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Let's say I'm a Level 7 Wizard and I have no spell slots of 1st level left, but I do have some of 3rd and 4th level.

Can I cast Shield as a reaction after getting hit by an attack by using a 4th level spell slot?

Preparing and Casting Spells:

The Wizard table shows how many spell slots you have to cast your spells of 1st level and higher. To cast one of these spells, you must expend a slot of the spell's level or higher. You regain all expended slots when you finish a long rest. (PHB p. 114)

By RAW I think is this possible: am I correct?

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    \$\begingroup\$ Warlocks would be in very deep trouble otherwise \$\endgroup\$ – András Feb 22 '17 at 20:29
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Yes. You can cast any spell using a slot of the same level as the spell, or higher.

From PHB page 201, Chapter 10: Spellcasting:

SPELL SLOTS

[...] When a character casts a spell, he or she expends a slot of that spell's level or higher, effectively "filling" a slot with the spell. [...]

and, later on the same page:

CASTING A SPELL AT A HIGHER LEVEL

When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. For instance, if Umara casts magic missile using one of her 2nd-level slots, that magic missile is 2nd level. Effectively, the spell expands to fill the slot it is put into.

Some spells, such as magic missile and cure wounds, have more powerful effects when cast at a higher level, as detailed in a spell's description.

Note: In your example, you had both 3rd and 4th level spells slots remaining. You can cast Shield with either a 3rd or 4th level slot but (as there is no enhanced effect for that spell when cast at higher levels - a Shield spell is a Shield spell no matter what level it's cast at), it would be best to cast it as third level spell - keeping the 4th level slot for future 4th-or-lower level spells. Shield doesn't benefit from higher slot levels, but some other level 1-3 spells you might want to cast do.

In other words, cast it with the lowest level slot that will achieve the desired effect, taking into account any possible enhancements from casting at higher slot levels

For example, as mentioned by Mark Cogan, when casting Dispel Magic or Counterspell at higher levels, each increase in slot level increases the level of spell which can be automatically dispelled/interrupted without a spellcasting ability roll.

Similarly, Fireball can be cast using any slot of level 3 or higher - but you don't need to use a 9th level slot against a bunch of kobolds unless you have no lower-level slots remaining.

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You are correct: any memorized spell can be cast with any slot of equal or greater level.

Usually this is done for a spell such as Magic Missile, for increased effectiveness. Shield gains no such benefit. However, the level at which you cast Shield is pertinent for its interactions with Dispel Magic: since a spell assumes the higher level for that casting, it is harder to dispel. I point this out since one of the key uses for Shield at higher levels is that it nullifies castings of Magic Missile. In a wizard duel, dispel magic is often tossed about.

Citation:

Casting a Spell at Higher Level, page 100

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    \$\begingroup\$ Casting at a higher level also makes a spell harder to counterspell, which in the case of shield seems far more valuable. \$\endgroup\$ – Miniman Feb 22 '17 at 15:01

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