Differences in Differences

Differences in Differences - Another technique often used...

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Another technique often used in policy oriented articles is the differences in differences. This comes from the idea of “experiments” which are either thrown up naturally or by policy. Ideally, an experiment involves randomly assigning participants to different “treatments” but most policy or natural experiments do not do this. However some natural experiments can approximate this by exogenously changing policy variables (like the required age at which students must stay in school) in different jurisdictions. Because there must be other differences in provinces, we also try to control for as many other variables as we can. To take an example, suppose that we plan to construct a garbage incineration plant in Waterloo. Local residents complain that this will lower property values. How can we tell is this is likely to be true? Well, we could look at other examples where similar plants were constructed. One example was in Andover, Mass. When results were published after a plant was built. We have a sample of housing prices both before and after the construction (1978 and 1981 in this example). Suppose the results were as follows: in 1981, average prices near the incinerator were $70,600 and far away were $101,300. So the average difference was $30,700. Now by itself this doesn’t prove that the incinerator lowered housing prices – other factors might also be at play. So next suppose we compare the prices near the incinerator both before (78) and after (81) the construction and we find that it was $63,700 before so they rose $6,900. Since prices rose, does this imply that the incinerator had no effect?
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Well now let us compare the increases in the near and far areas: Near Far Difference (Far- Near) 1978 63,700 82,500 18,800 1981 70.600 101,300 30,700 Difference (time 81 - 78) 6,900 18,800 -11,900 Thus we can look at the difference in differences between near and far over time and it tends to suggest that the incinerator did lower property values. How can we show this more analytically? We can set up a table with the treatment group (where something changes) and the control group (where this factor does not change). (note: Y = average price of a house) Treatment Control Difference Before Treatment Y TB Y CB Y TB - Y CB After Treatment Y TA Y CA Y TA - Y CA Difference Y TA - Y TB Y CA - Y CB (Y TA - Y TB ) - (Y CA - Y CB ) = (Y TA - Y CA ) -(Y TB - Y CB ) So the bottom right corner gives us the difference-in-differences estimator in its simplest form. There are 2 ways to look at this estimator: 1. Treatment effect = (Y TA - Y TB ) - (Y CA - Y CB ) -- the estimator takes into account that other factors change over time and effect both the treated and control groups so we get effect of treatment plus other factor (Y TA - Y TB ) - effects of other factors (Y CA - Y CB ) 2. Treatment effect = (Y TA - Y CA ) - (Y TB - Y CB ) -- the estimator accounts for differences existing between the treatment and control groups even before the treatment
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There are of course many conditions that must be met for this to work. In
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Differences in Differences - Another technique often used...

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