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Basic Econometrics - Unit 7/B Serial Correlation in Time Series Regressions

计量经济学基础第7/乙单元时间序列回归中的序列相关性

Strict exogeneity ⇒ OLS is unbiased.
严格的外生性⇒OLS是无偏的。
Contemporaneous exogeneity ⇒ OLS is consistent (provided the time series are weakly dependent).
同期外生性⇒OLS是一致的（前提是时间序列具有弱依赖性）。
Unbiasedness/Consistency did not require assumptions on error autocorrelation.
无偏性/一致性不需要假设误差自相关。
There are situations where the nature of the \(x _{jt}\) mean that serial correlation in \(u _t\) implies that \(u _t\) correlated with \(x _{jt}\).
有些情况下，\(x _{jt}\)的性质意味着，\(u _t\) 中的序列相关意味着，\(u _t\)与(x _{jt}\)相关。

Testing for serial correlation

测试序列相关性

Testing for AR(1) serial correlation with strictly exog. regressors
AR（1）序列相关性测试与exog.回归函数

Large sample justification (unobservable residuals)
大样本调整（不可观测残差）
Durbin-Watson test under classical assumptions
经典假设下的Durbin-Watson检验
- Under assumptions TS.1–TS.6, the Durbin-Watson test is an exact test (whereas the previous t-test is only valid asymptotically).
  在假设TS.1–TS.6下，Durbin-Watson检验是精确检验（而之前的t检验仅渐近有效）。

When strictly exogeneity does not hold, one or more \(x_{jt}\) might be correlated with with \(u {t-1}\).
当严格外生性不成立时，一个或多个\(x{jt}\)可能与\(u _{t-1}\)相关。
𝑡−test and DW test are not valid (even asymptotically)
𝑡−检验和DW检验无效（甚至是渐近的）
Example: lagged dependent variables as regressors: \(y_{t-1}\) and \(u {t-1}\) are obviously correlated.
例如：作为回归变量的滞后因变量：\(y{t-1}\)和\(u _{t-1}\)有明显的相关性。
Consider the AR(1) test.
考虑AR（1）测试
Testing for AR(1) serial correlation with general regressors
AR（1）序列相关性的一般回归检验
- The t-test for autocorrelation can be easily generalized to allow for the possibility that the explanatory variables are not strictly exogenous:
  自相关的t检验可以很容易地推广，以允许解释变量不是严格的外生变量的可能性：
- The test may be carried out in a heteroscedasticity robust way
  检验可以异方差稳健的方式进行
General Breusch-Godfrey test for AR(q) serial correlation
AR（q）序列相关的一般Breusch-Godfrey检验

用严格exog回归函数校正序列相关性

Under the assumption of AR(1) errors, one can transform the model so that it satisfies all GM-assumptions. For this model, OLS is BLUE.
在AR（1）误差的假设下，可以对模型进行变换，使其满足所有GM假设。对于这个模型，OLS是蓝色的。

Problem: The AR(1)-coefficient is not known and has to be estimated (FGLS)
问题：AR（1）-系数未知，必须估计（FGLS）

OLS后的序列相关稳健推理

In the presence of positive serial correlation, OLS standard errors overstate statistical significance
在正序列相关的情况下，OLS标准差夸大了统计显著性
One can compute serial correlation -robust std. errors after OLS
我们可以在OLS之后计算序列相关稳健标准误差
This is useful because FGLS requires strict exogeneity and assumes a very specific form of serial correlation (AR(1) or, generally, AR(q))
这是有用的，因为FGLS需要严格的外生性，并假设一种非常特殊的序列相关形式（AR（1）或，通常，AR（q））
Serial correlation-robust standard errors:
序列相关稳健标准误差：
Serial correlation-robust F-and t-tests are also available
序列相关稳健F检验和t检验也可用
Correction factor for serial correlation (Newey-West formula)
序列相关校正系数（Newey-West公式）

Discussion of serial correlation-robust standard errors
讨论序列相关稳健标准误差
- The formulas are also robust to heteroscedasticity; they are therefore called “heteroscedasticity and autocorrelation consistent” (=HAC)
  公式对异方差也很稳健，因此称为“异方差和自相关一致”（=HAC）
- For the integer g, values such as g=2 or g=3 are normally sufficient (there are more involved rules of thumb for how to choose g)
  对于整数g，像g=2或g=3这样的值通常就足够了（对于如何选择g，有更复杂的经验法则）
- Serial correlation-robust standard errors are only valid asymptotically; they may be severely biased if the sample size is not large enough
  序列相关稳健标准误差仅渐近有效；如果样本量不够大，它们可能会严重偏误
- The bias is the higher the more autocorrelation there is; if the series are highly correlated, it might be a good idea to difference them first
  偏差越大，自相关就越多；如果序列高度相关，最好先将它们区分开来
- Serial correlation-robust errors should be used if there is serial corr. and strict exogeneity fails (e.g. in the presence of lagged dep. var.)
  如果存在序列相关性且严格的外生性失败（例如存在滞后的dep.var），则应使用序列相关性稳健误差

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