How to interpret interaction between a dummy and a continuous variables in ols?

Question

I am trying to examine the association between maternal education and child school test scores using the following equation.

$$\small{TestScore_i = \beta_0 + \beta_1 EduYrMom_i + \beta_2 FemaleChild_i + \beta_3 EduYrMom_i * FemaleChild_i + \varepsilon_i}$$

where TestScores is the test score of the child, EduYrMom is the years of education of the child’s mother, FemaleChild is a binary value which equals 1 if the child is a girl. The following regression coefficients are obtained using Stata’s regress command.

    Model 1 Test scores of child

(1a)Years of education of mother    0.028
(1b)Years of education of mother *female child  0.031
Female (=1 if female)   -0.388
Effect of mother's education on female children---- 1a+ 1b  0.059*

Model 2 

(2a)Years of education of mother    0.039**
(2b)Years of education of mother *female child  -0.005
Female (=1 if female)   0.091
Effect of mother's education on female children---- 2a+ 2b  0.034

Model 3 
(3a)Years of education of mother    0.303***
(3b)Years of education of mother *female child  -0.047
Female (=1 if female)   -0.003
Effect of mother's education on female children---- 3a+ 3b  0.256**

note:  *** p<0.01, ** p<0.05, * p<0.1   

For model 1, it seems that mother’s education is positively and significantly associated with test scores of female children, but not for boys. For model 2, I find that mother’s education is significantly associated with test scores of boys only. For model 3, mother’s education is significantly and positively associated with the test scores of both boys and girls.

I was wondering whether my interpretations are correct. I am also not sure how to interpret $\beta_3$, the coefficient of the interaction term, which is insignificant for all the 3 models. I look forward to your suggestions.

Monzur

Answer 1

You can think of the slope estimate for an interaction ($\beta_{cts \times cat}$) between a continuous and a dummy-coded categorical predictor as the expected difference in the simple slope estimate for the continuous predictor between the group coded as zero and the group coded as one. E.g., in the context of the following model:

$$Y = \beta_0 + \beta_{cat} X_1 + \beta_{cts}X_2 + \beta_{cts \times cat} X_1 \times X_2 + \varepsilon$$

$$\beta_{cts \times cat} = \beta_{cts\ \tiny{simple slope}}\ \small{(when\ X_1\ = 1}) - \beta_{cts\ \tiny{simple slope}}\ \small{(when\ X_1\ = 0})$$

(assuming $X_1$ is dummy-coded). More generally, a significant categorical $\times$ continuous interaction can be understood to mean that the effect of the the continuous predictor depends on the level of categorical predictor. Or, for a one unit increase in $X_1$, the simple slope for $X_2$ will increase by $\beta_{cts \times cat}$.

Note that these relationships could just as easily, but perhaps less intuitively, be conversely phrased: "the simple effect of the categorical predictor depends on the continuous predictor."