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CHAPTER 6FORECASTING WITH MOVING AVERAGE (MA) MODELS

González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Figure 6.1 Time Series and Autocorrelation Functions of a Simulated White Noise Process

6.1 A Model with No Dependence: White Noise

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25 50 75 100 125 150

Ynoise=1+N(0,4)

0

20

40

60

80

100

-4 -2 0 2 4 6

Series: YnoiseSample 1 1000Observations 1000

Mean 0.957014Median 0.941058Maximum 7.235267Minimum -5.443433Std. Dev. 2.034487Skewness -0.029960Kurtosis 2.784223

Jarque-Bera 2.089597Probability 0.351763

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Figure 6.2 Autocorrelation Functions of Monthly Returns to Microsoft and the Dow Jones Index

González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

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86 88 90 92 94 96 98 00 02 04

Microsoft monthly returns

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86 88 90 92 94 96 98 00 02 04

Dow Jones monthly returns

3González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Figure 6.3 Simulated MA(1) Processes

6.3.1 MA(1) Process

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MA(1) with theta=0.5

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MA(1) with theta= 0.05

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MA(1) with theta=2.0

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MA(1) with theta=0.95

4González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Figure 6.4 Autocorrelation Functions of Simulated MA(1) Processes

05.0

95.0

5.0

2

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Figure 6.5 Percentage Changes in the 5-Year Treasury Note Yield

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55 60 65 70 75 80 85 90 95 00 05

Change in the 5-year Treasury-Note Yield (%)

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Table 6.1 Estimation Output: 5-Year Treasury Yield (Monthly Percentage Changes)

González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

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Table 6.2 December 2007-April 2008 Forecasts of 5-year Treasure Yield Changes

1h

12/2007 %276.6ˆ485.0160.0

ˆˆ1,

t

ttf

222|1 449.4ˆ tt

)449.4,276.6(

),()|(2

2|1|11

N

NIYf tttttt

2h

1/2008

%160.0ˆ2, tf

2

22

222|2

ˆ683.23

)485.01(449.4

)ˆ1(ˆ

Y

tt

)683.23,16.0()|( 2 NIYf tt

3h

2/2008

%160.0ˆ3, tf 22|3 ˆ683.23 Ytt )683.23,16.0()|( 3 NIYf tt

4h

3/2008

%160.0ˆ4, tf 22|4 ˆ683.23 Ytt )683.23,16.0()|( 4 NIYf tt

5h

4/2008

%160.0ˆ5, tf 22|5 ˆ683.23 Ytt )683.23,16.0()|( 5 NIYf tt

González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

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Figure 6.6 Multistep Forecast of Monthly Changes of 5-year Treasury Yield

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2007M01 2007M04 2007M07 2007M10 2008M01 2008M04

Actual data (5-year Treasure rate changes)ForecastLower bound, 95% confidence intervalUpper bound, 95% confidence interval

estimation sample forecasting sample

multistep forecast

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Figure 6.7 Simulated MA(2) Processes

6.3.2 MA(q) Process

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MA(2) with theta_1=1.70 and theta_2=0.72

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MA(2) with theta_1=-1 and theta_2=0.25

ttttY 21 25.02ttttY 21 72.07.12

(a) (b)

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Figure 6.8 Autocorrelation Functions of Simulated MA(2) Processes

ttttY 21 25.02

ttttY 21 72.07.12

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Figure 6.9 Autocorrelation Functions of MA Process ttttY 21 442

González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

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Figure 6.10 Microsoft Daily Stock Prices

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Stock Price (close) 20-day moving average

Jun/03 Mar/04 Dec/04

MICROSOFT daily stock pricesJan.8, 2003-April 13, 2005

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Figure 6.11 Daily Returns to Microsoft

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Daily returns (3-day moving average smoothed price)

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Daily returns (close price)

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Figure 6.12 Autocorrelograms of Microsoft Daily Returns based on the Closing Price (top) and on the 3-day Moving Average Price (bottom)