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Let's suppose we have the following regression model:

$$Y_i=\beta_0+\beta_1D_i+\beta_2D_iX_i+\epsilon_i$$ where $Y_i$ represents the test score of the i-th student, $D_i$ is a dummy variable that takes the value 1 if the student's family is an immigrant and 0 otherwise, and $X_i$ is also a dummy variable that takes 1 if the student's father has an above average income.

Now if I were to describe what the coefficients $B_0,B_1,B_2$ mean how would I go about doing that?

I believe $B_0$ is simply the slope. $B_1$ is the change in the mean test score ($Y_i$) when comparing a family who is an immigrant with a non-immigrant. $B_2$ is a bit more difficult. The answers I have here are that $B_2$ is just the effect of father's income on immigrant students' average test score.

I'm a bit confused as to how to correctly interpret these coefficients (especially when there is interaction) and help would be greatly appreciated!

$\bf{EDIT}$: Here is an original picture of the question since it appears as though my professor's model is a bit out of the ordinary. enter image description here

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    $\begingroup$ Your model is unusual. You don't have a term w/ only X & not also D, but then you have an interaction term. This is usually a bad thing to do, but perhaps this is intended as the data generating process. Are you sure you want it that way? Also, we don't usually add the $_i$s w/ capital letters. The caps indicate random variables, whereas lower case are realized values & the subscript indexes which realization out of the set is being referred to. $\endgroup$
    – gung - Reinstate Monica
    Feb 22, 2015 at 18:02
  • $\begingroup$ @gung I am not sure if this is unusual but this is how my professor gave it. I can attach a photo of the original problem in the edited question. $\endgroup$
    – nicefella
    Feb 22, 2015 at 18:05
  • $\begingroup$ Hmmm, it might help if you showed the exact question. You may also benefit from reading some related, existing answers. I have relevant answers here & here. $\endgroup$
    – gung - Reinstate Monica
    Feb 22, 2015 at 18:08

3 Answers 3

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You can lay out different scenarios in a table like this:

enter image description here

The 1 and 0 indicates if the regression estimates will be included (1) or excluded (0)

Since there isn't an indicator just for father's income, both non-immigrants with either rich or poor father were grouped together, represented just by the intercept, $\beta_0$.

Now, if we recast this into a 2x2 table, it'd be easier to look at:

enter image description here

Pay attention to which two groups that are only differ by $\beta_2$, and those are the groups whose mean difference $\beta_2$ is intended to capture. Hints: focus on the last column.

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You have 3 different types of students described by the regression equation.

  1. non-immigrants
  2. immigrants without fathers earning above average income
  3. immigrants with fathers earning above average income

Consider the value of $\beta_0+\beta_1D_i+\beta_2D_iX_i$ in each of the above cases as it will help you interpret the coefficients.

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  • $\begingroup$ Could you please go a little bit more in depth? I understand that $D_i$ is immigrant/non-immigrant but for 2/3 why do we not have non-immigrants without fathers earning above average income and non-immigrants with father earning above average income? Why is it that in the interaction term, we keep $D=1$ while we change $X$ but not vice versa (so why don't we do $D=0$ while we change $X$ $\endgroup$
    – nicefella
    Feb 22, 2015 at 22:26
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    $\begingroup$ This is the way the regression is set up, if a student is non-immigrant, $D_i=0$ and so $X_i$ does not enter into the equation for non-immigrants. I cannot tell you why the regression is set up this way, perhaps whoever wrote it wanted to focus more on interpreting coefficients instead of conventional specification. The regression you are thinking of is $$Y_i=\beta_0+\beta_1D_i+\beta_2X_iD_i+\beta_3X_i$$. In which case non-immigrants would be split into two further groups; those with and without fathers earning above average income. But this isn't whats given. $\endgroup$
    – Zachary Blumenfeld
    Feb 22, 2015 at 22:40
  • $\begingroup$ By systematically going through each type of student above and calculating $\beta_0+\beta_1D_i+\beta_2X_i$ you should be able to interpret the coefficients better. $\endgroup$
    – Zachary Blumenfeld
    Feb 22, 2015 at 22:43
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If your model were as follows: $$ y_i = B_0 + B_1*\text{Immigrant}_i +B_2*\text{UpperIncome}_i +B_3*\text{Immigrant}_i*\text{UpperIncome}_i + e_i, $$ where $\text{Immigrant}_i$ and $\text{UpperIncome}_i$ were 0,1 dummy variables, then $B_1$ would be the effect of immigrant status when $\text{UpperIncome}$ status = 0, and the effect of immigrant status is $B_1 + B_3$ when $\text{UpperIncome}$ status = 1. Similarly, $B_2$ would be the effect of $\text{UpperIncome}$ status = 1 when Immigrant status = 0, and the effect of UpperIncome status is $B2 + B3$ when Immigrant status = 1. If there is no interaction ($B_3 = 0$), then the effect of Immigrant status is independent of UpperIncome status and vice versa.

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    $\begingroup$ True, but given that that isn't the model, this is of only partial use to the OP. In addition, please note our policies regarding answering [self-study] questions (viz: provide hints only). $\endgroup$
    – gung - Reinstate Monica
    Feb 22, 2015 at 18:34

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