《计量经济学导论》ch18教学课件

1、Chapter 18Advanced TimeSeries TopicsWooldridge: Introductory Econometrics: A Modern Approach, 5e第1页，共18页。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-sta

2、tistic to test the hypothesis, 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 ba

3、sed on an AR(1) regressionAdvanced Time Series Topics第2页，共18页。Alternative Formulation of the Dickey-Fuller testCritical values for Dickey-Fuller testThe alternative representation is obtained by subtracting yt-1 from both sidesThe critical value is much more negative than it would be in a t-distribu

4、tionAdvanced Time Series Topics第3页，共18页。Example: Unit root 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 nu

5、ll hypothesis of a unit root cannot be rejectedInclude lagged differences of dependent variable.Advanced Time Series Topics第4页，共18页。Dickey-Fuller test for time series that have a time trendCritical values for Dickey-Fuller test with time trendThere are many other unit root tests Under the alternativ

6、e hypothesis of no unit root, the process is trend-stationaryAdvanced Time Series TopicsEven more negative第5页，共18页。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 su

7、ch regressions tends to be very highThis means that regression analysis involving time 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 se

8、ries that, individually, display unit root behavior, these time series are called co-integrated“Advanced Time Series Topics第6页，共18页。Example for time-series that are potentially cointegratedSpread between interest ratesInterest rates of 6-months bill and of 3-months billIndividually, it can not be re

9、jected 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 contradicts arbitrage arguments).If the spread

10、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第7页，共18页。General definition of cointegrationTwo I(1)-time series are said to be cointegrated if

11、 there exists a stable relationship between them in the sense thatTest for cointegration if the cointegration parameters 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

12、 the stable relationship is I(0)Advanced Time Series Topics第8页，共18页。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 residuals h

13、ave 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第9页，共18页。Critica

14、l values for cointegration testThe cointegration relationship may include a time trendIf 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 trendIn this case one should include a time trend in the first stage regre

15、s-sion but one has to use different critical values when testing residuals Even more negative than in Dickey-Fuller distributionAdvanced Time Series Topics第10页，共18页。Critical values for cointegration test including time trendExample: Cointegration between fertility and tax exemptionDF-tests suggest t

16、hat fertiliy and tax exemption 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第11页，共18

17、页。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 methodsAll concepts can be generalized to arbitrarily many time seriesCointegration is the leading methodology in

18、empirical macro/finance as it models equilibrium relationships between nonstationary variablesEstimation and inference is complicated and requires extra careDeviations from the long-term relationship directly feed back into the change of the variablesAdvanced Time Series Topics第12页，共18页。Forecasting

19、economic time seriesForecasting economic time series 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

20、 forecasting error is given by the conditional expectationHere, 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第13页，共18页。R

21、egression 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 many other variablesIf enough lags have been included, the model is dynamically com

22、plete 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第14页，共18页。Example: Forecasting the U.S. unemployment rateLagged inflation significantly helps to predict current unemploymen

23、tNote that these regressions are not meant as causal equations. The hope is that the linear regressions approximate well the conditional expectation. Advanced Time Series Topics第15页，共18页。Evaluating forecast quality of one-step-ahead forecastsOne can measure how good the forecasted values fit the act

24、ual 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 observations to calcula

25、te forecast errors There are different forecast evaluation measures, e.g.Advanced Time Series Topics第16页，共18页。Vector autoregressive 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 estima

26、ted 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 first series in the sense of Granger-Causality.VAR models work for arbitrarily many simultaneously observed time series. They are w