Chapter12
Autocorrelation,
What Happens
if Error Terms are Correlated
12.1 The Nature of Autocorrelation
1.Definition
（ 1） CLRM assumption,
No autocorrelation exist in dishurbances μi;
E(μiμj)＝ 0 i≠j
Autocorrelation means,E(μiμj)≠0 i≠j
（ 2） Autocorrelation is usually associated with time series
data,but it can also occur in cross-sectional data,which is
called spatial correlation.
（ 3） Autocorrelation can be positive as well as negative.
2,Patterns of autocorrelation
Figure 12-1,p379
3,Reasons of autocorrelation
(1) Inertia or sluggishness
Most economic time-series is inertia,
such as GDP,money supply,price indexes,
so successive observations are correlated.
（ 2） Model Specification Error(s)
? Some important variables that should be included in
the model are not included (underspecification)
? The model has the wrong functional form
e.g,a linear-in-variable(LIV) model is fitted
whereas a log-linear model should have been fitted.
（ 3） Cobweb phenomenon
The agriculture commodities often reflects
the Cobweb phenomenon,where supply reacts
to price with a lag of one time period because
supply decisions take time to implement,the
beginning of this year’s planting of crops
farmers are influenced by the price prevailing
last year
Supplyt=B1+B2Pt-1+μt
（ 4） Data Manipulation
Data smoothness can itself lead to a
systematic pattern in the disturbances,
thereby inducing autocorrelation.
12.2 Consequences of autocorrelation
(1)The OLS estimators are linear and unbiased
(2)The OLS estimators are not efficient
The error variance of OLS estimators is a biased
estimator of the true σ2
The estimated variances sometimes underestimate true
variances and standard errors,thereby inflating t values
（ 3） The t and F tests are not generally reliable.
（ 4） The conventionally computed R2 may be an unreliable
measure of true R2.
（ 5） Variances and standard errors of forecast may also be
inefficient.
12.3 Detecting Autocorrelation
Because the true ui are unobservable,we
have to rely on the ets obtained from the
standard OLS procedure to,learn”something
about the presence,or lack thereof,of
autocorrelation.
1,The Graphical Method,
～ Visual examine the OLS residuals,ets
(1) Plot residuals against time（ time-
sequence plot）, ei～ t
(2) Plot the residuals at time t against their
values lagged in one period; that is,plot et
against et-1, et～ et-1
2.The Runs Test
（ 1） Note down the sign(+or-)of the residuals
（ 2） Get the number of runs
A run,an uninterrupted sequence of one
symbol or attribute,such as“+” or“－”
The length of the run,the number of elements
in the run.
（ 3） Test the randomness of runs.
Let,N = total number of observations(=N1+N2)
N1=number of,+”symbols(i.e.,+residuals)
N2=number of,–”symbols(i.e.,-residuals)
k=number of runs
H0,the successive residuals are independent,or random.
? Compare the runs observed with the number of runs
expected in a strictly random sequence of same observations.
① If there are too many runs,it suggests negative serial
correlation (it means that the ets change sign frequently)
② If there are too few runs,it suggests positive autocorrelation.
Swed-Eisenhart critical runs test:
k ≥critical value (kright),reject H0,negative
serial correlation
k ≤critical value (kleft),reject H0,positive
autocorrelation
kleft ＜ k ＜ kright,accept H0
k ---Actual number of runs
3.The Durbin-Watson d Test
（ 1） Durbin-Watson d statistic
(12.5)
?
?
?
?
?
?
?
n
1t
2
t
n
2t
2
1tt
e
)e(e
d
（ 2） The assumptions underlying the d statistic.
? The regression model includes an intercept term.
? The X variables are nonstochastic;that is,their values
are fixed in repeated sampling.
? The disturbances μt are generated by the following
mechanism
μt=ρμt-1+vt -1≤ρ≤1 （ 12.7）
ρ－－ coefficient of autocorrelation
μt=ρμt-1+vt
－－（ Markov） first-order autoregression scheme,
AR(1) scheme
?The regression does not contain the lagged
value(s) of the dependent variable as one of
the explanatory variables.
That is,the test is not applicable to
models such as
Yt=B1+B2Xt+B3Yt-1+μt （ 12.8）
which is called autoregression models
（ 3） For a large sample size,
can be approximately expressed as
d≈ 2( 1- )
,the estimator of ρ,－ 1≤ ρ ≤1,0≤ d≤4;
? ρ=－ 1,d=4,negative serial correlation
? ρ=0,no autoregression
? ρ= 1,d=0 positive atuoregression
?
?
?
?
?
?
n
t
t
n
t
tt
e
ee
1
2
2
1
??p?
?
?
?
?
??
? n
1t
2
t
n
2t
2
1tt
e
)e(e
d
p?
（ 4） Durbin-Watson test
?Run the OLS regression and obtain the residuals et
?Compute d
?Find out the critical dL and dU from the Durbin-
Watson tables for the given sample size and the
given number of explanatory variables.
?Follow the decision rules to make a decision
dL ≤ d≤4－ dU,accept H0
0≤ d≤ dL,,positive autocorrelation,reject H0
4 － dL ≤ d≤4,negative autocorrelation,reject H0
（ 5） Drawback of the d test:
?If it falls in the indecisive zone,or region
of ignorance,We cannot conclude whether
or not autocorrelation does exist.
?If a d statistic lies in the indecisive zone,it
might be prudent to assume that
autocorrelation exists and proceed to
correct the condition.
Note:
?The d test should not be applied if the
assumptions underlying the test discussed
earlier do not hold.
?It should not be used to test for serial
correlation in autoregressive models like
the regression(12.8)
12.4 Remedial Measures
Two-variable model,Yt=B1+B2Xt+μt （ 12.12）
μt～ AR（ 1）, μt=ρμt-1+vt
－ 1≤p≤1 （ 12.7）
Transform (12.12)
Yt-1=B1+B2Xt-1+μt-1 （ 12.13）
ρYt-1=ρB1+ρB2Xt-1+ρμt-1 （ 12.14）
(Yt-ρYt-1)=B1(1-ρ)+B2(Xt-ρXt-1)+vt （ 12.15）
（ 12.16）
◆ （ 12.15）（ 12.16） are generalized difference equation.
t
*
t2
*
1
*
t vXBBY ???
Apply OLS to （ 12.16）,the estimators are
BLUE,the estimators are called generalized least
squares（ GLS） estimators,
To avoid the loss of one observation,the first
observation of Y and X are transformed as
follows:
)(Y1Y 12*1 ???
)(X1X 12*1 ???
12.5 How to Estimateρ
1,ρ=1:The First Difference Method
Suppose,ρ=1,that is,the error terms are
perfectly positively autocorrelated.
Form the first difference equation:
Yt-Yt-1=B2(Xt-Xt-1)+Vt
Or,△ Yt=B2△ Xt+Vt (12.18)
Note,This first difference equation has no
intercept.
Run the regress on the transformed model.
2,ρEstimated from Durbin-Watson d Statistic－
－ when the sample size is reasonable large.
d ≈ 2 (1- )
≈ 1- d / 2
3,Estimated from OLS Residuals,et －－ when
the sample size is reasonable large.
μt=ρμt-1+vt
et= et-1+vt
??
??
??
After you get ρ,then you can:
1,Transform the model;
2,Regress the transformed model by OLS
Autocorrelation,
What Happens
if Error Terms are Correlated
12.1 The Nature of Autocorrelation
1.Definition
（ 1） CLRM assumption,
No autocorrelation exist in dishurbances μi;
E(μiμj)＝ 0 i≠j
Autocorrelation means,E(μiμj)≠0 i≠j
（ 2） Autocorrelation is usually associated with time series
data,but it can also occur in cross-sectional data,which is
called spatial correlation.
（ 3） Autocorrelation can be positive as well as negative.
2,Patterns of autocorrelation
Figure 12-1,p379
3,Reasons of autocorrelation
(1) Inertia or sluggishness
Most economic time-series is inertia,
such as GDP,money supply,price indexes,
so successive observations are correlated.
（ 2） Model Specification Error(s)
? Some important variables that should be included in
the model are not included (underspecification)
? The model has the wrong functional form
e.g,a linear-in-variable(LIV) model is fitted
whereas a log-linear model should have been fitted.
（ 3） Cobweb phenomenon
The agriculture commodities often reflects
the Cobweb phenomenon,where supply reacts
to price with a lag of one time period because
supply decisions take time to implement,the
beginning of this year’s planting of crops
farmers are influenced by the price prevailing
last year
Supplyt=B1+B2Pt-1+μt
（ 4） Data Manipulation
Data smoothness can itself lead to a
systematic pattern in the disturbances,
thereby inducing autocorrelation.
12.2 Consequences of autocorrelation
(1)The OLS estimators are linear and unbiased
(2)The OLS estimators are not efficient
The error variance of OLS estimators is a biased
estimator of the true σ2
The estimated variances sometimes underestimate true
variances and standard errors,thereby inflating t values
（ 3） The t and F tests are not generally reliable.
（ 4） The conventionally computed R2 may be an unreliable
measure of true R2.
（ 5） Variances and standard errors of forecast may also be
inefficient.
12.3 Detecting Autocorrelation
Because the true ui are unobservable,we
have to rely on the ets obtained from the
standard OLS procedure to,learn”something
about the presence,or lack thereof,of
autocorrelation.
1,The Graphical Method,
～ Visual examine the OLS residuals,ets
(1) Plot residuals against time（ time-
sequence plot）, ei～ t
(2) Plot the residuals at time t against their
values lagged in one period; that is,plot et
against et-1, et～ et-1
2.The Runs Test
（ 1） Note down the sign(+or-)of the residuals
（ 2） Get the number of runs
A run,an uninterrupted sequence of one
symbol or attribute,such as“+” or“－”
The length of the run,the number of elements
in the run.
（ 3） Test the randomness of runs.
Let,N = total number of observations(=N1+N2)
N1=number of,+”symbols(i.e.,+residuals)
N2=number of,–”symbols(i.e.,-residuals)
k=number of runs
H0,the successive residuals are independent,or random.
? Compare the runs observed with the number of runs
expected in a strictly random sequence of same observations.
① If there are too many runs,it suggests negative serial
correlation (it means that the ets change sign frequently)
② If there are too few runs,it suggests positive autocorrelation.
Swed-Eisenhart critical runs test:
k ≥critical value (kright),reject H0,negative
serial correlation
k ≤critical value (kleft),reject H0,positive
autocorrelation
kleft ＜ k ＜ kright,accept H0
k ---Actual number of runs
3.The Durbin-Watson d Test
（ 1） Durbin-Watson d statistic
(12.5)
?
?
?
?
?
?
?
n
1t
2
t
n
2t
2
1tt
e
)e(e
d
（ 2） The assumptions underlying the d statistic.
? The regression model includes an intercept term.
? The X variables are nonstochastic;that is,their values
are fixed in repeated sampling.
? The disturbances μt are generated by the following
mechanism
μt=ρμt-1+vt -1≤ρ≤1 （ 12.7）
ρ－－ coefficient of autocorrelation
μt=ρμt-1+vt
－－（ Markov） first-order autoregression scheme,
AR(1) scheme
?The regression does not contain the lagged
value(s) of the dependent variable as one of
the explanatory variables.
That is,the test is not applicable to
models such as
Yt=B1+B2Xt+B3Yt-1+μt （ 12.8）
which is called autoregression models
（ 3） For a large sample size,
can be approximately expressed as
d≈ 2( 1- )
,the estimator of ρ,－ 1≤ ρ ≤1,0≤ d≤4;
? ρ=－ 1,d=4,negative serial correlation
? ρ=0,no autoregression
? ρ= 1,d=0 positive atuoregression
?
?
?
?
?
?
n
t
t
n
t
tt
e
ee
1
2
2
1
??p?
?
?
?
?
??
? n
1t
2
t
n
2t
2
1tt
e
)e(e
d
p?
（ 4） Durbin-Watson test
?Run the OLS regression and obtain the residuals et
?Compute d
?Find out the critical dL and dU from the Durbin-
Watson tables for the given sample size and the
given number of explanatory variables.
?Follow the decision rules to make a decision
dL ≤ d≤4－ dU,accept H0
0≤ d≤ dL,,positive autocorrelation,reject H0
4 － dL ≤ d≤4,negative autocorrelation,reject H0
（ 5） Drawback of the d test:
?If it falls in the indecisive zone,or region
of ignorance,We cannot conclude whether
or not autocorrelation does exist.
?If a d statistic lies in the indecisive zone,it
might be prudent to assume that
autocorrelation exists and proceed to
correct the condition.
Note:
?The d test should not be applied if the
assumptions underlying the test discussed
earlier do not hold.
?It should not be used to test for serial
correlation in autoregressive models like
the regression(12.8)
12.4 Remedial Measures
Two-variable model,Yt=B1+B2Xt+μt （ 12.12）
μt～ AR（ 1）, μt=ρμt-1+vt
－ 1≤p≤1 （ 12.7）
Transform (12.12)
Yt-1=B1+B2Xt-1+μt-1 （ 12.13）
ρYt-1=ρB1+ρB2Xt-1+ρμt-1 （ 12.14）
(Yt-ρYt-1)=B1(1-ρ)+B2(Xt-ρXt-1)+vt （ 12.15）
（ 12.16）
◆ （ 12.15）（ 12.16） are generalized difference equation.
t
*
t2
*
1
*
t vXBBY ???
Apply OLS to （ 12.16）,the estimators are
BLUE,the estimators are called generalized least
squares（ GLS） estimators,
To avoid the loss of one observation,the first
observation of Y and X are transformed as
follows:
)(Y1Y 12*1 ???
)(X1X 12*1 ???
12.5 How to Estimateρ
1,ρ=1:The First Difference Method
Suppose,ρ=1,that is,the error terms are
perfectly positively autocorrelated.
Form the first difference equation:
Yt-Yt-1=B2(Xt-Xt-1)+Vt
Or,△ Yt=B2△ Xt+Vt (12.18)
Note,This first difference equation has no
intercept.
Run the regress on the transformed model.
2,ρEstimated from Durbin-Watson d Statistic－
－ when the sample size is reasonable large.
d ≈ 2 (1- )
≈ 1- d / 2
3,Estimated from OLS Residuals,et －－ when
the sample size is reasonable large.
μt=ρμt-1+vt
et= et-1+vt
??
??
??
After you get ρ,then you can:
1,Transform the model;
2,Regress the transformed model by OLS
