1. 1 ECON 240C Lecture 8

2. 2 Part I. Economic Forecast Project Santa Barbara County Seminar Santa Barbara County Seminar  April 22, 2004 (April 17, 2003) URL: http://www.ucsb-efp.com URL: http://www.ucsb-efp.com URL: http://www.ucsb-efp.com URL: http://www.ucsb-efp.com

3. 3 Part II. Forecasting Trends

4. 4 Lab Two: LNSP500

5. 5 Note: Autocorrelated Residual

6. 6 Autorrelation Confirmed from the Correlogram of the Residual

7. 7 Visual Representation of the Forecast

8. 8 Numerical Representation of the Forecast

9. 9 One Period Ahead Forecast Note the standard error of the regression is 0.2237 Note the standard error of the regression is 0.2237 Note: the standard error of the forecast is 0.2248 Note: the standard error of the forecast is 0.2248 Diebold refers to the forecast error Diebold refers to the forecast error  without parameter uncertainty, which will just be the standard error of the regression  or with parameter uncertainty, which accounts for the fact that the estimated intercept and slope are uncertain as well

10. 10 Parameter Uncertainty Trend model: y(t) = a + b*t + e(t) Trend model: y(t) = a + b*t + e(t) Fitted model: Fitted model:

11. 11 Parameter Uncertainty Estimated error Estimated error

12. 12 Forecast Formula

13. 13 Forecast E t E t

14. 14 Forecast error Forecast = a + b*(t+1) + 0 Forecast = a + b*(t+1) + 0

15. 15 Variance in the Forecast Error

16. 16

17. 17 Variance of the Forecast Error 0.000501 +2*(-0.00000189)*398 + 9.52x10 -9 *(398) 2 +(0.223686) 2 0.000501 - 0.00150 + 0.001508 + 0.0500354 0.505444 SEF = (0.0505444) 1/2 = 0.22482

18. 18 Numerical Representation of the Forecast

19. 19 Evolutionary Vs. Stationary Evolutionary: Trend model for lnSp500(t) Evolutionary: Trend model for lnSp500(t) Stationary: Model for Dlnsp500(t) Stationary: Model for Dlnsp500(t)

20. 20 Pre-whitened Time Series

21. 21 Note: 0 008625 is monthly growth rate; times 12=0.1035

22. 22 Is the Mean Fractional Rate of Growth Different from Zero? Econ 240A, Ch.12.2 Econ 240A, Ch.12.2 where the null hypothesis is that  = 0. where the null hypothesis is that  = 0. (0.008625-0)/(0.045661/397 1/2 ) (0.008625-0)/(0.045661/397 1/2 ) 0.008625/0.002292 = 3.76 t-statistic, so 0.008625 is significantly different from zero 0.008625/0.002292 = 3.76 t-statistic, so 0.008625 is significantly different from zero

23. 23 Model for lnsp500(t) Lnsp500(t) = a +b*t +resid(t), where resid(t) is close to a random walk, so the model is: Lnsp500(t) = a +b*t +resid(t), where resid(t) is close to a random walk, so the model is: lnsp500(t) a +b*t + RW(t), and taking exponential lnsp500(t) a +b*t + RW(t), and taking exponential sp500(t) = e a + b*t + RW(t) = e a + b*t e RW(t) sp500(t) = e a + b*t + RW(t) = e a + b*t e RW(t)

24. 24 Note: The Fitted Trend Line Forecasts Above the Observations

25. 25

26. 26 Part III. Autoregressive Representation of a Moving Average Process MAONE(t) = WN(t) + a*WN(t-1) MAONE(t) = WN(t) + a*WN(t-1) MAONE(t) = WN(t) +a*Z*WN(t) MAONE(t) = WN(t) +a*Z*WN(t) MAONE(t) = [1 +a*Z] WN(t) MAONE(t) = [1 +a*Z] WN(t) MAONE(t)/[1 - (-aZ)] = WN(t) MAONE(t)/[1 - (-aZ)] = WN(t) [1 + (-aZ) + (-aZ) 2 + …]MAONE(t) = WN(t) [1 + (-aZ) + (-aZ) 2 + …]MAONE(t) = WN(t) MAONE(t) -a*MAONE(t-1) + a 2 MAONE(t-2) +.. =WN(t) MAONE(t) -a*MAONE(t-1) + a 2 MAONE(t-2) +.. =WN(t)

27. 27 MAONE(t) = a*MAONE(t-1) - a 2 *MAONE(t-2) + …. +WN(t) MAONE(t) = a*MAONE(t-1) - a 2 *MAONE(t-2) + …. +WN(t)

28. 28 Lab 4: Alternating Pattern in PACF of MATHREE

29. 29 Part IV. Significance of Autocorrelations x, x (u) ~ N(0, 1/T), where T is # of observations

30. 30 Correlogram of the Residual from the Trend Model for LNSP500(t)

31. 31 Box-Pierce Statistic Is normalized, 1.e. is N(0,1) The square of N(0,1) variables is distributed Chi-square

32. 32 Box-Pierce Statistic The sum of the squares of independent N(0, 1) variables is Chi-square, and if the autocorrelations are close to zero they will be independent, so under the null hypothesis that the autocorrelations are zero, we have a Chi-square statistic: that has K-p-q degrees of freedom where K is the number of lags in the sum, and p+q are the number of parameters estimated.

33. 33 Application to Lab Four: the Fractional Change in the Federal Funds Rate Dlnffr = lnffr-lnffr(-1) Dlnffr = lnffr-lnffr(-1) Does taking the logarithm and then differencing help model this rate?? Does taking the logarithm and then differencing help model this rate??

34. 34

35. 35

36. 36 Correlogram of dlnffr(t)

37. 37 How would you model dlnffr(t) ? Notation (p,d,q) for ARIMA models where d stands for the number of times first differenced, p is the order of the autoregressive part, and q is the order of the moving average part. Notation (p,d,q) for ARIMA models where d stands for the number of times first differenced, p is the order of the autoregressive part, and q is the order of the moving average part.

38. 38 Estimated MAThree Model for dlnffr

39. 39 Correlogram of Residual from (0,0,3) Model for dlnffr

40. 40 Calculating the Box-Pierce Stat

41. 41 EVIEWS Uses the Ljung-Box Statistic

42. 42 Q-Stat at Lag 5 (T+2)/(T-5) * Box-Pierce = Ljung-Box (T+2)/(T-5) * Box-Pierce = Ljung-Box (586/581)*1.25368 = 1.135 compared to 1.132(EVIEWS) (586/581)*1.25368 = 1.135 compared to 1.132(EVIEWS)

43. 43 GENR: chi=rchisq(3); dens=dchisq(chi, 3)

44. 44 Correlogram of Residual from (0,0,3) Model for dlnffr

45. 45