1 econ 240c lecture five. 2 3 4 5 outline w box-jenkins models w time series components model w...
Post on 20-Dec-2015
217 views
TRANSCRIPT
![Page 1: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/1.jpg)
1
Econ 240C
Lecture Five
![Page 2: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/2.jpg)
2
![Page 3: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/3.jpg)
3
![Page 4: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/4.jpg)
4
![Page 5: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/5.jpg)
5
Outline Box-Jenkins Models Time Series Components Model Autoregressive of order one
![Page 6: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/6.jpg)
6
Box and Jenkins Analysis The grand design A stationary time series, x(t), is modeled as
the ratio of polynomials in the lag operator times white noise, wn(t)
X(t) = A(z)/B(z) * wn(t) Example 1: A(z)=1=z0, B(z)=1, x(t) =wn(t) Example 2: A(z)=1, B(z) = (1-z), x(t) =rw(t)
![Page 7: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/7.jpg)
7
ARIMA models cont.
Example 3: A(z) =1, B(z) = (1-bz), x(t) =ARONE(t)
Historically, before Box and Jenkins, time series were modeled as higher order autoregressive processes
Example 4: A(z) = 1, B(z) = (1 –b1z –b2 z2), x(t) =ARTWO(t)
![Page 8: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/8.jpg)
8
ARIMA models cont.
To estimate higher order models, you have to estimate more parameters, i.e. tease more information out of the data, risking insignificant parameters
Box and Jenkins discovered that by using a ratio of polynomials you could get by with fewer paramemters: x(t) = [(1 + az)/(1-bz)] wn(t)
![Page 9: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/9.jpg)
9
Part I: Time Series Components Model The conceptual framework for inertial
(mechanical) time series models: Time series (t) = trend + cycle + seasonal +
residual We are familiar with trend models, e.g. Time series = a + b*t + e(t) , i.e. time series = trend + residual
where e(t) is i.i.d. N(0, )
![Page 10: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/10.jpg)
10
Time Series Components Model We also know how to deal with seasonality.
For example, using quarterly data we could add a dummy zero-one variable, D1 that takes on the value of one if the observation is for the first quarter and zero otherwise. Similarly, we could add dummy variables for second quarter observations and for third quarter observations: Time series = a + b*t + c1*D1 + c2*D2 + c3*D3 +
e(t)
![Page 11: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/11.jpg)
11
Time Series Components Models So we have: time
series = trend + seasonal + residual But how do we model cycles? Since macroeconomic variables are likely to
be affected by economic conditions and the business cycle, this is an important question.
The answer lies in Box-Jenkins or ARIMA models.
![Page 12: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/12.jpg)
12
Time Series Components Models ARIMA models are about modeling the
residual The simplest time series model is:
time series(t) = white noise(t) some other time series models are of the
form: time series = A(Z)*white noise(t), where A(Z) is a polynomial in Z, a dynamic multiplier for white noise.
For the white noise model, A(Z) = Z0 =1
![Page 13: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/13.jpg)
13
Time Series Components Models For the random walk, RW(t), RW(t) =
A(Z)*WN(t) where A(Z) = (1+Z+Z2 +Z3 + …)
For the autoregressive process of the first order, ARONE(t), ARONE(t) = A(Z) * WN(t) where A(Z) = (1+b*Z+b2*Z2 +b3*Z3 +…) and -1<b<1 , i.e. b is on the real number line and is less than one in absolute value
![Page 14: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/14.jpg)
14
Time Series Components Models So ARIMA models have a residual, white
noise, as an input, and transform it with the polynomial in lag, A(Z), to model time series behavior.
One can think of ARIMA models in terms of the time series components model, where the time series, y(t), for quarterly data is modeled as: y(t)=a+b*t+c1*D1+c2*D2 +c3*D3+A(Z)WN(t)
![Page 15: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/15.jpg)
15
Time Series Components Models But y(t), with a trend component, and a
seasonal component, is evolutionary, i.e. time dependent, on two counts. So first we difference the time series, y(t), to remove trend, and seasonally difference it to remove the seasonal component, making it stationary. Then we can model it as an ARMA model, i.e. an autoregressive- moving average time series.
![Page 16: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/16.jpg)
16
Time Series Components Models
Symbolically, we difference, , y(t) to remove trend, obtaining y(t)
Then we seasonally difference, S, y(t) to remove the seasonality, obtaining S y(t).
Now we can model this stationary time series, S y(t) as ARMA, e.g.
S y(t) = A(Z)*WN(t)
![Page 17: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/17.jpg)
17
Time Series Components Models After modeling S y(t) as an ARMA
process, we can recover the model for the original time series, y(t), by undoing the differencing and seasonal differencing.
This is accomplished by summation, i.e. integration, the inverse of differencing. Hence the name autoregressive integrated moving average, or ARIMA, for the model of y(t).
![Page 18: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/18.jpg)
18
Part II. Behavior of Autoregressive Processes of the
First Order From PowerPoint Lecture Three
![Page 19: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/19.jpg)
19
Model Three: Autoregressive Time Series of Order One
An analogy to our model of trend plus shock for the logarithm of the Standard Poors is inertia plus shock for an economic time series such as the ratio of inventory to sales for total business
Source: FRED http://research.stlouisfed.org/fred/
![Page 20: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/20.jpg)
20
Trace of Inventory to Sales, Total Business
1.30
1.35
1.40
1.45
1.50
1.55
1.60
92 93 94 95 96 97 98 99 00 01 02 03
RATIOINVSALE
Ratio of Inventory to Sales, Monthly, 1992:01-2003:01
![Page 21: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/21.jpg)
21
Behavior of ARONE Processes
So we have a typical trace of an ARONE How about the histogram? How about the correlogram?
![Page 22: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/22.jpg)
22
Histogram: Ratio of Inventory to Sales, Total Business
0
4
8
12
16
1.35 1.40 1.45 1.50 1.55
Series: RATIOINVSALESample 1992:01 2003:01Observations 133
Mean 1.449925Median 1.450000Maximum 1.560000Minimum 1.350000Std. Dev. 0.047681Skewness 0.027828Kurtosis 2.475353
Jarque-Bera 1.542537Probability 0.462426
Ratio of Inventory to Sales
![Page 23: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/23.jpg)
Ratio of Inventory to SalesSample: 1992:01 2003:01Included observations: 133
Autocorrelation Partial CorrelationAC PAC Q-Stat Prob
.|*******| .|*******|1 0.928 0.928 117.23 0.000 .|*******| .|* | 2 0.886 0.175 224.83 0.000 .|*******| .|* | 3 0.851 0.074 324.86 0.000 .|****** | *|. | 4 0.803 -0.087 414.54 0.000 .|****** | .|. | 5 0.764 0.019 496.51 0.000 .|****** | *|. | 6 0.715 -0.092 568.80 0.000 .|***** | .|. | 7 0.665 -0.048 631.90 0.000 .|***** | *|. | 8 0.611 -0.086 685.61 0.000 .|**** | .|. | 9 0.563 0.001 731.48 0.000 .|**** | .|. | 10 0.513 -0.038 769.87 0.000 .|**** | .|. | 11 0.462 -0.028 801.26 0.000 .|*** | .|. | 12 0.416 -0.006 826.93 0.000
![Page 24: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/24.jpg)
24
Part III. Characterizing Autoregressive Processes of the
First Order ARONE(t) = b*ARONE(t-1) + WN(t) Lag by one ARONE(t-1) = b*ARONE(t-2) + WN(t-1) Substitute for ARONE(t-1) ARONE(t) = b*[b*ARONE(t-2) + WN(t-1] +
WN(t) ARONE(t) = WN(t) + b*WN(t-1) +b2*ARONE(t-2)
![Page 25: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/25.jpg)
25
Characterize AR of 1st Order
Keep lagging and substituting to obtain ARONE(t) = WN(t) +b*WN(t-1) +
b2*WN(t-2) + ….. ARONE(t) = [1+b*Z+b2Z2+…] WN(t) ARONE(t) = A(Z)*WN(t)
![Page 26: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/26.jpg)
26
Characterize AR of 1st Order
ARONE(t) = WN(t) +b*WN(t-1) + b2*WN(t-2) + …..
Note that the mean function of an ARONE process is zero
m(t) = E ARONE(t) = E{WN(t) + b*WN(t-1) + b2*WN(t-2) + …..} where E WN(t) =0, and EWN(t-1) =0 etc.
m(t) = 0
![Page 27: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/27.jpg)
27
WN(t) and WN(t-1)
WN(t) WN(t-1)WN(1)WN(2) WN(1)WN(3) WN(2)WN(4) WN(3)WN(5) WN(4)WN(6) WN(5)
![Page 28: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/28.jpg)
28
Autocovariance of ARONE E{[ARONE(t) - EARONE(t)]*[ARONE(t-u)-
EARONE(t-u)]}=E{ARONE(t)*ARONE(t-u)] since EARONE(t) = 0 = EARONE(t-u)
So AR,AR(u) = E{ARONE(t)*ARONE(t-u)}
For u=1, i.e. lag one, AR,AR(1) = E{ARONE(t)*ARONE(t-1)}, and
use ARONE(t) = b*ARONE(t-1) + WN(t) and multiply byARONE(t-1)
![Page 29: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/29.jpg)
29
Autocovariance of ARONE ARONE(t)*ARONE(t-1) = b*[ARONE(t-1)]2
+ARONE(t-1)*WN(t) and take expectations, E E{ARONE(t)*ARONE(t-1) = b*[ARONE(t-
1)]2 +ARONE(t-1)*WN(t)} where the LHS E{ARONE(t)*ARONE(t-1) is
AR,AR(1) by definition and
b*E *[ARONE(t-1)]2 is b*AR,AR(0) , i.e. b* the variance by definition but how about
![Page 30: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/30.jpg)
30
Autocovariance of an ARONE
E{ARONE(t-1)*WN(t)} = ? Note that ARONE(t) = WN(t) +b*WN(t-1) +
b2*WN(t-2) + ….. And lagging by one, ARONE(t-1) = WN(t-1)
+b*WN(t-2) + b2*WN(t-3) + ….. So ARONE(t-1) depends on WN(t-1) and earlier
shocks, so that E{ARONE(t-1)*WN(t)} = 0, i.e. ARONE(t-1) is independent of WN(t).
![Page 31: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/31.jpg)
31
Autocovariance of ARONE In sum, AR,AR(1) = b*AR,AR(0)
or in general for an ARONE, AR,AR(u) = b*AR,AR(u-1) which can be confirmed by taking the formula :
ARONE(t) = b*ARONE(t-1) + WN(t), multiplying by ARONE(t-u) and taking expectations.
Note AR,AR (u) = AR,AR(u) / AR,AR(0)
![Page 32: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/32.jpg)
32
Autocorrelation of ARONE(t)
So dividing AR,AR(u) = b*AR,AR(u-1) by AR,AR(0) results in
AR,AR (u) = b* AR,AR (u-1), u>0
AR,AR (1) = b* AR,AR (0) = b
AR,AR (2) = b* AR,AR (1) = b*b = b2
etc.
![Page 33: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/33.jpg)
33
Autocorrelation Function of an Autoregressive Process of the
First OrderLag u Autocorrelation
0 1
1 b
2 b2
3 b3
![Page 34: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/34.jpg)
Autocorrelation of Autoregressive Time Series of First Order, b=0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
Lag
Au
toc
orr
ela
tio
n
![Page 35: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/35.jpg)
35
Part IV. Forecasting-Information Sources
The Conference Board publishes a monthly, Business Cycle Indicators
A monthly series followed in the popular press is the Index of Leading Indicators
![Page 36: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/36.jpg)
36
![Page 37: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/37.jpg)
37
![Page 38: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/38.jpg)
38
Leading Index Components Average Weekly Hours, manufacturing Initial claims For Unemployment Insurance Manufacturers’ New Orders
• Consumer goods
Vendor Performance Building Permits
• new private housing
Manufacturers’ New Orders• nondefense capital goods
![Page 39: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/39.jpg)
39
Leading Index (Cont.0 Stock Prices
• 500 common stocks
Money Supply M2 Interest Rate Spread
• 10 Treasury bonds - Federal Funds Rate
Index of Consumer Expectations
![Page 40: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/40.jpg)
40
Article About Leading Indices
http://www.tcb-indicators.org/GeneralInfo/bci4.pdf
BCI Web page: http://www.tcb-indicators.org/
![Page 41: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/41.jpg)
41
Revised Versus Old Leading Index
![Page 42: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/42.jpg)
42
Part V: Autoregressive of 1st Order The model for an autoregressive process of
the first order, ARONE(t) is: ARONE(t) = b*ARONE(t-1) + WN(t) or, using the lag operator,
ARONE(t) = b*ZARONE(t) + WN(t), i.e. ARONE(t) - b*ZARONE(t) = WN(t), and factoring out ARONE(t): [1 -b*Z]*ARONE(t) = WN(t)
![Page 43: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/43.jpg)
43
Autoregressive of 1st Order Dividing by [1 - b*Z], ARONE(t) = {1/[1 - b*Z]}WN(t)
where the reciprocal of [1 - bZ] is: {1/[1 - b*Z]} = (1+b*Z+b2*Z2 +b3*Z3 +…) which can be verified by multiplying [1 - b*Z] by (1+b*Z+b2*Z2 +b3*Z3 +…) to obtain 1.
![Page 44: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/44.jpg)
44
Autoregressive of the First Order Note that we can write ARONE(t) as
1/[1-b*Z]*WN(t), i.e. ARONE(t) ={1/B(Z)}* WN(t), where B(Z) = [1 - b*Z] is a first order polynomial in Z,
Or, we can write ARONE(t) as ARONE(t) = A(Z) * WN(t) where A(Z) = (1+b*Z+b2*Z2 +b3*Z3 +…) is a polynomial in Z of infinite order.
![Page 45: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/45.jpg)
45
Autoregresive of the First Order So 1/B(Z) = A(Z). A first order polynomial
in the denominator can approximate an infinite order polynomial in the numerator.
Box and Jenkins achieved parsimony, i.e. the use of only a few parameters which you need to estimate by modeling time series using the ratio of low order polynomials in the numerator and denominator:
ARMA(t) = {A(Z)/B(Z)}* WN(t)
![Page 46: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/46.jpg)
46
Autoregressive of the First Order For Now we will concentrate on the
denominator: ARONE(t) = {1/B(Z)}*WN(t), where the polynomial in the denominator, B(Z) = [1 - b*Z], captures autoregressive behavior of the first order.
Later, we will turn our attention to the numerator, where A(Z) captures moving average behavior.
Then we will combine A(Z) and B(Z).
![Page 47: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/47.jpg)
47
Puzzles
Annual data on output per hour; all persons, manufacturing
measure of productivity
![Page 48: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/48.jpg)
48
Fractional Changes: Productivity
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
50 55 60 65 70 75 80 85 90 95 00
DLNPROD
Fractional Changes in Productivity
![Page 49: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/49.jpg)
49
Histogram
0
5
10
15
20
25
-0.10 -0.05 0.00 0.05 0.10 0.15
Series: DLNPRODSample 1950 2000Observations 51
Mean 0.027297Median 0.032187Maximum 0.142921Minimum -0.099530Std. Dev. 0.040806Skewness -0.691895Kurtosis 6.957785
Jarque-Bera 37.35525Probability 0.000000
Fractional Changes in Productivity
![Page 50: 1 Econ 240C Lecture Five. 2 3 4 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one](https://reader030.vdocument.in/reader030/viewer/2022032800/56649d445503460f94a20501/html5/thumbnails/50.jpg)
Date: 04/15/03 Time: 16:59
Sample: 1949 2000Included observations: 51
Autocorrelation Partial CorrelationAC PAC Q-Stat Prob
**| . | **| . | 1 -0.237 -0.237 3.0243 0.082 . | . | .*| . | 2 -0.036 -0.097 3.0953 0.213 .*| . | .*| . | 3 -0.059 -0.098 3.2906 0.349 ***| . | ****| . | 4 -0.402 -0.481 12.605 0.013 . |**** | . |**** | 5 0.588 0.461 32.933 0.000 .*| . | .*| . | 6 -0.187 -0.132 35.033 0.000 . | . | .*| . | 7 -0.041 -0.139 35.136 0.000 . | . | .*| . | 8 -0.017 -0.125 35.155 0.000 **| . | . |** | 9 -0.221 0.205 38.310 0.000 . |*** | .*| . | 10 0.351 -0.107 46.419 0.000 . | . | . |*. | 11 -0.030 0.114 46.480 0.000