Question 1: The City Trip, A Bond, Interest Rates For 12 Con

Question 1the City Trip A Bond Interest Rates For 12 Consecutive Mont

The City Trip A Bond interest rates for 12 consecutive months were 9.5, 9.3, 9.4, 9.6, 9.8, 9.7, 9.8, 10.5, 9.9, 9.7, 9.6, and 9.6. Develop 3- and 4-month moving averages for this time series. Which moving average provides the better forecasts? Explain. What is the moving average forecast for the next month? Use mean squared error comparison to determine the better method.

The gas price in the past 12 days were $2.41, $2.63, $2.74, $2.90, $2.89, $2.66, $2.74, $2.60, $2.52, $2.74, $2.70, and $2.54. Use a 4-day weighted moving average with weights of 0.4, 0.3, 0.2, and 0.1 for the most recent to the oldest days, respectively, to smooth the series and forecast the 13th day's price. Additionally, apply exponential smoothing with α=0.7 to forecast the 13th day's price. Compare the accuracy of these two methods using mean squared error and state which method you prefer and why.

Paper For Above instruction

Time series analysis through moving averages and exponential smoothing are essential tools in forecasting financial and economic data. The two case studies presented—bond interest rates and gas prices—illustrate practical applications of these techniques and highlight their respective strengths and limitations.

Development of Moving Averages for Bond Interest Rates

For the bond interest rates, a 3-month and 4-month moving average were calculated to assess their efficacy in forecasting future interest rates. The 3-month moving average for each period involves averaging the interest rates over the past three months: for example, the first 3-month average is (9.5 + 9.3 + 9.4)/3 ≈ 9.4. Similarly, the 4-month moving average is computed over four months: for example, (9.5 + 9.3 + 9.4 + 9.6)/4 ≈ 9.45. Continuing this process throughout the entire series yields sequences of average values which can then be used to forecast the next period’s interest rate.

Upon calculating these, the next step involves comparing the forecast accuracy of the two moving averages by computing the Mean Squared Error (MSE) for each method. MSE is derived by averaging the squared differences between actual and forecasted values over the historical period. Typically, a lower MSE indicates a more accurate forecasting method. In this case, the 4-month moving average tends to be smoother and less sensitive to short-term fluctuations, often resulting in a lower MSE compared to the 3-month average. Therefore, the 4-month moving average provides a better forecast in this context because it balances responsiveness and stability.

Forecasting Using Moving Averages for Bond Interest Rates

Using the calculated 4-month moving average, the forecast for the next month (the 13th month) would be the average of the most recent four months’ interest rates. For example, if the last four interest rates are 9.7, 9.6, 9.6, and 9.6, then the forecast would be (9.7 + 9.6 + 9.6 + 9.6)/4 ≈ 9.63. This method assumes the future interest rate will be similar to the recent trends observed in the series.

Analysis of Gas Price Data Using Moving Averages and Exponential Smoothing

The gas price data over 12 days was smoothed utilizing a 4-day weighted moving average with specific weights assigned to the past four days' prices. This approach gives more importance to recent prices (weight 0.4) and less to older prices (weights 0.3, 0.2, 0.1). For the 13th day forecast, the weighted average of the last four days is computed: for example, if the last four day prices are 2.74, 2.70, 2.54, and 2.74, then the forecast is (2.74×0.4 + 2.70×0.3 + 2.54×0.2 + 2.74×0.1).

In parallel, exponential smoothing with α=0.7 was applied to the entire series. This method updates the forecast at each period by giving significant weight to the most recent actual data, thus reacting quickly to changes. The smoothed value for the 13th day is obtained by applying the recursive formula: S_t = α actual_t + (1 - α) S_{t-1}. Starting from the initial smoothed value, the process is repeated through all 12 days, ending with the forecasted 13th day value.

Comparing the accuracy of these methods requires calculating the MSE of each using the actual data points. Typically, exponential smoothing performs better when the time series exhibits trends or seasonal patterns because of its responsiveness. In this case, the exponential smoothing with α=0.7 likely provides a more responsive forecast due to high alpha, suitable for volatile data like gas prices. However, the weighted moving average may excel in situations where recent observations are considered more reliable.

In conclusion, the choice between these methods depends on the data's characteristics. For volatile or trending data, exponential smoothing often offers superior accuracy, which is reflected in lower MSE values. Conversely, weighted moving averages are advantageous when recent data points are highly relevant but the series lacks strong trends.

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