Answer: 0.8589%

The forecast under the log-linear model is given by: $$ \begin{aligned} \mathbb{E}_T(Y_{T+h}) &= e^{\beta_0 + \beta_1(Y_{T+h}) + \frac{\sigma^2}{2}} \\ \Rightarrow \mathbb{E}_T(I_3) &= e^{\beta_0 + \beta_1(3) + \frac{\sigma^2}{2}} \\ &= e^{-0.1567 + 0.00134(3) + \frac{0.0342^2}{2}} = 0.8589\% \end{aligned} $$ **Explanation:** 1. **Model Understanding:** This is a log-linear trend model where the natural logarithm of the interest rate is modeled as a linear function of time plus an error term. 2. **Forecast Formula:** For log-linear models, when forecasting the original variable (not the log), we need to account for the transformation bias. The correct forecast formula is: $$\mathbb{E}_T(I_{T+h}) = e^{\beta_0 + \beta_1(T+h) + \frac{\sigma^2}{2}}$$ The term $\frac{\sigma^2}{2}$ is the bias adjustment term needed because $E[e^X] \neq e^{E[X]}$ when X is normally distributed. 3. **Calculation Steps:** - $\beta_0 = -0.1567$ - $\beta_1 = 0.00134$ - $h = 3$ (forecast horizon) - $\sigma = 0.0342$ - $\sigma^2 = (0.0342)^2 = 0.00116964$ **Compute exponent:** $$-0.1567 + 0.00134 \times 3 + \frac{0.00116964}{2}$$ $$= -0.1567 + 0.00402 + 0.00058482$$ $$= -0.15209518$$ **Compute forecast:** $$e^{-0.15209518} = 0.8589$$ 4. **Interpretation:** The point forecast for the interest rate after 3 years is 0.8589%.

Author: Nikitesh Somanthe