经济代写|计量经济学代写Econometrics代考|ECON2271

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 $\rho$.

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

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

$$y_t=x_t^{\prime}(\boldsymbol{\beta}, \rho)+\varepsilon_t \equiv x_t(\boldsymbol{\beta})+\rho\left(y_{t-1}-x_{t-1}(\boldsymbol{\beta})\right)+\varepsilon_t,$$
，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

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$$H_0: y_t=\boldsymbol{X}t \boldsymbol{\beta}+u_t, \quad u_t \sim \operatorname{IID}\left(0, \sigma^2\right) .$$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:$$H_2: \quad y_t=\boldsymbol{X}t \boldsymbol{\beta}+\rho y{t-1}+\boldsymbol{X}{t-1} \boldsymbol{\gamma}+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right),$$

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