# What do "extrapolate" and "imputation" mean?

I am using an imputation estimator for Diff-in-Diff proposed by Borusyak, 2021. The way to calculate it is

Estimation proceeds in three steps:

1. Estimate a model for non-treated potential outcomes using the non-treated (i.e. never- treated or not-yet-treated) observations only. The benchmark model for diff-in-diff designs is a two-way fixed effect (FE) model:

Y_it = a_i + b_t + eps_it, but other FEs, controls, etc., are also allowed.

1. Extrapolate the model from Step 1 to treated observations, imputing non-treated potential outcomes Y_it(0), and obtain an estimate of the treatment effect tau_it = Y_it - Y_it(0) for each treated observation. (See What if imputation is not possible)
1. Take averages of estimated treatment effects corresponding to the estimand of interest.

Even I am using it but I do not fully understand the meaning of "extrapolate" and "imputation" words even after search from the dictionary. I decided to ask here rather than in English Learner because it relates more in economic-and-finance context. I may think that "extrapolate" because the author apply the coefficient of the regression on untreated for the treated to get the potential outcome of the treated if the treatment does not happen, but I totally get lost regarding the word "imputation"

Is there any intuitive way to explain these two words?

extrapolation - means making predictions or estimations outside the range of your original sample.

For example, if you estimate relationships between calorie intake and weight to be:

$$\hat{Y}=50+0.001X$$

Using sample where calorie intake ($$X$$) ranges from 1000-5000 per day, and weight of individuals ranges from 40 to 120 kg, it you try to use the same model to estimate the weight of someone eating 10000 calories (that is using model above we would have $$\hat{Y}=50+0.001 \cdot 10000 \implies \hat{Y}= 150$$ that estimate 150kg for someone eating 10000 calories would be extrapolation.

However, do not confuse this with any prediction as trying to estimate what weight a person eating 4000 would have, is not extrapolation anymore as 4000 calories and corresponding weight of 90kg is within range of data in our sample.

In your example of DiD the extrapolation involves estimating potential outcomes of treated individuals from the non-treated sample.

imputation - is replacement of missing data with substituted values.

For example, suppose you have data on GDP for 2011, 2012 and 2014 but you miss GDP for 2013. Instead of leaving that one observation missing you could impute the missing value for example by linear interpolation.

In your case above, as far as I can understand from the passage, the unobserved potential non-treated outcomes for the treated are being imputed.

• Thanks a heap, @1muflon1, it is a very clear explanation in "extrapolation" case. I am just thinking, whether they mean that they use the extrapolate method to imputate the potential treated outcome. I mean. They initially set the potential treated outcome as missing for all. Then they use the extrapolated method to get the value for potential treated outcome after treatment (imputation)? Sep 30, 2021 at 22:17
• I mean, they extrapolate to imputate, is it a correct thought? So, I try to add on your last sentence "the unobserved potential non-treated outcomes for the treated are being imputed by extrapolation method" Sep 30, 2021 at 22:19
• @NguyenLis I do not have time to read their paper but just based on that exert you provided it looks like yes they extrapolate from their untreated sample to impute (not imputate which is incorrect spelling) those values
– 1muflon1
Oct 1, 2021 at 12:00