1

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

2

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.

3

The most often seen form of the equation is a linear form:

p

ititit eybby

10

where:yt – the 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,p – autoregression rank,et – disturbance term.

4

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

5

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

6

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.

7

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)

8

)( p

p

bS

bZ

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

9

Some helpful information:

 

645,11,0 Z

960,105,0 Z

236,202,0 Z

576,201,0 Z

291,3001,0 Z

10

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.

11

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.

12

t yt

1 1.892 2.463 3.234 3.955 4.566 5.077 5.628 6.169 6.2610 6.5611 6.9812 7.3613 7.5314 7.8415 8.09

tttt eybybby 22110

yXXXb TT 1)(

p = 2ytmean = 5,570667

n = 13k = 2

Example 1

13

t yt yt-1 yt-2

1 1.89 - -2 2.46 1.89 -3 3.23 2.46 1.894 3.95 3.23 2.465 4.56 3.95 3.236 5.07 4.56 3.957 5.62 5.07 4.568 6.16 5.62 5.079 6.26 6.16 5.6210 6.56 6.26 6.1611 6.98 6.56 6.2612 7.36 6.98 6.5613 7.53 7.36 6.9814 7.84 7.53 7.3615 8.09 7.84 7.53

p = 2ytmean = 5.570667

n = 13k = 2

Calculations

14

b0

b = b1

b2

3,23

3,95

4,565,075,62

6,16

y = 6,26

6,56

6,98

7,36

7,537,848,09

1 2,46 1,89

1 3,23 2,46

1 3,95 3,231 4,56 3,951 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 6,981 7,53 7,361 7,84 7,53

15

13 73,58 67,63

XTX= 73,58 451,3932 420,84267,63 420,8423 393,5

5,523661 -5,28007 4,69762

(XTX)-1= -5,28007 5,811533 -5,307884,697623 -5,30788 4,87187

79,21

XTy= 479,6185446,1821

1,103369

b = 0,8049360,08338

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

08,08,01,1ˆ 21 tttt eyyy

16

t yt yt-1 yt-2 y^t yt - y

^t (yt - y

^t)

2 yt - ytmean (yt - ytmean)2

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 2,626565 4,56 3,95 3,23 4,552185 0,007815 6,11E-05 -1,01067 1,0214476 5,07 4,56 3,95 5,103229 -0,03323 0,001104 -0,50067 0,2506677 5,62 5,07 4,56 5,564609 0,055391 0,003068 0,049333 0,0024348 6,16 5,62 5,07 6,049848 0,110152 0,012134 0,589333 0,3473149 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 0,9787811 6,98 6,56 6,26 6,90571 0,07429 0,005519 1,409333 1,9862212 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 3,83898714 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

17

Variance

S2 = 0,012683

Standard error of the estimateS = 0,112619

Variance and covarince matrix0,070057 -0,06697 0,059581

D2(b) = -0,06697 0,073708 -0,067320,059581 -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

1,103369b = 0,804936

0,08338

18

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.

19

t yt yt-1

1 1,89 -2 2,46 1,893 3,23 2,464 3,95 3,235 4,56 3,956 5,07 4,567 5,62 5,07

8 6,16 5,62

9 6,26 6,1610 6,56 6,2611 6,98 6,5612 7,36 6,9813 7,53 7,3614 7,84 7,5315 8,09 7,84

REGLINP0,914 0,9040,0173 0,09850

99,573% 0,1202800,6 1240,241 0,172

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

The first-order parameter contributes to the prediction of Y

20

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 55,45 90,4 87,2 68,7 61,56 86,2 90,4 87,2 68,77 94,7 86,2 90,4 87,28 103,2 94,7 86,2 90,49 119 103,2 94,7 86,210 122,4 119 103,2 94,711 131,6 122,4 119 103,212 157,6 131,6 122,4 11913 181 157,6 131,6 122,414 217,8 181 157,6 131,615 244,1 217,8 181 157,6

21

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

55,4 - - -61,5 55,4 - -68,7 61,5 55,4 -87,2 68,7 61,5 55,490,4 87,2 68,7 61,586,2 90,4 87,2 68,794,7 86,2 90,4 87,2103,2 94,7 86,2 90,4119 103,2 94,7 86,2

122,4 119 103,2 94,7131,6 122,4 119 103,2157,6 131,6 122,4 119181 157,6 131,6 122,4

217,8 181 157,6 131,6244,1 217,8 181 157,6

Third-order autoregressive modelb3 b2 b1 b0

0,2903 -0,1987 1,1541 -11,04380,4485 0,5982 0,3569 10,79190,9753 9,7932 #N/D! #N/D!

Z b3 0,647227 Z 0,05 1,96The third-order parameter does not contribute to the prediction of Y

22

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

55,4 - - -61,5 55,4 - -68,7 61,5 55,4 -87,2 68,7 61,5 55,490,4 87,2 68,7 61,586,2 90,4 87,2 68,794,7 86,2 90,4 87,2103,2 94,7 86,2 90,4119 103,2 94,7 86,2

122,4 119 103,2 94,7131,6 122,4 119 103,2157,6 131,6 122,4 119181 157,6 131,6 122,4

217,8 181 157,6 131,6244,1 217,8 181 157,6

Second-order autoregressive modelb2 b1 b0

0,0220 1,1616 -7,15500,4000 0,3254 8,39270,9767 9,0609 #N/D!

Z b2 0,054917 Z 0,05 1,96The second-order parameter does not contribute to the prediction of Y

23

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

55,4 - - -61,5 55,4 - -68,7 61,5 55,4 -87,2 68,7 61,5 55,490,4 87,2 68,7 61,586,2 90,4 87,2 68,794,7 86,2 90,4 87,2103,2 94,7 86,2 90,4119 103,2 94,7 86,2

122,4 119 103,2 94,7131,6 122,4 119 103,2157,6 131,6 122,4 119181 157,6 131,6 122,4

217,8 181 157,6 131,6244,1 217,8 181 157,6

First-order autoregressive modelb1 b0

1,1729 -5,99240,0494 5,98940,9792 8,3118

Z b1 23,74814 Z 0,05 1,96The first-order parameter does contribute to the prediction of Y

24

Autogregressive Modeling

Used for ForecastingTakes Advantage of Autocorrelation

1st order - correlation between consecutive values

2nd order - correlation between values 2 periods apart

Autoregressive Model for pth order:

ipipiii eYbYbYbbY 22110

Random Error

25

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 p variables 4. Test significance of Bp

If null hypothesis rejected, this model is selected If null hypothesis not rejected, decrease p by 1 and

repeat your calculations