STAT 361 (Fall 2023) Assignment 3

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1. How is R2 related to the sample correlation coefficient? Recall the correlation coefficient

E{[X − E(X)][Y − E(Y )]} = q .

V ar(X)V ar(Y )

forrandomvariablesXandY,definedasρ= q

The sample correlation coefficient for the observed data x and y is

P[(xi − x)(yi − y)] ρˆ= qP(xi −x)2 P(yi −y)2.

Cov(X, Y )

V ar(X)V ar(Y )

Show that the R2 of the simple linear regression, model (1) of Chapter 2, is the square of the sample correlation coefficient between x and y,

22

R = ρˆ .

2. Consider the multiple regression model Y = Xβ + ε, where ε ∼ MVNn(0, σ2I). See descriptions of model forms (1) and (2) in Chapter 4.

(a) Show that the residual vector r = (I − P)Y, where P = X(XT X)−1XT , and show that 1

I − P is also a projection matrix.

(b) Let U = (βˆ , r)T . Find the joint distribution of the random vector U. It may be helpful

(XT X)−1XT ! to notice that U = (I − P)

Y. (c) Show that βˆ and r are independent.

Hint: For (b) and (c), properties of multivariate normal distribution may be useful.

3. Consider the “Savings.txt” data posted. It is an economic dataset collected in 48 different countries. The variable “sr” is ratio of savings (aggregate personal saving divided by dis- posable income). The variables “pop15” and “pop75” are percentages of population under 15 and over 75 respectively. The variable “dpi” is disposable income (per-capita, in dollars) while the variable “ddpi” is the rate (percent) of change in disposable income (per capita). (a) Draw scatter plot matrix for all the variables involved. Comment on the possible rela- tionships between variables, focus on those appear interesting to you.

(b) Fit a simple linear regression model with disposable income (“dpi”) as response and percentage of population under 15 as the only covariate. Describe the model clearly in mathematical form. Report and interpret the fitted model: is there a significant association between the variables, is this what you expect?

(c) Find the sample correlation coefficient between the two variables you studied in (b). How
is it related to R2 of the model you fitted in (b)?

(d) Fit a regression model with ratio of savings (Y , “sr”) as the response, and all other
variables as the covariates. Describe the model clearly in mathematical form, report and
discuss the fit of the model. Interpret the estimated coefficient for the rate of change in
disposable income.

(e) Present the analysis of variance table for the model in (c), i.e, the ANOVA table in the form of Table 1 of Section 4.4. The model you specified in (d) assumes that the error terms are i.i.d. normal with mean 0 and variance σ2. An estimate of σ, denoted by σˆ, can be extracted from your fitted model (supposed it’s named “fitd” in your code), by the R code “sigma(fitd)”. How is σˆ related to SS(Res), the residual sum of squares?