复旦管理学院考博资料——计量经济学Lecture 7.ppt

1、Econometrics (I)Lecture 7 Autocorrelation and Regression with Time Series Data,Dr. Sun Pei (孙霈) Associate Professor in Industrial Economics School of Management, Fudan University,2,Serially Correlated Disturbances,For time series regression model,3,Sources of Serial Correlation,Inertia in most econo

2、mic time series: GDP, Price indexes, etc Specification bias Omitted variables Incorrect functional forms Lagged dependent variables Data transformation: Quarterly data, first differencing, etc Nonstationarity: Mean, variance, and covariance of a time series do not change over time,4,The Lag Operator

3、,LYt = Yt-1, L2Yt = L(LYt) = Yt-2 Lq Yt = Yt-q , L0 = I, L0Yt = Yt The lag operator can be treated as a scalar Polynomials,5,First-order Autocorrelation: AR(1),Specification To ensure that AR(1) is a stationary stochastic process, Properties,6,First-order Autocorrelation: AR(1),AR(1) suggests that u

4、t has a long memory, but when s, the time interval, goes to infinity, the autocorrelation coefficient converges to zero. Var-Cov matrix AR (p),7,First-order Moving Average: MA(1),Specification Properties,8,First-order Moving Average: MA(1),MA(1) suggests that ut has a rather short memory: The errors

5、 are only interrelated in two successive periods Var-Cov matrix MA (q),9,Relations between AR and MA,AR (1) and MA () MA(1) and AR (),10,Consequences of Autocorrelation,The OLS estimators are still linear, unbiased, consistent, and asymptotically normal However, they are no longer efficient Normally

6、 underestimating the error variance Underestimating and inflating t-statistics The problems are not resolved by using large sample sizes,11,Detecting Autocorrelation,Informal/Graphical Methods Durbin-Watson Test Assumptions: The disturbances are generated by AR(1) process and are normally distribute

7、d. Also, the regression model does not include the lagged values of the dependent variable as one of the explanatory variables H0: Test statistic,12,Detecting Autocorrelation,Durbin-Watson Test The exact probability distribution of D.W. is unknown, so unlike t, F tests, there is no unique critical v

8、alue that will lead to the rejection of H0 However, we can derive a lower bound dL and an upper bound dU depending on T and k (Already tabulated) Decision rules Disadvantages,13,Detecting Autocorrelation,The Breusch-Godfrey Test It is a general test of the AR(p) and MA(q) processes Lagged dependent

9、variables can be included Procedures Estimate the original model by OLS to obtain the residuals Regress the residuals on the original Xs and Obtain R2 from this auxiliary regression It can be shown that , so reject the null hypothesis of no autocorrelation if the test statistic is sufficiently large

10、 Note that this is essentially a Lagrange Multiplier test,14,Correcting OLS Standard Errors,HAC (heteroscedasticity- and autocorrelaton-consistent) standard errors are developed by Newey and West (1987) It is an extension of heteroskedasticity-robust standard errors (White, 1980) They are only valid

11、 in large samples, and we can get the estimators of No need to know the structure of autocorrelation and easy to execute via standard econometric packages,15,GLS Estimation,In the case of AR(1),16,GLS Estimation,After the transformation (quasi-differencing) For the estimation to be feasible, Then it

12、 becomes FGLS by plugging the estimated AR(1) coefficient into the transformed regression model. The FGLS estimators are biased but consistent,17,FGLS Estimation,Cochrane-Orcutt estimation omits the first observation, whereas Prais-Winsten estimation keeps the first one. It makes no difference in la

13、rge samples, but since the size of many time series samples is small, the differences are notable in these applications. Iterative scheme: When the FGLS estimators are found using , a new set of residuals can be computed, and a estimator would be easy to obtain, thus leading to the new set FGLS esti

14、mators. We can repeat the process many times, until the estimate of changes very little from the last iteration.,18,Antidumping Filings and Chemical Imports,The effects of antidumping filings by U.S. chemical industries on Chinese imports Were imports unusually high in the period immediately precedi

15、ng the initial filing? befile6 = 1 during the six months before filing Did imports change noticeably after an antidumping filing? affile6 =1 during the six months after filing Was there a significant reduction in Chinese imports after the decision in favor of the U.S. industry? afdec6 =1 during the

16、six months after the positive decision Dependent variable: chnimp = volume of Chinese imports Control variables chempi = chemical production index (control for overall demand) gas = volume of gasoline production (control for demand) rtwex = exchange rate index (strength of dollar against other curre

17、ncies) Monthly data from February 1978 through December 1988,19,Antidumping Filings and Chemical Imports,20,Regression with Time Series Data,A sequence of random variables indexed by time is called a stochastic process or a time series process. Normally, we obtain one and only one possible outcome,

18、or realization, of the process. Finite distributed lag (FDL) models Impact propensity/multiplier: Immediate change in Y due to a one-unit temporary increase in X at time t Long-run propensity/multiplier: Long-run change in Y given a permanent one-unit increase in X,21,OLS Assumptions and Properties,

19、Linearity and no perfect collinearity Zero conditional mean Strict exogeneity: The error at time t is uncorrelated with each explanatory variables in every time period. In cross-sectional data, random sampling suggests that ui is automatically independent of the explanatory variables other than i, b

20、ut in time series regression, random sampling is almost never appropriate. Contemporaneous exogeneity,22,OLS Assumptions and Properties,Linearity, no perfect collinearity, and strictly exogeneity lead to unbiased OLS estimators in time series regression Zero conditional mean: What causes the violati

21、on of the strict exogeneity assumption? Changes in the error term today can cause future changes in the explanatory variable. That is, there may exist feedback from Y on future values of X. Whereas, strictly exogenous explanatory variables cannot react to what has happened to Y in the past. Rainfall

22、 and output vs. Police officers per capita and murder rate,23,OLS Assumptions and Properties,Homoskedasticity (likely to be violated in time series data) and No Serial Correlation The classical linear model assumptions for time series data are more restrictive than those for cross-sectional data: St

23、rict exogeneity and no autocorrelation In cross-sectional data, random sampling ensures that ui and uj are independent for any two observations i and j. All the above assumptions lead to the BLUE. Plus the normality assumption, standard inference (t, F tests) can be undertaken in time series data.,2

24、4,Effects of Personal Tax Exemption on Fertility Rates,Variables of interest gfr: General fertility rate, the No. of children born to very 1,000 women of child-bearing age pe: Average real dollar value of the personal tax exemption Control variables ww2 = 1 during the years 1941 through 1945 pill =

25、1 from year 1963 on, when the birth control pill was made available Yearly data from 1913 through 1984,25,Regression result Result with a FDL model Multicollinearity between pet, pet-1 and pet-2 makes it hard to estimate the individual effect However, the three variables are jointly significant. The

26、 estimated LRP = 0.101, but is it statistically significant?,Effects of Personal Tax Exemption on Fertility Rates,26,Effects of Personal Tax Exemption on Fertility Rates,Reparameterization Running the new regression yields Even though are not significant, the estimate of LRP is our primary interest,

27、27,Trending Variables in OLS Regression,Many time series have a tendency of growing or declining over time. Nothing about trending variables necessarily violates the classical linear model assumptions. However, in many cases, two time series processes appear to be correlated only because they are bo

28、th trending over time for reasons related to other unobserved factors: Spurious regression problem Formulation:,28,29,Trending Variables in OLS Regression,Adding a time trend to solve the omitted variable problem: “Detrending” Housing investment and prices (1947-82, U.S.) Puerto Rican Employment Rat

29、e and U.S. minimum wage and GNP (1950-97),30,Stationary Time Series,Stationary stochastic process For every collection of time indices , the joint distribution of is the same as the joint distribution of No restriction on how xt and xt-1 are related to one another, but it requires the nature of any

30、correlation between adjacent terms be the same across all time periods Stationarity is an aspect of the underlying stochastic process, so it is very hard to determine whether a single realization was generated by a stationary process or not. However, a process with a time trend is clearly nonstation

31、ary,31,Covariance Stationary Process,A stochastic process xt: t = 1, 2, with a finite second moment E(xt2) is covariance stationary if E (xt) is constant Var (xt) is constnat For any t, s1, Cov (xt, xt+s) depends only on s and not on t If a stationary process has a finite second moment, it must be c

32、ovariance stationary, but the converse is certainly not true. If we allow the relationship between two variables to change arbitrarily in each time period, we cannot hope to learn much about how a change in one variable affects the other if we only have a single realization We assume a certain form

33、of stationarity in that OLS estimators do not change over time,32,Weakly Dependent Time Series,A covariance stationary time series is weakly dependent if the correlation between xt and xt+s goes to zero “sufficient quickly” as s. It is also called asymptotically uncorrelated. Examples: MA(1) and AR(

34、1) The central limit theorem for time series data requires stationarity and some form of weak dependence. A trending series, though certainly nonstationary, can be weakly dependent. A series that is stationary about its trend, as well as weakly dependent, is called a tread-stationary process. It can

35、 be used in OLS regression provided that time trend is included in the model,33,Asymptotic Properties of OLS with Time Series Data,Stationarity and weak dependence (xt,Yt); t = 1, 2, is stationary and weakly dependent, so that the law of large numbers and the central limit theorem can be applied to

36、sample averages Contemporaneous exogeneity Along with the usual assumptions of linearity and no perfect collinearity, the OLS estimators are consistent:,34,Examples,Static model: Under weak dependence, So it does not rule out correlation between Xs and past error terms or Ys, for example, Finite dis

37、tributed lag model AR(1) model A model with a lagged dependent variable cannot satisfy the strict exogeneity assumption However, the OLS estimators of the AR(1) model are consistent, though biased. The bias can be large if sample size is small or if 1 is close to 1.,35,HAC Standard Errors and FGLS R

38、evisited,FGLS estimators are consistent if data are weakly dependent and explanatory variables are strictly exogenous. (OLS just requires weak dependency and contemporaneous exogeneity.) It can be shown that the weakest condition for the consistency of FGLS is that ut is uncorrelated with xt-1, xt,

39、and xt+1 HAC standard errors can be poorly behaved in the presence of small sample size and severe serial correlation (OLS estimators are too inefficient) Computing HAC standard errors after FGLS estimation may be a good approach,36,Highly Persistent (Strongly Dependent) Time Series,Random Walk It c

40、an be shown that So it is neither stationary nor weakly dependent Random walk is a special case of what is known as a unit root process Policy relevance,37,38,Highly Persistent (Strongly Dependent) Time Series,A series can be trending but not highly persistent (trend stationary), and highly persiste

41、nt series (interest rates, inflation rates, unemployment rates) have no obvious trends. Random walk with a drift,39,40,Highly Persistent (Strongly Dependent) Time Series,Weakly dependent process are said to be integrated of order zero, or I(0). Nothing needs to be done to such series before using them in regression analysis. Unit root processes are said to be integrated of order