- A well-known result is that, in designs where the errors are within-group correlated and where a variable of interest does not vary within the group, the conventional OLS estimates of standard errors are seriously downward biased: this produces t-statistics that are too large and, accordingly, leads analysts to over-reject the null hypothesis of no treatment effect (Moulton, 1990). To the best of our knowledge, though, none of the research cited in Section 2 addresses this issue: most studies use heteroscedasticity-robust standard errors, but do not allow for any dependence between different individuals.
Paper not yet in RePEc: Add citation now
Allegretto, S. A., A. Dube, and M. Reich (2011). Do minimum wages really reduce teen employment ? Accounting for heterogeneity and selectivity in state panel data. Industrial Relations: A Journal of Economy and Society 50(2), 205–240.
- Angrist, J. D. and J.-S. Pischke (2008). Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton University Press.
Paper not yet in RePEc: Add citation now
Arellano, M. and O. Bover (1995). Another look at the instrumental variable estimation of error-components models. Journal of Econometrics 68(1), 29–51.
Arellano, M. and S. Bond (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. The Review of Economic Studies 58(2), 277–297.
- As Donald and Lang (2007) observe, standard errors cannot be estimated with this approach for a two-by-two DiD design, as the second step is an exactly-identified regression, with 4 coefficients (2 time effects, 1 group effect and 1 policy effect) being estimated from 4 data points. Clearly, the same is true for other types of DiD where the second step is an exactly-identified regression. What might not be immediately clear is that this situation also holds when we apply the two-step to the unrestricted equation 10. The argument runs as All the control variables x in the equation 10 vary within-cell.
Paper not yet in RePEc: Add citation now
Bazen, S. and V. Marimoutou (2016). Federal Minimum Wage Hikes Do Reduce Teenage Employment: The Time Series Effects of Minimum Wages in the US Revisited.
Bertrand, M., E. Duflo, and S. Mullainathan (2004). How Much Should We Trust Differencesin -Differences Estimates? Quarterly Journal of Economics 119(1).
- Bloom, H. S. (1995). Minimum Detectable Effects: A Simple Way to Report the Statistical Power of Experimental Designs. Evaluation Review 19(5), 547–556.
Paper not yet in RePEc: Add citation now
Brewer, M., T. F. Crossley, and R. Joyce (2018). Inference with Difference-in-Differences Revisited. Journal of Econometric Methods 7(1).
Brown, C., C. Gilroy, and A. Kohen (1983). Time-Series Evidence of the Effect of the Minimum Wage on Youth Employment and Unemployment. The Journal of Human Resources 18(1), 3–31.
- Bryan, M., A. Salvatori, and M. Taylor (2013). The Impact of the National Minimum Wage on Employment Retention, Hours and Job Entry. Technical report, Research Report for the Low Pay Commission. Institute for Social and Economic Research, University of Essex.
Paper not yet in RePEc: Add citation now
Cameron, A. C., J. B. Gelbach, and D. L. Miller (2008). Bootstrap-Based Improvements for inference with clustered errors. The Review of Economics and Statistics 90(3), 414–427.
Campolieti, M., T. Fang, and M. Gunderson (2005). Minimum wage impacts on youth employment transitions, 1993–1999. Canadian Journal of Economics/Revue canadienne d’eÃŒÂconomique 38(1), 81–104.
- Card, D. and A. B. Krueger (1995). Myth and Measurement: The New Economics of the Minimum Wage. Princeton University Press.
Paper not yet in RePEc: Add citation now
Card, D. and A. B. Krueger (2000). Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania: reply. American Economic Review, 1397–1420.
- Coats, D. (2007). The National Minimum Wage: Retrospect and Prospect. Work Foundation.
Paper not yet in RePEc: Add citation now
- Cohen, J. (1994). The Earth is Round (p ≤ .05). American Psychologist 49(12), 997.
Paper not yet in RePEc: Add citation now
Currie, J. and B. C. Fallick (1996). The Minimum Wage and the Employment of Youth Evidence from the NLSY. The Journal of Human Resources 31(2), pp. 404–428.
Dickens, R. and M. Draca (2005). The Employment Effects of the October 2003 Increase in the National Minimum Wage. Technical report, Research Report for the Low Pay Commission. Centre for Economic Performance, London School of Economics and Political Science.
- Dickens, R., R. Riley, and D. Wilkinson (2012). Re-examining the impact of the National Minimum Wage on earnings, employment and hours: the importance of recession and firm size. Technical report, Research Report for the Low Pay Commission.
Paper not yet in RePEc: Add citation now
- Dolton, P., C. R. Bondibene, and J. Wadsworth (2012). Employment, Inequality and the UK National Minimum Wage over the Medium-Term. Oxford Bulletin of Economics and Statistics 74(1), 78–106.
Paper not yet in RePEc: Add citation now
Dolton, P., C. R. Bondibene, and M. Stops (2015). Identifying the employment effect of invoking and changing the minimum wage: A spatial analysis of the UK. Labour Economics 37, 54–76.
Donald, S. G. and K. Lang (2007). Inference with Difference-in-Differences and Other Panel Data. The Review of Economics and Statistics 89(2), 221–233.
Dube, A., T. W. Lester, and M. Reich (2010). Minimum wage effects across state borders: Estimates using contiguous counties. The Review of Economics and Statistics 92(4), 945– 964.
- Elhorst, J. P. (2010). Spatial Panel Data Models.
Paper not yet in RePEc: Add citation now
Finn, D. (2005). The National Minimum Wage in the United Kingdom.
- For the variant where we estimate the impact of a 1% rise in the NMW on job retention, the amended model is: yigts = δts + αg + βgdgsÉt + x igstγ + igst i = 1, ..., N; g = C, B, T, A; t = 2000, ..., 2011 and the second stage in the Donald and Lang two-step estimator is: ̂c = δts + αg + βgdcÉt + c (15) c = 1, ..., 80; t = 2000, ..., 2011 where βT is the (weighted) average effect of a 1% NMW rise on the probability of remaining employed. A second concern about inference in DiD studies, as initially noted by Bertrand et al.
Paper not yet in RePEc: Add citation now
Hansen, C. B. (2007). Generalized Least Squares Inference in Panel and Multilevel Models with Serial Correlation and Fixed Effects. Journal of Econometrics 140(2), 670–694.
Holtz-Eakin, D., W. Newey, and H. S. Rosen (1988). Estimating vector autoregressions with panel data. Econometrica: Journal of the Econometric Society, 1371–1395.
- In our study, we define a cell as the interaction of group, year and transition-type, giving us 96 cells (4 groups, 12 years of data, and 2 transition types). The first stage regression is then: yic = x icγ + X c=1 Icc + ic (11) i = 1, ..., N; c = 1, ..., 96 where Ic is a dummy variable which identifies the c-th cell, and x are the controls that vary within-cell.18 In the second stage, the coefficients associated with the cell membership dummies c are regressed on the cell-invariant variables. In our example, this second step is: ̂c = δts + αgt + βgtdc + c (12) c = 1, ..., 96; t = 2000, ..., 2011 where dc is a dummy indicating whether the c-th cell is affected by a minimum wage uprating.
Paper not yet in RePEc: Add citation now
Ioannidis, J. P. (2005). Why Most Published Research Findings Are False. PLoS medicine 2(8), e124.
- Liang, K.-Y. and S. L. Zeger (1986). Longitudinal Data Analysis Using Generalized Linear Models. Biometrika 73(1), 13–22.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (1998). The National Minimum Wage: First Report of the Low Pay Commission.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (2000). The National Minimum Wage. The story so far; Second Report of the Low Pay Commission.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (2003). The National Minimum Wage. Fourth Report of the Low Pay Commission.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (2006). National Minimum Wage. Low Pay Commission Report 2006.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (2007). National Minimum Wage. Low Pay Commission Report 2007.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (2008). National minimum wage: Low Pay Commission Report 2008.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (2009). National minimum wage: Low Pay Commission Report 2009.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (2013). National Minimum Wage. Low Pay Commission Report 2013.
Paper not yet in RePEc: Add citation now
- Low Pay Commission (2016). National Minimum Wage. Low Pay Commission Report Autumn 2016.
Paper not yet in RePEc: Add citation now
Machin, S. and J. Wilson (2004). Minimum Wages in a low-wage labour market: Care homes in the UK. The Economic Journal 114(494), C102–C109.
Machin, S., A. Manning, and L. Rahman (2003). Where the minimum wage bites hard: Introduction of minimum wages to a low wage sector. Journal of the European Economic Association 1(1), 154–180.
- McShane, B. B., D. Gal, A. Gelman, C. Robert, and J. L. Tackett (2017). Abandon Statistical Significance. arXiv preprint arXiv:1709.07588.
Paper not yet in RePEc: Add citation now
Meer, J. and J. West (2015). Effects of the minimum wage on employment dynamics. Journal of Human Resources.
Moulton, B. R. (1990). An Illustration of a Pitfall in Estimating the Effects of Aggregate Variables on Micro Units. The Review of Economics and Statistics, 334–338.
Neumark, D. and W. Wascher (1992). Employment effects of minimum and subminimum wages: panel data on state minimum wage laws. ILR Review 46(1), 55–81.
Neumark, D. and W. Wascher (2004). Minimum wages, labor market institutions, and youth employment: a cross-national analysis. ILR Review 57(2), 223–248.
Neumark, D., J. I. Salas, and W. Wascher (2014). Revisiting the Minimum WageEmployment Debate: Throwing Out the Baby with the Bathwater? ILR Review 67(3 suppl), 608–648.
Neumark, D., M. Schweitzer, and W. Wascher (2004). Minimum Wage Effects throughout the Wage Distribution. Journal of Human Resources 39(2), 425–450.
- Northern Ireland Statistics and Research Agency, Central Survey Unit, Office for National Statistics, Social Survey Division (2018). Quarterly Labour Force Survey, 1992-2018: Secure Access. 13th Edition. UK Data Service. [data collection]. http://doi.org/10.5255/ UKDA-SN-6727-14.
Paper not yet in RePEc: Add citation now
- Office for Budget Responsibility (2015). Economic and Fiscal Outlook. Cm 9088. July. The Stationary Office.
Paper not yet in RePEc: Add citation now
- Office for National Statistics (2018). Annual Survey of Hours and Earnings, 1997-2018: Secure Access. [data collection]. 13th Edition. UK Data Service. SN: 6689 http://doi.org/10. 5255/UKDA-SN-6689-12.
Paper not yet in RePEc: Add citation now
Sabia, J. J., R. V. Burkhauser, and B. Hansen (2012). Are the Effects of Minimum Wage Increases Always Small-New Evidence from a Case Study of New York State. Industrial & Labour Relation Review 65, 350.
See https://tinyurl.com/yy3bgzm2 Card, D. (1992). Using regional variation in wages to measure the effects of the federal minimum wage. Industrial & Labor Relations Review 46(1), 22–37.
Spiegelhalter, D. (2017). Trust in numbers. Journal of the Royal Statistical Society: Series A (Statistics in Society) 180(4), 948–965.
- Sterne, J. A., G. D. Smith, and D. Cox (2001). Sifting the evidence-what’s wrong with significance tests? Another comment on the role of statistical methods. BMJ 322(7280), 226–231.
Paper not yet in RePEc: Add citation now
Stewart, M. B. (2002). Estimating the impact of the minimum wage using geographical wage variation. Oxford Bulletin of Economics and Statistics 64(supplement), 583–605.
Stewart, M. B. (2004a). The employment effects of the National Minimum Wage. The Economic Journal 114(494), C110–C116.
Stewart, M. B. (2004b). The Impact of the Introduction of the U.K. Minimum Wage on the Employment Probabilities of Low-Wage Workers. Journal of the European Economic Association 2(1), 67–97.
- The first concern relates to the grouped error structure. In DiD designs, the error term igts is unlikely to be iid, because an individual may have unobservable characteristics that are correlated with other individuals of the same group, or may be affected by common group shocks. In the case of these studies of the minimum wage, members of the treatment group are all located at the bottom of the wage distribution, and so it is highly plausible that they may have some common unobservable characteristics (low ability, low skills, etc.) or are influenced by the same economic shocks. A comprehensive specification of equation (1) which includes common group shocks Õgts is: yigts = δts + αgt + βgtdgsÉt + x igtsγ + Õgts + ξigts (10) i = 1, ..., N; g = C, B, T, A; s = 0, 1; t = 2000, ..., 2011 and igts = Õgts + ξigts.
Paper not yet in RePEc: Add citation now
- The two-step estimator consists in retrieving estimates in two stages: in the first step, the dependent variable is regressed on dummies that identify cell membership and all the variables which vary within cells. In the second stage, the set of parameters associated with the cell membership are regressed on the variables which do not vary within cells. In the Donald and Lang (2007) two-step estimator, the concept of cell or cluster is essential: errors within a cell are allowed to be correlated, but shocks between cells are assumed to be independent.
Paper not yet in RePEc: Add citation now
White, H. (1980). A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity. Econometrica: Journal of the Econometric Society, 817–838.
Yuen, T. (2003). The Effect of Minimum Wages on Youth Employment in Canada A Panel Study. Journal of Human Resources 38(3), 647–672.
Ziliak, S. T. and D. N. McCloskey (2004). Size matters: the standard error of regressions in the American Economic Review. The Journal of Socio-Economics 33(5), 527–546.
- Zilio, F. (2018). Essays in the microeconometric evaluation of public policies. PhD thesis, University of Essex. A Appendix A:Inference in Difference-in-Differences with Grouped Errors A broad literature has raised concerns about the accuracy of the inference in DiD designs when using the naı̈ve estimates of the standard errors provided by OLS.
Paper not yet in RePEc: Add citation now