The finite distributed lag (FDL) model
allow for one or more variables to affect \({{y}_{t}}\) with a lag. For example, the model
\({{y}_{t}}={{\alpha }_{0}}+{{\delta
}_{0}}{{z}_{t}}+{{\delta }_{1}}{{z}_{t-1}}+{{\delta }_{2}}{{z}_{t-2}}+{{u}_{t}}\)
(1)
which is an FDL of order two.
To interpret the coefficients in
(1), lets \({{z}_{t}}\)
is a constant (or equal to \(c\)
for all time period before \(t\) ) .
At time \(t\) , let \(z\) increases by one unit to \(c+1\) an then it reverts back at time \(t+1\) . (We assume he
increase in \(z\) is
temporary);
\(..,{{z}_{t-2}}=c,{{z}_{t-1}}=c,{{z}_{t}}=c+1,{{z}_{t+1}}=c,{{z}_{t+2}}=c,...\)
(2)
If we focus the effect of \({{z}_{t}}\) on \({{y}_{t}}\) and ceteris paribus, and we set the error term
each period to zero,
\({{y}_{t-1}}={{\alpha }_{0}}+{{\delta }_{0}}c+{{\delta
}_{1}}c+{{\delta }_{2}}c\)
\({{y}_{t}}={{\alpha }_{0}}+{{\delta }_{0}}(c+1)+{{\delta
}_{1}}c+{{\delta }_{2}}c\)
\({{y}_{t+1}}={{\alpha }_{0}}+{{\delta }_{0}}(c+1)+{{\delta
}_{1}}(c+1)+{{\delta }_{2}}c\)
\({{y}_{t+2}}={{\alpha }_{0}}+{{\delta }_{0}}(c+1)+{{\delta
}_{1}}(c+1)+{{\delta }_{2}}(c+1)\)
(3)
and so on.
With the permanent increase in \({{z}_{t}}\) after one period, \({{y}_{t}}\) has
increased by \({{\delta
}_{0}}+{{\delta }_{1}}\). After two periods, \({{y}_{t}}\) has increased by \({{\delta }_{0}}+{{\delta
}_{1}}+{{\delta }_{2}}\). There is no further changes in \({{y}_{t}}\) after two
period.
This show that the sum of the coefficients on current lagged
\({{z}_{t}}\), \({{\delta }_{0}}+{{\delta
}_{1}}+{{\delta }_{2}}\) is the long-run change in \({{y}_{t}}\) given a permanent increase in \({{z}_{t}}\). It’s
called the long-run propensity (LRP)
or long-run multiplier. The LRP is
often interest in distributed lag models.
\(LRP={{\delta }_{0}}+{{\delta
}_{1}}+{{\delta }_{2}}\) (4)
In Eq(1), if \({{z}_{t}}\) permanently
increase by one unit, then, after two years, \({{y}_{t}}\) will have changed by \({{\delta }_{0}}+{{\delta
}_{1}}+{{\delta }_{2}}\). This model assume that there are no further
changes after two years. Whether this is a case is an empirical matter.
If we regress the Eq(1), we do not enough information from
Stata output to obtain the standard error (SE) for the estimated \(LRP={{\delta }_{0}}+{{\delta }_{1}}+{{\delta }_{2}}\) to
calculate the value of \(t\)-statistics
for LRP. We cannot just simply sum up all the SE for the LRP, \(\left( se\left( {{{\hat{\delta
}}}_{0}} \right)+se\left( {{{\hat{\delta }}}_{1}} \right)+se\left(
{{{\hat{\delta }}}_{2}} \right) \right)\) .
To get the value of SE for the LRP, we will use some trick.
Lets we donate
\(\theta
={{\delta }_{0}}+{{\delta }_{1}}+{{\delta }_{2}}\) (5)
as for LRP.
And then, write \({{\delta }_{0}}\) in
term of \(\theta \),
\({{\delta }_{1}}\)
and \({{\delta }_{2}}\);
\({{\delta
}_{0}}=\theta -{{\delta }_{1}}-{{\delta }_{2}}\) (6)
Next, we substitute
for \({{\delta
}_{0}}\), \({{\delta
}_{1}}\) and \({{\delta
}_{1}}\) in the Eq(1);
\({{y}_{t}}={{\alpha
}_{0}}+\left( \theta -{{\delta }_{1}}-{{\delta }_{2}} \right){{z}_{t}}+{{\delta
}_{1}}{{z}_{t-1}}+{{\delta }_{2}}{{z}_{t-2}}+{{u}_{t}}\)
\(={{\alpha
}_{0}}+\theta {{z}_{t}}-{{\delta }_{1}}{{z}_{t}}-{{\delta
}_{2}}{{z}_{t}}+{{\delta }_{1}}{{z}_{t-1}}+{{\delta }_{2}}{{z}_{t-2}}+{{u}_{t}}\)
\(={{\alpha }_{0}}+\theta {{z}_{t}}-{{\delta }_{1}}\left(
{{z}_{t-1}}-{{z}_{t}} \right)+{{\delta }_{2}}\left( {{z}_{t-2}}-{{z}_{t}}
\right)+{{u}_{t}}\) (6)
From the last equation Eq(6), now
can obtain the \(\hat{\theta
}\) and also its value of SE by
regressing \({{y}_{t}}\)
on \({{z}_{t}}\), \(\left( {{z}_{t-1}}-{{z}_{t}}
\right)\)and \(\left(
{{z}_{t-2}}-{{z}_{t}} \right)\).
For this regression, we need the
value of coefficient and SE for variable \({{z}_{t}}\) for the LRP.
Estimation with Stata
For the estimation , we use the fertil3.dta to estimate the model;
\(gf{{r}_{t}}={{\beta
}_{0}}+{{\beta }_{1}}p{{e}_{t}}+{{\beta }_{2}}p{{e}_{t-1}}+{{\beta
}_{3}}p{{e}_{t-2}}+{{\beta }_{4}}ww{{2}_{t}}+{{\beta
}_{5}}pil{{l}_{t}}+{{u}_{t}}\) (7)
where
\(gf{{r}_{t}}\) = general fertility rate (children born per
1,000 women child bearing age)
\(p{{e}_{t}}\) = real dollar value personal tax exemption
\(ww{{2}_{t}}\) = United States involved in World War 2 during
1941-1945. 1= year involved, 0 = otherwise.
\(pil{{l}_{t}}\) = the year of introduced birth control pill
in year start from 1963 . 1= year start 1963, 0 = otherwise (before 1963).
To estimate
the Eq(7);
reg
gfr pe L.pe L2.pe ww2 pill
To calculate the estimated value
of LRP as in Eq(5);
display _b[pe]+_b[L1.pe]+_b[L2.pe]
To estimate the value of LRP along with their SE and the
- statistics, we need first to generate the value of \(\left( p{{e}_{t-1}}-p{{e}_{t}}
\right)\) and \(\left(
p{{e}_{t-2}}-p{{e}_{t}} \right)\)
gen dif1 = L.pe-pe
gen dif2 = L2.pe-pe
And now, we already to estimate the LRP as in Eq(6);
reg gfr pe dif1 dif2 ww2 pill
The regression result give \(\hat{\theta }=0.1007\) and must be same as \(\left( {{\delta
}_{0}}+{{\delta }_{1}}+{{\delta }_{2}} \right)\approx 0.1007\).
The SE for \(\hat{\theta
}\) or \(\left(
{{\delta }_{0}}+{{\delta }_{1}}+{{\delta }_{2}} \right)=0.03\) .
Therefore, the \(t\) -statistic for \(\hat{\theta }\) is
about 3.38, so \(\hat{\theta
}\) is statistically different from zero at small significance level.
Even though none of the \({{\hat{\delta }}_{j}}\)
is individually significant, the LRP is very significant.
