1. 1 Econ 240C Lecture Five

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5. 5 Outline w Box-Jenkins Models w Time Series Components Model w Autoregressive of order one

6. 6 Box and Jenkins Analysis w The grand design w A stationary time series, x(t), is modeled as the ratio of polynomials in the lag operator times white noise, wn(t) w X(t) = A(z)/B(z) * wn(t) w Example 1: A(z)=1=z 0, B(z)=1, x(t) =wn(t) w Example 2: A(z)=1, B(z) = (1-z), x(t) =rw(t)

7. 7 ARIMA models cont. w Example 3: A(z) =1, B(z) = (1-bz), x(t) =ARONE(t) w Historically, before Box and Jenkins, time series were modeled as higher order autoregressive processes w Example 4: A(z) = 1, B(z) = (1 –b 1 z –b 2 z 2 ), x(t) =ARTWO(t)

8. 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)

9. 9 Part I: Time Series Components Model w The conceptual framework for inertial (mechanical) time series models: w Time series (t) = trend + cycle + seasonal + residual w We are familiar with trend models, e.g. w Time series = a + b*t + e(t), i.e.  time series = trend + residual where e(t) is i.i.d. N(0,   )

10. 10 Time Series Components Model w We also know how to deal with seasonality. For example, using quarterly data we could add a dummy zero-one variable, D 1 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 + c 1 *D 1 + c 2 *D 2 + c 3 *D 3 + e(t)

11. 11 Time Series Components Models w So we have: time series = trend + seasonal + residual w But how do we model cycles? w Since macroeconomic variables are likely to be affected by economic conditions and the business cycle, this is an important question. w The answer lies in Box-Jenkins or ARIMA models.

12. 12 Time Series Components Models w ARIMA models are about modeling the residual w The simplest time series model is: time series(t) = white noise(t) w 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. w For the white noise model, A(Z) = Z 0 =1

13. 13 Time Series Components Models w For the random walk, RW(t), RW(t) = A(Z)*WN(t) where A(Z) = (1+Z+Z 2 +Z 3 + …) w For the autoregressive process of the first order, ARONE(t), ARONE(t) = A(Z) * WN(t) where A(Z) = (1+b*Z+b 2 *Z 2 +b 3 *Z 3 +…) and -1<b<1, i.e. b is on the real number line and is less than one in absolute value w

14. 14 Time Series Components Models w 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. w 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+c 1 *D 1 +c 2 *D 2 +c 3 *D 3 +A(Z)WN(t)

15. 15 Time Series Components Models w 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.

16. 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)

17. 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. w 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).

18. 18 Part II. Behavior of Autoregressive Processes of the First Order w From PowerPoint Lecture Three

19. 19 Model Three: Autoregressive Time Series of Order One w 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 w Source: FRED http://research.stlouisfed.org/fred/ http://research.stlouisfed.org/fred/

20. 20 Trace of Inventory to Sales, Total Business

21. 21 Behavior of ARONE Processes w So we have a typical trace of an ARONE w How about the histogram? w How about the correlogram?

22. 22 Histogram: Ratio of Inventory to Sales, Total Business

23. Ratio of Inventory to Sales Sample: 1992:01 2003:01 Included observations: 133 AutocorrelationPartial Correlation AC PAC Q-Stat Prob.|*******|.|*******|10.9280.928117.230.000.|*******|.|* |20.8860.175224.830.000.|*******|.|* |30.8510.074324.860.000.|****** | *|. |40.803-0.087414.540.000.|****** |.|. |50.7640.019496.510.000.|****** | *|. |60.715-0.092568.800.000.|***** |.|. |70.665-0.048631.900.000.|***** | *|. |80.611-0.086685.610.000.|**** |.|. |90.5630.001731.480.000.|**** |.|. |100.513-0.038769.870.000.|**** |.|. |110.462-0.028801.260.000.|*** |.|. |120.416-0.006826.930.000

24. 24 Part III. Characterizing Autoregressive Processes of the First Order w ARONE(t) = b*ARONE(t-1) + WN(t) w Lag by one w ARONE(t-1) = b*ARONE(t-2) + WN(t-1) w Substitute for ARONE(t-1) w ARONE(t) = b*[b*ARONE(t-2) + WN(t-1] + WN(t) w ARONE(t) = WN(t) + b*WN(t-1) +b 2 *ARONE(t-2)

25. 25 Characterize AR of 1st Order w Keep lagging and substituting to obtain w ARONE(t) = WN(t) +b*WN(t-1) + b 2 *WN(t-2) + ….. w ARONE(t) = [1+b*Z+b 2 Z 2 +…] WN(t) w ARONE(t) = A(Z)*WN(t)

26. 26 Characterize AR of 1st Order w ARONE(t) = WN(t) +b*WN(t-1) + b 2 *WN(t-2) + ….. w Note that the mean function of an ARONE process is zero w m(t) = E ARONE(t) = E{WN(t) + b*WN(t-1) + b 2 *WN(t-2) + …..} where E WN(t) =0, and EWN(t-1) =0 etc. w m(t) = 0

27. 27 WN(t) and WN(t-1)

28. 28 Autocovariance of ARONE w 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 w use ARONE(t) = b*ARONE(t-1) + WN(t) w and multiply byARONE(t-1)

29. 29 Autocovariance of ARONE w ARONE(t)*ARONE(t-1) = b*[ARONE(t-1)] 2 +ARONE(t-1)*WN(t) w and take expectations, E w 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

30. 30 Autocovariance of an ARONE w E{ARONE(t-1)*WN(t)} = ? w Note that ARONE(t) = WN(t) +b*WN(t-1) + b 2 *WN(t-2) + ….. w And lagging by one, ARONE(t-1) = WN(t-1) +b*WN(t-2) + b 2 *WN(t-3) + ….. w 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).

31. 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 : w 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)

32. 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 = b 2 w etc.

33. 33 Autocorrelation Function of an Autoregressive Process of the First Order

35. 35 Part IV. Forecasting-Information Sources w The Conference Board publishes a monthly, Business Cycle Indicators w A monthly series followed in the popular press is the Index of Leading Indicators

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38. 38 Leading Index Components w Average Weekly Hours, manufacturing w Initial claims For Unemployment Insurance w Manufacturers’ New Orders Consumer goods w Vendor Performance w Building Permits new private housing w Manufacturers’ New Orders nondefense capital goods

39. 39 Leading Index (Cont.0 w Stock Prices 500 common stocks w Money Supply M2 w Interest Rate Spread 10 Treasury bonds - Federal Funds Rate w Index of Consumer Expectations

40. 40 Article About Leading Indices w http://www.tcb- indicators.org/GeneralInfo/bci4.pdf w BCI Web page: http://www.tcb- indicators.org/

41. 41 Revised Versus Old Leading Index

42. 42 Part V: Autoregressive of 1st Order w The model for an autoregressive process of the first order, ARONE(t) is: w ARONE(t) = b*ARONE(t-1) + WN(t) w 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)

43. 43 Autoregressive of 1st Order w Dividing by [1 - b*Z], w ARONE(t) = {1/[1 - b*Z]}WN(t) where the reciprocal of [1 - bZ] is: {1/[1 - b*Z]} = (1+b*Z+b 2 *Z 2 +b 3 *Z 3 +…) which can be verified by multiplying [1 - b*Z] by (1+b*Z+b 2 *Z 2 +b 3 *Z 3 +…) to obtain 1.

44. 44 Autoregressive of the First Order w 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, w Or, we can write ARONE(t) as ARONE(t) = A(Z) * WN(t) where A(Z) = (1+b*Z+b 2 *Z 2 +b 3 *Z 3 +…) is a polynomial in Z of infinite order.

45. 45 Autoregresive of the First Order w So 1/B(Z) = A(Z). A first order polynomial in the denominator can approximate an infinite order polynomial in the numerator. w 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: w ARMA(t) = {A(Z)/B(Z)}* WN(t)

46. 46 Autoregressive of the First Order w 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. w Later, we will turn our attention to the numerator, where A(Z) captures moving average behavior. w Then we will combine A(Z) and B(Z).

47. 47 Puzzles w Annual data on output per hour; all persons, manufacturing w measure of productivity

48. 48 Fractional Changes: Productivity

49. 49 Histogram

50. Date: 04/15/03 Time: 16:59 Sample: 1949 2000 Included observations: 51 AutocorrelationPartial Correlation AC PAC Q-Stat Prob **|. | **|. |1-0.237-0.2373.02430.082. |. |.*|. |2-0.036-0.0973.09530.213.*|. |.*|. |3-0.059-0.0983.29060.349 ***|. | ****|. |4-0.402-0.48112.6050.013. |**** |. |**** |50.5880.46132.9330.000.*|. |.*|. |6-0.187-0.13235.0330.000. |. |.*|. |7-0.041-0.13935.1360.000. |. |.*|. |8-0.017-0.12535.1550.000 **|. |. |** |9-0.2210.20538.3100.000. |*** |.*|. |100.351-0.10746.4190.000. |. |. |*. |11-0.0300.11446.4800.000