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Chapter 13

Chapter 13. Time Series Analysis and Index Numbers. A time series represents a variable observed across time. The time increment can be years, quarters, months, weeks, or even days.

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Chapter 13

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  1. Chapter 13 Time Series Analysis and Index Numbers

  2. A time series represents a variable observed across time. The time increment can be years, quarters, months, weeks, or even days. Example 13.1: The number of employees from 1997 to 2004 at Video-Comp recorded in the following table represents time series data. Components of a Time Series

  3. Components of a Time Series (cont.) • The values of the time series can be presented in a table or illustrated using a scatter diagram.

  4. Components of a Time Series (cont.) • The components of a time series are • Trend (TR) • Seasonal Variation (S) • Cyclical Variation (C) • Irregular Activity (I) • The purpose of time series analysis is to describe a particular data set by estimating the various components that make up this time series.

  5. Trend (TR) • Steady increase or decrease in the time series. • Reflects any long-term growth or decline in the observations. • A trend may be due to inflation, increases in the population, increases in personal income, market growth or decline, or change in technology. • Usually follows a straight line (linear trend), but can also be curvilinear (quadratic trend).

  6. 11.0 – 10.0 – 9.0 – 8.0 – 7.0 – 6.0 – 5.0 – 4.0 – 3.0 – 2.0 – 1.0 – Trend Line Number of employees (thousands) | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 t Linear Trend • In linear trend, the rate of change in Y from one time period to the next is relative constant. • The linear trend line is TR = b0 + b1t

  7. Yt Yt t t b2< 0 (b) b2 < 0 (a) Yt Yt t t b2 > 0 (d) b2 > 0 (c) Curvilinear Trend • In curvilinear trend, the time series appears to be slowing down or accelerating as time increases. • For example, TR = b0 + b1t + b2t2 represents a quadratic trend.

  8. Seasonality (S) • Seasonal variation refers to periodic increases or decreases that occur within a calendar year in a time series. • The key is that these movements in the time series follow the same pattern each year. • For example, sales are always high in December. • Monthly or quarterly data. Annual data can not be used to examine seasonality. • Two types of seasonal variation • Additive seasonality • Multiplicative seasonality (usually the case)

  9. 40 – 35 – 30 – 25 – 20 – 15 – 10 – Power consumption (millions kwh) | | | | | | | | | Jan Jul Dec Jan Jul Dec Jan Jul Dec 2002 2003 2004 Illustration of Seasonal Variation

  10. 4 – 3 – 2 – 1 – Linear trend Sales of Wildcat sailboats (millions of dollars) | July 2001 | July 2002 | July 2003 | July 2004 t Illustration of Seasonal Variation (cont.) • A time series containing trend and seasonal variation.

  11. Yt Trend 2000 – 1500 – 1000 – 500 – 100 units 100 units Actual time series Units sold 100 units | Winter 2002 | Winter 2003 | Winter 2004 t Types of Seasonal Variation • Additive Seasonal Variation • The amount of variation due to seasonality does not depend on the level of yt.

  12. Yt 2000 – 1500 – 1000 – 500 – 250 units Trend 180 units Units sold Actual time series 100 units | Winter 2002 | Winter 2003 | Winter 2004 t Types of Seasonal Variation (cont.) • Multiplicative Seasonal Variation • In multiplicative seasonal variation, the seasonal fluctuation is proportional to the trend level for each observation.

  13. P1 P2 Z1 Z2 Cyclical activity V1 V2 t Cyclical Variation • Cyclical variation describes a gradual cyclical movement about the trend. • It is generally attributable to business and economic conditions. • The length of the cycle is the period of that cycle and is measured from one peak to the next.

  14. Irregular Activity (Noise) • Irregular activity consists of what is “left over” after accounting for the effect of any trend, seasonality, or cyclical activity.

  15. Combining the Components • Additive Structure • If the seasonality is additive, each yt is the sum of its four components. • yt = TRt+ St+ Ct + It • Multiplicative Structure • If the seasonality is believed to be multiplicative, then each yt is the product of its four components. • yt = TRt×St×Ct×It • The time series is usually multiplicative and multiplicative structure is assumed in KPK Macro.

  16. Decomposing Time Series Data: Four-Step Procedure (Multiplicative Model) • Determine a seasonal index, St, for each time period. • Four seasonal indexes for quarterly data • Twelve seasonal indexes for monthly data • Deseasonalize the data • Determine the trend components using deseasonalized data • Determine the cyclical components.

  17. Example 13.2: Decomposing Time Series Data – Sales Data Y(t) = S(t) * TR(t) * C(t) * I(t)

  18. Example 13.2 (cont.): Time Series Components – Sales Data

  19. Measuring Seasonality: Finding Seasonal Indexes • Find the centered moving averages • Find and center the moving totals by summing the observations for 4 (quarterly data) or 12 (monthly data) consecutive time periods. • Find and center the averages. • Divide each yt by its corresponding centered moving average. • Put the ratios into a table and find the mean for each period. Adjust these so they sum to 4 (quarterly data) or 12 (monthly data). • For each column in the table, determine the means of these ratios. These are the unadjusted seasonal indexes. • Sum the column averages. Divide 4 (quarterly data) or 12 (monthly data) by this total. Now multiply each column average by the quotient to obtain the seasonal indexes.

  20. Find the seasonal indexes for the sales data. Solution: See Excel Output Page 1 Example 13.2 (cont.): Sales Data • Using Excel: KPK Data Analysis > Time Series Analysis > Decomposition > “Specify the parameters” > OK.

  21. Deseasonalizing the Data • Once we remove the seasonality from the data, we obtain deseasonalized data. Deseasonalized data contains the trend, cyclical activity, and irregular activity. • We deseasonalize the data by dividing each data point by the corresponding seasonal index. • Example 13.3: Deseasonalize the data given in Example 13.2. • Solution: See Excel Output Page 2

  22. T(T + 1) 2 T(T + 1)(2T + 1) 6 t = = ∑t2 = 1 + 4 + ... + T2 = ∑t = 1 + 2 + ... + T = T + 1 2 ∑t T Measuring Trend • Linear Trend Alternatively, Where, A = ∑Dt and B = ∑tDt

  23. Measuring Trend (cont.) • Example 13.4: Find the trend line for the deseasonalized sales data. Using Excel: Tools > Data Analysis > Regression Analysis See Excel Output Page 2

  24. Forecasting: Extending the Trend Line • Example 13.5: Using the trend line, estimate the sales for the first quarter of Year 2004.

  25. Measuring Cyclical Activity • Usually annual data, using multiplicative model. In case of sales data, we have to use deseasonalized data to find the cyclical components. • Ratio of data to trend. Using sales data, • For a small irregular activity, we can ignore this component. • Then,

  26. Measuring Cyclical Activity (cont.) • Example 13.6: Estimate the cyclical components for Sales Data. • See Excel Output Page 3, which is reproduced below. * 3-Point Moving Averages

  27. Measuring Irregular Activity • Example 13.7: Estimate the irregular components for Sales Data. • See Excel Output Page 4, which is reproduced below.

  28. All Components Together (Sales Data)

  29. Some Exercises • Example 13.6.3: You are using a Time Series Model: (TR+S+C+I) and the estimate of the model for (TR+S+C) is 149.410 for data point 4. If the original observation at data point 4 is 150.840, what is the value of the (I) term? • Example 13.6.4: You are using a Time Series Model: (TR)(S)(C)(I) and the estimate of the model for (TR)(S)(C) is 144.37 for data point 18. If the original observation at data point 18 is 155.28, what is the value of the (I) term for this data point? • Example 13.6.5: Exercise 13.48 (Page 575 of the textbook).

  30. Index Numbers • Index Number • Measures the change in an item (such as price) across two or more time periods. • Usually a benchmark value is set at 100, and the value before and after that year is calculated as a percentage of the baseline. • Example 13.7.1: Finding index numbers from total annual profits (millions of dollars) data for XYZ Company. Base Year = 1999 Profits in 2002 were 146.48% of the profits in 1999.

  31. Price Indexes • Compare prices from one year to the base year. • The most popular price index is Consumer Price Index (CPI). • CPI combines large number of prices for consumer goods and family services into a single index. • A price index that includes more than one item is an aggregate price index. • Index is calculated for a reference year using a base year. • For example, using 2000 as base year, determine price index for 2005. 2005 is, therefore, the reference year.

  32. Calculating an Aggregate Price Index • There are two methods • Simple Aggregate Price Index = • Weighted Aggregate Price Index = • Example 3.7.2: Compute price indexes for 2000 using 1990 as the base year.

  33. Calculating an Aggregate Price Index (cont.) Problem: Does not consider the quantity of each item that is typically purchased.

  34. Calculating an Aggregate Price Index (cont.) How to select Q?

  35. Calculating an Aggregate Price Index (cont.)

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