Studenmund (2006) Chapter 9

1
Lecture 9
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
Decomposition of time series
Xt Trend seasonal random
4
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
5
Finite Distributed Lag Models
Forward Distributed Lag Effect (with order q)
. Ctq?0?1Ydtq?1Ydt?t?Ct?0?1Ydt?1Ydt-
q?t
6
Backward Distributed Lag Effect
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
?1 .. ?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)
Autoregressive The regression of ?t can be
explained by itself
lagged one period.
?(RHO) the first-order autocorrelation
coefficient or coefficient of
autocovariance
10
TaxPay2007 ?? TaxPay2006 u2007 ? ?t
?? ?t-1 ut
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
If ?t ?1 ?t-1 ?2 ?t-2 ?3 ?t-3 ut
it is AR(2), third-order autoregressive
12
The current error term tends to have the same
sign as the previous one.
13
The current error term tends to have the opposite
sign from the previous.
The current error term tends to be randomly
appeared from the previous.
14
(No Transcript)
15
The consequences of serial correlation
BLUE
Therefore, when AR(1) is existing in the
regression, The estimation will not be BLUE
16
Example
If E(?t?t-1) ? 0, and ?t ? ?t-1 ut , then
The AR(1) variance is not the smallest
17
Autoregressive scheme
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, n24
DW 0.9107
? DW lt dL
20
Durbin-Watson Autocorrelation test
From OLS regression result where d or DW
0.9107
21
Durbin-Watson test
22
Durbin-Watson test(Cont.)
23
Durbin-Watson test(Cont.)
II. H0 ? 0 no
negative autocorrelation H1 ? lt 0
yes, negative autocorrelation
if dL ? (4 - d)? du or 4 - du ? d ? 4 - dL
gt inconclusive
if dL ? (4 - d)? du or 4 - du gt d gt 2
gt not reject H0
24
Durbin-Watson test(Cont.)
II. H0 ? 0 No
autocorrelation H1 ? ? 0
two-tailed test for auto correlation
either positive or negative AR(1)
If du lt d lt 4 - du gt not reject H0
25
(No Transcript)
26
(No Transcript)
27
The assumptions underlying the d(DW) statistics
1. Intercept term must be included.
2. Xs are nonstochastic
3. Only test AR(1) ?t ??t-1 ut
where ut iid (0, ?u2)
28
Lagrange Multiplier (LM) Test or called Durbins
m test Or Breusch-Godfrey (BG) test of
higher-order autocorrelation
(3) compute the BG-statistic (n-p)R2
(4) compare the BG-statistic to the ?2p (p is
of degree-order)
29
Remedy 1. First-difference transformation
30
3. Cochrane-Orcutt Two-step procedure (CORC)
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
32
Cochrane-Orcutt Iterative procedure(Cont.)
33
Example Studenmund (2006) Exercise 14 and Table
9.1, pp.342-344
(1)
34
(2)
Give a new name for the residual series
Run regression of the current residual on the
lagged residual
Obtain the estimated ?(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
(5)(9)
The Cochrane-Orcutt Iterative procedure in the
EVIEWS
38
The result of the Iterative procedure
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 lt ?
lt 1
?Yt-1 ??0 ??1 Xt-1 ??t-1
(2)
(1) - (2) gt (Yt - ?Yt-1) ?0 (1 - ?) ?1 (Xt -
?Xt-1) (?t - ??t-1)
40
To avoid the loss of the first observation, the
first observation of Y1 and X1 should be
transformed as
41
6. Durbins Two-step method
42
Including this lagged term of Y
Obtain the estimated ?(rho)

43
Limitation of Durbin-Watson Test
Lagged Dependent Variable and Autocorrelation
44
Therefore reject H0 ? 0 (no autocorrelation)