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The rules for Wild Arcana say:

When casting a spell in this way, you treat your caster level as 2 levels higher for the purpose of any effect dependent on level.

The Contingency rules say:

The spell to be brought into effect by the contingency must be one that affects your person and be of a spell level no higher than one-third your caster level (rounded down, maximum 6th level).

Let us say I am a Wizard CL 13. Will I be able to use (maximum) 4th or 5th spells with contingency? I am not sure whether this counts as an effect.

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Yes, because the spell's "Description" is part of its Effect.

If you look through Pathfinder's core rules on magic, in the "Caster Level" section, you can find the following text:

In the event that a class feature or other special ability provides an adjustment to your caster level, that adjustment applies not only to effects based on caster level (such as range, duration, and damage dealt), but also to your caster level check to overcome your target's spell resistance and to the caster level used in dispel checks (both the dispel check and the DC of the check).

(Thanks to VoromirKadien for the correction)

The "effect" of a spell includes the description and all values that are based on caster level, including range, targets, area, and duration. Therefore, the level-limitation rule in Contingency counts as part of its effect, and so Wild Arcana applies.

For the purposes of the spell effect, your caster level is 13 + 2 = 15. So your Contingency allows spells up to level 15/3 rounded down, or 5th level.

Note: When you apply Wild Arcana to Contingency, it does not modify the companion spell, its effects, or saving throws. You would have to use Wild Arcana again to separately cast the companion spell, in order to get these benefits.

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    \$\begingroup\$ Just wanted to point out "In the event that a class feature or other special ability provides an adjustment to your caster level, that adjustment applies not only to effects based on caster level (such as range, duration, and damage dealt), but also to your caster level check to overcome your target's spell resistance and to the caster level used in dispel checks (both the dispel check and the DC of the check)." Taken from the very page you linked to. So it would last 15 days. \$\endgroup\$ – Voromir Kadien May 14 '18 at 6:32

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