1. Lecture 27 Distributed Lag Models
Economics 310
Lecture 27
Distributed Lag Models

2. 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.

3. 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?

4. Role of “Time” or “lag” in Economics

5. Demonstration of distributed Lag
Effect of 1 unit sustained increase in X Y
time 1 2

6. Example Distributed Lag Model

7. Reasons for Lags Psychological Reasons Technological Reasons
Institutional Reasons

8. Estimation of Distributed Lag Models

9. Problems of Ad-hoc Estimation
No a priori guide to length of lag.
Longer lags => less degrees of freedom
Multicollinearity
Data mining

10. Koyck Lag

11. Properties of Koyck Lag

12. Table of Mean & Median Lags

13. 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.

14. Gasoline Consumption Example of Koyck Lag

15. Koyck Lags Economic rational for Koyck model
Adaptive Expectations
Partial Adjustment
Estimation of Autoregressive models
Method of Instrumental Variables
Detecting autocorrelation
Durbin h-test

16. Adaptive Expectation Model

17. 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.

18. Partial Adjustment model

19. 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

20. Estimating Koyck model
Model can be estimated by maximum likelihood. This is difficult.
Simple method of estimation is instrumental variables.

21. Instrumental Variable Estimation

22. Instrumental Variable Estimation Continued

23. 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.

24. Testing autoregressive model for autocorrelation

25. Adaptive expectations example

26. 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

27. 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 = R-SQUARE ADJUSTED =
VARIANCE OF THE ESTIMATE-SIGMA**2 =
STANDARD ERROR OF THE ESTIMATE-SIGMA =
SUM OF SQUARED ERRORS-SSE=
MEAN OF DEPENDENT VARIABLE =
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR DF P-VALUE CORR. COEFFICIENT AT MEANS
INT
SALES E
SALESLAG E
INVESTLG E
CONSTANT
|_stop

28. True model