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The covariance between the 10-year money supply growth rates and the inflation rate is 0.007668, and the variance of the money supply growth rates is 0.02320. An investment analyst wants to explain the inflation rates using the money supply growth rates and predict the inflation rate when the money supply rate is 25%. The 10-year means for the money supply growth rate and inflation rate are 9% and 3%, respectively. The predicted inflation rate is closest to:

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NNikitesh

Last updated: February 2, 2026 at 10:22

A

7.234%

B

8.289%

C

6.345%

D

8.756%

Explanation:

Explanation

This is a linear regression prediction problem where:

Dependent variable (Y) = Inflation rate
Independent variable (X) = Money supply growth rate

Step 1: Calculate the slope coefficient (β)

\hat{\beta} = \frac{\hat{\sigma}_{XY}}{\hat{\sigma}_X^2} = \frac{0.007668}{0.02320} = 0.33051

Where:

$\hat{\sigma}_{XY} = 0.007668$ (covariance between money supply growth and inflation)
$\hat{\sigma}_X^2 = 0.02320$ (variance of money supply growth)

Step 2: Calculate the intercept (β₀)

\hat{\beta}_0 = \bar{Y} - \hat{\beta}\bar{X} = 0.03 - 0.33051 \times 0.09 = 0.0002541

Where:

$\bar{Y} = 0.03$ (mean inflation rate = 3%)
$\bar{X} = 0.09$ (mean money supply growth rate = 9%)

Step 3: Form the regression equation

\hat{Y} = 0.0002541 + 0.33051X

Step 4: Predict inflation rate when X = 25% (0.25)

\hat{Y} = 0.0002541 + 0.33051 \times 0.25 = 0.0828816 \approx 8.289\%

Step 5: Compare with options

A. 7.234% ❌
B. 8.289% ✅ Correct
C. 6.345% ❌
D. 8.756% ❌

Key Concepts:

Simple linear regression: $Y = \beta_0 + \beta_1X + \epsilon$
Slope coefficient: $\beta_1 = \frac{Cov(X,Y)}{Var(X)}$
Intercept: $\beta_0 = \bar{Y} - \beta_1\bar{X}$
Prediction: Substitute X value into regression equation

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