第10章 序列相关性-32页文档

for non-stationary time series, the R2 and adjR2 may be invalid.
The consequences of serial correlation (Autocorrelation)
OLS estimators will not be efficient. The variance of
There must be a intercept term in the regression model;
It only can be used to detect the first order serial
correlation. That is, ut=r ut-1+vt, -1r1.
4
2
0
e_t
-2
-4
-4
-2
0
2
4
e_{t-1}
Runs test
First, get the sign of the residuals, et, for example, (---------)(++++++++)(----)(++)(---), that is, there are 9 negative signs, followed by 8 positive signs and so on.
What is Serial correlation (Autocorrelation)?
The assumption that errors corresponding to different observations are uncorrelated often breaks down in time-series studies.
ut=r ut-1+vt
Second-order serial correlation will be
ut=r1ut-1+r2ut-2+vt
The consequences of serial correlation (Autocorrelation)
OLS estimators will be still unbiased and consistent. take the simple regression as an example
In this chapter, we only deal with the problem of first-order serial correlation, in which errors in one time period are correlated directly with errors in the ensuing period. For example,
19. So, if the runs in our model 8 or ≥19, we
should reject the null hypothesis H0.
The number of runs in our model is 5<8, so we
reject the H0, which mean there is serial correlation in our model.
There is no lagged dependent variable as explanatory variable. Ct=b0+b1Yt+b2Ct-1+ut
Durbin-Watson Test
We can rewrite the Durbin-Watson d-stat as
d 2 1 rˆ
第10章 序列相关性
Serial Correlation / Autocorrelation
Main Contents
What is Serial correlation (Autocorrelation)? The consequences of serial correlation How to detect the serial correlation? Corrections for serial correlation
n
etet1
w here,
rˆ =
t2 n
e
2 t
t1
r d-value
-1
4
0
2
1
0
Durbin-Watson Test
Reject H0,
Positive serial correlation
Can not Accept H0, identify. there is no serial
Y= b0 + b1 X +u We know the OLS estimator of b1 is
bˆ1

Xi X Yi Xi X2
b1
+
Xi X ui Xi X 2

E
bˆ1

E
b1
+

Xi X Xi X
If d<dL or d>4-dL, there are positive and negative serial correlation respectively.
If dL<d<dU, or 4-dU<d<4-dL, then we can’t detect the serial correlation.
The same signs in the parentheses are called a run.
Let N is the number of observations, and N1 is the number of positive signs of the residuals, and N2 is the number of negative signs. And k’ is the number of runs.
– tsset year /* to describe the data is time series */
– estat dwatson /* must using after reg */
– dwstat /*the out of dated command*/
Durbin-Watson Test
However, OLSestimateofthevarianceofbˆ1 is
2
2. Xi X
So, in this case, OLSestimates of the variances of the partial coefficients are biased.
The consequences of serial correlation (Autocorrelation)
OLS estimators will be biased.
n1 nt

var bˆ1 varb1+

Xt X Xt X
ut
2

var

xtut xt2 2

xt2 varut +2
xtxt+j cov ut,ut+j
t-statistics and F-statistic will be misleading when there are serial correlation in error terms ut.
The variance and standard error of the predicted value will be invalid.
Durbin-Watson Test: Procedure
First regress Y on Xs, and get the residuals et. Calculate the DW d-stat. May be given by
How to detect the serial correlation?
Time-sequence plot Runs test Durbin-Watson test
Time sequence plot
4
2
0
e_t
-2
-4
1960
1970
1980 year
1990
2000
Example: Real wages and productivity( Example 10-1)
t1 j1
xt2 2
n1 nt
2 TSSx +2 2 TSSx2
rjxtxt+j,where, varut 2,cov ut,ut+j rj2,TSSx xt2.
t1 j1
Ifthereexistsfirst orderserial correlation, ie. ut rut1+vt.
Runs test
Swed and Eisenhart give us a table of critical values.
H0: the residual e is stochastic, that is, there is no serial correlation.
How to test? If the number of run in your model is less than or equal the critical value n1(table A6a), and larger than or equal to the critical value n2(A-6b), then we can reject the null hypothesis, H0, means there exists serial correlation.
Runs test (example)
If the signs of the residual is (---------)(++++++++)(----)(++)(---)
9
பைடு நூலகம்
8
423
Then, N1=8+2=10, N2=9+4+3=16, N=26, k’=5, then the critical value at 5% significance is 8 and
Durbin-Watson Test
Durbin and Watson put forward an d statistic (DW).
n
e t e t 1 2
d t2 n
e
2 t
t 1
In most software, d- value will be provided with R2, adjR2(Eviews), in STATA, using command
When the error terms from different (usually adjacent) time periods are correlated, we say that the error term is serially correlated. That is,
Cov(ui, uj)0, i.e. E(ui, uj) 0 for i j.
Patterns of serial correlation
Reasons of serial correlation
Inertia or sluggishness Model specification errors (omitted variables)
What is Serial correlation (Autocorrelation)?
correlation.
Reject H0, Can not identify. Negative
serial correlation
0
dL
dU
2
4-dU 4-dL
4
If the Durbin-Watson d-stat lies in (du, 4- du), there is no serial correlation.
ui
2



b1
The consequences of serial correlation (Autocorrelation)
The R2 and adj-R2 are still consistent if the
time series is stationary (that’s r <1). Or else,