# Finite Distributed Lags

Finite distributed lag models contain independent variables and their lags as regressors.

Vi = a + воXt + ві%і-1 + в2Xt-2 + … PqXt-q + 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 – ut-i = – 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 — ut-1 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 distributed-lag effect on unemployment – not all of the effect will take place instantaneously-lags are added to the model to produce:

Aut = a + во gt + Plgt-1 + P2gt-2 +——— + Pq gt-q + et (9.4) U. S. 1985:3 – 2009:3  Figure 9.3: Time-Series graphs of Okun data Figure 9.4: Multiple time-series graphs of Okun data produced using View>Multiple graphs>Time-series. This uses the scatters command.

 D. Unemployment Rate

 Real GDP growth

 0.4 –

 0 –

 L992

 1998 2004 2010

 19Si

 L992

 1998

 2004 2010  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, gt-1, gt-2, and gt-3. 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 1986:1-2009:3 (T = 95)
Dependent variable: d_u

 Coefficient Std. Error t-ratio p-value const 0.580975 0.0538893 10.7809 0.0000 g -0.202053 0.0330131 -6.1204 0.0000 g- -0.164535 0.0358175 -4.5937 0.0000 g-2 -0.0715560 0.0353043 -2.0268 0.0456 g-3 0.00330302 0.0362603 0.0911 0.9276

 Mean dependent var Sum squared resid R F(4, 90) Log-likelihood Schwarz criterion p 0.027368 S. D. dependent var 2.735164 S. E. of regression 0.652406 Adjusted R2 42.23065 P-value(F) 33.71590 Akaike criterion -44.66241 Hannan-Quinn 0.358631 Durbin-Watson

Notice that the t-ratio 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. Figure 9.6: Notice that the lagged variables used in the model are added to the list of available series. They also receive ID numbers.

The AIC reported by gretl has fallen to -59.42303, indicating a marginal improvement in the model.