Time-Series Analysis

Time-series analysis is a type of statistical analysis of observational data that vary over regular units of time. In political science, the data used are typically aggregate descriptors, such as unemployment rates, levels of presidential approval, the incidence of terrorist attacks, or the percentages of Democratic and Republican Party identifiers in the electorate. The time intervals over which these data are gathered are usually years, quarters, or months, but data gathered over much shorter periods also may be employed. For example, political economists studying variations in currency exchange rates or volatility in stock markets may use data gathered every day, every minute, or even every second. Although there are exceptions, time-series analysts normally do not generate their data de novo but rather rely on data gathered by government statistical agencies or private entities, such as public opinion polling agencies. In the following, the basic features and recent advances in this field are presented.

Historically, time-series analysis was closely integrated with other aspects of econometrics. Researchers typically began by specifying a model where the dependent (Yt) and independent (Xkt) variables as well as the stochastic error term (εt), were subscripted with t to denote variation over time. Equation 1 is an example with a single predictor variable (Xt). The coefficients β0 and β1 in this equation were assumed to be time invariant and were estimated using ordinary least squares (OLS) regression:

A standard set of postestimation diagnostics was performed, with a Durbin-Watson test used to detect the presence of first-order autocorrelation in the residuals. Autocorrelated residuals prompted analysts to infer the existence of autocorrelation in the error process, that is, εt = ρε t − 1 + vt, where vt is a well-behaved Gaussian error term that is ∼N(0,σ2) by assumption. In turn, autocorrelated errors were treated as a nuisance, rather than as an indicator of model misspecification. This nuisance could be eliminated by a variant of generalized least squares (the Cochrane-Orcutt transformation), with the parameter p being estimated from the data. The resulting model (Equation 2) involves first-order partial differences of all left- and right-hand-side variables, including the error process:

The model thus contains a common factor restriction (1 – ρL), where L is a backshift operator. Political scientists using this technique often appeared to be unaware that the model had dynamic properties with effects of all predictor variables being distributed over time by the presence of a lagged endogenous variable Yt − 1, with the rate of decay of these effects being governed by the parameter ρ.

Starting in the 1970s, analysts became increasingly aware of the restrictive nature of the Cochrane-Orcutt transformation, and many began to address the “nuisance” of autocorrelated errors by inserting a lagged endogenous variable on the right-hand side of their model (Equation 3).

This specification avoided the common factor restriction associated with the Cochrane-Orcutt transformation, but it still imposed a possibly theoretically unattractive uniform dynamic on the effects of all predictor variables, with the rate of decay of those effects being governed by the parameter λ associated with the lagged endogenous variable. In addition, the lagged endogenous variable heightened the possibility that parameter estimates would be biased and inconsistent. Analysts recognizing this possibility typically tested model residuals for first-order correlation using statistics such as Durbin's H (the Durbin-Watson test being inapplicable for models with lagged endogenous variables).

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