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# Chap19 time series-analysis_and_forecasting

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### Chap19 time series-analysis_and_forecasting

1. 1. Statistics for Business and Economics 6th Edition Chapter 19 Time-Series Analysis and Forecasting Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-1
2. 2. 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 Statistics for Business and integrated moving average models Economics, 6e © 2007 Pearson Chap 19-2 Education, Inc. 
3. 3. 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-3
4. 4. 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 0  Let p1 be the price in a second period  The price index for this second period is  p1   Statistics for Business and 100 p  Economics, 6e © 2007 Pearson  0  Education, Inc. Chap 19-4
5. 5. Index Numbers: Example  Airplane ticket prices from 1995 to 2003: Index Year Price (base year = 2000) 1995 272 85.0 1996 288 90.0 1997 295 92.2 1998 311 97.2 1999 322 100.6 2000 320 100.0 2001 348 108.8 2002 366 114.4 I1996 P1996 288 = 100 = (100) = 90 P2000 320 Base Year: P2000 320 I2000 = 100 = (100) = 100 P2000 320 I2003 Statistics for384 Business and 2003 120.0 Economics, 6e © 2007 Pearson Education, Inc. P2003 384 = 100 = (100) = 120 P2000 320 Chap 19-5
6. 6. Index Numbers: Interpretation P1996 288 = × 100 = (100 ) = 90 P2000 320  Prices in 1996 were 90% of base year prices P2000 320 = × 100 = (100 ) = 100 P2000 320  Prices in 2000 were 100% of base year prices (by definition, since 2000 is the base year) P 384 I2003 = 2003 × 100 = (100 ) = 120 P2000 320 Statistics for Business and  Prices in 2003 were 120% of base year prices I1996 I2000 Economics, 6e © 2007 Pearson Education, Inc. Chap 19-6
7. 7. 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Weighted aggregate price indexes Laspeyres Index Chap 19-7
8. 8. Unweighted Aggregate Price Index  Unweighted aggregate price index for period t for a group of K items:  K  p ti  ∑ = 100 iK1     ∑ p 0i   i=1  i = item t = time period K = total number of items K ∑p i=1 ti = sum of the prices for the group of items at time t 0i = sum of the prices for the group of items in time period 0 K ∑p i=1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-8
9. 9. Unweighted Aggregate Price Index: Example Automobile Expenses: Monthly Amounts (\$): Index Year Lease payment Fuel Repair Total (2001=100) 2001 260 45 40 345 100.0 2002 280 60 40 380 110.1 2003 305 55 45 405 117.4 2004 310 50 50 410 118.8 I2004 ∑P = 100 ∑P 2004 2001 410 = (100) = 118.8 345  Unweighted total expenses were 18.8% Statistics for Business and higher in Economics, 6e © 2007 Pearson2004 than in 2001 Chap 19-9 Education, Inc.
10. 10. 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 Statisticsquantities for Business and  Economics, 6e © 2007 Pearson Education, Inc. Chap 19-10
11. 11. Laspeyres Price Index  Laspeyres price index for time period t:    ∑ q0ip ti  =  100 iK1    ∑ q0ip0i   i=1  K q0i = quantity of item i purchased in period 0 p0i = price of item i in time period 0 Statistics for Business and p ti = price of item i in period t Economics, 6e © 2007 Pearson Education, Inc. Chap 19-11
12. 12. Laspeyres Quantity Index  Laspeyres quantity index for time period t:  K   ∑ qtip0i  =  100 iK1    ∑ q0ip0i   i=1  p 0i = price of item i in period 0 q0i = quantity of item i in time period 0 Statistics for Business and qti = quantity of item i in period t Economics, 6e © 2007 Pearson Education, Inc. Chap 19-12
13. 13. 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-13
14. 14. The Runs Test for Randomness (continued) 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 Statistics for Business and a function of R and n  Economics, 6e © 2007 Pearson Education, Inc. Chap 19-14
15. 15. The Runs Test for Randomness (continued)  If the alternative is a two-sided hypothesis on nonrandomness,   the significance level must be doubled if it is less than 0.5 if the significance level, α, read from the table is greater than 0.5, the appropriate significance level for the test against the twosided alternative is 2(1 - α) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-15
16. 16. Counting Runs Sales Median Time --+--++++-----++++ Runs: 1 2 3 4 5 6 Statistics for Business and n = 18 and there Economics, 6e © 2007 Pearson are R = 6 runs Chap 19-16 Education, Inc.
17. 17. Runs Test Example n = 18 and there are R = 6 runs  Use Appendix Table 14  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 α = 0.05 Statistics for Business and Economics, 6e © 2007 Pearson Chap 19-17 Education, Inc. 
18. 18. Runs Test: Large Samples   Given n > 20 observations Let R be the number of sequences above or below the median Consider the null hypothesis H0: The series is random  If the alternative hypothesis is positive association between adjacent observations, the decision rule is: n R − −1 2 Reject H0 if z= < −z α 2 n − 2n Statistics for Business and 4(n − 1) Economics, 6e © 2007 Pearson Education, Inc. Chap 19-18
19. 19. Runs Test: Large Samples (continued) Consider the null hypothesis H0: The series is random  If the alternative is a two-sided hypothesis of nonrandomness, the decision rule is: Reject H0 if n R − −1 2 z= < − z α/2 2 n − 2n 4(n − 1) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. or n R − −1 2 z= > z α/2 2 n − 2n 4(n − 1) Chap 19-19
20. 20. Example: Large Sample Runs Test  A filling process over- or under-fills packages, compared to the median OOO U OO U O UU OO UU OOOO UU O UU OOO UUU OOOO UU OO UUU O U OO UUUUU OOO U O UU OOO U OOOO UUU O UU OOO U OO UU O U OO UUU O UU OOOO UUU OOO n = 100 (53 overfilled, 47 underfilled) R = 45 runs Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-20
21. 21. Example: Large Sample Runs Test (continued)   A filling process over- or under-fills packages, compared to the median n = 100 , R = 45 Z= n 100 −1 45 − −1 −6 2 2 = = = −1.206 2 2 n − 2n 100 − 2(100) 4.975 4(n − 1) 4(100 − 1) R− Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-21
22. 22. Example: Large Sample Runs Test H0: Fill amounts are random (continued) H1: Fill amounts are not random Test using α = 0.05 Rejection Region α/2 = 0.025 Rejection Region α/2 = 0.025 − 1.96 0 1.96 Statistics for Business and is not less than -z = -1.96, Since z = -1.206 .025 Economics, 6e © 2007 Pearson reject H we do not 0 Chap 19-22 Education, Inc.
23. 23. 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: 2001 2002 2003 2004 2005 Sales: 75.3 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. 74.2 78.5 79.7 80.2 Chap 19-23
24. 24. Time-Series Plot A time-series plot is a two-dimensional plot of time series data U.S. Inflation Rate 2001 1999 1997 1995 1993 1991 1989 1987 1985 1983 1981 1975 the horizontal axis corresponds to the time periods Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.  1979 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 1977 the vertical axis measures the variable of interest Inflation Rate (%)  Year Chap 19-24
25. 25. Time-Series Components Time Series Trend Component Seasonality Component Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Cyclical Component Irregular Component Chap 19-25
26. 26. 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. d rd tren Up w a Time 19-26 Chap
27. 27. Trend Component (continued)   Trend can be upward or downward Trend can be linear or non-linear Sales Sales Time Statistics for Business and Downward linear trend Economics, 6e © 2007 Pearson Education, Inc. Time Upward nonlinear trend Chap 19-27
28. 28. 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 Statistics for Business and Economics, 6e © 2007 Pearson Time (Quarterly) Education, Inc. Chap 19-28
29. 29. 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Year Chap 19-29
30. 30. 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 19-30
31. 31. Time-Series Component Analysis    Used primarily for forecasting Observed value in time series is the sum or product of components Additive Model X t = Tt + S t + C t × It  Multiplicative model (linear in log form) X t = Tt S t C tIt where Tt = Trend value at period t St = Seasonality value for period t Statistics for Business and Ct = Cyclical value at time t Economics, 6e © 2007 Pearson It = Irregular (random) value for period t Chap 19-31 Education, Inc.
32. 32. 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 19-32
33. 33. (2m+1)-Point Moving Average    A series of arithmetic means over time Result depends upon choice of m (the number of data values in each average) Examples:     For a 5 year moving average, m = 2 For a 7 year moving average, m = 3 Etc. Replace each xt with m 1 Statistics for Business and X X* = ∑ t+ j (t = m + 1,m + 2,,n − m) t 2m + 1 j= −m Economics, 6e © 2007 Pearson Education, Inc. Chap 19-33
34. 34. Moving Averages  Example: Five-year moving average  First average: * x5 =  x1 + x 2 + x 3 + x 4 + x 5 5 Second average: x* = 6 x2 + x3 + x 4 + x5 + x6 5  etc. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-34
35. 35. Example: Annual Data 1 2 3 4 5 6 7 8 9 10 11 Statistics etc… Sales 23 40 25 27 32 48 33 37 37 50 40 for Business etc… Annual Sales 60 50 … 40 Sales Year 30 20 10 0 1 and Economics, 6e © 2007 Pearson Education, Inc. 2 3 4 5 6 7 8 9 10 11 Year Chap 19-35 …
36. 36. 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 etc… 29.4 = 23 + 40 + 25 + 27 + 32 5  Each 10 50 Statistics for Business and moving average is for a 11 40 Economics, 6e © 2007consecutive block of (2m+1) years Pearson Chap 19-36 Education, Inc.
37. 37. Annual vs. Moving Average The 5-year moving average smoothes the data and shows the underlying trend Annual vs. 5-Year Moving Average 60 50 40 Sales  30 20 10 0 1 2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. 3 4 5 6 7 8 9 10 11 Year Annual 5-Year Moving Average Chap 19-37
38. 38. Centered Moving Averages (continued)  Let the time series have period s, where s is even number   i.e., s = 4 for quarterly data and s = 12 for monthly data To obtain a centered s-point moving average series Xt*:  Form the s-point moving averages x  * t +.5 = s/2 ∑ j = − (s/2)+1 x t+ j s s s s (t = , + 1, + 2,, n − ) 2 2 2 2 Form the centered s-point moving averages x *−.5 + x *+.5 t Statistics for Business andt x = 2 Economics, 6e © 2007 Pearson * t Education, Inc. s s s (t = + 1, + 2,, n − ) 2 2 2 Chap 19-38
39. 39. Centered Moving Averages   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 9 40.13 Statistics for Business and 8.5 39.25 Economics, 6e © 2007 Pearson 9.5 41.00 Education, Inc. Chap 19-39
40. 40. Calculating the Ratio-to-Moving Average   Now estimate the seasonal impact Divide the actual sales value by the centered moving average for that period xt 100 * xt Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-40
41. 41. Calculating a Seasonal Index Quarter Sales Centered Moving Average Ratio-toMoving Average 1 23 2 40 3 25 29.88 83.7 4 27 32.00 84.4 5 32 34.00 94.1 6 48 36.25 132.4 7 33 38.13 86.5 8 37 39.00 94.9 9 37 40.13 92.2 10 50 etc… etc… Statistics for 40 Business … and 11 … Economics, 6e © 2007… Pearson … … … Education, Inc. x3 25 100 * = (100) = 83.7 x3 29.88 Chap 19-41
42. 42. Calculating Seasonal Indexes (continued) Quarter Sales Centered Moving Average Ratio-toMoving Average 1 23 2 40 Fall 3 25 29.88 83.7 4 27 32.00 84.4 5 32 34.00 94.1 6 48 36.25 132.4 Fall 7 33 38.13 86.5 8 37 39.00 94.9 9 37 40.13 92.2 10 50 etc… etc… Statistics for Business … and Fall 11 40 … Economics, 6e © 2007… Pearson … … … Education, Inc. 1. Find the median of all of the same-season values 2. Adjust so that the average over all seasons is 100 Chap 19-42
43. 43. Interpreting Seasonal Indexes  Suppose we get these seasonal indexes: Seasonal Season 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 Statistics for Business and Σ 2007 -- four seasons, so must sum to 4 Economics, 6e © = 4.000Pearson Chap 19-43 Education, Inc.
44. 44. Exponential Smoothing  A weighted moving average    Weights decline exponentially Most recent observation weighted most Used for smoothing and short term forecasting (often one or two periods into the future) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-44
45. 45. Exponential Smoothing (continued)  The weight (smoothing coefficient) is α     Subjectively chosen Range from 0 to 1 Smaller α gives more smoothing, larger α gives less smoothing The weight is:   Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 for forecasting Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-45
46. 46. Exponential Smoothing Model  Exponential smoothing model ˆ x1 = x1 ˆ ˆ x t = α x t −1 + (1− α )x t where: (0 < α < 1; t = 1,2,, n) ˆ x t = exponentially smoothed value for period t ˆ x t -1 = exponentially smoothed value already computed for period i - 1 xt = observed Statistics for Business and value in period t Economics, 6e © α = weight (smoothing coefficient), 0 < α < 1 2007 Pearson Education, Inc. Chap 19-46
47. 47. Exponential Smoothing Example  ˆ ˆ Suppose we use weight α = .2 x t = 0.2 x t −1 + (1− 0.2)x t Time Period (i) Sales (Yi) Forecast from prior period (Ei-1) Exponentially Smoothed Value for this period (Ei) 1 23 -23 2 40 23 (.2)(40)+(.8)(23)=26.4 3 25 26.4 (.2)(25)+(.8)(26.4)=26.12 4 27 26.12 (.2)(27)+(.8)(26.12)=26.296 5 32 26.296 (.2)(32)+(.8)(26.296)=27.437 6 48 27.437 (.2)(48)+(.8)(27.437)=31.549 7 33 31.549 (.2)(48)+(.8)(31.549)=31.840 8 37 31.840 (.2)(33)+(.8)(31.840)=32.872 9 37 32.872 (.2)(37)+(.8)(32.872)=33.697 Statistics for Business and 10 50 33.697 (.2)(50)+(.8)(33.697)=36.958 Economics, 6e © 2007 Pearson etc. etc. etc. etc. Education, Inc. ˆ x1 = x1 since no prior information exists Chap 19-47
48. 48. Sales vs. Smoothed Sales  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 60 50 40 Sales  30 20 10 0 1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. 2 3 4 5 6 7 Time Period Sales 8 9 10 Smoothed Chap 19-48
49. 49. 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 ˆ ˆ x n +h = x n (h = 1,2,3 ) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-49
50. 50. Exponential Smoothing in Excel  Use tools / data analysis / exponential smoothing  The “damping factor” is (1 - α) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-50
51. 51. Forecasting with the Holt-Winters Method: Nonseasonal Series   To perform the Holt-Winters method of forecasting: ˆ Obtain estimates of level x t and trend T t as ˆ x1 = x 2 T2 = x 2 − x1 ˆ ˆ x t = α(x t −1 + Tt −1 ) + (1− α)x t (0 < α < 1; t = 3,4,, n) ˆ ˆ Tt = βTt −1 + (1− β)(x t − x t −1 ) (0 < β < 1; t = 3,4,, n) Where α and β 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 Statistics for Business and ˆ x n +h = ˆ Economics, 6e © 2007 Pearson x n + hTn Chap 19-51 Education, Inc. 
52. 52. 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, α, a trend factor, β, and a multiplicative seasonal factor, γ Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-52
53. 53. Forecasting with the Holt-Winters Method: Seasonal Series (continued)  The recursive estimates are based on the following equations xt Ft −s (0 < α < 1) ˆ ˆ Tt = βTt −1 + (1− β)(x t − x t −1 ) (0 < β < 1) ˆ ˆ x t = α(x t −1 + Tt −1 ) + (1− α) xt Ft = γFt −s + (1− γ ) ˆ xt (0 < γ < 1) ˆ Where x t is the smoothed level of the series, Tt is the smoothed trend Statistics for Businessthe smoothed seasonal adjustment for the series of the series, and Ft is and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-53
54. 54. 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 ˆ ˆ xn+h = (x t + hTt )Ft +h−s where the seasonal Statistics for Business and factor, Ft, is the one generated for the most recent Economics, 6e © 2007 Pearsonseasonal time period Chap 19-54 Education, Inc.
55. 55. Autoregressive Models   Used for forecasting Takes advantage of autocorrelation    1st order - correlation between consecutive values 2nd order - correlation between values 2 periods apart pth order autoregressive model: x t = γ + φ1x t −1 + φ2 x t −2 +  + φp x t −p + εt Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Random Error Chap 19-55
56. 56. 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 −1 + φ2 x t −2 +  + φp x t −p + εt  where  γ, φ1 φ2, . . .,φp are fixed parameters  εt are random variables that have  mean 0 Statistics for Business and constant variance Economics, 6e © 2007 Pearson one another and are uncorrelated with Education, Inc.   Chap 19-56
57. 57. Autoregressive Models (continued)  The parameters of the autoregressive model are estimated through a least squares algorithm, as the values of γ, φ1 φ2, . . .,φp for which the sum of squares SS = n (x t − γ − φ1x t −1 − φ2 x t −2 −  − φp x t −p )2 ∑ t =p +1 is a minimum Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-57
58. 58. 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: ˆ ˆ ˆ ˆ x t = γ + φ1x t −1 + φ2 x t −2 +  + φp x t −p + εt  Standing at time n, we obtain forecasts of future values of the series from ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ x t +h = γ + φ1x t +h−1 + φ2 x t +h−2 +  + φp x t +h−p (h = 1,2,3,) ˆ n+ j WhereBusinessxand is the forecast of Xt+j standing at time n and Statistics for for j > 0,  ˆ for j 0 x + j is Pearson Economics,≤6e, © n2007 simply the observed value of Xt+j Education, Inc. Chap 19-58
59. 59. 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 Units 1999 2000 2001 2002 2003 2004 2005 2006 Business 4 3 2 3 2 2 4 6 and Statistics for Economics, 6e © 2007 Pearson Education, Inc. Chap 19-59
60. 60. Autoregressive Model: Example Solution  Develop the 2nd order table  Use Excel to estimate a regression model Excel Output Coefficients Intercept 3.5 X Variable 1 0.8125 X Variable 2 -0.9375 Year xt xt-1 99 00 01 02 03 04 05 06 4 3 2 3 2 2 4 6 -4 3 2 3 2 2 4 Statistics 3.5 Business and 0.9375x ˆ x t = for + 0.8125x t −1 − t −2 Economics, 6e © 2007 Pearson Education, Inc. xt-2 --4 3 2 3 2 2 Chap 19-60
61. 61. Autoregressive Model Example: Forecasting Use the second-order equation to forecast number of units for 2007: ˆ x t = 3.5 + 0.8125x t −1 − 0.9375x t −2 ˆ x 2007 = 3.5 + 0.8125(x 2006 ) − 0.9375(x 2005 ) = 3.5 + 0.8125(6) − 0.9375(4) = 4.625 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-61
62. 62. 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 StatisticsUse model for forecasting for Business and Economics, 6e © 2007 Pearson Education, Inc.  Chap 19-62
63. 63. 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 Statistics for Business and Economics, 6e © 2007 Pearson Chap 19-63 Education, Inc. 