Chapter 2 Finite Distributed Lags Models
2.1 Personal Exemption on Fertility Rates
Lesson: using lags in the explanatory variables
Our interest the general fertility rate, which is the number of children born to every 1,000 women of childbearing age. PE is the average real dollar value of the personal tax exemption and two binary variables for World War II and the introduction of the birth control pill in 1963.
Our contemporenous model is: \[ gfr_t=\beta_0 + \beta_1 pe_t + \beta_2 ww2_t + \beta_3 pill_t + u_t \]
Where
- gfr: the fertility rate or births per 1,000 women at time t
- pe: the personal tax exemptions at time t
- ww2: a binary for World War II
- pill: a binary for birth control pill
Set Time Series
cd "/Users/Sam/Desktop/Econ 645/Data/Wooldridge"
use fertil3.dta, clear
tsset year, yearly
reg gfr pe i.ww2 i.pill if year < 1985/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1913 to 1984
delta: 1 year
Source | SS df MS Number of obs = 72
-------------+---------------------------------- F(3, 68) = 20.38
Model | 13183.6215 3 4394.54049 Prob > F = 0.0000
Residual | 14664.2739 68 215.651087 R-squared = 0.4734
-------------+---------------------------------- Adj R-squared = 0.4502
Total | 27847.8954 71 392.223879 Root MSE = 14.685
------------------------------------------------------------------------------
gfr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
pe | .08254 .0296462 2.78 0.007 .0233819 .1416981
1.ww2 | -24.2384 7.458253 -3.25 0.002 -39.12111 -9.355684
1.pill | -31.59403 4.081068 -7.74 0.000 -39.73768 -23.45039
_cons | 98.68176 3.208129 30.76 0.000 92.28003 105.0835
------------------------------------------------------------------------------World War II reduced the fertility rate by 24.24 births per 1,000 women of child bearing age, while the introduction of the birth control pill reduced the number of birth by 31.59 births per 1,000 women of childbearing age. The personal tax exemption. A one dollar increase in the average real value of the personal tax exemption increases 0.083 births per 1,000 women of child bearing age or a 12 dollar increase in the average real personal tax exemption increases 1 child per 1,000 women of child bearing ago
Next lets add lags in the average real value of the personal tax exemption. Our Finite Distributed Lag model is: \[ gfr_t=\beta_0 + \beta_1 pe_t + \beta_2 pe_{t-1} + \beta_3 pe_{t-2} + \beta_4 ww2_t + \beta_5 pill_t + u_t \]
Source | SS df MS Number of obs = 70
-------------+---------------------------------- F(5, 64) = 12.73
Model | 12959.7886 5 2591.95772 Prob > F = 0.0000
Residual | 13032.6443 64 203.635067 R-squared = 0.4986
-------------+---------------------------------- Adj R-squared = 0.4594
Total | 25992.4329 69 376.701926 Root MSE = 14.27
------------------------------------------------------------------------------
gfr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
pe | .0726718 .1255331 0.58 0.565 -.1781094 .323453
pe_1 | -.0057796 .1556629 -0.04 0.970 -.316752 .3051929
pe_2 | .0338268 .1262574 0.27 0.790 -.2184013 .286055
1.ww2 | -22.1265 10.73197 -2.06 0.043 -43.56608 -.6869196
1.pill | -31.30499 3.981559 -7.86 0.000 -39.25907 -23.35091
_cons | 95.8705 3.281957 29.21 0.000 89.31403 102.427
------------------------------------------------------------------------------Our personal exemption variables are no longer statistically significiant, but let’s test the joint significance. We might have multicollinearity that bias our standard errors for our PE variables.
( 1) pe = 0
( 2) pe_1 = 0
( 3) pe_2 = 0
F( 3, 64) = 3.97
Prob > F = 0.0117
( 1) pe_1 = 0
( 2) pe_2 = 0
F( 2, 64) = 0.05
Prob > F = 0.9480Our PE variables are jointly significant, but our lags are jointly insignificiant so we’ll use our static model.
For the estimated LRP: \[ .073-.0058+0.034 \approx .101, \] but we lack a standard error. We’ll need to estimate the standard error. Let \[ \theta_0=\delta_0+\delta_1+\delta_2 \] denote the LRP and write \[ \delta_0 \] in terms of \[ \theta_0 \, , \delta_1 \, and \, \delta_2 \, where \, \delta_0 = \theta_0 - \delta_1 - \delta_2 \] Next substitute for \[ \delta_0 \] \[ gfr_t = \alpha_0 + \delta_0 pe_t + \delta_1 pe_{t-1} + \delta_2 pe_{t-2} + ... \] to get \[ gfr_t = \alpha_0 + (\theta_0 - \delta_1 - \delta_2)pe_t + \delta_1 pe_{t-1} + \delta_2 pe_{t-2} + ... \] \[ gfr_t = \alpha_0 + \theta_0 pe_t + \delta_1 (pe_{t-1} - pe_t) + \delta_2 (pe_{t-2} - pe_t) + \]
We can estimate \[ \hat{\theta}_{0} \] and its standard error. We can regress gfr_t onto pe_t, (pe_(t-1) - pe_t), and (pe_t-2 - pe_t), ww2, and pill
(1 missing value generated)
(2 missing values generated)
Source | SS df MS Number of obs = 70
-------------+---------------------------------- F(5, 64) = 12.73
Model | 12959.7886 5 2591.95772 Prob > F = 0.0000
Residual | 13032.6443 64 203.635067 R-squared = 0.4986
-------------+---------------------------------- Adj R-squared = 0.4594
Total | 25992.4329 69 376.701926 Root MSE = 14.27
------------------------------------------------------------------------------
gfr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
pe | .1007191 .0298027 3.38 0.001 .0411814 .1602568
pe_1_1 | -.0057796 .1556629 -0.04 0.970 -.316752 .3051929
pe_2_1 | .0338268 .1262574 0.27 0.790 -.2184013 .286055
1.ww2 | -22.1265 10.73197 -2.06 0.043 -43.56608 -.6869196
1.pill | -31.30499 3.981559 -7.86 0.000 -39.25907 -23.35091
_cons | 95.8705 3.281957 29.21 0.000 89.31403 102.427
------------------------------------------------------------------------------\(\hat{\theta} \approx 0.101\) and significant, so our LRP has an effect on general fertility rates.