We use time series for two purposes: to understand (model) past values of the data, and to forecast future data values (assuming the same patterns and model hold)
Time Series Explanation
The features we look for in a time series are:
Trend - the most slowly changing component of a time series. Trends are often caused by population growth or change, technological change, etc. and can increase or decrease over time.
Cycle - the next most slowly changing component appearing as recurrent wave-like patterns in the series The main cause of cyclical behaviour is the Business Cycle.
Seasonality - a more rapidly changing component than the cycle, and shows up as an equally spaced regular repeating pattern (for example, in data collected every 3 months - i.e. quarterly but can also be present in other data collected monthly, weekly, daily or hourly).
Random - always present and consists of small variations about the overall trend, cycle and seasonal components that are unpredictable. It can be thought of as 'what is left' after the other components (trend, cycle and seasonal) have been removed.
EXAMPLE 1
The graph below displays time series of New Zealand's CPI and Food Price Index (FPI) over the years since 1959. Although both indices fluctuate up and down there is an obvious increasing trend. Overall the trend looks steeper since 1974 but there are some short periods with lower annual average change than before 1974.
EXAMPLE 2
The next graph is for one item, apples, in the Fresh Fruit and Vegetable Subgroup. The graph is of monthly data, with a clear annual peak that is almost always in January (early summer in New Zealand, when only apples kept in cool storage are available) and the lowest prices are in the June quarter months when new apples are available. This repeated annual pattern is a seasonal pattern and is shown by many fresh fruit and vegetables.
Index numbers can be used in any situation when relative rather than absolute change is desired. The index numbers in a time series are calculated relative to the observation at a particular reference time.
END TIME SERIES EXPLANATION
By checking for the trend and seasonality key points can be read off time series graphs.
Interpreting Time Series
EXAMPLE
New Zealand
In the graph below of the value/price of New Zealand Merchandise Exports there is an obvious upward trend, and also regular seasonal variation: a peak early each year, and a trough at the end. But each year is also different in detail from the previous one.
EXERCISE
New Zealand
What can we observe in the following time series example of Retail Sales for three New Zealand regional councils ($1000/Quarter)?
There is an obvious upward trend and also a regular seasonal variation with a peak and a trough in each year.
In the next section we see how to measure the upward trend and identify the seasonal variation using moving averages.
END INTERPRETING TIME SERIES
In order to see the trend in a time series better we can smooth the data by taking averages of a number of observed values at adjacent time periods.
Moving Averages
This is called taking "moving averages". The simplest moving average is a 3-point moving average where we replace each value in the original series by the mean of the value and the two points either side of it. Increasing the number of points in the moving-average results in a smoother trend line but a balance needs to be achieved between smoothing and losing information. Moving-averages create a problem at the two ends of the graph so some adjustment has to be made at these points.
EXAMPLE
United States
The following table demonstrates the construction of a 3-point moving average for the US Consumers Price Index.
| Annual Percentage Increase in CPI US and 3-point Moving Average |
|||
|---|---|---|---|
| Year | % increase in CPI (yt) |
3-point moving total |
3-point moving average (mt) |
| 1985 | 4.1 | - | - |
| 1986 | 3.5 | 9.20 | 3.07 |
| 1987 | 1.6 | 9.20 | 3.07 |
| 1988 | 4.1 | 11.00 | 3.67 |
| 1989 | 5.3 | 14.30 | 4.77 |
| 1990 | 4.9 | 15.70 | 5.23 |
| 1991 | 5.5 | 14.00 | 4.67 |
| 1992 | 3.6 | 12.20 | 4.07 |
| 1993 | 3.1 | 9.50 | 3.17 |
| 1994 | 2.8 | - | - |
EXERCISE
The table below gives the sales (in millions of dollars) for a supermarket chain over a four-year period. You can use the spreadsheet download below for this exercise, the spreadsheet contains a completed answer.
| Months | 2008 | 2009 | 2010 | 2011 | 2012 |
|---|---|---|---|---|---|
| Jan-Apr | 48.6 | 51.8 | 55.2 | 54.9 | 58.7 |
| May-Aug | 52.4 | 54.1 | 57.6 | 58.9 | 59.5 |
| Sept-Dec | 56.1 | 58.9 | 59.7 | 62.9 | 64.5 |
NOTE 1: There are three period effects in this exercise matching the three 4 month periods. To estimate these we would compute the differences of the data values from the estimated trend values and average the differences for each period. If you got approximate values of -2.2(Jan-April), -0.4(May-August) and 2.6(Sept-Dec) then to forecast sales for 2013(Jan-April) you add the approximate average period for Jan-April to the forecasted 3-point moving average; that is, 62.6+(-2.2) that gives 60.4.
NOTE 2: Seasonal data are very common with price indexes. In this case there is a moving average of four required which introduces some additional problems. These are discussed in the links provided in the next section on Exploring Time Series.
END MOVING AVERAGES
You should find out more about time series and the components; trend, seasonality and random.
Exploring Time Series
You can learn more about time series (including those of variables other than price changes) by exploring either of the following teaching resources. You will need internet access for these resources although both will work offline once downloaded or installed to your computer.
END EXPLORING TIME SERIES
One of the most common uses of the CPI is to adjust a time series to take account of inflation.
Deflating Time Series Indexes
EXAMPLE
The table below gives the weekly income for New Zealand workers for the quarters December 2004 to December 2006 together with the CPI for the same periods. We are interested in knowing what the time series looks like when inflation is taken into account (that is, the time series is deflated) in order to decide whether income has really increased (relative to the reference/base period) and if so by how much.
For each quarter the Average Weekly Earning is divided by the CPI for that quarter (relative to the first quarter) x 100.
For December, 2005 the Average Weekly Earning = $807.56 and the CPI (relative to December 2004) = 103.2.
The deflated value of the Weekly Earnings at December 2004 prices is:
| Year | Month | Average Weekly Earnings |
The CPI | Earnings at Dec. 2004 prices | |
|---|---|---|---|---|---|
| 2004 | DEC | 782.00 | 100.0 | 782.00/100.0 x 100 = | 782.0 |
| 2005 | MAR | 794.83 | 100.4 | 794.83/100.4 x 100 = | 791.7 |
| JUN | 801.89 | 101.4 | 801.89/101.4 x 100 = | 790.8 | |
| SEP | 795.95 | 102.5 | 795.95/102.5 x 100 = | 776.5 | |
| DEC | 807.56 | 103.2 | 807.56/103.2 x 100 = | 782.5 | |
| 2006 | MAR | 826.08 | 103.8 | 826.08/103.8 x 100 = | 795.8 |
| JUN | 830.46 | 105.4 | 830.46/105.4 x 100 = | 787.9 | |
| SEP | 843.73 | 106.1 | 843.73/106.1 x 100 = | 795.2 | |
| DEC | 855.68 | 105.9 | 855.68/105.9 x 100 = | 808.0 |
Table 1: Weekly income for New Zealand workers - Statistics New Zealand
The plot below shows two series -- the top one is that of the raw average weekly incomes and the one below it is the deflated series showing earnings relative to Dec. 2004.
The calculations in the example can be generalised. There is no need to re-reference the time series initially so that the index number for the period in which prices are expressed in the deflated series is equal to 100 (or 1000). The new monetary earnings can be calculated directly by:
Deflating Exercises
In following exercises the final answers are to two decimal places.
EXERCISE 1
1. Deflate the weekly earnings for 1990 in the table below to 1988 price levels using the CPI.
For exercise 1 and 2 use the downloadable Excel spreadsheet below to answer the questions. Click 'Check' after you have entered all values in each table, any correct responses will be highlighted with a green border.
Excel Download| Weekly | Earnings ($) | CPI (Reference 1988) | 1990 Deflated Earnings ($) |
|---|---|---|---|
| 1988 | 394.60 | 100 | |
| 1989 | 448.30 | 126 | |
| 1990 | 550.00 | 160 |
The correct answers are: $, $, $
EXERCISE 2
The partial table below gives average wages of Public Servants between 1947-58, together with the CPI with 1947 as the reference year. Again, use the downloadable Excel spreadsheet above to help with this question.
| Year | Wage ($/hour) | CPI |
|---|---|---|
| 1947 | 1.19 | 100.0 |
| 1948 | 1.33 | 108.0 |
| 1949 | 1.44 | 109.7 |
| 1950 | 1.57 | 116.0 |
| ... | ... | ... |
| ... | ... | ... |
| 1958 | 2.45 | 171.8 |
Although the apparent wage increase from 1947 to 1958 more than doubled, what was the real wage increase in terms of relative purchasing power?
In this exercise the apparent wage increase is $2.45-$1.19 which equals $1.26. This is an increase of $2.45 divided by $1.19 that equals 2.059 or a 105.9% increase.
The real wage increase using the deflated value is $1.43-$1.19 which equals $0.24. The purchase power increased by $1.43 divided by $1.19 that equals 2.202 or a 20.2% increase.
These two calculations could instead be done as:
EXERCISE 3
The accommodation fees per week at a student residential college for several years over the period 1970 to 1989 together with CPI relative to a reference/base in 1970 are in the table below. Calculate the weekly fees for the years 1975, 1980, 1985 and 1989 in terms of 1970 levels.
| Year | Weekly Fee ($) | CPI | Weekly Fee ($) at 1970 level |
|---|---|---|---|
| 1970 | 14.00 | 100 | 14.00 |
| 1975 | 22.75 | 163 | |
| 1980 | 46.50 | 325 | |
| 1985 | 92.50 | 572 | |
| 1989 | 136.65 | 843 |
The correct answers are: $, $, $ and $. As an example:
EXERCISE 4
The table below shows the starting annual salary for new employees in an organisation in a country with high inflation. The period is 2000 to 2014 and the CPI is adjusted so that 2000 is the reference/base year.
| Year | Salary ($) | CPI | Salary with Reduction ($) |
|---|---|---|---|
| 2000 | 42000 | 100 | |
| 2002 | 42500 | 126 | |
| 2004 | 43000 | 190 | |
| 2006 | 45000 | 277 | |
| 2008 | 47000 | 435 | |
| 2010 | 48500 | 572 | |
| 2012 | 52000 | 797 | |
| 2014 | 54000 | 843 |
The correct answers from top to bottom are: , , , , , , , and .
The correct answer is 'No'. The answers show a downward trend.
END DEFLATING TIME SERIES