chap19 time series-analysis_and_forecasting
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Chap 19-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 19
Time-Series Analysis and Forecasting
Statistics for Business and Economics
6th Edition
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-2
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
integrated moving average models
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-3
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 19-4
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 p0 denote the price in the base period
Let p1 be the price in a second period The price index for this second period is
0
1
p
p100
Single Item Price Index
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-5
Index Numbers: Example
Airplane ticket prices from 1995 to 2003:
90320
288(100)
P
P100I
2000
19961996
Year PriceIndex
(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
2003 384 120.0
100320
320(100)
P
P100I
2000
20002000
120320
384(100)
P
P100I
2000
20032003
Base Year:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-6
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 prices
Index Numbers: Interpretation
90)100(320
288100
P
PI
2000
19961996
100)100(320
320100
P
PI
2000
20002000
120)100(320
384100
P
PI
2000
20032003
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-7
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
Weighted aggregate
price indexes
Laspeyres Index
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-8
Unweighted Aggregate Price Index
Unweighted aggregate price index for period t for a group of K items:
K
1i0i
K
1iti
p
p100
= sum of the prices for the group of items at time t
= sum of the prices for the group of items in time period 0
i = item
t = time period
K = total number of items
K
1i0i
K
1iti
p
p
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-9
Unweighted total expenses were 18.8% higher in 2004 than in 2001
Automobile Expenses:Monthly Amounts ($):
Year Lease payment Fuel Repair TotalIndex
(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
Unweighted Aggregate Price Index: Example
118.8345
410(100)
P
P100I
2001
20042004
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-10
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 quantities
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-11
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 t
Laspeyres price index for time period t:
0iq
K
1i0i0i
K
1iti0i
pq
pq100
ti
0i
p
p
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-12
Laspeyres Quantity Index
K
1i0i0i
K
1i0iti
pq
pq100
= price of item i in period 0
= quantity of item i in time period 0
= quantity of item i in period t
Laspeyres quantity index for time period t:
ti
0i
q
q
0ip
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-13
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 19-14
The Runs Test for Randomness
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 a function of R and n
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-15
The Runs Test for Randomness
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, , read from the table is greater than 0.5, the appropriate significance level for the test against the two-sided alternative is 2(1 - )
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-16
Counting Runs
Sales
Time
- - + - - + + + + - - - - - + + + +
Runs: 1 2 3 4 5 6
n = 18 and there are R = 6 runs
Median
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-17
Runs Test Example
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 = 0.05
n = 18 and there are R = 6 runs
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-18
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:
α20 z
1)4(n2nn
12n
Rz if H Reject
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-19
Runs Test: Large Samples
Consider the null hypothesis H0: The series is random
If the alternative is a two-sided hypothesis of nonrandomness, the decision rule is:
α/22α/220 z
1)4(n2nn
12n
Rzorz
1)4(n2nn
12n
Rz if H Reject
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-20
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 19-21
Example: Large Sample Runs Test
A filling process over- or under-fills packages, compared to the median
n = 100 , R = 45
1.2064.975
6
1)4(1002(100)100
12
10045
1)4(n2nn
12n
RZ
22
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-22
1.960
Rejection Region /2 = 0.025
Since z = -1.206 is not less than -z.025 = -1.96, we do not reject H0
1.96
Rejection Region /2 = 0.025
H0: Fill amounts are random
H1: Fill amounts are not random
Test using = 0.05
Example: Large Sample Runs Test
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-23
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 74.2 78.5 79.7 80.2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-24
Time-Series Plot
the vertical axis measures the variable of interest
the horizontal axis corresponds to the time periods
U.S. Inflation Rate
0.002.004.006.008.00
10.0012.0014.0016.00
197
5
197
7
197
9
198
1
198
3
198
5
198
7
198
9
199
1
199
3
199
5
199
7
199
9
200
1
Year
Infl
ati
on
Rat
e (
%)
A time-series plot is a two-dimensional plot of time series data
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-25
Time-Series Components
Time Series
Cyclical Component
Irregular Component
Trend Component
Seasonality Component
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-26
Upward trend
Trend Component
Long-run increase or decrease over time (overall upward or downward movement)
Data taken over a long period of time
Sales
Time
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-27
Downward linear trend
Trend Component
Trend can be upward or downward Trend can be linear or non-linear
Sales
Time Upward nonlinear trend
Sales
Time
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-28
Seasonal Component
Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly
Sales
Time (Quarterly)
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-29
Cyclical Component
Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to
trough
Sales1 Cycle
Year
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-30
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 19-31
Time-Series Component Analysis
Used primarily for forecasting Observed value in time series is the sum or product of
components Additive Model
Multiplicative model (linear in log form)
where Tt = Trend value at period t
St = Seasonality value for period t
Ct = Cyclical value at time t
It = Irregular (random) value for period t
ttttt ICSTX
ttttt ICSTX
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-32
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 designatednumber of consecutivetime series values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-33
(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
mjjt
*t m)n,2,m1,m(tX
12m
1X
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-34
Moving Averages
Example: Five-year moving average
First average:
Second average:
etc.
5
xxxxxx 54321*
5
5
xxxxxx 65432*
6
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-35
Example: Annual Data
Year Sales
1
2
3
4
5
6
7
8
9
10
11
etc…
23
40
25
27
32
48
33
37
37
50
40
etc…
Annual Sales
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sa
les
…
…
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-36
Calculating Moving Averages
Each moving average is for a consecutive block of (2m+1) years
Year Sales
1 23
2 40
3 25
4 27
5 32
6 48
7 33
8 37
9 37
10 50
11 40
Average Year
5-Year Moving Average
3 29.4
4 34.4
5 33.0
6 35.4
7 37.4
8 41.0
9 39.4
… …
5
322725402329.4
etc…
Let m = 2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-37
Annual vs. 5-Year Moving Average
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sal
es
Annual 5-Year Moving Average
Annual vs. Moving Average
The 5-year moving average smoothes the data and shows the underlying trend
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-38
Centered Moving Averages
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
Form the centered s-point moving averages
(continued)
s/2
1(s/2)jjt
*.5t )
2
sn,2,
2
s1,
2
s,
2
s(txx
)2
sn,2,
2
s1,
2
s(t
2
xxx
*.5t
*.5t*
t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-39
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
2.5 28.75
3.5 31.00
4.5 33.00
5.5 35.00
6.5 37.50
7.5 38.75
8.5 39.25
9.5 41.00
Centered Period
Centered Moving
Average
3 29.88
4 32.00
5 34.00
6 36.25
7 38.13
8 39.00
9 40.13
etc…
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-40
Calculating the Ratio-to-Moving Average
Now estimate the seasonal impact Divide the actual sales value by the centered
moving average for that period
*t
t
x
x100
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-41
Calculating a Seasonal Index
Quarter Sales
Centered Moving Average
Ratio-to-Moving Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
83.729.88
25(100)
x
x100
*3
3
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-42
Calculating Seasonal Indexes
Quarter Sales
Centered Moving Average
Ratio-to-Moving Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
1. Find the median of all of the same-season values
2. Adjust so that the average over all seasons is 100
Fall
Fall
Fall
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-43
Interpreting Seasonal Indexes
Suppose we get these seasonal indexes:
SeasonSeasonal
Index
Spring 0.825
Summer 1.310
Fall 0.920
Winter 0.945
= 4.000 -- four seasons, so must sum to 4
Spring sales average 82.5% of the annual average sales
Summer sales are 31.0% higher than the annual average sales
etc…
Interpretation:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-44
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 19-45
Exponential Smoothing
The weight (smoothing coefficient) is Subjectively chosen Range from 0 to 1 Smaller gives more smoothing, larger
gives less smoothing The weight is:
Close to 0 for smoothing out unwanted cyclical and irregular components
Close to 1 for forecasting
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-46
Exponential Smoothing Model
Exponential smoothing model
where: = exponentially smoothed value for period t = exponentially smoothed value already
computed for period i - 1 xt = observed value in period t = weight (smoothing coefficient), 0 < < 1
11 xx ˆ
n),1,2,t1;(0)x(1xx t1tt ααα ˆˆ
tx̂
1-tx̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-47
Exponential Smoothing Example Suppose we use weight = .2
Time Period
(i)
Sales(Yi)
Forecast from prior
period (Ei-1)
Exponentially Smoothed Value for this period (Ei)
1
2
3
4
5
6
7
8
9
10
etc.
23
40
25
27
32
48
33
37
37
50
etc.
--
23
26.4
26.12
26.296
27.437
31.549
31.840
32.872
33.697
etc.
23
(.2)(40)+(.8)(23)=26.4
(.2)(25)+(.8)(26.4)=26.12
(.2)(27)+(.8)(26.12)=26.296
(.2)(32)+(.8)(26.296)=27.437
(.2)(48)+(.8)(27.437)=31.549
(.2)(48)+(.8)(31.549)=31.840
(.2)(33)+(.8)(31.840)=32.872
(.2)(37)+(.8)(32.872)=33.697
(.2)(50)+(.8)(33.697)=36.958
etc.
= x1 since no prior information exists
1x̂
t1tt 0.2)x(1x0.2x ˆˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-48
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
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10Time Period
Sa
les
Sales Smoothed
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-49
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
)1,2,3(hxx nhn ˆˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-50
Exponential Smoothing in Excel
Use tools / data analysis /
exponential smoothing
The “damping factor” is (1 - )
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-51
To perform the Holt-Winters method of forecasting: Obtain estimates of level and trend Tt as
Where and 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
Forecasting with the Holt-Winters Method: Nonseasonal Series
tx̂
12221 xxTxx ˆ
n),3,4,t1;α(0α)x(1)Txα(x t1t1tt ˆˆ
nnhn hTxx ˆˆ
n),3,4,t1;β(0)xxβ)((1βTT 1tt1tt ˆˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-52
Assume a seasonal time series of period s
The Holt-Winters method of forecasting uses a set of recursive estimates from historical series
These estimates utilize a level factor, , a trend factor, , and a multiplicative seasonal factor,
Forecasting with the Holt-Winters Method: Seasonal Series
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-53
The recursive estimates are based on the following equations
Forecasting with the Holt-Winters Method: Seasonal Series
1)α(0F
xα)(1)Txα(x
st
t1t1tt
ˆˆ
1)(0x
x)(1FF
t
tstt γγγ
ˆ
1)β(0)xxβ)((1βTT 1tt1tt ˆˆ
Where is the smoothed level of the series, Tt is the smoothed trend of the series, and Ft is the smoothed seasonal adjustment for the series
tx̂
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-54
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
shttthn )FhTx(x ˆˆ
where the seasonal factor, Ft, is the one generated for the most recent seasonal time period
Forecasting with the Holt-Winters Method: Seasonal Series
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-55
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:
Random Error
tptp2t21t1t xxxx εφφφγ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-56
Autoregressive Models
Let Xt (t = 1, 2, . . ., n) be a time series A model to represent that series is the autoregressive
model of order p:
where , 1 2, . . .,p are fixed parameters
t are random variables that have mean 0 constant variance and are uncorrelated with one another
tptp2t21t1t xxxx εφφφγ
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-57
Autoregressive Models
The parameters of the autoregressive model are estimated through a least squares algorithm, as the values of , 1 2, . . .,p for which the sum of squares
is a minimum
2ptp2t21t1
n
1ptt )xxx(xSS
φφφγ
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-58
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:
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
tptp2t21t1t xxxx εφφφγ ˆˆˆˆ
)1,2,3,(hxxxx phtp2ht21ht1ht ˆˆˆˆˆˆˆˆ φφφγ
jnx ˆ
jnx ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-59
Autoregressive Model: Example
Year Units
1999 4 2000 3 2001 2 2002 3 2003 2 2004 2 2005 4 2006 6
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.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-60
Autoregressive Model: Example Solution
Year 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 2
CoefficientsIntercept 3.5X Variable 1 0.8125X Variable 2 -0.9375
Excel Output
Develop the 2nd order table
Use Excel to estimate a regression model
2t1tt 0.9375x0.8125x3.5x ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-61
Autoregressive Model Example: Forecasting
Use the second-order equation to forecast number of units for 2007:
4.625
0.9375(4)0.8125(6)3.5
)0.9375(x)0.8125(x3.5x
0.9375x0.8125x3.5x
200520062007
2t1tt
ˆ
ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-62
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
Use model for forecasting
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-63
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