<ol>
<li><p>The Stata outputs say that you did not omit the intercept. No worries about the intercept. The <code>_cons</code> row is for the intercept.</p>
</li>
<li><p>As you know, Stat's <code>xtreg ..., fe</code> will give you identical coefficient estimates except the intercept. The constant terms are different between <code>areg</code> and <code>xtreg</code>. That's because <code>areg</code> imposes the constraint that <span class="math-container">$\gamma_1 = 0$</span> and <code>xtreg, fe</code> that <span class="math-container">$n^{-1} \sum_{i=1}^n \gamma_i = 0$</span>, which is to say that <code>areg</code>'s <span class="math-container">$\alpha$</span> is <span class="math-container">$\alpha_1$</span> and <code>xtreg, fe</code>'s <span class="math-container">$\alpha$</span> is <span class="math-container">$n^{-1} \sum_{i=1}^n \alpha_i$</span>, where <span class="math-container">$\alpha_i = \alpha + \gamma_i$</span>.</p>
</li>
<li><p>The statistics for the <code>pri_ove_run</code> variable are not displayed in a pleasant way; <span class="math-container">$1.04\times 10^{-12}$</span> and <span class="math-container">$5.17\times 10^{-12}$</span> are hard to read. You can divide the variable by, e.g., <span class="math-container">$10^{12}$</span> (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 <span class="math-container">$t$</span> value is reasonable); still, it would be a good idea to investigate into the variable in your data set.</p>
</li>
<li><p>(On excluding the intercept) In general, it is not a good idea to exclude the intercept by using Stata's <code>nocons</code> option unless you know what you are doing (see 6 below for an example). For the model <span class="math-container">$y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + u$</span> (with all good assumptions), for example, if you do <code>reg y x1 x2, nocons</code>, it means you are imposing the constraint that <span class="math-container">$\beta_0=0$</span>, which means that <span class="math-container">$E(y)=\beta_1 E(x_1) + \beta_2 E(x_2)$</span>. If everyone agrees that <span class="math-container">$E(y)=\beta_1 E(x_1) + \beta_2 E(x_2)$</span> (e.g., you know that <span class="math-container">$E(y)= E(x_1)= E(x_2)=0$</span>), this constraint is acceptable, but otherwise your regression will yield an inconsistent slope estimator. Even if <span class="math-container">$\beta_0$</span> is really 0, the gain from imposing the constraint (i.e., from excluding the intercept) is usually trivial.</p>
</li>
<li><p>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 <span class="math-container">$X$</span> variables show little variability across <span class="math-container">$i$</span> or over <span class="math-container">$t$</span>, then strange things can happen. We should always watch out for collinearity, though it is often alright anyway.</p>
</li>
<li><p>There are some cases you should exclude the intercept. For example, for the panel model <span class="math-container">$y_{it} = \alpha_i + \beta x_{it} + u_{it}$</span> (without period dummies), the first difference (FD) regression is OLS of the model <span class="math-container">$\Delta y_{it} = \beta \Delta x_{it} + \Delta u_{it}$</span>, where <span class="math-container">$\Delta y_{it} = y_{it} - y_{it-1}$</span>, etc., and <span class="math-container">$\Delta \alpha_i = 0$</span>. The differenced equation has no intercept, and thus you should do <code>reg d.y d.x, nocons vce(cluster TYPE2)</code> for the FD regression. If you omit the <code>nocons</code> 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.</p>
</li>
</ol>

CC BY-SA 4.0
2021-06-11T02:49:47.283Z
Thank you so much, @chan1142. Can I ask a couple of questions? (1). What are $\gamma_1$ and $\alpha_1$ in your second bullet then?
Phil Nguyen
CC BY-SA 4.0
2021-06-11T07:10:58.467Z
In your model, the individual effects are $\alpha + \gamma_i$ (your notations). $\gamma_1$ is for $i=1$, and $\alpha_i$ is defined as $\alpha + \gamma_i$ so $\alpha_1 = \alpha + \gamma_1$. Here I presumed that the reference case is $i=1$.
chan1142
CC BY-SA 4.0
2021-06-17T22:28:55.477Z
@chan1141, from your experience, whether including ```nocons``` will change coefficients of other independent variables ?
Phil Nguyen
CC BY-SA 4.0
2021-06-18T10:52:35.667Z
It depends on how other dummy variables are handled by the `nocons` option. Without `nocons`, `i.yr` includes dummy variables except for the reference year. With `nocons`, if all year dummies are included (including one for the reference year), then $\beta$ won't change. But if the intercept is excluded and still the dummy for the base year is excluded, then the results can change. In your model there are also individual fixed effects (`a(TYPE2)`) and it all depends on how Stata handles the `nocons` option. You can find it out by running the command.
chan1142
CC BY-SA 4.0
2021-06-09T11:30:50.297Z
2021-06-09T11:30:50.297Z
chan1142
<p>Recently, I run the regression for the generalised DID following <a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">this paper</a>:</p>
<p><span class="math-container">$Y_{it}$</span> = <span class="math-container">$\alpha$</span> + <span class="math-container">$\beta$</span> <span class="math-container">$(Leniency Law)_{kt}$</span> + <span class="math-container">$\delta$$X_{ikt}$</span> + <span class="math-container">$\theta$$_t$</span> + <span class="math-container">$\gamma$$_i$</span> +<span class="math-container">$\epsilon$$_{it}$</span> <em>(1)</em></p>
<p>I accidentally ran the regression without intercept (<span class="math-container">$\alpha$</span>), 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?</p>
<p><strong>Update</strong>: Adding the output of regressing for collinearity suspicion from @chan1142</p>
<p><a href="https://i.sstatic.net/2Qv00.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2Qv00.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-06-08T07:43:15.850Z
Did you use `xtreg, fe`? Or, what was your command (stata or R)? Please explain exactly how you ran the regression without $\alpha$.
chan1142
CC BY-SA 4.0
2021-06-08T07:47:40.723Z
@chan1142 yes, it should be **xtreg y x1 x2 i.year, fe** or **areg y x1 x2 i.year, a(type)** where **type** is firm identification
Phil Nguyen
CC BY-SA 4.0
2021-06-08T07:52:35.713Z
If you ran `xtreg, fe`, it's not an issue. You've done it correctly. (In your model, $\alpha$ and $\gamma_i$ are not separately identified and `Stata` reports $\hat\alpha$ as the sample mean of the estimates of $\alpha+\gamma_i$. Stata does everything for you correctly.)
chan1142
CC BY-SA 4.0
2021-06-08T07:57:43.750Z
I am wondering if it is the case. Because when I run the regression **xtreg y x1 x2 i.year, fe**  , the result did not show me the intercept, that is why I am so reluctant
Phil Nguyen
CC BY-SA 4.0
2021-06-08T08:02:24.800Z
Then, I would first worry about collinearity among the explanatory variables. There can be some strange things involved. I wonder if you are willing to show the regression outputs (Stata outputs).
chan1142
CC BY-SA 4.0
2021-06-08T08:05:25.497Z
@chan1142 I did add the result, thank you so much for having a look on that, much appreciated
Phil Nguyen
CC BY-SA 4.0
2021-06-08T08:09:25.140Z
Thanks. I see the `_cons` row. That's the intercept; the dummy variable for the reference group (probably TYPE2 = 1) is omitted in order to avoid the dummy variable trap. That said, I see that `pri_ove_ern` is almost omitted. I would look into that variable.
chan1142
CC BY-SA 4.0
2021-06-08T08:11:38.290Z
@chan1142 thank you so much for your spot out, can I ask why you say ```pri_ove_ern``` is almost omitted, is it because the very low coefficient? Much appreciated
Phil Nguyen
CC BY-SA 4.0
2021-06-08T08:16:07.363Z
Yes. The estimate and the standard error are very small. If they are small because the scale is huge (you can see it by reading its standard deviation), that's OK, but I would divide the variable by, for example 10^12 or similar so that the reported numbers are nice.
chan1142
CC BY-SA 4.0
2021-06-08T08:20:39.980Z
@chan1142 what a hands-on suggestion, so, normally we will have a look at coefficient and standard error to have a suspicion for multicollinearity, isn't it? And I am sorry that I did not catch the idea of "**divide the variable by, for example 10^12**" I am wondering if you could please help me to clarify it more?
Phil Nguyen
CC BY-SA 4.0
2021-06-08T08:27:06.113Z
You change the unit of measurement, by which you change the scale of the coefficient. It’s only a matter of cosmetics.
chan1142
CC BY-SA 4.0
2021-06-08T08:27:55.157Z
I see, thank you so much for your help and suggestion so far
Phil Nguyen
CC BY-SA 4.0
2021-06-08T08:47:17.027Z
@chan1142 you could make your comments into an answer.
Jesper Hybel
<econometrics><mathematical-economics>
Why excluding intercept is dangerous if there is no literature back up in DID setting?
Phil Nguyen
# Why excluding intercept is dangerous if there is no literature back up in DID setting?

(Asked by: username_0 on 2021-06-08)

Recently, I run the regression for the generalised DID following [this paper](https://academic.oup.com/rfs/article/32/7/2587/5079300):

$Y\_{it}$ = $\alpha$ + $\beta$ $(Leniency Law)\_{kt}$ + $\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 @username_1

[![enter image description here](https://i.sstatic.net/2Qv00.png)](https://i.sstatic.net/2Qv00.png)

**Tags:** econometrics, mathematical-economics

> **Comments:**
> **username_1**: Did you use `xtreg, fe`? Or, what was your command (stata or R)? Please explain exactly how you ran the regression without $\alpha$.  
> **username_0**: @username_1 yes, it should be **xtreg y x1 x2 i.year, fe** or **areg y x1 x2 i.year, a(type)** where **type** is firm identification  
> **username_1**: If you ran `xtreg, fe`, it's not an issue. You've done it correctly. (In your model, $\alpha$ and $\gamma_i$ are not separately identified and `Stata` reports $\hat\alpha$ as the sample mean of the estimates of $\alpha+\gamma_i$. Stata does everything for you correctly.)  
> **username_0**: I am wondering if it is the case. Because when I run the regression **xtreg y x1 x2 i.year, fe**  , the result did not show me the intercept, that is why I am so reluctant  
> **username_1**: Then, I would first worry about collinearity among the explanatory variables. There can be some strange things involved. I wonder if you are willing to show the regression outputs (Stata outputs).  
> **username_0**: @username_1 I did add the result, thank you so much for having a look on that, much appreciated  
> **username_1**: Thanks. I see the `_cons` row. That's the intercept; the dummy variable for the reference group (probably TYPE2 = 1) is omitted in order to avoid the dummy variable trap. That said, I see that `pri_ove_ern` is almost omitted. I would look into that variable.  
> **username_0**: @username_1 thank you so much for your spot out, can I ask why you say ```pri_ove_ern``` is almost omitted, is it because the very low coefficient? Much appreciated  
> **username_1**: Yes. The estimate and the standard error are very small. If they are small because the scale is huge (you can see it by reading its standard deviation), that's OK, but I would divide the variable by, for example 10^12 or similar so that the reported numbers are nice.  
> **username_0**: @username_1 what a hands-on suggestion, so, normally we will have a look at coefficient and standard error to have a suspicion for multicollinearity, isn't it? And I am sorry that I did not catch the idea of "**divide the variable by, for example 10^12**" I am wondering if you could please help me to clarify it more?  
> **username_1**: You change the unit of measurement, by which you change the scale of the coefficient. It’s only a matter of cosmetics.  
> **username_0**: I see, thank you so much for your help and suggestion so far  
> **username_2**: @username_1 you could make your comments into an answer.  

---

## 1 answers

### Answer by username_1 on 2021-06-09. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you so much, @username_1. Can I ask a couple of questions? (1). What are $\gamma_1$ and $\alpha_1$ in your second bullet then?  
> **username_1**: In your model, the individual effects are $\alpha + \gamma_i$ (your notations). $\gamma_1$ is for $i=1$, and $\alpha_i$ is defined as $\alpha + \gamma_i$ so $\alpha_1 = \alpha + \gamma_1$. Here I presumed that the reference case is $i=1$.  
> **username_0**: @chan1141, from your experience, whether including ```nocons``` will change coefficients of other independent variables ?  
> **username_1**: It depends on how other dummy variables are handled by the `nocons` option. Without `nocons`, `i.yr` includes dummy variables except for the reference year. With `nocons`, if all year dummies are included (including one for the reference year), then $\beta$ won't change. But if the intercept is excluded and still the dummy for the base year is excluded, then the results can change. In your model there are also individual fixed effects (`a(TYPE2)`) and it all depends on how Stata handles the `nocons` option. You can find it out by running the command.  

---

