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# Multiple Regression Model

Consider the multiple regression of the price of the US Dollar index on inflation rates and real interest rates. The estimated regression model is expressed as:

$$P=81-276INF+902IR$$

Where:

• P = Price of USDX.
• INF = Inflation rate.
• IR = Real interest rate.

The estimated regression model is commonly interpreted using the slope coefficients. The slope coefficients show the expected change in the price of the USDX for a one-unit change in inflation rates or real interest rates.

From the above model, we can deduce the following:

i. The price of the USDX is $81 when both inflation and real interest rates are 0%. ii. A 1% increase in the inflation rate (keeping the real interest rate constant) decreases the price of the USDX by$276.

iii. A 1% increase in the real rate of interest (keeping the inflation rate constant) increases the price of the USDX by \$902.

## Cheatsheet for Evaluating a Multiple Regression Model

The following is a summarized guideline for assessing a multiple regression model.

1. Check whether the model is correctly specified.

• Investigate functional form misspecification, time series model misspecification, and nonstationarity.

2. Check whether the individual coefficients are statistically significant.

• By performing t-tests on individual coefficients.
• For a two-tailed test of the regression coefficient, reject the null hypothesis if the t-statistic is greater than the upper critical t-value or lower than the lower critical t-value. The conclusion is that the regression coefficient is statistically significantly different from the null hypothesis value at the given significance level.

3. Check the validity of the model (statistical significance)

• Perform an F-test: It is a one-tailed test that checks for the overall fit of the model.
• Use R2: It is the percentage of variation in the dependent variable that is explained by the independent variables. Adjusted R2 is preferred as it adjusts for the number of independent variables.

4. Assess the presence of heteroskedasticity.

• Check for conditional heteroskedasticity by carrying out a Breusch-pagan chi-square test.
• If present, use white-corrected standard errors to correct the standard errors of the estimated regression coefficients.

5. Examine if there is autocorrelation (serial correlation).

• Perform the Durbin-Watson test.
• If present, adjust coefficient standard errors using the Hansen method.

6. Check for multicollinearity

• Compare T-test, F-test, and R2 to check for confliction.
• Check for high correlation among independent variables.
• If present, use stepwise regression to systematically eliminate variables from the regression until multicollinearity is minimized.

## Question

Adil Suleman, CFA, wishes to establish the possible drivers of a company’s percentage return on capital (ROC). Suleman identifies performance measures such as the profit margin (%), sales, and debt ratio as possible drivers of ROC.

He obtains the following results from the regression of ROC on profit margin (%), sales, and the debt ratio.

$$\small{\begin{array}{lc}\hline{}&\textbf{Regression Statistics}\\ \hline\text{Multiple R} & 0.79\\ \text{R Square} & 0.62\\ \text{Adjusted R Square} & 0.57\\ \text{Standard Error} & 1.20\\ \text{Observations} & 25\\ \hline\end{array}}$$

The most accurate interpretation of the multiple R-squared for the above model is that:

A. Explained variation in the dependent variable is 21% of the total variation.

B. The correlation between predicted and actual values of the dependent variable is 0.89.

C. The correlation between predicted and actual values of the dependent variable is 0.79.

### Solution

The multiple R-squared for the regression is 0.79; thus, the model explains 79% of the variation in the dependent variable. The correlation between the predicted and actual values of the dependent variable is the square root of the multiple R-squared:

$$\text{Correlation}=\sqrt{0.79}≈0.89$$

LOS 2 (o) Evaluate and interpret a multiple regression model and its results.

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