A recession causes unemployment to rise. We are experiencing an increase in the unemployment rate, so we must be in a recession.”

This is called affirming the consequent.

https://en.wikipedia.org/wiki/Affirming_the_consequent

It is the innapropriate use of the converse of a true statement.

Example,

Proposition.

If you own cat, then you have a pet.

Valid: You own a cat, therefore you must have a pet.

Converse: If you have a pet, you own a cat.

Invalid: You have a pet, therefore you must own a cat.

Converses and inverses (negation of proposition and conclusion) are only valid when the proposition is essentially a definition or follows an If and only If pattern. (necessary and sufficient)

For example:

Definition. Daytime is the period of time whne the sun is above the horizon.
Proposition form: If the sun is above the horizon, then it must be daytime.
Converse (valid) : If it is daytime the sun must be above the horizon.
Inverse (Valid): If the sun is NOT above the Horizon then it is NOT daytime.

Or

Proposition: A regular polygon is an equilateral triangle if and only if its internal angles are all 60 degrees.
Since this is an if and only oif statement all the following are valid:

VALID
Using converse: My regular polygon's internal angles are 60 degrees, therefore it is an equilateral traingle.
Using inverse: My regular polygon is NOT an equilateral triangle, therefore NOT all its internal angles are 60 degrees.

Finally you should know aboutthe CONTRAPOSITIVE. It is the combined converse and inverse. If the proposition is true, then the Contrapositive is always true, even when the COVERSE and INVERSE are FALSE.

Example

Proposition: If you own a cat you have a pet. (assumed true)
Converse: If you have a pet, then you own a cat (False)
Inverse: If you do NOT own a cat you do NOT have a pet (False)
Contrapositive (converse and inverse together) : If you do NOT have a pet, You do NOT own a cat (Also true)

Clearly begging the question. The conclusion of the argument is used as the premise. I'm not quite sure but I think it's also circular reasoning.

I don't think is a begging the question case here. It seems to contain both an content and a formal fallacy. The formal fallacy that I identify here is affirming the consequent .

Logical form: If P then Q
Q
Therefore P.

Standard form: P1: Any recession causes unemployment to rise,
P2: We are experiencing an increase in the unemployment rate
Cl: Therefore, so we must be in a recession.

Modus ponens :
P1: Any recession causes unemployment to rise
P2: We are in a recession
Cl: Therefore, the recession is the cause of unemployment rate increase. - (still, an informal fallacy is present)

The fallacy simply reflects the failure to see other reasons for why an unemployment increase rate came about.

The content fallacy I see here is confusion of the cause and effect or causal reductionism . It does not mean that if there is a recession the unemployment rate increases. The reverse could also be true.

But I'm not entirely sure about the formal fallacy. Could this qualify as a false conversion ? (i.e., All P are Q, therefore all Q are P).