7052 autoregressive models (1)

7/31/2019 7052 Autoregressive Models (1)

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Another useful model is autoregressive model.

Frequently, we find that the values of a series of financial

data at particular points in time are highly correlated with

the value which precede and succeed them.

Autoregressive models


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Autoregressive models

Models with lagged variable

Dependent variable is a function of itself at the

previous moment of period or time.

),...,,( ,21 tptttt yyyfy

The creation of an autoregressive model generates a new

predictor variable by using the Y variable lagged 1 or more

periods.


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The most often seen form of the equation is a linear

form:

p

ititit eybby 10

where:

ytthe dependent variable values at the moment t,

yt-i

(i = 1, 2, ..., p)the dependent variable values at the

moment t-i,

bo, bi (i=1,..., p)regression coefficient,

pautoregression rank,

etdisturbance term.


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pb

b

b

b

.

.

.

.

1

0

n

p

p

y

y

y

y.

.

.

.

.

2

1

pnnn

pp

pp

yyy

yyy

yyy

X

21

....

...

....

....

21

11

1

1

...1


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A first-order autoregressive model is concerned with only the

correlation between consecutive values in a series.

A second-order autoregressive model considers the effect of

relationship between consecutive values in a series as well as

the correlation between values two periods apart.

tttt eybybby 22110

ttt eybby 110


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The selection of an appropriate autoregressive model is

not an easy task.

Once a model is selected and OLS method is used to

obtain estimates of the parameters, the next step would be

to eliminate those parameters which do not contribute

significantly.


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0;0 pH

(The highest-order parameter does not contribute to the

prediction of Yt)

0;1 pH

(The highest-order parameter is significantly meaningful)


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)(p

p

bS

b

Z

using an alpha level of significance, the decision rule is

to reject H0 if ZZ or if ZZ

and not to reject H0 if ZZZ


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Some helpful information:

645,11,0

Z

960,105,0 Z

236,202,0 Z

576,201,0 Z

291,3001,0 Z


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If the null hypothesis is NOT rejected we may conclude that

the selected model contains too many estimated parameters.

The highest-order term then be deleted an a new

autoregressive model would be obtained through least-

squares regression. A test of the hypothesis that the new

highest-order term is 0 would then be repeated.


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This testing and modeling procedure continues until we

reject H0. When this occurs, we know that our highest-order

parameter is significant and we are ready to use this model.


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t yt

1 1.89

2 2.46

3 3.23

4 3.95

5 4.566 5.07

7 5.62

8 6.16

9 6.26

10 6.56

11 6.9812 7.36

13 7.53

14 7.84

15 8.09

tttt eybybby 22110

yXXXb TT 1)(

p = 2

ytmean = 5,570667

n = 13

k = 2

Example 1


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t yt yt-1 yt-2

1 1.89 - -

2 2.46 1.89 -

3 3.23 2.46 1.89

4 3.95 3.23 2.46

5 4.56 3.95 3.236 5.07 4.56 3.95

7 5.62 5.07 4.56

8 6.16 5.62 5.07

9 6.26 6.16 5.62

10 6.56 6.26 6.16

11 6.98 6.56 6.2612 7.36 6.98 6.56

13 7.53 7.36 6.98

14 7.84 7.53 7.36

15 8.09 7.84 7.53

p = 2

ytmean = 5.570667

n = 13

k = 2

Calculations


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b0

b = b1

b2

3,23

3,95

4,56

5,07

5,62

6,16

y = 6,26

6,56

6,98

7,36

7,537,84

8,09

1 2,46 1,89

1 3,23 2,46

1 3,95 3,23

1 4,56 3,95

1 5,07 4,56

1 5,62 5,07

X = 1 6,16 5,62

1 6,26 6,16

1 6,56 6,26

1 6,98 6,56

1 7,36 6,98

1 7,53 7,36

1 7,84 7,53


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13 73,58 67,63

XTX= 73,58 451,3932 420,842

67,63 420,8423 393,5

5,523661 -5,28007 4,69762

(XTX)-1= -5,28007 5,811533 -5,30788

4,697623 -5,30788 4,87187

79,21

XTy= 479,6185

446,1821

1,103369

b = 0,804936

0,08338

)25,0()27,0()26,0(

08,08,01,1 21 tttt eyyy


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t yt yt-1 yt-2 y^

t yt - y^t (yt - y

^t)

2 yt - ytmean yt - ytmean)

1 1,89 - - - - - - -

2 2,46 1,89 - - - - - -

3 3,23 2,46 1,89 3,2411 -0,0111 0,000123 -2,34067 5,47872

4 3,95 3,23 2,46 3,908428 0,041572 0,001728 -1,62067 2,62656

5 4,56 3,95 3,23 4,552185 0,007815 6,11E-05 -1,01067 1,021447

6 5,07 4,56 3,95 5,103229 -0,03323 0,001104 -0,50067 0,250667

7 5,62 5,07 4,56 5,564609 0,055391 0,003068 0,049333 0,002434

8 6,16 5,62 5,07 6,049848 0,110152 0,012134 0,589333 0,347314

9 6,26 6,16 5,62 6,530372 -0,27037 0,073101 0,689333 0,47518

10 6,56 6,26 6,16 6,655891 -0,09589 0,009195 0,989333 0,97878

11 6,98 6,56 6,26 6,90571 0,07429 0,005519 1,409333 1,98622

12 7,36 6,98 6,56 7,268797 0,091203 0,008318 1,789333 3,20171413 7,53 7,36 6,98 7,609693 -0,07969 0,006351 1,959333 3,838987

14 7,84 7,53 7,36 7,778216 0,061784 0,003817 2,269333 5,149874

15 8,09 7,84 7,53 8,041921 0,048079 0,002312 2,519333 6,34704

0,126831 31,70494


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Variance

S2 = 0,012683

Standard error of the estimate

S = 0,112619

Variance and covarince matrix

0,070057 -0,06697 0,059581

D2(b) = -0,06697 0,073708 -0,06732

0,059581 -0,06732 0,061791

Standard errors of the coefficients

D(b0) = 0,264684

D(b1) = 0,271493

D(b2) = 0,248577

Indetermination coefficient

0,004

Determination coefficient

R2 = 0,996

Goodness of fit

2

1,103369

b = 0,804936

0,08338


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Z b2= 0,33543 Z 0,05= 1,96

The second-order parameter does not contribute to the prediction of Y

Calculations

We have to estimate the parameters of the first-order

autoregressive model:

ttt eybby

110

and then check if Beta1 is statistically significant.


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t yt yt-1

1 1,89 -

2 2,46 1,89

3 3,23 2,464 3,95 3,23

5 4,56 3,95

6 5,07 4,56

7 5,62 5,07

8 6,16 5,62

9 6,26 6,1610 6,56 6,26

11 6,98 6,56

12 7,36 6,98

13 7,53 7,36

14 7,84 7,53

15 8,09 7,84

REGLINP

0,914 0,904

0,0173 0,0985099,573% 0,120

2800,6 12

40,241 0,172

Z b1= 52,921 Z 0,05= 1,96

The first-order parameter contributes to the prediction of Y


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Example 2

Y - annual income taxesYear Yt Yt-1 Yt-2 Yt-3

1 55,4 - - -

2 61,5 55,4 - -

3 68,7 61,5 55,4 -

4 87,2 68,7 61,5 55,4

5 90,4 87,2 68,7 61,56 86,2 90,4 87,2 68,7

7 94,7 86,2 90,4 87,2

8 103,2 94,7 86,2 90,4

9 119 103,2 94,7 86,2

10 122,4 119 103,2 94,7

11 131,6 122,4 119 103,212 157,6 131,6 122,4 119

13 181 157,6 131,6 122,4

14 217,8 181 157,6 131,6

15 244,1 217,8 181 157,6


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income taxes

Yt Yt-1 Yt-2 Yt-3

55,4 - - -

61,5 55,4 - -

68,7 61,5 55,4 -

87,2 68,7 61,5 55,4

90,4 87,2 68,7 61,5

86,2 90,4 87,2 68,7

94,7 86,2 90,4 87,2

103,2 94,7 86,2 90,4

119 103,2 94,7 86,2

122,4 119 103,2 94,7

131,6 122,4 119 103,2

157,6 131,6 122,4 119

181 157,6 131,6 122,4

217,8 181 157,6 131,6

244,1 217,8 181 157,6

Third-order autoregressive model

b3 b2 b1 b0

0,2903 -0,1987 1,1541 -11,0438

0,4485 0,5982 0,3569 10,7919

0,9753 9,7932 #N/D! #N/D!

Z b3 0,647227 Z 0,05 1,96

The third-order parameter does not contribute to the prediction of Y


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income taxes

Yt Yt-1 Yt-2 Yt-3

55,4 - - -

61,5 55,4 - -

68,7 61,5 55,4 -

87,2 68,7 61,5 55,4

90,4 87,2 68,7 61,5

86,2 90,4 87,2 68,7

94,7 86,2 90,4 87,2

103,2 94,7 86,2 90,4

119 103,2 94,7 86,2

122,4 119 103,2 94,7131,6 122,4 119 103,2

157,6 131,6 122,4 119

181 157,6 131,6 122,4

217,8 181 157,6 131,6

244,1 217,8 181 157,6

Second-order autoregressive modelb2 b1 b0

0,0220 1,1616 -7,1550

0,4000 0,3254 8,3927

0,9767 9,0609 #N/D!

Z b2 0,054917 Z 0,05 1,96

The second-order parameter does not contribute to the prediction of Y


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income taxes

Yt Yt-1 Yt-2 Yt-3

55,4 - - -

61,5 55,4 - -

68,7 61,5 55,4 -87,2 68,7 61,5 55,4

90,4 87,2 68,7 61,5

86,2 90,4 87,2 68,7

94,7 86,2 90,4 87,2

103,2 94,7 86,2 90,4

119 103,2 94,7 86,2

122,4 119 103,2 94,7

131,6 122,4 119 103,2

157,6 131,6 122,4 119

181 157,6 131,6 122,4

217,8 181 157,6 131,6

244,1 217,8 181 157,6

First-order autoregressive model

b1 b0

1,1729 -5,9924

0,0494 5,9894

0,9792 8,3118

Z b1 23,74814 Z 0,05 1,96

The first-order parameter does contribute to the prediction of Y


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Autogregressive Modeling

Used for Forecasting

Takes Advantage of Autocorrelation

1st order - correlation between consecutivevalues

2nd order - correlation between values 2periods apart

Autoregressive Model for pthorder:

ipipiii eYbYbYbbY 22110

Random

Error


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Autoregressive Modeling Steps

1. Choose p:

2. Form a series of lag predictor variables

Yi-1 , Yi-2, Yi-p 3. Use Excel to run regression model using

all pvariables

4. Test significance of Bp If null hypothesis rejected, this model is selected If null hypothesis not rejected, decrease pby 1 and

repeat your calculations

7052 autoregressive models (1)

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