Why excluding intercept is dangerous if there is no literature back up in DID setting?

Question

Recently, I run the regression for the generalised DID following this paper:

$Y_{it}$ = $\alpha$ + $\beta$ $(Leniency Law)_{kt}$ + $\delta$$X_{ikt}$ + $\theta$$_t$ + $\gamma$$_i$ +$\epsilon$$_{it}$ (1)

I accidentally ran the regression without intercept ($\alpha$), and my senior friend told me that it is really dangerous when running such an equation without intercept if there is no good literature backup. I am wondering why it is so dangerous in this case?

Update: Adding the output of regressing for collinearity suspicion from @chan1142

Answer 1

1. The Stata outputs say that you did not omit the intercept. No worries about the intercept. The _cons row is for the intercept.

2. As you know, Stat's xtreg ..., fe will give you identical coefficient estimates except the intercept. The constant terms are different between areg and xtreg. That's because areg imposes the constraint that $\gamma_1 = 0$ and xtreg, fe that $n^{-1} \sum_{i=1}^n \gamma_i = 0$, which is to say that areg's $\alpha$ is $\alpha_1$ and xtreg, fe's $\alpha$ is $n^{-1} \sum_{i=1}^n \alpha_i$, where $\alpha_i = \alpha + \gamma_i$.

3. The statistics for the pri_ove_run variable are not displayed in a pleasant way; $1.04\times 10^{-12}$ and $5.17\times 10^{-12}$ are hard to read. You can divide the variable by, e.g., $10^{12}$ (which means the unit is upscaled by the factor of one trillion - does it make sense?). Then the reported estimate will be 1.04XXXX and the reported standard error will be 5.17XXXX, which look nicer. Though it's only a matter of cosmetics, I believe it important too. BTW, at first I thought that there might be collinearity issues involved, but now I believe it's only about unit of measurement (because the $t$ value is reasonable); still, it would be a good idea to investigate into the variable in your data set.

4. (On excluding the intercept) In general, it is not a good idea to exclude the intercept by using Stata's nocons option unless you know what you are doing (see 6 below for an example). For the model $y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + u$ (with all good assumptions), for example, if you do reg y x1 x2, nocons, it means you are imposing the constraint that $\beta_0=0$, which means that $E(y)=\beta_1 E(x_1) + \beta_2 E(x_2)$. If everyone agrees that $E(y)=\beta_1 E(x_1) + \beta_2 E(x_2)$ (e.g., you know that $E(y)= E(x_1)= E(x_2)=0$), this constraint is acceptable, but otherwise your regression will yield an inconsistent slope estimator. Even if $\beta_0$ is really 0, the gain from imposing the constraint (i.e., from excluding the intercept) is usually trivial.

5. Things can be quite complicated for models with many dummy variables (and interaction terms). In your model, you have the period dummies and the individual fixed effects. If some of $X$ variables show little variability across $i$ or over $t$, then strange things can happen. We should always watch out for collinearity, though it is often alright anyway.

6. There are some cases you should exclude the intercept. For example, for the panel model $y_{it} = \alpha_i + \beta x_{it} + u_{it}$ (without period dummies), the first difference (FD) regression is OLS of the model $\Delta y_{it} = \beta \Delta x_{it} + \Delta u_{it}$, where $\Delta y_{it} = y_{it} - y_{it-1}$, etc., and $\Delta \alpha_i = 0$. The differenced equation has no intercept, and thus you should do reg d.y d.x, nocons vce(cluster TYPE2) for the FD regression. If you omit the nocons option, it means your original model contains a linear time trend (yes, a linear trend, not time dummies)! This is also related with Anderson and Hsiao's instrumental variable estimation of dynamic panel data models.