《计量经济学导论》cha.ppt

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Further Issues Using OLS with Time Series Data
Wooldridge: Introductory Econometrics: A Modern Approach, 5e
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The assumptions used so far seem to be too restricitive
Strict exogeneity, homoscedasticity, and no serial correlation are very demanding requirements, especially in the time series context
Statistical inference rests on the validity of the normality assumption
Much weaker assumptions are needed if the sample size is large
A key requirement for large sample analysis of time series is that the time series in question are stationary and weakly dependent
Stationary time series
Loosely speaking, a time series is stationary if its stochastic properties and its temporal dependence structure do not change over time
Analyzing Time Series:Further Issues Using OLS
Stationary stochastic processes
Covariance stationary processes
A stochastic process is stationary, if for every collection of indices the joint distribution of , is the same as that of
for all integers .
A stochastic process is covariance stationary, if its expected value, its variance, and its covariances are constant over time:
1) , 2) , and 3) .
Analyzing Time Series:Further Issues Using OLS
Weakly dependent time series
Discussion of the weak dependence property
An implication of weak dependence is that the correlation between , and must converge to zero if grows to infinity
For the LLN and the CLT to hold, the individual observations must not be too strongly related to each other; in particular their relation must become weaker (and this fast enough) the farther they are apart
Note that a series may be nonstationary but weakly dependent
A stochastic process is weakly dependent , if is „almost independent“ of if grows to infinity (for all ).
Analyzing Time Series:Further Issues Using OLS
Examples for weakly dependent time series
Moving average process of order one (MA(1))
Autoregressive process of order one (AR(1))
The process is weakly dependent because observations that are more than one time period apart have nothing in common and are therefore uncorrelated.
The process is a short moving average of an . series et
The process carries over to a certain extent the value of the previous period (plus random shocks from an . series et)
If the stability condition holds, the process is weakly dependent because serial correlation converges to zero as the distance between observations grows to infinity.
Analyzing Time Series:Further Issues Using OLS
Asymptotic properties of OLS
Assumption ‘ (Linear in parameters)
Same as assumption but now the dependent and independent variables are assumed to be stationary and weakly dependent
Assumption ‘ (No perfect collinearity)
Same as assumption
Assumption ‘ (Zero conditional mean)
Now the explanatory variables are assumed to be only contempo-raneously exogenous rather than strictly exogenous, .
The explanatory variables of the same period are uninformative about the mean of the error term
Analyzing Time Series:Further Issues Using OLS
Theorem (Consistency of OLS)
Why is it important to relax the strict exogeneity assumption?
Strict exogeneity is a serious restriction beause it rules out all kinds of dynamic relationships between explanatory variables and the error term
In particular, it rules out feedback from the dep. var. on future values of the explanat. variables (which is very common in economic contexts)
Strict exogeneity precludes the use of lagged dep. var. as regressors
Important note: For consistency it would even suffice to assume that the explanatory variables are merely contemporaneously uncorrelated with the error term.
Analyzing Time Series:Further Issues Using OLS
Why do lagged dependent variables violate strict exogeneity?
OLS estimation in the presence of lagged dependent variables
Under contemporaneous exogeneity, OLS is consistent but biased
This is the simplest possible regression model with a lagged dependent variable
Contemporanous exogeneity:
Strict exogeneity:
Strict exogeneity would imply that the error term is uncorre-lated with all yt, t=1, … , n-1
This leads to a contradiction because:
Analyzing Time Series:Further Issues Using OLS
Assumption ‘ (Homoscedasticity)
Assumption ‘ (No serial correlation)
Theorem (Asymptotic normality of OLS)
Under assumptions ‘ – ‘, the OLS estimators are asymptotically normally distributed. Further, the usual OLS standard errors, t-statistics and F-statistics are asymptotically valid.
The errors are contemporaneously homoscedastic
Conditional on the explanatory variables in periods t and s, the errors are uncorrelated
Analyzing Time Series:Further Issues Using OLS
Example: Efficient Markets Hypothesis (EMH)
The EMH in a strict form states that information observable to the market prior to week t should not help to predict the return during week t. A simplification assumes in addition that only past returns are considered as relevant information to predict the return in week implies that
A simple way to test the EMH is to specify an AR(1) model. Under the EMH assumption,‘ holds so that an OLS regression can be used to test whether this week‘s returns depend on last week‘s.
There is no evidence against the EMH. Including more lagged returns yields similar results.
Analyzing Time Series:Further Issues Using OLS