# 经济代写|计量经济学代写Econometrics代考|Find2022

## 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
$$u_t=\varepsilon_t+\rho \varepsilon_{t-1}+\rho^2 \varepsilon_{t-2}+\rho^3 \varepsilon_{t-3}+\cdots$$

# 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

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$$y_{t i}=x_{t i}(\boldsymbol{\beta})+u_{t i}, \quad t=1, \ldots, T, i=1, \ldots, n .$$

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

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

$|\rho|<1$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
$$u_t=\varepsilon_t+\rho \varepsilon_{t-1}+\rho^2 \varepsilon_{t-2}+\rho^3 \varepsilon_{t-3}+\cdots$$

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