计量经济学导论ch18 课件

1、?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Chapter 18,Advanced Time,Series Topics,Wooldridge: Introductory Econometrics:,A Modern Approach, 5e,?,2013 Cengage Learning. All Rights Reserved. M

2、ay not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Testing for unit roots,For the validity of regression analysis, it is crucial to know whether,or not dependent or independent variables are highly persistent,Dickey-Fuller test,One can use the t

3、-statistic to test the hypothesis, but under the null, it,has not got the t-distribution but the Dickey-Fuller distribution,The Dickey-Fuller distribution has to be looked up in tables,Under the null hypothesis, the process has a unit root.,Under the alternative, it is a stable AR(1) process,The tes

4、t is based on an AR(1) regression,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Alternative Formulation of the Dickey-Fuller test,Critical values for Dickey-Fuller t

5、est,The alternative representation is obtained,by subtracting y,t-1,from both sides,The critical value is much,more negative than it would,be in a t-distribution,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicl

6、y accessible website, in whole or in part.,Example: Unit root test for three-month T-Bill rates,Augmented Dickey-Fuller test,The augmented Dickey-Fuller test allows for more serial correlation,The critical values and the rejection rule are the same as before,The t-statistic is -2.46. As consequence,

7、 the null,hypothesis of a unit root cannot be rejected,Include lagged differences of dependent variable.,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Dickey-Fuller

8、test for time series that have a time trend,Critical values for Dickey-Fuller test with time trend,There are many other unit root tests ,Under the alternative hypothesis of no unit root, the process is trend-stationary,Advanced Time,Series Topics,Even more negative,?,2013 Cengage Learning. All Right

9、s Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Spurious regression,Regressing one I(1)-series on another I(1)-series may lead to extre-,mely high t-statistics even if the series are completely independent,Similarly, the R-square

10、d of such regressions tends to be very high,This means that regression analysis involving time series that have a,unit root may generally lead to completely misleading inferences,Cointegration,Fortunately, regressions with I(1)-variables are not always spurious,If there is a stable relationship betw

11、een time series that, individually,display unit root behavior, these time series are called ?co,-,integrated“,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Example f

12、or time-series that are potentially cointegrated,Spread between,interest rates,Interest rates of 6-months,bill and of 3-months bill,Individually, it can not be rejected that,the interest rates have a unit root.,It is unlikely that the spread has a unit root because this would mean the interest,rates

13、 can move arbitrarily far away from each other with no tendendency to come,back together (this is implausible as it contradicts arbitrage arguments).,If the spread is an I(0) variable, there is a stable relationship between the interest rates:,Mean spread between 6-months and 3-months interest rate,

14、Temporary deviation from stable relationship,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,General definition of cointegration,Two I(1)-time series are said to be co

15、integrated if there,exists a stable relationship between them in the sense that,Test for cointegration if the cointegration parameters are known,Form residuals of the known cointegration relationship:,Test whether the residuals have a unit root,If the unit root can be rejected, are cointegrated,with

16、,The deviation from the,stable relationship is I(0),Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Example: Cointegration between interest rates (cont.),Testing for c

17、ointegration if the parameters are unknown,If the potential relationship is unknown, it can be estimated by OLS,After that, one tests whether the regression residuals have a unit root,If the unit root is rejected, this means that are cointegrated,Due to the pre-estimation of parameters, critical val

18、ues are different,The Dickey-Fuller strongly rejects a,unit root in the spread. This means,the interest rates are cointegrated.,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or

19、in part.,Critical values for cointegration test,The cointegration relationship may include a time trend,If the two series have differential time trends (=drifts in this case), the,deviation between them may still be I(0) but with a linear time trend,In this case one should include a time trend in th

20、e first stage regres-sion,but one has to use different critical values when testing residuals,Even more negative than in Dickey-Fuller distribution,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible w

21、ebsite, in whole or in part.,Critical values for cointegration test including time trend,Example: Cointegration between fertility and tax exemption,DF-tests suggest that fertiliy and tax exemption have unit roots,Regressing fertility on tax exemption and a time trend and carrying out a,cointegration

22、 test suggests there is no evidence for cointegration,This means that the regression in levels is probably spurious,Even more negative,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in wh

23、ole or in part.,Error correction models,One can show that when variables are cointegrated, their short-term,dynamics are related in a so-called error correction representation:,Summary of cointegration methods,All concepts can be generalized to arbitrarily many time series,Cointegration is the leadi

24、ng methodology in empirical macro/finance as,it models equilibrium relationships between nonstationary variables,Estimation and inference is complicated and requires extra care,Deviations from the long-,term relationship directly,feed back into the change,of the variables,Advanced Time,Series Topics

25、,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Forecasting economic time series,Forecasting economic time series is of great practical importance,In forecasting, one is not interested in modeli

26、ng causal relationships,but in predicting future outcomes using currently available information,One can show that the forecasting rule with the minimum expected,squared forecasting error is given by the conditional expectation,Here, we only consider one-step-ahead forecasts, multiple-step-ahead,fore

27、casts are similar but more complicated (and also less precise),One-step ahead,forecast of y,All information avai-,lable up to period t,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in wh

28、ole or in part.,Regression based forecast models,Typical forecast models predict a variable in a linear regression using,lagged values of the variable and lagged values of other variables,One may include the lagged value of arbitrarily many other variables,If enough lags have been included, the mode

29、l is dynamically complete,and there is no serial corr. in the error (but may be heteroscedasticity),OLS inference methods can be used if the error is conditionally normal,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to

30、a publicly accessible website, in whole or in part.,Example: Forecasting the U.S. unemployment rate,Lagged inflation significantly helps,to predict current unemployment,Note that these regressions are not meant as causal,equations. The hope is that the linear regressions,approximate well the conditi

31、onal expectation.,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Evaluating forecast quality of one-step-ahead forecasts,One can measure how good the forecasted value

32、s fit the actual,values over the whole sample (= in-sample criteria, e.g. R-squared),It is better, however, to evaluate the forecasting performance when,forecasting out-of-sample values (= out-of-sample criteria),For this purpose, use first n observations for estimation, and the,remaining m observat

33、ions to calculate forecast errors,There are different forecast evaluation measures, e.g.,Advanced Time,Series Topics,?,2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.,Vector autoregressive models (

34、VAR),VAR models model a collection (= vector) of time series as linear functions of their own past,values and the past values of other series. They can be estimated eq. by eq. using OLS.,Using an F-test, one can test whether the past values of a series helps to predict the values,of another series. If this is the case, the other series is caused by the firs