Consider a nine-year moving average used to smooth a time series that was first recorded in 1984.
Which year serves as the first centered value in the smoothed series?
How many years of values in the series are lost when computing all the nine-year moving averages?
To find the first centered value in the smoothed series, we need to calculate the moving average for the year 1988.
The first centered value is obtained by taking the average of the five values before and the four values after the given year. In this case, the five values before 1988 would be from 1984 to 1988, and the four values after 1988 would be from 1988 to 1991.
Therefore, the first centered value in the smoothed series is for the year 1988.
To compute all the nine-year moving averages, we lose eight years of values at the beginning and eight years of values at the end. This is because when calculating the moving average, we need a minimum of nine years of data for each average.
Since the time series was first recorded in 1984, the first nine-year moving average can be calculated for the year 1992 (1984 + 8 years). This implies that we lose data from 1984 to 1991 (eight years) at the beginning.
Similarly, if the time series continues up to the present year, the last nine-year moving average can be calculated for the year 2016. Hence, we lose data from 2017 to present (eight years) at the end.
In summary, when computing all the nine-year moving averages, we lose a total of sixteen years of values - eight years at the beginning and eight years at the end.
To determine the first centered value in the smoothed series using a nine-year moving average, we need to compute the moving average for each year, starting from 1984.
Since the moving average is centered, the first centered value would be calculated using a centered window of nine years. This means that the first centered value will be the average of the years 1984 to 1992.
To compute all the nine-year moving averages, we need to lose the values from the first four and last four years of the original time series. These years are lost in order to have a centered window of nine years for each moving average calculation.
To determine the first centered value in the smoothed series, we need to calculate the number of years for the moving average to become centered.
Since we are using a nine-year moving average, the centered value will be in the middle of the average, surrounded by four years on each side. To have a balanced nine-year moving average, we need to have an equal number of years before and after the centered value.
Let's calculate the number of years needed before the first centered value:
- The first recorded year in the time series is in 1984.
- To have four years before the first centered value, we need to subtract four years from 1984 (1984 - 4 = 1980).
- Therefore, the first centered value will be in the year 1980.
To determine the number of lost years when computing all the nine-year moving averages, we need to consider the length of the moving average.
For a nine-year moving average, we compute the average using a window of nine consecutive years. As a result, the first four years and the last four years of the series cannot be included in the moving average calculation since we don't have a full window of nine years.
Therefore, when computing all the nine-year moving averages, we lose eight years of values - the four years at the beginning and the four years at the end of the series.