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Econ 240 C. Lecture 12. 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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**Econ 240 C**Lecture 12**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 • 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**Post-Midterm Topics**• Distributed lag Models • Exponential smoothing • Intervention models • Autoregressive conditional heteroskedasticity ARCH • Vector autoregression VAR**?**Output Y(t) Dynamic Relationship Input X(t)**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)**Resid(t)**+ Dynamic relationship C(Z) Input X(t) + Output Y(t)**Pre-midterm**Resid(t) + Dynamic relationship C(Z) Input X(t) + Output Y(t)**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)**Part II: Estimation of Distributed Lag Models**• Why not use simple OLS regression of SP500 on consumer sentiment in levels?**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?**Resid(t)**+ Dynamic relationship C(Z) Input X(t) consumer sentiment [c0 + c1 Z + c2 Z2 + …] + Output Y(t) sp500**Regress sp500 on a distributed lag of consumer sentiment**• sp500(t) = c0consen(t) + c1consen(t-1) + c2consen(t-2) + … + resid(t)**Distributed Lag Model of SP500 on Consumer Sentiment**• Why are not the t-statistics more significant**Distributed Lag Model of SP500 on Consumer Sentiment**• Fixup: Taking logarithms is not enough**Correlogram of natural logarithm of the Index**of Consumer Sentiment**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**Resid(t)**+ Dynamic relationship C(Z) Input X(t) D lnconsumer sentiment [c0 + c1 Z + c2 Z2 + …] + Output Y(t) D lnsp500**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!**Correlogram of fractional changes in**consumer sentiment**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**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]**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**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]**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:**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:**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**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]**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.**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**Resid(t)**+ Dynamic relationship C(Z) Input X(t) Mortgage rate [c0 + c1 Z + c2 Z2 + …] + Output Y(t) starts**Three Possibilities**• Regress starts on mortgage rate and lagged values**Three Possibilities**• Regress starts on mortgage rate and lagged values strike one • Regress dstarts on dmort and lagged values

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