1. Econ 240 C Lecture 11

2. Outline • Benchmark Forecasts • Project One • Post-Midterm Topics • Distributed Lag Models: Lab 6 • Distributed Lag Models: Lab 7 • Appendix: Moving Averages

3. Benchmark Forecasts • “Naïve Forecasts” • Time series is level: use mean • Time series is trended: use trend to forecast, e.g. Lab 2 • Time series is random walk: best forecast for next period is this period • ARMA model forecasts • Also uses the past to extrapolate the future

4. Takehome One 2009

9. Project One: 2008

10. 3 month treasury bill rate

11. 5 year Treasury

14. A measure of the term structure

16. Questions: Takehome One 1. You should try this so that you know at least one way of obtaining time series from FRED. If you have difficulty, an Excel file called Takeone, is available on the class page. 2. Generate a time series called term that is the difference between GS5 and TB3MS. 3. Is term stationary, i.e. are GS5 and TB3ms co-integrated? 4. Is term normally distributed? 5. Estimate your best autoregressive model for term. 6. Estimate your best ARMA model for term through April 2007 and see how well a forecast for this model fits the next 12 months. 7. Re-estimate your best model for term through April 2008 and forecast for the remaining months of 2008.

17. Post-Midterm Topics • Distributed lag Models • Exponential smoothing • Intervention models • Autoregressive conditional heteroskedasticity ARCH • Vector autoregression VAR

18. Part I: Distributed Lag Models

19. ? Output Y(t) Dynamic Relationship Input X(t)

20. Distributed Lag • Y(t) = c0x(t) + c1x(t-1) + c2x(t-2) + … + resid(t) • Y(t) = c0 x(t) + c1 Z x(t) + c2 Z2 x(t) + … resid(t) • Y(t) = [c0 + c1 Z + c2 Z2 + …] x(t) + resid(t)

21. Y(t) = C(Z) x(t) + resid(t)

22. Resid(t) A(z)/B(z) Dynamic relationship + C(Z) Input X(t) + Output Y(t)

23. Pre-midterm Resid(t) A(z)/B(z) Dynamic relationship + C(Z) Input X(t) + Output Y(t)

24. Dynamic Model Building • Process • Identification of C(Z) • How many lags? • Which lags • Use cross-correlation function to answer specification of lags • Identification of resid(t) • Resid(t) captures part of y(t) • Use the univariate ARMA model for y(t) as a starting point for modeling resid(t)

25. Part II: Estimation of Distributed Lag Models

26. Part II: Estimation of Distributed Lag Models • Why not use simple OLS regression of SP500 on consumer sentiment in levels?

27. Part II: Estimation of Distributed Lag Models • Example : The Index of Consumer Sentiment and the Standard & Poors 500 Index • Does consumer sentiment affect the stock market?

28. Resid(t) A(z)/B(z) Dynamic relationship + C(Z) Input X(t) Consumer sentiment + Output Y(t) SP500

29. Resid(t) A(z)/B(z) Dynamic relationship + C(Z) Input X(t) Consumer sentiment + [c0 + c1 Z + c2 Z2 + …] Output Y(t) SP500

30. Regress sp500 on a distributed lag of consumer sentiment • sp500(t) = c0consen(t) + c1consen(t-1) + c2consen(t-2) + … + resid(t)

31. Regression of SP500 on Consen

32. Distributed Lag Model of SP500 on Consumer Sentiment • Why are not the t-statistics more significant

34. Correlogram of Consumer Sentiment

35. Distributed Lag Model of SP500 on Consumer Sentiment • Fixup: Taking logarithms is not enough

38. Correlogram of natural logarithm of the Index of Consumer Sentiment

39. Distributed Lag Model of SP500 on Consumer Sentiment • Fixup: Taking logarithms is not enough • First difference after taking natural logarithm to obtain the fractional change in the Index of Consumer Sentiment

40. Resid(t) A(z)/B(z) Dynamic relationship + C(Z) Input X(t) lnConsumer sentiment D + [c0 + c1 Z + c2 Z2 + …] Output Y(t) lnSP500 D

41. Regression of dlnsp500 on distributed lag of dlncons

42. Distributed Lag Model of Fractional Changes in SP500 on Fractional Changes in Consumer Sentiment • Why does this work? Because fractional changes in consumer sentiment are orthogonal!

44. Correlogram of fractional changes in consumer sentiment

45. Contemporary correlation • dlnsp500(t) = c0 dlncons(t) + c1 dlncons(t-1) + c2 dlncons(t-2) + c3 dlncons(t-3) + resid(t) • multiply by dlcons(t) and take expectations

46. E{dlnsp500(t)*dlncons(t) = c0 [dlncons(t)]2 + c1 dlncons(t-1)*dlncons(t) + c2 dlncons(t-2)*dlncons(t) + c3 dlncons(t-3)*dlmcons(t) + resid(t)*dlncons(t)} • cov dlnsp500(t) dlncons(t) = c0 var dlncons(t) + 0 • c0 = cov[dlny*dlnx]/var[dlnx]

47. Correlation at lag one • dlnsp500(t) = c0 dlncons(t) + c1 dlncons(t-1) + c2 dlncons(t-2) + c3 dlncons(t-3) + resid(t) • multiply by dlcons(t-1) and take expectations

48. E{dlnsp500(t)*dlncons(t-1) = c0 [dlncons(t)*dlncons(t-1)] + c1 dlncons(t-1)*dlncons(t-1) + c2 dlncons(t-2)*dlncons(t-1) + c3 dlncons(t-3)*dlmcons(t-1) + resid(t)*dlncons(t-1)} • crosscov dlnsp500(t) dlncons(t-1) = c1 autocov dlncons(t-1)*dlncons(t-1)) + 0 • c1 = cov[dlny(t)*dlnx(t-1)]/var[dlnx]

49. Cross-correlation function

50. Part III: Definitions of Cross-Covariance and Cross-Correlation • In general, the cross-covariance function where y depends on current and lagged values of x: