Statistics for Business and Economics 7th Edition Chapter 16 Time-Series Analysis and Forecasting Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-1 Chapter Goals After completing this chapter, you should be able to:  Compute and interpret index numbers       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 Index Numbers  Index numbers allow relative comparisons over time  Index numbers are reported relative to a Base Period Index  Base period index = 100 by definition  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  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  Let p denote the price in the base period  Let p1 be the price in a second period  The price index for this second period is  p1  100   p0  Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-4 Index Numbers: Example  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 P2001 288 I2001 100 (100) 90 P2005 320 Base Year: P2005 320 I2005 100 (100) 100 P2005 320 I2008 P2008 384 100 (100) 120 P2005 320 Ch 16-5 Index Numbers: Interpretation  Prices in 2001 were 90% of base year prices I2005 P2005 320  100  (100) 100 P2005 320  Prices in 2005 were 100% of base year prices (by definition, since 2005 is the base year) I2008 P 384  2008 100  (100) 120 P2005 320  Prices in 2008 were 120% of base year prices P2001 288 I2001  100  (100) 90 P2005 320 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-6 Aggregate Price Indexes  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:  K    p ti  100 iK1      p0i   i1  K  p ti i = item t = time period K = total number of items = sum of the prices for the group of items at time t i1 = sum of the prices for the group of items in time period K p 0i i1 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 I2010 P  100 P 2004 2001  410 (100) 118.8 345 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 Forecasting Time Period (t + 1)  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 xˆ nh xˆ n (h 1,2,3 ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-49 Exponential Smoothing in Excel  Use Data / Data Analysis / exponential smoothing  The “damping factor” is (1 - ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-50 Forecasting with the Holt-Winters Method: Nonseasonal Series   To perform the Holt-Winters method of forecasting: Obtain estimates of level xˆ t and trend Tt as xˆ x T2 x  x1 xˆ t (1 α)(xˆ t   Tt  )  αx t (0  α  1; t 3,4, , n) Tt (1 β)Tt   β(xˆ t  xˆ t  ) (0  β  1; t 3,4,, n)   Where  and  are smoothing constants whose values are fixed between and Standing at time n , we obtain the forecasts of future values, Xn+h of the series by xˆ nh xˆ n  hTn Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-51 Forecasting with the Holt-Winters Method: Seasonal Series  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)  The recursive estimates are based on the following equations xt Ft  s (0  α  1) Tt (1 β)Tt   β(xˆ t  xˆ t  ) (0  β  1) xˆ t (1 α)(xˆ t   Tt  )  α xt Ft (1 γ )Ft  s  γ xˆ t (0  γ  1) Where xˆ t 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)  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 xˆ nh (xˆ t  hTt )Ft h s 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   Autoregressive Models Used for forecasting Takes advantage of autocorrelation    1st order - correlation between consecutive values 2nd order - correlation between values periods apart pth order autoregressive model: x t γ  φ1x t   φ2 x t     φp x t  p  εt Random Error Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-55 Autoregressive Models (continued)  Let Xt (t = 1, 2, , n) be a time series  A model to represent that series is the autoregressive model of order p: x t γ  φ1x t   φ2 x t     φp x t  p  εt  where  , 1 2, ,p are fixed parameters  t are random variables that have  mean  constant variance  and are uncorrelated with one another Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-56 Autoregressive Models (continued)  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 n SS   (x t  γ  φ1x t   φ2 x t     φp x t  p )2 t p 1 is a minimum Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-57 Forecasting from Estimated Autoregressive Models   Consider time series observations x1, x2, , xt Suppose that an autoregressive model of order p has been fitted to these data: x t γˆ  φˆ1x t   φˆ2 x t     φˆp x t  p  εt  Standing at time n, we obtain forecasts of future values of the series from xˆ t h γˆ  φˆ1xˆ t h  φˆ2 xˆ t h    φˆp xˆ t h p  (h 1,2,3,) ˆ Where for j > 0, x n j is the forecast of X t+j standing at time n and ˆ n j for j  , x is simply the observed value of X t+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 3 2 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-59 Autoregressive Model: Example Solution  Develop the 2nd order table  Use Excel to estimate a regression model Excel Output Coefficients Intercept 3.5 X Variable 0.8125 X Variable -0.9375 Year xt xt-1 2002 2003 2004 2005 2006 2007 2008 2009 3 2 -4 3 2 xt-2 3 2 xˆ t 3.5  0.8125x t   0.9375x t  Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-60 Autoregressive Model Example: Forecasting Use the second-order equation to forecast number of units for 2010: xˆ t  3.5  0.8125x t   0.9375x t  xˆ 2010  3.5  0.8125(x 2009 )  0.9375(x 2008 )  3.5  0.8125(6)  0.9375(4)  4.625 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-61 Autoregressive Modeling Steps  Choose p  Form a series of “lagged predictor” variables xt-1 , xt-2 , … ,xt-p  Run a regression model using all p variables  Test model for significance  Use model for forecasting Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-62 Chapter Summary      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    Moving averages Exponential smoothing Addressed autoregressive models for forecasting Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 16-63 ... Ch 16- 2 16. 1 Index Numbers  Index numbers allow relative comparisons over time  Index numbers are reported relative to a Base Period Index  Base period index = 100 by definition  Used for. . .Chapter Goals After completing this chapter, you should be able to:  Compute and interpret index numbers       Weighted and unweighted price index Weighted quantity index Test for. .. Prentice Hall Ch 16- 3 Single Item Price Index Consider observations over time on the price of a single item  To form a price index, one time period is chosen as a base, and the price for every period