translated by damien from Marco Avarucci

Basic Econometrics - Unit 6 Basic Regression Analysis with Time Series

 计量经济学基础第6单元时间序列基本回归分析

The nature of time series data

时间序列数据的性质

• Temporal ordering of observations; may not be arbitrarily reordered Typical features: serial correlation/dependence of observations How should we think about the randomness in time series data?
   观测值的时间顺序；不能任意重新排序典型特征：观测值的序列相关性/依赖性我们应该如何看待时间序列数据中的随机性？

• The outcome of economic variables (e.g. GNP, Dow Jones) is uncertain; they should therefore be modeled as random variables
   经济变量（如国民生产总值、道琼斯指数）的结果是不确定的，因此应将其建模为随机变量

• Time series are sequences of r.v. (= stochastic processes)
   时间序列是r.v.（=随机过程）的序列

• “Sample” = the one realized path of the time series out of the many possible paths the stochastic process could have taken
   “样本”=随机过程可能采取的许多可能路径中，时间序列的一条实现路径

Examples of time series regression models

 时间序列回归模型实例

• Static models
   静态模型

  • In static time series models, the current value of one variable is modeled as the result of the current values of explanatory variables
     在静态时间序列模型中，一个变量的当前值被建模为解释变量当前值的结果

• Examples for static models
   静态模型示例

• Finite distributed lag models
   有限分布滞后模型

  • In finite distributed lag models, the explanatory variables are allowed to influence the dependent variable with a time lag
     在有限分布滞后模型中，允许解释变量以时滞影响因变量

• Example for a finite distributed lag model
   有限分布滞后模型示例

  • The fertility rate may depend on the tax value of a child, but for biological and behavioral reasons, the effect may have a lag
     生育率可能取决于孩子的纳税价值，但由于生理和行为原因，其影响可能有滞后性

Finite sample properties of OLS under classical assumptions

 经典假设下OLS的有限样本性质

• Assumption TS.1 (Linear in parameters)
   假设TS.1（参数线性）

• Assumption TS.2 (No perfect collinearity)
   假设TS.2（无完全共线）

“In the sample (and therefore in the underlying time series process), no independent variable is constant nor a perfect linear combination of the others.”
 “在样本中（因此在基本的时间序列过程中），没有自变量是常数，也没有其他变量的完美线性组合。

• Notion
   概念

• Assumption TS.3 (Zero conditional mean)
   假设TS.3（零条件平均值）

\[E(u _t | X) = 0 \quad t = 1,2, \cdots, n\]

• Discussion of assumption TS.3
   关于假设TS.3的讨论

• (Contemporaneous) Exogeneity is not enough to ensure unbiasdness.
   （同期）外生不足以确保公正。

• TS.3 rules out feedback from the dep. variable on future values of the explanatory variables; \( y _{t-j}, j >0 \) cannot be included as regressors.
   TS.3排除了来自解释变量未来值的部门变量的反馈；\（y{t-j}，j>0\）不能作为回归变量。

  • \( mrdrte _t = \beta _0 + \beta _1 polpc _t + \beta _2 mrdrte _{t-1} + u _t = \beta _0 + \beta _1 x _{1,t} + \beta _2 x _{2,t} + u _t \)

  • \( E(u _t

    x _{2, t+1}) \neq 0 \)

• Example (inflation and federal fund rate)
   *示例（通货膨胀和联邦基金利率）**

  • \( inf _t = \beta _0 + \beta _1 ffrate _t + \beta _1 ffrate _{t-1} + \beta _2 ffrate _{t-2} + u _t .\)

  • \( u _t >0 \) might lead FED to increase the \( ffrate \) the next period. Then \( ffrate _{t+1}\) and \( u _t \) are correlated, violating strict exogeneity.

• Theorem 10.1 (Unbiasedness of OLS)
    定理10.1（OLS的无偏性）

\[TS.1 - TS.3 \quad \Rightarrow \quad E(\hat{\beta} _j) = \beta _j, \quad j = 0,1, \cdots, k\]

• Assumption TS.4 (Homoscedasticity)
   假设TS.4（同方差）

  • A sufficient condition is that the volatility of the error is independent of the explanatory variables and that it is constant over time.
     一个充分条件是，误差的波动性与解释变量无关，并且随时间的变化是常数。

  • In the time series context, homoscedasticity may also be easily violated, e.g. if the volatility of the dep. variable depends on regime changes.
     在时间序列的背景下，同方差也可能很容易被破坏，例如，如果dep.变量的波动性取决于制度的变化。

• Assumption TS.5 (No serial correlation)
   假设TS.5（无序列相关）

• Discussion of assumption TS.5
   假设TS.5的讨论

  • Why was such an assumption not made in the cross-sectional case?
     为什么在横截面情况下没有作出这样的假设？

  • When \(Corr(u _t, u _s) \neq 0\) for some \(𝑡≠𝑠 \), we say that the error exibith serial correlation (autocorrelation)
     当\(Corr(u _t, u _s) \neq 0\)用于某些\(𝑡≠𝑠 \)时，我们说错误存在位序列相关（自相关）

  • Three types of correlation:
     三种类型的相关性：

  \[1) Corr(x _{jt}, x _{hs}), \quad 2) Corr(x _{js}, u _t), \quad 3) Corr(u _t, u _s)\]

• Theorem 10.2 (OLS sampling variances)
    定理10.2（OLS抽样方差）

• Theorem 10.3 (Unbiased estimation of the error variance)
   定理10.3（误差方差的无偏估计）

\[TS.1 - TS.5 \quad \Rightarrow \quad E(\hat{\sigma} ^2) = \sigma ^2\]

• Theorem 10.4 (Gauss-Markov Theorem)
   定理10.4（高斯-马尔可夫定理）

  • Under assumptions TS.1–TS.5, the OLS estimators have the minimal variance of all linear unbiased estimators of the regression coefficients
     在假设TS.1–TS.5下，OLS估计量具有回归系数的所有线性无偏估计量的最小方差

  • This holds conditional as well as unconditional on the regressors
     这对回归者既有条件的，也有无条件的

• Assumption TS.6 (Normality)
   假设TS.6（正态）

  \[u _t \sim \text{iid} \ N(0, \sigma ^2) \quad \text{ indenpent of } X\]

This assumption implies TS.3 - TS.5.
 这一假设意味着TS.3-TS.5。

• Theorem 10.5 (Normal sampling distributions)
   定理10.5（正态抽样分布）

  • Under assumptions TS.1 – TS.6, the OLS estimators have the usual nor-mal distribution (conditional on X ). The usual F-and t-tests are valid.
     在假设TS.1-TS.6下，OLS估计具有通常的正态分布（以X为条件）。通常的F检验和t检验是有效的。

• Using dummy explanatory variables in time series
   在时间序列中使用虚拟解释变量

  • Period: 1913-1984
     时间：1913-1984

  • During World War II, the fertility rate was temporarily lower
     二战期间，生育率暂时较低

  • It has been permanently lower since the introduction of the pill in 1963
     自1963年推出避孕药以来，该数值一直处于较低水平

• If we assume that the full set of CLM assumptions hold, we can reject \( H _0: \beta _{pe} = 0\) against the two-sided alternative at less than the 1% significance level \(( t _{pe} = 2.77 )\)
   如果我们假设全套CLM假设成立，我们可以在低于1%显著性水平的情况下\(( t _{pe} = 2.77 )\)，对双边替代方案拒绝 \( H _0: \beta _{pe} = 0\)

• The estimated effect is practically large (probably too large). A $100 increase in \(pe \) increases the estimated fertility rate by 8.3 children per thousand women.
   估计的影响实际上很大（可能太大）。每千名妇女生育率增加100美元，估计增加8.3个孩子。

• Fertility rates were much lover, on average, during WWII and after the introduction of the birth control pill.
   平均而言，在二战期间和避孕药问世后，生育率要低得多。

Trends and Seasonality

 趋势和季节性

• Characterizing Trending Data
   描述趋势数据

  • Many series tend to increase over time, at least on average. They typically have up and down periods, but the overall trend is up. (Example: GDP)
     许多系列往往随着时间的推移而增加，至少在平均水平上是这样。它们通常有上升和下降的时期，但总体趋势是上升的。（例如：GDP）

  • Other variables tend to decline over time (such as the rate of traffic fatalities).
     其他变量往往随着时间的推移而下降（如交通事故死亡率）。

  • We can find spurious relations among trending variables that have nothing to do with each other.
     我们可以在趋势变量之间找到彼此无关的虚假关系。

• Time series with trends
   有趋势的时间序列

• Modelling a linear time trend
   线性时间趋势建模

• Modelling an exponential time trend
   指数时间趋势建模

• Example for a time series with an exponential trend
   具有指数趋势的时间序列示例

• Using trending variables in regression analysis
   在回归分析中使用趋势变量

  • If trending variables are regressed on each other, a spurious relationship may arise just because they grows over time
     如果趋势变量相互回归，就可能因为它们随时间增长而产生虚假关系

  • In this case, it is important to include a trend in the regression
     在这种情况下，在回归中包含趋势是很重要的

• Example: Housing investment and prices
   例如：住房投资和价格

• The are other factors, captured in the time trend, that affect invpc
   时间趋势中捕获的其他因素也会影响invpc