Last updated: 2019-05-30

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File Version Author Date Message
Rmd 7ab13db John Blischak 2019-05-30 Add another summation property.
html 95b7acd John Blischak 2019-05-30 Build site.
Rmd e2719a5 John Blischak 2019-05-30 Explore property of the summation operator.

Exploring some of the properties of the summation operator described in Appendix A of Introductory Econometrics by Jeffrey Woolridge.

Equation A.7

The sum of squared deviations is the sum of the squared \(x_i\) minus \(n\) times the square of \(\bar{x}:\)

\[ \sum_{i=1}^{n} (x_i - \bar{x})^2 = \sum_{i=1}^{n} x_i^2 - n(\bar{x})^2 \]

First, confirming this computationally with an example.

n <- 10
x <- rpois(n, lambda = 5)
xbar <- mean(x)
sum((x - xbar)^2)
[1] 24.1
sum(x^2) - n * xbar^2
[1] 24.1

Second, visually exploring this relationship.

plot(x)
abline(h = xbar, col = "red", lty = "dashed")
for (i in 1:n) lines(x = c(i, i), y = c(x[i], xbar), col = "blue")

Third, the algebraic proof from the book:

\[ \begin{align*} \sum_{i=1}^{n} (x_i - \bar{x})^2 \\ = \sum_{i=1}^{n} (x_i^2 - 2x_i\bar{x} + \bar{x}^2) \\ = \sum_{i=1}^{n} x_i^2 - 2\bar{x}\sum_{i=1}^{n} x_i + n(\bar{x})^2 \\ = \sum_{i=1}^{n} x_i^2 - 2n(\bar{x})^2 + n(\bar{x})^2 \\ = \sum_{i=1}^{n} x_i^2 - n(\bar{x})^2 \end{align*} \]

The key step for me is the substitution of \(n\bar{x}\) for \(\sum_{i=1}^{n} x_i\). This relationship is obtained by rearranging the equation for the mean:

\[ \bar{x} = (1/n) \sum_{i=1}^{n} x_i\]

Equation A.8

\[ \begin{align*} \sum_{i=1}^{n} (x_i - \bar{x}) (y_i - \bar{y}) \\ = \sum_{i=1}^{n} x_i (y_i - \bar{y}) \\ = \sum_{i=1}^{n} (x_i - \bar{x}) y_i \\ = \sum_{i=1}^{n} x_i y_i - n(\bar{x} \cdot \bar{y}) \end{align*} \]

First, confirming this computationally with an example.

n <- 10
x <- rpois(n, lambda = 5)
y <- rpois(n, lambda = 5)
xbar <- mean(x)
ybar <- mean(y)
sum((x - xbar) * (y - ybar))
[1] 43.8
sum(x * (y - ybar))
[1] 43.8
sum((x - xbar) * y)
[1] 43.8
sum(x * y) - n * (xbar * ybar)
[1] 43.8

Second, visually exploring this relationship.

op <- par(mfrow = c(1, 2))
plot(x, y)
abline(h = xbar, col = "red", lty = "dashed")
for (i in 1:n) lines(x = c(x[i], x[i]), y = c(y[i], xbar), col = "blue")
plot(x, y)
abline(v = ybar, col = "red", lty = "dashed")
for (i in 1:n) lines(x = c(x[i], ybar), y = c(y[i], y[i]), col = "purple")

par(op)

sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 18.04.2 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/atlas/libblas.so.3.10.3
LAPACK: /usr/lib/x86_64-linux-gnu/atlas/liblapack.so.3.10.3

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.3.0.9000 Rcpp_1.0.1           digest_0.6.19       
 [4] rprojroot_1.2        backports_1.1.4      git2r_0.25.2        
 [7] magrittr_1.5         evaluate_0.13        stringi_1.4.3       
[10] fs_1.3.1             whisker_0.3-2        rmarkdown_1.13      
[13] tools_3.6.0          stringr_1.4.0        glue_1.3.1          
[16] xfun_0.7             yaml_2.2.0           compiler_3.6.0      
[19] htmltools_0.3.6      knitr_1.23