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Econ 240C

Lecture Five

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Outline Box-Jenkins Models Time Series Components Model Autoregressive of order one

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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)

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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)

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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)

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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, )

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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)

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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.

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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

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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

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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)

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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.

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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)

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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).

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Part II. Behavior of Autoregressive Processes of the

First Order From PowerPoint Lecture Three

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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/

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Trace of Inventory to Sales, Total Business

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92 93 94 95 96 97 98 99 00 01 02 03

RATIOINVSALE

Ratio of Inventory to Sales, Monthly, 1992:01-2003:01

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Behavior of ARONE Processes

So we have a typical trace of an ARONE How about the histogram? How about the correlogram?

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Histogram: Ratio of Inventory to Sales, Total Business

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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

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

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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)

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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)

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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

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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)

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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)

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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

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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).

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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)

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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.

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Autocorrelation Function of an Autoregressive Process of the

First OrderLag u Autocorrelation

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Autocorrelation of Autoregressive Time Series of First Order, b=0.9

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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

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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

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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

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Article About Leading Indices

http://www.tcb-indicators.org/GeneralInfo/bci4.pdf

BCI Web page: http://www.tcb-indicators.org/

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Revised Versus Old Leading Index

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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)

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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.

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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.

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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)

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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).

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Puzzles

Annual data on output per hour; all persons, manufacturing

measure of productivity

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Fractional Changes: Productivity

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DLNPROD

Fractional Changes in Productivity

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Histogram

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-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

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