《計量經(jīng)濟學導論》ch10課件

1、Chapter 10 Basic Regression Analysis with Time Series Data Wooldridge: Introductory Econometrics: A Modern Approach, 5e第1頁，共26頁。The nature of time series dataTemporal ordering of observations; may not be arbitrarily reorderedTypical features: serial correlation/nonindependence of observationsHow sho

2、uld we think about 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 variablesTime series are sequences of r.v. (= stochastic processes)Randomness does not come from sampling from a populationSample“ =

3、the one realized path of the time series out of the many possible paths the stochastic process could have takenAnalyzing Time Series:Basic Regression Analysis第2頁，共26頁。Example: US inflation and unemployment rates 1948-2019Here, there are only two time series. There may be many more variables whose pa

4、ths over time are observed simultaneously.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 Analysis第3頁，共26頁。Examples of time series regression modelsStatic modelsIn stat

5、ic time series models, the current value of 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

6、conviction rate, unemployment rate, and fraction of young males in the population.Analyzing Time Series:Basic Regression Analysis第4頁，共26頁。Finite distributed lag modelsIn finite distributed lag models, the explanatory variables are allowed to influence the dependent variable with a time lagExample fo

7、r a finite distributed lag modelThe fertility rate may depend 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

8、Regression Analysis第5頁，共26頁。Interpretation of 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 t

9、he coefficient of the corresponding lag.Effect of 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 depen

10、dent variable.Analyzing Time Series:Basic Regression Analysis第6頁，共26頁。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

11、effect of all relevant lagged effects. It does not vanish (if the initial shock is a per-manent one).Analyzing Time Series:Basic Regression Analysis第7頁，共26頁。Finite sample properties of OLS under classical assumptionsAssumption TS.1 (Linear in parameters)Assumption TS.2 (No perfect collinearity)In th

12、e sample (and therefore 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 definit

13、ion of the explanatory variables is general, e.g. they may be lags or functions of other explanatory variables.Analyzing Time Series:Basic Regression Analysis第8頁，共26頁。NotationAssumption TS.3 (Zero conditional mean)The mean value of the unobserved factors is unrelated to the values of the explanatory

14、 variables 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 Analysis第9頁，共26頁。Discussion of assumption TS.3Strict exogeneity is stronger than c

15、ontemporaneous 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 variables,

16、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 Series:Basi

17、c Regression Analysis第10頁，共26頁。Theorem 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, homoscedasticity may also be easily v

18、iolated, 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 Analysis第11頁，共26頁。Assumption TS.5 (No serial correlation)Discussion of assumption TS

19、.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 random sampling assumption if sampling a

20、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 over time Analyzing Time Series:

21、Basic Regression Analysis第12頁，共26頁。Theorem 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 explanatory variables is not easy to understand. It effect

22、ively 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 Analysis第13頁，共26頁。Theorem 10.4 (Gauss-Markov Theorem)Unde

23、r 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 (Normal sampling distributions)Under assumptions TS.1 TS.6, t

24、he 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 Analysis第14頁，共26頁。Example: Static Phillips curveDiscussion of CLM assumptionsContrary to theory, the es

25、timated 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 varies over time. TS.1:The error term contains factors such as monetar

26、y shocks, income/demand shocks, oil price shocks, supply shocks, or exchange rate shocksTS.2:Analyzing Time Series:Basic Regression Analysis第15頁，共26頁。Discussion of CLM assumptions (cont.)TS.3:For example, past unemployment shocks may lead to future demand shocks which may dampen inflationFor example

27、, 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 influences persist over time (they cannot be explained by unemploy

28、ment)QuestionableEasily violatedAnalyzing Time Series:Basic Regression Analysis第16頁，共26頁。Example: Effects of inflation and deficits on interest ratesDiscussion of CLM assumptionsA linear relationship might be restrictive, but it should be a good approximation. Perfect collinearity will seldomly be a

29、 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 Analysis第17頁，共26頁。Discussion of CLM assumptions (

30、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 violated if higher deficits lead to

31、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 violatedAnalyzing Time Series:Basic Regres

32、sion Analysis第18頁，共26頁。Using 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 tDummy for World War

33、II years (1941-45)Dummy for availabity of con-traceptive pill (1963-present)Analyzing Time Series:Basic Regression Analysis第19頁，共26頁。Time series with trendsExample for a time series with a linear upward trend Analyzing Time Series:Basic Regression Analysis第20頁，共26頁。Modelling a linear time trendModel

34、ling 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 con

35、stant percentage per time unit Analyzing Time Series:Basic Regression Analysis第21頁，共26頁。Example for a time series with an exponential trendAbstracting from random deviations, the time series has a constant growth rateAnalyzing Time Series:Basic Regression Analysis第22頁，共26頁。Using trending variables i

36、n regression analysisIf trending 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

37、indexIt looks as if investment and prices are positively relatedAnalyzing Time Series:Basic Regression Analysis第23頁，共26頁。Example: 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 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 Analysis