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经济代写|计量经济学代写Econometrics代考|Modeling Time-Series/Cross-Section Data

Many data sets have both a time-series and a cross-section dimension. For example, they might contain 40 years of data on 20 countries, or 132 quarters of data on 50 states. The advantage of such data sets is that the sample size is usually quite large (for the above examples, $40 \times 20=800$ and $132 \times 50=$ 6600 ), which means that they should potentially be very informative about the parameters to be estimated. The disadvantage is that it is necessary to take the two-dimensional nature of the data into account. A particular type of time-series/cross-section data arises when the same sample of individuals, households, or firms is observed at two or more points in time. Data of this type are often referred to as panel data. A panel data set generally consists of a fairly small number of temporal observations on a large number of cross section units. The imbalance between the two dimensions of the sample may make it necessary to use special techniques and can make reliance on standard asymptotic theory inappropriate.

If we let $t$ index the time dimension of the data and $i$ index the crosssection dimension, we can write a univariate nonlinear regression model for time-series/cross-section data as
$$
y_{t i}=x_{t i}(\boldsymbol{\beta})+u_{t i}, \quad t=1, \ldots, T, i=1, \ldots, n .
$$
There are $T$ time periods and $n$ cross-sectional units, for a total of $n T$ observations. If we were willing to assume that the $u_{t i}$ ‘s are homoskedastic and independent, we could simply estimate (9.71) by NLS. But often this will not be a realistic assumption. The variance of $u_{t i}$ might well vary systematically with $t$ or $i$ or both of them. Moreover, it seems plausible that the error terms $u_{t i}$ and $u_{t j}$ will be correlated for some $i \neq j$ if certain shocks affect several cross-sectional units at the same point in time. Similarly, it seems plausible that the error terms $u_{t i}$ and $u_{s i}$ will be correlated for some $t \neq s$ if certain shocks affect the same cross-section unit at more than one point in time. Whether any of these failures of the i.i.d. assumption will occur for any given data set is difficult to say a priori. But if they do occur, and we simply use NLS, we will obtain an estimated covariance matrix that is inconsistent and may lead to serious errors of inference. In some circumstances, we may even obtain inconsistent parameter estimates.

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

The phenomenon of serial correlation, in which successive residuals appear to be correlated with each other, is very often encountered in models estimated with time-series data. As a result, testing for serial correlation and estimating models that take account of it are both topics which have been studied for a very long time by econometricians, and the literature is consequently vast. Happily, the results we have already obtained about NLS, GNLS, and ML allow us to handle most of the problems associated with serial correlation in a straightforward way.

Although error terms may fail to be independent in any sort of model, lack of independence is most often observed in models estimated with timeseries data. In particular, observations that are close in time often have error terms which appear to be correlated, while observations that are far apart in time rarely do. We say appear to be correlated because misspecification of the regression function may lead residuals to be correlated across observations, even when the actual error terms are not. In any case, whether the appearance of serial correelation in time-series models is genuine or not, one particularly simple model of serial correlation has become very popular. In this model, the error terms $u_t$ are assumed to follow the first-order autoregressive, or AR(1), process
$$
u_t=\rho u_{t-1}+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right), \quad|\rho|<1 .
$$
This stochastic process says that the error at time $t, u_t$, is equal to some fraction $\rho$ of the error at time $t-1$ (with the sign changed if $\rho<0$ ), plus a new error term or innovation $\varepsilon_t$ that is homoskedastic and independent of all past and future innovations. Thus in each period part of the error term is the last period’s error term, shrunk somewhat toward zero and possibly changed in sign, and part is the innovation $\varepsilon_t$.

The condition that $|\rho|<1$ is called a stationarity condition. It ensures that the variance of $u_t$ tends to a limiting value, $\sigma^2$, rather than increasing without limit as $t$ gets large. By substituting successively for $u_{t-1}, u_{t-2}, u_{t-3}$, and so on in (10.01), we see that
$$
u_t=\varepsilon_t+\rho \varepsilon_{t-1}+\rho^2 \varepsilon_{t-2}+\rho^3 \varepsilon_{t-3}+\cdots
$$

经济代写|计量经济学代写Econometrics代考|Find2022

计量经济学代考

经济代写|计量经济学代写Econometrics代考|建模时间序列/横断面数据

.


许多数据集既有时间序列又有截面维度。例如,它们可能包含20个国家40年的数据,或50个州132个季度的数据。这样的数据集的优点是,样本量通常相当大(对于上面的例子,$40 \times 20=800$和$132 \times 50=$ 6600),这意味着它们可能会提供关于要估计的参数的大量信息。缺点是必须考虑到数据的二维性质。当在两个或两个以上的时间点观察相同的个人、家庭或公司样本时,就产生了一种特定类型的时间序列/截面数据。这种类型的数据通常称为面板数据。面板数据集通常由大量截面单元上相当少量的时间观测数据组成。样本的两个维度之间的不平衡可能使使用特殊技术成为必要,也可能使依赖标准渐近理论变得不合适


如果我们让$t$索引数据的时间维度,$i$索引横截面维度,我们可以为时间序列/横截面数据编写一个单变量非线性回归模型为
$$
y_{t i}=x_{t i}(\boldsymbol{\beta})+u_{t i}, \quad t=1, \ldots, T, i=1, \ldots, n .
$$
有$T$个时间段和$n$个横截面单位,总共有$n T$个观测值。如果我们愿意假设$u_{t i}$的是同方差和独立的,我们可以通过NLS简单地估计(9.71)。但这往往不是一个现实的假设。$u_{t i}$的方差很可能与$t$或$i$或两者都有系统差异。此外,如果某些冲击在同一时间点影响多个截面单元,误差项$u_{t i}$和$u_{t j}$对于某些$i \neq j$将是相关的,这似乎是合理的。同样,如果某些冲击在多个时间点上影响同一截面单元,误差项$u_{t i}$和$u_{s i}$对于某些$t \neq s$将是相关的,这似乎是合理的。对于任何给定的数据集,i.i.d.假设的这些错误是否会发生,很难从先验上进行判断。但如果它们确实发生了,而我们只是简单地使用NLS,我们将得到一个估计的协方差矩阵,这是不一致的,可能导致严重的推断错误。在某些情况下,我们甚至可能得到不一致的参数估计

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

.


序列相关现象,即连续残差似乎彼此相关,在用时间序列数据估计的模型中经常遇到。因此,检验序列相关性和考虑到序列相关性的估计模型都是计量经济学家研究了很长时间的课题,因此文献非常丰富。令人高兴的是,我们已经获得的关于NLS、GNLS和ML的结果允许我们以一种直接的方式处理与序列相关的大多数问题


虽然误差项可能在任何类型的模型中都不是独立的,但在用时间序列数据估计的模型中最常观察到缺乏独立性。特别是,在时间上接近的观测结果往往有误差项,这些误差项似乎是相关的,而在时间上相差很远的观测结果则很少如此。我们说似乎是相关的,因为错误的回归函数可能导致残差在不同的观察之间是相关的,即使实际的误差项并不相关。无论如何,无论序列相关在时间序列模型中的出现是否真实,一个特别简单的序列相关模型已经变得非常流行。在这个模型中,假设误差项$u_t$遵循一阶自回归,或AR(1),过程
$$
u_t=\rho u_{t-1}+\varepsilon_t, \quad \varepsilon_t \sim \operatorname{IID}\left(0, \omega^2\right), \quad|\rho|<1 .
$$
这个随机过程说,在$t, u_t$时刻的误差,等于在$t-1$时刻误差的某个部分$\rho$(如果$\rho<0$则符号改变),加上一个同方差的新误差项或创新$\varepsilon_t$,它独立于所有过去和未来的创新。因此,在每个周期中,误差项的一部分是上一个周期的误差项,它向零收缩了一些,并且可能在符号上发生了变化,部分是创新$\varepsilon_t$ .

$|\rho|<1$的条件被称为平稳条件。它确保$u_t$的方差趋于一个极限值$\sigma^2$,而不是随着$t$变大而无限制地增加。通过在(10.01)中依次替换$u_{t-1}, u_{t-2}, u_{t-3}$,我们看到
$$
u_t=\varepsilon_t+\rho \varepsilon_{t-1}+\rho^2 \varepsilon_{t-2}+\rho^3 \varepsilon_{t-3}+\cdots
$$

经济代写|博弈论代写Game Theory代考

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