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I Want to know how to use time and country dummies in fixed effects model?

user27825
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  • Hi and welcome to the site! The question as it stands is very broad and it is unclear what exactly you want to know. Could you please provide more details about your data, your model, your questions etc.? – COOLSerdash Jul 09 '13 at 17:59

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Say you have 2 countries and a general model of the form $$y_{it} = \alpha + \beta_{1}X_{it} + \beta_{2}C_{1} + \nu_{i} + \epsilon_{it}$$ where $X$ varies by country and time, $\alpha$ is a constant, $\nu_{i}$ is a country specific fixed effect, and $C_{1}$ is a dummy for country 1. The dummy for country 2 is omitted as otherwise you will fall into the dummy variable trap.
Now you have two options to keep the dummy in a FE regression.

1) First differences
Interact the dummy with your time variable $t$ and write the model as $$y_{it} = \alpha + \beta_{1}X_{it} + \beta_{2}(t\cdot C_{1}) + \nu_{i} + \epsilon_{it}$$ Then taking first differences gives you $$\Delta y_{it} = \beta_{1}\Delta X_{i} + \beta_{2}C_{1} + \Delta \epsilon_{it}$$ Note that your country dummy is still there. This holds because when you take the first difference for the dummy you have: $$t\cdot C_{1} - (t-1)\cdot C_{1} = C_{1}[t - (t-1)] = C_{1}\cdot 1 = C_{1}$$ since $[t - (t-1)] = 1$.

2) Fixed effects estimation
Say you have 2 years in your data, generate year dummies (denote them as $D_{1}$ and $D_{2}$ for period 1 and 2, respectively) and choose year 1 as base year. Using the fixed effects estimator, write the model as $$y_{it} = \nu_{i} + \beta_{1}X_{it} + \beta_{2}D_{2} + \beta_{3}(D_{2}\cdot C_{1}) + \epsilon_{it}$$ All fixed effects are absorbed in $\nu_{i}$, whilst $C_{1}$ remains in the model due to its interaction with the time dummy. Since the period $D_{1}$ is chosen as base, the coefficient $\beta_{3}$ measures the difference between the partial effect of $C_{1}$ on the outcome in period 2 and its partial effect in period 1.

This way you will not be able to retrieve a partial effect but only the difference in partial effects - still better than nothing and depending on your question at hand this actually might be very interesting. Given the question as it stands I also had to write the answer in a rather general way though. For more information on this issue see the corresponding chapter in Wooldridge (2010) "Econometric Analysis of Cross Section and Panel Data".

Andy
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    I can't follow your intuition: `αi is a country specific constant, νi is a country specific fixed effect, and C1 is a dummy for country 1` ? – Metrics Jul 09 '13 at 19:30
  • Do you mean the notation or one of the methods? In the FE model normally the constant and the dummies for time-invariant characteristics would also go into the fixed effect but for the sake of making the argument I wrote them separately. – Andy Jul 09 '13 at 19:37
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    I can understand why some people didn't like the question as it is not very well posed. Yet it is not irrelevant. Sometimes you want to keep a time-invariant variable in a FE regression and I answered this question with two methods. I put some effort into this answer, so if I should have made a mistake along the lines it would be nice to know about it and not just receive this -1 – Andy Jul 09 '13 at 19:56
  • I am not the one to downvote and there is no doubt that you put an effort on answering the OP's question.As to your answer,-`vi` is irrelevant in the model and `αi` should be just `α` since you already included country dummy. Your first difference model is also incorrect. When you use first difference, country effects cancel out, so you can't estimate that. Please update that. – Metrics Jul 09 '13 at 21:18
  • Didn't mean to imply it was you, sorry if it sounded this way. About the constant you are right, I edited that. The differenced model is correct. The country dummy stays in there because it is interacted with the time variable and differencing this interaction only leaves the country dummy, as explained. If you prefer a reference for this method, have a look at Ichino, A. and Maggi, G. (2000) "Work Environment And Individual Background: Explaining Regional Shirking Differentials In A Large Italian Firm" in the Quarterly Journal of Economics – Andy Jul 10 '13 at 06:57
  • I read the model in the paper and thanks for the reference. But, country dummy in your model is unobservable (as a general rule) and it can't be treated as same as observable in the model (5) in the paper. So, you should be explicit that country dummy refers to time invariant observable characteristics of a country. It would be nice if you give the reference in the answer. – Metrics Jul 10 '13 at 11:48