Download `Simple` Univariate Time Series Methods

‘Simple’Univariate
TimeSeriesMethods
Dr. Geoffrey Ducanes
UP School of Economics
July 12 – Morning Sessions
‘Simple’ Univariate Time Series Methods
•  Moving Average Methods
•  Exponential Smoothing Methods
•  Trend Forecasting Methods
Autoregressive Distributive Lag Models
‘Simple’ Univariate Time Series
Methods
•  Simple: used to make quick forecasts (e.g. daily,
weekly, or monthly production, sales, revenues,
etc.)
•  ‘Easily’ implementable in a Worksheet Program,
such as Excel
•  Uses only the historical values of the variable to be
forecasted
•  Already allows analysts to distinguish between the
basic underlying pattern in data (signal) and
random fluctuations (noise)
‘Simple’ Univariate Time Series
Methods
•  Method to be used depends on
characteristics of data, whether
•  Stationary (use Moving Average or Single
Exponential Smoothing)
•  With Linear Trend (use Double Moving Average
or Double Exponential Smoothing)
•  With Seasonality and Trend (use Holt-Winters)
Moving Average
•  Arithmetic mean of the n most recent
observations
•  Equal weights assigned to each of the n data
points
•  Analyst chooses number of periods (n) in a
moving average
•  The smaller the number of observations in
computing the moving average, the more weight
is given to more recent periods.
•  The greater the number of observations in
computing the moving average, the less weight is
given to more recent periods.
•  Best suited to stationary data.
Moving Average
•  ​𝑦 ↓𝑡+1 =(​​𝑦↓𝑡 +​𝑦↓𝑡−1 +​𝑦↓𝑡−2 +…+​𝑦↓𝑡
−𝑛+1 /𝑛 )
•  where
•  ​𝑦 ↓𝑡+1 = forecast value for next
period
•  ​𝑦↓𝑡 = actual value at time period t
•  n = number of terms in the
moving average
Example
•  Open file ‘Forecasting Simple
Smoothing.xls’
Double Moving Average
•  Used when the time series data have a
linear trend.
Double Moving Average
•  1stMA=​𝑀𝐴↓𝑡 =(​​𝑦↓𝑡 +​𝑦↓𝑡−1 +​𝑦↓𝑡−2 +…+​𝑦↓𝑡−𝑛+1 /
𝑛 )
•  2ndMA=​𝑀𝐴↓𝑡 ′=(​​𝑀𝐴↓𝑡 +​𝑀𝐴↓𝑡−1 +​𝑀𝐴↓𝑡−2 +…+​
𝑀𝐴↓𝑡−𝑛+1 /𝑛 )
•  ​𝑎↓𝑡 =​2​𝑀𝐴↓𝑡 −𝑀𝐴↓𝑡 ’
•  ​𝑏↓𝑡 =​2/𝑛−1 ​(​𝑀𝐴↓𝑡 −𝑀𝐴↓𝑡 ’)
•  ​𝑦 ↓𝑡+𝑥 =​𝑎↓𝑡 +​𝑏↓𝑡 x
•  where
–  ​𝑦 ↓𝑡+𝑥 =forecastvalue
–  x=numberofperiodsaheadtobeforecast
Example
•  Open file ‘Forecasting Simple
Smoothing.xls’
Exponential Smoothing
•  Depends on three pieces of data
–  Most recent actual
–  Most recent forecast
–  Smoothing constant (𝛼)
•  If time series appears to evolve quite
smoothly, give greater weight to more
recent values
•  If time series quite erratic, less weight to
most recent actual values is desirable
Exponen8alSmoothing
•  ​𝑦 ↓𝑡+1 =​𝛼​𝑦↓𝑡 +𝛼(1−𝛼)​𝑦↓𝑡−1 +𝛼(1−𝛼)​2𝑦↓𝑡−2 + 𝛼 (1−𝛼)​3𝑦↓𝑡−3
…↓ •  where
–  ​𝑦 ↓𝑡+1 =newsmoothedvalueortheforecastvalueofthenextperiod
–  𝛼=smoothingconstant
–  ​𝑦↓𝑡 =actualvalueofseriesinperiodt
• 
Notethatthisisatrueweightedaveragesince
–  𝛼+𝛼(1−𝛼) +𝛼(1−𝛼)2+𝛼 (1−𝛼)3 +…=1
•  ​𝑦 ↓𝑡+1 =​𝛼​𝑦↓𝑡 +(1−𝛼) ​𝑦 ↓𝑡 ↓ •  Newes>mate=𝛼(Newdata)+(1−𝛼)(Previouses>mate)
Example
•  Open file ‘Data Set Annualb.xls’ and import into
Eviews
•  Double click on the variable ‘temp_ann’ so that
the spreadsheet for the variable appears
•  [Verify that ‘temp_ann’ is stationary]
•  Select Proc on the button bar for that variable
and then Exponential Smoothing followed by
Simple Exponential Smoothing
•  Select Single and specify the estimation sample
period as 1989 – 2010
•  Give a name to the forecasted variable: e.g.
temp_asm1
Double Exponential Smoothing
•  Used for forecasting data that have a
linear trend
•  Requires less data than DMA and
computationally more efficient.
Double Exponential Smoothing
•  ​𝑦 ↓𝑡+𝑥 =​𝑎↓𝑡 +​𝑏↓𝑡 𝑥↓ •  where
–  ​𝑦 ↓𝑡+𝑥 = forecast value x periods into the future
–  𝑎
​ ↓𝑡 = the difference between the simple (A’) and the double (A’’)
smoothed values
–  ​𝑏↓𝑡 = an adjustment factor similar to a slope in a time series
–  x = number of periods ahead to be forecast
• 
• 
• 
• 
​𝐴↓𝑡↑′ =𝛼​𝑦↓𝑡 +(1−𝛼)​𝐴↓𝑡−1↑′ ​𝐴↓𝑡↑′′ =𝛼​𝐴↓𝑡↑′ +(1−𝛼)​𝐴↓𝑡−1↑′′ ​𝑎↓𝑡 = 2​𝐴↓𝑡↑′ −​𝐴↓𝑡↑′′ ​𝑏↓𝑡 =​𝛼/1−𝛼 ​(𝐴↓𝑡↑′ −​𝐴↓𝑡↑′′ )
Example
•  Again, use the file ‘Data Set Annualb’
•  Double click on the variable ‘tot_elec_cons’ so
that the spreadsheet for the variable appears
•  [Verify that ‘tot_elec_cons’ has an approximately
linear trend]
•  Select Proc on the button bar for that variable
and then Exponential Smoothing followed by
Simple Exponential Smoothing
•  Select Double and specify the estimation sample
period as 1989 – 2010
•  Give a name to the forecasted variable: e.g.
tot_elsm2
Holt-Winters’ Seasonal Exponential
Smoothing
•  Allows for both trend and seasonal
patterns of the data to be taken into
account as the smoothing process is
applied.
•  Seasonality can be specified as either
additive or multiplicative
Example
•  Open file ‘gdp sectoral quarterly.xls’ and import into Eviews
•  Double click on the variable ‘utilities’ so that the
spreadsheet for the variable appears
•  [Verify that ‘utilities’ is seasonal apart from having a trend]
•  Select Proc on the button bar for that variable and then
Exponential Smoothing followed by Simple Exponential
Smoothing
•  Select Holt-Winters-Additive (or Holt-Winters-Multiplicative)
and specify the estimation sample period as 1998q1 –
2014q4
•  Give a name to the forecasted variable: e.g. utilsm3
Trend Forecasting
•  Modeling and forecasting a variable as a
function of time or some transformation of
time
o t
o ln(t)
o exp(t)
o t and t^2
•  Choose model that leads to least
systematic error
Example
•  Use the file ‘Data Set Annualb’
•  Model 1
o  On the command window, type:
smpl 1989 2009
ls gdp_val c @trend
o  On the equation dialog box, click on forecast. Check
forecast graph, forecast evaluation, and Insert actual
for out-of-sample observations and for forecast sample
type 2010 2020
o  Copy output and paste into Word
o  Click on View, then click on Actual, Fitted, Residual, then
Actual, Fitted, Residual Graph
o  Copy output and paste into Word
Example
•  Model 2
o  On the command window, type:
smpl 1989 2010
ls gdp_val c @trend @trend^2
o  On the equation dialog box, click on forecast. Check
forecast graph, forecast evaluation, and Insert actual
for out-of-sample observations and for forecast sample
type 2010 2020
o  Copy output and paste into Word
o  Click on View, then click on Actual, Fitted, Residual, then
Actual, Fitted, Residual Graph
o  Copy output and paste into Word
•  Compare results for Model 1 and Model 2
Exercise
1.  Using observations only up to 2010, use Single Exponential
Smoothing and Double Exponential Smoothing to forecast
Commercial Electricity Consumption from 2011 to 2020.
Which yields the more accurate forecast? (Data to be
used: ‘Data Set Annualb’.)
2.  Using observations only up to 2011q4, use Holt-WintersMultiplicative to forecast Manufacturing Value Added from
2012q1 to 2020q4. (Data to be used: ‘gdp sectoral
quarterly’).
3.  Using observations only from 1990 to 2010, use trend
forecasting to forecast Total Electricity Consumption from
2011 to 2020. [Note that this requires choosing which the
most appropriate trend function to use.] (Data to be used:
‘Data Set Annualb’.)