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.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45819738507270813, "perplexity": 11035.17195826187}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986688826.38/warc/CC-MAIN-20191019040458-20191019063958-00380.warc.gz"}