2019年整理【管理】国际经济(英文版)(3)

1、I. Time Averages of Statio nary Stochastic ProcessesRecall the gen erati on of ten observati ons from the uniform distributi on that we discussed in Lecture Two. If we index these observations by time we have a strictly stationary series, x. This can be considered as one sample or representation of

2、such a time series and we can use Eviews or TSP to gen erate more. Four such series are listed in the followi ng table and their traces are plotted in the graph.Ten Observati ons from the Uni form Distributi on, TSPobservatio n nu mberxX2X3X4OBSERVATIONMEAN10.5195170.2569350.8752710.6133300.45301120

3、.2874540.2597740.5118870.5172890.31528130.0161750.3010960.5166780.3815420.24309840.6566970.4034850.7258830.4784690.45290750.6327710.1287270.7689750.4266490.39142460.1508530.6176340.6432690.0519120.29273470.4646750.4762410.6055790.9120760.49171480.9380780.0107120.8879360.1317480.39369590.6430250.2103

4、340.8084050.4702600.426405100.7777340.6845300.1131320.9870600.512491Time Av0.508698Four Samples of a Time Series : Independent Unifrorm Distributionmea nFor each observati on nu mber, the values draw n for the four time series can be averaged to obtain an average value for that observation. The time

5、 series of such means is listed in the last column of the table and is labeled observation mean. This time series is an estimate of the function , m(t):m(t) = E x(t).Usually, one does not have the luxury of being able to gen erate multiple samples of the time series, a feasible opti on in this examp

6、le because of simulati on. But if the time series is stati on ary, it is possible to average over the observations for a single time series to obtain an estimate of the mean. This estimate, presumed the same at each observati on, provides an alter native estimate of the mean function and is labeled

7、time average in the table. The next figure compares these alter native estimates of the mea n fun cti on. The mea n at each observati on, OBMEAN, has a lot of variatio n because there are only four observati ons to average.X-X3-X2 -X48 101.0 一246Mean Func tion Es timates Compared: Time Av erage and

8、Observ ation Av erage-OBMEAN -TIMEAVII. Evolutionary Stochastic ProcessesAn evolutionary stochastic process can be simulated using Eviews or TSP. For example, use GENR to gen erate a trend variable:GENR TREND = TREND(1)and the n gen erate an evoluti onary process using two of the uni variate process

9、es we used above:GENR EVOL = X + X2*TREND .This process and its component are listed in the table below and its trace is plotted:Evoluti onary ProcessObservati onNumberTRENDXX2EVOL10.00000000.5195170.2569350.51951721.0000000.2874540.2597740.54722732.0000000.0161750.3010960.61836643.0000000.6566970.4

10、034851.86715354.0000000.6327710.1287271.14767965.0000000.1508530.6176343.23902176.0000000.4646750.4762413.32212387.0000000.9380780.0107121.01306298.0000000.6430250.2103342.325694109.0000000.7777340.6845306.938505An Ev olutionary Stoc hastic Proc ess: Trends in the Mean and Variance7 -EVOL,EVOL0.2674

11、536SMPL range: 6 -10We can divide this time series into two halves, the first five observations and the second five, using the sample(SMPL) comma nd. The descriptive statistics for the two halves are listed in the following tables. Note that both the sample mean and the sample variance are higher in

12、 the sec ond half.SMPL ran ge: 1 -5Number of observatio ns: 5Time-EVOLAn Ev olutionary Stoc hastic Proc ess: Trends in the Mean and Variance7 -SeriesMea nS.D.Maximum MinimumEVOL0.93998840.57820161.86715300.5195170Covaria nceCorrelati on1.0000000Number of observatio ns: 5SeriesMea nS.D.Maximum Minimu

13、mEVOL3.3676809 2.2022365 6.93850501.0130620Covaria nceCorrelati onEVOL,EVOL3.87987661.0000000It is clear that taking time averages over a realization of an evolutionary process will not be satisfactory. This means that if we need to analyze an evolutionary process in order to apply Box- Jenkins mode

14、lli ng, it is n ecessary to tran sform the process and make it approximately stati on ary. If the process, x(t), is trended in the mean, differencing the time series may remove the trend:y(t) = ? x(t) = x( t) - x(t-1)If the times series is trended in varianee, using a logarithmic transformation of t

15、he time series, w(t), and then differencing may yield an approximately stationary time series:z(t) = ? ln w(t).III. Autocovaria nee and Autocorrelati on FunctionsThe autocovariance functionof a time series, x(t), is the variance and covariance of theseries with itself, hence the n ame, and is deno t

16、ed x,x (t,u) and defi ned as:x,x(t, u) = Ex(t) - Ex(t)x(t-u) - Ex(t-u).where u is an index of lag. Recall that for an evolutionary time series, the variance may be trending over time and thus the autocovariance function will, in general, depend upon time. The value of the autocovaria nce function at

17、 lag zero( u = 0) is the varia nce of the series:x,x(t, 0) = Ex(t) - Ex(t)x(t) - Ex(t).The autocovariance function divided by the variance, i.e. standardized, is called the autocorrelation function , ?x,x (t,u):;x,x (t, u) = x,x(t, u)/ x,x(t, 0).A plot of the autocorrelation function against lag is

18、called an autocorrelogram.If the time series is covaria nee stati on ary, the n the autocovaria nee function and autocorrelation function depend only on lag, u, and not time, t. Covariance stationarity is a restriction on some of the moments of the series and thus is not as restrictive a property as

19、 strict stati on arity.Ten observatio ns are gen erated from the Uni form distributio n, followi ng the procedure discussed in Lecture Two, and are reproduced in the table below. The sample mean can be calculated by sum ming the values for x and dividi ng by ten. To calculate the sample varia nce, s

20、ubtract the mea n from x. I n turn, these values can be squared and summed equalli ng 0.733725, which divided by ten is the estimate of the varia nce. These values for the mean and varia nce can be compared with those obta ined using the COVA comma nd and reported below the table.SMPL range: 1 -10Nu

21、mber of observatio ns: 10SeriesMea nS.D.Maximum Minimumobservatio n nu mberxMea nx - Meanx - Mean, squared10.5195170.5086980.0108190.00011720.2874540.508698-0.2212440.04894930.0161750.508698-0.4925230.24257940.6566970.5086980.1479990.02190450.6327710.5086980.1240730.01539460.1508530.508698-0.3578450

22、.12805370.4646750.508698-0.0440230.00193880.9380780.5086980.4293800.18436790.6430250.5086980.1343270.018044100.7777340.5086980.2690360.072380Ten Observati ons from the Uni form Distributi on, TSPSum5.085980.733725X0.50869780.28552590.93807800.0161750Covaria nceCorrelati onX,X0.07337251.0000000It is

23、possible to illustrate the calculation of the autocovarianee function at lag one,x,x(1),us ing the data for this series of ten observati on s gen erated from the un iform distributio n.The column of figures for x(t) minus the mean can be lagged by one, as illustrated in the table below. The product

24、of these two colu mns of figures can be summed, yield ing 0.098311, which divided by ten is an estimate of the autocorrelati on fun cti on at lag one, x,x (1) = 0.009831. Note one observation is lost in the lagging process so that estimates of the autocovariance function at higher lags becomes in ac

25、curate if the sample is small. The value of the estimated autocorrelati on function at lag one, ?x,x (1) is:xx (1) = x,x (1) x,x (0) = 0.009831/0.0735725 = 0.134.Note that the sample autocorrelati on at lag one is small as expected since the observati ons were in depe ndent by con structi on. The ca

26、lculati on of the autocorrelati on fun cti on is done swiftly using Eviews or TSP. In TSP, go to the Comma nd Menu and select time series. From time series select autocorrelogram(IDENT), i.e. the comma nd IDENT for the variable x with, for example, three correlations. This is reproduced below after

27、the table. In Eviews, after selecting the time series and ope ning the select ion, choose correlogram from the view menu.Ten Observati ons from the Uni form Distributi on, TSP| 1 0.1340.134| 2 -0.275 -0.298observatio n nu mberxx - Meanx - Mean, lagged onecross-product10.5195170.01081920.287454-0.221

28、2440.010819-0.00239430.016175-0.492523-0.2212440.10896840.6566970.147999-0.492523-0.07289350.6327710.1240730.1479990.01836360.150853-0.3578450.124073-0.04439970.464675-0.044023-0.3578450.01575380.9380780.429380-0.044023-0.01890390.6430250.1343270.4293800.057677100.7777340.2690360.1343270.036139Sum5.

29、085980.098311IDENT XSMPL range: 1 -10Number of observatio ns: 10Autocorrelati onsPartial Autocorrelati ons ac pac|Q-Statistic (3 lags) 1.285S.E. of Correlations 0.31622226262| 3 0.1870.307IV. The Normal Ran dom VariableA normal random variable has a range from minus infinity to plus infinity, mean .

30、L：: and variance c2. Its density function is:f(x) = 1心2)*exp-(1/2)*(x -)/ 于.The standardized variate z, z = (x -J)/ 二 has mean zero and varianee one. The density anddistribution functions for the standardized normal variable are plotted in the attached graph, see page 11. Observe that the no rmal de

31、n sity fun cti on is symmetric, as was the uniform den sity. However the no rmal den sity fun cti on is much more peaked around the mean.Normal Density and Distribution FunctionsuDen sity Fun cti onDistributi on FunctionIt is possible to gen erate a sample of 100 in depe ndent observati ons from a s

32、ta ndardized no rmal distributio n using Eviews or TSP. An example using Eviews was developed in Lab One. For TSP, CREATE a file of 100 un dated observati ons and use the GENR comma nd to gen erate a variable, y = nrnd. A histogram of the gen erated sample den sity follows.122110000.80.40.2Stan dardized Normal Variate8 80.60Sample of 100 Obs erv ations: Normal DistributionV. White NoiseIf we index the