Rows: 165
Columns: 13
$ country.name <chr> "Afghanistan", "Albania", "Algeria", "Angola", "Ar…
$ inf.mort <dbl> 73.4, 14.3, 22.8, 106.8, 12.7, 15.3, 3.8, 3.4, 32.…
$ life.expect <dbl> 59.32795, 77.24059, 74.07000, 51.05932, 75.64905, …
$ sanit.access <dbl> 29.9, 91.5, 86.8, 47.6, 95.2, 89.5, 100.0, 100.0, …
$ adol.fert <dbl> 93.7132, 20.3584, 11.0822, 178.4360, 63.2980, 26.0…
$ edu.expend <dbl> 4.08791, NA, NA, NA, 4.98632, 3.14385, 5.10608, 5.…
$ adult.lit <dbl> 31.74112, 96.84530, NA, NA, NA, 99.74442, NA, NA, …
$ prim.edu.fem <dbl> NA, NA, 95.26551, 39.77358, 110.72836, NA, NA, 96.…
$ birth.rate <dbl> 37.636, 12.651, 24.921, 47.018, 18.018, 13.652, 13…
$ log.gdp.per.capita <dbl> 6.433550, 8.397917, 8.602894, 8.527884, 9.502481, …
$ sanit.access.factor <chr> "low", "high", "high", "low", "high", "high", "hig…
$ sanit.access.num <int> 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1,…
$ gdp.per.capita <dbl> 622.3797, 4437.8120, 5447.4040, 5053.7386, 13392.9…ADEC 7320: Module 1 Discussion
I. Introduction
I chose the data set “World Development Indicators, 2011” (OpenIntro) which is composed of data from the World Bank. It is cross-sectional because it collects indicators from the world’s countries at one point in time. Some of the variables that are included in the data set are:
- infant mortality (deaths per 1000 live births)
- life expectancy (years)
- sanitation access (% of population)
- adolescent fertility rate (per 1000 women)
- birth rate (per 1000 people)
- adult literacy rate
- GDP per capita (U.S. dollars)
II. Data selection
The dependent variable for this analysis is GDP per capita. Since the data is incomplete for the categories for the education variables (adult literary, education spending, female education), I will not include these in the analysis, and instead use infant mortality, life expectancy, birth rate, and sanitation access.
\[ GDP\;per\;capita = \beta_0\; + \;\beta_1Infant\;Mortality \; +\;\beta_2Life\;Expectancy \;+\;\beta_3Sanitation\;+\;\beta_4Birth\;Rate\ \]
III. Multivariate Regression
Applying the lm() command and backwards elimination, we have the following :
Results
===========================================================================================================================================
gdp.per.capita
(1) (2) (3) (4) (5)
-------------------------------------------------------------------------------------------------------------------------------------------
inf.mort 284.579* -105.535 259.093* 261.475*
(152.933) (127.354) (148.969) (143.158)
life.expect 1,741.744*** 1,269.412*** 1,853.415*** 1,777.310***
(416.314) (332.525) (397.974) (407.258)
sanit.access 80.077 52.967 178.673* 91.518
(89.955) (89.452) (91.208) (85.850)
birth.rate -119.287 55.960 -342.102 -175.747
(272.698) (257.893) (280.984) (257.280)
Constant -118,563.700*** -79,607.960*** 12,132.810 -118,752.000*** -123,880.300***
(32,888.890) (25,561.190) (10,805.920) (32,576.400) (30,482.930)
Observations 163 163 163 165 163
R2 0.375 0.361 0.306 0.376 0.374
Adjusted R2 0.359 0.349 0.292 0.365 0.362
Residual Std. Error 16,551.800 (df = 158) 16,679.490 (df = 159) 17,389.600 (df = 159) 16,454.950 (df = 161) 16,509.660 (df = 159)
F Statistic 23.681*** (df = 4; 158) 29.957*** (df = 3; 159) 23.320*** (df = 3; 159) 32.364*** (df = 3; 161) 31.672*** (df = 3; 159)
-------------------------------------------------------------------------------------------------------------------------------------------
Notes: ***Significant at the 1 percent level.
**Significant at the 5 percent level.
*Significant at the 10 percent level.
Results
===================================================================================================================
gdp.per.capita
(1) (2) (3) (4)
-------------------------------------------------------------------------------------------------------------------
inf.mort 259.093* -218.418* 217.520
(148.969) (114.762) (135.746)
life.expect 1,853.415*** 1,377.004*** 1,933.981***
(397.974) (290.505) (379.470)
birth.rate -175.747 7.062 -530.835**
(257.280) (236.293) (260.941)
Constant -118,752.000*** -82,271.740*** 32,121.100*** -127,153.800***
(32,576.400) (25,080.720) (3,633.609) (30,115.540)
Observations 165 165 165 165
R2 0.376 0.364 0.292 0.374
Adjusted R2 0.365 0.357 0.283 0.367
Residual Std. Error 16,454.950 (df = 161) 16,557.470 (df = 162) 17,474.100 (df = 162) 16,427.840 (df = 162)
F Statistic 32.364*** (df = 3; 161) 46.452*** (df = 2; 162) 33.432*** (df = 2; 162) 48.472*** (df = 2; 162)
-------------------------------------------------------------------------------------------------------------------
Notes: ***Significant at the 1 percent level.
**Significant at the 5 percent level.
*Significant at the 10 percent level.
Call:
lm(formula = gdp.per.capita ~ inf.mort + life.expect, data = WDI2011)
Residuals:
Min 1Q Median 3Q Max
-20901 -11028 -3787 6640 83373
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -127153.8 30115.5 -4.222 4.02e-05 ***
inf.mort 217.5 135.7 1.602 0.111
life.expect 1934.0 379.5 5.097 9.54e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 16430 on 162 degrees of freedom
Multiple R-squared: 0.3744, Adjusted R-squared: 0.3667
F-statistic: 48.47 on 2 and 162 DF, p-value: < 2.2e-16
Results
===========================================================================================
gdp.per.capita
(1) (2) (3)
-------------------------------------------------------------------------------------------
inf.mort 217.520 -424.810***
(135.746) (54.152)
life.expect 1,933.981*** 1,369.430***
(379.470) (141.641)
Constant -127,153.800*** -81,584.130*** 26,016.640***
(30,115.540) (9,957.242) (2,068.678)
Observations 165 165 165
R2 0.374 0.364 0.274
Adjusted R2 0.367 0.361 0.270
Residual Std. Error 16,427.840 (df = 162) 16,506.650 (df = 163) 17,641.530 (df = 163)
F Statistic 48.472*** (df = 2; 162) 93.477*** (df = 1; 163) 61.540*** (df = 1; 163)
-------------------------------------------------------------------------------------------
Notes: ***Significant at the 1 percent level.
**Significant at the 5 percent level.
*Significant at the 10 percent level. Given that we cannot improve the \(R^2\) value any more, the best regression equation that we have is :
\[ GDP\;per\;capita = \beta_0\;+\; \beta_1Infant\;Mortality \; +\;\beta_2Life\;Expectancy \]
IV. Matrix Algebra Application
Only keeping the variables of Infant Mortality and Life Expectancy, the following confirms the previous answer using the formula \(A = (X^TX)^{-1}X^TY\) :
Betas Linear.Model
int -127153.78 -127153.78
inf.mort 217.52 217.52
life.expect 1933.98 1933.98V. Estimating standard error
Applying the R code from our R pubs document, code to my data set, we have the following:
[1] 165[1] 269873795 int inf.mort life.expect
int 906945928 -3860413 -11404547
inf.mort -3860413 18427 47826
life.expect -11404547 47826 143998 int inf.mort life.expect
30115.5430 135.7464 379.4702 The standard errors of the coefficients match what the lm() function provides. So far we’ve learned that the slope of a linear equation \(\beta_1\) equals \(cov(x,y)/var(x)\). In calculating the standard errors of the additional variables, I have tried to think of it as each variable having it’s own slope. Given the formulas for matrix multiplication, it would make sense that the standard error for the entire equation would need to be spread among the variables that the equation is composed of. I’m not sure, however, if this reasoning is correct. Taking \((X^{T}X)^{-1}\) would summarize the variables in the equation that comprise the relationship of the linear regression. Would multiplying these numbers by the variance \(\sigma^2\) term provide each variable’s standard error - what are your thoughts?