Here's a plot showing the federal rate on 3 -month Treasury...

Here's a plot showing the federal rate on 3 -month Treasury bills from 1950 to 1980 , and a regression model fit to the relationship between the Rate (in \%) and Years Since 1950 (www.gpoaccess gov/eop/). a) What is the correlation between Rate and Year? b) Interpret the slope and intercept. c) What does this model predict for the interest rate in the year $2000 ?$ d) Would you expect this prediction to have been accurate? Explain.

Here's a plot showing the federal rate on 3 -month Treasury bills from 1950 to 1980 , and a regression model fit to the relationship between the Rate (in \%) and Years Since 1950 (www.gpoaccess gov/eop/).
a) What is the correlation between Rate and Year?
b) Interpret the slope and intercept.
c) What does this model predict for the interest rate in the year $2000 ?$
d) Would you expect this prediction to have been accurate? Explain.

Key Concepts

Correlation

Correlation measures the strength and direction of the linear relationship between two variables. In the context of regression analysis, it helps in determining whether an increase in one variable (e.g., year) is associated with an increase or decrease in the other variable (e.g., interest rate), and how closely the data points cluster around a line of best fit.

Slope

The slope in a regression analysis represents the rate of change in the dependent variable (interest rate) for every one-unit increase in the independent variable (years since 1950). It quantifies the relationship between the two variables and indicates whether the interest rate increases or decreases over time.

Intercept

The intercept is the value of the dependent variable when the independent variable is zero. In a time-based regression model, it provides a baseline measurement, representing the predicted interest rate at the starting point of the study period. It is useful for anchoring the regression line, though its practical interpretability depends on the context of the data.

Prediction

Prediction in a regression context involves using the estimated model to forecast future outcomes based on values of the independent variable. This entails substituting a specific value, such as a year, into the model equation to predict the corresponding interest rate.

Extrapolation

Extrapolation occurs when predictions are made outside the range of the data used to fit the model. Although mathematically valid, extrapolation carries risks as the underlying trends may change beyond the observed period, making such predictions less reliable. In this context, predicting interest rates for a year far beyond the available data (e.g., the year 2000) involves extrapolation and should be treated with caution.

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