Finite Distributed Lags
Finite distributed lag models contain independent variables and their lags as regressors.
Vi = a + воXt + ві%і1 + в2Xt2 + … PqXtq + Є (9.1)
for t = q + 1,… ,T. The particular example considered here is an examination of Okun’s Law. In this model the change in the unemployment rate from one period to the next depends on the rate of growth of output in the economy.
ut – uti = – y(gt – gw) (9.2)
where ut is the unemployment rate, gt is GDP growth, and gw is the normal rate of GDP growth. The regression model is
Aut = a + во gt + et (9.3)
where A is the difference operator, a = yGn, and во = — Y. An error term has been added to the model. The difference operator, Au = ut — ut1 for all = 2,3,…, T. Notice that when you take the difference of a series, you will lose an observation.
Recognizing that changes in output are likely to have a distributedlag effect on unemployment – not all of the effect will take place instantaneouslylags are added to the model to produce:
Aut = a + во gt + Plgt1 + P2gt2 +——— + Pq gtq + et (9.4)


Figure 9.3: TimeSeries graphs of Okun data
















Figure 9.5: Change in unemployment and real GDP growth. This uses the scatters command.
The differences of the unemployment rate are taken and the series plotted in Figure 9.5 below. and this will produce a single graph that looks like those in Figure 9.4 of POE4. To estimate a finite distributed lag model in gretl is quite simple using the lag operators. Letting q = 3 and
1 diff u
2 ols d_u const g(0 to 3)
This syntax is particularly pleasing. First, the diff varname function is used to add the first difference of any series that follow; the new series is called d_varname. Next, the contemporaneous and lagged values of g can be succinctly written g(0 to 3). That tells gretl to use the variable named g and to include g, gt1, gt2, and gt3. When the lagged values of g are used in the regression, they are actually being created and added to the dataset. The names are g_number. The number after the underline tells you the lag position. For instance, g_2 is g lagged two time periods. The new variables are given ID numbers and added to the variable list in the main gretl window as shown in Figure 9.6.
The regression output that uses the new variables is:
OLS, using observations 1986:12009:3 (T = 95)
Dependent variable: d_u
Coefficient 
Std. Error 
tratio 
pvalue 

const 
0.580975 
0.0538893 
10.7809 
0.0000 
g 
0.202053 
0.0330131 
6.1204 
0.0000 
g[66] 
0.164535 
0.0358175 
4.5937 
0.0000 
g2 
0.0715560 
0.0353043 
2.0268 
0.0456 
g3 
0.00330302 
0.0362603 
0.0911 
0.9276 
Mean dependent var Sum squared resid R[67] F(4, 90) Loglikelihood Schwarz criterion p 
0.027368 S. D. dependent var 2.735164 S. E. of regression 0.652406 Adjusted R2 42.23065 Pvalue(F) 33.71590 Akaike criterion 44.66241 HannanQuinn 0.358631 DurbinWatson
Notice that the tratio on g_3 is not significantly different from zero at 10%. We drop it and
reestimate the model with only 2 lagged values of g. For comparison, the sample is held constant.
The AIC reported by gretl has fallen to 59.42303, indicating a marginal improvement in the model.
If you are using the GUI rather than a gretl script to estimate the model, you have the opportunity to create the lagged variables through a dialog box. The specify model dialog and the lag order dialog are shown in Figure 9.7 below.
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