第10章 序列相关性-32页文档

What is Serial correlation (Autocorrelation)?
The assumption that errors corresponding to different observations are uncorrelated often breaks down in time-series studies.
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
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.
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.
Runs test (example)
If the signs of the residual is (---------)(++++++++)(----)(++)(---)
9
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
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.
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,
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,
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.
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
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.
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.
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.
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
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
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
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.
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)
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.
– tsset year /* to describe the data is time series */
– estat dwatson /* must using after reg */
– dwstat /*the out of dated command*/
Durbin-Watson Test
Durure
First regress Y on Xs, and get the residuals et. Calculate the DW d-stat. May be given by
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
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.
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)
Patterns of serial correlation
Reasons of serial correlation
Inertia or sluggishness Model specification errors (omitted variables)
What is Serial correlation (Autocorrelation)?