Econ 140
Autocorrelation
Lecture 18
Lecture 18
1
Today’s plan
Econ 140
• Definition and implications
• How to test for first-order autocorrelation
• Durbin-Watson test as an introduction to the presence of
first-order serial correlation in the model
• How to to estimate allowing for autocorrelation: GLS generalized least squares as a means of allowing for nonconstant variance.
• Phillips curve as an example: Note, the original Phillips
curve has no lagged variables included in it. Consideration
of the model: understand wage inflation and
unemployment; or just prediction of inflation?
Lecture 18
2
Definitions and implications
Econ 140
• Autocorrelation is a time-series phenomenon
• 1st-order autocorrelation implies that neighboring
observations are correlated
– the observations aren’t independent draws from the
sample
Lecture 18
3
Definitions and implications (2)
Econ 140
• In terms of the Gauss-Markov (or BLUE) theorem:
xt
ˆ
b   ctYt where ct 
 xt2
the model is Yt = a + bXt + et
• The model is still linear and unbiased if autocorrelation
exists:

E bˆ   ct a  bX t   b
Lecture 18
4
Definitions and implications (3)
Econ 140
• Autocorrelation will affect the variance:

V bˆ  s t cs ct st
• if s = t, then we would have:
2 2
c
 t
• but if s  t, and Y observations are not independent, we
have nonzero covariance terms:
2s t cs ct st
Lecture 18
5
Definitions and implications (4)
Econ 140
• Think of a numerical example to demonstrate this
• assuming t = 3: if we wanted to estimate b̂ :
bˆ  c1Y1  c2Y2  c3Y3   ctYt
• We want to consider the efficiency, or the variance of b̂

V bˆ   2  ct2  2s t cs ct st
– If BLUE: all covariance terms are zero.
– If covariance terms are nonzero: we no longer have
minimum variance
– minimum variance is defined as  2  ct2
Lecture 18
6
Summary of implications
Econ 140
1) Estimates are linear and unbiased
2) Estimates are not efficient. We no longer have minimum
variance
3) Estimated variances are biased either positively or
negatively
4) Unreliable t and F test results
5) Computed variances and standard errors for predictions
are biased
• Main idea: autocorrelation affects the efficiency of
estimators
Lecture 18
7
How does autocorrelation occur?Econ 140
Autocorrelation occurs through one of the following avenues:
1) Intertia in economic series through construction
– With regards to unemployment this was called
hysteresis: this means that certain sections of society
who are prone to unemployment
2) Incorrect model specification
– There might be missing variables or we might have
transformed the model to create correlation across
variables (more on this later)
Lecture 18
8
How does autocorrelation occur?Econ 140
3) Cobweb phenomenon
– agents respond to information with lags
4) Data manipulation
– example: constructing annual information based on
quarterly data
Lecture 18
9
Graphical results
Econ 140
• With no autocorrelation in the error term: we would expect
all errors to be randomly dispersed around zero within
reasonable boundaries
• Simply graphing the estimated errors against time indicates
the possibility of autocorrelation:
– we look for patterns of errors over time
– patterns can be positive, negative, or zero
• Graph error vs. time, we have positive correlation in the
error term
– errors from one time period to the next tend to move in
1 direction, with a positive slope
Lecture 18
10
Phillips Curve
Econ 140
• L_18.xls : Phillips Curve data
– Can calculate predicted wage inflation using the
observed unemployment rate and the estimated
regression coefficients
– Can then calculate the estimated error of the regression
equation
– Can also calculate the error lagged one time period
Lecture 18
11
Durbin-Watson statistic
Econ 140
• We will use the Durbin-Watson statistic to test for
autocorrelation
• This is computed by looking over T-1 observations where
t = 1, …T
eˆt  eˆt 1 

d
T
2
e
ˆ
t 2 t
T
t 2
Lecture 18
2
12
Durbin-Watson statistic (2)
Econ 140
• The assumptions behind the Durbin-Watson statistic are:
1) You must include an intercept in the regression
2) Values of X are independent and fixed
3) Disturbances, or errors, are generated by:
et  et 1  v
• this says that errors in this time period are correlated
with errors in the last time period and some random
error vt
•  is the coefficient of autocorrelation and is bounded
-1    1
T
•  can be calculated as ̂  t 2 eˆt eˆt 1
Lecture 18

T
2
e
ˆ
t 2 t
13
How to estimate 
Econ 140
• This estimation matters because it will be used in the
model correction
•  can be estimated by this equation: ˆ  1  d2
• Once we have the Durbin-Watson statistic d, you can
obtain an estimate for 
Lecture 18
14
How to estimate  (2)
Econ 140
• How the test works:
• The values for d range between 0 and 4 with 2 as the
midpoint
d
– using ˆ  1  , d  4 :
2
 value d value
-1
4
Autocorrelation?
Perfect negative autocorrelation
0
2
Zero autocorrelation
1
0
Perfect positive autocorrelation
Lecture 18
15
How to estimate  (3)
Econ 140
• We can represent this in the following figure:
indeterminate
indeterminate
0
2
4
dL
H1
Reject null
dU
4-dU 4-dL
H0:  =0
Cannot reject null
H1
Reject null
• dL represents the D-W upper bound
• dU represents the D-W lower bound
• The mirror image of dL and dU are 4- dL and 4-dU
Lecture 18
16
Procedure
Econ 140
Table on the second handout for today is the Durbin-Watson
statistical table and an additional table for this analysis
1) Run model: Yt  a  bX t  et
2) Compute: eˆt  Yt  Yˆt
3) Compute d statistic
4) Find dL and dU from the tables K’ is the number of
parameters minus the constant and T is the number of
observations
5) Test to see if autocorrelation is present
Lecture 18
17
Example (2)
Econ 140
• Returning to L18.xls
d  0.331
d L  1.475
K ' 1
observations  47
0.331
0
1.475
dL
H1
Reject null
Lecture 18
2
4
4-dU 4-dL
dU
H0:  =0
Accept null
H1
Reject null
18
Generalized least squares
• What can we do about autocorrelation?
• Recall that our model is: Yt = a + bXt + et
• We also know: et = et-1 + vt
Econ 140
(1)
• We will have to estimate the model using generalized least
squares (GLS)
Lecture 18
19
Generalized least squares (2)
Econ 140
• Let’s take our model and lag it by one time period:
Yt-1 = a + bXt-1 + et-1
• Multiplying by :
Yt-1 = a + bXt-1 + et-1
(2)
• Subtracting our (2) from (1), we get
Yt - Yt-1 = a(1-) + b(Xt-1 -  Xt-1) + vt
where
vt = et - et-1
Lecture 18
20
Generalized least squares (3)
Econ 140
• Now we need an estimate of  : we can transform the
variables such that:
where:
Yt*  a * bX t*  et*
(3)
Yt*  Yt  Yt 1
• Estimating equation (3) allows us to estimate without firstorder autocorrelation
Lecture 18
21
Estimating 
Econ 140
• There are several approaches
• One way is by using a short cut:
d
ˆ  1 
2
thinking back to the Durbin-Watson statistic,
2


e

e
ˆt ˆt 1

d
T
2
e
ˆ
t 2 t
T
t 2
we can rewrite the expression for d as:
d
Lecture 18


T
2
t 2 eˆt
T
2
t 2 eˆt
2


T
t 2 eˆt eˆt 1
T 2
2 eˆt



T
2
t 2 eˆt 1
T
2
t 2 eˆt
22
Estimating  (2)
Econ 140
• Collecting like terms, we have: 2  2 ̂  d
• Solving for , we can get an estimate in terms of d:
d
ˆ  1 
2
• Since earlier we defined  as:

̂ 

T
t 2 eˆt eˆt 1
T 2
2 eˆt
– we can use this to get a more precise estimate
• There are three or four other methods in the text
Lecture 18
23