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Estimating Effects over Time for Single and Multiple Units

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Abstract

Criminology can easily be characterized by its investigation of change. We need to understand the conditions that facilitate the change in order to inform policy makers on how to reduce crime or improve social welfare. Yet, much of the published research in our field relies on cross-sectional data. As most criminological research questions are inherently dynamic, criminologists have more recently adopted the methods of analyzing changes over time. This chapter introduces a set of methodological choices to estimate the effects of changes in an independent variable on a dependent variable. It begins by outlining several methodological options when the scholar has repeated the measures of a single unit. Then, several other options for a scenario in which the data include many units that are repeatedly measured, are discussed. The chapter concludes with a discussion designed to guide the readers’ methodological decisions while analyzing dynamic data.

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Notes

  1. 1.

    If the model includes a lagged dependent variable, the Durbin–Watson statistic will be biased toward no autocorrelation. Turn to Greene (2008: 646) for an adjustment to the Durbin–Watson statistic for the lagged dependent variable. The other two tests produce unbiased estimates regardless of whether the lagged dependent variable is included in the model.

  2. 2.

    Autocorrelation can also be problematic in panel data. See Greene (2008) for more detail.

  3. 3.

    This approached used by Cantor and Land (1985) and the others has been criticized by Greenberg (2001). Greenberg explains that nonstationary crime rates are more likely explained by variables that also follow nonstationary processes (unemployment is stationary). Greenberg (2001) shows that while crime and divorce are each nonstationary, when combined they follow a stationary process, suggesting that they are cointegrated (Banerjee et al. 1993). By using error correction models that account for cointegration instead of relying on first- or second-differencing, long term effects can be estimated. See Britt (2001) for continued discussion in this debate.

  4. 4.

    Most standard statistical packages run ARIMA models.

  5. 5.

    \(\mathit{ACF}(k) = \frac{{\sum }_{i=1}^{N}\left ({Y }_{i}-\bar{Y }\right )\left ({Y }_{i+k}-\bar{Y }\right )} {{\sum }_{i=1}^{N}{\left ({Y }_{i}-\bar{Y }\right )}^{2}} \left ( \frac{N} {N-k}\right )\) (Cook and Campbell 1979; McDowall et al. 1980).

  6. 6.

    Some of correlations in the ACF can still be large in a stationary series; however, they will follow a pattern consistent with an autoregressive or moving average process.

  7. 7.

    For simplicity, the ARIMA(1,0,0) is written as if the series needed no differencing. The text could also refer a differenced ARIMA(1,d,0) process.

  8. 8.

    Most standard statistical packages will produce ACF and PACF tables. For example, the Stata command corrgram will produce a table the autocorrelations, partial autocorrelations, and the Portmantequ (Q) statistics for time series data. The commands AC and PAC produce a graphical version of the ACF and PACF values.

  9. 9.

    See Cook and Campbell (1979: 247–250) for a more thorough discussion of mixed models.

  10. 10.

    By repeated substitution, autoregressive processes can take a moving average form (Greene 2008).

  11. 11.

    He finds that Granger causality is unidirectional from money stock to income, dismissing arguments that changes in income lead to changes in money stock.

  12. 12.

    This system works better if the variables are stationary. This can be easily determined using the augmented Dickey–Fuller test, which is found in most statistical packages. If the data are nonstationary, it is usually recommended that the analyst estimate Vector Error Correction Models (Stata Press 2005; Greenberg 2001). However, others suggest that there is still value in estimating VAR if the short-term dynamics are the main focus of the analysis (Brandt and Williams 2007).

  13. 13.

    This notation is borrowed from Brandt and Williams (2007).

  14. 14.

    Stata has a post estimation command vargranger that performs pairwise Granger causality tests after running VAR (Stata Press 2005).

  15. 15.

    This decomposition is done by first inverting the VAR to its Vector Moving Average representation (VMA) and then decomposing the error covariance matrix. This method is sensitive to the ordering of the variables and all orderings should be considered (Brandt and Williams 2007). The Stata post estimation command fevd (forecast-error variance decompositions) calculates these values.

  16. 16.

    This method also relies on inverting the VAR to its VMA representation and then decomposing the error variance matrix. Once again, all orderings should be considered (Brandt and Williams 2007). The Stata post estimation command irf (impulse-response functions) produce these graphs.

  17. 17.

    The notation for all equations is borrowed from Greene (2008).

  18. 18.

    We could easily add another term, { T} ′ t γ, to represent those variables that change over time, but are stable across units.

  19. 19.

    Note that we could also add a time specific constant that captures variation that is constant across units, but varies over time.

  20. 20.

    Exceptions are found in the more recent population heterogeneity versus state dependency literature, which will be discussed below.

  21. 21.

    Their method differs from that used by Apel and colleagues (2008). Here they incorporated IV estimated with a generalized-method of moments framework.

  22. 22.

    Multilevel modeling can be accomplished using a variety of statistical software packages such as Stata, SAS and SPSS. The packages HLM, MLwinN, aML, and WINBUGS were explicitly designed to estimate multilevel models.

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Authors and Affiliations

  1. Department of Criminology and Criminal Justice, University of Maryland, College Park, MD, USA

    Laura Dugan

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  1. Laura Dugan
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Editors and Affiliations

  1. College of Criminology, Florida State University, West Call Street 643, Tallahassee, 32306, U.S.A.

    Alex R. Piquero

  2. Inst. Criminology, Hebrew University of Jerusalem, Jerusalem, 91905, Israel

    David Weisburd

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Dugan, L. (2010). Estimating Effects over Time for Single and Multiple Units. In: Piquero, A., Weisburd, D. (eds) Handbook of Quantitative Criminology. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77650-7_35

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