计量经济学导论ch18

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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 18Advanced TimeSeries TopicsWooldridge:Introductory Econometrics:A Modern Approach,5e 2013 Cengage Learning.All Rights Reserved.May not be scanned

2、,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Testing for unit rootsFor the validity of regression analysis,it is crucial to know whether or not dependent or independent variables are highly persistentDickey-Fuller testOne can use the t-statistic to test the hy

3、pothesis,but under the null,it has not got the t-distribution but the Dickey-Fuller distributionThe Dickey-Fuller distribution has to be looked up in tablesUnder the null hypothesis,the process has a unit root.Under the alternative,it is a stable AR(1)processThe test is based on an AR(1)regressionAd

4、vanced 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 testCritical values for Dickey-Fuller testThe alternative representation is obtained

5、 by subtracting yt-1 from both sidesThe critical value is much more negative than it would be in a t-distributionAdvanced 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:Unit r

6、oot test for three-month T-Bill ratesAugmented Dickey-Fuller testThe augmented Dickey-Fuller test allows for more serial correlationThe critical values and the rejection rule are the same as beforeThe t-statistic is-2.46.As consequence,the null hypothesis of a unit root cannot be rejectedInclude lag

7、ged 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 test for time series that have a time trendCritical values for Dickey-Fulle

8、r test with time trendThere are many other unit root tests Under the alternative hypothesis of no unit root,the process is trend-stationaryAdvanced Time Series TopicsEven more negative 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessibl

9、e website,in whole or in part.Spurious regressionRegressing one I(1)-series on another I(1)-series may lead to extre-mely high t-statistics even if the series are completely independentSimilarly,the R-squared of such regressions tends to be very highThis means that regression analysis involving time

10、 series that have a unit root may generally lead to completely misleading inferencesCointegrationFortunately,regressions with I(1)-variables are not always spuriousIf there is a stable relationship between time series that,individually,display unit root behavior,these time series are called co-integ

11、rated“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 for time-series that are potentially cointegratedSpread between interest ratesInterest rates of 6-months bill and

12、 of 3-months billIndividually,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 can move arbitrarily far away from each other with no tendendency to come back together(this is implausible as it co

13、ntradicts 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 rateTemporary deviation from stable relationshipAdvanced Time Series Topics 2013 Cengage Learning.All Rights Reserved.May not b

14、e scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.General definition of cointegrationTwo I(1)-time series are said to be cointegrated if there exists a stable relationship between them in the sense thatTest for cointegration if the cointegration parameters

15、 are knownForm residuals of the known cointegration relationship:Test whether the residuals have a unit rootIf the unit root can be rejected,are cointegratedwithThe deviation from the stable relationship is I(0)Advanced Time Series Topics 2013 Cengage Learning.All Rights Reserved.May not be scanned,

16、copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Cointegration between interest rates(cont.)Testing for cointegration if the parameters are unknownIf the potential relationship is unknown,it can be estimated by OLSAfter that,one tests whether the regression

17、 residuals have a unit rootIf the unit root is rejected,this means that are cointegrated Due to the pre-estimation of parameters,critical values are differentThe Dickey-Fuller strongly rejects a unit root in the spread.This means the interest rates are cointegrated.Advanced Time Series Topics 2013 C

18、engage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Critical values for cointegration testThe cointegration relationship may include a time trendIf the two series have differential time trends(=drifts in this case

19、),the deviation between them may still be I(0)but with a linear time trendIn this case one should include a time trend in the first stage regres-sion but one has to use different critical values when testing residuals Even more negative than in Dickey-Fuller distributionAdvanced Time Series Topics 2

20、013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Critical values for cointegration test including time trendExample:Cointegration between fertility and tax exemptionDF-tests suggest that fertiliy and tax e

21、xemption have unit rootsRegressing fertility on tax exemption and a time trend and carrying out a cointegration test suggests there is no evidence for cointegrationThis means that the regression in levels is probably spuriousEven more negativeAdvanced Time Series Topics 2013 Cengage Learning.All Rig

22、hts Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Error correction modelsOne can show that when variables are cointegrated,their short-term dynamics are related in a so-called error correction representation:Summary of cointegration m

23、ethodsAll concepts can be generalized to arbitrarily many time seriesCointegration is the leading methodology in empirical macro/finance as it models equilibrium relationships between nonstationary variablesEstimation and inference is complicated and requires extra careDeviations from the long-term

24、relationship directly feed back into the change of the variablesAdvanced 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.Forecasting economic time seriesForecasting economic time serie

25、s is of great practical importanceIn forecasting,one is not interested in modeling causal relationships,but in predicting future outcomes using currently available informationOne can show that the forecasting rule with the minimum expected squared forecasting error is given by the conditional expect

26、ationHere,we only consider one-step-ahead forecasts,multiple-step-ahead forecasts are similar but more complicated(and also less precise)One-step ahead forecast of yAll information avai-lable up to period tAdvanced Time Series Topics 2013 Cengage Learning.All Rights Reserved.May not be scanned,copie

27、d or duplicated,or posted to a publicly accessible website,in whole or in part.Regression based forecast modelsTypical forecast models predict a variable in a linear regression using lagged values of the variable and lagged values of other variablesOne may include the lagged value of arbitrarily man

28、y other variablesIf enough lags have been included,the model 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 normalAdvanced Time Series Topics 2013 Cengage Learning.All Rights Reserved.May

29、not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Forecasting the U.S.unemployment rateLagged inflation significantly helps to predict current unemploymentNote that these regressions are not meant as causal equations.The hope is that the linea

30、r regressions approximate well the conditional 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 forecastsOne can measu

31、re how good the forecasted values 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

32、remaining m observations 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 autoregressiv

33、e models(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.I

34、f this is the case,the other series is caused by the first series in the sense of Granger-Causality.VAR models work for arbitrarily many simultaneously observed time series.They are widely used in practice.They are also well suited to model systems of variables that are related to each other through

35、 cointegration relationships.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.Further topics in forecasting time seriesMultiple-step forecasts are possible,but necessarily less

36、 preciseForecasts may make use of deterministic trends,but the error made by extrapolating time trends too far into the future may be largeSimilarly,seasonal patterns may be incorporated into forecastsForecasting integrated time series is either based on regressions in levels,or on adding predicted changes(which are I(0)to base levelsIt is possible to calculate confidence intervals for the point forecasts Forecast intervals for nonintegrated series converge to the uncondi-tional variance,whereas for integrated series,they are unboundedAdvanced Time Series Topics