I. Lab#5 Includes Modeling the Index of New Private Housing Units Authorized by Local Building

4-20-2011 Lab Four 1 Econ 240C

I. ARMA Models This exercise involves modeling New Private Housing Units Started, in thousands. This is a monthly series beginning in January of 1959 and extending through February 2011 for this data set. . We will plot, identify and model the series. We will estimate a model and use the model to forecast housing starts for March 2011, noting the standard error of our forecasts. This series may be obtained from FRED in the Business and fiscal category.. The file is an EViews file called Houst..

Open EViews File Menu/Open: Houst( in the Lab Four Folder) Workfile Window: Sample: 1959:01 2011:02 Workfile Window: Select Houst Workfile Menu: VIEW: • open selection Group Window: VIEW: • line graph Group Window: VIEW: • histogram and stats Group Window: VIEW: • correlogram: level, 36 lags Group Window: VIEW: • correlogram: level, 72 lags Group Window: VIEW: •unit root test: constant, no trend, 1 lag Note: there appears to be a cyclical pattern in the time series with a period in the business cycle range.. There is no trend and the series may be stationary. No at the 10% level, no at the 5% level. I decided to begin with an autoregressive model of order two to see how well it modeled the series. Object Menu/New/Equation houst c ar(1) ar(2) Equation Object Window Menu: VIEW: • actual, fitted, residuals: graph Equation Object Window Menu: VIEW: • residual tests: correlogram, 36 lags Note: the standard error in the autocorrelations is about 0.04, i.e the reciprocal of the square root of the number of observations.. The autocorrelation at lags 2 and 4 are more than two standard deviations from zero. Also, the Breusch-Godfrey test for serial correlation in the residuals is significant at the 5 % level. The high Q statistic begins at lag 2. I decided to add an MA(2) term to the model. 4-20-2011 Lab Four 2 Econ 240C

Equation Object Window Menu: Estimate: starts c ar(1) ar(2) ma(2) Equation Object Window Menu: VIEW: • actual, fitted, residuals: graph Equation Object Window Menu: VIEW: • residual tests: correlogram, 36 lags Note: The autocorrelations at lag 4 is still significant. We could keep fishing for a model but maybe pre-whitening is a better choice. Note also that this time series has been deseasonalized, as indicated on Fred, though not perfectly. Genr: dhoust = houst – houst (-1) Workfile Menu: Group Window: VIEW: • line graph Group Window: VIEW: • histogram and stats Group Window: VIEW: • correlogram: level, 36 lags Group Window: VIEW: • correlogram: level, 72 lags Group Window: VIEW: •unit root test: constant, no trend, 1 lag Note: dhoust is stationary. From the partial autocorrelation function, there are negative spikes at lags one and two, so an ARTWO model with negative coefficients seems appropriate. Equation Object Window Menu: Note the complex roots. Equation Object Window Menu: VIEW: • residual tests: histogram, serial correlation test, normality tests. Note: Some residual correlation but the serial correlation test is not significant. Adding ma(4) ma(13) and ma(24) terms helps clean up the seasonal residual, but we will go with the ARTWO model. Note the intercept is insignificant, so we will ignore it. Note: The error, N(t), is approximately orthogonal and is kurtotic. The estimated model is: dhoust(t) = 0 + res(t) res(t) = - 0.345 res(t-1) – 0.127 res(t-2) + N(t) or (1+ 0.345Z + 0.127Z2 ) res(t) = N(t) so dhoust(t) = {1 / (1+ 0.345Z + 0.127Z2 ) } N(t)

II. Forecasting 4-20-2011 Lab Four 3 Econ 240C

We could use the AR(2) model for forecasting. The standard error of the regression is 111.. The estimated AR(2) is: dhoust(t) = 0 + res(t), res(t) = -0.345 res(t-1) - 0.127 res(t-2) + N(t), or combining equations: [dhoust(2011:03) - 0] = - 0.345 [dhoust(2011:02) - 0] - 0.127 [houst(2011:01) - 0] + N(t), where dhoust(2011:02) is -139 and dhoust(2011:01) is 96. The one period ahead forecast is the expectation at 2011.02:

E2011:02 [dhoust(2011:03) - 0] = - 0.345 [dhoust(2011:02) - 0] - 0.127 [dhoust(2011:01) - 0], and the forecast is E2011:02 [dhoust(2011:03) - 0] = 35.8, with an ser of 111. Using EViews to calculate the forecast it is 33.2 because the program uses the constant even though it is insignificant, with an sef of 111. The forecast of housing starts is just the last observed value plus the forecast of the change, i.e. E2011:02 houst(2011:03) = 479 +33 = 512.. The forecast error is the difference between the observed value and the forecast, or N(t). The variance in this one period ahead forecast error is VAR[N(t)] or the standard error of the regression, 111.0, squared. Eviews forecasts 512 with a sef of 111.0.: We will wait and see what the repoted value for March is. Workfile Window: Procs: Expand Range: 1959:01 2011:03 Equation Object Window Menu: Estimate: dhoust c ar(1) ar(2) Equation Object Window Menu: Forecast: sample range for the forecast: 2011:03 2011:03 forecast name: dhoust; method: dynamic The forecast for the rest of 2011 is: Equation Object Window Menu: Forecast: sample range for the forecast: 2011:03 2011:12 forecast name: dhoust; method: dynamic; output: graph Note: the forecast series are stored in the workfile window and may be selected and plotted.

II. Forecasting Airline Passengers 4-20-2011 Lab Four 4 Econ 240C

This lab exercise uses the TSP file BJPASS , examines the monthly series for non- stationarity, prewhitens it, specifies alternative models, including a seasonal model, estimates these models, and diagnoses the adequacy of the models in the sense that they leave residuals that approximate white noise. A model is then used to forecast airline passengers.

Open EViews File Menu/Open: BJPASS copy ( in the Lab Five Folder) Workfile Window: Sample: 1949:01 1960:12 Workfile Window: Select bjpass Workfile Menu: VIEW: • open selection Group Window: VIEW: • line graph Note: the trend in mean, the trend in variance and the seasonal pattern Group Window: VIEW: • histogram and stats Group Window: VIEW: • correlogram: level, 36 lags Note: the spike at lag one in the partial autocorrelation function and the 12 month cycle in the autocorrelation function Workfile Menu: GENR lnbjpass =LOG(bjpass) Note: the functional relatioship in this transformation can be illustrated graphically as follows: Quick Menu/Graph Series List Window: lnbjpass bjpass Graph Options Window graph options: scatter diagram type: scatter diagram Workfile Menu: Select lnbjpass Workfile Menu: VIEW: • open selection Group Window: VIEW: • line graph Note: the trend in variance has been removed Group Window: VIEW: • histogram and stats Group Window: VIEW: • correlogram: level, 36 lags 4-20-2011 Lab Four 5 Econ 240C

Note: the spike at lag one in the partial autocorrelation function and the 12 month cycle in the autocorrelation function remain. Workfile Menu: GENR sdlnbjpa =lnbjpass-lnbjpass(-12) Workfile Menu: Select sdlnbjpa Workfile Menu: VIEW: • open selection Note: we lose the first 12 observations Group Window: VIEW: • line graph Group Window: VIEW: • histogram and stats Group Window: VIEW: • correlogram: level, 36 lags Note: the spike at lag one in the partial autocorrelation function remains. Workfile Menu: GENR dsdlnbjp =sdlnbjpa-sdlnbjpa(-1) Workfile Menu: Select dsdlnbjp Workfile Menu: VIEW: • open selection Note: we lose another observation, 13 in total Group Window: VIEW: • line graph Note: this looks much more like noise Group Window: VIEW: • histogram and stats Group Window: VIEW: • correlogram: level, 36 lags Note: the remaining structure is a negative spike at lag one and a negative spike at lag 12 in the autocorrelation function. This suggests a model of ma(1)

ma(12), i.e. dsdbjp= c +wn(t) -a1wn(t-1) ) -a12wn(t-12). An alternative model is considered below. Object Menu/New/Equation dsdlnbjp c ma(1) ma(12) Equation Object Window Menu: VIEW: • actual, fitted, residuals: graph Equation Object Window Menu: VIEW: • residual tests: correlogram, 36 lags Equation Object Window Menu: VIEW: • residual tests: histogram: normality test

12 Note: an alternative specification is: dsdbjp= c +(1- a1Z)(1- a12Z )wn(t) or 4-20-2011 Lab Four 6 Econ 240C

dsdbjp= c +wn(t) -a1wn(t-1) ) -a12wn(t-12) + a1 a12wn(t-13). Object Menu/New/Equation dsdlnbjp c ma(1) sma(12) Equation Object Window Menu: VIEW: • actual, fitted, residuals: graph Equation Object Window Menu: VIEW: • residual tests: correlogram, 36 lags Equation Object Window Menu: VIEW: • residual tests: histogram: normality test Note: the latter specification is marginally better with no significant Q statistics and smaller Akaike and Schwartz criteria and a smaller standard error. To forecast 12 months into 1961: Procs Menu/expand range: 1949:01 1961:12 Equation Object Window Menu: FORECAST: sample range for the forecast: 1961:01 1961:12 forecast name: dsdlnbjf method: dynamic output: do graph Workfile Window: Sample: 1958:01 1961:12 Workfile Window: Select dsdlnbjp dsdlnbjf Workfile Menu: VIEW: • open selection Group Window: VIEW: • line graph Double click on the graph: select lines and symbols. Note: using the difference equations, the forecast of the logarithm of passengers can be synthesized: Workfile Window: Sample: 1960:12 1960:12 Workfile Window: GENR: sdlnbjpf = sdlnbjpa Workfile Window: Sample: 1961:01 1961:12 Workfile Window: GENR: sdlnbjpf = sdlnbjpf(-1) + dsdlnbjpf Workfile Window: Sample: 1961:01 1961:12 Workfile Window: GENR: lnbjpasf=lnbjpass(-12)+sdlnbjpf Workfile Window: GENR: bjpassf=exp(lnbjpasf) Workfile Window: Sample: 1949:01 1961:12 4-20-2011 Lab Four 7 Econ 240C

Workfile Window: Select bjpass bjpassf Workfile Menu: VIEW: • open selection Group Window: VIEW: • line graph Exercise: Due April 27 2011 Plot bjpass from Jan. 1949- Dec. 1960 and its forecast for the 12 months of 1961 plus a ~ 95% confidence interval, i.e. plus or minus two standard errors of the forecast. 4-20-2011 Lab Four 8 Econ 240C