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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: 2010

  5. Canadian exchange rate

  6. Takehome One 2009

  7. Takehome One 2008

  8. 5 year Treasury

  9. A measure of the term structure

  10. 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.

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

  12. Part I: Distributed Lag Models

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

  14. 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)

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

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

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

  18. 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)

  19. Part II: Estimation of Distributed Lag Models

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

  21. 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?

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

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

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

  25. Regression of SP500 on Consen

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

  27. Correlogram of Consumer Sentiment

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

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

  30. 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

  31. 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

  32. Regression of dlnsp500 on distributed lag of dlncons

  33. 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!

  34. Correlogram of fractional changes in consumer sentiment

  35. 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

  36. 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]

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