Economics 310
Lecture 27
Distributed Lag Models
Type of Models


If the regression model includes not only the
current but also the the lagged (past) values
of the explanatory variables (the X’s) it is
called a distributed-lag model.
If the model includes one or more lagged
values of the dependent variable among its
explanatory variables, it is called an
autoregressive model. This model is know
as a dynamic model.
Key Questions




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What is the role of lags in economics?
What are the reasons for the lags?
Is there any theoretical justification for the
commonly used lagged models in empirical
econometrics?
What is the relationship between
autoregressive and distributed lag models?
What are the statistical estimation problems?
Role of “Time” or “lag” in
Economics
Distribute d Lag Model
yt     0 X t  1 X t 1  ...   K X t  K   t
 0  the short - run or impact multiplier
(  0  1 ) or (  0  1   2 ) are examples of interim
or intermedia te multiplier s.
K
 .
i 0
 i* 
i
 long - run or total distribute d lag multiplier .
i
 standardiz ed coefficien t. Share of total impact.
K

i 0
i
Demonstration of distributed
Lag
Effect of 1 unit sustained increase in X
Y
yt     0 X t  1 X t 1   2 X t 2  t
2
1
0
0
1
2
time
Example Distributed Lag
Model
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.22076461
R Square
0.04873701
Adjusted R Square
0.03447825
Standard Error
3.02573705
Observations
475
ANOVA
df
Regression
Residual
Total
Intercept
mg
mg-1
mg-2
mg-3
mg-4
mg-5
mg-6
7
467
474
SS
MS
219.0471264 31.29245
4275.424556 9.155085
4494.471682
Coefficients Standard Error
3.10859938
0.362060615
0.25615126
0.424810093
-0.3547323
0.859144959
0.04661922
0.955379154
-0.03928199
0.960509863
0.19367237
0.953796304
-0.62968985
0.857586208
0.72688165
0.424265971
t Stat
8.585853
0.602978
-0.41289
0.048797
-0.0409
0.203054
-0.73426
1.713269
F
Significance F
3.41804 0.001418212
P-value
1.35E-16
0.546816
0.679877
0.961102
0.967395
0.839181
0.46316
0.087327
Lower 95%
2.39713063
-0.578623619
-2.042998759
-1.83075265
-1.926735984
-1.68058911
-2.314893275
-0.106823999
Reasons for Lags
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
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Psychological Reasons
Technological Reasons
Institutional Reasons
Estimation of Distributed Lag
Models
Infinite Lag
yt     0 X t  1 X t 1  ....  t
Not enough data to estimate. Need Restrictio ns
Finite Lag
yt     0 X t  1 X t 1  ...   K X t  K  t
Problems of Ad-hoc Estimation


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No a priori guide to length of lag.
Longer lags => less degrees of freedom
Multicollinearity
Data mining
Koyck Lag
Use restrictio n to estimate infinite lag.
Assume :  k   0k k  1,2,3,.....
0  ,  1


i 0
i 0
 i  0  i 
0
1 
yt     0 X t   0X t 1   02 X t 2  ....  t
yt 1  
  0X t 1   02 X t 2  ....  t 1
Subtractin g the second from the first, we get
y t  yt 1   (1   )   0 X t  ( t  t 1 )
or
yt   (1   )   0 X t  yt 1  ( t  t 1 )
Properties of Koyck Lag
Median lag  
log( 2)
log(  )

Mean lag 
 i
i 0

i
 i
i 0


1 
Table of Mean & Median Lags
lamda
0.15
0.3
0.45
0.6
0.75
0.9
Median Lag 0.365368 0.575717 0.868053 1.356915 2.409421 6.578813
Mean Lag
0.176471 0.428571 0.818182
1.5
3
9
Problems with koyck Model



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We converted a distributed lag model to
autoregressive model.
Lag dependent variable on RHS may
not be independent of new error
Error term is MA(1).
Model does not satisfy conditions for
Durbin-Watson d-test. Must use Durbin
h-test.
Gasoline Consumption
Example of Koyck Lag
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.988322853
R Square
0.976782062
Adjusted R Square 0.973879819
Standard Error
0.268219836
Observations
19
ANOVA
df
Regression
Residual
Total
Intercept
Relative Price
Lag Consumption
2
16
18
SS
MS
F
Significance F
48.42568707 24.21284 336.5612 8.4448E-14
1.151070089 0.071942
49.57675716
Coefficients Standard Error
t Stat
P-value
Lower 95%
6.860131612
1.534694078 4.470032 0.000387 3.606726238
-2.29831002
0.384178333
-5.9824 1.91E-05 -3.11273153
0.791345188
0.059796617 13.23395 4.92E-10 0.66458205
Koyck Lags

Economic rational for Koyck model
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Estimation of Autoregressive models
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Adaptive Expectations
Partial Adjustment
Method of Instrumental Variables
Detecting autocorrelation

Durbin h-test
Adaptive Expectation Model
Basic mod el : yt    X te   t
Adjustment process
X
e
t
 X te1    ( X t  X te1 ) or
X te  X t  (1   ) X te1 , if we lag and substitute
X te  X t  (1   )X t 1  (1   ) 2 X te 2
With succesive lagging and substituti ng, we get
X te  X t  (1   )X t 1  (1   ) 2 X t  2  ....
Substituti ng back into the basic model, we get
y t    X t   (1   ) X t 1   (1   ) 2 X t  2  ...   t
This is the Koyck model with  0   and   (1   )
Our estimating equation t herefore is
y t    X t  (1   ) yt 1  (  t  (1   ) t 1 )
Facts about Adaptive
Expectation model


Expected value of the independent
variable is weighted average of the
present and all past values of X.
The estimating equation has a MA(1)
process error term.
Partial Adjustment model
let ytd  the desired level of Y in period t and is a function of
X t , i.e. ytd    X t   t , one adjusts the actual level of y
according to the adjustment process
(y t  yt 1 )   ( ytd  yt 1 ), 0    1
substituti ng the first equation into the second, we get
yt    X t  (1   ) yt 1  t
Properties of partial
adjustment model
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

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Estimating equation looks like Koyck but is
different as far as estimation is concerned
Error term is well behaved
In the limit the lagged dependent variable is
uncorrelated with the error term
model can be estimated consistently by OLS
Estimating Koyck model


Model can be estimated by maximum
likelihood. This is difficult.
Simple method of estimation is
instrumental variables.
Instrumental Variable
Estimation
For each RHS variable in our estimating equation, we need
a variable Z with the properties that Z is correlated with the
RHS variable, but uncorrelat ed with the error term .
Plim[(Z t  E ( Z t ))( X t  E ( X t ))]  0 and
P lim[ (Z t  E ( Z t ))(  t  E ( t ))]  0
For the Koyck model, we may use 1 as instrument for itself
and X as instrument for X. For y t -1 we need some other vari able
as the instrument . Choices included X t -1 and yˆ t 1 where
yˆ t 1  d 0  d1 X t  d 2 X t 1
Instrumental Variable
Estimation Continued
For our Koyck style model, multiple the equation by
the instrument al variable and sum across all observatio ns.
We get the following normal equations that must be solved
for our parameter estimates.
y  b n  b  X  b y
 X y  b  X  b  X  b X y
Z y  b Z  b Z X  b Z y
t
1
t
t
t
t
2
t
1
1
t
t
3
t -1
2
t
2
2
t
3
t
t
3
t -1
t
t -1
Properties of IV estimators



Estimators are consistent
Estimators are asymptotically unbiased.
Parameter estimates will not be as
efficient as the maximum likelihood
estimates, but are easier to do.
Testing autoregressive model
for autocorrelation
If we have the model,
y t   1   2 X t   3 yt 1   t
We test for autocorrel ation with the Durbin h - statistic
n
h  ˆ
1 - n[var(b 3 )]
If we estimate  , the autocorrel ation coefficien t as
d
ˆ  1 - , where d is the tradition al Durbin - Watson
2
statistic. The Durbin h is now
d
n
h  (1 - )
~ N (0,1)
2 1 - n[v âr(b 3 )]
Note h does not exist when n[v âr(b 3 )]  1.
Adaptive expectations
example
Investmentt    1 Interest te   2 Salest   t
( Interest te  Interest te1 )   ( Interest t  Interest te1 )
(1  (1   ) L) Interest te  Interestt
Where L  lag operator, i.e. Lxt  xt 1
replaced expected interest in the first equation w ith
Interestt
Interest te 
gives
(1  (1   ) L)
Interest t
Investmentt    1
  2 Salest   t
(1  (1   ) L)
multiplyin g through by (1  (1   ) L) gives
Investmentt    1Interest t   2 sales t   2 (1   ) sales t 1
 (1 -  )Investmen t t -1  (  t  (1   )  t 1 )
Shazam commands to estimate
adaptive expectations model
file output c:\mydocu~1\koyck.out
sample 1 30
read (c:\mydocu~1\koyck.prn) invest int sales
sample 2 30
genr saleslag=lag(sales)
genr investlg=lag(invest)
genr intlag=lag(int)
inst invest int sales saleslag investlg (int intlag sales saleslag)
stop
Results of IV estimation of
model
|_inst invest int sales saleslag investlg (int intlag sales saleslag)
INSTRUMENTAL VARIABLES REGRESSION - DEPENDENT VARIABLE = INVEST
4 INSTRUMENTAL VARIABLES
2 POSSIBLE ENDOGENOUS VARIABLES
29 OBSERVATIONS
R-SQUARE =
0.9810
R-SQUARE ADJUSTED =
0.9779
VARIANCE OF THE ESTIMATE-SIGMA**2 =
10.229
STANDARD ERROR OF THE ESTIMATE-SIGMA =
3.1984
SUM OF SQUARED ERRORS-SSE=
245.51
MEAN OF DEPENDENT VARIABLE =
85.817
VARIABLE
ESTIMATED
NAME
COEFFICIENT
INT
-2.3341
SALES
0.44316
SALESLAG -0.14122
INVESTLG -0.41223
CONSTANT
117.54
|_stop
STANDARD
T-RATIO
ERROR
24 DF
0.2323
-10.05
0.2833E-01
15.64
0.3504E-01 -4.030
0.7292E-01 -5.653
4.148
28.34
PARTIAL STANDARDIZED ELASTICITY
P-VALUE CORR. COEFFICIENT AT MEANS
0.000-0.899
-0.3363
-0.1357
0.000 0.954
0.6131
0.2655
0.000-0.635
-0.1917
-0.0795
0.000-0.756
-0.4883
-0.4199
0.000 0.985
0.0000
1.3696
True model
Investmentt  200  4 Interestte  0.4Sales  t