## Exam 1

1. #### Question

Theory: Consider a linear regression of y on x. It is usually estimated with which estimation technique (three-letter abbreviation)?
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?
independent / zero expectation / normally distributed / identically distributed / homoscedastic
Application: Using the data provided in linreg.csv estimate a linear regression of y on x. What are the estimated parameters?
Intercept:
Slope:
In terms of significance at 5% level:
x and y are not significantly correlated / y increases significantly with x / y decreases significantly with x
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?
Code: Please upload your code script that reads the data, fits the regression model, extracts the quantities of interest, and generates the diagnostic plots.

#### 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.
Code: The analysis can be replicated in R using the following code.
```## data