I think the \(\alpha\) parameter needs to be set better in the model:

knitr::include_graphics("RStudio/r-rstudio-1-3-console.png")

\[ \begin{eqnarray} Y_{i} &=& \alpha + \beta x_{i} + \varepsilon_{i} \\ Z_{i} &=& \gamma + \beta x_{i} + \varepsilon_{i} \end{eqnarray} \]

I think the paper by Eloyan and Ghosh (2013) is awesome! I like to cite it by (Eloyan and Ghosh 2013).

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summary(cars)
##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00

The mean distance of each car is 42.98 miles.

The equation was

model = lm( mpg ~ cyl + carb, data = mtcars)
tab = broom::tidy(model, conf_int = TRUE)
knitr::kable(tab)
term estimate std.error statistic p.value
(Intercept) 37.812739 2.0540223 18.409118 0.0000000
cyl -2.625023 0.3755924 -6.989020 0.0000001
carb -0.526146 0.4152914 -1.266932 0.2152603
# DT::datatable(tab)
pander::pander(tab)
term estimate std.error statistic p.value
(Intercept) 37.81 2.054 18.41 1.532e-17
cyl -2.625 0.3756 -6.989 1.102e-07
carb -0.5261 0.4153 -1.267 0.2153

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References

Eloyan, Ani, and Sujit K Ghosh. 2013. “A Semiparametric Approach to Source Separation Using Independent Component Analysis.” Computational Statistics & Data Analysis 58. Elsevier: 383–96.