Question

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/). Dependent variable is: Rate R-squared \(=77.4 \% \quad s=1.239\) \(\begin{array}{ll}\text { Variable } & \text { Coefficient } \\ & 0.640282\end{array}\) \(\begin{array}{ll}\text { Intercept } & \text { u.to } 40 \\\ \text { Year }-1950 & 0.247637\end{array}\) 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.

Short answer

a) 0.88. b) Slope: +0.2476%/year, intercept unclear. c) Prediction: Intercept + 12.38%. d) Likely not accurate; the model is outdated by 2000.

Step-by-step solution

Find the Correlation

To find the correlation between the Rate and Year, we use the formula for R-squared: \[ R^2 = r^2 \]Given that \( R^2 = 77.4\% \), convert this to a decimal: \[ R^2 = 0.774 \]To find the correlation \( r \), take the square root:\[ r = \sqrt{0.774} \approx 0.88 \]Thus, the correlation is approximately 0.88.

Interpret the Slope

The slope, given as 0.247637, indicates the change in the Rate per year since 1950. Specifically, for each additional year, the Rate is expected to increase by approximately 0.2476%.

Interpret the Intercept

The intercept is given as a placeholder term here (\( u.to \, 40 \)) which generally means 'unknown to 40'. Traditionally, the intercept would show the expected Rate when the Year is 1950. However, the exact intercept value isn't directly provided.

Predict the Rate for Year 2000

Using the linear regression model formula:\[ \text{Rate} = \text{Intercept} + \text{Slope} \times (\text{Year} - 1950) \]Substituting values for the year 2000:\[ \text{Rate} = \text{Intercept} + 0.247637 \times (2000 - 1950) \]\[ \text{Rate} = \text{Intercept} + 0.247637 \times 50 \]\[ \text{Rate} = \text{Intercept} + 12.38185 \]Since the exact intercept is not provided, we can only express the rate in terms of 'Intercept + 12.38185'.

Evaluate Prediction Accuracy

Considering this model was made based on data from 1950 to 1980, predicting values for 2000 may be inaccurate as economic conditions, policies, and global events can significantly change the dynamics affecting federal rates. Thus, the model is likely not reliable for predicting the 2000 rate.

Key concepts

Correlation Coefficient

The correlation coefficient, represented by the symbol \( r \), is a numerical measure that expresses the strength and direction of a linear relationship between two variables. In this exercise, we are exploring the correlation between the Rate of 3-month Treasury bills and the Year. The correlation coefficient can range from -1 to 1, where:

• -1 indicates a perfect negative linear relationship.
• 0 indicates no linear relationship.
• 1 indicates a perfect positive linear relationship.

In our solution, we found the correlation coefficient to be approximately 0.88. This number is derived from the given \( R^2 \), or coefficient of determination, of 77.4%. To find \( r \), we took the square root of \( R^2 \) (0.774), which revealed a correlation of about 0.88.
This indicates a strong positive linear relationship between the Rate and Year, suggesting that as years increase, the Rate tends to increase as well.

Slope Interpretation

The slope in a regression analysis is a crucial concept as it explains how much the dependent variable (in this case, the Rate) is expected to change per unit increase in the independent variable (the Year). Here, the slope is given as 0.247637. This means that for each additional year since 1950, the Rate is predicted to increase by approximately 0.2476%.
In simpler terms, the slope tells us how steep the line is in the regression plot. The positive slope of 0.247637 confirms a positive relationship between the variables, implying that over the years, the interest rate has generally followed an upward trend. Understanding the slope gives insight into the rate of change, which in real-world terms, can reflect things like economic growth or inflation trends over time.

Model Prediction Accuracy

It's essential to assess the accuracy of any model before relying heavily on its predictions. In this exercise, we used historical data from 1950 to 1980 to create a model predicting the interest rate for the year 2000. Evaluating this prediction involves considering several external factors that could influence accuracy.

• The period difference: The model was built from data that is 20 years prior to the predicted year.
• Economic and policy changes: Significant changes in economic policies, unforeseen events, and global financial conditions can all greatly affect interest rates.

Given these considerations, the prediction accuracy is questionable. The model's R-squared value of 77.4% signifies reasonably high model fitting quality, but extrapolating it to a point (year 2000) outside the original data range is risky. It's better used as a rough estimate than a precise prediction, and it highlights how assumptions and historical data limitations can impact model reliability.