Inquire about .regress y if x == 0

[Yao Zhao]

I saw a code which is kind of like

regress y if x == 0

Both x and y are dummy variables. What is this regression? This is my 1st time to see a regression (reg y x) without x.


Many thanks in advance!

[Joseph Coveney]

It looks as if whoever wrote it wants something like the mean, standard error of the mean and confidence interval for y when x is zero. Unless the author wants something else that's specifically returned by -regress- (like test if the mean is different from zero or the variance or SD), it would seem more straightforward at least to me to have used -ci means-.

sysuse auto
regress mpg if !foreign
ci means mpg if !foreign

[Carlo Lazzaro]

Yao:
as an aside to Joseph's helpful insight, you can easily find yourself that:

sysuse auto.dta
. regress mpg if !foreign

      Source |       SS           df       MS      Number of obs   =        52
-------------+----------------------------------   F(0, 51)        =      0.00
       Model |           0         0           .   Prob > F        =         .
    Residual |  1147.44231        51  22.4988688   R-squared       =    0.0000
-------------+----------------------------------   Adj R-squared   =    0.0000
       Total |  1147.44231        51  22.4988688   Root MSE        =    4.7433

------------------------------------------------------------------------------
         mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   19.82692    .657777    30.14   0.000     18.50638    21.14747
------------------------------------------------------------------------------

*obtains the very same results of:*

. mean mpg if !foreign

Mean estimation                   Number of obs   =         52

--------------------------------------------------------------
             |       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
         mpg |   19.82692    .657777      18.50638    21.14747
--------------------------------------------------------------

[Yao Zhao]

Thank you above all. Now I am clear. But let me ask this question in another way, given both y and x are dummy vars, what's the difference between

regress y if x == 0

regress y x

[Marcos Almeida]

First, if yvar is binary, - logistic - or - logit - should preferrably be used instead of - regress - command. I'm thinking about modeling, I mean, adding new xvars.

Second, the "if" condition will select the zero category of xvar. For example, being it "diabetics", we'd have selected those who aren't diabetics.

[Carlo Lazzaro]

Yao:
Marcos already replied helpfully to your last query.
Marcos is also right in underlying that, provided the regressand (y) is a two-level categorical variable, there's no room for -regress-.
Possibly, who created the example was trying to investigate something that has basically no effect in practical statistics.

[Yao Zhao]

Let me add the background of these two codes. The dependent variable y here is whether you receive the offer, i.e., take-up rate. The x is the treatment indicator.

To form Table 1 of this paper (https://academic.oup.com/qje/article/133/3/1561/4768294), the author's do file include the two codes above.

For example, I want to know the mean value of y given x = 0. Then I can do

summarize y if x == 0

(this is I know),or

regress y if x == 0

(this is what I don't know) or

regress y x

In fact, I guess if y and x are both dummy, then doing regression in this case is similar to do tabulate. Am I right?

[Carlo Lazzaro]

Yao:
thanks for providing further clarifications.
Taking a glance to the paper, it seems that authors expressed the y in percentage and then (legally) go -regress- (because y is actually not a dummy).
As far as the second part of your question is concerned:
- both -summarize- and -regress- (and -mean-) conditional on -if- qualifier clause, will give you back the same sample estimate of the mean (provided that a zero value for -x- does exist).
Conversely:

regress y x

will include all the values of the x predictor.

[Yao Zhao]

But I am still confused about the function of regression without x.

reg y if x == 0

When I analyze this code, I still feel very confused. Is the outcome above the same as ".reg y x" but then you just drop the constant row? I checked them and found that the coefficient is the same but the SE is different.

When I see the regression word, then I image reg y x immediately. This is my 1st time to see reg y only (without x). So I'm confused.

[Carlo Lazzaro]

Yao:
I think we should consider two different scenarios, then.
1)

. sysuse auto
(1978 Automobile Data)

. regress price if foreign==0

      Source |       SS           df       MS      Number of obs   =        52
-------------+----------------------------------   F(0, 51)        =      0.00
       Model |           0         0           .   Prob > F        =         .
    Residual |   489194801        51  9592054.92   R-squared       =    0.0000
-------------+----------------------------------   Adj R-squared   =    0.0000
       Total |   489194801        51  9592054.92   Root MSE        =    3097.1

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   6072.423   429.4911    14.14   0.000     5210.184    6934.662
------------------------------------------------------------------------------

.

2)

. sysuse auto
(1978 Automobile Data)

. regress price

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(0, 73)        =      0.00
       Model |           0         0           .   Prob > F        =         .
    Residual |   635065396        73  8699525.97   R-squared       =    0.0000
-------------+----------------------------------   Adj R-squared   =    0.0000
       Total |   635065396        73  8699525.97   Root MSE        =    2949.5

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   6165.257   342.8719    17.98   0.000     5481.914      6848.6
------------------------------------------------------------------------------

. mean price

Mean estimation                   Number of obs   =         74

--------------------------------------------------------------
             |       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
       price |   6165.257   342.8719      5481.914      6848.6
--------------------------------------------------------------

While in the 1st scenario I condition the regression on the categorical predictor level=0 (which does not mean that x=0), in the 2nd scenario I use -regress- as another way to estimate the mean of the dataset sample.

[Yao Zhao]

Carlo:

You prove that the 2 codes are different if y is not a dummy variable. Thanks! But also see this:

sysuse auto
 generate fprice=1 if price > 6000
replace fprice = 0 if missing(fprice)
regress fprice if foreign == 0
regress fprice foreign

Are they the same?

[Wouter Wakker]

reg y x

This is the familiar model of y = a + bx + e where a is a constant and e is the error term.

reg y

Now the model is y = a + e. You're only estimating a constant here, which is the mean of y.

reg y if x ==0

This is the mean of y for all observations where x is 0.

If we take the cars example:

. sysuse auto, clear
(1978 Automobile Data)

. 
. * model: y = a + bx + e
. * a = mean price for domestic cars
. * a + b = mean price for foreign cars
. reg price foreign

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(1, 72)        =      0.17
       Model |  1507382.66         1  1507382.66   Prob > F        =    0.6802
    Residual |   633558013        72  8799416.85   R-squared       =    0.0024
-------------+----------------------------------   Adj R-squared   =   -0.0115
       Total |   635065396        73  8699525.97   Root MSE        =    2966.4

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     foreign |   312.2587   754.4488     0.41   0.680    -1191.708    1816.225
       _cons |   6072.423    411.363    14.76   0.000     5252.386     6892.46
------------------------------------------------------------------------------

. 
. * model: y = a + e
. * a = mean price for domestic cars
. reg price if foreign == 0

      Source |       SS           df       MS      Number of obs   =        52
-------------+----------------------------------   F(0, 51)        =      0.00
       Model |           0         0           .   Prob > F        =         .
    Residual |   489194801        51  9592054.92   R-squared       =    0.0000
-------------+----------------------------------   Adj R-squared   =    0.0000
       Total |   489194801        51  9592054.92   Root MSE        =    3097.1

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   6072.423   429.4911    14.14   0.000     5210.184    6934.662
------------------------------------------------------------------------------

The coefficients you get from reg price foreign are the same as reg price if foreign == 0 and reg price if foreign == 1. The standard errors are different because they are calculated over the whole sample in the first case and over sub-samples in the latter.

Hope this helps.

[Yao Zhao]

Thank you so much!