# Fixed effects in first-difference model with percentage changes?

I'm estimating the following panel data model (years $t$, regions $i$) in Stata: $$\%Δy_{ti} = \beta_0 + \beta_1 \%Δx_{ti} + \beta_2 \%Δcontrols_{ti} \dots + e_{ti}$$

As far as I know, this is a first-difference model with the only distinction being that I use % change ($\%Δ$) instead of absolute change ($Δ$). There are about 125 regions $i$ and 3 three-year periods $t$.

My questions:

1. I understood that region-fixed effects shouldn't be included as this part is already taken care of by this differencing method. Is this right? I'm a bit worried about this, since adding those fixed effects leads to a loss of significance of all coefficients (thus the choice is of huge impact).
2. I deliberately chose for the $\%Δ$ of the variables instead of the $Δ$, as this would yield elasticities in my results, and it felt like this would put the changes in perspective. Is this indeed better than looking at just $Δ$?
• If you really do xtreg y x controls with no options at all, know that you are actually computing a random-effects (RE) model, since this is the default model parameter. May 18, 2017 at 1:13

First, assuming you do xtreg y x controls, fe, let's answer with a simple specification. Say you have $\forall \{t,i\}\in[1,T]\times[1,I]$ the following model specification

$y_{ti} = \beta_0 + X_{ti}\beta + e_{ti}= \beta_0 + X_{ti}\beta + \alpha_i + \varepsilon_{ti}$

where arbitrarily $\alpha_1$ is not specified (which is different from being $0$), and $\varepsilon_{ti}$ is the error component varying both over regions and over time, said to be idiosyncratic, .

The non-relative (and traditional) first-difference would have been

$y_{ti}-y_{(t-1)i} = (\beta_0 -\beta_0)+(X_{ti}-X_{(t-1)i})\beta + (\alpha_i-\alpha_i) + (\varepsilon_{ti}-\varepsilon_{(t-1)i})$

Instead of this, you choose to compute the following relative first-difference.

$-1 + \frac{y_{ti}}{y_{(t-1)i}} = \left(-1 + \frac{\beta_0}{\beta_0}\right)+\left(-1 + \frac{X_{ti}}{X_{(t-1)i}}\right)\beta + \left(-1 + \frac{\alpha_i}{\alpha_i}\right) + \left(-1 + \frac{\varepsilon_{ti}}{\varepsilon_{(t-1)i}}\right)$

or in continuous terms

$\ln\left(\frac{y_{ti}}{y_{(t-1)i}}\right) = \ln\left(\frac{\beta_0}{\beta_0}\right) + \ln\left(\frac{X_{ti}}{X_{(t-1)i}}\right)\beta + \ln\left(\frac{\alpha_i}{\alpha_i}\right) + \ln\left(\frac{\varepsilon_{ti}}{\varepsilon_{(t-1)i}}\right)$

1. All these regions fixed-effects are removed by the first difference. So, this is right, region-fixed effects are already taken care of by the differencing method.

$\%\Delta y_{ti} = \%\Delta X_{ti}\beta + \%\Delta \varepsilon_{ti}$

$\%\Delta y_{ti} - \overline{\%\Delta y_{i} } + \overline{\overline{\%\Delta y}} = \left(\%\Delta X_{ti} - \overline{\%\Delta X_{i} } + \overline{\overline{\%\Delta X}} \right)\beta + \left(\%\Delta \varepsilon_{ti} - \overline{\%\Delta \varepsilon_{i} } + \overline{\overline{\%\Delta \varepsilon}} \right)$

Where $\overline{Z_i} \forall i$ stand for within region over-time averages, and $\overline{\overline{Z}}$ stand for a grand average over time and regions.

1. If your question of research is about estimating short-run elasticities (between two years) controling for effects that are entailed in all regions over time (actually all what the FE model is about since it puts out of $\beta$ any region and region-time common denominator), your choice of using relative variations as input to an FE estimation procedure is fair.

Note that you should check beforehand with a Durbin–Wu–Hausman test how such using an FE (vs RE) won't return consitent but inefficient estimates.

• Thank you very much Kanak! I was wondering if you could explain the difference between relative variations expressed in percentage change versus the log of the change. I interpreted percentage changes as generally more accurate since the ln(change) only approaches the percentage change (for small % changes). But I understood that the log of the change has advantages in terms of symmetry (?). How do you choose between those two methods? May 20, 2017 at 12:02
• Percent change cannot be used in arithmetic operations, so it's not proper in statistical analysis. This is because of the asymmetry. See fharrell.com/2017/04/… May 20, 2017 at 12:26
• Even if the percentage changes are relatively large (up to ~30% for one of the variables), causing the logarithmic approximation of the percentage change to be relatively inaccurate? May 20, 2017 at 12:54
• @Michiel. It is very common to work with regressions involving percentage changes, actually to estimate partial elasticities. The choice of expressing relative variations, say, of quantity $A$ between two consecutive years, as $-1+A_t/A_{t-1}$ or as $\ln(A_t/A_{t-1})$ is only driven by the discrete or continuous nature of $A$. Indeed, (equivalently) discrete quantities increase in a $1+r$ fashion, while continuous ones in $e^r$, where $r$ is a growth rate involved over $dt$ (assuming that the growth occurs over time). That being said, elasticities are economical objects, not medical. May 20, 2017 at 20:28
• @Michiel. Also, it is very common to read that, to compute a percentage change, one needs to do something like $100\times(-1+A_t/A_{t-1}) = \%\Delta$ or $100\times\ln(A_t/A_{t-1})=\%\Delta$ and then append $\%$ as if it were a unit. But $\%$ is dimensionless and multiplying by $100$ like so is (wrong) a misunderstanding of what a percentage really is. A percentage is just a notation, as the scientific one. E.g. $0.05 = 5\times10^{-2} = 5\%$ and this is not incorrect to say that the current year is $201700\%$. And a percentage can be representative of a change, be it continuous or discrete. May 20, 2017 at 20:42