A story by James R. Hagerty entitled With Buyers Sidelined, Home Prices Slide published in the Thursday October 25, 2007 edition of the Wall Street Journal contained data on so-called fundamental housing indicators in major real estate markets across the US. The author argues that…prices are generally falling and overdue loan payments are pilling up. Thus, we shall consider data presented in the article on

Y = Percentage change in average price from July 2006 to July 2007 (based on the S&P/Case-Shiller national housing index); and

\(x\) = Percentage of mortgage loans 30 days or more overdue in latest quarter (based on data from Equifax and Moody’s).Fit the following model to the data: \(Y=\beta_{0}+\beta_{1}x+e\). Complete the following tasks:

setwd("~/Data 621/Discussion2")
indicators <- read.delim("indicators.txt")
model <- lm(PriceChange ~ LoanPaymentsOverdue, data = indicators)
  1. Find a 95% confidence interval for the slope of the regression model, \(\beta_{1}\) . On the basis of this confidence interval decide whether there is evidence of a significant negative linear association.
confint(model, "LoanPaymentsOverdue")
##                         2.5 %    97.5 %
## LoanPaymentsOverdue -6.759119 0.3731191
summary(model)
## 
## Call:
## lm(formula = PriceChange ~ LoanPaymentsOverdue, data = indicators)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -5.873 -2.890 -1.074  2.811  6.746 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)  
## (Intercept)            8.982      5.764   1.558   0.1578  
## LoanPaymentsOverdue   -3.193      1.546  -2.065   0.0728 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.385 on 8 degrees of freedom
##   (8 observations deleted due to missingness)
## Multiple R-squared:  0.3476, Adjusted R-squared:  0.2661 
## F-statistic: 4.263 on 1 and 8 DF,  p-value: 0.07282

The confidence interval does show evidence that there is a negative association. Based on the coefficient of the LoadPaymentsOverdue, it reaffrims that the association is negative but the relationship isn’t strong as the p-value is denoted as <= 0.1.

  1. Use the fitted regression model to estimate E ( Y | X =4). Find a 95% confidence interval for E ( Y | X =4). Is 0% a feasible value for E ( Y | X =4)? Give a reason to support your answer.
predict(model, data.frame("LoanPaymentsOverdue" = 4), interval="predict")
##         fit       lwr      upr
## 1 -3.789726 -14.48141 6.901961

\(E(Y|X = 4) = -3.789726\)

The 95 CI for E(Y|X=4) is [-14.48141, 6.901961].

0% is definitely a feasible value for \(E (Y | X =4)\) as the 95% CI spans 0.