QBUS 6840 Lecture 5 Forecasting with Exponential Smoothing
QBUS 6840 Lecture 5
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Forecasting with Exponential Smoothing
The University of School
Exponential smoothing
Simple Exponential Smoothing
Trend Corrected Exponential Smoothing (Holt’s Linear Trend
Online Textbook Sections 7.1-7.2
(https://otexts.org/fpp2/expsmooth.html); and/or
BOK Sec 8.1-8.3
https://otexts.org/fpp2/expsmooth.html
Objectives
Be able to conceptually explain the Simple Exponential
Smoothing (SES)
Fully understand how Trend Corrected Exponential Smoothing
(TCES or Holt’s Linear Trend Method) works
Fully understand the component form and the error form of
SES and TCES, and their statistical models as well
Fully understand SES statistical model
Be able to conduct forecasts and their variances for both
Conceptually know how model parameters can be learned or
determined
S&P 500 index
Daily return series 1996-2012
Forecasting the standard variation (called volatility in financial
econometrics) of asset returns is of major interest in banking
sector, risk management, etc. Can you forecast the volatility of
tomorrow’s S&P 500 return?
Exponential smoothing methods
Exponential smoothing is the most basic, yet very successful
forecasting method, developed in the 1950s. The idea of
exponential smoothing has motivated the most successful
forecasting methods being used nowadays.
In simple terms, exponential smoothing forecasts are weighted
averages of previous observations. The weights decay
exponentially as we go further into the past.
Useful when parameters or components are changing with
Simple exponential smoothing (SES)
Näıve Method
ŶT+1|1:T = YT
Overall Average Method
ŶT+1|1:T =
(YT + YT−1 + · · ·+ Y1)
Something between two extremes
ŶT+1|1:T = αYT + α(1− α)YT−1 + α(1− α)2YT−2 + · · ·+ α(1− α)T−1Y1
αYT + (1− α)[αYT−1 + α(1− α)YT−2 + · · ·+ α(1− α)T−2Y1]
αYT + (1− α)ŶT |1:T−1
Simple exponential smoothing (SES)
Näıve Method
ŶT+1|1:T = YT
Overall Average Method
ŶT+1|1:T =
(YT + YT−1 + · · ·+ Y1)
Something between two extremes
ŶT+1|1:T = αYT + α(1 − α)YT−1 + α(1 − α)2YT−2 + · · + α(1 − α)T−1Y1
αYT + (1− α)[αYT−1 + α(1− α)YT−2 + · · ·+ α(1− α)T−2Y1]
αYT + (1− α)ŶT |1:T−1
Simple exponential smoothing (SES)
Näıve Method
ŶT+1|1:T = YT
Overall Average Method
ŶT+1|1:T =
(YT + YT−1 + · · ·+ Y1)
Something between two extremes
ŶT+1|1:T = αYT + α(1 − α)YT−1 + α(1 − α)2YT−2 + · · + α(1 − α)T−1Y1
]= αYT + (1 − α)[αYT−1 + α(1 − α)YT−2 + · · + α(1 − α)T−2Y1
αYT + (1− α)ŶT |1:T−1
Simple exponential smoothing (SES)
Näıve Method
ŶT+1|1:T = YT
Overall Average Method
ŶT+1|1:T =
(YT + YT−1 + · · ·+ Y1)
Something between two extremes
ŶT+1|1:T = αYT + α(1 − α)YT−1 + α(1 − α)2YT−2 + · ·+ α(1 − α)T−1Y1
]= αYT + (1 − α)[αYT−1 + α(1 − α)YT−2 + · · + α(1 − α)T−2Y1
= αYT + (1 − α)ŶT |1:T−1
Simple exponential smoothing (SES)
SES for forecasting in the Weighted Average Form
Ŷt+1|1:t = αYt + (1− α)Ŷt|1:t−1
The forecast at time t + 1 is equal to a weighted average
between the most recent observation Yt and the most recent
forecast Ŷt|1:t−1.
Two Alternative Forms
The Component Form
lt = αYt + (1− α)lt−1, 0 ≤ α ≤ 1.
Ŷt+1|1:t = lt .
lt is called the level (or the smoothed value) of the series at
time t. We first calculate the level lt , then use it as the
forecast Ŷt+1|1:t .
The Error Correction Form
Define εt = Yt − lt−1, or Yt+1 = lt + εt+1
lt = αYt + (1− α)lt−1 = lt−1 + α(Yt − lt−1)
= lt−1 + αεt
where εt = Yt − Ŷt|1:t−1 is the forecast error at time t.
Explanation: Simple exponential smoothing
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
Ŷt|1:t−1 = lt−1
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = lt
Explanation: Simple exponential smoothing
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
Ŷt|1:t−1 = lt−1
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = lt
Explanation: Simple exponential smoothing
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
Ŷt|1:t−1 = lt−1
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = lt
Explanation: Simple exponential smoothing
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
Ŷt|1:t−1 = lt−1
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = lt
Explanation: Simple exponential smoothing
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
Ŷt|1:t−1 = lt−1
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = lt
Explanation: Simple exponential smoothing
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
Ŷt|1:t−1 = lt−1
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = lt
S&P 500 Closing Price (Lecture05 Example01.py)
Exponential smoothing with α = 0.05
S&P 500 Closing Price (Lecture05 Example01.py)
Exponential smoothing with α = 0.1
Visitor arrivals in Australia
Exponential smoothing with α = 0.1
Visitor arrivals in Australia
Exponential smoothing with α = 0.5
Simple exponential smoothing
Specify an initial value l0 (an estimate or a guess, e.g., the average
of Y1,Y2,Y3).
l1 = αY1 + (1− α)l0
l2 = αY2 + (1− α)l1 = αY2 + (1− α)αY1 + (1− α)2l0
l3 = αY3+(1−α)l2 = αY3+(1−α)αY2+(1−α)2αY1+(1−α)3l0
Simple exponential smoothing
l4 = αY4 + (1− α)l3
= αY4 + (1− α)αY3 + (1− α)2αY2 + (1− α)3αY1 + (1− α)4l0
lt =αYt + (1− α)lt−1
=αYt + (1− α)αYt−1 + (1− α)2αYt−2 + . . .+ (1− α)t−1αY1
+ (1− α)t l0
A WMA smoother (is moving?)
SES is sometimes called exponentially weighted moving average
Choice of initial level
We left l0 unspecified above. How should we set it?
Use the average of very initial observations, i.e., Y1,Y2,Y3
etc., or even simply Y1
Take l0 as a parameters, and use an algorithm to estimate it.
How to set α?
Use expert’s knowledge
Take α as a parameters, and use an algorithm to estimate it.
We come back to this later.
Simple exponential smoothing (SES)
Some notes
SES can be used for both smoothing and forecasting
Useful for forecasting when the time series has no clear trend
or seasonal patterns, hence it should be used for seasonally
adjusted data
Weights all previous observations in smoothing.
Weights decrease exponentially.
Weights add to 1 (always, check it!).
Simple exponential forecasting: Statistical Model
SES considered so far can only produce point forecasts. How
can we produce forecast intervals? We need a statistical
The basic model:
lt = αYt + (1− α)lt−1, 0 ≤ α ≤ 1.
Yt+1 = lt + εt+1, with εt+1 ∼ N(0, σ2).
We assume all εt ’s are independent of each other
The level is the hidden underlying mechanism, where the next
observation is “noised” current level.
What is about when α = 1?
Simple exponential forecasting: Statistical Model
SES considered so far can only produce point forecasts. How
can we produce forecast intervals? We need a statistical
The basic model:
lt = αYt + (1− α)lt−1, 0 ≤ α ≤ 1.
Yt+1 = lt + εt+1, with εt+1 ∼ N(0, σ2).
We assume all εt ’s are independent of each other
The level is the hidden underlying mechanism, where the next
observation is “noised” current level.
What is about when α = 1?
Simple exponential forecasting: Statistical Model
SES considered so far can only produce point forecasts. How
can we produce forecast intervals? We need a statistical
The basic model:
lt = αYt + (1− α)lt−1, 0 ≤ α ≤ 1.
Yt+1 = lt + εt+1, with εt+1 ∼ N(0, σ2).
We assume all εt ’s are independent of each other
The level is the hidden underlying mechanism, where the next
observation is “noised” current level.
What is about when α = 1?
Simple exponential forecasting: Statistical Model
SES considered so far can only produce point forecasts. How
can we produce forecast intervals? We need a statistical
The basic model:
lt = αYt + (1− α)lt−1, 0 ≤ α ≤ 1.
Yt+1 = lt + εt+1, with εt+1 ∼ N(0, σ2).
We assume all εt ’s are independent of each other
The level is the hidden underlying mechanism, where the next
observation is “noised” current level.
What is about when α = 1?
Simple exponential forecasting
Formal statistical model: Estimating Parameters
Recall the basic model:
Yt+1 = lt + εt+1
lt = αYt + (1− α)lt−1.
We can chose α (and l0) by minimising
(Yt − lt−1)2
(Yt − αYt−1 − (1− α)lt−2)2.
Simple exponential smoothing
A big picture note
The basic SES model:
Measurement equation: Yt+1 = lt + εt+1
Transition equation: lt = αYt + (1− α)lt−1
The measurement equation represents the observation Yt+1 as
a function of a hidden/unobservable lt . The transition
equation describes how the hidden lt evolves over time.
Many state-of-the-art forecasting methods have this form:
state space models, stochastic volatility models, recurrent
neural network models
Simple exponential smoothing: Analysis
Error correction formulation
The basic model for the level can be rewritten in terms of
lt = αYt + (1− α)lt−1
= lt−1 + α(Yt − lt−1)
= lt−1 + αεt .
The next observation is
Yt+1 = lt + εt+1 = lt−1 + αεt + εt+1 = · · ·
= l0 + αε1 + · · ·+ αεt + εt+1
Yt+2 = lt+1 + εt+2 = lt + αεt+1 + εt+2
= lt−1 + αεt + αεt+1 + εt+2 = · · ·
= l0 + αε1 + · · ·+ αεt + αεt+1 + εt+2
Simple exponential smoothing: Analysis
Forecast equations
The forecast is defined as the average over all possible
uncertainty [Carefully understand the meaning of this!]
Ŷt+h|1:t := E(Yt+h|Y1:t), lt = lt−1 + αεt
We have already observed up to time t, so lt is certain.
Uncertainty occurs after this time point, e.g., εt+1
Ŷt+1|1:t =E(lt + εt+1|Y1:t) [Note: Yt+1 = lt + εt+1]
=E(lt) + E(εt+1|Y1:t) = lt + E(εt+1) = lt + 0 = lt
where we have used the assumption εt+1 ∼ N(0, σ2), i.e.,
E(εt+1) = 0.
So, similarly
Ŷt+2|1:t = E(lt+1 + εt+2|Y1:t) = lt+1
Is this correct? It is wrong as lt+1 contains uncertainty
Simple exponential smoothing: Analysis
Forecast equations
The forecast is defined as the average over all possible
uncertainty [Carefully understand the meaning of this!]
Ŷt+h|1:t := E(Yt+h|Y1:t), lt = lt−1 + αεt
We have already observed up to time t, so lt is certain.
Uncertainty occurs after this time point, e.g., εt+1
Ŷt+1|1:t =E(lt + εt+1|Y1:t) [Note: Yt+1 = lt + εt+1]
=E(lt) + E(εt+1|Y1:t) = lt + E(εt+1) = lt + 0 = lt
where we have used the assumption εt+1 ∼ N(0, σ2), i.e.,
E(εt+1) = 0.
So, similarly
Ŷt+2|1:t = E(lt+1 + εt+2|Y1:t) = lt+1
Is this correct?
It is wrong as lt+1 contains uncertainty
Simple exponential smoothing: Analysis
Forecast equations
The forecast is defined as the average over all possible
uncertainty [Carefully understand the meaning of this!]
Ŷt+h|1:t := E(Yt+h|Y1:t), lt = lt−1 + αεt
We have already observed up to time t, so lt is certain.
Uncertainty occurs after this time point, e.g., εt+1
Ŷt+1|1:t =E(lt + εt+1|Y1:t) [Note: Yt+1 = lt + εt+1]
=E(lt) + E(εt+1|Y1:t) = lt + E(εt+1) = lt + 0 = lt
where we have used the assumption εt+1 ∼ N(0, σ2), i.e.,
E(εt+1) = 0.
So, similarly
Ŷt+2|1:t = E(lt+1 + εt+2|Y1:t) = lt+1
Is this correct? It is wrong as lt+1 contains uncertainty
Simple exponential smoothing: Analysis
Forecast equations
The forecast is defined as the average over all possible
uncertainty [Carefully understand the meaning of this!]
Ŷt+h|1:t := E(Yt+h|Y1:t), lt = lt−1 + αεt
We have already observed up to time t, so lt is certain.
Uncertainty occurs after this time point, e.g., �t+1
Ŷt+1|1:t =E(lt + εt+1|Y1:t) [Note: Yt+1 = lt + εt+1]
=E(lt) + E(εt+1|Y1:t) = lt + E(εt+1) = lt + 0 = lt
where we have used the assumption εt+1 ∼ N(0, σ2), i.e.,
E(εt+1) = 0.
Ŷt+2|1:t = E(lt+1 + εt+2|Y1:t) = E(lt + αεt+1|Y1:t) = lt
Ŷt+h|1:t = E(lt+h−1 + εt+h|Y1:t) = E(lt+h−2 + αεt+h−1|Y1:t) = lt
Hence the forecast is always lt after time t.
Simple exponential smoothing: Analysis
Forecast equations
The forecast is defined as the average over all possible
uncertainty [Carefully understand the meaning of this!]
Ŷt+h|1:t := E(Yt+h|Y1:t), lt = lt−1 + αεt
We have already observed up to time t, so lt is certain.
Uncertainty occurs after this time point, e.g., �t+1
Ŷt+1|1:t =E(lt + εt+1|Y1:t) [Note: Yt+1 = lt + εt+1]
=E(lt) + E(εt+1|Y1:t) = lt + E(εt+1) = lt + 0 = lt
where we have used the assumption εt+1 ∼ N(0, σ2), i.e.,
E(εt+1) = 0.
Ŷt+2|1:t = E(lt+1 + εt+2|Y1:t) = E(lt + αεt+1|Y1:t) = lt
Ŷt+h|1:t = E(lt+h−1 + εt+h|Y1:t) = E(lt+h−2 + αεt+h−1|Y1:t) = lt
Hence the forecast is always lt after time t. 23 / 45
Simple exponential smoothing
Variance for interval forecasts
Recall the model again:
Yt+1 = lt + εt+1, lt = lt−1 + αεt , εt ∼ N(0, σ2).
Consider the variance of the new observation
V(Yt+1|Y1:t) = V(lt + εt+1|Y1:t)
= V(lt) + V(εt+1|Y1:t)
= 0 + σ2 = σ2. Why?
V(Yt+2|Y1:t) = V(lt+1 + εt+2|Y1:t) = V(lt + αεt+1 + εt+2|Y1:t)
= V(lt) + V(αεt+1) + V(εt+2|Y1:t)
= 0 + α2V(εt+1) + V(εt+2|Y1:t)
= α2σ2 + σ2 = σ2(1 + α2)
Simple exponential smoothing
Variance for interval forecasts
V(Yt+3|Y1:t) = V(lt+2 + εt+3|Y1:t)
= V(lt+1 + αεt+2 + εt+3|Y1:t)
= V(lt + αεt+1 + αεt+2 + εt+3|Y1:t)
= σ2(1 + 2α2)
V(Yt+h|Y1:t) = V(lt+h−1 + εt+h|Y1:t)
= V(lt+h−2 + αεt+h−1 + εt+h|Y1:t)
αεt+h−i + εt+h|Y1:t)
= σ2(1 + (h − 1)α2)
Simple exponential smoothing
Forecasting: collecting the results
Ŷt+h|1:t = E(Yt+h|Y1:t) = lt
V(Yt+h|Y1:t) = σ2(1 + (h − 1)α2)
What happens as h increases?
Alcohol related assaults in NSW
Forecasting
Are the forecasts reasonable?
Explanation: Including Trend Information
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = ltbt−3
Ŷt|1:t−1 = lt−1 + bt−1 × 1
lt = αYt + (1− α)Ŷt|1:t−1
bt = β(lt − lt−1) + (1− β)bt−1
Ŷt+1|1:t = lt + bt × 1Ŷt+1|1:t = lt + bt
Explanation: Including Trend Information
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = ltbt−3
Ŷt|1:t−1 = lt−1 + bt−1 × 1
lt = αYt + (1− α)Ŷt|1:t−1
bt = β(lt − lt−1) + (1− β)bt−1
Ŷt+1|1:t = lt + bt × 1Ŷt+1|1:t = lt + bt
Explanation: Including Trend Information
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = ltbt−3
Ŷt|1:t−1 = lt−1 + bt−1 × 1
lt = αYt + (1− α)Ŷt|1:t−1
bt = β(lt − lt−1) + (1− β)bt−1
Ŷt+1|1:t = lt + bt × 1Ŷt+1|1:t = lt + bt
Explanation: Including Trend Information
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = ltbt−3
Ŷt|1:t−1 = lt−1 + bt−1 × 1
lt = αYt + (1− α)Ŷt|1:t−1
bt = β(lt − lt−1) + (1− β)bt−1
Ŷt+1|1:t = lt + bt × 1Ŷt+1|1:t = lt + bt
Explanation: Including Trend Information
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = ltbt−3
Ŷt|1:t−1 = lt−1 + bt−1 × 1
lt = αYt + (1− α)Ŷt|1:t−1
bt = β(lt − lt−1) + (1− β)bt−1
Ŷt+1|1:t = lt + bt × 1Ŷt+1|1:t = lt + bt
Explanation: Including Trend Information
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = ltbt−3
Ŷt|1:t−1 = lt−1 + bt−1 × 1
lt = αYt + (1− α)Ŷt|1:t−1
bt = β(lt − lt−1) + (1− β)bt−1
Ŷt+1|1:t = lt + bt × 1Ŷt+1|1:t = lt + bt
Explanation: Including Trend Information
t − 4 t − 3 t − 2 t − 1 t t + 1 t + 2
lt = αYt + (1− α)lt−1
Ŷt+1|1:t = ltbt−3
Ŷt|1:t−1 = lt−1 + bt−1 × 1
lt = αYt + (1− α)Ŷt|1:t−1
bt = β(lt − lt−1) + (1− β)bt−1
Ŷt+1|1:t = lt + bt × 1Ŷt+1|1:t = lt + bt
Trend corrected exponential smoothing (TCES)
lt = αYt + (1− α)(lt−1 + bt−1), 0 ≤ α ≤ 1
(= αYt + (1− α)Ŷt |1:t−1))
bt = β(lt − lt−1) + (1− β)bt−1, 0 ≤ β ≤ 1
Ŷt+1|1:t = lt + bt×1.
Trend corrected exponential smoothing
lt = αYt + (1− α)(lt−1 + bt−1), 0 ≤ α ≤ 1
(= αYt + (1− α)Ŷt|1:t))
bt = β(lt − lt−1) + (1− β)bt−1, 0 ≤ β ≤ 1
Ŷt+1|1:t = lt + bt .
Trend corrected exponential smoothing (TCES)
Yt+1 = lt + bt + εt+1
lt = αYt + (1− α)(lt−1 + bt−1)
bt = β(lt − lt−1) + (1− β)bt−1
εt+1 ∼ N(0, σ2)
We can choose α and β even l0 and b0 by minimising
SSE(α, β, l0, b0) =
(Yt − lt−1 − bt−1)2
If you have provided l0 and b0, you may only need count the error
from t = 2.
Visitor arrivals in Australia: Lecture05 Example02.py
Original series (2010-2016)
Visitor arrivals in Australia: Lecture05 Example02.py
Seasonally adjusted series (2010-2016)
Trend corrected exponential smoothing (TCES)
Error correction formulation
The basic model for the trend corrected exponential smoothing can
be written in many ways. We can express all the components in
terms of errors:
lt = αYt + (1− α)(lt−1 + bt−1)
= lt−1 + bt−1 + α(Yt − lt−1 − bt−1)
= lt−1 + bt−1 + αεt
Yt+1 = lt−1 + bt−1 + bt + αεt + εt+1
= lt + bt + εt+1
Trend corrected exponential smoothing (TCES)
Error correction formulation
bt = β(lt − lt−1) + (1− β)bt−1
= bt−1 + β(lt − lt−1 − bt−1)
= bt−1 + βαεt from lt = lt−1 + bt−1 + αεt
(= bt−1 + βα(Yt − lt−1 − bt−1))
Trend corrected exponential smoothing
Error correction formulation
lt = lt−1 + bt−1 + αεt
bt = bt−1 + βαεt
Yt+1 = lt + bt + εt+1
(= lt−1 + 2bt−1 + α(1 + β)εt + εt+1)
εt ∼ N(0, σ2)
Trend corrected exponential smoothing (TCES)
Forecasting equations
Ŷt+1|1:t : = E(Yt+1|Y1:t)
= E(lt + bt + εt+1|Y1:t)
(= αYt + (1− α)(lt−1 + bt−1))
(= lt−1 + 2bt−1 + α(1 + β)εt)
Ŷt+2|1:t = E(lt+1 + bt+1 + εt+2|Y1:t)
= E(lt + 2bt + α(1 + β)εt+1 + εt+2|Y1:t)
= lt + 2bt
Trick: We iteratively expand the formula until we arrive the time
point all are known.
Trend corrected exponential smoothing (TCES)
Forecasting equations
Ŷt+3|1:t = E(lt+2 + bt+2 + εt+3|Y1:t)
= E((lt+1 + bt+1 + αεt+2) + (bt+1 + βαεt+2) + εt+3)
= E(lt+1 + 2bt+1 + α(1 + β)εt+2 + εt+3)
= E((lt + bt + αεt+1) + 2(bt + βαεt+1) + α(1 + β)εt+2 + εt+3)
= E(lt + 3bt + α(1 + 2β)εt+1 + α(1 + β)εt+2 + εt+3)
= lt + 3bt
Ŷt+h|1:t = lt + hbt
Trend corrected exponential smoothing (TCES)
Variance for interval forecasts
V(Yt+1|Y1:t) = V(lt + bt + εt+1|Y1:t)
V(Yt+2|Y1:t) = V(lt+1 + bt+1 + εt+2|Y1:t)
= V(lt + 2bt + α(1 + β)εt+1 + εt+2|Y1:t)
= σ2(1 + α2(1 + β)2)
Trend corrected exponential smoothing (TCES)
Variance for interval forecasts
V(Yt+3|Y1:t) = V(lt+2 + bt+2 + εt+3|Y1:t)
= V(lt+1 + 2bt+1 + α(1 + β)εt+2 + εt+3|Y1:t)
= V(lt + 3bt + α(1 + 2β)εt+1 + α(1 + β)εt+2 + εt+3|Y1:t)
= V(α(1 + 2β)εt+1 + α(1 + β)εt+2 + εt+3|Y1:t)
= α2(1 + 2β)2σ2 + α2(1 + β)2σ2 + σ2
= σ2(1 + α2(1 + β)2 + α2(1 + 2β)2)
V(Yt+h|Y1:t) = V
lt + hbt + α
(1 + iβ)εt+i + εt+h|Y1:t
h(h − 1)(2h − 1) + (βh + 1)(h − 1)
(I used the formula for the sum of an arithmetic progression to get
to the last step (optional))
Trend corrected exponential smoothing (TCES)
Forecasting: collecting the results
Ŷt+h|1:t = lt + hbt
V(Yt+h|Y1:t) = σ2
h(h − 1)(2h − 1) + (βh + 1)(h − 1)
What happens as h increases?
Alcohol related assaults in NSW
Forecasting the seasonally adjusted series (last 12 months)
Comparison
What criteria should we use to compare SES and TCES?
Alcohol related assaults in NSW
Forecasting the seasonally adjusted series (last 12 months)
Alcohol related assaults in NSW
Forecasting the seasonally adjusted series (last 12 months)
One month ahead forecasts
RMSE 70.9 63.5
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