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?



    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.


    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).

    lm(formula = y ~ x, data = d)
         Min       1Q   Median       3Q      Max 
    -0.55258 -0.15907 -0.02757  0.15782  0.74504 
                 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
    d <- read.csv("linreg.csv")
    ## regression
    m <- lm(y ~ x, data = d)
    ## visualization
    plot(y ~ x, data = d)
    ## diagnostic plots