1 / 83

# Econ 240 C

Econ 240 C. Lecture 3. Synthesis. White noise. White noise. 1. output. input. Random walk. 1/(1 – z). White noise. output. input. First order autoregressive. White noise. 1/(1 – bz). input. output. Simulated Random walk. Eviews, sample 1 1000, gen wn = nrnd

Télécharger la présentation

## Econ 240 C

E N D

### Presentation Transcript

1. Econ 240 C Lecture 3

2. Synthesis White noise White noise 1 output input Random walk 1/(1 – z) White noise output input First order autoregressive White noise 1/(1 – bz) input output

3. Simulated Random walk • Eviews, sample 1 1000, gen wn = nrnd • EViews, sample 1 1, gen rw = wn • Sample 2 1000, gen rw = rw(-1) + wn

4. Simulated First Order Autoregressive Process • Eviews, sample 1 1000, gen wn = nrnd • EViews, sample 1 1, gen arone = wn • Sample 2 1000, gen arone = b*arone(-1) + wn

5. Systematics • b =1, random walk • b = 0.9 • b = 0.5 • b = 0.1 • b = 0, white noise

6. Arone(t) = 0.9*arone(t-1) = wn(t)

7. Arone(t) = 0.5*arone(t-1) + wn(t)

8. Arone(t) = 0.1*arone(t-1) + wn(t)

9. Moving Averages

10. Bloomberg

11. Time Series Concepts • Analysis and Synthesis

12. Analysis • Model a real time seies in terms of its components

13. Total Returns to Standard and Poors 500, Monthly, 1970-2003 Source: FRED http://research.stlouisfed.org/fred/

14. 9 8 7 6 5 4 0 100 200 300 400 500 Trace of ln S&P 500(t) Logarithm of Total Returns to Standard & Poors 500 LNSP500 TIME

15. Model • Ln S&P500(t) = a + b*t + e(t) • time series = linear trend + error

16. Dependent Variable: LNSP500 Method: Least Squares Sample(adjusted): 1970:01 2003:02 Included observations: 398 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C 4.049837 0.022383 180.9370 0.0000 TIME 0.010867 9.76E-05 111.3580 0.0000 R-squared 0.969054 Mean dependent var 6.207030 Adjusted R-squared 0.968976 S.D. dependent var 1.269965 S.E. of regression 0.223686 Akaike info criterion -0.152131 Sum squared resid 19.81410 Schwarz criterion -0.132098 Log likelihood 32.27404 F-statistic 12400.61 Durbin-Watson stat 0.041769 Prob(F-statistic) 0.000000

17. Time Series Components Model • Time series = trend + cycle + seasonal + error • two components, trend and seasonal, are time dependent and are called non-stationary

18. Synthesis • The Box-Jenkins approach is to start with the simplest building block to a time series, white noise and build from there, or synthesize. • Non-stationary components such as trend and seasonal are removed by differencing

19. First Difference • Lnsp500(t) - lnsp500(t-1) = dlnsp500(t)

20. Time series • A sequence of values indexed by time

21. Stationary time series • A sequence of values indexed by time where, for example, the first half of the time series is indistinguishable from the last half

22. Stochastic Stationary Time Series • A sequence of random values, indexed by time, where the time series is not time dependent

23. Summary of Concepts • Analysis and Synthesis • Stationary and Evolutionary • Deterministic and Stochastic • Time Series Components Model

24. White Noise Synthesis • Eviews: New Workfile • undated 1 1000 • Genr wn = nrnd • 1000 observations N(0,1) • Index them by time in the order they were drawn from the random number generator

25. Synthesis • Random Walk • RW(t) -RW(t-1) = WN(t) = dRW(t) • or RW(t) = RW (t-1) + WN(t) • lag by one: RW(t-1) = RW(t-2) + WN(t-1) • substitute: RW(t) = RW(t-2) + WN(t) + WN(t-1) • continue with lagging and substitutingRW(t) = WN(t) + WN (t-1) + WN (t-2) + ...

26. Part I • Modeling Economic Time Series

27. Total Returns to Standard and Poors 500, Monthly, 1970-2003 Source: FRED http://research.stlouisfed.org/fred/

28. Analysis (Decomposition) • Lesson one: plot the time series

29. Model One: Random Walks • we can characterize the logarithm of total returns to the Standard and Poors 500 as trend plus a random walk. • Ln S&P 500(t) = trend + random walk = a + b*t + RW(t)

30. 9 8 7 6 5 4 0 100 200 300 400 500 Trace of ln S&P 500(t) Logarithm of Total Returns to Standard & Poors 500 LNSP500 TIME

31. Analysis(Decomposition) • Lesson one: Plot the time series • Lesson two: Use logarithmic transformation to linearize

32. Ln S&P 500(t) = trend + RW(t) • Trend is an evolutionary process, i.e. depends on time explicitly, a + b*t, rather than being a stationary process, i. e. independent of time • A random walk is also an evolutionary process, as we will see, and hence is not stationary

33. Model One: Random Walks • This model of the Standard and Poors 500 is an approximation. As we will see, a random walk could wander off, upward or downward, without limit. • Certainly we do not expect the Standard and Poors to move to zero or into negative territory. So its lower bound is zero, and its model is an approximation.

More Related