review of time series (econ403)
TRANSCRIPT
Review of Basic Time-Series Econometrics
ECON403
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16-1
Time-Series Data
n Numerical data ordered over timen The time intervals can be annually, quarterly,
daily, hourly, etc.n The sequence of the observations is importantn Example:
Year: 2005 2006 2007 2008 2009Sales: 75.3 74.2 78.5 79.7 80.2
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16.3
Time-Series Plot
n the vertical axis measures the variable of interest
n the horizontal axis corresponds to the time periods
A time-series plot is a two-dimensional plot of time series data
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0.00
5.00
10.00
15.00
1975
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
2007
U.S. Inflation Rate
Time-Series Components
Time Series
Cyclical Component
Irregular Component
Trend Component
Seasonality Component
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Trend Component
n Long-run increase or decrease over time (overall upward or downward movement)
n Data taken over a long period of time
Sales
Time Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 16-5
Trend Component
n Trend can be upward or downwardn Trend can be linear or non-linear
Downward linear trend
Sales
Time Upward nonlinear trend
Sales
Time
(continued)
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Seasonal Component
n Short-term regular wave-like patternsn Observed within 1 yearn Often monthly or quarterly
Sales
Time (Quarterly)
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
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Year n
Year n+1
Cyclical Component
n Long-term wave-like patternsn Regularly occur but may vary in lengthn Often measured peak to peak or trough to
trough
Sales1 Cycle
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Irregular Component
n Unpredictable, random, “residual”fluctuations
n Due to random variations of n Naturen Accidents or unusual events
n “Noise” in the time series
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Time-Series Component Analysisn Used primarily for forecastingn Observed value in time series is the sum or product of
components n Additive Model
n Multiplicative model (linear in log form)
where Tt = Trend value at period tSt = Seasonality value for period tCt = Cyclical value at time tIt = Irregular (random) value for period t
ttttt ICSTX ´++=
ttttt ICSTX =
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Moving Averages:Smoothing the Time Series
n Calculate moving averages to get an overall impression of the pattern of movement over time
n This smooths out the irregular component
Moving Average: averages of a designatednumber of consecutivetime series values
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16.4
(2m+1)-Point Moving Average
n A series of arithmetic means over timen Result depends upon choice of m (the
number of data values in each average) n Examples:
n For a 5 year moving average, m = 2n For a 7 year moving average, m = 3n Etc.
n Replace each xt with
å-=
+ -++=+
=m
mjjt
*t m)n,2,m1,m(tX
12m1X !
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Moving Averages
n Example: Five-year moving average
n First average:
n Second average:
n etc.
5xxxxxx 54321*
5++++
=
5xxxxxx 65432*
6++++
=
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Example: Annual Data
…
Year Sales
1234567891011etc…
2340252732483337375040etc…
Annual Sales
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sale
s
…
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Example: Quarter Data
…
Quarter Sales
1234567891011etc…
2340252732483337375040etc…
0
20
40
60
1 2 3 4 5 6 7 8 9 10 11
Sale
s
Year
Annual Sales
…
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Calculating Moving Averages
n Each moving average is for a consecutive block of (2m+1) years
Year Sales1 232 403 254 275 326 487 338 379 3710 5011 40
Average Year
5-Year Moving Average
3 29.44 34.45 33.06 35.47 37.48 41.09 39.4… …
5322725402329.4 ++++
=
etc…
§ Let m = 2
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Annual vs. Moving Average
n The 5-year moving average smoothes the data and shows the underlying trend
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
Sale
s
Annual 5-Year Moving Average
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Centered Moving Averages
n Let the time series have period s, where s is even number n i.e., s = 4 for quarterly data and s = 12 for monthly data
n To obtain a centered s-point moving averageseries Xt
*:
n Form the s-point moving averages
n Form the centered s-point moving averages
(continued)
å+-=
++ -++==s/2
1(s/2)jjt
*.5t )
2sn,2,
2s1,
2s,
2s(txx !
)2sn,2,
2s1,
2s(t
2xxx*.5t
*.5t*
t -++=+
= +- !
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Centered Moving Averagesn Used when an even number of values is used in the moving
averagen 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
Average2.5 28.753.5 31.004.5 33.005.5 35.006.5 37.507.5 38.758.5 39.259.5 41.00
Centered Period
Centered Moving
Average3 29.884 32.005 34.006 36.257 38.138 39.009 40.13
etc…
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Calculating the Ratio-to-Moving Average
n Now estimate the seasonal impactn Divide the actual sales value by the centered
moving average for that period
*t
t
xx100
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Calculating a Seasonal Index
Quarter Sales
Centered Moving Average
Ratio-to-Moving Average
1234567891011…
2340252732483337375040…
29.8832.0034.0036.2538.1339.0040.13
etc………
83.784.494.1132.486.594.992.2etc………
83.729.8825(100)
xx100 *3
3 ==
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Calculating Seasonal Indexes
Quarter Sales
Centered Moving Average
Ratio-to-Moving Average
1234567891011…
2340252732483337375040…
29.8832.0034.0036.2538.1339.0040.13
etc………
83.784.494.1132.486.594.992.2etc………
1. Find the mean of all of the same-season values
2. Adjust so that the average over all seasons is 100
Fall
Fall
Fall
(continued)
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Interpreting Seasonal Indexes
n Suppose we get these seasonal indexes:
Season Seasonal Index
Spring 0.825
Summer 1.310
Fall 0.920
Winter 0.945
S = 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:
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Exponential Smoothing
n A weighted moving averagen Weights decline exponentiallyn Most recent observation weighted most
n Used for smoothing and short term forecasting (often one or two periods into the future)
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16.5
Exponential Smoothing Model§ Exponential smoothing model
where:= exponentially smoothed value for period t= exponentially smoothed value already
computed for period i - 1xt = observed value in period ta = weight (smoothing coefficient), 0 < a < 1
11 xx =ˆ
tx̂1-tx̂
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n),1,2,t1;(0 !=<< α
Exponential Smoothing
n The weight (smoothing coefficient) is an Subjectively chosenn Range from 0 to 1n Smaller a gives more smoothing, larger a
gives less smoothingn The weight is:
n Close to 0 for smoothing out unwanted cyclical and irregular components
n Close to 1 for forecasting
(continued)
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Exponential Smoothing Examplen Suppose we use weight a = .2
Time Period
(i)Sales
(Yi)Forecast
from prior period (Ei-1)
Exponentially Smoothed Value for this period (Ei)
12345678910etc.
23402527324833373750etc.
--23
26.426.1226.29627.43731.54931.84032.87233.697
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.
= x1since no prior information exists
1x̂
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Sales vs. Smoothed Sales
n Fluctuations have been smoothed
n 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
Sale
s
Sales Smoothed
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Forecasting Time Period (t + 1)
n The smoothed value in the current period (t) is used as the forecast value for next period (t + 1)
n At time n, we obtain the forecasts of future values, Xn+h of the series
)1,2,3(hxx nhn !==+ˆˆ
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Your turn
n Take your tourism data n Calculate Seasonal Index using 12 Month
Moving Averagen Submit your calculation and answer (12 monthly
seasonal index) before leaving
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