1、Chapter 10 Basic Regression Analysis with Time Series Data Wooldridge:Introductory Econometrics:A Modern Approach,5eThe nature of time series dataTemporal ordering of observations;may not be arbitrarily reorderedTypical features:serial correlation/nonindependence of observationsHow should we think a
2、bout the randomness in time series data?The outcome of economic variables(e.g.GNP,Dow Jones)is uncertain;they should therefore be modeled as random variables Time series are sequences of r.v.(=stochastic processes)Randomness does not come from sampling from a population Sample“=the one realized path
3、 of the time series out of the many possible paths the stochastic process could have takenAnalyzing Time Series:Basic Regression AnalysisExample: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 simult
4、aneously.Time series analysis focuses on modeling the dependency of a variable on its own past,and on the present and past values of other variables.Analyzing Time Series:Basic Regression AnalysisExamples of time series regression modelsStatic modelsIn static time series models,the current value of
5、one variable is modeled as the result 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
6、 of young males in the population.Analyzing Time Series:Basic Regression AnalysisFinite 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 dep
7、end on 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 AnalysisInterpretation of the effects in finite d
8、istributed 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 permanent shock:If the
9、re 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 AnalysisGraphical illust
10、ration 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.It does not vanish(if the initial shock is a per-man
11、ent one).Analyzing Time Series:Basic Regression AnalysisFinite sample properties of OLS under classical assumptionsAssumption TS.1(Linear in parameters)Assumption TS.2(No perfect collinearity)In the sample(and therefore in the underlying time series process),no independent variable is constant nor a
12、 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 variables is general,e.g.they may be lags or functions of other explanatory variables.A
13、nalyzing Time Series:Basic Regression AnalysisNotationAssumption TS.3(Zero conditional mean)The mean value of the unobserved factors is unrelated to the values of the explanatory variables in all periodsThe values of all explanatory variables in period number tThis matrix collects all the informatio
14、n on the complete time paths of all explanatory variablesAnalyzing Time Series:Basic Regression AnalysisDiscussion of assumption TS.3Strict exogeneity is stronger than contemporaneous exogeneityTS.3 rules out feedback from the dep.variable on future values of the explanatory variables;this is often
15、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 variables,one should include these values as contemporaneous regressorsThe mean of the error term is unrelated to the values of the explanatory va
16、riables of all periodsThe mean of the error term is unrelated to the explanatory variables of the same periodExogeneity:Strict exogeneity:Analyzing Time Series:Basic Regression AnalysisTheorem 10.1(Unbiasedness of OLS)Assumption TS.4(Homoscedasticity)A sufficient condition is that the volatility of
17、the error is independent of the explanatory variables and that it is constant over timeIn the time series context,homoscedasticity 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 variab
18、les in any of the periodsAnalyzing Time Series:Basic Regression AnalysisAssumption TS.5(No serial 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
19、,omitted factors are correlated over timeThe assumption may also serve as substitute for the random 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.
20、g.states)Conditional on the explanatory variables,the un-observed factors must not be correlated over time Analyzing Time Series:Basic Regression AnalysisTheorem 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
21、cross-sectional case The conditioning on the values of the explanatory 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(becau
22、se of the sums).Analyzing Time Series:Basic Regression AnalysisTheorem 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 regressorsA
23、ssumption TS.6(Normality)Theorem 10.5(Normal 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 AnalysisEx
24、ample:Static 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 unem
25、ployment varies 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 AnalysisDiscussion of CLM assumptions(cont.)TS.3:For example,past unemployment shocks may lead
26、 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-change rate influe
27、nces persist over time(they cannot be explained by unemployment)QuestionableEasily violatedAnalyzing Time Series:Basic Regression AnalysisExample:Effects of inflation and deficits on interest ratesDiscussion of CLM assumptionsA linear relationship might be restrictive,but it should be a good approxi
28、mation.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 GDPAnalyzing Time Series:Basic Regression Analys
29、isDiscussion 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 and lead to higher inflation in future periodsTS.4:TS.5:Assumption is violate
30、d 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 by inflation and the evolution of deficits)QuestionableEasily violatedAnalyz
31、ing Time Series:Basic Regression AnalysisUsing dummy explanatory variables in time seriesInterpretationDuring World War II,the fertility rate was temporarily lowerIt has been permanently lower since the introduction of the pill in 1963Children born per 1,000 women in year tTax exemption in year tDum
32、my for World War II years(1941-45)Dummy for availabity of con-traceptive pill(1963-present)Analyzing Time Series:Basic Regression AnalysisTime series with trendsExample for a time series with a linear upward trend Analyzing Time Series:Basic Regression AnalysisModelling a linear time trendModelling
33、an exponential time trendAbstracting from random deviations,the dependent variable increases by a constant amount per time unit Alternatively,the expected value of the dependent variable is a linear function of timeAbstracting from random deviations,the dependent vari-able increases by a constant pe
34、rcentage per time unit Analyzing Time Series:Basic Regression AnalysisExample for a time series with an exponential trendAbstracting from random deviations,the time series has a constant growth rateAnalyzing Time Series:Basic Regression AnalysisUsing trending variables in regression analysisIf trend
35、ing variables are regressed on each other,a spurious re-lationship may arise 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 an
36、d prices are positively relatedAnalyzing Time Series:Basic Regression AnalysisExample: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 th
37、e independent variables have trends;their effect 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 AnalysisA Detrending interpretation of regressions with a time trendIt
38、 turns out that 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 th
39、e dependent variable 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 fit
40、Analyzing Time Series:Basic Regression AnalysisModelling 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 result 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 Analysis
