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Econ 240C

Econ 240C. Lecture 12. Part I: Forecasting Time Series. Housing Starts Labs 5 and 7. Capacity Utilization, Mfg. First quarter of 1972-First Quarter 2003 Estimate for a sub-sample 1972.1-2000.4 Test the forecast for sub-sample 2001.1-2003.1. Identification Process. Trace Histogram

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Econ 240C

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  1. Econ 240C Lecture 12

  2. Part I: Forecasting Time Series • Housing Starts • Labs 5 and 7

  3. Capacity Utilization, Mfg. First quarter of 1972-First Quarter 2003 Estimate for a sub-sample 1972.1-2000.4 Test the forecast for sub-sample 2001.1-2003.1

  4. Identification Process • Trace • Histogram • Correlogram • Dickey-Fuller Test

  5. Estimation and Validation Process • First Model: cumfn c ar(1) ar(2) • goodness of fit • correlogram of residuals • histogram of residuals • Second Model: cumfn c ar(1) ar(2) ma(8) • goodness of fit • correlogram of residuals • histogram of residuals

  6. Additional Validation • Forecasting within the sample range for the subsample beyond the data range used for estimation

  7. Estimation for the whole Period • 1972.1-2003.1

  8. Correlogram of Residuals from Estimated Model, 1972.1-2003.1

  9. Augmenting Data Range for Forecasting • Workfile Window in EVIEWS • PROCS menu

  10. Equation Window; Forecast Instructions

  11. Generating Forecast and Upper and Lower Bounds for the 95% Confidence Interval, 2003.2-2004.4

  12. Compare the Observed to the Forecast Series with Bounds

  13. Spreadsheet: Observed, Forecast and Bounds

  14. Observed Series, Forecast, and 95% Confidence Interval

  15. AdditionalProspective Model Validation • How well do the forecasts for 2003-2004 compare to future observations? • Do future observations lie within the confidence bounds?

  16. Part II. Augmented Dickey-Fuller Tests • A second order autoregressive process • lecture 7-Powerpoint

  17. Lecture 7-Part III. Autoregressive of the Second Order • ARTWO(t) = b1 *ARTWO(t-1) + b2 *ARTWO(t-2) + WN(t) • ARTWO(t) - b1 *ARTWO(t-1) - b2 *ARTWO(t-2) = WN(t) • ARTWO(t) - b1 *Z*ARTWO(t) - b2 *Z*ARTWO(t) = WN(t) • [1 - b1 *Z - b2 *Z2] ARTWO(t) = WN(t)

  18. Triangle of Stable Parameter Space: Heuristic Explanation 1 b2 (-1, 0) (1, 0) b1 = 0 -1 Draw a line from the vertex, for (b1=0, b2=1), though the end points for b1, i.e. through (b1=1, b2=-1) and (b1=-1, b2=0),

  19. Triangle of Stable Parameter Space • If we are along the right hand diagonal border of the parameter space then we are on the boundary of stability, I.e. there must be a unit root, and from: • [1 - b1 *Z - b2 *Z2] ARTWO(t) = WN(t) • ignoring white noise shocks, • [1 - b1 *Z - b2 *Z2] = [1 -Z][1 + c Z], where multiplying the expressions on the right hand side(RHS), noting that c is a parameter to be solved for and setting the RHS equal to the LHS:

  20. [1 - b1 *Z - b2 *Z2] = [1 + (c - 1)Z -c Z2], so - b1 = c - 1, and - b2 = -c, or • b1 = 1 - c , (line2) • b2 = c , (line 3) • and adding lines 2 and 3: b1 + b2 = 1, so • b2 = 1 - b1 , the formula for the right hand boundary

  21. b1 + b2 = 1 is the condition for a unit root for a second order process • ARTWO(t) = b1 *ARTWO(t-1) + b2 *ARTWO(t-2) + WN(t) • add b2 *ARTWO(t-1) - b2 *ARTWO(t-1) to RHS • ARTWO(t) = (b1 + b2)*ARTWO(t-1) - b2 *ARTWO(t-1) + b2 *ARTWO(t-2) + WN(t)

  22. ARTWO(t) = (b1 + b2)*ARTWO(t-1) - b2 *[ARTWO(t-1) -*ARTWO(t-2)] + WN(t) • ARTWO(t) = (b1 + b2)*ARTWO(t-1) - b2 *DARTWO(t-1) + WN(t) • subtract ARTWO(t-1) from both sides • D ARTWO(t) = (b1 + b2 - 1)*ARTWO(t-1) - b2 *DARTWO(t-1) + WN(t)

  23. Example • Capacity utilization manufacturing

  24. ARTWO(t) = (b1 + b2)*ARTWO(t-1) - b2 *[ARTWO(t-1) -*ARTWO(t-2)] + WN(t) • ARTWO(t) = (b1 + b2)*ARTWO(t-1) - b2 *DARTWO(t-1) + WN(t) • subtract ARTWO(t-1) from both sides • D ARTWO(t) = (b1 + b2 - 1)*ARTWO(t-1) - b2 *DARTWO(t-1) + WN(t)

  25. Part III. Lab Seven • New privately owned housing units and the 30 year conventional mortgage rate, April 1974-March 2003 • starts(t) = c0mort(t) + c1mort(t-1) + c2mort(t-2) + … + resid(t) • starts(t) = c0mort(t) + c1 Zmort(t) + c2 Z2 mort(t) + … + resid(t) • starts(t) = [c0 + c1 Z + c2 Z2 + …] mort(t) + resid(t)

  26. Resid(t) + Dynamic relationship C(Z) Input mort(t) + Output starts(t)

  27. Identification Process for Mortrate • Trace • Histogram • Correlogram • Dickey-Fuller Test

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