Time Series

1
Time Series Forecasting Basics Regression
Based Trend Models
Topics Components of a Time Series Measures of
Forecast Error Testing for Randomness Autocorrelat
ion Trend Fitting with Excel Tool Assessing Trend
Models Forecasting with Trend Models Implementatio
ns in StatTools
2
Trend Component

• When observations, Yt, increase or decrease
  regularly through time.

3
Seasonal Component

• When observations, Yt, exhibit a regularly
  wavelike pattern through time.

4
Cyclic Component

• When wave pattern is irregular and time between
  peaks exceeds 1 year

5
Random Component

• When Yt varies randomly

6
Forecast Error
7
Measures of Forecast Error
Usually Based on historical time series
observations Reported in Stat-Tools Software
8
Measures of Forecast Error
9
Measures of Forecast Error
10
Interpreting Measures of Forecast Error

• Good Models give small values for MAE, RMSE, and
  MAPE
• When comparing several forecast models choose
  model with lowest values of all 3
• MAPE easier to interpret

11
Building Forecast Models

• Requires identification and appropriate modeling
  of Trend and Seasonal Components
• Left over variation assumed to be random

12
Residual Analysis

• If Forecast model is successful residuals should
  be truly random
• Check with plot of residuals vs. time
• Follow-up with Runs Test and Correlogram

13
Classical Departures from Randomness Trend
14
Classical Departures from Randomness
Non-constant Variance
15
Classical Departures from Randomness Seasonality
16
Classical Departures from Randomness Meandering
Pattern
17
Runs Test for Randomness

• Choose a base value such as the mean of the
  series
• Define a run as a consecutive series of
  observations that remain on one side of this base
  level

18
Runs Test for Randomness

• If too many or too few runs in series, conclude
  series is not random
• StatTools output gives p-value for Runs test
• If p-value lt a (say 0.05) conclude series not
  random

19
Lagged Series and Autocorrelation

• Autocorrelation exists when successive series
  observations are correlated with one another
• To lag by 1 time unit, we shift the series to
  make Yt in new series Yt-1 in original series

20
Lagged Series and Autocorrelation

• Lag can equal 1, 2,.k time units
• Correlation between lagged series and original
  series measures autocorrelation

21
Correlogram

• This is a chart showing autocorrelations between
  original series and lagged series for lag 1,
  2,.k
• Available in StatTools

22
Interpreting Correlograms

• Any autocorrelation gt 2 std. errors in magnitude
  suggests non-randomness
• StatTools automatically highlights lags with
  significant positive autocorrelations

23
Interpreting Correlograms
24
Interpreting Correlograms
25
Trend Modeling Problem Scenario

• A company wants to forecast sales ( units sold)
  based on monthly time series data collected over
  the last 5 years. They believe sales have been
  growing by a constant percentage
• Is there a good trend model?

26
Exploring the Best Trend Model with Excel
27
Exploring the Best Trend Model with Excel
28
Interpretation of Exponential Model

• Rewrite equation Y 5779e0.026t in log form
  Log(Y) 0.026t log(5779)
• Coefficient of t Unit Sales grew at a constant
  rate of 2.6 per month

29
Interpretation of Linear Model

• Equation Y 342.7t 3706.4
• Coefficient of t Unit Sales grew at a constant
  rate of 343 units, on average, per month

30
Computing Forecast Error for Exponential Model

• Transform Sales to Ln(sales) and regress it (as Y
  variable) against time in months
• Let slope b and intercept a
• Compute Forecast Sales for each month Ft
  eaebt

31
Assessing Forecast Error for Exponential Model

• Compute Forecast Error for each month Et Yt
  Ft
• Compute MAE, RMSE, MAPE from the Et values
  applying formulas in Excel

32
Interpreting Forecast Error for Exponential Model

• In each of the past 60 months the exponential
  model forecast the Sales with a mean error of
  8.3
• The monthly forecast was off by 1103 units on
  average

33
Computing and Assessing Forecast Error for Linear
Model

• Compute Forecast Sales for each month Ft
  342.7t 3706.4
• Compute Forecast Error for each month Et Yt
  Ft
• Compute MAE, RMSE, MAPE from the Et values
  applying formulas in Excel

34
Interpreting Forecast Error for Linear Model

• In each of the past 60 months the Linear model
  forecast the Sales with a mean error of 10.1
• The monthly forecast was off by 1169 units on
  average

35
Forecast for Future Time Periods (Exponential
Model)

• Plug in appropriate value for t in the equation
  Ft eaebt
• E.g. for first month in 6th year, use t 61
• F61 5779e0.02661 28,226

36
Runs Test in StatTools

• Name the data set in the usual way
• Place the cursor anywhere in the spreadsheet and
  click on the Time Series Forecasting icon (4th
  from left)
• Select Runs test for Randomness from drop down
  menu

37
Runs Test in StatTools

• Select the variable of interest by clicking in
  the box next to it
• Accept the default radio button for Cut-off
  value then click O.K
• StatTools will insert new worksheet with
  Runs-Test output

38
Correlogram in StatTools

• Name the data set in the usual way
• Place the cursor anywhere in the spreadsheet and
  click on the Time Series Forecasting icon (4th
  from left)
• Select Autocorrelation from drop down menu

39
Correlogram in StatTools

• Select the variable of interest by clicking in
  the box next to it
• Accept the defaults for Number of Lags and
  Create Autocorrel Chart then click O.K
• StatTools will insert new worksheet with
  Correlogram