Chapter 3 Time Trends
3.1 Housing Investment and Housing Prices
Lesson: Importance of time trends
We look at annual housing investement and a housing price index from 1947 to 1988.
Set Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1947 to 1988
delta: 1 yearOur model is the natural log of invpc or real per capita housing investment ($1,000) and price be the housing price index where 1982 = 1 (or 100). We’ll assume constant elasticity with our log-log model.
\[ linvpc_t = \beta_0 + \beta_1 lprice_t + u_t \]
Source | SS df MS Number of obs = 42
-------------+---------------------------------- F(1, 40) = 10.53
Model | .254364468 1 .254364468 Prob > F = 0.0024
Residual | .966255566 40 .024156389 R-squared = 0.2084
-------------+---------------------------------- Adj R-squared = 0.1886
Total | 1.22062003 41 .02977122 Root MSE = .15542
------------------------------------------------------------------------------
linvpc | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lprice | 1.240943 .3824192 3.24 0.002 .4680452 2.013841
_cons | -.5502345 .0430266 -12.79 0.000 -.6371945 -.4632746
------------------------------------------------------------------------------Our output shows that a 1 percent increase in the pricing index increases the investment per capita by 1.24 percent (or elastic response). Note: both investment and price have upward trends that we need to account for.
\[ invpc_t = \beta_0 + \beta_1 lprice_t + \beta_2 t + u_t \]
Source | SS df MS Number of obs = 42
-------------+---------------------------------- F(2, 39) = 10.08
Model | .415945108 2 .207972554 Prob > F = 0.0003
Residual | .804674927 39 .02063269 R-squared = 0.3408
-------------+---------------------------------- Adj R-squared = 0.3070
Total | 1.22062003 41 .02977122 Root MSE = .14364
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linvpc | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lprice | -.3809612 .6788352 -0.56 0.578 -1.754035 .9921125
t | .0098287 .0035122 2.80 0.008 .0027246 .0169328
_cons | -.9130595 .1356133 -6.73 0.000 -1.187363 -.6387557
------------------------------------------------------------------------------After accounting for the upward linear trend for both investment per capita and the housing price index, an increase in price is coefficient is now negative, but it is not statistically significant. We probably have serial correlation which will bias our standard errors.
Source | SS df MS Number of obs = 41
-------------+---------------------------------- F(1, 40) = 11.10
Model | .170666637 1 .170666637 Prob > F = 0.0019
Residual | .614779847 40 .015369496 R-squared = 0.2173
-------------+---------------------------------- Adj R-squared = 0.1977
Total | .785446484 41 .019157231 Root MSE = .12397
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u | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u |
L1. | .4632581 .1390204 3.33 0.002 .1822874 .7442288
------------------------------------------------------------------------------3.2 Personal Exemption on Fertility Rates
Lesson: Importance of non-linear time trends
We’ll look at fertility again, but we’ll account for a quadratic time trend. First, we’ll estimate a linear time trend, and then we’ll try a quadratic time trend.
Set Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1913 to 1984
delta: 1 yearLinear Trend
Source | SS df MS Number of obs = 72
-------------+---------------------------------- F(4, 67) = 32.84
Model | 18441.2357 4 4610.30894 Prob > F = 0.0000
Residual | 9406.65967 67 140.397905 R-squared = 0.6622
-------------+---------------------------------- Adj R-squared = 0.6420
Total | 27847.8954 71 392.223879 Root MSE = 11.849
------------------------------------------------------------------------------
gfr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
pe | .2788778 .0400199 6.97 0.000 .1989978 .3587578
1.ww2 | -35.59228 6.297377 -5.65 0.000 -48.1619 -23.02266
1.pill | .9974479 6.26163 0.16 0.874 -11.50082 13.49571
t | -1.149872 .1879038 -6.12 0.000 -1.524929 -.7748145
_cons | 111.7694 3.357765 33.29 0.000 105.0673 118.4716
------------------------------------------------------------------------------Quadratic Trend Fertility is declining but at a decreasing negatively rate (eventually a positive \(t^2\) takes over t)
Source | SS df MS Number of obs = 72
-------------+---------------------------------- F(5, 66) = 35.09
Model | 20236.3981 5 4047.27961 Prob > F = 0.0000
Residual | 7611.49734 66 115.325717 R-squared = 0.7267
-------------+---------------------------------- Adj R-squared = 0.7060
Total | 27847.8954 71 392.223879 Root MSE = 10.739
------------------------------------------------------------------------------
gfr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
pe | .3478126 .0402599 8.64 0.000 .2674311 .428194
1.ww2 | -35.88028 5.707921 -6.29 0.000 -47.27651 -24.48404
1.pill | -10.11972 6.336094 -1.60 0.115 -22.77014 2.530696
t | -2.531426 .3893863 -6.50 0.000 -3.308861 -1.753991
|
c.t#c.t | .0196126 .004971 3.95 0.000 .0096876 .0295377
|
_cons | 124.0919 4.360738 28.46 0.000 115.3854 132.7984
------------------------------------------------------------------------------The fertility rate exhibits upward and downward trends between 1913 and 1984. Interestingly, pe remains fairly robust after adding time trends. The linear time trend shows that general fertility rates are falling by 1.15 childs per 1,000 women of childbearing age for each additional year. However, the quadratic shows that the trend is negative but decreasing rate and will eventually become positive.
3.3 Puerto Rican Employment
The last linear trend is for employment-population ratio and the importance of minimum wage coverage. We’ll add a time trend along with mincov and usgnp.
Set the Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1950 to 1987
delta: 1 yearAdd linear trend
Source | SS df MS Number of obs = 38
-------------+---------------------------------- F(3, 34) = 62.78
Model | .270948453 3 .090316151 Prob > F = 0.0000
Residual | .048910064 34 .001438531 R-squared = 0.8471
-------------+---------------------------------- Adj R-squared = 0.8336
Total | .319858518 37 .008644825 Root MSE = .03793
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lprepop | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lmincov | -.1686949 .0442463 -3.81 0.001 -.2586142 -.0787757
lusgnp | 1.057351 .1766369 5.99 0.000 .6983813 1.41632
t | -.0323542 .0050227 -6.44 0.000 -.0425616 -.0221468
_cons | -8.696298 1.295764 -6.71 0.000 -11.32961 -6.062988
------------------------------------------------------------------------------The employment-population ratio does not have much of a downward or upward trend, but usgnp does. Тhe linear trend of usgnp is about 3% per year, so an estimate of 1.06 in our model mean that when usgnp increases by 1% above its long-term trend, employment-population ratio increases by about 1.06%