Whether a Pokémon is Shiny depends on its Original Trainer's Trainer ID number and secret ID number and on the Pokémon's personality value.
The first two numbers give you the Trainer Shiny Value (TSV), which is a number between 1 and 4096.
Similarly, the Personality Value gives you the Pokemon/Egg shiny value, a.k.a. the Egg Shiny Value (ESV), again, 1 of 4096. It is set when a Pokémon appears in the wild, when an Egg is first received or when the Pokemon is first received from an NPC.
If the ESV matches your TSV, the Pokemon will be a Shiny Pokemon.
So no, if you reset your trainer ID (and/or the secret ID) you are just changing your TSV, and you will always get another number between 1 and 4096.
The chance to have a TSV matching the ESV is still 1/4096.
If you want some math, from Bulbapedia:
The Shininess is calculated with this formula (these are all 16-bit numbers):
S = IDTrainer ⊕ IDSecret ⊕ p1 ⊕ p2
p is the Pokémon's personality value. The high half of p is referred to as p1, while the low half is referred to as p2.
In Generation VI, the Pokémon is Shiny if S < 16. This is a 16 in 65536 chance of a Pokémon being Shiny.