# Dummy regression with two dichotomous dummy variables

I want to run a regression model with two independent dummy variables each dichotomous variable for example market states with two levels named negative and positive, market volatility with two levels high and low.

When I run regression model it results in three situations, e.g. negative and high (0,0), negative and low (0,1), positive and high (1,0). But where is fourth situation positive and low (1,1)?

Is this

$$y = a + b$$1$$x$$1+ $$b$$2$$x$$2+ $$e$$

the regression equation?

• You need an interaction term. Or you can just create three dummies (neglow, poshigh, poslow). – Jeremy Miles Oct 26 '19 at 5:55
• To answer the OPs question, the OP doesn't need an interaction term, although I would advise including one for different reasons, in addition to probably modeling Y as a function of the true measures of volatility (presumed a continuous measure) and market state (again, probably some continuous measure that's been categorized). – LSC Oct 26 '19 at 11:56

Assuming negative is the base level for X1 so that X1=0 is negative market state and that X2=0 is high volatility making high the reference volatility, then your mean Y for the four states is:

y=a+b1X1+ b2X2+ e

=>

E(Y)= a + b1X1 + b2X2

X1,X2 the E(Y) =...

0,0 then E(Y)= a = mean of Y for negative market state with high volatility

0,1 then E(Y)=a+b2 = mean of Y for negative market state with low volatility

1,0 then E(Y)=a+b1 = mean of Y for positive market state with high volatility

1,1 then E(Y)=a+b1+b2 = mean of Y for positive market state with low volatility

This is seen by plugging in 0/1 for X1/X2 and simplifying. You can use some algebra to help interpret the beta coefficients, too (just subtract the two equations of interest to isolate the particular beta estimate).

I will recommend, though, that you consider modeling true numerical variables as such rather than arbitrarily categorized variables as this can introduce material problems in to the modeling and resulting estimates. For example, volatility is probably an underlying continuous variable that someone said "x>X is high", which loses information and is arbitrary. The same might be said for market state, although I'm not sure of the true underlying variable used to generate the dummy variable.

I would also recommend an interaction term between X1 and X2, as a previous commentator noted, but my reasoning is different from their apparent reasoning. They seem to imply you needed an interaction to find the fourth mean, but that is incorrect. I think it is reasonable to assume (at least allow for) the change in E(Y) between market states is different depending on the current volatility level (definition of interaction, could also be interpreted as the change in E(Y) between volatility states depends on the market state, possibly context dependent interpretation).

The hypothesized interaction model would generally look like this:

E(Y)= a + b1X1 + b2X2 + b3(X1*X2)