SES: x StepAhead. a:minimize Rmse. F0:mean. OutSample. Slacked. DES: a&B. holt Linear: small period ahead. Damped T.

TES: Winters: Seasonality: SI visually or by Acf. Additive/multiplicative. Deseason. MultiSeasonality. a B ¥. Outliers!

We normally use Multiplicative. We only use additive qhen the data set has unstable variability, e.g. Hetero.

**Single Exponential Smoothing**

**One-step ahead** forecast is the weighted average of current value and past forecast

Ft(1) = a(Current Value)+ (1-a) Past Forecast = aXt+ (1-a) Ft-1(1)

Ft(1) = Ft-1(1) + a [ Xt – Ft-1(1) ]

- I.e. previous forecast plus a constant times previous forecast error

To apply this we need to choose the smoothing weight α

◦The closer a is to 1, the more reactive the forecast is to changes

**Recursive function:**

◦F[t](1) = αX[t]+ (1-α) F[t-1](1),

◦F[t-1](1) = αX[t]-1+ (1-α) F[t-2](1) , etc

**Backward substitute:**

◦Ft(1) = αX[t] + (1-α)αX[t]-1 + (1-α)2 αX[t-2] + (1-α)3 αX[t-3] +…

Today’s level = α* Today’s value +(1-a)*Yesterday’s Level

Tomorrow’s forecast = Today’s level

L[t] = α X[t] + (1- α) L[t]-1

F[t](k) = L[t] for all k

The level represents the systematic part of the series

**Starting Values** – need F0(1) to start process. Possible Choices:

◦Data Mean

◦Backcasting

**SES is identical to ARIMA(0,1,1) model.**

Parameter is chosen to **minimize** either the root mean square (**RMS**), mean absolute or mean absolute percentage **one step (F(1)) **ahead forecast error.

**Out of Sample Testing: training data **is called **In-Sample **data, while the **test** data is called **Out-of-Sample** data.

When comparing methods out of sample be sure to check how the out of sample forecast is computed and what information is **assumed** **known**. In some automatic programs – exponential smoothing is applied one step ahead out of sample so that it uses more data than other methods.

**
Double Exponential Smoothing
** Double Exponential Smoothing factors-in the trends in the so it models

**trending**data better; but only as the trend remains, so it

**ages out**quickly.

Linear Trend Model (regression) Yt=b0+b1t is inflexible. Assumes a *constant* trend b1 per period throughout the data.

}Basic idea – introduce a trend estimate that **changes over time**.

}Similar to single exponential smoothing but **two** equations.

}Issue is to choose two smoothing rates, a and b.

}Referred to as Holt’s Linear Trend Model

}Trend dominates **after a few periods **in forecasts so forecasts are only good for a short term.

}The model: Separate smoothing equations for **level** and **trend**

◦**Level Equation**

Lt = a(Current Value)

+ (1 – a) (Level + Trend Adjustment)t-1

Lt = aXt + (1 – a) (Lt-1 + T t-1)

◦**Trend Equation**

Tt = b(Lt – Lt-1) + (1 – b) Tt-1

◦**Forecasting Equation **

Ft(k) = Lt + k Tt

**Damped Trend Models**

}Problem with a trend model is that **trend dominates **forecast in a couple of periods.

}Approach – introduce **trend damping **parameter f

◦**Level Equation**

Lt = aXt + (1 – a) (Lt-1 + fT t-1)

◦**Trend Equation**

Tt = b(Lt – Lt-1) + (1 – b) fTt-1

◦**Forecasting Equation **

◦Available in SAS ETS, R, Textbook’s forecast package for Excel and http://www.peerforecaster.com

**Seasonality**

}A persistent pattern that occurs at regularly spaced time intervals

◦quarterly, monthly, weekly, daily

}Data may exhibit several levels of seasonality simultaneously

}May be modeled as multiplicative or additive

}Should be included in systematic part of forecasting model

}Detected visually or through ACF

Double Exponential smoothing (with trend factor) doesn’t model seasonality

The Holt-Winters Model: Factors-in seasonality:

**Level Equation**:

}Lt=a(Current Value/Seasonal Adjustmentt-p)

+ (1-a)(Levelt-1 + Trendt-1)

}Lt = a(Deseasonalized Current Value)

+ (1-a)(Levelt-1 + Trendt-1)

}Lt = a(Xt/It-p) + (1-a)(Lt-1 + Tt-1)

where It-p = Seasonal component

}Generalizes Double Exponential Smoothing by including (multiplicative) seasonal indicators.

}Separate smoothing equations for level, trend and seasonal indicators.

}Allows trend and seasonal pattern to change over time

}Must estimate **three** smoothing parameters

}Equations more complicated but implemented with software

}One of the best methods for **short term seasonal **forecasts

**Trend Equation**:

}Same as double exponential smoothing method

}Tt = b(Change in level in the last period)

+ (1 – b) (Trend Adjustment)t-1

}Tt = b(Lt – Lt-1) + (1 – b) Tt-1

**Seasonal Equation**:

}It = g(Current Value/Current Level)

+ (1-g)(Seasonal Adjustment)t-p

}It = g(Xt/Lt) + (1-g)It-p

where ** p is the length of the seasonality **(i.e. p months) so that t-p is the same season in the previous year.

Note this model assumes the same g for every season*.*

**Forecasting equations**:

}Ft(k) = (Lt + kTt)It-p+k for k=1,2, …, p

}Ft(k) = (Lt + kTt)It-2p+k for k=p+1,p+2, …, 2p

}Lt = a(Xt/It-p) + (1-a)(Lt-1 + Tt-1) Level Equation

}Tt = b(Lt – Lt-1) + (1-b)Tt-1 Trend Equation

}It = g(Xt/Lt) + (1- g)It-p Seasonal Factor Equation

}

**Forecasting equations**:

}Ft(k) = (Lt + kTt)It-p+k for k=1,2, …, p

}Ft(k) = (Lt + kTt)It-2p+k for k=p+1,p+2, …, 2p

}

}

}Can add damped trend to this model too.

}Additive version also available but multiplicative model is preferable. Note the HW model combines additive trend with multiplicative seasonality.

}Missing values cannot be skipped, they must be estimated.

}Outliers have a big impact and could be handled like missing values

}This is a special case of a **“state space model”(chapter 5)**

}Different computer packages give different estimates and forecasts.

}Early reference: Chatfield and Yar “Holt-Winters forecasting: some practical issues”, The Statistician, 1988, 129-140.

}

}Plot data

◦determine patterns

seasonality, trend, outliers

}Fit model

}Check residuals

◦Any information present?

Plots or ACF functions

}Adjust

}Produce forecasts

}Calibrate on **hold out **sample

◦Multiple one step ahead

◦k-step ahead (where is k is the **practical forecast **horizon)

}Important issue is how frequently to recalibrate the model

◦Possible choices

Every period

Quarterly

Annually

◦The point here is that the model can be determined by analysts, programmed into a forecasting system with fixed parameters and **recalibrated as needed**.

**SES is ARIMA(0,1,1):
**

*short forecast lags behind actual*

*DES plot fix ses trend bad>dump.*

*Winters (TES): plot/Acf deseasonlize*

*(anual 4cast)? Why? Damped too.*

*Addtv L +T + I vs multiplicative: L +T * I. can decide b4 modeling?*

* Missing values and outliers to replace with estimts.*

*Procedure: Plot & acf dtct trnd outlir set params1 fit calc rmse recalibrate by diff params & models.*