1. Lecture #9 Autocorrelation Serial Correlation
Studenmund (2006) Chapter 9
Autocorrelation
Serial Correlation
Objectives
The nature of autocorrelation
The consequences of autocorrelation
Testing the existence of autocorrelation
Correcting autocorrelation

2. Time Series Data Time series process of economic variables
e.g., GDP, M1, interest rate, exchange rate,
imports, exports, inflation rate, etc.
Realization
An observed time series data set generated from a time series process
Remark:
Age is not a realization of time series process.
Time trend is not a time series process too.

3. Cyclical or seasonal random
Decomposition of time series
Xt = Trend + seasonal + random
Trend Xt
time
Cyclical or
seasonal
random

4. Example: Static Phillips curve model inflatt = 0 + 1unemployt + t
Static Models
Ct = 0 + 1Ydt + t
Subscript “t” indicates time. The regression is a contemporaneous relationship, i.e., how does current consumption (C) be affected by current Yd?
Example: Static Phillips curve model
inflatt = 0 + 1unemployt + t
inflat: inflation rate
unemploy: unemployment rate
Contemporaneous relation:
a. Equilibrium relation
b. No dynamic effects, i.e., immediate effect only

5. Ct+1=0+0Ydt+1+1Ydt+tCt=0 +0Ydt+1Ydt-1+t
Finite Distributed Lag Models
Economic action
at time t
Effect
Ct =0+0Ydt+t
Effect
at time t+1
Ct+1=0+0Ydt+1+1Ydt+tCt=0 +0Ydt+1Ydt-1+t
Effect
at time t+2
Effect
at time t+q ….
Forward Distributed Lag Effect (with order q) ….
Ct+q=0+1Ydt+q+…+1Ydt+tCt=0+1Ydt+…+1Ydt-q+t

6. Backward Distributed Lag Effect
Economic action
at time t
Effect
at time t-1
Backward Distributed Lag Effect
Effect
at time t-2
Effect
at time t-3
Effect
at time t-q ….
Yt= 0+0Zt+1Zt-1+2Zt-2+…+2Zt-q+t
Initial state: zt = zt-1 = zt-2 = c

7. C = 0 + 0Ydt + 1Ydt-1 + 2Ydt-2 + t
Long-run propensity (LRP) = (0 + 1 + 2)
Permanent unit change in C for 1 unit permanent (long-run) change in Yd.
Distributed Lag model in general:
Ct = 0 + 0Ydt + 1Ydt-1 +…+ qYdt-q
+ other factors + t
LRP (or long run multiplier) = 0 +  q

8. Time Trends Linear time trend
Yt = 0 + 1t + t Constant absolute change
Exponential time trend
ln(Yt) = 0 + 1t + t Constant growth rate
Quadratic time trend
Yt = 0 + 1t + 2t2 + t Accelerate change
For advances on time series analysis and modeling , welcome to take ECON 3670

9. Definition: First-order of Autocorrelation, AR(1)
If Cov (t, s) = E (t s)  where t  s
Yt = 0 + 1 X1t + t t = 1,……,T
and if
t =  t-1 + ut
where <  < ( : RHO)
and ut ~ iid (0, u2) (white noise)
This scheme is called first-order autocorrelation and denotes as AR(1)
Autoregressive : The regression of t can be explained by itself lagged one period.
(RHO) : the first-order autocorrelation coefficient
or ‘coefficient of autocovariance’

10. Example of serial correlation:
u1990
… … … ….
… … ….
u2002
u2003
u2004
u2005
u2006
u2007
Year Consumptiont = 0 + 1 Incomet + errort
Example of serial correlation:
TaxPay2006
TaxPay2007
Error term
represents
other factors
that affect
consumption
The current year Tax Pay may be determined by previous year rate
TaxPay2007 =  TaxPay u2007
 t =  t-1 + ut
ut ~ iid(0, u2)

11. If t = 1 t-1 + ut
it is AR(1), first-order autoregressive
If t = 1 t-1 + 2 t-2 + ut
it is AR(2), second-order autoregressive
High order
autocorrelation
If t = 1 t-1 + 2 t-2 + 3 t-3 + ut
it is AR(2), third-order autoregressive
If t = 1 t-1 + 2 t-2 + …… + n t-n + ut
it is AR(n), nth-order autoregressive
……………………………………………….
Autocorrelation AR(1) :
Cov (t  t-1) > => 0 <  < positive AR(1)
Cov (t t-1) < => -1 <  < negative AR(1)
-1 <  < 1

12. time i ^ x
Positive autocorrelation
time i ^ x
Positive autocorrelation
time i ^
Cyclical: Positive autocorrelation x
The current error
term tends to have
the same sign as
the previous one.

13. Negative autocorrelation
time i ^ x
The current error term tends to have the opposite sign from the previous.
No autocorrelation x
time i ^
The current error term tends to be randomly appeared from the previous.

14. The meaning of  : The error term t at time t is a linear
The meaning of  : The error term t at time t is a linear combination of the current and past disturbance.
0 <  < 1
-1 <  < 0
The further the period is in the past, the smaller is the weight of that error term (t-1) in determining t
 = 1
The past is equal importance to the current.
 > 1
The past is more importance than the current.

15. The consequences of serial correlation:
The estimated coefficients are still unbiased.
E(k) = k ^
BLUE ^
2. The variances of the k is no longer the smallest
3. The standard error of the estimated coefficient, Se(k)
becomes large ^
Therefore, when AR(1) is existing in the regression,
The estimation will not be “BLUE”

16. The AR(1) variance is not the smallest
Example:
Two variable regression model: Yt = 0 + 1X1t + t
The OLS estimator of 1, ^
 x y
 xt2
If E(t t-1) = 0
then Var (1) = 2
===> 1 =
If E(tt-1)  0, and t =  t-1 + ut , then
Var (1)AR1=  2 ^
   xt xt xt xt+2
 xt2  xt  xt xt2
-1 <  < 1
+ ….
If  = 0, zero autocorrelation, than Var(1)AR1 = Var(1) ^
If   0, autocorrelation, than Var(1)AR1 > Var(1)
The AR(1) variance is not the smallest

17. Autoregressive scheme:
t =  t-1 + ut ==> t = [ t-2 + ut-1] + ut
==> t-2 =  t-3 + ut => t = 2 [ t-3 + ut-2] + ut-1 + ut
==> t-1 =  t-2 + ut  t = 2  t-2 + ut-1 + ut
t = 3 t-3 + 2 ut-2 + ut-1 + ut
E(t t-1) = 2
1 - 2
E(t t-3) = 2 2
E(t t-2) = 2
…………….
E(t t-k) = k-1 2
It means the more periods in the past,
the less effect on current period
k-1  becomes smaller and smaller

18. How to detect autocorrelation ?
DW* or d*

19. 5% level of significance,
k = 1,
n=24
k is the number of independent variables (excluding the intercept)
dL = 1.27
du = 1.45
DW* =
 DW* < dL

20. Durbin-Watson Autocorrelation test
From OLS regression result: where d or DW* =
Check DW Statistic Table (At 5% level of significance, k’ = 1, n=24)
dL = 1.27
du = 1.45
1.27
1.45 2 dL du
DW*
0.9107
Reject H0 region
H0 : no autocorrelation
 = 0
H1 : yes, autocorrelation exists.
or  > 0
positive autocorrelation

21. Durbin-Watson test OLS : Y = 0 + 1 X2 + …… + k Xk + t
obtain t , DW-statistic(d) ^
Assuming AR(1) process: t =  t-1 + ut
I H0 :  ≤ no positive autocorrelation
H1 :  > yes, positive autocorrelation
-1 <  < 1
Compare d* and dL, du (critical values)
DW*
if d* < dL ==> reject H0
if d* > du ==> not reject H0
if dL  d*  du ==> this test is inconclusive

22. ^ d  2(1-r) Durbin-Watson test(Cont.) DW =  2 (1 - )  (t - t-1)2
==> ≈ 1 - 
==>  ≈ 1- ^ d 2
Since -1    1 ^
implies 0  d  4
1.27
1.45 2 dL du 4
(4-dL)
(4-dU)
2.73
2.55

23. Durbin-Watson test(Cont.)
II. H0 :  ≥ no negative autocorrelation
H1 :  < yes, negative autocorrelation
we use (4-d) (when d is greater than 2)
if (4 - d) < dL
or dL < d < ==> reject H0
if dL  (4 - d) du
or du  d  4 - dL ==> inconclusive
if dL  (4 - d) du
or du > d > ==> not reject H0
1.27
1.45 2 dL du 4
(4 - dL)
(4-dU)
2.73
2.55

24. Durbin-Watson test(Cont.)
II. H0 :  = No autocorrelation
H1 :   two-tailed test for auto correlation
either positive or negative AR(1)
If d < dL
or d > 4 - dL
==> reject H0
If du < d < 4 - du ==> not reject H0
If dL  d  du
or 4 - du  d  4 - dL
==> inconclusive

25. For example :
UMt = CAPt CAPt Tt ^
(15.6) (2.0) (3.7) (10.3)
R2 = F =  = SSR = DW = n = 68 _
(i) K = 3 (number of independent variable)
Observed
(ii) n = 68 , = significance level
0.05
(iii) dL = , du =
dL = , du =
Reject H0, positive autocorrelation exists
(excluding intercept)

26. negative autocorrelation
H0 :  = 0
positive autocorrelation
H1 :  > 0 dL du 2
DW (d)
4-du
4-dL 4
inconclusive
reject H0
H0 :  = 0
negative autocorrelation
H1 :  < 0
not reject
reject H0
not reject
inconclusive
1% & 5%
Critical values
0.23

27. The assumptions underlying the d(DW) statistics :
1. Intercept term must be included.
2. X’s are nonstochastic
3. Only test AR(1) : t = t-1 + ut where ut ~ iid (0, u2)
4. Not include the lagged dependent variable,
Yt = 0+ 1 Xt1 + 2 Xt2 + …… + kXtk +  Yt-1 + t
(autoregressive model)
5. No missing observation
N.A N.A.
N.A N.A. 95
... Y X
missing

28. Lagrange Multiplier (LM) Test or called Durbin’s m test
Or Breusch-Godfrey (BG) test of higher-order autocorrelation ^
Test Procedures:
(1) Run OLS and obtain the residuals t. ^
(2) Run t against all the regressors in the model
plus the additional regressors, t-1, t-2, t-3,…, t-p.
t = 0 + 1 Xt + t-1 + t-2 + t-3 + … + t-p + u
Obtain the R2 value from this regression.
(3) compute the BG-statistic: (n-p)R2
(4) compare the BG-statistic to the 2p (p is # of degree-order)
(5) If BG > 2p, reject Ho,
it means there is a higher-order autocorrelation
If BG < 2p, not reject Ho,
it means there is a no higher-order autocorrelation

29. D D Remedy: 1. First-difference transformation Yt = 0 + 1 Xt + t
Yt-1 = 0 + 1 Xt-1 + t assume  = 1
==> Yt - Yt-1 = 0 - 0 + 1 (Xt - Xt-1) + (t - t-1)
==> Yt = 1 DXt + t D
no intercept
2. Add a trend (T)
Yt = 0 + 1 Xt + 2 T + t
Yt-1 = 0 + 1 Xt-1 + 2 (T -1) + t-1
==> (Yt - Yt-1) = (0 - 0) + 1 (Xt - Xt-1) + 2 [T- (T -1)] + (t - t-1)
==> DYt = 1 DXt + 2*1 + ’t
==> Yt = 2* + 1 DXt + ’t D
If 1* > 0 => an upward trend in Y ^
(2 > 0)

30. 3. Cochrane-Orcutt Two-step procedure (CORC)
(1). Run OLS on
Yt = 0 + 1 Xt + t
and obtains t ^
Generalized
Least Squares
(GLS)
method
(2). Run OLS on
t =  t-1 + ut ^
and obtains 
Where u~(0, )
(3). Use the  to transform the variables : ^
Yt* = Yt -  Yt-1
Xt* = Xt -  Xt-1
Yt = 0 + 1 Xt + t
-)  Yt-1 = 0  + 1  Xt-1 + t-1 ^
(Yt - Yt-1)= 0(1-) +1(Xt - Xt-1) + (t -t-1) ^
(4). Run OLS on
Yt* = 0* + 1* Xt* + ut

31. 4. Cochrane-Orcutt Iterative Procedure
(5). If DW test shows that the autocorrelation still existing, than
it needs to iterate the procedures from (4). Obtains the t*
(6). Run OLS
t* =  t-1* + ut’ ^
  ( )
DW2 2
and obtains  which is the second-round estimated 
Xt** = Xt -  Xt  Yt-1 = 0  + 1 Xt-1 + t-1
(7). Use the  to transform the variable ^
Yt** = Yt -  Yt Yt = 0 + 1 Xt + t

32. Cochrane-Orcutt Iterative procedure(Cont.)
(8). Run OLS on
Yt** = 0** + 1** Xt** + t**
Where is ^
(Yt -  Yt-1) = 0 (1 - ) + 1 (Xt -  Xt-1) + (t -  t-1)
(9). Check on the DW3 -statistic, if the autocorrelation is still existing, than go into third-round procedures and so on.
Until the estimated ’s differs a little ^
( -  < 0.01).

33. Example: Studenmund (2006) Exercise 14 and Table 9.1, pp.342-344
(1)
Low DW statistic
Obtain the
Residuals
(Usually after you run regression, the residuals will be immediately stored in this icon

34. (2)
Give a new name
for the residual series
Run regression of the current residual on the lagged residual
Obtain the estimated ρ(“rho”) ^

35. (3)
Transform the Y* and X*
New series are created,
but each first observation
is lost.

36. (4)
Run the
transformed
regression
Obtain the estimated result
which is improved

37. The is the EVIEWS’ Command to run the iterative procedure
(5)~(9)
The Cochrane-Orcutt Iterative procedure in the EVIEWS
The is the EVIEWS’ Command to run the iterative procedure

38. The result of the Iterative procedure
This is the
estimated ρ
Each
variable is
transformed
The DW
is improved

39. Generalized least Squares (GLS) 5. Prais-Winsten transformation
Yt = 0 + 1 Xt + t t = 1,……,T (1)
Assume AR(1) : t = t-1 + ut <  < 1
Yt-1 = 0 + 1 Xt-1 + t (2)
(1) - (2) => (Yt - Yt-1) = 0 (1 - ) + 1 (Xt - Xt-1) + (t - t-1)
GLS => Yt* = 0* + 1* Xt* + ut

40. To avoid the loss of the first observation, the first observation of Y1* and X1* should be transformed as :
Edit the
figure here
To restore
the first
observation
Y1* = 2 (Y1) ^
X1* = 2 (X1) ^
but Y2* = Y2 -  Y1 ; X2* = X2 -  X1 ^
Y3* = Y3 -  Y2 ; X3* = X3 -  X2
…...
Yt* = Yt -  Yt-1 ; Xt* = Xt -  Xt-1

41. 6. Durbin’s Two-step method :
Since (Yt - Yt-1) = 0 (1 - ) + 1 (Xt - Xt-1) + ut
=> Yt = 0* + 1 Xt - 1 Xt-1 + Yt-1 + ut
Yt = 0 + 1 Xt + t
6. Durbin’s Two-step method :
I. Run OLS =>
this specification
Yt = 0* + 1* Xt - 2* Xt-1 + 3* Yt-1 + ut
Obtain 3* as an estimated  (RHO) ^
II. Transforming the variables :
Yt* = Yt - 3* Yt as Yt* = Yt -  Yt-1
and Xt* = Xt - 3* Xt as Xt* = Xt -  Xt-1 ^
III. Run OLS on model :
Yt* = 0 + 1 Xt* + ’t
and 1 = 1 ^
where 0 = 0 (1 - )

42. Including this
lagged term of Y
Obtain the estimated
ρ(“rho”) ^

43. Limitation of Durbin-Watson Test:
Lagged Dependent Variable and Autocorrelation
Yt = 0 + 1 X1t + 2 X2t + …… + k Xk.t + 1 Yt-1 +t
DW statistic will often be closed to 2 or
DW does not converge to 2 (1 - ) ^
DW is not reliable
Durbin-h Test:
Compute h* =   ^
1 - n*Var (1) n
Compare h* to Z where Zc ~ N (0,1) normal distribution
If |h*| > Zc => reject H0 :  = 0 (no autocorrelation)

44. Durbin-h Test: n ^ Compute h* =   1 - n*Var (1) h* = 4.458 > Z
Therefore reject
H0 :  = 0 (no autocorrelation)