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Unit5 Time Series Forecasting & Index Numbers.ppt

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This document provides an overview of time series forecasting techniques. It discusses the components of time series data including trend, seasonal, cyclical, and irregular fluctuations. It also describes various smoothing techniques for time series forecasting such as naive methods, averaging models including simple averages and moving averages, and exponential smoothing. Examples are provided to illustrate how to calculate forecasts using a four-month moving average and a four-month weighted moving average. Measurement methods for forecasting error are also defined.

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Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-1 Business Statistics, 4e by Ken Black Chapter 16 Time Series Forecasting & Index Numbers Discrete Distributions
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-2 Learning Objectives • Gain a general understanding of time series forecasting techniques. • Understand the four possible components of time-series data. • Understand stationary forecasting techniques. • Understand how to use regression models for trend analysis. • Learn how to decompose time-series data into their various elements and to forecast by using decomposition techniques.. • Understand the nature of autocorrelation and how to test for it. • Understand autoregression in forecasting.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-3 Time-Series Forecasting • Time-series data: data gathered on a given characteristic over a period of time at regular intervals • Time-series techniques – Attempt to account for changes over time by examining patterns, cycles, trends, or using using information about previous time periods – Naive Methods – Averaging – Smoothing – Decomposition • Forecast error: Error = Xactual - Xforecast
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-4 Bond Yields of Three-Month Treasury Bills Year Average Yield 1 14.03% 2 10.69% 3 8.63% 4 9.58% 5 7.48% 6 5.98% 7 5.82% 8 6.69% 9 8.12% 10 7.51% 11 5.42% 12 3.45% 13 3.02% 14 4.29% 15 5.51% 16 5.02% 17 5.07% 0% 2% 4% 6% 8% 10% 12% 14% 16% 0 5 10 15 20 Year Average Yield
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-5 Composite Time Series Data 1 2 3 4 5 6 7 8 9 10 11 12 13 Year
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-6 Components of Time Series Data Trend Irregular Seasonal Cyclical
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-7 Components of Time Series Data 1 2 3 4 5 6 7 8 9 10 11 12 13 Year Seasonal Cyclical Trend Irregular fluctuations
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-8 Measurement of Forecasting Error  et = Xt - Ft  Mean Absolute Deviation (MAD)  Mean Square Error (MSE)  Mean Percentage Error (MPE)  Mean Absolute Percentage Error (MAPE)  Mean Error (ME)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-9 Nonfarm Partnership Tax Returns: Actual and Forecast with  = .7 Year Actual Forecast Error 1 1402 2 1458 1402.0 56.0 3 1553 1441.2 111.8 4 1613 1519.5 93.5 5 1676 1584.9 91.1 6 1755 1648.7 106.3 7 1807 1723.1 83.9 8 1824 1781.8 42.2 9 1826 1811.3 14.7 10 1780 1821.6 -41.6 11 1759 1792.5 -33.5
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-10 Mean Absolute Deviation: Nonfarm Partnership Forecasted Data MAD i e     number of forecasts 674 5 10 67 45 . . Year Actual Forecast Error |Error| 1 1402.0 2 1458.0 1402.0 56.0 56.0 3 1553.0 1441.2 111.8 111.8 4 1613.0 1519.5 93.5 93.5 5 1676.0 1584.9 91.1 91.1 6 1755.0 1648.7 106.3 106.3 7 1807.0 1723.1 83.9 83.9 8 1824.0 1781.8 42.2 42.2 9 1826.0 1811.3 14.7 14.7 10 1780.0 1821.6 -41.6 41.6 11 1759.0 1792.5 -33.5 33.5 674.5
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-11 Mean Square Error: Nonfarm Partnership Forecasted Data MSE i e     2 55864 2 10 5586 42 number of forecasts . . Year Actual Forecast Error Error2 1 1402 2 1458 1402.0 56.0 3136.0 3 1553 1441.2 111.8 12499.2 4 1613 1519.5 93.5 8749.7 5 1676 1584.9 91.1 8292.3 6 1755 1648.7 106.3 11303.6 7 1807 1723.1 83.9 7038.5 8 1824 1781.8 42.2 1778.2 9 1826 1811.3 14.7 214.6 10 1780 1821.6 -41.6 1731.0 11 1759 1792.5 -33.5 1121.0 55864.2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-12 Mean Percentage Error: Nonfarm Partnership Forecasted Data MPE i i e X            100 318 10 318% number of forecasts . . Year Actual Forecast Error Error % 1 1402 2 1458 1402.0 56.0 3.8% 3 1553 1441.2 111.8 7.2% 4 1613 1519.5 93.5 5.8% 5 1676 1584.9 91.1 5.4% 6 1755 1648.7 106.3 6.1% 7 1807 1723.1 83.9 4.6% 8 1824 1781.8 42.2 2.3% 9 1826 1811.3 14.7 0.8% 10 1780 1821.6 -41.6 -2.3% 11 1759 1792.5 -33.5 -1.9% 31.8%
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-13 Mean Absolute Percentage Error: Nonfarm Partnership Forecasted Data MAPE i i e X              100 40 3 10 4 03% number of forecasts . . Year Actual Forecast Error|Error %| 1 1402 2 1458 1402.0 56.0 3.8% 3 1553 1441.2 111.8 7.2% 4 1613 1519.5 93.5 5.8% 5 1676 1584.9 91.1 5.4% 6 1755 1648.7 106.3 6.1% 7 1807 1723.1 83.9 4.6% 8 1824 1781.8 42.2 2.3% 9 1826 1811.3 14.7 0.8% 10 1780 1821.6 -41.6 2.3% 11 1759 1792.5 -33.5 1.9% 40.3%
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-14 Mean Error for the Nonfarm Partnership Forecasted Data ME i e     number of forecasts 524 3 10 52 43 . . Year Actual Forecast Error 1 1402.0 2 1458.0 1402.0 56.0 3 1553.0 1441.2 111.8 4 1613.0 1519.5 93.5 5 1676.0 1584.9 91.1 6 1755.0 1648.7 106.3 7 1807.0 1723.1 83.9 8 1824.0 1781.8 42.2 9 1826.0 1811.3 14.7 10 1780.0 1821.6 -41.6 11 1759.0 1792.5 -33.5 524.3
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-15 Smoothing Techniques • Naive Forecasting Models • Averaging Models – Simple Averages – Moving Averages – Weighted Moving Averages • Exponential Smoothing
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-16 Naive Forecasting Simplest of the naive forecasting models t t t t F X F X where t t      1 1 1 : the forecast for time period the value for time period - We sold 532 pairs of shoes last week, I predict we’ll sell 532 pairs this week.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-17 Simple Average Model t t t t t n F X X X X n          1 2 3  The monthly average last 12 months was 56.45, so I predict 56.45 for September. Month Year Cents per Gallon Month Year Cents per Gallon January 2 61.3 January 3 58.2 February 63.3 February 58.3 March 62.1 March 57.7 April 59.8 April 56.7 May 58.4 May 56.8 June 57.6 June 55.5 July 55.7 July 53.8 August 55.1 August 52.8 September 55.7 September October 56.7 October November 57.2 November December 58.0 December
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-18 Moving Average • Updated (recomputed) for every new time period • May be difficult to choose optimal number of periods • May not adjust for trend, cyclical, or seasonal effects t t t t t n F X X X X n          1 2 3  Update me each period.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-19 Demonstration Problem 16.1: Four-Month Moving Average May May June June F Error F Error                 1056 1345 1381 1191 4 124325 1259 124325 1575 1345 1381 1191 1259 4 1294 00 1361 1294 00 67 00 . . . . . . Months Shipments 4-Mo Moving Average Forecast Error January 1056 February 1345 March 1381 April 1191 May 1259 1243.25 15.75 June 1361 1294.00 67.00 July 1110 1298.00 -188.00 August 1334 1230.25 103.75 September 1416 1266.00 150.00 October 1282 1305.25 -23.25 November 1341 1285.50 55.50 December 1382 1343.25 38.75
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-20 Demonstration Problem 16.1: Four-Month Moving Average 1000 1100 1200 1300 1400 1500 0 2 4 6 8 10 12 Time Shipments Shipments 4-Mo Moving Average
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-21 Weighted Moving Average Forecasting Model t t t t t t t t n t n i i t t n F W X W X W X W X W                  1 1 2 2 3 3 1 
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-22 Demonstration Problem 16.2: Four- Month Weighted Moving Average                 May May June June F Error F Error                 4 1191 2 1381 1 1345 1 1056 8 124088 1259 124088 1813 4 1259 2 1191 1 1381 1 1345 8 1268 00 1361 1268 00 9300 . . . . . . Months Shipments 4-Mo Weighted Moving Average Forecast Error January 1056 February 1345 March 1381 April 1191 May 1259 1240.88 18.13 June 1361 1268.00 93.00 July 1110 1316.75 -206.75 August 1334 1201.50 132.50 September 1416 1272.00 144.00 October 1282 1350.38 -68.38 November 1341 1300.50 40.50 December 1382 1334.75 47.25
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-23 Exponential Smoothing   t t t t t t F X F F F X where           1 1 1    : the forecast for the next time period (t+1) the forecast for the present time period (t) the actual value for the present time period = a value between 0 and 1  is the exponential smoothing constant
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-24 Demonstration Problem 16.3:  = 0.2  = 0.2 Year Housing Units (1,000) F e |e| e2 1984 1750 -- -- -- -- 1985 1742 1750.0 -8.0 8.0 64.0 1986 1805 1748.4 56.6 56.6 3203.6 1987 1620 1759.72 -139.7 139.7 19521.7 1988 1488 1731.776 -243.8 243.8 59426.7 1989 1376 1683.021 -307.0 307.0 94261.8 1990 1193 1621.617 -428.6 428.6 183712.2 1991 1014 1535.893 -521.9 521.9 272372.6 1992 1200 1431.515 -231.5 231.5 53599.0 1993 1288 1385.212 -97.2 97.2 9450.1 1994 1457 1365.769 91.2 91.2 8323.0 1995 1354 1384.016 -30.0 30.0 900.9 1996 1477 1378.012 99.0 99.0 9798.5 1997 1474 1397.81 76.2 76.2 5804.9 1998 1617 1413.048 204.0 204.0 41596.4 1999 1666 1453.838 212.2 212.2 45012.6 2746.9 807048.2 MAD 183.1 MSE 53803.2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-25 Demonstration Problem 16.3:  = 0.2 1000 1200 1400 1600 1800 2000 1983 1988 1993 1998 2003 Year Housing Units (1,000) Actual Predicted
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-26 Demonstration Problem 16.3:  = 0.8  = 0.8 Year Housing Units (1,000) F e |e| e2 1984 1750 -- -- -- -- 1985 1742 1750.0 -8.0 8.0 64.0 1986 1805 1743.6 61.4 61.4 3770.0 1987 1620 1792.72 -172.7 172.7 29832.2 1988 1488 1654.544 -166.5 166.5 27736.9 1989 1376 1521.309 -145.3 145.3 21114.6 1990 1193 1405.062 -212.1 212.1 44970.2 1991 1014 1235.412 -221.4 221.4 49023.4 1992 1200 1058.282 141.7 141.7 20083.9 1993 1288 1171.656 116.3 116.3 13535.8 1994 1457 1264.731 192.3 192.3 36967.3 1995 1354 1418.546 -64.5 64.5 4166.2 1996 1477 1366.909 110.1 110.1 12120.0 1997 1474 1454.982 19.0 19.0 361.7 1998 1617 1470.196 146.8 146.8 21551.3 1999 1666 1587.639 78.4 78.4 6140.4 1856.6 291437.8 MAD 123.8 MSE 19429.2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-27 Demonstration Problem 16.3:  = 0.8 1000 1200 1400 1600 1800 2000 1983 1988 1993 1998 2003 Year Housing Units (1,000) Actual Predicted
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-28 Trend Analysis • Linear Trend • Quadratic Trend • Holt’s Two Parameter Exponential Smoothing
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-29 Average Hours Worked per Week by Canadian Manufacturing Workers Period Hours Period Hours Period Hours Period Hours 1 37.2 11 36.9 21 35.6 31 35.7 2 37.0 12 36.7 22 35.2 32 35.5 3 37.4 13 36.7 23 34.8 33 35.6 4 37.5 14 36.5 24 35.3 34 36.3 5 37.7 15 36.3 25 35.6 35 36.5 6 37.7 16 35.9 26 35.6 7 37.4 17 35.8 27 35.6 8 37.2 18 35.9 28 35.9 9 37.3 19 36.0 29 36.0 10 37.2 20 35.7 30 35.7
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-30 Excel Regression Output using Linear Trend Regression Statistics Multiple R 0.782 R Square 0.611 Adjusted R Square 0.5600 Standard Error 0.509 Observations 35 ANOVA df SS MS F Significance F Regression 1 13.4467 13.4467 51.91 .00000003 Residual 33 8.5487 0.2591 Total 34 21.9954 Coefficients Standard Error t Stat P-value Intercept 37.4161 0.17582 212.81 .0000000 Period -0.0614 0.00852 -7.20 .00000003 i ti i t Y X X where Y        0 1 37 416 0 0614    :  . . data value for period i time period i i Y X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-31 Excel Graph of Hours Worked Data with a Trend Line 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.0 0 5 10 15 20 25 30 35 Time Period Work Week
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-32 Excel Regression Output using Quadratic Trend Regression Statistics Multiple R 0.8723 R Square 0.761 Adjusted R Square 0.747 Standard Error 0.405 Observations 35 ANOVA df SS MS F Significance F Regression 2 16.7483 8.3741 51.07 1.10021E-10 Residual 32 5.2472 0.1640 Total 34 21.9954 Coefficients Standard Error t Stat P-value Intercept 38.16442 0.21766 175.34 2.61E-49 Period -0.18272 0.02788 -6.55 2.21E-07 Period2 0.00337 0.00075 4.49 8.76E-05 i ti ti i ti t t Y X X X X X where Y           0 1 2 2 2 2 38164 0183 0 003     :  . . . data value for period i time period the square of the i period i i th Y X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-33 Excel Graph of Hourly Data with Quadratic Trend Line 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.0 0 5 10 15 20 25 30 35 Period Work Week
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-34 Time Series: Decomposition Basis for analysis is the Multiplicative Model Y = T · C · S · I where: T = trend component C = cyclical component S = seasonal component I = irregular component
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-35 Household Appliance Shipment Data Year Quarter Shipments Year Quarter Shipments 1 1 4009 4 1 4595 2 4321 2 4799 3 4224 3 4417 4 3944 4 4258 2 1 4123 5 1 4245 2 4522 2 4900 3 4657 3 4585 4 4030 4 4533 3 1 4493 2 4806 3 4551 4 4485 Shipments in $1,000,000.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-36 Graph of Household Appliance Shipment Data 3900 4050 4200 4350 4500 4650 4800 4950 0 4 8 12 16 20 Quarter Shipments
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-37 Development of Four- Quarter Moving Averages QuarterShipments 4 Qtr M.T. 2 Yr M.T. 4 Qtr Centered M.A. Ratios of Actual Values to M.A. 1 1 4009 2 4321 16,498 3 4224 16,612 33,110 4139 102.06% 4 3944 16,813 33,425 4178 94.40% 2 1 4123 17,246 34,059 4257 96.84% 2 4522 17,332 34,578 4322 104.62% 3 4657 17,702 35,034 4379 106.34% 4 4030 17,986 35,688 4461 90.34% 3 1 4493 17,880 35,866 4483 100.22% 2 4806 18,335 36,215 4527 106.17% 3 4551 18,437 36,772 4597 99.01% 4 4485 18,430 36,867 4608 97.32% 4 1 4595 18,296 36,726 4591 100.09% 2 4799 18,069 36,365 4546 105.57% 3 4417 17,719 35,788 4474 98.74% 4 4258 17,820 35,539 4442 95.85% 5 1 4245 17,988 35,808 4476 94.84% 2 4900 18,263 36,251 4531 108.13% 3 4585 4 4533 S·I(100) T·C
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-38 Ratios of Actuals to Moving Averages 1 2 3 4 5 Q1 96.84% 100.22% 100.09% 94.84% Q2 104.62% 106.17% 105.57% 108.13% Q3 102.06% 106.34% 99.01% 98.74% Q4 94.40% 90.34% 97.32% 95.85%
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-39 Eliminate the Max and Min for each Qtr Eliminate the maximum and the minimum for each quarter. Average the remaining ratios for each quarter. 1 2 3 4 5 Q1 96.84% 100.22% 100.09% 94.84% Q2 104.62% 106.17% 105.57% 108.13% Q3 102.06% 106.34% 99.01% 98.74% Q4 94.40% 90.34% 97.32% 95.85% S · I
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-40 Computation of Average of Seasonal Indexes 1 2 3 4 5 Average Q1 96.84% 100.09% 98.47% Q2 106.17% 105.57% 105.87% Q3 102.06% 99.01% 100.53% Q4 94.40% 95.85% 95.13%
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-41 Final Adjustments of Seasonal Indexes Average Final Adjusted Seasonal Indexes Q1 98.47% 98.47% Q2 105.87% 105.87% Q3 100.53% 100.54% Q4 95.13% 95.13% Total 400.00% 400.00% Adjustments are unnecessary since the four averages sum to 400.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-42 Deseasonalized House Appliance Date Year Quarter Shipments (T*C*S*I) Seasonal Indexes (S) Deseasonalized Data (T*C*I) 1 1 4009 98.47% 4,071 2 4321 105.87% 4,081 3 4224 100.53% 4,202 4 3944 95.12% 4,146 2 1 4123 98.47% 4,187 2 4522 105.87% 4,271 3 4657 100.53% 4,632 4 4030 95.12% 4,237 3 1 4493 98.47% 4,563 2 4806 105.87% 4,540 3 4551 100.53% 4,527 4 4485 95.12% 4,715 4 1 4595 98.47% 4,666 2 4799 105.87% 4,533 3 4417 100.53% 4,393 4 4258 95.12% 4,476 5 1 4245 98.47% 4,311 2 4900 105.87% 4,628 3 4585 100.53% 4,561 4 4533 95.12% 4,765
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-43 Autocorrelation (Serial Correlation)  Autocorrelation occurs in data when the error terms of a regression forecasting model are correlated.  Potential Problems • Estimates of the regression coefficients no longer have the minimum variance property and may be inefficient. • The variance of the error terms may be greatly underestimated by the mean square error value. • The true standard deviation of the estimated regression coefficient may be seriously underestimated. • The confidence intervals and tests using the t and F distributions are no longer strictly applicable.  First-order autocorrelation occurs when there is correlation between the error terms of adjacent time periods.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-44 Durbin-Watson Test H Ha 0 0 0 : :       D t t where e e e t n t t n        2 2 2 1 1 : n = the number of observations If D > do not reject H (there is no significant autocorrelation). If D < , reject H (there is significant autocorrelation). If , the test is inconclusive. U 0 L 0 L U d d d d ,   D
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-45 Durbin-Watson Test for the Oil and Gas Well Drilling Example H Ha 0 0 0 : :       371 . 118 . 1036 3516 . 384 1 1 2 2 2          n t t n t e e e t t D For k = 1, n = 21, and = .05, D = 0.367 < , reject H (there is significant autocorrelation). . L L 0 d d   122 .
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-46 Overcoming the Autocorrelation Problem • Addition of Independent Variables • Transforming Variables – First-differences approach – Percentage change from period to period – Use autoregression
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-47 Autoregression Model  Y b b Y b Y t t      0 1 1 2 2  Y b b Y b Y b Y t t t        0 1 1 2 2 3 3 Autoregression Model with two lagged variables Autoregression Model with three lagged variables
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-48 Index Numbers • A ratio of a measure taken during one time frame to that same measure taken during another time frame, usually denoted as the base period • Simple Index Numbers • Unweighted Aggregate Price Indexes • Weighted Aggregate Price Index Numbers – Laspeyres Price Index – Paasche Price Index
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-49 Simple Index Numbers   i i I X X where     0 100 : the quantity, price, or cost in the base year the quantity, price, or cost in the year of interest the index number of the year of interest 0 i i X X I The motivation for using an index number is to reduce data to an easier-to- use, more convenient form.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-50 Index Numbers for Business Starts in the U. S. Year Starts Index 1985 249,770 100.0 1986 253,092 101.3 1987 233,710 93.6 1988 199,091 79.7 1989 181,645 72.7 1990 158,930 63.6 1991 155,672 62.3 1992 164,086 65.7 1993 166,154 66.5 1994 188,387 75.4 1995 168,158 67.3 1996 170,475 68.3 1997 166,740 66.8 1998 155,141 62.1 1999 151,016 60.5
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-51 Unweighted Aggregate Price Index Numbers   i i i i I P P P P I where i i       0 0 100 0 : the price of an item in the year of interest ( ) the price of an item in the base year ( ) the index number for the year of interest ( )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-52 Unweighted Aggregate Price Index for Basket of Food Items Year 1990 1995 2000 Eggs (dozen) 0.78 0.86 1.06 Milk (1/2 gallon) 1.14 1.39 1.59 Bananas (per lb) 0.36 0.46 0.49 Potatoes (per lb) 0.28 0.31 0.36 Sugar (per lb) 0.35 0.42 0.43 Total 2.91 3.44 3.93 Base 1990 100.00 118.21 135.05 1995 84.59 100.00 114.24 2000 74.05 87.53 100.00
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-53 Weighted Aggregate Price Index Numbers • Computed by multiplying quantity weights and item prices in determining the market basket worth for a given year • Also called value indexes • Laspeyres - uses base period weights • Paasche - use current period weights
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-54 Laspeyres Price Index   L i I P Q P Q    0 0 0 100 Laspeyres Price Index uses base period weights
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-55 Laspeyres Price Index: 1990 Base Year 1990 Quantity Price 1990 1995 2000 Eggs (dozen) 45 0.78 0.86 1.06 Milk (1/2 gallon) 60 1.14 1.39 1.59 Bananas (per lb) 12 0.36 0.46 0.49 Potatoes (per lb) 55 0.28 0.31 0.36 Sugar (per lb) 36 0.35 0.42 0.43 Sum of Products 135.82 159.79 184.26 Index Values 100.00 117.65 135.66
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-56 Paasche Price Index   p i i i I P Q P Q    0 100 Paasche Price Index uses current period weights
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-57 Paasche Price Index: 199 Base Year 1999 2000 Price Quantity Price Quantity Syringes (dozen) 6.70 150 6.95 135 Cotton swabs (box) 1.35 60 1.45 65 Patient record forms (pad) 5.10 8 6.25 12 Children's Tylenol (bottle) 4.50 25 4.95 30 Computer paper (box) 11.95 6 13.20 8 Thermometers 7.90 4 9.00 2 Numerator 1342.60 1379.60 Denominator 1342.60 1299.85 Index 100.00 106.14
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-58 Important Indexes • Consumer Price Index (CPI) • Producer Price Index (PPI) • Dow Jones Industrial Average (DJIA)

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Unit5 Time Series Forecasting & Index Numbers.ppt

  • 1. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-1 Business Statistics, 4e by Ken Black Chapter 16 Time Series Forecasting & Index Numbers Discrete Distributions
  • 2. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-2 Learning Objectives • Gain a general understanding of time series forecasting techniques. • Understand the four possible components of time-series data. • Understand stationary forecasting techniques. • Understand how to use regression models for trend analysis. • Learn how to decompose time-series data into their various elements and to forecast by using decomposition techniques.. • Understand the nature of autocorrelation and how to test for it. • Understand autoregression in forecasting.
  • 3. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-3 Time-Series Forecasting • Time-series data: data gathered on a given characteristic over a period of time at regular intervals • Time-series techniques – Attempt to account for changes over time by examining patterns, cycles, trends, or using using information about previous time periods – Naive Methods – Averaging – Smoothing – Decomposition • Forecast error: Error = Xactual - Xforecast
  • 4. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-4 Bond Yields of Three-Month Treasury Bills Year Average Yield 1 14.03% 2 10.69% 3 8.63% 4 9.58% 5 7.48% 6 5.98% 7 5.82% 8 6.69% 9 8.12% 10 7.51% 11 5.42% 12 3.45% 13 3.02% 14 4.29% 15 5.51% 16 5.02% 17 5.07% 0% 2% 4% 6% 8% 10% 12% 14% 16% 0 5 10 15 20 Year Average Yield
  • 5. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-5 Composite Time Series Data 1 2 3 4 5 6 7 8 9 10 11 12 13 Year
  • 6. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-6 Components of Time Series Data Trend Irregular Seasonal Cyclical
  • 7. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-7 Components of Time Series Data 1 2 3 4 5 6 7 8 9 10 11 12 13 Year Seasonal Cyclical Trend Irregular fluctuations
  • 8. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-8 Measurement of Forecasting Error  et = Xt - Ft  Mean Absolute Deviation (MAD)  Mean Square Error (MSE)  Mean Percentage Error (MPE)  Mean Absolute Percentage Error (MAPE)  Mean Error (ME)
  • 9. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-9 Nonfarm Partnership Tax Returns: Actual and Forecast with  = .7 Year Actual Forecast Error 1 1402 2 1458 1402.0 56.0 3 1553 1441.2 111.8 4 1613 1519.5 93.5 5 1676 1584.9 91.1 6 1755 1648.7 106.3 7 1807 1723.1 83.9 8 1824 1781.8 42.2 9 1826 1811.3 14.7 10 1780 1821.6 -41.6 11 1759 1792.5 -33.5
  • 10. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-10 Mean Absolute Deviation: Nonfarm Partnership Forecasted Data MAD i e     number of forecasts 674 5 10 67 45 . . Year Actual Forecast Error |Error| 1 1402.0 2 1458.0 1402.0 56.0 56.0 3 1553.0 1441.2 111.8 111.8 4 1613.0 1519.5 93.5 93.5 5 1676.0 1584.9 91.1 91.1 6 1755.0 1648.7 106.3 106.3 7 1807.0 1723.1 83.9 83.9 8 1824.0 1781.8 42.2 42.2 9 1826.0 1811.3 14.7 14.7 10 1780.0 1821.6 -41.6 41.6 11 1759.0 1792.5 -33.5 33.5 674.5
  • 11. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-11 Mean Square Error: Nonfarm Partnership Forecasted Data MSE i e     2 55864 2 10 5586 42 number of forecasts . . Year Actual Forecast Error Error2 1 1402 2 1458 1402.0 56.0 3136.0 3 1553 1441.2 111.8 12499.2 4 1613 1519.5 93.5 8749.7 5 1676 1584.9 91.1 8292.3 6 1755 1648.7 106.3 11303.6 7 1807 1723.1 83.9 7038.5 8 1824 1781.8 42.2 1778.2 9 1826 1811.3 14.7 214.6 10 1780 1821.6 -41.6 1731.0 11 1759 1792.5 -33.5 1121.0 55864.2
  • 12. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-12 Mean Percentage Error: Nonfarm Partnership Forecasted Data MPE i i e X            100 318 10 318% number of forecasts . . Year Actual Forecast Error Error % 1 1402 2 1458 1402.0 56.0 3.8% 3 1553 1441.2 111.8 7.2% 4 1613 1519.5 93.5 5.8% 5 1676 1584.9 91.1 5.4% 6 1755 1648.7 106.3 6.1% 7 1807 1723.1 83.9 4.6% 8 1824 1781.8 42.2 2.3% 9 1826 1811.3 14.7 0.8% 10 1780 1821.6 -41.6 -2.3% 11 1759 1792.5 -33.5 -1.9% 31.8%
  • 13. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-13 Mean Absolute Percentage Error: Nonfarm Partnership Forecasted Data MAPE i i e X              100 40 3 10 4 03% number of forecasts . . Year Actual Forecast Error|Error %| 1 1402 2 1458 1402.0 56.0 3.8% 3 1553 1441.2 111.8 7.2% 4 1613 1519.5 93.5 5.8% 5 1676 1584.9 91.1 5.4% 6 1755 1648.7 106.3 6.1% 7 1807 1723.1 83.9 4.6% 8 1824 1781.8 42.2 2.3% 9 1826 1811.3 14.7 0.8% 10 1780 1821.6 -41.6 2.3% 11 1759 1792.5 -33.5 1.9% 40.3%
  • 14. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-14 Mean Error for the Nonfarm Partnership Forecasted Data ME i e     number of forecasts 524 3 10 52 43 . . Year Actual Forecast Error 1 1402.0 2 1458.0 1402.0 56.0 3 1553.0 1441.2 111.8 4 1613.0 1519.5 93.5 5 1676.0 1584.9 91.1 6 1755.0 1648.7 106.3 7 1807.0 1723.1 83.9 8 1824.0 1781.8 42.2 9 1826.0 1811.3 14.7 10 1780.0 1821.6 -41.6 11 1759.0 1792.5 -33.5 524.3
  • 15. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-15 Smoothing Techniques • Naive Forecasting Models • Averaging Models – Simple Averages – Moving Averages – Weighted Moving Averages • Exponential Smoothing
  • 16. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-16 Naive Forecasting Simplest of the naive forecasting models t t t t F X F X where t t      1 1 1 : the forecast for time period the value for time period - We sold 532 pairs of shoes last week, I predict we’ll sell 532 pairs this week.
  • 17. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-17 Simple Average Model t t t t t n F X X X X n          1 2 3  The monthly average last 12 months was 56.45, so I predict 56.45 for September. Month Year Cents per Gallon Month Year Cents per Gallon January 2 61.3 January 3 58.2 February 63.3 February 58.3 March 62.1 March 57.7 April 59.8 April 56.7 May 58.4 May 56.8 June 57.6 June 55.5 July 55.7 July 53.8 August 55.1 August 52.8 September 55.7 September October 56.7 October November 57.2 November December 58.0 December
  • 18. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-18 Moving Average • Updated (recomputed) for every new time period • May be difficult to choose optimal number of periods • May not adjust for trend, cyclical, or seasonal effects t t t t t n F X X X X n          1 2 3  Update me each period.
  • 19. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-19 Demonstration Problem 16.1: Four-Month Moving Average May May June June F Error F Error                 1056 1345 1381 1191 4 124325 1259 124325 1575 1345 1381 1191 1259 4 1294 00 1361 1294 00 67 00 . . . . . . Months Shipments 4-Mo Moving Average Forecast Error January 1056 February 1345 March 1381 April 1191 May 1259 1243.25 15.75 June 1361 1294.00 67.00 July 1110 1298.00 -188.00 August 1334 1230.25 103.75 September 1416 1266.00 150.00 October 1282 1305.25 -23.25 November 1341 1285.50 55.50 December 1382 1343.25 38.75
  • 20. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-20 Demonstration Problem 16.1: Four-Month Moving Average 1000 1100 1200 1300 1400 1500 0 2 4 6 8 10 12 Time Shipments Shipments 4-Mo Moving Average
  • 21. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-21 Weighted Moving Average Forecasting Model t t t t t t t t n t n i i t t n F W X W X W X W X W                  1 1 2 2 3 3 1 
  • 22. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-22 Demonstration Problem 16.2: Four- Month Weighted Moving Average                 May May June June F Error F Error                 4 1191 2 1381 1 1345 1 1056 8 124088 1259 124088 1813 4 1259 2 1191 1 1381 1 1345 8 1268 00 1361 1268 00 9300 . . . . . . Months Shipments 4-Mo Weighted Moving Average Forecast Error January 1056 February 1345 March 1381 April 1191 May 1259 1240.88 18.13 June 1361 1268.00 93.00 July 1110 1316.75 -206.75 August 1334 1201.50 132.50 September 1416 1272.00 144.00 October 1282 1350.38 -68.38 November 1341 1300.50 40.50 December 1382 1334.75 47.25
  • 23. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-23 Exponential Smoothing   t t t t t t F X F F F X where           1 1 1    : the forecast for the next time period (t+1) the forecast for the present time period (t) the actual value for the present time period = a value between 0 and 1  is the exponential smoothing constant
  • 24. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-24 Demonstration Problem 16.3:  = 0.2  = 0.2 Year Housing Units (1,000) F e |e| e2 1984 1750 -- -- -- -- 1985 1742 1750.0 -8.0 8.0 64.0 1986 1805 1748.4 56.6 56.6 3203.6 1987 1620 1759.72 -139.7 139.7 19521.7 1988 1488 1731.776 -243.8 243.8 59426.7 1989 1376 1683.021 -307.0 307.0 94261.8 1990 1193 1621.617 -428.6 428.6 183712.2 1991 1014 1535.893 -521.9 521.9 272372.6 1992 1200 1431.515 -231.5 231.5 53599.0 1993 1288 1385.212 -97.2 97.2 9450.1 1994 1457 1365.769 91.2 91.2 8323.0 1995 1354 1384.016 -30.0 30.0 900.9 1996 1477 1378.012 99.0 99.0 9798.5 1997 1474 1397.81 76.2 76.2 5804.9 1998 1617 1413.048 204.0 204.0 41596.4 1999 1666 1453.838 212.2 212.2 45012.6 2746.9 807048.2 MAD 183.1 MSE 53803.2
  • 25. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-25 Demonstration Problem 16.3:  = 0.2 1000 1200 1400 1600 1800 2000 1983 1988 1993 1998 2003 Year Housing Units (1,000) Actual Predicted
  • 26. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-26 Demonstration Problem 16.3:  = 0.8  = 0.8 Year Housing Units (1,000) F e |e| e2 1984 1750 -- -- -- -- 1985 1742 1750.0 -8.0 8.0 64.0 1986 1805 1743.6 61.4 61.4 3770.0 1987 1620 1792.72 -172.7 172.7 29832.2 1988 1488 1654.544 -166.5 166.5 27736.9 1989 1376 1521.309 -145.3 145.3 21114.6 1990 1193 1405.062 -212.1 212.1 44970.2 1991 1014 1235.412 -221.4 221.4 49023.4 1992 1200 1058.282 141.7 141.7 20083.9 1993 1288 1171.656 116.3 116.3 13535.8 1994 1457 1264.731 192.3 192.3 36967.3 1995 1354 1418.546 -64.5 64.5 4166.2 1996 1477 1366.909 110.1 110.1 12120.0 1997 1474 1454.982 19.0 19.0 361.7 1998 1617 1470.196 146.8 146.8 21551.3 1999 1666 1587.639 78.4 78.4 6140.4 1856.6 291437.8 MAD 123.8 MSE 19429.2
  • 27. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-27 Demonstration Problem 16.3:  = 0.8 1000 1200 1400 1600 1800 2000 1983 1988 1993 1998 2003 Year Housing Units (1,000) Actual Predicted
  • 28. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-28 Trend Analysis • Linear Trend • Quadratic Trend • Holt’s Two Parameter Exponential Smoothing
  • 29. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-29 Average Hours Worked per Week by Canadian Manufacturing Workers Period Hours Period Hours Period Hours Period Hours 1 37.2 11 36.9 21 35.6 31 35.7 2 37.0 12 36.7 22 35.2 32 35.5 3 37.4 13 36.7 23 34.8 33 35.6 4 37.5 14 36.5 24 35.3 34 36.3 5 37.7 15 36.3 25 35.6 35 36.5 6 37.7 16 35.9 26 35.6 7 37.4 17 35.8 27 35.6 8 37.2 18 35.9 28 35.9 9 37.3 19 36.0 29 36.0 10 37.2 20 35.7 30 35.7
  • 30. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-30 Excel Regression Output using Linear Trend Regression Statistics Multiple R 0.782 R Square 0.611 Adjusted R Square 0.5600 Standard Error 0.509 Observations 35 ANOVA df SS MS F Significance F Regression 1 13.4467 13.4467 51.91 .00000003 Residual 33 8.5487 0.2591 Total 34 21.9954 Coefficients Standard Error t Stat P-value Intercept 37.4161 0.17582 212.81 .0000000 Period -0.0614 0.00852 -7.20 .00000003 i ti i t Y X X where Y        0 1 37 416 0 0614    :  . . data value for period i time period i i Y X
  • 31. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-31 Excel Graph of Hours Worked Data with a Trend Line 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.0 0 5 10 15 20 25 30 35 Time Period Work Week
  • 32. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-32 Excel Regression Output using Quadratic Trend Regression Statistics Multiple R 0.8723 R Square 0.761 Adjusted R Square 0.747 Standard Error 0.405 Observations 35 ANOVA df SS MS F Significance F Regression 2 16.7483 8.3741 51.07 1.10021E-10 Residual 32 5.2472 0.1640 Total 34 21.9954 Coefficients Standard Error t Stat P-value Intercept 38.16442 0.21766 175.34 2.61E-49 Period -0.18272 0.02788 -6.55 2.21E-07 Period2 0.00337 0.00075 4.49 8.76E-05 i ti ti i ti t t Y X X X X X where Y           0 1 2 2 2 2 38164 0183 0 003     :  . . . data value for period i time period the square of the i period i i th Y X
  • 33. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-33 Excel Graph of Hourly Data with Quadratic Trend Line 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.0 0 5 10 15 20 25 30 35 Period Work Week
  • 34. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-34 Time Series: Decomposition Basis for analysis is the Multiplicative Model Y = T · C · S · I where: T = trend component C = cyclical component S = seasonal component I = irregular component
  • 35. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-35 Household Appliance Shipment Data Year Quarter Shipments Year Quarter Shipments 1 1 4009 4 1 4595 2 4321 2 4799 3 4224 3 4417 4 3944 4 4258 2 1 4123 5 1 4245 2 4522 2 4900 3 4657 3 4585 4 4030 4 4533 3 1 4493 2 4806 3 4551 4 4485 Shipments in $1,000,000.
  • 36. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-36 Graph of Household Appliance Shipment Data 3900 4050 4200 4350 4500 4650 4800 4950 0 4 8 12 16 20 Quarter Shipments
  • 37. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-37 Development of Four- Quarter Moving Averages QuarterShipments 4 Qtr M.T. 2 Yr M.T. 4 Qtr Centered M.A. Ratios of Actual Values to M.A. 1 1 4009 2 4321 16,498 3 4224 16,612 33,110 4139 102.06% 4 3944 16,813 33,425 4178 94.40% 2 1 4123 17,246 34,059 4257 96.84% 2 4522 17,332 34,578 4322 104.62% 3 4657 17,702 35,034 4379 106.34% 4 4030 17,986 35,688 4461 90.34% 3 1 4493 17,880 35,866 4483 100.22% 2 4806 18,335 36,215 4527 106.17% 3 4551 18,437 36,772 4597 99.01% 4 4485 18,430 36,867 4608 97.32% 4 1 4595 18,296 36,726 4591 100.09% 2 4799 18,069 36,365 4546 105.57% 3 4417 17,719 35,788 4474 98.74% 4 4258 17,820 35,539 4442 95.85% 5 1 4245 17,988 35,808 4476 94.84% 2 4900 18,263 36,251 4531 108.13% 3 4585 4 4533 S·I(100) T·C
  • 38. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-38 Ratios of Actuals to Moving Averages 1 2 3 4 5 Q1 96.84% 100.22% 100.09% 94.84% Q2 104.62% 106.17% 105.57% 108.13% Q3 102.06% 106.34% 99.01% 98.74% Q4 94.40% 90.34% 97.32% 95.85%
  • 39. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-39 Eliminate the Max and Min for each Qtr Eliminate the maximum and the minimum for each quarter. Average the remaining ratios for each quarter. 1 2 3 4 5 Q1 96.84% 100.22% 100.09% 94.84% Q2 104.62% 106.17% 105.57% 108.13% Q3 102.06% 106.34% 99.01% 98.74% Q4 94.40% 90.34% 97.32% 95.85% S · I
  • 40. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-40 Computation of Average of Seasonal Indexes 1 2 3 4 5 Average Q1 96.84% 100.09% 98.47% Q2 106.17% 105.57% 105.87% Q3 102.06% 99.01% 100.53% Q4 94.40% 95.85% 95.13%
  • 41. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-41 Final Adjustments of Seasonal Indexes Average Final Adjusted Seasonal Indexes Q1 98.47% 98.47% Q2 105.87% 105.87% Q3 100.53% 100.54% Q4 95.13% 95.13% Total 400.00% 400.00% Adjustments are unnecessary since the four averages sum to 400.
  • 42. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-42 Deseasonalized House Appliance Date Year Quarter Shipments (T*C*S*I) Seasonal Indexes (S) Deseasonalized Data (T*C*I) 1 1 4009 98.47% 4,071 2 4321 105.87% 4,081 3 4224 100.53% 4,202 4 3944 95.12% 4,146 2 1 4123 98.47% 4,187 2 4522 105.87% 4,271 3 4657 100.53% 4,632 4 4030 95.12% 4,237 3 1 4493 98.47% 4,563 2 4806 105.87% 4,540 3 4551 100.53% 4,527 4 4485 95.12% 4,715 4 1 4595 98.47% 4,666 2 4799 105.87% 4,533 3 4417 100.53% 4,393 4 4258 95.12% 4,476 5 1 4245 98.47% 4,311 2 4900 105.87% 4,628 3 4585 100.53% 4,561 4 4533 95.12% 4,765
  • 43. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-43 Autocorrelation (Serial Correlation)  Autocorrelation occurs in data when the error terms of a regression forecasting model are correlated.  Potential Problems • Estimates of the regression coefficients no longer have the minimum variance property and may be inefficient. • The variance of the error terms may be greatly underestimated by the mean square error value. • The true standard deviation of the estimated regression coefficient may be seriously underestimated. • The confidence intervals and tests using the t and F distributions are no longer strictly applicable.  First-order autocorrelation occurs when there is correlation between the error terms of adjacent time periods.
  • 44. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-44 Durbin-Watson Test H Ha 0 0 0 : :       D t t where e e e t n t t n        2 2 2 1 1 : n = the number of observations If D > do not reject H (there is no significant autocorrelation). If D < , reject H (there is significant autocorrelation). If , the test is inconclusive. U 0 L 0 L U d d d d ,   D
  • 45. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-45 Durbin-Watson Test for the Oil and Gas Well Drilling Example H Ha 0 0 0 : :       371 . 118 . 1036 3516 . 384 1 1 2 2 2          n t t n t e e e t t D For k = 1, n = 21, and = .05, D = 0.367 < , reject H (there is significant autocorrelation). . L L 0 d d   122 .
  • 46. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-46 Overcoming the Autocorrelation Problem • Addition of Independent Variables • Transforming Variables – First-differences approach – Percentage change from period to period – Use autoregression
  • 47. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-47 Autoregression Model  Y b b Y b Y t t      0 1 1 2 2  Y b b Y b Y b Y t t t        0 1 1 2 2 3 3 Autoregression Model with two lagged variables Autoregression Model with three lagged variables
  • 48. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-48 Index Numbers • A ratio of a measure taken during one time frame to that same measure taken during another time frame, usually denoted as the base period • Simple Index Numbers • Unweighted Aggregate Price Indexes • Weighted Aggregate Price Index Numbers – Laspeyres Price Index – Paasche Price Index
  • 49. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-49 Simple Index Numbers   i i I X X where     0 100 : the quantity, price, or cost in the base year the quantity, price, or cost in the year of interest the index number of the year of interest 0 i i X X I The motivation for using an index number is to reduce data to an easier-to- use, more convenient form.
  • 50. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-50 Index Numbers for Business Starts in the U. S. Year Starts Index 1985 249,770 100.0 1986 253,092 101.3 1987 233,710 93.6 1988 199,091 79.7 1989 181,645 72.7 1990 158,930 63.6 1991 155,672 62.3 1992 164,086 65.7 1993 166,154 66.5 1994 188,387 75.4 1995 168,158 67.3 1996 170,475 68.3 1997 166,740 66.8 1998 155,141 62.1 1999 151,016 60.5
  • 51. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-51 Unweighted Aggregate Price Index Numbers   i i i i I P P P P I where i i       0 0 100 0 : the price of an item in the year of interest ( ) the price of an item in the base year ( ) the index number for the year of interest ( )
  • 52. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-52 Unweighted Aggregate Price Index for Basket of Food Items Year 1990 1995 2000 Eggs (dozen) 0.78 0.86 1.06 Milk (1/2 gallon) 1.14 1.39 1.59 Bananas (per lb) 0.36 0.46 0.49 Potatoes (per lb) 0.28 0.31 0.36 Sugar (per lb) 0.35 0.42 0.43 Total 2.91 3.44 3.93 Base 1990 100.00 118.21 135.05 1995 84.59 100.00 114.24 2000 74.05 87.53 100.00
  • 53. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-53 Weighted Aggregate Price Index Numbers • Computed by multiplying quantity weights and item prices in determining the market basket worth for a given year • Also called value indexes • Laspeyres - uses base period weights • Paasche - use current period weights
  • 54. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-54 Laspeyres Price Index   L i I P Q P Q    0 0 0 100 Laspeyres Price Index uses base period weights
  • 55. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-55 Laspeyres Price Index: 1990 Base Year 1990 Quantity Price 1990 1995 2000 Eggs (dozen) 45 0.78 0.86 1.06 Milk (1/2 gallon) 60 1.14 1.39 1.59 Bananas (per lb) 12 0.36 0.46 0.49 Potatoes (per lb) 55 0.28 0.31 0.36 Sugar (per lb) 36 0.35 0.42 0.43 Sum of Products 135.82 159.79 184.26 Index Values 100.00 117.65 135.66
  • 56. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-56 Paasche Price Index   p i i i I P Q P Q    0 100 Paasche Price Index uses current period weights
  • 57. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-57 Paasche Price Index: 199 Base Year 1999 2000 Price Quantity Price Quantity Syringes (dozen) 6.70 150 6.95 135 Cotton swabs (box) 1.35 60 1.45 65 Patient record forms (pad) 5.10 8 6.25 12 Children's Tylenol (bottle) 4.50 25 4.95 30 Computer paper (box) 11.95 6 13.20 8 Thermometers 7.90 4 9.00 2 Numerator 1342.60 1379.60 Denominator 1342.60 1299.85 Index 100.00 106.14
  • 58. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16-58 Important Indexes • Consumer Price Index (CPI) • Producer Price Index (PPI) • Dow Jones Industrial Average (DJIA)