This document provides an overview of time series analysis techniques including moving average (MA) models, exponential smoothing, and ARMA models. It describes the key components of MA models including the MA(q) notation and theoretical properties. Exponential smoothing is presented as a weighted moving average for smoothing and short-term forecasting. The ARMA model is introduced as combining autoregressive and moving average terms to model a time series.

1. Time Series Analysis

2. Contents
• Moving average (MA) models
• Exponential Smoothing
• ARMA Model

3. Moving Average Model
• Moving average (MA) models account for the possibility of a relationship between a variable and the
residuals from previous periods.
• MA(q) is moving average with q legs:
• The 1st order moving average model, denoted by MA(1) is:
• The 2nd order moving average model, denoted by MA(2) is:
• The qth order moving average model, denoted by MA(q) is:
yt=µ +εt+ 𝑖=1
𝑞
𝜃𝑖 𝜀 𝑡 − 𝑖
yt=µ +εt+Ɵ1εt-1 + Ɵ2εt-2 + ………+ Ɵqεt-q
yt=µ +εt+Ɵ1εt-1
yt=µ +εt+Ɵ1εt-1 + Ɵ2εt-2

4. Theoretical Properties of a Time Series with an MA(1) Model
•Mean is E(yt) = μ
•Variance is Var(yt) = σ 2(1 + θ1
2)
•Autocovarience at lag 1 is:
Var(Yt) = E(yt-μ)2
= E[(εt-θ1εt-1)2]
= E[(ε2
t+ 2θ1εtεt-1 +θ2
1ε2
t-1)]
= E[εt
2] + 2θ1E[εtεt-1 ] + θ2
1E[ε2
t-1]
= σ 2 + 2θ1 (0)+ θ1
2 σ 2
= σ 2(1 + θ1
2)
E(Yt) = E(μ+εt+θ1εt-1)
= μ + 0 + (θ1)(0)
= μ
Covar = E(yt-μ) (yt-1-μ)
= E[(εt-θ1εt-1) (εt-1-θ1εt-2)]
= E[(εtεt-1 + θ1εt-1εt-1 + θ1εtεt-2 +θ2
1εt-1 εt-2)]
= 0+ θ1E[ε2
t-1 ] + 0 + 0
= θ1σ 2

5. Theoretical Properties of a Time Series with an MA(1) Model
•Autocovarience at lag 2 is:
•For MA(1) , the autocovarience at higher lags (k>1) is 0.
•The autocorrelation function is: =
cov[yt,yt−k]
𝑣𝑎𝑟[𝑦𝑡]
=
θ1
σ 2
(1 + θ1
2)σ 2
•The autocorrelation of MA(q) series is non zero only for lags k≤q and is zero for all higher lags.
Cov = E(yt-μ) (yt-2-μ)
= E[(εt-θ1εt-1) (εt-2-θ1εt-3)]
= E[(εtεt-2 + θ1εt-1εt-2 + θ1εtεt-3 +θ2
1εt-1 εt-3)]
= 0+ 0 + 0 + 0
= 0
ρ1=
1 𝑘 = 0
θ1
(1 + θ1
2)
𝑘 = 1
0 𝑘 > 1

6. Example 1 Suppose that an MA(1) model is yt = 10 + εt + .7εt-1 .The coefficient θ1= 0.7(calculated by yule walker
method). Because this is an MA(1), the theoretical ACF will have nonzero values only at lags 1.The theoretical ACF is
given by
ρ1=
θ1
(1 + θ1
2)
=
0.7
1+0.72 = 0.4698
A plot of this ACF follows.

7. Determining the order MA(q)
Autocorrelation is:
The order of MA(q) model is last significant value observed from autocorrelation
function plot.

8. A “spike” at lag 1 followed by generally non-significant values for lags past 1. Note that the sample ACF does
not match the theoretical pattern of the underlying MA(1), which is that all autocorrelations for lags past 1 will be
0. A different sample would have a slightly different sample ACF shown above, but would likely have the same
broad features.

9. Moving Average Model : Forecasting
• Used for smoothing
• A series of arithmetic means over time
• Result dependent upon choice of L (length of period for computing means)
• Examples:
• For a 5 year moving average, L = 5
• For a 7 year moving average, L = 7
• Etc.
• Example: Five-year moving average
• First average:
• Second average:
• etc.
5
YYYYY
MA(5) 54321 

5
YYYYY
MA(5) 65432 


10. Example: Annual Data
Year Sales
1
2
3
4
5
6
7
8
9
10
11
etc…
23
40
25
27
32
48
33
37
37
50
40
etc…
Annual Sales
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sales

11. Calculating Moving average
• Each moving average is for a consecutive block of 5 years
Year Sales
1 23
2 40
3 25
4 27
5 32
6 48
7 33
8 37
9 37
10 50
11 40
Average
Year
5-Year
Moving
Average
3 29.4
4 34.4
5 33.0
6 35.4
7 37.4
8 41.0
9 39.4
… …
5
54321
3


5
3227254023
29.4


etc…

12. Annual vs. Moving Average
Annual vs. 5-Year Moving Average
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Year
Sales
Annual 5-Year Moving Average
• The 5-year moving average
smoothes the data and shows
the underlying trend.

13. Exponential Smoothing
• A weighted moving average.
• Used for smoothing and short term forecasting .
• The weight (smoothing coefficient) is α -
• Subjectively chosen
• Range from 0 to 1
• Weights decline exponentially
• Most recent observation weighted most
• Smaller W gives more smoothing, larger W gives less smoothing
• The weight is:
• Close to 0 for smoothing out unwanted cyclical and irregular components
• Close to 1 for forecasting

14. Exponential Smoothing Model
 Exponential smoothing model
S1=X1
St=αXt+(1-α)St-1 For t = 2, 3, 4, …
where:
St = exponentially smoothed value for period t
St-1 = exponentially smoothed value already computed for period t - 1
xt = observed value in period t
α = weight (smoothing coefficient), 0 < α < 1

15. Deriving the Exponential Smoothing Formula

16. Exponential Smoothing Example
• Suppose we use weight α = .2
Time
Period
(t)
Sales
(Xt)
Forecast
from prior
period (st-1)
Exponentially Smoothed
Value for this period (St)
1
2
3
4
5
6
7
8
9
10
etc.
23
40
25
27
32
48
33
37
37
50
etc.
--
23
26.4
26.12
26.296
27.437
31.549
31.840
32.872
33.697
etc.
23
(.2)(40)+(.8)(23)=26.4
(.2)(25)+(.8)(26.4)=26.12
(.2)(27)+(.8)(26.12)=26.296
(.2)(32)+(.8)(26.296)=27.437
(.2)(48)+(.8)(27.437)=31.549
(.2)(48)+(.8)(31.549)=31.840
(.2)(33)+(.8)(31.840)=32.872
(.2)(37)+(.8)(32.872)=33.697
(.2)(50)+(.8)(33.697)=36.958
etc.
1)1( 

tt
t
SX
S

s1 = x1 since
no prior
information
exists

17. Sales vs. Smoothed Sales
• Fluctuations have
been smoothed
• NOTE: the smoothed
value in this case is
generally a little low,
since the trend is
upward sloping and the
weighting factor is
only .2 0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10
Time Period
Sales
Sales Smoothed

18. ARMA Model
• Autoregressive moving average model combines both p autoregressive terms and q moving
average terms.
• Where εt ⁓ WN(0,σ2)
• Φi and Ɵi are the parameters of the process
• Special Cases: q=0 Autoregressive model(p)
p=0 Moving Average model(q)
Model :
yt=µ+ Φ1yt-1+ Φ2yt-2+…+Φpyt-p + εt + Ɵ1εt-1 + Ɵ2εt-2 + ………+ Ɵqεt-q

19. The models will be specified in terms of the lag operator L. In these terms then the AR(p) model is given by

20. Thank You