Econ 240 C

1. Econ 240 C Lecture 12

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. Project One

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

8. Part I: Distributed Lag Models

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

10. 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)

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

12. Resid(t) + Dynamic relationship C(Z) Input X(t) + Output Y(t)

13. Pre-midterm Resid(t) + Dynamic relationship C(Z) Input X(t) + Output Y(t)

14. 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)

15. Part II: Estimation of Distributed Lag Models

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

17. Part II: Estimation of Distributed Lag Models • Example from Lab 6: The Index of Consumer Sentiment and the Standard & Poors 500 Index • Does consumer sentiment affect the stock market?

18. Resid(t) + Dynamic relationship C(Z) Input X(t) consumer sentiment [c0 + c1 Z + c2 Z2 + …] + Output Y(t) sp500

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

20. Regression of SP500 on Consen

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

23. Correlogram of Consumer Sentiment

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

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

28. 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

29. Resid(t) + Dynamic relationship C(Z) Input X(t) D lnconsumer sentiment [c0 + c1 Z + c2 Z2 + …] + Output Y(t) D lnsp500

30. Regression of dlnsp500 on distributed lag of dlncons

31. 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!

33. Correlogram of fractional changes in consumer sentiment

34. 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

35. 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]

36. 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

37. 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]

38. Cross-correlation function

39. 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:

40. Part III: Definitions of Cross-Covariance and Cross-Correlation • If y and x are covariance stationary, then the cross-correlation function depends on lag only:

41. Part III: Definitions of Cross-Covariance and Cross-Correlation • Definition of cross-correlation • rx,y(u) = gx,y(u)/sx sy • note: (sy/ sx) * rx,y(u) = gx,y(u)/sx2 is the cross covariance function divided by the variance of x and reveals the distributed lag of y on lagged values of x

42. 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]

43. Cross-correlation function

44. What happens if the input(x, dlncons) is not orthogonal? • Then we need to make it orthogonal. This is the Box-Jenkins procedure for estimating distributed lags and we will study it in Lab 7.

45. Distributed Lag: Lab 7 • ARIMA model for input • X(t) = A(z)/B(z) wnx(t) • Y(t) = h(z) x(t) +resy(t) • [B(z)/A(z)] y(t) =h(z)* [B(z)/A(z)] x(t) + [B(z)/A(z)] resy(t) • Or w(t) = h(z)*wnx(t) + resw(t) • Where wnx(t) is orthogonal

46. Resid(t) + Dynamic relationship C(Z) Input X(t) Mortgage rate [c0 + c1 Z + c2 Z2 + …] + Output Y(t) starts

47. Three Possibilities • Regress starts on mortgage rate and lagged values

49. Three Possibilities • Regress starts on mortgage rate and lagged values strike one • Regress dstarts on dmort and lagged values