lectures 11 and 12

Chap 19-*Prof. Miruna Mazurencu Marinescu, Ph.D, MBALectures 10 and 11

Index Numbers, Time-Series Analysis and ForecastingStatistics for Business and Economics

Prof. Miruna Mazurencu Marinescu, Ph.D, MBA

Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Chapter GoalsAfter completing this chapter, you should be able to: Compute and interpret index numbersWeighted and unweighted price indexWeighted quantity indexTest for randomness in a time seriesIdentify the trend, seasonality, cyclical, and irregular components in a time seriesUse smoothing-based forecasting models, including moving average and exponential smoothingApply autoregressive models and autoregressive integrated moving average models


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Index NumbersIndex numbers allow relative comparisons over timeIndex numbers are reported relative to a Base Period IndexBase period index = 100 by definitionUsed for an individual item or measurement


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Consider observations over time on the price of a single itemTo 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 priceLet p0 denote the price in the base periodLet p1 be the price in a second periodThe price index for this second period isSingle Item Price Index


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Index Numbers: ExampleAirplane ticket prices from 1995 to 2003:Base Year:

YearPriceIndex (base year = 2000)199527285.0199628890.0199729592.2199831197.21999322100.62000320100.02001348108.82002366114.42003384120.0


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Prices in 1996 were 90% of base year prices

Prices in 2000 were 100% of base year prices (by definition, since 2000 is the base year)

Prices in 2003 were 120% of base year pricesIndex Numbers: Interpretation


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Aggregate Price IndexesAn aggregate index is used to measure the rate of change from a base period for a group of itemsAggregate Price IndexesUnweighted aggregate price indexWeighted aggregate price indexesLaspeyres Index


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Unweighted Aggregate Price IndexUnweighted aggregate price index for period t for a group of K items:= sum of the prices for the group of items at time t= sum of the prices for the group of items in time period 0i = itemt = time periodK = total number of items


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Unweighted total expenses were 18.8% higher in 2004 than in 2001Unweighted Aggregate Price Index: Example

Automobile Expenses:Monthly Amounts ($):YearLease paymentFuelRepairTotalIndex (2001=100)20012604540345100.020022806040380110.120033055545405117.420043105050410118.8


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Weighted Aggregate Price IndexesA weighted index weights the individual prices by some measure of the quantity soldIf the weights are based on base period quantities the index is called a Laspeyres price indexThe 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 periodThe 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


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Laspeyres Price Index= quantity of item i purchased in period 0

= price of item i in time period 0= price of item i in period tLaspeyres price index for time period t:


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Laspeyres Quantity Index= price of item i in period 0

= quantity of item i in time period 0= quantity of item i in period tLaspeyres quantity index for time period t:


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Time-Series DataNumerical data ordered over timeThe time intervals can be annually, quarterly, daily, hourly, etc.The sequence of the observations is importantExample:Year:2001 2002 2003 2004 2005Sales: 75.3 74.2 78.5 79.7 80.2


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Time-Series Plotthe vertical axis measures the variable of interest

the horizontal axis corresponds to the time periodsA time-series plot is a two-dimensional plot of time series data


Chart1

9.1277890467

5.7620817844

6.5026362039

7.5907590759

11.3496932515

13.4986225895

10.3155339806

6.1606160616

3.2124352332

4.3172690763

3.5611164581

1.8587360595

3.6496350365

4.1373239437

4.8182586644

5.4032258065

4.208110176

3.0102790015

2.9935851746

2.5605536332

2.8340080972

2.9527559055

2.2944550669

1.5576323988

2.2085889571

3.3613445378

2.8455284553

1.581027668

Inflation Rate

Year

Inflation Rate (%)

U.S. Inflation Rate

Sheet1

YearInflation RateAnnual CPI-U

197449.3

19759.1353.8

19765.7656.9

19776.5060.6

19787.5965.2

197911.3572.6

198013.5082.4

198110.3290.9

19826.1696.5

19833.2199.6

19844.32103.9

19853.56107.6

19861.86109.6

19873.65113.6

19884.14118.3

19894.82124

19905.40130.7

19914.21136.2

19923.01140.3

19932.99144.5

19942.56148.2

19952.83152.4

19962.95156.9

19972.29160.5

19981.56163

19992.21166.6

20003.36172.2

20012.85177.1

20021.58179.9

Sheet1

Inflation Rate

U.S. Inflation Rate

Sheet2

Sheet3

Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Time-Series ComponentsTime SeriesCyclical ComponentIrregular ComponentTrend ComponentSeasonality Component


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Upward trendTrend ComponentLong-run increase or decrease over time (overall upward or downward movement)Data taken over a long period of timeSalesTime


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Downward linear trendTrend ComponentTrend can be upward or downwardTrend can be linear or non-linearSalesTime Upward nonlinear trendSalesTime (continued)


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Seasonal ComponentShort-term regular wave-like patternsObserved within 1 yearOften monthly or quarterlySalesTime (Quarterly) WinterSpringSummerFallWinterSpringSummerFall


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Cyclical ComponentLong-term wave-like patternsRegularly occur but may vary in lengthOften measured peak to peak or trough to troughSales1 CycleYear


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Irregular ComponentUnpredictable, random, residual fluctuationsDue to random variations of NatureAccidents or unusual eventsNoise in the time series


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Time-Series Component AnalysisUsed primarily for forecastingObserved value in time series is the sum or product of components Additive Model

Multiplicative model (linear in log form)whereTt = Trend value at period tSt = Seasonality value for period tCt = Cyclical value at time t It = Irregular (random) value for period t


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Smoothing the Time SeriesCalculate moving averages to get an overall impression of the pattern of movement over timeThis smooths out the irregular component

Moving Average: averages of a designatednumber of consecutivetime series values


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*(2m+1)-Point Moving AverageA series of arithmetic means over timeResult depends upon choice of m (the number of data values in each average) Examples: For a 5 year moving average, m = 2For a 7 year moving average, m = 3Etc.Replace each xt with


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Moving AveragesExample: Five-year moving average

First average:

Second average:

etc.


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Example: Annual Data

YearSales1234567891011 etc2340252732483337375040 etc


Chart4

23

40

25

27

32

48

33

37

37

50

40

Annual

Year

Sales

Annual Sales

Sheet1

YearSales

123

240

32529.4

42734.4

53233

64835.4

73337.4

83741

93739.4

105041

1140

Sheet1

00

00

00

00

00

00

00

00

00

00

00

Annual

5-Year Moving Average

Year

Sales

Annual vs. 5-Year Moving Average

Sheet2

Sheet3

Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Calculating Moving AveragesEach moving average is for a consecutive block of (2m+1) yearsetcLet m = 2

YearSales12324032542753264873383793710501140

Average Year5-Year Moving Average329.4434.4533.0635.4737.4841.0939.4


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Annual vs. Moving Average The 5-year moving average smoothes the data and shows the underlying trend


Chart2

23

40

2529.4

2734.4

3233

4835.4

3337.4

3741

3739.4

50

40

Annual


Year

Sales


Chart1

23

40

2529.4

2734.4

3233

4835.4

3337.4

3741

3739.4

5041

40

Annual


Year

Sales


Sheet1

YearSales

123

240

32529.4

42734.4

53233

64835.4

73337.4

83741

93739.4

1050

1140

Sheet1

Annual


Year

Sales


Sheet2

Sheet3

Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Centered Moving AveragesLet the time series have period s, where s is even number i.e., s = 4 for quarterly data and s = 12 for monthly dataTo obtain a centered s-point moving average series Xt*:Form the s-point moving averages

Form the centered s-point moving averages(continued)


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Centered Moving AveragesUsed when an even number of values is used in the moving averageAverage periods of 2.5 or 3.5 dont match the original periods, so we average two consecutive moving averages to get centered moving averagesetc

Average Period4-Quarter Moving Average2.528.753.531.004.533.005.535.006.537.507.538.758.539.259.541.00

Centered PeriodCentered Moving Average329.88432.00534.00636.25738.13839.00940.13


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Calculating the Ratio-to-Moving AverageNow estimate the seasonal impactDivide the actual sales value by the centered moving average for that period


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Calculating a Seasonal Index

QuarterSalesCentered Moving AverageRatio-to-Moving Average1234567891011234025273248333737504029.8832.0034.0036.2538.1339.0040.13 etc 83.784.494.1132.486.594.992.2 etc


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Calculating Seasonal IndexesFind the median of all of the same-season values Adjust so that the average over all seasons is 100FallFallFall(continued)

QuarterSalesCentered Moving AverageRatio-to-Moving Average1234567891011234025273248333737504029.8832.0034.0036.2538.1339.0040.13 etc 83.784.494.1132.486.594.992.2 etc


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Interpreting Seasonal IndexesSuppose we get these seasonal indexes: = 4.000 -- four seasons, so must sum to 4Spring sales average 82.5% of the annual average salesSummer sales are 31.0% higher than the annual average salesetcInterpretation:

SeasonSeasonal IndexSpring0.825Summer1.310Fall0.920Winter0.945


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*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


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Autoregressive ModelsUsed for forecastingTakes advantage of autocorrelation1st order - correlation between consecutive values2nd order - correlation between values 2 periods apartpth order autoregressive model:Random Error


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Autoregressive ModelsLet Xt (t = 1, 2, . . ., n) be a time seriesA model to represent that series is the autoregressive model of order p:

where , 1 2, . . .,p are fixed parameterst are random variables that have mean 0constant variance and are uncorrelated with one another(continued)


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Autoregressive ModelsThe parameters of the autoregressive model are estimated through a least squares algorithm, as the values of , 1 2, . . .,p for which the sum of squares

is a minimum(continued)


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Forecasting from Estimated Autoregressive ModelsConsider time series observations x1, x2, . . . , xt Suppose that an autoregressive model of order p has been fitted to these data:

Standing at time n, we obtain forecasts of future values of the series from

Where for j > 0, is the forecast of Xt+j standing at time n and for j 0 , is simply the observed value of Xt+j


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Autoregressive Model: Example Year Units 1999 4 2000 3 2001 2 2002 3 2003 2 2004 2 2005 4 2006 6The 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.


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Autoregressive Model: Example SolutionYear xt xt-1 xt-2 99 4 -- -- 00 3 4 -- 01 2 3 4 02 3 2 3 03 2 3 2 04 2 2 3 05 4 2 2 06 6 4 2Excel Output Develop the 2nd order table Use Excel to estimate a regression model


Sheet6

SUMMARY OUTPUT

Regression Statistics

Multiple R0.667413042

R Square0.4454401687

Adjusted R Square0.0757336145

Standard Error1.7985917684

Observations6

ANOVA

dfSSMSFSignificance F

Regression27.7952029523.8976014761.20484790870.4129739079

Residual39.7047970483.2349323493

Total517.5

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept3.41697416974.87663076060.70068338930.5339558989-12.102655935718.9366042752-12.102655935718.9366042752

X Variable 1-1.32656826572.9081572319-0.45615424470.6792756863-10.58163118567.9284946543-10.58163118567.9284946543

X Variable 20.21586715870.34115383990.63275605740.5718314052-0.86983763711.3015719544-0.86983763711.3015719544

RESIDUAL OUTPUT

ObservationPredicted YResidualsStandard Residuals

11.6273062731-1.6273062731-1.1680490269

22.1808118081-1.1808118081-0.8475639198

31.62730627310.37269372690.2675123622

41.62730627311.37269372690.9852930567

54.7084870849-0.7084870849-0.5085383519

63.22878228781.77121771221.2713458797

Sheet6

0

0

0

0

0

0

X Variable 1

Residuals

X Variable 1 Residual Plot

Sheet7

0

0

0

0

0

0

X Variable 2

Residuals


Sheet8

00

10

20

30

40

50

Y

Predicted Y

X Variable 1

Y

X Variable 1 Line Fit Plot

Sheet10

00

10

20

30

40

50

Y

Predicted Y

X Variable 2

Y


Sheet1

SUMMARY OUTPUT



R Square0.3289313574



Observations6

ANOVA


Regression10.11392745240.11392745241.96064209550.2340371757

Residual40.23242886130.0581072153

Total50.3463563137


Intercept0.33583794830.17446235351.92498806530.1265429144-0.14854820240.8202240991-0.14854820240.8202240991

X Variable 10.08068543940.05762301891.40022930110.2340371757-0.07930204060.2406729195-0.07930204060.2406729195

RESIDUAL OUTPUTantilog(.33583795) =2.1668954035

antilog(.08068544) =1.2041634459


10.3358379483-0.0348079527-0.1614427158

20.41652338780.28244661661.3100152506

30.4972088272-0.1961788316-0.9098967597

40.5778942667-0.276864271-1.2841237817

50.65857970610.18651833390.8650904192

60.73926514560.03888610480.1803575874

Sheet1

0

0

0

0

0

0

X Variable 1

Residuals


Sheet9

0.30102999570

0.69897000430

0.30102999570

0.30102999570

0.845098040

0.77815125040

Y

Predicted Y

X Variable 1

Y


Sheet11

SUMMARY OUTPUT



R Square0.0014239943

Adjusted R Square-0.3314346743


Observations5

ANOVA


Regression10.03018867920.03018867920.00427807490.9519646302

Residual321.16981132087.0566037736

Total421.2


Intercept4.26415094342.39273255911.78212601620.172748462-3.350599092311.8789009791-3.350599092311.8789009791

X Variable 10.03773584910.5769390510.0654069940.9519646302-1.79834342461.8738151227-1.79834342461.8738151227

RESIDUAL OUTPUT


14.33962264150.66037735850.2870540488

24.4528301887-2.4528301887-1.0662007525

34.3396226415-2.3396226415-1.016991487

44.33962264152.66037735851.1564177393

54.52830188681.47169811320.6397204515

Sheet11

0

0

0

0

0

X Variable 1

Residuals


Sheet12

50

20

20

70

60

Y

Predicted Y

X Variable 1

Y


Sheet2

Sheet3

YearSales1stCodedSalesSales^2log(sales)

199420240.3010299957

19955215250.6989700043

1996252240.3010299957

1997223240.3010299957

19987247490.84509804

19996756360.7781512504

0

Sheet3

0

0

0

0

0

0

Sales

SUMMARY OUTPUT



R Square0.1581699346



Observations7

ANOVA


Regression12.03361344542.03361344540.93944099380.3769372438

Residual510.82352941182.1647058824

Total612.8571428571


Intercept1.29411764711.98680770190.65135525990.5435635839-3.81312579696.401361091-3.81312579696.401361091

X Variable 10.64705882350.6675887510.96924764320.3769372438-1.06902988932.3631475364-1.06902988932.3631475364

RESIDUAL OUTPUT


13.8823529412-0.8823529412-0.6569518078

23.2352941176-1.2352941176-0.9197325309

32.58823529410.41176470590.3065775103

43.2352941176-1.2352941176-0.9197325309

52.5882352941-0.5882352941-0.4379678719

62.58823529411.41176470591.0511228925

73.88235294122.11764705881.5766843387

0

0

0

0

0

0

0

X Variable 1

Residuals


30

20

30

20

20

40

60

Y

Predicted Y

X Variable 1

Y


SUMMARY OUTPUT



R Square0.4788961039



Observations6

ANOVA


Regression26.14583333333.07291666671.37850467290.3761720123

Residual36.68752.2291666667

Total512.8333333333


Intercept3.53.5014877790.99957510090.3911779502-7.64330729914.643307299-7.64330729914.643307299

X Variable 10.81250.8346344010.97348012380.4021141357-1.84368165753.4686816575-1.84368165753.4686816575

X Variable 2-0.93750.834634401-1.12324629670.3431108215-3.59368165751.7186816575-3.59368165751.7186816575

RESIDUAL OUTPUT


12.1875-0.1875-0.1621266379

22.31250.68750.5944643391

34.0625-2.0625-1.7833930173

42.3125-0.3125-0.2702110632

53.250.750.6485065518

64.8751.1250.9727598276

0

0

0

0

0

0

X Variable 1

Residuals


0

0

0

0

0

0

X Variable 2

Residuals


22.1875

32.3125

24.0625

22.3125

43.25

64.875

Y

Predicted Y

X Variable 1

Y


22.1875

32.3125

24.0625

22.3125

43.25

64.875

Y

Predicted Y

X Variable 2

Y


SUMMARY OUTPUT



R Square0.4196428571



Observations5

ANOVA


Regression34.71.56666666670.2410256410.8655519358

Residual16.56.5

Total411.2


Intercept213.20984481360.15140223280.9043408499-165.8462736521169.8462736521-165.8462736521169.8462736521

X Variable 1120.50.7048327648-24.412300601626.4123006016-24.412300601626.4123006016

X Variable 2-0.52.9580398915-0.16903085090.8933992419-38.085299457837.0852994578-38.085299457837.0852994578

X Variable 301.732050807601-22.007697889622.0076978896-22.007697889622.0076978896

RESIDUAL OUTPUT


12.50.50.3922322703

24-2-1.5689290811

32.5-0.5-0.3922322703

4310.7844645406

5510.7844645406

0

0

0

0

0

X Variable 1

Residuals


0

0

0

0

0

X Variable 2

Residuals


0

0

0

0

0

X Variable 3

Residuals


30

20

20

40

60

Y

Predicted Y

X Variable 1

Y


30

20

20

40

60

Y

Predicted Y

X Variable 2

Y


30

20

20

40

60

Y

Predicted Y

X Variable 3

Y


14

234

3234

33234

42323

52232

64223

76422

Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Autoregressive Model Example: ForecastingUse the second-order equation to forecast number of units for 2007:


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Autoregressive Modeling StepsChoose p Form a series of lagged predictor variables xt-1 , xt-2 , ,xt-pRun a regression model using all p variablesTest model for significanceUse model for forecasting


Prof. Miruna Mazurencu Marinescu, Ph.D, MBAChap 19-*Chapter SummaryDiscussed weighted and unweighted index numbersAddressed components of the time-series modelAddressed time series forecasting of seasonal data using a seasonal indexPerformed smoothing of data seriesMoving averagesAddressed autoregressive models for forecasting


lectures 11 and 12

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