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There are many academic sources, books and articles, introducing forward interest rate curve. For example, those authors define $f(\tau)=f(\tau;\beta_0,\beta_1,\beta_2,\lambda)$ as a function of time to maturity $\tau$, dependent on parameters to be estimated. Such approach is use while introducing, e.g., Nelson-Siegel or Svensson model.

However, given the spot interst rate structure $R(\tau)$, the forward rate $f(\tau)$, estimated right now, should also have one more argument $t$, i.e. $f(\tau)$ is in fact $f(t;\tau)$, since it's the rate for the period $[t;t+\tau]$, implied from the spot term structure $R$.

Please, let me know, if I'm missing something. Where is $t$ in the forward rate curve definition?

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You've correctly identified that the forward curve indeed has two time indices -- one for when we observe it, and one for the future date at which the forward rate applies.

I would personally take the view that the parameters are where the "observation time" index comes into the picture. In practical terms, this could mean that at time $t$ we calibrate $\Theta(t) = \beta_{0; t}, \beta_{1, t}, \beta_{2, t}, \lambda_t$ so that $f$, which is otherwise not varying with $t$, is "close to" the observed forward curve.

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Those are instantaneous forwards, spanning $[τ,τ+dτ]$. They are a complete description of the rates economy since all other forwards can be implied from these.

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