It's a cause as even Calvin agrees
John 3:16 (KJV 1900) — 16 For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.
The efficient (primary) cause of our eternal salvation the Scripture uniformly proclaims to be the mercy and free love of the heavenly Father towards us; the material cause to be Christ, with the obedience by which he purchased righteousness for us; and what can the formal or instrumental cause be but faith? John includes the three in one sentence when he says, “God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish but have everlasting life,” (John 3:16).
That is not the word of an Arminian or a provisionist but of
John Calvin, Institutes of the Christian Religion (Bellingham, WA: Logos Bible Software, 1997).
Similarly The Calvinist Theologian Berkouwer states
3. TO FAITH. Faith is the mediate or instrumental cause of sanctification as well as of justification. It does not merit sanctification any more than it does justification, but it unites us to Christ and keeps us in touch with Him as the Head of the new humanity, who is the source of the new life within us, and also of our progressive sanctification, through the operation of the Holy Spirit
L. Berkhof, Systematic Theology (Grand Rapids, MI: Wm. B. Eerdmans publishing co., 1938), 537.
There is not just a correlation between faith and justification, faith and regeneration , faith and salvation
There is a causal relationship even Calvinist theologians have affirmed