## 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

#### 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 y increases significantly with x (at 5% level).
```Call:
lm(formula = y ~ x, data = d)
Residuals:
Min       1Q   Median       3Q      Max
-0.50503 -0.17149 -0.01047  0.13726  0.69840
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.005094   0.023993  -0.212    0.832
x            0.558063   0.044927  12.421   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2399 on 98 degrees of freedom
Multiple R-squared:  0.6116,	Adjusted R-squared:  0.6076
F-statistic: 154.3 on 1 and 98 DF,  p-value: < 2.2e-16
```
Code: The analysis can be replicated in R using the following code.
```## data