Objectives: This lab practices part 1 operations with matrices which are relevant to regression analysis
and explore properties of different coding schemes. Part 2 focuses on model building and standard
model diagnostics.
Format of answer: Your answers (statistical figures and verbal description) should be submitted as
hardcopy. Add a running title with the following information: Lab03, your name and page numbers.
You may use this document as template. Copy the requested statistical figures into your document.
Trial and error answers will lead to a deduction of points. Label each answer properly with the bold
task headings. You are expected to hand in professionally formatted answers: use a fixed pitch font,
like Courier New, for any code the use mathematical type-setting when equations are required.
Copy and paste figures into your document. Make sure that each figure has a proper caption
describing its content.
Part 1: Matrix Operations (5 points)
Task 1: Manual matrix operations and regression analysis with matrices [1.5 points]
You are given a vector of the dependent variable and the design matrix
manually the vector regression coefficients .
The analytical equation for a inverse matrix is
Type your solution with Word’s equation editor. (0.5 points)
[b] Write your own OLS function using the dependent vector and the associated design matrix as
input. Your function should return the vector of the estimated regression coefficients. Repeat the
analysis from task 1 [a] using your function and compare the estimated regression coefficients with
those in task 1 [a]. (0.5 point)
[c] Use 's matrix operations to calculate for a dependent variable , the design matrix
and the diagonal weights matrix
the weighted regression
coefficients with the formula . (0.5 points)
[d] Compare the estimated regression coefficients from task 1 [b] with those from task 1 [c]. Explain why
they are identically. Hint: what is the effect of the weights matrix . [0.5 points]
Task 2: Coding schemes of categorical variables (3.5 points)
Provide the syntax code of your answers. You can either use the lm(…) or your coded ordinary least
squares function for this task
[a] Enter the matrix and the design matrices to separate matrix objects into and
show these object in your answer (0.5 points):
and are given in the indicator coding scheme ( codes it as contrasts(factor) <-
"contr.treatment") whereas and are given in the centered coding scheme ( codes it as
contrasts(factor) <- "contr.sum" and Hamilton p 99 calls it effect coding). In and
the last category is suppressed, whereas in and the second category is suppressed.
[b] Calculate group means of the observations , and as well as the
global mean for all observations . (0.5 points)
[c] Find the four sets of estimated regression parameters by regressing on the four design matrices ,
, and with your linear regression function from task 1 [b] and enter these estimates into the
table below (see columns Assign Estimated Regression Coefficients). (0.5 points)
Hints: (i) in the centered coding scheme the coefficient for the missing category can be calculated as the
negative sum of the two other estimated parameters, i.e., . (ii) For the cornered
coding scheme the values for the dashed cells cannot be calculated from the regression results.
Assign Estimated
Regression Coefficients
Give Expressions for the Means in Terms of
the Estimate Regression Coefficients
Model Coding
y~X1 cornered ─ ─
y~X2 cornered ─ ─
ECON6306 Applied Econometrics Michael Tiefelsdorf
3 Lab03: Matrix Operations Model Diagnostics
y~X3 centered
y~X4 centered
[d] For each design matrix the global mean and group means , and can be expressed
as a function of the estimated regression coefficients. (0.5 point)
Find the expressions for the means and write them into columns labels by “Give Expressions…”.
[e] Which coding scheme has a more intuitive interpretation? Justify your answer. (0.5 points)
[f] Argue, based on the four different models which however give identical predictions , whether it
make more sense to test individual regression coefficients with a t-tests or whether a simultaneous
partial F-test of all coefficients associated with the factor is more appropriate? (0.5 points)
Part 2: Model Building and Diagnostics (5 points)
Open the CPS1985 data-frame. with data("CPS1985",package="AER"). Assign new row-names
with the statement rownames(CPS1985) <- 1:nrow(CPS1985) to the data-frame. Study the
description of the variable experience in the associated online help.
Task 3: Multicollinearity diagnostics (3 points)
[a] For the variables ~log(wage)+education+age+experience generate a scatterplot matrix.
(0.5 points)
Based on the definition of the variables and the scatterplot matrix, which variables do you expect to be
multicollinear? Justify your decisions.
[b] Estimate the model log(wage)~education+experience and calculate the variance inflation
factors. Fully interpret the estimated model and the VIF. (1 point)
[c] Estimate the augmented model log(wage)~education+experience+age and show the
output. (1.5 points)
Address the following points:
i. What do the VIF tell you?
ii. What happened to the significances of the t-tests for the estimated regression parameters of
the augmented model and why?
iii. Why does the global F-test still remain significant?
Task 4: Refined model specification (1 point)
[a] Estimate the model: log(wage)~education+experience+gender+occupation+union
and fully interpret the estimated regression model. (0.5 point)
[b] Test whether the factor occupation is significant and if necessary refine the model specification.
(0.25 points)
[c] Investigate the model with car::residualPlots( ). Discuss the output and if advisable refine
the model. (0.25 point)
4 Lab03: Matrix Operations Model Diagnostics
Task 5: Case statistics of the final model (1 point)
[a] Generate the following plots and interpret them for your final model. (0.75 points)
i. Identify the two most extreme observations with a car::qqPlot( ) and interpret it.
ii. Identify potential extreme observations with a car::influenceIndexPlot( ) and
interpret the plots.
iii. Identify the two most extreme observation with a car::avPlots( ) and interpret the plots.
[b] Inspect the two most extreme observations in the data-frame. by examining their records. (0.25
points)
i. Discuss their attributes and argue if they are representative of the underlying population.
ii. Drop them from the data-frame. and show your code of doing so.