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To guarantee the accuracy of the DID estimate, the composition of individuals of the two groups is assumed to remain unchanged over time. When using a DID model, various issues that may compromise the results, such as [[autocorrelation]]<ref>{{cite journal |first1=Marianne |last1=Bertrand |first2=Esther |last2=Duflo | first3=Sendhil | last3=Mullainathan |year=2004 |title=How Much Should We Trust Differences-In-Differences Estimates? |journal=[[Quarterly Journal of Economics]] |volume=119 |issue=1 |pages=249–275 |doi=10.1162/003355304772839588|s2cid=470667 |url=http://www.nber.org/papers/w8841.pdf }}</ref> and [[Ashenfelter dip]]s, must be considered and dealt with.
==Implementation==
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where <math>\widehat{E}(\dots \mid \dots )</math> stands for conditional averages computed on the sample, for example, <math>T=1</math> is the indicator for the after period, <math>S=0</math> is an indicator for the control group. Note that <math>\hat{\beta}_1</math> is an estimate of the counterfactual rather than the impact of the control group. The control group is often used as a proxy for the [[counterfactual]] (see, [[Synthetic control method]] for a deeper understanding of this point). Thereby, <math>\hat{\beta}_1</math> can be interpreted as the impact of both the control group and the intervention's (treatment's) counterfactual. Similarly, <math>\hat{\beta}_2</math>, due to the parallel trend assumption, is also the same differential between the treatment and control group in <math> T=1 </math>. The above descriptions should not be construed to imply the (average) effect of only the control group, for <math>\hat{\beta}_1</math>, or only the difference of the treatment and control groups in the pre-period, for <math>\hat{\beta}_2</math>. As in [[David Card|Card]] and [[Alan Krueger|Krueger]], below, a first (time) difference of the outcome variable <math>(\Delta Y_i = Y_{i,1} - Y_{i,0})</math> eliminates the need for time-trend (i.e., <math>\hat{\beta}_1</math>) to form an unbiased estimate of <math>\hat{\beta}_3</math>, implying that <math>\hat{\beta}_1</math> is not actually conditional on the treatment or control group.<ref>{{cite journal |first1=David |last1=Card |first2=Alan B. |last2=Krueger |year=1994 |title=Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania |journal=[[American Economic Review]] |volume=84 |issue=4 |pages=772–793 |jstor=2118030 }}</ref> Consistently, a difference among the treatment and control groups would eliminate the need for treatment differentials (i.e., <math>\hat{\beta}_2</math>) to form an unbiased estimate of <math>\hat{\beta}_3</math>. This nuance is important to understand when the user believes (weak) violations of parallel pre-trend exist or in the case of violations of the appropriate counterfactual approximation assumptions given the existence of non-common shocks or confounding events. To see the relation between this notation and the previous section, consider as above only one observation per time period for each group, then
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