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经济代写|计量经济学代写Econometrics代考|Testing for Serial Correlation

A very substantial fraction of all the literature in econometrics has been devoted to the problem of testing for serial correlation in the error terms of regression models. The largest part of that fraction has dealt with testing the null hypothesis that the errors for a linear regression model are serially independent against the alternative that they follow an $\mathrm{AR}(1)$ process. Although serial correlation is certainly a widespread phenomenon with time-series data, so that testing for it is clearly important, the amount of effort devoted to this problem seems somewhat disproportionate. As we will see, asymptotic tests for serial correlation can readily be derived as applications of the GaussNewton regression. Only when it is possible to make inferences that are exact in finite samples is there any reason to make use of more specialized and difficult procedures.

Suppose we wish to test the null hypothesis that the errors $u_t$ in the model
$$
y_t=x_t(\boldsymbol{\beta})+u_t
$$
are serially independent against the alternative that they follow an $\operatorname{AR}(1)$ process. As we have already seen, for observations $t=2, \ldots, n$, this alternative model can be written as
$$
y_t=x_t^{\prime}(\boldsymbol{\beta}, \rho)+\varepsilon_t \equiv x_t(\boldsymbol{\beta})+\rho\left(y_{t-1}-x_{t-1}(\boldsymbol{\beta})\right)+\varepsilon_t,
$$
where $\varepsilon_t$ is assumed to be $\operatorname{IID}\left(0, \omega^2\right)$. As we saw in Chapter 6 , any restrictions on the parameters of a nonlinear regression function can be tested by running a Gauss-Newton regression evaluated at estimates that are root- $n$ consistent under the null hypothesis. These would typically, but not necessarily, be restricted NLS estimates. Thus, in this case, the restriction that $\rho=0$ can be tested by regressing $y_t-x_t^{\prime}$ on the derivatives of the regression function $x_t^{\prime}(\boldsymbol{\beta}, \rho)$ with respect to all of the parameters, where both $x_t^{\prime}$ and its derivatives are evaluated at the estimates of the parameter vector $[\boldsymbol{\beta} \vdots \rho]$ under the null.

经济代写|计量经济学代写Econometrics代考|Common Factor Restrictions

If the regression function is misspecified, the residuals may display serial correlation even when the error terms are in fact serially independent. This might happen if a variable that was itself serially correlated, or a lagged dependent variable, were incorrectly omitted from the regression function. In such a case, we can in general make valid inferences only by eliminating the misspecification rather than by “correcting” the model for AR(1) errors or some other simple error process. If we simply do the latter, as used to be done all too frequently in applied work, we may well end up with a seriously misspecified model.

There is no universally effective way to avoid misinterpreting misspecification of the regression function as the presence of serially correlated errors. Model specification is an art as much as a science, and with the short samples typical of time-series data we can never expect to detect all forms of misspecification. Nevertheless, there is one family of tests that has been shown to be very effective in detecting misspecification in models which appear to have errors that follow a low-order AR process. These are tests of what are, for reasons that will be apparent shortly, generally called common factor restrictions. The basic idea of testing common factor restrictions, although not the terminology, may be found in Sargan (1964). More recent references include Hendry and Mizon (1978), Mizon and Hendry (1980), and Sargan (1980a). An illuminating example is provided by Hendry (1980), who presents a grossly misspecified model that yields apparently sensible results after “correction” for $\mathrm{AR}(1)$ errors and then shows that a test for common factor restrictions would detect the misspecification.

In order to fix ideas, we will assume for the moment that the model to be tested is a linear regression model that apparently has $\operatorname{AR}(1)$ errors. It is natural to think of there being threc nested models in this case. The first of these is the original linear regression model with error terms assumed to be serially independent,
$$
H_0: y_t=\boldsymbol{X}t \boldsymbol{\beta}+u_t, \quad u_t \sim \operatorname{IID}\left(0, \sigma^2\right) . $$ The second is the nonlinear model that results when the errors $u_t$ of (10.85) are assumed to follow the AR(1) process $u_t=\rho u{t-1}+\varepsilon_t$,
$$
H_1: y_t=\boldsymbol{X}t \boldsymbol{\beta}+\rho\left(y{t-1}-\boldsymbol{X}{t-1} \boldsymbol{\beta}\right)+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right) . $$ The third is the linear model that results when the nonlinear restrictions on (10.86) are relaxed: $$ H_2: \quad y_t=\boldsymbol{X}_t \boldsymbol{\beta}+\rho y{t-1}+\boldsymbol{X}_{t-1} \boldsymbol{\gamma}+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right),
$$
where $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ are both $k$-vectors. We encountered $H_2$ previously, in Section $10.3$, where it was used to obtain an initial consistent estimate of $\rho$.

经济代写|计量经济学代写Econometrics代考|ECON2271

计量经济学代考

经济代写|计量经济学代写Econometrics代考|序列相关性检验

.


在计量经济学的所有文献中,有相当大一部分致力于检验回归模型误差项中的序列相关性问题。该部分的最大部分处理了零假设的检验,即线性回归模型的误差与它们遵循$\mathrm{AR}(1)$过程的替代方案是串行独立的。尽管序列相关性是时间序列数据的普遍现象,因此对其进行测试显然很重要,但在这个问题上投入的精力似乎有些不成比例。正如我们将看到的,序列相关的渐近检验可以很容易地作为高斯牛顿回归的应用得到。只有当有可能在有限样本中作出精确的推论时,才有理由使用更专门和更困难的程序


假设我们希望检验零假设,即模型
$$
y_t=x_t(\boldsymbol{\beta})+u_t
$$
中的错误$u_t$与它们遵循$\operatorname{AR}(1)$过程的替代方案是串行独立的。正如我们已经看到的,对于观察$t=2, \ldots, n$,这个替代模型可以写成
$$
y_t=x_t^{\prime}(\boldsymbol{\beta}, \rho)+\varepsilon_t \equiv x_t(\boldsymbol{\beta})+\rho\left(y_{t-1}-x_{t-1}(\boldsymbol{\beta})\right)+\varepsilon_t,
$$
,其中$\varepsilon_t$假设为$\operatorname{IID}\left(0, \omega^2\right)$。正如我们在第6章中所看到的,对非线性回归函数参数的任何限制都可以通过运行高斯-牛顿回归来测试,在零假设下的估计值为root- $n$一致。这些通常(但不一定)是受限的NLS估计。因此,在本例中,可以通过在回归函数$x_t^{\prime}(\boldsymbol{\beta}, \rho)$对所有参数的导数上回归$y_t-x_t^{\prime}$来测试$\rho=0$的限制,其中$x_t^{\prime}$和它的导数都是在null下的参数向量$[\boldsymbol{\beta} \vdots \rho]$的估计值上求值。

经济代写|计量经济学代写Econometrics代考|公共因素限制

.


如果回归函数指定错误,即使误差项实际上是序列独立的,残差也可能显示序列相关。如果一个变量本身是序列相关的,或者一个滞后的因变量被错误地从回归函数中遗漏了,就可能发生这种情况。在这种情况下,我们通常只能通过消除错误规范而不是通过为AR(1)错误或其他一些简单的错误过程“纠正”模型来做出有效的推断。如果我们只是做后者,就像在应用工作中经常做的那样,我们很可能以一个严重错误指定的模型而告终


没有一种普遍有效的方法可以避免将回归函数的错误规范误解为存在序列相关错误。模型规范既是一门科学,也是一门艺术,对于时间序列数据的典型短样本,我们永远不能期望检测到所有形式的错误规范。尽管如此,有一类测试已被证明非常有效地检测模型中的错误规范,这些模型似乎具有遵循低阶AR过程的错误。这些测试通常被称为共同因素限制,原因很快就会清楚。Sargan(1964)给出了测试公共因素限制的基本思想,尽管没有给出术语。最近的参考文献包括亨德利和米森(1978)、米森和亨德利(1980)和萨根(1980a)。Hendry(1980)提供了一个具有启发意义的例子,他提出了一个严重错误指定的模型,在对$\mathrm{AR}(1)$错误进行“修正”之后,该模型产生了明显合理的结果,然后表明对公共因素限制的测试将检测到错误说明


为了修正想法,我们暂时假设要测试的模型是一个明显有$\operatorname{AR}(1)$错误的线性回归模型。在这种情况下,很自然地认为有三个嵌套模型。第一个是原始线性回归模型,误差项假设是序列独立的,
$$
H_0: y_t=\boldsymbol{X}t \boldsymbol{\beta}+u_t, \quad u_t \sim \operatorname{IID}\left(0, \sigma^2\right) . $$第二个是非线性模型,当(10.85)的误差$u_t$假设遵循AR(1)过程$u_t=\rho u{t-1}+\varepsilon_t$,
$$
H_1: y_t=\boldsymbol{X}t \boldsymbol{\beta}+\rho\left(y{t-1}-\boldsymbol{X}{t-1} \boldsymbol{\beta}\right)+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right) . $$第三个是线性模型,当(10.86)的非线性限制放松时:$$ H_2: \quad y_t=\boldsymbol{X}t \boldsymbol{\beta}+\rho y{t-1}+\boldsymbol{X}{t-1} \boldsymbol{\gamma}+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right),
$$
其中$\boldsymbol{\beta}$和$\boldsymbol{\gamma}$都是$k$ -vector。我们之前在$10.3$部分遇到过$H_2$,在那里它被用来获得$\rho$的初始一致估计。

经济代写|博弈论代写Game Theory代考

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