《計量經(jīng)濟學導論》ch10

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1、計量經(jīng)濟學導論ch10計量經(jīng)濟學導論ch10The nature of time series dataTemporal ordering of observations;may not be arbitrarily reorderedTypical features:serial correlation/nonindependence of observationsHow should we think about the randomness in time series data?The outcome of economic variables(e.g.GNP,Dow Jones)is

2、 uncertain;they should therefore be modeled as random variablesTime series are sequences of r.v.(=stochastic processes)Randomness does not come from sampling from a populationSample“=the one realized path of the time series out of the many possible paths the stochastic process could have takenAnalyz

3、ing Time Series:Basic Regression AnalysisThe nature of time series dataExample:US inflation and unemployment rates 1948-2003Here,there are only two time series.There may be many more variables whose paths over time are observed simultaneously.Time series analysis focuses on modeling the dependency o

4、f a variable on its own past,and on the present and past values of other variables.Analyzing Time Series:Basic Regression AnalysisExample:US inflation and unemExamples of time series regression modelsStatic modelsIn static time series models,the current value of one variable is modeled as the result

5、 of the current values of explanatory variablesExamples for static modelsThere is a contemporaneous relationship between unemployment and inflation(=Phillips-Curve).The current murderrate is determined by the current conviction rate,unemployment rate,and fraction of young males in the population.Ana

6、lyzing Time Series:Basic Regression AnalysisExamples of time series regresFinite distributed lag modelsIn finite distributed lag models,the explanatory variables are allowed to influence the dependent variable with a time lagExample for a finite distributed lag modelThe fertility rate may depend on

7、the tax value of a child,but for biological and behavioral reasons,the effect may have a lagChildren born per 1,000 women in year tTax exemption in year tTax exemption in year t-1Tax exemption in year t-2 Analyzing Time Series:Basic Regression AnalysisFinite distributed lag modelsCInterpretation of

8、the effects in finite distributed lag modelsEffect of a past shock on the current value of the dep.variableEffect of a transitory shock:If there is a one time shock in a past period,the dep.variable will change temporarily by the amount indicated by the coefficient of the corresponding lag.Effect of

9、 permanent shock:If there is a permanent shock in a past period,i.e.the explanatory variable permanently increases by one unit,the effect on the dep.variable will be the cumulated effect of all relevant lags.This is a long-run effect on the dependent variable.Analyzing Time Series:Basic Regression A

10、nalysisInterpretation of the effects Graphical illustration of lagged effectsFor example,the effect is biggest after a lag of one period.After that,the effect vanishes(if the initial shock was transitory).The long run effect of a permanent shock is the cumulated effect of all relevant lagged effects

11、.It does not vanish(if the initial shock is a per-manent one).Analyzing Time Series:Basic Regression AnalysisGraphical illustration of laggFinite sample properties of OLS under classical assumptionsAssumption TS.1(Linear in parameters)Assumption TS.2(No perfect collinearity)In the sample(and therefo

12、re in the underlying time series process),no independent variable is constant nor a perfect linear combination of the others.“The time series involved obey a linear relationship.The stochastic processes yt,xt1,xtk are observed,the error process ut is unobserved.The definition of the explanatory vari

13、ables is general,e.g.they may be lags or functions of other explanatory variables.Analyzing Time Series:Basic Regression AnalysisFinite sample properties of OLNotationAssumption TS.3(Zero conditional mean)The mean value of the unobserved factors is unrelated to the values of the explanatory variable

14、s in all periodsThe values of all explanatory variables in period number tThis matrix collects all the information on the complete time paths of all explanatory variablesAnalyzing Time Series:Basic Regression AnalysisNotationThe mean value of the Discussion of assumption TS.3Strict exogeneity is str

15、onger than contemporaneous exogeneityTS.3 rules out feedback from the dep.variable on future values of the explanatory variables;this is often questionable esp.if explanatory variables adjust“to past changes in the dependent variable If the error term is related to past values of the explanatory var

16、iables,one should include these values as contemporaneous regressorsThe mean of the error term is unrelated to the values of the explanatory variables of all periodsThe mean of the error term is unrelated to the explanatory variables of the same periodExogeneity:Strict exogeneity:Analyzing Time Seri

17、es:Basic Regression AnalysisDiscussion of assumption TS.3TTheorem 10.1(Unbiasedness of OLS)Assumption TS.4(Homoscedasticity)A sufficient condition is that the volatility of the error is independent of the explanatory variables and that it is constant over timeIn the time series context,homoscedastic

18、ity may also be easily violated,e.g.if the volatility of the dep.variable depends on regime changesThe volatility of the errors must not be related to the explanatory variables in any of the periodsAnalyzing Time Series:Basic Regression AnalysisTheorem 10.1(Unbiasedness of Assumption TS.5(No serial

19、correlation)Discussion of assumption TS.5Why was such an assumption not made in the cross-sectional case?The assumption may easily be violated if,conditional on knowing the values of the indep.variables,omitted factors are correlated over timeThe assumption may also serve as substitute for the rando

20、m sampling assumption if sampling a cross-section is not done completely randomlyIn this case,given the values of the explanatory variables,errors have to be uncorrelated across cross-sectional units(e.g.states)Conditional on the explanatory variables,the un-observed factors must not be correlated o

21、ver time Analyzing Time Series:Basic Regression AnalysisAssumption TS.5(No serial corTheorem 10.2(OLS sampling variances)Theorem 10.3(Unbiased estimation of the error variance)Under assumptions TS.1 TS.5:The same formula as in the cross-sectional case The conditioning on the values of the explanator

22、y variables is not easy to understand.It effectively means that,in a finite sample,one ignores the sampling variability coming from the randomness of the regressors.This kind of sampling variability will normally not be large(because of the sums).Analyzing Time Series:Basic Regression AnalysisTheore

23、m 10.2(OLS sampling varTheorem 10.4(Gauss-Markov Theorem)Under assumptions TS.1 TS.5,the OLS estimators have the minimal variance of all linear unbiased estimators of the regression coefficientsThis holds conditional as well as unconditional on the regressorsAssumption TS.6(Normality)Theorem 10.5(No

24、rmal sampling distributions)Under assumptions TS.1 TS.6,the OLS estimators have the usual nor-mal distribution(conditional on ).The usual F-and t-tests are valid.independently ofThis assumption implies TS.3 TS.5Analyzing Time Series:Basic Regression AnalysisTheorem 10.4(Gauss-Markov TheExample:Stati

25、c Phillips curveDiscussion of CLM assumptionsContrary to theory,the estimated Phillips Curve does not suggest a tradeoff between inflation and unemploymentA linear relationship might be restrictive,but it should be a good approximation.Perfect collinearity is not a problem as long as unemployment va

26、ries over time.TS.1:The error term contains factors such as monetary shocks,income/demand shocks,oil price shocks,supply shocks,or exchange rate shocksTS.2:Analyzing Time Series:Basic Regression AnalysisExample:Static Phillips curveDiscussion of CLM assumptions(cont.)TS.3:For example,past unemployme

27、nt shocks may lead to future demand shocks which may dampen inflationFor example,an oil price shock means more inflation and may lead to future increases in unemploymentTS.4:TS.5:Assumption is violated if monetary policy is more nervous“in times of high unemploymentTS.6:Assumption is violated if ex-

28、change rate influences persist over time(they cannot be explained by unemployment)QuestionableEasily violatedAnalyzing Time Series:Basic Regression AnalysisDiscussion of CLM assumptions Example:Effects of inflation and deficits on interest ratesDiscussion of CLM assumptionsA linear relationship migh

29、t be restrictive,but it should be a good approximation.Perfect collinearity will seldomly be a problem in practice.TS.1:The error term represents other factors that determine interest rates in general,e.g.business cycle effectsTS.2:Interest rate on 3-months T-billGovernment deficit as percentage of

30、GDPAnalyzing Time Series:Basic Regression AnalysisExample:Effects of inflation Discussion of CLM assumptions(cont.)TS.3:For example,past deficit spending may boost economic activity,which in turn may lead to general interest rate risesFor example,unobserved demand shocks may increase interest rates

31、and lead to higher inflation in future periodsTS.4:TS.5:Assumption is violated if higher deficits lead to more uncertainty about state finances and possibly more abrupt rate changes TS.6:Assumption is violated if business cylce effects persist across years(and they cannot be completely accounted for

32、 by inflation and the evolution of deficits)QuestionableEasily violatedAnalyzing Time Series:Basic Regression AnalysisDiscussion of CLM assumptions Using dummy explanatory variables in time seriesInterpretationDuring World War II,the fertility rate was temporarily lowerIt has been permanently lower

33、since the introduction of the pill in 1963Children born per 1,000 women in year tTax exemption in year tDummy for World War II years(1941-45)Dummy for availabity of con-traceptive pill(1963-present)Analyzing Time Series:Basic Regression AnalysisUsing dummy explanatory variabTime series with trendsEx

34、ample for a time series with a linear upward trend Analyzing Time Series:Basic Regression AnalysisTime series with trendsExampleModelling a linear time trendModelling an exponential time trendAbstracting from random deviations,the dependent variable increases by a constant amount per time unit Alter

35、natively,the expected value of the dependent variable is a linear function of timeAbstracting from random deviations,the dependent vari-able increases by a constant percentage per time unit Analyzing Time Series:Basic Regression AnalysisModelling a linear time trendAExample for a time series with an

36、 exponential trendAbstracting from random deviations,the time series has a constant growth rateAnalyzing Time Series:Basic Regression AnalysisExample for a time series withUsing trending variables in regression analysisIf trending variables are regressed on each other,a spurious re-lationship may ar

37、ise if the variables are driven by a common trendIn this case,it is important to include a trend in the regressionExample:Housing investment and pricesPer capita housing investmentHousing price indexIt looks as if investment and prices are positively relatedAnalyzing Time Series:Basic Regression Ana

38、lysisUsing trending variables in reExample:Housing investment and prices(cont.)When should a trend be included?If the dependent variable displays an obvious trending behaviourIf both the dependent and some independent variables have trendsIf only some of the independent variables have trends;their e

39、ffect on the dep.var.may only be visible after a trend has been substractedThere is no significant relationship between price and investment anymoreAnalyzing Time Series:Basic Regression AnalysisExample:Housing investment anA Detrending interpretation of regressions with a time trendIt turns out tha

40、t the OLS coefficients in a regression including a trend are the same as the coefficients in a regression without a trend but where all the variables have been detrended before the regressionThis follows from the general interpretation of multiple regressionsComputing R-squared when the dependent va

41、riable is trendingDue to the trend,the variance of the dep.var.will be overstatedIt is better to first detrend the dep.var.and then run the regression on all the indep.variables(plus a trend if they are trending as well)The R-squared of this regression is a more adequate measure of fitAnalyzing Time

42、 Series:Basic Regression AnalysisA Detrending interpretation ofModelling seasonality in time seriesA simple method is to include a set of seasonal dummies:Similar remarks apply as in the case of deterministic time trendsThe regression coefficients on the explanatory variables can be seen as the resu

43、lt of first deseasonalizing the dep.and the explanat.variablesAn R-squared that is based on first deseasonalizing the dep.var.may better reflect the explanatory power of the explanatory variables=1 if obs.from december=0 otherwiseAnalyzing Time Series:Basic Regression AnalysisModelling seasonality in time 感謝聆聽