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I am trying to create a time series visualization of unemployment rates for every county in the continental US using ArcMap 10.3.1.

I have data for every month going back to January 1990, but when I try to join it to my county shapefile in ArcMap, I get an error message because the spreadsheet has too many columns.

How can I join this spreadsheet to a shapefile such that I can symbolize the same shapefile by each field for which I have data?

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The maximum number of columns you can have in a Shapefile is 255.

See Geoprocessing considerations for shapefile output

The maximum number of fields is 255. A conversion to shapefile will convert the first 255 fields if this limit is exceeded.

If using ArcMap you could look at using a File Geodatabase which has a limit of 655,346 fields per table (which should be enough for what you're doing).

See FAQ: Is there a limit to the number of fields that can be added to a table or feature class in a geodatabase?

The number of allowable fields for any feature class or table in a geodatabase depends on the limitations of the database in which the geodatabase is stored. While file geodatabases have a limit of 655346 fields per table, the limit for personal geodatabases is governed by the .mdb file format, which has a limit of 255 fields per table. Therefore, personal geodatabases stored in an Access database may have a feature class or table with no more than 255 fields. The answer for feature classes or tables stored in an ArcSDE Geodatabase varies depending on the underlying DBMS.

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