5-5-2003 Economics 240C Mr. PhillipsMidterm 11. (15 min) The time series components model conceptually decomposes a time series into four components.a. List all four components. Trend, cycle, seasonal, randomb. Using the Box-Jenkins (ARMA) modeling technique, why don’t we modeltrend like we model it when we use the linear regression modeling technique? We prewhiten the series by first differencing to remove time dependence.c. If a time series has a trend component, how do we handle it if we use an ARMA model? We remove it using differencing, see part b above.2. (15 min) A second order autoregressive time series can be described or expressed as: x(t) = b1 x(t-1) + b2 x(t-2) + wn(t), where wn(t) is the white noise shock at time t. The autocorrelation function for this second order stochastic process is: (u) = b1 (u-1) + b2 (u-2), u2.a. What is (0) equal to? One.b. Do you notice any correspondence between the formula for the time seriesand the formula for its autocorrelation function? Explain. Yes. They both have the identical quadratic form and equation.c. What is a practical implication or consequence of this correspondence between the formula for the time series and the formula for its autocorrelation function? The time series and the autocovariance function both cycle at the same frequency.3. (15 min) The monthly exchange rate for the Euro, in $ per Euro, is available at FRED and is plotted in Fig. 3-1 from January 1999 through April 2003. Its histogram follows in Fig.3-2, and its correlogram in Fig 3-3.a. You have certain expectations for the behavior of a price (the price of Euros in dollars). Does the evidence from the trace, the histogram, and the correlogram support your expectations? Yes, no, or maybe? Why? I expecta random walk, and the trace looks like it could be, although the histogramand correlogram (low # of 0.884 at lag one) leave some doubt.5-5-2003 Economics 240C Mr. PhillipsMidterm 2b. A unit root test is reported Table 3.1. Does this support that the exchange rate for the Euro is evolutionary or stationary? Evolutionary.0.80.91.01.11.299:01 99:07 00:01 00:07 01:01 01:07 02:01 02:07 03:01EXUSEUFigure 3.1: Price of the Euro in $, Jan 1999-April 200302468100.85 0.90 0.95 1.00 1.05 1.10 1.15Series: EXUSEUSample 1999:01 2003:04Observations 52Mean 0.966873Median 0.953300Maximum 1.159100Minimum 0.852500Std. Dev. 0.082114Skewness 0.376424Kurtosis 1.982310Jarque-Bera 3.472028Probability 0.176221Figure 3.2: Histogram of the Euro Exchange Rate5-5-2003 Economics 240C Mr. PhillipsMidterm 3Table 3-1: Dickey-Fuller Test for a Unit RootADF Test Statistic -0.581805 1% Critical Value* -2.6081 5% Critical Value -1.9471 10% Critical Value -1.6191*MacKinnon critical values for rejection of hypothesis of a unit root.Augmented Dickey-Fuller Test EquationDependent Variable: D(EXUSEU)Method: Least SquaresSample(adjusted): 1999:02 2003:04Included observations: 51 after adjusting endpointsVariable Coefficient Std. Error t-Statistic Prob. EXUSEU(-1) -0.002015 0.003463 -0.581805 0.5633R-squared 0.003123 Mean dependent var -0.001429Adjusted R-squared 0.003123 S.D. dependent var 0.023975S.E. of regression 0.023937 Akaike info criterion -4.6073295-5-2003 Economics 240C Mr. PhillipsMidterm 4Sum squared resid 0.028650 Schwarz criterion -4.569450Log likelihood 118.4869 Durbin-Watson stat 1.2467844. (15 min) The capacity utilization rate for manufacturing is a quarterly time series available from the first quarter of 1972 though the first quarter of 2003 at FRED. A plot of this time series is reproduced from the March 2003 issue of Business Cycle Indicators.a. Based on the behavior of this time series in the recoveries from the recessions of 1970, 1974, 1980, 1982, and 1991, do you think we are recovering from the recession that began in March 2001? Yes, the series tends to trough at the end of the recession.A Dickey-Fuller test, with a constant but no trend, indicates the time series is stationary. The following second order autoregressive model was estimated, and is reproduced in Table 4-1. The residuals were approximately orthogonal, with a Q-statistic of 10.1 at lag 7, insignificant at the 5% level.TABLE 4.1: Model for Capacity Utilization, ManufacturingDependent Variable: CUMFNMethod: Least SquaresSample(adjusted): 1972:3 2003:1Included observations: 123 after adjusting endpointsConvergence achieved after 3 iterationsVariable Coefficient Std. Error t-Statistic Prob. C 79.57255 1.146075 69.43050 0.0000AR(1) 1.513502 0.073061 20.71562 0.0000AR(2) -0.601902 0.073945 -8.139836 0.0000R-squared 0.928353 Mean dependent var 79.94309Adjusted R-squared 0.927159 S.D. dependent var 4.126982S.E. of regression 1.113836 Akaike info criterion 3.077585Sum squared resid 148.8758 Schwarz criterion 3.146175Log likelihood -186.2715 F-statistic 777.4370Durbin-Watson stat 1.891267 Prob(F-statistic) 0.000000Inverted AR Roots .76 -.17i .76+.17i5-5-2003 Economics 240C Mr. PhillipsMidterm 5b. Based on the two estimated autoregressive parameters (b1 and b2), do they lie within the triangle of stability? Yes.c. Do these two estimated autoregressive parameters lie in the region that produces complex roots and cycles? Supplementary information hint: EVIEWS calculates the roots from the estimated autoregressive parameters and prints them with the estimated model. Yes.5. (15 min) In Lab Five, you modeled and forecasted the monthly series, New Private Housing Starts, thousands of units, seasonally adjusted rate (SAR). This series was updated to March of 2003. It begins January of 1946. A second order model was estimated and the results are displayed in Table 5.1. A plot of actual, fitted, and residual is also displayed as Figure 5.1. The observations for February and March 2003 are 1644 and 1780, respectively.Table 5.1 Second Order Autoregressive Model for Housing StartsDependent Variable: STARTSMethod: Least SquaresSample(adjusted): 1946:03 2003:03Included observations: 685 after adjusting endpointsConvergence achieved after 3 iterationsVariable Coefficient Std. Error t-Statistic Prob. C 1514.569 75.65255 20.02007 0.0000AR(1) 0.732833 0.037460 19.56316 0.0000AR(2) 0.213187 0.037406 5.699335 0.0000R-squared 0.875722 Mean dependent var 1493.305Adjusted R-squared0.875357 S.D. dependent var 301.8431S.E. of regression 106.5651 Akaike info criterion 12.17976Sum squared resid7744873. Schwarz criterion