# Feasibility of Interaction Term X_it*Z_t in unbalanced panel with time dummies

I have a data set on autombiles that has a time dimension, cross-sectional dimension with different levels, and multiple observations per cross-sectional unit. The dataset is quasi a panel that is unbalanced, i.e. I have a various number of time observations for different cross-sectional units.

A structure of my dataset is similar to that below (a simplified version):

Time Brand Model   Product_i  Y_it  X_it  Z_t

t1   Audi  A3      version1   y_11  x_11  z_1
t1   Audi  A3      version2   y_21  x_21  z_1
t1   Audi  A3      version3   y_31  x_31  z_1

t1   Audi  A4      version1   y_41  x_41  z_1
t1   Audi  A4      version2   y_51  x_51  z_1

t1   BMW   1       version1   y_61  x_61  z_1
t1   BMW   1       version2   y_71  x_71  z_1
t1   BMW   1       version3   y_81  x_81  z_1

t1   VW    Golf    version1   ...         z_1
t1   VW    Golf    version2   ...         z_1
...
t2   .................................... z_2
...

• Brands and Models repeat over time
• Products do not repeat over time
• Time dimension is at monthly level over 13 years (but unbalanced over models)
• Y_it varies at the product and time level.
• X_it: some varies over product and time; but! some varies only over brands and time
• Z_t varies only over time

I want to estimate a simple linear model where Y_it and Z_it are continuous variables, and X_it can contain both contin. and dichotom. variables for each product in each time period.

$$Y_{it} = const + \beta*X_{it} + \gamma*Z_t + \theta*(X_{it}\times Z_t) + error_{it}$$

My main interest is in $\theta$.

I want to control for time-invariant unobservable variables for cross-sectional units and for time-related unobservable variables (time dummies) with fixed effects in a form of dummy variables.

My general question is "How can I decide what fixed effects and at what level I may include and what variation is then left in explaining the $X_{it}\times Z_t$ relationship?"

More detailed:

• Am I allowed to include Cross-sectional Fixed effects and Time fixed effects simultaneously in case I have a variable that varies only over time?
• How can I decide at what level I should have the cross-sectional fixed effects - for Brand or Brand-Model?
• May I/Should I consider Brand*Time dummies?

Any suggestions / clarifications / links to a relevant reading are appreciated.