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.