计量经济学(英文PPT)Chapter 21 Time Series Econometrics ⅠStationarity Unit roots and Cointegration

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k


k
0
covariance at lag k variance
obviously when k 0, 1 0
1 1 k
plot against k, the graph we obtain is known as the population correlogram k
• Returning to the example given in figure 21.8,the value of the Q statistic up to lag 25 is about 793,the LB statistic is about 891,both are highly significant, the probability of obtaining such a high
• We can rewrite the functions above as,
Yt ( 1)Yt1 ut
(21.9.1)
• or,
• Yt Yt 1 ut
(-1.96*0.1066,1.96*0.1066) or (0.2089,0.2089)
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• In figure 21.8,the left two lines of dots represent the 95% confidence interval.
the joint hypothesis test of k H0 : all the k are simultaneo usly equal to zero.
• This can be done by using the Q statistic developed by Box and Pierce,
m
Q n k k 1
• Where n is the sample size, and m is the lag length.
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• Q is approximately (in large sample) distributed as chi-square distribution with m df. If Q ,df 2 we can reject the
§ 21.9 The Unit Root Test
• Consider the model as below:
Y Y u t
t 1
t
(21.3.4)
Where ut is a random error term with zero mean, constant v ariance and is serially uncorrelat ed. That is, ut is a white noise. It is an AR(1) model, if the coefficien t of Y t-1 is 1, then Y t has the unit root problem, that is, the time series Y t is nonstation ary.
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Stochastic time series Yt has following properties:
Mean E(Yt)
Variance var(Yt) E(Yt - )2 2 Covariance k E[(Yt )(Y t k )] where k,the covariance at lag k, or autocovariance , is the
• For example, in figure 21.8, n=88,the variance is 1/88, and the
standard error is 1 0.1066 88
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ˆ k
~
N (0, 1 ) n
(21.8.5)
The 95% confidence interval for k is
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Example : pollen swimming in the water is a Ito process. stationary stochastic process: If a random process’s mean and variance are constant over
time and the value of covariance between the two time periods only depends on the distance or lag between the two time periods and not the actual time at which the covariance is computed, then it is stationary. actually, such process is known as weakly stationary stochastic process or wide sense stochastic process
ˆk
(Yt Y )(Ytk Y ) n
(21.8.2)
The sample variance
ˆ0
(Yt Y )2 n
(21.8.3)
The sample autocorrelation function is
ˆ k
ˆk ˆ0
(21.8.4)
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• In Figure 21.6,we have show this correlogram up to 30 lags. Bartlett has show that if a time series is purely random, that is ,it exhibits white noise, its sample autocorrelation coefficients are approximately normally distributed with zero mean and variance equal to 1n,n is the sample size.
first step in the analysis of any time series, getting a overall impression.
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§21.3 Stochastic Processes
Any time series data can be seen as a result generated by stochastic or random process; a concrete set of data can be seen as a particular realization of the underlying stochastic process, which is akin to the relationship between population and sample.
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• So if we do the regression, we will get
• Yt Yt 1 ut
(21.4.1)
• And then if we find that is in fact 1,we can
say the variance Yt has one unit root.
• A time series with an unit root is a random walk (time series). Random walk is an example of nonstationary time series.
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• Assume that:
E(ut) , E(ut )2 2 , and each ut are serially uncorrelat ed. So time series Yt is a random walk, if : Y t Yt1 ut (utis the random stock)
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• Why ? How can we show that ?
var(Y ) E(Yt E(Yt ))2 E(Yt t)2 E( ut t)2 E((ut ))2 E[(u1 ) (u2 ) ut ]2 E(u1 )2 E(u2 )2 E(ut )2 (E(ut )(us ) 0) t 2
Generally, we only have a realization of a stochastic process,
we can only compute the sample autocorrelation function k
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• The sample covariance at lag k
covariance between the values of Yt and Y t k,that is , between two Y values k period apart
k=0, covariance is v time serial is stationary, its mean 、variance and covariance are constant over time.
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• An improvement of the Box-Pierce Q statistic is the LB
statistic developed by Ljung-Box.

LB

n(n

m
2) (
2 k
)
k 1 n k
~
2 m
• The LB statistic has been found to have better small-sample properties than the Q statistic.