1 Lecture Eleven Econ 240C

2 Outline Review Stochastic Time Series –White noise –Random walk –ARONE: –ARTWO –ARTHREE –ARMA(2,2) –MAONE*SMATWELVE

3 Outline Forecasting –Federal: Federal Philidelphia –State: CA Department of Finance –Local UCSB: tri-counties Chapman College: Orange County UCLA: National, CA

4 Memory Lane White Noise = wn(t)

5 Nov 14, 2003 April 29,2005

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8 Memory lane Random walk = [1 – z] -1 wn(t)

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13 Memory Lane ARONE = [1 – bz] -1 wn(t) ARONE = [1 – 0.62z] -1 wn(t)

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16 Realchpvti(t) – 21.8 = Res(t) Res (t) = [1 – 0.62z] -1 wn(t) Realchpvti(t) – 21.8 = [1 – 0.62z] -1 wn(t) Realchpvti(t) – 21.8 =.62[Realchpvti(t-1) – 21.8] + wn(t) Realchpvti(t) – 21.8 =.62*62[Realchpvti(t-2) – 21.8] + wn(t) wn(t-1)

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19 Memory Lane ARTWO = [1- b 1 z – b 2 z 2 ] -1 wn(t) dstarts = [ z – 0.21 z 2 ] -1 wn(t) [dstarts – 3.24] = [ z – 0.21 z 2 ] -1 wn(t)

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23 Memory Lane ARTHREE = [1- b 1 z – b 2 z 2 – b 3 z 3 ] -1 wn(t) Dlnffr(t) = [ z 2 – 0.135z 3 ] -1 wn(t) –Constant is insignificantly different from zero

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27 Memory Lane ARMA(2,2) = [1 + a 1 z + a 2 z 2 ]*[1 – b 1 z –b 2 z 2 ] wn(t) [ Starts – ] = [1 – z 2 ][1 – 0.653z – 0.319z 2 ] -1 wn(t)

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29 Memory Lane (1-z)(1-z 12 ) ln bjpass(t) = MAONE*SMATWELVE (1-z)(1-z 12 ) ln bjpass(t) = (1 –a 1 z)(1 – a 12 z 12 ) wn(t) (1-z)(1-z 12 ) ln bjpass(t) = (1 –0.377 z)(1 – z 12 ) wn(t) –Constant not significantly different from zero

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33 Outline-Summary Review Stochastic Time Series –White noise –Random walk: weekly price of gold –ARONE: real change in private inventories, quarterly –ARTWO: monthly change in private housing starts, single units –ARTHREE: monthly fractional change in federal funds rate –ARMA(2,2): private housing starts, single structures, monthly –MAONE*SMATWELVE: airline passengers, monthly All data from Fred except price of gold, airline passengers –Freddy, we hardly knew ye

34 Process Identification –Spreadsheet Any funny numbers? –Plot (trace) Any time dependence? –Trend in variance? –Trend in mean? –Seasonality? If so, prewhiten –Log transform –First difference –Seasonal difference

35 Process for Pre-whitened Series Identification –Spreadsheet Are transformations correct? –Plot Is it close to white noise? –Histogram Is it single peaked? –Correlogram Is there a small amount of prominent structure? PACF: order of AR terms ACF: order of MA terms Postulate alternative ARMA models –Augmented Dickey-Fuller tests Is it stationary?

36 Process: Estimation Estimate a trial ARMA model –Are the estimated parameters significant? –Record ser Validation –Actual, fitted, residual: Does the model fit the data? Do the residuals look white? –Correlogram of the residuals Are they orthogonal? If not, modify the model –Histogram of the residuals Are they normal?

37 Forecasting with an acceptable model Hand calculate a one period ahead forecast Estimate the model, leaving some data to check the forecast Forecast for the test period using competing models Plot the series, its forecast, and ~ 95% confidence interval Recolor if necessary –Could show fractional changes, forecast, upper, lower –Also original series, forecast, upper*, lower* * exponentiated upper and lower from above

38 Outline Forecasting –Federal: Federal Philidelphia –State: CA Department of Finance –Local UCSB: tri-counties Chapman College: Orange County UCLA: National, CA

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52 Economic Application The Term Structure of Interest Rates

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58 The big gap

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61 Term Structure Ratio = treas20yr/bill3mth

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63 Questions Did the Fed drive short-term interest rates down? Did the Fed drive long-term interest rates down, and create a bubble in housing prices?

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69 Conclusion About Term Structure The federal funds rate is affecting the 3 month bill rate and vice versa The federal funds rate is not affecting the 20 year bond rate

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