Long-run Multiplier Estimation with Stata (Time Series)

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.