Refer again to the gasoline sales time series data in Table...

Refer again to the gasoline sales time series data in Table 18.1 a. Using a weight of $1 / 2$ for the most recent observation, $1 / 3$ for the second most recent observation, and $1 / 6$ for third most recent observation, compute a three-week weighted moving average for the time series. b. Compute the MSE for the weighted moving average in part (a). Do you prefer this weighted moving average to the unweighted moving average? Remember that the MSE for the unweighted moving average is 10.22 c. Suppose you are allowed to choose any weights as long as they sum to $1 .$ Could you always find a set of weights that would make the MSE at least as small for a weighted moving average than for an unweighted moving average? Why or why not?

Refer again to the gasoline sales time series data in Table 18.1
a. Using a weight of $1 / 2$ for the most recent observation, $1 / 3$ for the second most recent observation, and $1 / 6$ for third most recent observation, compute a three-week weighted moving average for the time series.
b. Compute the MSE for the weighted moving average in part (a). Do you prefer this weighted moving average to the unweighted moving average? Remember that the MSE for the unweighted moving average is 10.22
c. Suppose you are allowed to choose any weights as long as they sum to $1 .$ Could you always find a set of weights that would make the MSE at least as small for a weighted moving average than for an unweighted moving average? Why or why not?

Key Concepts

Weight Optimization

Weight optimization involves selecting or adjusting the weights in a weighted moving average to achieve the best forecast performance, typically by minimizing an error metric such as MSE. This concept addresses whether it is always possible to choose weights that improve or at least match the performance of a standard unweighted moving average.

Mean Squared Error (MSE)

Mean Squared Error (MSE) is a common metric for evaluating the performance of forecasting models. It is calculated as the average of the squared differences between the predicted values and the actual observations. A lower MSE indicates that the forecasting model has less error and is generally considered more accurate.

Weighted Moving Average

The weighted moving average is a variation of the simple moving average where different weights are assigned to data points, usually giving more significance to more recent observations. This method aims to produce forecasts that are more responsive to recent changes in the data, enhancing the prediction accuracy for short-term trends.

Moving Average

A moving average is a statistical method used to smooth out fluctuations in time series data by calculating the average of a subset of data points over a defined time period. This technique helps in highlighting longer-term trends or cycles by minimizing the impact of short-term variations.

Time Series Data

Time series data consist of observations collected sequentially over time. This data is used to identify trends, seasonal effects, and patterns that can inform forecasts and decision-making processes.

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