ken black qa ch17
TRANSCRIPT
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Business Statistics, 5th ed.
by Ken Black
Chapter 17
Time SeriesForecasting &Index Numbers
Discrete Distributions
PowerPoint presentations prepared by Lloyd Jaisingh,Morehead State University
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Learning Objectives
Gain a general understanding of time series forecastingtechniques.
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 variouselements and to forecast by using decomposition techniques..
Understand the nature of autocorrelation and how to test for it.
Understand autoregression in forecasting.
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Time-Series Forecasting
Time-series data: data gathered on a givencharacteristic over a period of time at regularintervals
Time-series techniquesAttempt to account for changes over time by
examining patterns, cycles, trends, or usinginformation about previous time periodsNaive MethodsAveragingSmoothing
Decomposition Forecast error: Error = Xactual - Xforecast
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Time Series Components
Trendlong term general direction Cycles (Cyclical effects)patterns ofhighs and lows through which data moveover time periods usually of more than ayear.
Seasonal effectsshorter cycles, whichusually occur in time periods of less thanone year.
Irregular fluctuationsrapid changes orbleeps in the data, which occur in evenshorter time frames than seasonal effects.
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Components of Time Series Data
1 2 3 4 5 6 7 8 9 10 11 12 13
Year
Seasonal
Cyclical
Trend
Irregularfluctuations
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Time Series Components
Stationary time-series - data that containno trend, cyclical, or seasonal effects.
Error of individual forecast etthe
difference between the actual value xt andthe forecast of that value Ft.
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Measurement of Forecasting Error
et = Xt - FtMean Absolute Deviation (MAD)
Mean Square Error (MSE)
Mean Percentage Error (MPE)
Mean Absolute Percentage Error (MAPE)
Mean Error (ME)
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Nonfarm
PartnershipTax
Returns:
Actual andForecast
with = .7
Year Actual Forecast Error
1 14022 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 -41.611 1759 1792.5 -33.5
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Mean Absolute Deviation: Nonfarm
Partnership Forecasted Data
MADie
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 93.55 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
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Mean Square Error: Nonfarm
Partnership Forecasted Data
MSEi
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 8749.75 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 14.7 214.610 1780 1821.6 -41.6 1731.0
11 1759 1792.5 -33.5 1121.0
55864.2
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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%
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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%
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Mean Error for the Nonfarm
Partnership Forecasted Data
MEie
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 93.55 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
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Smoothing Techniques
Naive Forecasting Models
Averaging ModelsSimple Averages
Moving AveragesWeighted Moving Averages
Exponential Smoothing
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Naive Forecasting
Simplest of the
naive forecasting
models
t t
t
t
F X
F
X
where t
t
1
11
: the forecast for time period
the value for time period -
We sold 532 pairs of shoes last
week, I predict well
sell 532 pairs this week.
It assumes that the more recent time periods
of data represent the best forecasts.
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Simple Average Model
t
t t t t n
FX 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
GallonJanuary 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
The forecast for time
period t is the average of
the values for a given
number of previous time
periods.
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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 seasonaleffects
t
t t t t n
FX X X X
n
1 2 3
Update me each period.
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Demonstration Problem 17.1:
Four-Month Moving Average
May
May
June
June
F
Error
F
Error
1056 1345 1381 1191
4124325
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 1230.25 103.75September 1416 1266.00 150.00
October 1282 1305.25 -23.25
November 1341 1285.50 55.50
December 1382 1343.25 38.75
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Demonstration Problem 17.1:
Four-Month Moving Average
1000
1100
1200
1300
1400
1500
0 2 4 6 8 10 12
Time
Shipment
s
Shipments 4-Mo Moving Average
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Weighted Moving Average
Forecasting Model
t
t t t t t t t n t n
i
i t
t nFW X W X W X W X
W
1 1 2 2 3 3
1
A moving average in which some time periods are
weighted differently than others.
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Demonstration Problem 17.2: Four-
Month Weighted Moving Average
May
May
June
June
F
Error
F
Error
4 1191 2 1381 1 1345 1 1056
8
1240881259 124088
1813
4 1259 2 1191 1 1381 1 1345
8
126800
1361 1268009300
..
.
.
..
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
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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 exponentialsmoothing constant
Used to weight data from previous time periods withexponentially decreasing importance in the forecast
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Demonstration Problem 17.3: = 0.2
= 0.2
Year
Housing Units
(1,000) F e |e| e2
1990 1193 -- -- -- --
1991 1014 1193.0 -179 179 32041
1992 1200 1157.2 42.8 42.8 1832
1993 1288 1165.8 122.2 122.2 14933
1994 1457 1190.2 266.8 266.8 71182
1995 1354 1243.6 110.4 110.4 12188
1996 1477 1265.7 211.3 211.3 44648
1997 1474 1307.9 166.1 166.1 27589
1998 1617 1341.1 275.9 275.9 76121
1999 1641 1396.3 244.7 244.7 59878
2000 1569 1445.2 123.8 123.8 15326
2001 1603 1470.0 133.0 133.0 17689
2002 1705 1496.6 208.4 208.4 43431
2003 1848 1538.3 309.7 309.7 95914
2004 1956 1600.2 355.8 355.8 126594
2005 2068 1671.4 396.6 396.6 157292
3146.5 796657
MAD 209.8
MSE 53110
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Demonstration Problem 17.3: = 0.8
= 0.8
Year
Housing Units
(1,000) F e |e| e2
1990 1193 -- -- -- --1991 1014 1193.0 -179 179 64.01992 1200 1049.8 150.2 150.2 3770.01993 1288 1170.0 118.0 118.0 29832.2
1994 1457 1264.4 192.6 192.6 27736.91995 1354 1418.5 -64.5 64.5 21114.61996 1477 1366.9 110.1 110.1 44970.21997 1474 1455.0 19.0 19.0 49023.4
1998 1617 1470.2 146.8 146.8 20083.91999 1641 1587.6 53.4 53.4 13535.82000 1569 1630.3 -61.3 61.3 36967.32001 1603 1581.3 21.7 21.7 4166.22002 1705 1598.7 106.3 106.3 12120.0
2003 1848 1683.7 164.3 164.3 361.72004 1956 1815.1 140.9 140.9 21551.32005 2068 1927.8 140.2 140.2 6140.4
1668.3 228896
MAD 111.2
MSE 15245.9
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Trend Analysis
Linear Trend
Quadratic Trend
Holts Two Parameter Exponential
Smoothing
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Average Hours Worked per Week by
Canadian Manufacturing Workers
Period Hours Period Hours Period Hours Period Hours1 37.2 11 36.9 21 35.6 31 35.72 37.0 12 36.7 22 35.2 32 35.53 37.4 13 36.7 23 34.8 33 35.64 37.5 14 36.5 24 35.3 34 36.35 37.7 15 36.3 25 35.6 35 36.56 37.7 16 35.9 26 35.67 37.4 17 35.8 27 35.68 37.2 18 35.9 28 35.9
9 37.3 19 36.0 29 36.010 37.2 20 35.7 30 35.7
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Excel Regression Output
using Linear Trend
Regression StatisticsMultiple 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 00614
:
. .
data value for period i
time period
i
i
Y
X
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Excel Graph of Hours Worked Data
with a Trend Line
34.5
35.0
35.5
36.0
36.537.0
37.5
38.0
0 5 10 15 20 25 30 35
Time Period
WorkWeek
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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
238164 0183 0 003
:
. . .
data value for period i
time period
the square of the i period
i
i
th
Y
X
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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
WorkWee
k
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MINITAB Regression Output
using Linear Trend
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MINITAB Regression Output
using Quadratic Trend
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Time Series: Decomposition
Basis for analysis is the Multiplicative Model
Y = T C S I
where:T= trend componentC= cyclical componentS = seasonal component
I= irregular component
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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 48063 4551
4 4485
Shipments in $1,000,000.
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Graph of Household Appliance Shipment
Data
3900
4050
4200
4350
4500
4650
4800
4950
0 4 8 12 16 20Quarter
Shipments
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Development
of Four-
Quarter
MovingAverages
Quarter Shipments
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
SI(100)
TC
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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%
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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
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Computation of Average
of Seasonal Indexes
1 2 3 4 5 AverageQ1 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%
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Final Adjustments of Seasonal Indexes
Average
Final Adjusted
Seasonal
IndexesQ1 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 unnecessarysince the fouraverages sum to400.
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Deseasonalized House Appliance Date
Year QuarterShipments(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% 4,6324 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
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Autocorrelation (Serial Correlation)
Autocorrelation occurs in data when the error terms of aregression forecasting model are correlated.
Potential Problems
Estimates of the regression coefficients no longer have the minimumvariance property and may be inefficient.
The variance of the error terms may be greatly underestimated by themean square error value.
The true standard deviation of the estimated regression coefficientmay 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 correlationbetween the error terms of adjacent time periods.
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Durbin-Watson Test
H
Ha
0 0
0
:
:
D
t t
where
e e
e
t
n
tt
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.
U0
L0
L U
d
d
d d
,
D
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Durbin-Watson Test for the Oil and
Gas Well Drilling Example
H
Ha
0 0
0
:
:
6897.
4394.14
9590.9
1
1
2
2
2
n
t t
n
t
e
ee ttD
.
ation).autocorreltsignificanis(thereHreject,
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Overcoming the
Autocorrelation Problem
Addition of Independent Variables
Transforming VariablesFirst-differences approach
Percentage change from period to period
Use autoregression
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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
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Index Numbers
A ratio of a measure taken during one timeframe to that same measure taken duringanother time frame, usually denoted as thebase period
Simple Index Numbers
Unweighted Aggregate Price Indexes
Weighted Aggregate Price Index Numbers
Laspeyres Price IndexPaasche Price Index
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Simple Index Numbers
i
i
IX
Xwhere
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.
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Index Numbers for Business Starts in the U. S.
Year Starts Index
1987 81463 100.0
1988 62845 77.1
1989 62449 76.7
1990 63912 78.5
1991 70605 86.7
1992
69848 85.7
1993 62399 76.6
1994 50845 62.4
1995 50516 62.0
1996 53200 65.3
1997 53819 66.1
1998 44197 54.31999 376390 46.2
2000 35219 43.2
2001 39719 48.8
2002 38155 46.8
i A
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Unweighted Aggregate
Price Index Numbers
i
i
i
i
IPP
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 ( )
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Unweighted Aggregate Price Index for
Basket of Food Items
Year
1995 2000 2005
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
Base1995 100.00 118.21 135.05
2000 84.59 100.00 114.24
2005 74.05 87.53 100.00
W i h d A
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Weighted Aggregate
Price Index Numbers
Computed by multiplying quantity weightsand 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
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Laspeyres Price Index
Li
I PQ
P Q
0
0 0
100
LaspeyresPrice Index
uses baseperiodweights
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Laspeyres Price Index: 1995 Base Year
1995
Quantity
Price
1995 2005
Eggs (dozen) 45 0.78 1.06
Milk (1/2 gallon) 60 1.14 1.59
Bananas (per lb) 12 0.36 0.49
Potatoes (per lb) 55 0.28 0.36
Sugar (per lb) 36 0.35 0.43
Sum of Products 135.82 184.26
Index Values 100.00 135.66
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Paasche Price Index
pi i
i
I P QP Q
0100
PaaschePrice Index
usescurrentperiodweights
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Paasche Price Index: 2005 Base Year
2005 2006
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
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Important Indexes
Consumer Price Index (CPI)
Producer Price Index (PPI)
Dow Jones Industrial Average (DJIA)
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