Question
========

**Theory:** Consider a linear regression of `y` on `x`. It is usually estimated with
which estimation technique (three-letter abbreviation)?

##ANSWER1##

This estimator yields the best linear unbiased estimator (BLUE) under the assumptions
of the Gauss-Markov theorem. Which of the following properties are required for the
errors of the linear regression model under these assumptions?

##ANSWER2##

**Application:** Using the data provided in [linreg.csv](linreg.csv) estimate a
linear regression of `y` on `x`. What are the estimated parameters?

Intercept: ##ANSWER3##

Slope: ##ANSWER4##

In terms of significance at 5% level:

##ANSWER5##

**Interpretation:** Consider various diagnostic plots for the fitted linear
regression model. Do you think the assumptions of the Gauss-Markov theorem are
fulfilled? What are the consequences?

##ANSWER6##

**Code:** Please upload your code script that reads the data, fits the regression model,
extracts the quantities of interest, and generates the diagnostic plots.

##ANSWER7##


Answerlist
----------
* 
* independent
* zero expectation
* normally distributed
* identically distributed
* homoscedastic
* 
* 
* `x` and `y` are not significantly correlated
* `y` increases significantly with `x`
* `y` decreases significantly with `x`
* 
* 

Solution
========

**Theory:** Linear regression models are typically estimated by ordinary least squares (OLS).
The Gauss-Markov theorem establishes certain optimality properties: Namely, if the errors
have expectation zero, constant variance (homoscedastic), no autocorrelation and the
regressors are exogenous and not linearly dependent, the OLS estimator is the best linear
unbiased estimator (BLUE).

**Application:** The estimated coefficients along with their significances are reported in the
summary of the fitted regression model, showing that `x` and `y` are not significantly correlated (at 5% level).


```

Call:
lm(formula = y ~ x, data = d)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.55258 -0.15907 -0.02757  0.15782  0.74504 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.007988   0.024256  -0.329    0.743
x           -0.031263   0.045420  -0.688    0.493

Residual standard error: 0.2425 on 98 degrees of freedom
Multiple R-squared:  0.004811,	Adjusted R-squared:  -0.005344 
F-statistic: 0.4738 on 1 and 98 DF,  p-value: 0.4929
```

**Interpretation:** Considering the visualization of the data along with the diagnostic plots suggests
that the assumptions of the Gauss-Markov theorem are reasonably well fulfilled.

![](visualizations-1.svg)

**Code:** The analysis can be replicated in R using the following code.

```
## data
d <- read.csv("linreg.csv")
## regression
m <- lm(y ~ x, data = d)
summary(m)
## visualization
plot(y ~ x, data = d)
abline(m)
## diagnostic plots
plot(m)
```

Meta-information
================
exname: Linear regression
extype: cloze
exsolution: OLS|01001|-0.008|-0.031|100|nil|nil
exclozetype: string|mchoice|num|num|schoice|essay|file
extol: 0.01