Review of Time series (ECON403)

transcript

Review of Basic Time-Series Econometrics

ECON403

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

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

U.S. Inflation Rate

Time-Series Components

Time Series

Cyclical Component

Irregular Component

Trend Component

Seasonality Component

Trend Component

n Long-run increase or decrease over time (overall upward or downward movement)

n Data taken over a long period of time

Trend Component

n Trend can be upward or downwardn Trend can be linear or non-linear

Downward linear trend

Time Upward nonlinear trend

(continued)

Seasonal Component

n Short-term regular wave-like patternsn Observed within 1 yearn Often monthly or quarterly

Time (Quarterly)

Winter

Spring

Summer

Winter

Spring

Summer

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

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

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 =

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

(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

+ -++=+

*t m)n,2,m1,m(tX

12m1X !

Moving Averages

n Example: Five-year moving average

n First average:

n Second average:

n etc.

5xxxxxx 54321*

5xxxxxx 65432*

Example: Annual Data

Year Sales

1234567891011etc…

2340252732483337375040etc…

Annual Sales

1 2 3 4 5 6 7 8 9 10 11

Example: Quarter Data

Quarter Sales

1234567891011etc…

2340252732483337375040etc…

1 2 3 4 5 6 7 8 9 10 11

Annual Sales

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

Annual vs. Moving Average

n The 5-year moving average smoothes the data and shows the underlying trend

Annual vs. 5-Year Moving Average

1 2 3 4 5 6 7 8 9 10 11

Annual 5-Year Moving Average

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)

++ -++==s/2

1(s/2)jjt

*.5t )

2sn,2,

2s(txx !

)2sn,2,

2xxx*.5t

t -++=+

= +- !

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…

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

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

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

(continued)

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:

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)

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

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)

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.

26.426.1226.29627.43731.54931.84032.87233.697

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

= x1since no prior information exists

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

1 2 3 4 5 6 7 8 9 10Time Period

Sales Smoothed

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

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

Review of Time series (ECON403)

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