Introduction

• Forecasts produced using exponential smoothing methods are weighted averages of past observations, with the weights decaying exponentially as the observations get older. The more recent the observation the higher the weight. This generates reliable forecasts quickly and for a wide range of time series.
• Content
  • Different exponential smoothing methods
    • These methods are mostly selected based on the key components of the Time Series like trend, seasonality etc. These models generate point forecasts.
    • Simple Exponential Smoothing, Holt's linear & damped trend, Holt Winter method are the example of the methods
  • Statistical models that underlie exponential smoothing methods
    • These models generate point forecasts as well as prediction intervals. This statistical framework allows for model selection between competing models.

Simple Exponential Smoothing

• Suitable for forecasting data with no clear trend or seasonality.

  • This method can be used to after removing trend and seasonality from the data.

• Comparison with other methods

  • With naïve method, all forecasts for the future are equal to the last observed value of the series. It assumes that the most recent observation is the only important one, and all previous observations provide no information for the future. This is equivalent to weighted average where all of the weight is given to the last observation.
  • With average method, all future forecasts are equal to a simple average of the observed data. It assumes that all observations are of equal importance, and gives them equal weights.
  • We need something in between these two extremes. It may be sensible to attach larger weights to more recent observations than to observations from the distant past.

• Simple Exponential Smoothing

  • Forecasts are calculated using weighted averages, where the weights decrease exponentially as observations come from further in the past — the smallest weights are associated with the oldest observations.

    

  • For any α (between 0 and 1), the weights decrease exponentially with time, hence the name “exponential smoothing.” For small α (close to 0), more weight is given to observations from the more distant past. For large α (i.e., close to 1), more weight is given to the more recent observations. For the extreme case where α=1, forecasts are equal to the naïve forecasts.

  • Two equivalent forms of simple exponential smoothing (which leads to equation 8.1 above):

    • Weighted Average Form

      

      ℓ0 is the first fitted value at time t1 (which we need to estimate).

      • Is the level of the time series?

    • Component Form

      8-ets.pdf

      • Just a note. For better understanding, please refer to the above PDF. It says, Simple Exponential Smoothing is modelling the level of a time series. So, we can write

        

        • Level of a time series is the average value of the time series.

      • Component form representations of exponential smoothing methods comprise a forecast equation and a smoothing equation for each of the components included in the method.

      • For SES, the only component included is the level, ℓ(t). Other methods which will be considered later may also include trend b(t) and seasonal component s(t).

      • The component form of simple exponential smoothing:

        

      • The forecast equation shows that the forecast value at time (t+1) is the estimated level at time t. The smoothing equation for the level (usually referred to as the level equation) gives the estimated level of the series at each period.

    • Flat Forecasts

      

    • Optimization

      • For simple exponential smoothing, we need to select the values of α and ℓ0. All forecasts can be computed from the data once we know those values.

        • These two values can be estimated by minimizing the SSE (sum of squared errors). This is a non-linear minimization problem and optimization tool is needed to solve it

          

  

Forecasts are for period 2018 to 2022. One step ahead fitted values are plotted for 1960 to 2017. The prediction interval is measured using a method discussed later in the chapter (SES doesn't generate PI). PI indicates there is a lot of uncertainty in the future exports over 5 year forecast period.

Methods with Trend

Holt’s linear trend method

• Extension of Simple Exponential Smoothing.

• Used to forecast on data with trend (SES can be used only using data with no trend and seasonality).

• Involves a forecast equation and two smoothing equations.

  

• The smoothing parameters, α, β*, initial values ℓ0 and b0 are estimated by minimising the SSE for the one-step training errors

• Challenge with this approach: The forecasts generated by Holt’s linear method display a constant trend (increasing or decreasing) indefinitely into the future. Empirical evidence indicates that these methods tend to over-forecast, especially for longer forecast horizons.

Damped Trend Methods

• This method includes a parameter that “dampens” the trend to a flat line some time in the future. Methods that include a damped trend have proven to be very successful, and are arguably the most popular individual methods when forecasts are required automatically for many series.

  

  

Methods with Trend & Seasonality (?)

• Holt & Winter extended Holt's method to capture seasonality. Thus, " Holt-Winters seasonal method" is applicable for data with trend and seasonality.
• The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations, one each for
  • level ℓt (Corresponding smoothing parameter: α)
  • trend bt (Smoothing parameter: β∗)
  • seasonal component st (Smoothing parameter: γ)