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# Econ 240 C

Econ 240 C. Lecture 11. Outline. Benchmark Forecasts Project One Post-Midterm Topics Distributed Lag Models: Lab 6 Distributed Lag Models: Lab 7 Appendix: Moving Averages. Benchmark Forecasts. “Naïve Forecasts” Time series is level: use mean

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## Econ 240 C

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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. Project One: 2008

5. 3 month treasury bill rate

6. 5 year Treasury

7. A measure of the term structure

8. 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.

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

10. Part I: Distributed Lag Models

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

12. 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)

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

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

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

16. 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)

17. Part II: Estimation of Distributed Lag Models

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

19. 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?

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

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

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

23. Regression of SP500 on Consen

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

25. Correlogram of Consumer Sentiment

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

30. 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!

31. Correlogram of fractional changes in consumer sentiment

32. 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

33. 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]

34. 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

35. 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]

36. 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:

37. 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:

38. 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

39. 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]

40. 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 6.

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