Once you identify and develop an appropriate exponential smoothing model with the starting
values for parameter(s)= 0.5, what are the following numerical values:
1. The within-sample forecast for January 2023.
2. The out-of-sample forecast for October 2023.
3. The out-of-sample forecast for June 2024.
4. The MAPE.
5. The MAE.
Critically think for a way to optimise alpha and beta (if there is no beta, you can input ‘0’ for
question 7) via the MSE, and report the following values after your optimisation:
6. Alpha.
7. Beta.
8. The MAPE.
9. The within-sample forecast for January 2023.
10. The out-of-sample forecast for June 2024.
解决计算这几个问题

Question

Once you identify and develop an appropriate exponential smoothing model with the starting 
values for parameter(s)= 0.5, what are the following numerical values:
1. The within-sample forecast for January 2023.
2. The out-of-sample forecast for October 2023.
3. The out-of-sample forecast for June 2024.
4. The MAPE.
5. The MAE.
Critically think for a way to optimise alpha and beta (if there is no beta, you can input ‘0’ for 
question 7) via the MSE, and report the following values after your optimisation:
6. Alpha.
7. Beta.
8. The MAPE.
9. The within-sample forecast for January 2023.
10. The out-of-sample forecast for June 2024.
解决计算这几个问题

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To develop the exponential smoothing model, we start with the formula for simple exponential smoothing: $ F_t = \alpha Y_{t-1} + (1 - \alpha) F_{t-1} $, where $ F_t $ is the forecast for the current period, $ Y_{t-1} $ is the actual value from the previous period, and $ \alpha $ is the smoothing constant (0.5 in this case). ‖ ‖ step 2 ⋮ For the within-sample forecast for January 2023, we apply the smoothing formula using the actual values from December 2022 and the forecast from November 2022. Assuming we have these values, we calculate $ F_{Jan2023} $. ‖ ‖ step 3 ⋮ For the out-of-sample forecast for October 2023, we continue applying the smoothing formula iteratively from the last known actual value up to October 2023. This involves using the forecast from September 2023 and the actual value from August 2023. ‖ ‖ step 4 ⋮ For the out-of-sample forecast for June 2024, we repeat the process, using the forecast from May 2024 and the actual value from April 2024. ‖ ‖ step 5 ⋮ To calculate the Mean Absolute Percentage Error (MAPE), we use the formula: $ MAPE = \frac{1}{n} \sum_{i=1}^{n} \left| \frac{Y_i - F_i}{Y_i} \right| \times 100 $, where $ Y_i $ are the actual values and $ F_i $ are the forecasted values. ‖ ‖ step 6 ⋮ For the Mean Absolute Error (MAE), we use the formula: $ MAE = \frac{1}{n} \sum_{i=1}^{n} |Y_i - F_i| $. ‖ ‖ step 7 ⋮ To optimize alpha and beta (if applicable), we can use the Mean Squared Error (MSE) as the objective function: $ MSE = \frac{1}{n} \sum_{i=1}^{n} (Y_i - F_i)^2 $. We adjust $ \alpha $ and $ \beta $ to minimize this value. ‖ ‖ step 8 ⋮ After optimization, we report the new values of $ \alpha $ and $ \beta $, along with the updated MAPE. ‖ ‖ step 9 ⋮ The within-sample forecast for January 2023 is recalculated using the optimized parameters. ‖ ‖ step 10 ⋮ The out-of-sample forecast for June 2024 is also recalculated using the optimized parameters. ‖ ∻Answer∻ ⚹ The specific numerical values for the forecasts, MAPE, and MAE depend on the actual data provided, which is not included here. Please input the actual values to compute the final results. ⚹ ∻Key Concept∻ ⚹ Exponential smoothing is a forecasting technique that uses weighted averages of past observations to predict future values. ⚹ ∻Explanation∻ ⚹ The answer outlines the steps to develop an exponential smoothing model, calculate forecasts, and evaluate the model's accuracy using MAPE and MAE. The optimization of parameters is crucial for improving forecast accuracy. ⚹◊How do you calculate the Mean Absolute Error (MAE) using the formula $MAE = \frac{1}{n} \sum_{t=1}^{n} |A_t - F_t|$, where $A_t$ is the actual value and $F_t$ is the forecasted value?⍭ Generate me a similar question◊

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