Transcript Economics 213 - University of Connecticut

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





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
Distribut ed Lag Model
yt     0 X t  1 X t 1  ...   K X t  K   t
 0  t heshort- run or impactmult iplier
(  0  1 ) or (  0  1   2 ) are examplesof int erim
or int ermediat e mult ipliers.
K
 .
 long- run or t ot aldist ribut ed lag mult iplier.
 
i
i 0
*
i
i
 st andardized coefficient . Share of t ot alimpact.
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



Psychological Reasons
Technological Reasons
Institutional Reasons
Estimation of Distributed Lag
Models
Infinit eLag
yt     0 X t  1 X t 1  ....  t
Not enough dat a t o est imat e. Need Rest rict ions
Finit eLag
yt     0 X t  1 X t 1  ...   K X t  K   t
Problems of Ad-hoc Estimation




No a priori guide to length of lag.
Longer lags => less degrees of freedom
Multicollinearity
Data mining
Koyck Lag
Use restriction to estimateinfinitelag.
Assume :  k   0k k  1,2,3,.....
0  ,  1
0
1 
i 0
i 0
yt     0 X t   0X t 1   02 X t 2  ....  t


 i  0  i 
yt 1  
  0X t 1   02 X t 2  ....   t 1
Subtracting thesecond from thefirst, 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
Medianlag  
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




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



Estimation of Autoregressive models


Adaptive Expectations
Partial Adjustment
Method of Instrumental Variables
Detecting autocorrelation

Durbin h-test
Adaptive Expectation Model
Basic model : yt    X te   t
Adjustmentprocess
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
Withsuccesive lagging and substituting, we get
X te  X t  (1   )X t 1  (1   ) 2 X t  2  ....
Substituting back into thebasic model, we get
y t     X t    (1   ) X t 1    (1   ) 2 X t  2  ...   t
T hisis theKoyckmodel with  0    and   (1   )
Our estimatingequation thereforeis
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  thedesired levelof Y in period t and is a functionof
X t , i.e. ytd    X t   t , one adjusts theactuallevelof y
accordingto theadjustmentprocess
(yt  yt 1 )   ( ytd  yt 1 ), 0    1
substituting thefirst equationinto thesecond, we get
yt     X t  (1   ) yt 1  t
Properties of partial
adjustment model




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 variablein our estimat ingequation,we need
a variableZ with thepropertiesthat Zis correlatedwith the
RHS variable,but uncorrelated with theerror term.
P lim[(Zt  E ( Z t ))(X t  E ( X t ))]  0 and
P lim[(Zt  E ( Z t ))(  t  E ( t ))]  0
For theKoyckmodel, we may use 1 as instrumentfor itself
and X as instrumentfor X. For y t -1 we need some other vari
able
as theinstrument. Choicesincluded 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 Koyckstylemodel,multipletheequation by
theinstrumental variableand sum acrossall observations.
We get thefollowingnormalequationsthatmust be solved
for our parameterestimates.
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 themodel,
y t   1   2 X t   3 yt 1   t
We test for autocorrelation withtheDurbin h - statistic
n
h  ˆ
1 - n[var(b3 )]
If we est imate , theautocorrelationcoefficient as
d
ˆ  1 - , where d is the traditional Durbin - Wat son
2
statistic. T heDurbin h is now
d
n
h  (1- )
~ N (0,1)
2 1 - n[vaˆr(b3 )]
Note h does not exist when n[vaˆr(b3 )]  1.
Adaptive expectations
example
Investm entt    1 Interestte   2 Salest   t
( Interestte  Interestte1 )   ( Interestt  Interestte1 )
(1  (1   ) L) Interestte  Interestt
Where L  lag operator, i.e. Lxt  xt 1
replacedexpectedinterestin thefirst equation with
Interestt
Interestte 
gives
(1  (1   ) L)
Interestt
Investm entt    1
  2 Salest   t
(1  (1   ) L)
multiplying throughby (1  (1   ) L) gives
Investm entt    1Interestt   2 salest   2 (1   ) salest 1
 (1-  )Investment 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 4Interestte  0.4Sales t