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时间序列回归模型

Time series regression modelsattempt to explain the current response using the response history (autoregressive dynamics) and the transfer of dynamics from relevant predictors (or otherwise). Theoretical frameworks for potential relationships among variables often permit different representations of the system.

Use time series regression models to analyzetime series data, which are measurements that you take at successive time points. For example, use time series regression modeling to:

  • 检查当前和过去的失业率以及过去通货膨胀率对当前通货膨胀率的线性影响。

  • Forecast GDP growth rates by using an ARIMA model and include the CPI growth rate as a predictor.

  • Determine how a unit increase in rainfall, amount of fertilizer, and labor affect crop yield.

You can start a time series analysis by building a design matrix (Xt），其中可以包括当前和过去的预测因素观察结果。您还可以使用自回归（AR）组件来补充回归组件，以解释响应的可能性（yt）动态。例如，在回归部分中包括过去对通货膨胀率的过去测量，以解释当前的通胀率。AR术语解释了回归部分无法解释的动力学，这必然在计量经济学应用中指定。此外，AR术语还会吸收残留的自相关，简化创新模型，并通常改善预测性能。然后，将普通最小二乘（OLS）应用于多线性回归（MLR）模型：

y t = X t β + u t .

If a residual analysis suggests classical linear model assumption departures such as that heteroscedasticity or autocorrelation (i.e., nonspherical errors), then:

  • You can estimate robustHAC(heteroscedasticity and autocorrelation consistent) standard errors (for details, seehac).

  • 如果您知道创新协方差矩阵（至少最多达到缩放因素），那么您可以申请generalized least squares(GLS). Given that the innovation covariance matrix is correct, GLS effectively reduces the problem to a linear regression where the residuals have covarianceI.

  • If you do not know the structure of the innovation covariance matrix, but know the nature of the heteroscedasticity and autocorrelation, then you can applyfeasible generalized least squares(FGLS). FGLS applies GLS iteratively, but uses the estimated residual covariance matrix. FGLS estimators are efficient under certain conditions. For details, see[1], Chapter 11.

There are time series models that model the dynamics more explicitly than MLR models. These models can account for AR and predictor effects as with MLR models, but have the added benefits of:

  • 考虑移动平均值（MA）效果。包括MA术语以减少AR滞后的数量，从而有效减少初始化模型所需的观察次数。

  • Easily modeling seasonal effects. In order to model seasonal effects with an MLR model, you have to build an indicator design matrix.

  • Modeling nonseasonal and seasonal integration for unit root nonstationary processes.

这些模型也不同于MLR，因为它们依赖于分布假设（即它们使用最大可能性进行估计）。流行类型的时间序列回归模型包括：

  • 自回归综合移动平均值与外源预测因子（Arimax）。这是一个线性包含预测因子（外源或其他方式）的Arima模型。有关详细信息，请参阅arima或者Arimax（P，D，Q）模型.

  • Regression model with ARIMA time series errors. This is an MLR model where the unconditional disturbance process (ut) is an ARIMA time series. In other words, you explicitly modelutas a linear time series. For details, seeregARIMA.

  • Distributed lag model(DLM). This is an MLR model that includes the effects of predictors that persist over time. In other words, the regression component contains coefficients for contemporaneous and lagged values of predictors. Econometrics Toolbox™ does not contain functions that model DLMs explicitly, but you can useregARIMA或者fitlm使用适当构建的预测指标（设计）矩阵来分析DLM。

  • 转换功能(autoregressive distributed lag) model. This model extends the distributed lag framework in that it includes autoregressive terms (lagged responses). Econometrics Toolbox does not contain functions that model DLMs explicitly, but you can use thearimafunctionality with an appropriately constructed predictor matrix to analyze an autoregressive DLM.

您在使用哪种模型上做出的选择取决于您的分析目标以及数据的属性。

参考

[1] Greene，W。H.计量经济学分析. 6th ed. Englewood Cliffs, NJ: Prentice Hall, 2008.

See Also

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