Economics 310

Lecture 27Distributed 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

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SUMMARY OUTPUT

Regression StatisticsMultiple R 0.22076461R Square 0.04873701Adjusted R Square 0.03447825Standard Error 3.02573705Observations 475

ANOVAdf SS MS F Significance F

Regression 7 219.0471264 31.29245 3.41804 0.001418212Residual 467 4275.424556 9.155085Total 474 4494.471682

Coefficients Standard Error t Stat P-value Lower 95%Intercept 3.10859938 0.362060615 8.585853 1.35E-16 2.39713063mg 0.25615126 0.424810093 0.602978 0.546816 -0.578623619mg-1 -0.3547323 0.859144959 -0.41289 0.679877 -2.042998759mg-2 0.04661922 0.955379154 0.048797 0.961102 -1.83075265mg-3 -0.03928199 0.960509863 -0.0409 0.967395 -1.926735984mg-4 0.19367237 0.953796304 0.203054 0.839181 -1.68058911mg-5 -0.62968985 0.857586208 -0.73426 0.46316 -2.314893275mg-6 0.72688165 0.424265971 1.713269 0.087327 -0.106823999

Reasons for Lags Psychological Reasons Technological Reasons Institutional Reasons

Estimation of Distributed Lag Models

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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 StatisticsMultiple R 0.988322853R Square 0.976782062Adjusted R Square 0.973879819Standard Error 0.268219836Observations 19

ANOVAdf SS MS F Significance F

Regression 2 48.42568707 24.21284 336.5612 8.4448E-14Residual 16 1.151070089 0.071942Total 18 49.57675716

Coefficients Standard Error t Stat P-value Lower 95%Intercept 6.860131612 1.534694078 4.470032 0.000387 3.606726238Relative Price -2.29831002 0.384178333 -5.9824 1.91E-05 -3.11273153Lag Consumption 0.791345188 0.059796617 13.23395 4.92E-10 0.66458205

Koyck Lags Economic rational for Koyck model

Adaptive Expectations Partial Adjustment

Estimation of Autoregressive models Method of Instrumental Variables

Detecting autocorrelation Durbin h-test

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Partial Adjustment model

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model can be estimated consistently by OLS

Estimating Koyck model Model can be estimated by

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Simple method of estimation is instrumental variables.

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Shazam commands to estimate adaptive expectations model

file output c:\mydocu~1\koyck.outsample 1 30read (c:\mydocu~1\koyck.prn) invest int salessample 2 30genr 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 ofmodel

|_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 STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 24 DF P-VALUE CORR. COEFFICIENT AT MEANS INT -2.3341 0.2323 -10.05 0.000-0.899 -0.3363 -0.1357 SALES 0.44316 0.2833E-01 15.64 0.000 0.954 0.6131 0.2655 SALESLAG -0.14122 0.3504E-01 -4.030 0.000-0.635 -0.1917 -0.0795 INVESTLG -0.41223 0.7292E-01 -5.653 0.000-0.756 -0.4883 -0.4199 CONSTANT 117.54 4.148 28.34 0.000 0.985 0.0000 1.3696 |_stop

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