# What is the difference between these two regression formulations?

I am trying to look at the impact of flooding risk on property values. I am using a repeated sales method for property transactions to control for time invariant characterestics. Assume I control for all relevant fixed effects and time trends. I have the following regression equations:

$$log(y) = \beta*Risk*Year + .....$$

where $$y$$ is sale price for property, Year is a continuous year variable i.e. 2016.5 for 6th month in 2016. The interpretation of $$\beta$$ is the discount in price appreciation trend of properties at flood risk, relative to the control group i.e. properties not at risk. Risk is therefore a dummy variable.

My question is what's the difference between the prior specification and the following, and which one would be more appropriate to use.

$$log(y) = \beta_0*Risk + \beta_1*Year + \beta*Risk*Year + .....$$

In your first equation, it looks like you are interested in an interaction effect between risk and year.

In that case, a good rule for interaction models, is to also include the variables themselves, as you do in the second equation. So, the second equation would be more appropriate.

The difference in the two models, is that the second one also looks at the overall effects of "risk" itself across all years on average, and it looks at the overall effect of "time" itself, regardless of risk or anything else. The first equation only looks at the interaction between "risk" and time, not the effects of each variable in isolation.

Since you are interested in risk, your coefficient of interest from the second equation would be both $$\beta_0$$ and $$\beta_1$$.

However, your interpretation of $$\beta$$ in the first equation, does not seem quite correct to me, as it seems you are only focusing on risk there and not on year.

In your second equation, $$\beta_0$$ would refer to the effect of being at flood risk, vs. not being. Then, $$\beta_1$$ should give the effect each year, kind of like a time fixed effect. Your interaction effect $$\beta$$, would then be interpreted as a combination of risk and time.

However, its unusual to have "Year" be continuous like you have it. Normally it should be a categorical variables, to be used as a series of dummies, even if you look at months within the year.

For more, you should look up the general interpretation of interaction effects as well as time fixed effects in any standard textbook.

As a side note, your equations don't seem to have an intercept, which is a bit unusual.