1986), and a review of time-series methods for econometricians is Granger and Watson (1984).
The $\operatorname{AR}(1)$ process (10.01) is actually a special case of the $p^{\text {th }{\text {-order }}}$ autoregressive, or $\operatorname{AR}(p)$, process $$u_t=\rho_1 u{t-1}+\rho_2 u_{t-2}+\cdots+\rho_p u_{t-p}+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right),$$
in which $u_t$ depends on up to $p$ lagged values of itself, as well as on $\varepsilon_t$. The $\operatorname{AR}(p)$ process (10.35) can be expressed more compactly as
$$\left(1-\rho_1 L-\rho_2 L^2-\cdots-\rho_p L^p\right) u_t=\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right)$$ where $L$ denotes the lag operator. The lag operator $L$ has the property that when $L$ multiplies anything with a time subscript, this subscript is lagged one period. Thus
$$L u_t=u_{t-1}, \quad L^2 u_t=u_{t-2}, \quad L^p u_t=u_{t-p},$$
and so on. The expression in parentheses in (10.36) is a polynomial in the lag operator $L$, with coefficients 1 and $-\rho_1, \ldots,-\rho_p$. If we define $A(L, \rho)$ as being equal to this polynomial, $\rho$ representing the vector $\left[\begin{array}{l:l:l:l}\rho_1 & \rho_2 & \cdots & \rho_p\end{array}\right]$, we can write (10.36) even more compactly as
$$A(L, \boldsymbol{\rho}) u_t=\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right) .$$

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|估计AR的回归模型

$$y_t=x_t(\boldsymbol{\beta})+u_t, \quad u_t=\rho u_{t-1}+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right) .$$

$$y_t=x_t(\boldsymbol{\beta})+\rho\left(y_{t-1}-x_{t-1}(\boldsymbol{\beta})\right)+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right),$$
，这也是一个非线性回归模型，但误差项(根据假设)是序列不相关的。由于(10.12)是一个具有良好误差项的非线性回归模型，用非线性最小二乘估计它，然后用高斯-牛顿回归对它进行推断似乎是很自然的。回归函数就是
$$x_t^{\prime}(\boldsymbol{\beta}, \rho)=x_t(\boldsymbol{\beta})+\rho\left(y_{t-1}-x_{t-1}(\boldsymbol{\beta})\right),$$
，它依赖于$\rho$和$\boldsymbol{\beta}$。
(10.12)有两个潜在问题。首先，回归函数$x_t^{\prime}(\boldsymbol{\beta}, \rho)$必然依赖于$y_{t-1}$，不管$x_t(\boldsymbol{\beta})$是否依赖于因变量的滞后值。正如我们在第5章中看到的，只要满足某些规律性条件，这种依赖性并不妨碍非线性最小二乘具有理想的渐近性质。可以证明，只要$x_t(\boldsymbol{\beta})$满足$5.1$和$5.2$定理的正则性条件和$|\rho|<1$所具有的平稳性条件，对于(10.12)确实是这样。但是，如果平稳性条件不成立，关于非线性最小二乘的标准结果，特别是定理5.2，渐近正态性定理，将不再适用于(10.12)

(10.12)的第二个问题是如何处理第一个观察结果。假设我们没有$y_0$以及评估$x_0(\boldsymbol{\beta})$所需的所有外生变量和预定变量的数据，因为如果我们这样做了，样本就不会从$t=1$对应的观察开始。因此我们不能评估$x_1^{\prime}(\boldsymbol{\beta}, \rho)$，它依赖于$y_0$和$x_0(\boldsymbol{\beta})$。这个问题最简单的解决方案就是删除第一个观察值，要求(10.12)只保存观察值2到$n$。删除一个观察结果在渐近上不会产生任何差异，因此我们可以在样本量相当大的情况下安全地这样做

## 经济代写|计量经济学代写Econometrics代考|高阶AR过程

$$\left(1-\rho_1 L-\rho_2 L^2-\cdots-\rho_p L^p\right) u_t=\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right)$$，其中$L$表示滞后操作符。滞后运算符$L$的属性是，当$L$乘以任何带有时间下标的内容时，这个下标将滞后一个周期。因此
$$L u_t=u_{t-1}, \quad L^2 u_t=u_{t-2}, \quad L^p u_t=u_{t-p},$$

$$A(L, \boldsymbol{\rho}) u_t=\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right) .$$

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