Answer: $Y_{(T+h,T)} = \beta_0 + \beta_1 \text{TIME}_{(T+h)}$

**Explanation:** When forecasting at time $T$ for a future time $T+h$, we need to consider: 1. **Forecast notation**: $Y_{(T+h,T)}$ represents the forecast made at time $T$ for time $T+h$ 2. **Time variable**: The TIME variable should be $TIME_{(T+h)}$ since we're forecasting for that specific future time 3. **Error term**: Since $\varepsilon_t$ is independent zero-mean random noise, the optimal forecast for $\varepsilon_{T+h}$ is zero (its expected value) Let's analyze each option: **Option A**: Incorrect - Uses $Y_t$ instead of $Y_{(T+h,T)}$ and includes $\varepsilon_t$ which should be omitted in a forecast **Option B**: Incorrect - Uses $TIME_T$ instead of $TIME_{(T+h)}$; this would be forecasting based on current time rather than future time **Option C**: **Correct** - Properly uses $Y_{(T+h,T)}$ notation, includes $TIME_{(T+h)}$, and correctly omits the error term since its expected value is zero **Option D**: Incorrect - Includes $\varepsilon_{(T+h)}$ which should not be in the forecast equation since we cannot know future error terms The key insight is that in forecasting, we use the expected value of the model, and since $E[\varepsilon_{T+h}] = 0$, the error term drops out of the forecast equation.

Author: Nikitesh Somanthe