Statistics for Business and Economics 7 th Edition

Chapter Goals After completing this chapter, you should be able to: n Compute and interpret index numbers n n n Weighted and unweighted price index Weighted quantity index Test for randomness in a time series Identify the trend, seasonality, cyclical, and irregular components in a time series Use smoothing-based forecasting models, including moving average and exponential smoothing Apply autoregressive models and autoregressive integrated moving average models Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -2

16. 1 n n Index Numbers Index numbers allow relative comparisons over time Index numbers are reported relative to a Base Period Index n Base period index = 100 by definition n Used for an individual item or measurement Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -3

Single Item Price Index Consider observations over time on the price of a single item n To form a price index, one time period is chosen as a base, and the price for every period is expressed as a percentage of the base period price n Let p 0 denote the price in the base period n Let p 1 be the price in a second period n The price index for this second period is Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -4

Index Numbers: Example n Airplane ticket prices from 2000 to 2008: Index Year Price (base year = 2005) 2000 272 85. 0 2001 288 90. 0 2002 295 92. 2 2003 311 97. 2 2004 322 100. 6 2005 320 100. 0 2006 348 108. 8 2007 366 114. 4 2008 384 120. 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Base Year: Ch. 16 -5

Index Numbers: Interpretation n Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Prices in 2001 were 90% of base year prices Prices in 2005 were 100% of base year prices (by definition, since 2005 is the base year) Prices in 2008 were 120% of base year prices Ch. 16 -6

Aggregate Price Indexes n An aggregate index is used to measure the rate of change from a base period for a group of items Aggregate Price Indexes Unweighted aggregate price index Weighted aggregate price indexes Laspeyres Index Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -7

Unweighted Aggregate Price Index § Unweighted aggregate price index for period t for a group of K items: i = item t = time period K = total number of items = sum of the prices for the group of items at time t = sum of the prices for the group of items in time period 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -8

Unweighted Aggregate Price Index: Example Automobile Expenses: Monthly Amounts ($): Index Year Lease payment Fuel Repair Total (2007=100) 2007 260 45 40 345 100. 0 2008 280 60 40 380 110. 1 2009 305 55 45 405 117. 4 2010 310 50 50 410 118. 8 n Unweighted total expenses were 18. 8% higher in 2010 than in 2007 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -9

Weighted Aggregate Price Indexes § A weighted index weights the individual prices by some measure of the quantity sold § If the weights are based on base period quantities the index is called a Laspeyres price index § The Laspeyres price index for period t is the total cost of purchasing the quantities traded in the base period at prices in period t , expressed as a percentage of the total cost of purchasing these same quantities in the base period § The Laspeyres quantity index for period t is the total cost of the quantities traded in period t , based on the base period prices, expressed as a percentage of the total cost of the base period quantities Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -10

Laspeyres Price Index § Laspeyres price index for time period t: = quantity of item i purchased in period 0 = price of item i in time period 0 = price of item i in period t Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -11

Laspeyres Quantity Index § Laspeyres quantity index for time period t: = price of item i in period 0 = quantity of item i in time period 0 = quantity of item i in period t Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -12

16. 2 n n n The Runs Test for Randomness The runs test is used to determine whether a pattern in time series data is random A run is a sequence of one or more occurrences above or below the median Denote observations above the median with “+” signs and observations below the median with “” signs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -13

The Runs Test for Randomness (continued) n n Consider n time series observations Let R denote the number of runs in the sequence The null hypothesis is that the series is random Appendix Table 14 gives the smallest significance level for which the null hypothesis can be rejected (against the alternative of positive association between adjacent observations) as a function of R and n Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -14

The Runs Test for Randomness (continued) n If the alternative is a two-sided hypothesis on nonrandomness, n n the significance level must be doubled if it is less than 0. 5 if the significance level, , read from the table is greater than 0. 5, the appropriate significance level for the test against the twosided alternative is 2(1 - ) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -15

Counting Runs Sales Median Time --+--++++-----++++ Runs: 1 2 3 4 5 6 n

Runs Test Example n = 18 and there are R = 6 runs n Use Appendix Table 14 n n = 18 and R = 6 the null hypothesis can be rejected (against the alternative of positive association between adjacent observations) at the 0. 044 level of significance Therefore we reject that this time series is random using = 0. 05 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -17

Runs Test: Large Samples n n Given n > 20 observations Let R be the number of sequences above or below the median Consider the null hypothesis H 0: The series is random n If the alternative hypothesis is positive association between adjacent observations, the decision rule is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -18

Runs Test: Large Samples (continued) Consider the null hypothesis H 0: The series is random n If the alternative is a two-sided hypothesis of nonrandomness, the decision rule is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -19

Example: Large Sample Runs Test n A filling process over- or under-fills packages, compared to the median OOO U O UU OOOO UUU OOOO UUU O U OO UUUUU OOO U OOOO UUU OOO UU O U OO UUU OOOO UUU OOO n = 100 (53 overfilled, 47 underfilled) R = 45 runs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -20

Example: Large Sample Runs Test (continued) n n A filling process over- or under-fills packages, compared to the median n = 100 , R = 45 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -21

Example: Large Sample Runs Test (continued) H 0: Fill amounts are random H 1: Fill amounts are not random Test using = 0. 05 Rejection Region /2 = 0. 025 Since z = -1. 206 is not less than -z. 025 = -1. 96, we do not reject H 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -22

16. 3 n n 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: Year: 2005 2006 2007 2008 2009 Sales: 75. 3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 74. 2 78. 5 79. 7 80. 2 Ch. 16 -23

Time-Series Plot A time-series plot is a two-dimensional plot of time series data n the vertical axis measures the variable 15. 00 of interest U. S. Inflation Rate n the horizontal axis 5. 00 corresponds to the time 0. 00 periods Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 10. 00 Ch. 16 -24

Time-Series Components Time Series Trend Component Seasonality Component Copyright © 2010 Pearson Education, Inc.

Trend Component n Long-run increase or decrease over time (overall upward or downward movement) n Data taken over a long period of time Sales Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall d ren t d r a pw U Time Ch. 16 -26

Trend Component (continued) n n Trend can be upward or downward Trend can be linear or non-linear Sales Time Downward linear trend Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Time Upward nonlinear trend Ch. 16 -27

Seasonal Component n n n Short-term regular wave-like patterns Observed within 1 year Year n+1 Often monthly or quarterly Year n Sales Summer Winter Summer Spring Winter Fall Spring Time (Quarterly) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -28

Cyclical Component n n n 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -29

Irregular Component n n Unpredictable, random, “residual” fluctuations Due to random variations of n n n Nature Accidents or unusual events “Noise” in the time series Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -30

Time-Series Component Analysis n Used primarily forecasting Observed value in time series is the sum or product of components Additive Model n Multiplicative model (linear in log form) n n 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -31

16. 4 n n Moving Averages: 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -32

(2 m+1)-Point Moving Average n n n A series of arithmetic means over time Result depends upon choice of m (the number of data values in each average) Examples: n n For a 5 year moving average, m = 2 For a 7 year moving average, m = 3 Etc. Replace each xt with Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -33

Moving Averages n Example: Five-year moving average n First average: n Second average: n

Example: Annual Data Year 1 2 3 4 5 6 7 8 9 10

Calculating Moving Averages § Let m = 2 Year Sales Average Year 5 -Year Moving Average 1 23 3 29. 4 2 40 4 34. 4 3 25 5 33. 0 4 27 6 35. 4 5 32 7 37. 4 6 48 8 41. 0 7 33 9 39. 4 8 37 … … 9 37 10 50 11 40 etc… n Each moving average is for a consecutive block of (2 m+1) years Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -36

Annual vs. Moving Average n The 5 -year moving average smoothes the data and

Centered Moving Averages (continued) n Let the time series have period s, where s is even number n n i. e. , s = 4 for quarterly data and s = 12 for monthly data To obtain a centered s-point moving average series Xt*: n Form the s-point moving averages n Form the centered s-point moving averages Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -38

Centered Moving Averages n n Used when an even number of values is used in the moving average Average periods of 2. 5 or 3. 5 don’t match the original periods, so we average two consecutive moving averages to get centered moving averages Average Period 4 -Quarter Moving Average Centered Period Centered Moving Average 2. 5 28. 75 3 29. 88 3. 5 31. 00 4 32. 00 4. 5 33. 00 5 34. 00 5. 5 6 36. 25 6. 5 35. 00 etc… 37. 50 7 38. 13 7. 5 38. 75 8 39. 00 8. 5 39. 25 9 40. 13 9. 5 41. 00 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -39

Calculating the Ratio-to-Moving Average n n Now estimate the seasonal impact Divide the actual sales value by the centered moving average for that period Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -40

Calculating a Seasonal Index Quarter Sales Centered Moving Average 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 37 50 40 … 29. 88 32. 00 34. 00 36. 25 38. 13 39. 00 40. 13 etc… … … Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ratio-to. Moving Average 83. 7 84. 4 94. 1 132. 4 86. 5 94. 9 92. 2 etc… … … Ch. 16 -41

Calculating Seasonal Indexes (continued) Fall Quarter Sales Centered Moving Average 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 37 50 40 … 29. 88 32. 00 34. 00 36. 25 38. 13 39. 00 40. 13 etc… … … Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ratio-to. Moving Average 83. 7 84. 4 94. 1 132. 4 86. 5 94. 9 92. 2 etc… … … 1. Find the median of all of the same -season values 2. Adjust so that the average over all seasons is 100 Ch. 16 -42

Interpreting Seasonal Indexes n Suppose we get these seasonal indexes: Seasonal Index § Interpretation: Spring 0. 825 Spring sales average 82. 5% of the annual average sales Summer 1. 310 Summer sales are 31. 0% higher than the annual average sales Fall 0. 920 etc… Winter 0. 945 = 4. 000 -- four seasons, so must sum to 4 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -43

16. 5 Exponential Smoothing n n A weighted moving average n Weights decline exponentially n Most recent observation weighted most Used for smoothing and short term forecasting (often one or two periods into the future) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -44

Exponential Smoothing (continued) n The weight (smoothing coefficient) is n n Subjectively chosen Range from 0 to 1 Smaller gives more smoothing, larger gives less smoothing The weight is: n n Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 forecasting Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -45

Exponential Smoothing Model § Exponential smoothing model where: = exponentially smoothed value for period t = exponentially smoothed value already computed for period i - 1 xt = observed value in period t = weight (smoothing coefficient), 0 < < 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -46

Exponential Smoothing Example n Suppose we use weight =. 2 Time Period (i) 1 2 3 4 5 6 7 8 9 10 etc. Sales (Yi) 23 40 25 27 32 48 33 37 37 50 etc. Forecast from prior period (Ei-1) Exponentially Smoothed Value for this period (Ei) -23 26. 4 26. 12 26. 296 27. 437 31. 549 31. 840 32. 872 33. 697 etc. 23 (. 2)(40)+(. 8)(23)=26. 4 (. 2)(25)+(. 8)(26. 4)=26. 12 (. 2)(27)+(. 8)(26. 12)=26. 296 (. 2)(32)+(. 8)(26. 296)=27. 437 (. 2)(48)+(. 8)(27. 437)=31. 549 (. 2)(48)+(. 8)(31. 549)=31. 840 (. 2)(33)+(. 8)(31. 840)=32. 872 (. 2)(37)+(. 8)(32. 872)=33. 697 (. 2)(50)+(. 8)(33. 697)=36. 958 etc. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall = x 1 since no prior information exists Ch. 16 -47

Sales vs. Smoothed Sales n n Fluctuations have been smoothed NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only. 2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -48

Forecasting Time Period (t + 1) n n The smoothed value in the current period (t) is used as the forecast value for next period (t + 1) At time n, we obtain the forecasts of future values, Xn+h of the series Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -49

Exponential Smoothing in Excel n Use Data / Data Analysis / exponential smoothing n

Forecasting with the Holt-Winters Method: Nonseasonal Series n n To perform the Holt-Winters method of forecasting: Obtain estimates of level and trend Tt as Where and are smoothing constants whose values are fixed between 0 and 1 Standing at time n , we obtain the forecasts of future values, Xn+h of the series by Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -51

Forecasting with the Holt-Winters Method: Seasonal Series n n n Assume a seasonal time series of period s The Holt-Winters method of forecasting uses a set of recursive estimates from historical series These estimates utilize a level factor, , a trend factor, , and a multiplicative seasonal factor, Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -52

Forecasting with the Holt-Winters Method: Seasonal Series (continued) n The recursive estimates are based on the following equations Where is the smoothed level of the series, Tt is the smoothed trend of the series, and Ft is the smoothed seasonal adjustment for the series Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -53

Forecasting with the Holt-Winters Method: Seasonal Series (continued) n n After the initial procedures generate the level, trend, and seasonal factors from a historical series we can use the results to forecast future values h time periods ahead from the last observation Xn in the historical series The forecast equation is where the seasonal factor, Ft, is the one generated for the most recent seasonal time period Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -54

16. 6 n n Autoregressive Models Used forecasting Takes advantage of autocorrelation n 1 st order - correlation between consecutive values 2 nd order - correlation between values 2 periods apart pth order autoregressive model: Random Error Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -55

Autoregressive Models (continued) n Let Xt (t = 1, 2, . . . , n) be a time series A model to represent that series is the autoregressive model of order p: n where n n , 1 2, . . . , p are fixed parameters n t are random variables that have n mean 0 n constant variance n and are uncorrelated with one another Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -56

Autoregressive Models (continued) n The parameters of the autoregressive model are estimated through a least squares algorithm, as the values of , 1 2, . . . , p for which the sum of squares is a minimum Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -57

Forecasting from Estimated Autoregressive Models n n Consider time series observations x 1, x 2, . . . , xt Suppose that an autoregressive model of order p has been fitted to these data: Standing at time n, we obtain forecasts of future values of the series from Where for j > 0, for j 0 , is the forecast of Xt+j standing at time n and is simply the observed value of Xt+j Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -58

Autoregressive Model: Example The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last eight years. Develop the second order autoregressive model. Year 2002 2003 2004 2005 2006 2007 2008 2009 Units 4 3 2 2 4 6 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -59

Autoregressive Model: Example Solution § Develop the 2 nd order table § Use Excel to estimate a regression model Excel Output Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Year 2002 2003 2004 2005 2006 2007 2008 2009 xt 4 3 2 2 4 6 xt-1 -4 3 2 2 4 xt-2 --4 3 2 2 Ch. 16 -60

Autoregressive Model Example: Forecasting Use the second-order equation to forecast number of units for

Autoregressive Modeling Steps n n n Choose p Form a series of “lagged predictor” variables xt-1 , xt-2 , … , xt-p Run a regression model using all p variables n Test model for significance n Use model forecasting Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -62

Chapter Summary n n n Discussed weighted and unweighted index numbers Used the runs test to test for randomness in time series data Addressed components of the time-series model Addressed time series forecasting of seasonal data using a seasonal index Performed smoothing of data series n n n Moving averages Exponential smoothing Addressed autoregressive models forecasting Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16 -63