1. 1 ECON 240C Lecture 8

2. 2 Part I. Economic Forecast Project Santa Barbara County Seminar –April 17, 2003 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 Note: The Fitted Trend Line Forecasts Above the Observations

10. 10 One Period Ahead Forecast Note the standard error of the regression is 0.2237 Note: the standard error of the forecast is 0.2248 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

11. 11 Parameter Uncertainty Trend model: y(t) = a + b*t + e(t) Fitted model:

12. 12 Parameter Uncertainty Estimated error

13. 13 Forecast Formula

14. 14 Forecast E t

15. 15 Forecast Forecast = a + b*(t+1) + 0

16. 16 Variance in the Forecast Error

17. 17

18. 18 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 = (0.505444) 1/2 = 0.22482

19. 19 Numerical Representation of the Forecast

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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 where the null hypothesis is that  = 0. (0.008625-0)/(0.045661/397 1/2 ) 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 + RW(t), and taking expontial sp500(t) = e a + b*t + RW(t) = e a + b*t e RW(t)

24. 24 Part III. Autoregressive Representation of a Moving Average Process MAONE(t) = WN(t) + a*WN(t-1) MAONE(t) = WN(t) +a*Z*WN(t) MAONE(t) = [1 +a*Z] WN(t) MAONE(t)/[1 - (-aZ)] = WN(t) [1 + (-aZ) + (-aZ) 2 + …]MAONE(t) = WN(t) MAONE(t) -a*MAONE(t-1) + a 2 MAONE(t-2) +.. =WN(t)

25. 25 MAONE(t) = a*MAONE(t-1) - a 2 *MAONE(t-2) + …. +WN(t)

26. 26 Lab 4: Alternating Pattern in PACF of MATHREE

27. 27 Part IV. Significance of Autocorrelations x, x (u) ~ N(0, 1/T), where T is # of observations

28. 28 Correlogram of the Residual from the Trend Model for LNSP500(t)

29. 29 Box-Pierce Statistic Is normalized, 1.e. is N(0,1) The square of N(0,1) variables is distributed Chi-square

30. 30 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.

31. 31 Application to Lab Four: the Fractional Change in the Federal Funds Rate Dlnffr = lnffr-lnffr(-1) Does taking the logarithm and then differencing help model this rate??

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34. 34 Correlogram of dlnffr(t)

35. 35 How would you model dlnff(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.

36. 36 Estimated MAThree Model for dlnffr

37. 37 Correlogram of Residual from (0,0,3) Model for dlnffr

38. 38 Calculating the Box-Pierce Stat

39. 39 EVIEWS Uses the Ljung-Box Statistic

40. 40 Q-Stat at Lag 5 (T+2)/(T-5) * Box-Pierce = Ljung-Box (586/581)*1.25368 = 1.135 compared to 1.132(EVIEWS)

41. 41 GENR: chi=rchisq(3); dens=dchisq(chi, 3)

42. 42 Correlogram of Residual from (0,0,3) Model for dlnffr

43. 43