1. Econ 240C Lecture 18

2. Review • 2002 Final • Ideas that are transcending p. 15 • Economic Models of Time Series • Symbolic Summary

15. Review • 2. Ideas That Are Transcending

16. Use the Past to Predict the Future • A. Applications • Trend Analysis • linear trend • quadratic trend • exponential trend • ARIMA Models • autoregressive models • moving average models • autoregressive moving average models

17. Use Assumptions To Cope With Constraints • A. Applications • 1. Limited number of observations: simple exponential smoothing • assume the model: (p, d, q) = (0, 1, 1) • 2. No or insufficient identifying exogenous variables: interpreting VAR impulse response functions • assume the error structure is dominated by one pure error or the other, e.g assume b1 = 0, then e1 = edcapu

18. Standard VAR (lecture 17) • dcapu(t) = (a1 + b1 a2)/(1- b1 b2) +[ (g11+ b1 g21)/(1- b1 b2)] dcapu(t-1) + [ (g12+ b1 g22)/(1- b1 b2)] dffr(t-1) + [(d1+ b1 d2 )/(1- b1 b2)] x(t) + (edcapu(t) + b1 edffr(t))/(1- b1 b2) • But if we assume b1 =0, • thendcapu(t) = a1 +g11 dcapu(t-1) + g12 dffr(t-1) + d1 x(t) + edcapu(t) +

19. Use Assumptions To Cope With Constraints • A. Applications • 3. No or insufficient identifying exogenous variables: simultaneous equations • assume the error structure is dominated by one error or the other, tracing out the other curve

20. Simultaneity • There are two relations that show the dependence of price on quantity and vice versa • demand: p = a - b*q +c*y + ep • supply: q= d + e*p + f*w + eq

21. Shift in demand with increased income, may trace out i.e. identify or reveal the demand curve price supply demand quantity

22. Review • 2. Ideas That Are Transcending

23. Reduce the unexplained sum of squares to increase the significance of results • A. Applications • 1. 2-way ANOVA: using randomized block design • example: minutes of rock music listened to on the radio by teenagers Lecture 1 Notes, 240 C • we are interested in the variation from day to day • to get better results, we control for variation across teenager

27. Reduce the unexplained sum of squares to increase the significance of results • A. Applications • 2. Distributed lag models: model dependence of y(t) on a distributed lag of x(t) and • model the residual using ARMA

28. Lab 7 240 C

30. Reduce the unexplained sum of squares to increase the significance of results • A. Applications • 3. Intervention Models: model known changes (policy, legal etc.) by using dummy variables, e.g. a step function or pulse function

31. Lab 8 240 C

33. Model with no Intervention Variable

35. Add seasonal difference of differenced step function

37. Review • 2002 Final • Ideas that are transcending • Economic Models of Time Series • Symbolic Summary

38. Time Series Models • Predicting the long run: trend models • Predicting short run: ARIMA models • Can combine trend and arima • Differenced series • Non-stationary time series models • Andrew Harvey “structural models using updating and the Kalman filter • Artificial neural networks

39. The Magic of Box and Jenkins • Past patterns of time series behavior can be captured by weighted averages of current and lagged white noise: ARIMA models • Modifications (add-ons) to this structure • Distributed lag models • Intervention models • Exponential smoothing • ARCH-GARCH

40. Economic Models of Time Series • Total return to Standard and Poors 500

41. Model One: Random Walks • Last time we characterized 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)

42. 9 8 7 6 5 4 0 100 200 300 400 500 Lecture 3, 240 C: Trace of ln S&P 500(t) Logarithm of Total Returns to Standard & Poors 500 LNSP500 TIME

43. The First Difference of ln S&P 500(t) • D ln S&P 500(t)=ln S&P 500(t) - ln S&P 500(t-1) • D ln S&P 500(t) = a + b*t + RW(t) - {a + b*(t-1) + RW(t-1)} • D ln S&P 500(t) = b + D RW(t) = b + WN(t) • Note that differencing ln S&P 500(t) where both components, trend and the random walk were evolutionary, results in two components, a constant and white noise, that are stationary.

44. Trace of ln S&P 500(t) – ln S&P(t-1)

45. Histogram of ln S&P 500(t) – ln S&P(t-1)

46. Cointegration Example • The Law of One Price • Dark Northern Spring wheat • Rotterdam import price, CIF, has a unit root • Gulf export price, fob, has a unit root • Freight rate ambiguous, has a unit root at 1% level, not at the 5% • Ln PR(t)/ln[PG(t) + F(t)] = diff(t) • Know cointegrating equation: • 1* ln PR(t) – 1* ln[PG(t) + F(t)] = diff(t) • So do a unit root test on diff, which should be stationary; check with Johansen test