How to formulate that a factor may influence a variable or the changes in that variable?

Question

I wonder how to formulate how that multiple factors (for instance an announcement of Central Bank A, B and C) may influence a variable. For example, consider the following formula:

$$\Delta I^n_t = f (A_t, B_t, C_t)$$ where $I^n_t$ refers to the interest rate of a security with maturity $n$ at time $t$ and $A_t, B_t, C_t$ refer to announcements of central bank A, B and C on time $t$ respectively.

The above formula indicates that changes in variable $I$ are a function of the announcements of the central banks'. However, how can one introduce this equation? Let me show what my concerns are by the following sentences:

Changes in $I$ may be due to announcements of Central bank A. However, announcements of central bank B and C may also influence $I$

My concern is the last part of the sentence, "influence $I$". Should this indeed be: "influence $I$" or should it be "influence $\Delta I$"? I doubt about this because when the announcements influence $I$ then this should automatically lead to a change in $I$ right?. On the other hand, it is not in line with my equation above. Furthermore, isn't it the case that when some factor influences the changes in some variable, that this variable actually influences the speed of changes in this variable (like a second derivative)?

Answer 1

"may also influence $\Delta I$" seems better to me. Of course for all $s \geq t$ the value of $I_s$ is also influenced by $A_t, B_t, C_t$, but this is an indirect effect. One could say it is fully encoded in $I_{t+1}$.

Answer 2

An alternative representation is the following:

$$ I^n_t = I^n_{t-1} + f(A_t, B_t, C_t) $$

You can then iterate using backward induction, to get the following expression:

$$I^n_t = I^n_0 + \sum_{j=1}^{t}f(A_j, B_j, C_j) $$

Thus, you can interpret $I^n_t$ as the accumulated effect of all the historical policy actions of the three central banks, between time 0 and time $t$, plus the initial interest rate. Moreover, you could stop at any arbitrary initial period:

$$I^n_t = I^n_{t_0} + \sum_{j={t_0}+1}^{t}f(A_j, B_j, C_j) $$

In this case, the value of the interest rate reflects the interest of a given period plus the accumulated effect of all subsequent policy changes of the three central banks.

Or even more extreme, you could iterate infinitely, in which case interest rates are simply an infinite accumulation of policy factors:

$$I^n_t = \sum_{j=0}^{\infty}f(A_{t-j}, B_{t-j}, C_{t-j}) $$

Without further information on $f$, nothing much can be said, in particular regarding the stability/convergence of the variable. Yet, as central banks react to $I$, the interest will not diverge, as central banks will attempt to bring it towards "equilibrium" levels.