[SOLVED] STAT 321 Winter 2025 Assignment 3

Assignment 3

STAT 321 Winter 2025

Due: Friday March 14, 5:00PM

1.  (8  points)  Suppose we have a dataset with continuous response  Y ∈ Rn ,  and two

continuous predictors x ∈ Rn  and z ∈ Rn. Suppose the predictors are both centered:

Define (βˆ0 ,βˆx ,βˆz ), the estimate of the unknown regression coefficients in the (multiple)

linear regression model

Yi  = β0 + βxxi + βz zi + ϵi.

Define (β0, βx ) and (β0, βz ), the estimates of the unkown regression coefficients in the (simple) linear regression models

Yi  = β0 + βxxi + ϵi ,     Yi  = β0 + βz zi + ϵi

respectively.

Show that if the sample correlation between x and z is 0, βˆ
x = β˜
x, βˆ
z = β˜
z, and
{s.e.(β(ˆ)x + β(ˆ)z )}2  = {s.e.(β(ˆ)x )}2 + {s.e.(β(ˆ)z )}2 .

2.  (15 points) In the next two problems, you will work with data in 321divorce . csv, located on Learn under Assignments/Datasets. The data contains information about annual divorce rates among American women.  More information about the data can be found by installing the faraway package and running the following line of code:

?faraway:: divusa

Fit an MLR model with divorce as the reponse and all other variates as the predictors.

(a) Report the variance inflation factors  (VIFs) for this model.   Is  there any evi- dence of extreme multicollinearity?  Draw and interpret a scatter plot of the two predictors with the greatest sample correlation.

(b)  Plot the fitted values against the residuals, and plot each predictor variate against the residuals. Describe at least two patterns in the plots which do not agree with the MLR modelling assumptions.

(c)  Use at least one plot to check the normality assumption for the errors.

(d)  Use at least one plot to check for high leverage and high influence years. For the three highest leverage years, report their predictors and explain why these are high leverage points.

(e)  Use at least one plot to check for autocorrelation in the errors.

3.  (10 points) Using the 321divorce . csv dataset from Problem 2, implement the follow- ing variable selection methods to determine the best model, with response divorce and all other variates as possible continuous predictors.

(a) Backwards elimination (with level α = 0.05)

(b) Adjusted R2

(c) AIC

4.  (10 points) In this problem, similar to Assignment 2 Problem 5, you will simulate new random response vectors to inspect properties of our MLR confidence intervals. Use the predictor matrix X  ∈ R30 ×3 from Assignment 2, Problem 5 (the data can also be loaded directly from 321galapagos . csv, located on Learn under Assignments/Datasets), and set the model parameters to

(a)  Simulate a new random response vector Y  = Xβ  + ϵ where the entries of ϵ are mutually independent and satisfy

ϵi  = 50 · Ti ,     Ti  ~ t(3),     i = 1, . . . , 30.

Fit a linear regression model with response Y  and predictor matrix X  and create a QQ-plot of the residuals. Describe the pattern in the QQ-plot.

(b)  Repeat the procedure in part (a) 1000 times, and for each replication store the

estimated regression coefficients, Report the sample means of

Compare these to the theoretical mean we  derived in class for the regression coefficient estimator.

(c)  Repeat the procedure in part  (a)  1000  times,  and for each replication store a 99% confidence interval for β1 .  Report the proportion of replications in which the confidence interval contains the true expected response.  Do these intervals actually cover the true parameter in 99% of replications?

Additional instructions:

– If you are unsure how to use an R function, its documentation can be viewed by typing a question mark and then the function name (i.e. ?function) into the RStudio console.

– Unless otherwise specified, you may use results from lecture without additional justification. Results from the reference textbooks can be used, but you should show all steps for full marks.

– This assignment will be graded out of 43 points. Per the course outline, in normal circumstances it will count for 5% of your final grade.

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