# Coefficient of Determination with Weighted Coefficients

I want to compare coefficients of determination (i.e., $$R^2$$) of the following two models. $$fe_{t+h}$$ is a forecast error in $$t+h$$ and $$\mathbf{\varepsilon}_t$$ is a shock of interest occurring between $$t$$ and $$t+h$$. The idea is to interpret the coefficient of determination as the share of forecast error variance (FEVD) which can be explained by my shock of interest.

Model 1 \begin{align*} fe_{t+h} = \beta_0 \mathbf{\varepsilon}_{t} + ... + \beta_h\mathbf{\varepsilon}_{t+h} + u_t \end{align*}

Model 2 \begin{align*} fe_{t+h} = F(z_t) \left(\beta_0^I \mathbf{\varepsilon}_{t} + ... + \beta_h^I \mathbf{\varepsilon}_{t+h}\right) + \left( 1-F(z_t)\right) \left(\beta_0^{II}\mathbf{\varepsilon}_{t} + ... + \beta_h^{II} \mathbf{\varepsilon}_{t+h}\right) + u_t \end{align*}

where $$F(z_t)$$ is a smooth logistic function with $$F(z_t)\in [0,1]$$ which weights the coefficients $$\beta_0, ..., \beta_h$$ between regimes 1 and 2. In my model at hand, $$h$$ is equal to 15.

Is it ok to use the non-adjusted $$R^2$$? I am aware that typically, $$R^2$$ will rise with the number of additional explanatory variables. However, in my case I have no additional variables but only weighted coefficients.

If you want to explain the fraction of variance explained, use $$R^2$$. If you want to compare $$R^2$$'s but adjust (very weakly) for model size, use adjusted $$R^2$$'s. If you want to compare these models, use an $$F$$-test since Model 1 is nested in Model 2.