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Time-Series Analysis and Forecasting – Part III To read at home Time-Series Data Numerical data ordered over time The time intervals can be annually, quarterly, daily, hourly, etc. The sequence of the observations is important Example 13 Year: 2005 2006 2007 2008 2009 Sales: 75.3 74.2 78.5 79.7 80.2 Time-Series Plot A time-series plot is a two-dimensional plot of time series data 2001 1999 1997 1995 1993 1991 1989 1987 1985 1983 1981 1979 1977 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 1975 the horizontal axis corresponds to the time periods U.S. Inflation Rate Inflation Rate (%) the vertical axis measures the variable of interest Year Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-3 The problem of comparability of levels of time series Jointing (смыкание) of time series Since time series is formed during the long period of time, its levels are frequently incomparable Reasons of the incomparability 1. Change of prices. 2. Different methods of calculation of the same indicator. 3. Change of «borders» (organizational, administrative) The method of jointing time series is often used to ensure the comparability of data. It is necessary to have a transitional link (переходное звено) for jointing time series. Transitional link – is the period of time, for which the investigated indicator was calculated using the old method (in old borders) and the new method (in new borders). A transitional coefficient for this transitional is calculated, the transitional coefficient spreads over all the previous period of time Example 14 Production of oil, mln t 2005 2006 2007 2008 2009 Before merger 6600 6700 6900 - - After merger - - 7500 7800 7900 Transitional coefficient 7500 К 1,087 6900 y06 6700 1,087 7283 y05 6600 1,087 7174 Production of oil, mln t 2005 2006 2007 2008 2009 Before merger 6600 6700 6900 - - After merger - - 7500 7800 7900 Comparable series 7174 7283 7500 7800 7900 Analysis of the main tendency of time series The levels of time series are formed under the influence of lots of factors. They can be divided into 5 groups Time-Series Components Time Series Trend Component Secular Component Seasonality Component Cyclical Component Irregular Component 1. Determining (Определяющие) factors have a constant and strong influence on the examined indicator. They determine the main tendency (the trend) of time series The trend component Trend Component Long-run increase or decrease over time (overall upward or downward movement) Data taken over a long period of time Sales Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Time Chap 19-17 Trend Component (continued) Trend can be upward or downward Trend can be linear or non-linear Sales Sales Time Downward linear trend Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Time Upward nonlinear trend Chap 19-18 Real fluctuations around the trend There are 4 kinds of charge or variations involved in time series analysis. They are: 2.Secular trend Ut. In the secular trends the value of variable tends to increase or decrease over a long period of time. The steady increase of the cost of living recorded by the Сonsumer price index is an example of secular trend. From year to year, the cost of living varies a great deal, but if we examine long-period, we see that the trend is toward a steady increase. 3. Cyclical fluctuation Vt. The most common example of cyclical fluctuation is the business cycle. Over time, there are years when the business cycle hits a peak above the trend line. At other times, business activity is likely to slump, hitting a low point below the trend line. The time between hitting peaks and falling to low points is at least one year and it can be as many as 15 or 20 years. Cyclical Component Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to trough 1 Cycle Sales Year Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-22 4. Seasonal variation St involves pattern of change within year that tend to be repeated from year to year. For example the consumption of drinks, juices, ice cream and other. Seasonal factors give rise to oscillations relative to the main tendency Seasonal Component Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly Sales Summer Winter Summer Spring Winter Spring Fall Fall Time (Quarterly) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-24 variation εt. The value of variable may be completely unpredictable changing in random manner. For example, the Iraqi situation in 1990, the ruble devaluation in 1998 and the others. Random factors cause the random fluctuations of levels of series (for example, weather factor) 5. Irregular Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-25 5. Irregular variation εt. The value of variable may be completely unpredictable changing in random manner. For example, the Iraqi situation in 1990, the ruble devaluation in 1998 and the others. Thus each value of time series could be presented as follows: yt Ut Vt St t. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-26 Irregular Component Unpredictable, random, “residual” fluctuations Due to random variations of Nature Accidents or unusual events “Noise” in the time series Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-27 Time-Series Component Analysis Used primarily for forecasting Observed value in time series is the sum or product of components Additive Model Xt Tt St Ct It Multiplicative model (linear in log form) Xt TtStCtIt where Tt = Trend value at period t St = Seasonality value for period t Ct = Cyclical value at time t It = Irregular (random) value for period t Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-28 Smoothing the Time Series Calculate moving averages to get an overall impression of the pattern of movement over time This smooths out the irregular component Moving Average: averages of a designated number of consecutive time series values Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-29 Method of interval enlargement Method of interval enlargement consists in replacement of initial levels of series by the average values, which are calculated for the enlarged intervals Example 15 Month yt 1 5.1 2 5.4 3 5.2 4 5.3 5 5.6 6 5.8 7 5.6 8 5.9 9 6.1 10 6.0 11 5.9 12 6.2 Quarterly sums Average monthly value (per quarter) 15.7 5.23 16.7 5.57 17.6 5.87 18.1 6.03 The End of Part III To be continued Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-33