What Multiple Regression is and How It Works

Example: Multiple Regression in Investment World

James Chase, an investment analyst, wants to determine the impact of inflation rates and real rates of interest on the price of the US Dollar index (USDX).

Chase uses the multiple regression model below:

$$ P=b_0+b_1INF+b_2IR+\epsilon_t $$

Where:

\(P\) = Price of USDX.

\(INF\) = Inflation rate.

\(IR\) = Real rate of interest.

\(\epsilon_t\) = Error term.

The regression of the price of USDX on inflation and real interest rates generates the following results:

$$ \begin{array}{c|c|c|c|c} & \textbf{Coefficients} & \textbf{Standard} & \textbf{t Stat} & \textbf{P-value} \\ & & \textbf{Error} & & \\ \hline \text{Intercept} & 81 & 7.9659 & 10.1296 & 0.0000 \\ \hline \text{Inflation rates} & -276 & 233.0748 & -1.1833 & 0.2753 \\ \hline \text{Real interest Rates} & 902 & 279.6949 & 3.2266 & 0.0145 \end{array} $$

Chase can express the multiple regression equation as follows:

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

The regression coefficient estimate of the inflation rate is negative. This indicates that an increase in the inflation rates causes a decrease in the price of the US Dollar index (USDX).

Furthermore, the positive real rate of interest coefficient implies that an increase in the price of USDX accompanies the real interest rate.

The t-statistic indicates that only the real interest rate variable is significant at the 5% significance level.

The intercept term is defined as the value of the dependent variable when the independent variables are zero. On the other hand, each slope coefficient is the estimated change in the value of the dependent variable for a one-unit change in the value of the respective independent variable, keeping the other independent variables constant. Slope coefficients are also called partial slope coefficients.

Continuing with this example:

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

Where:

\(P\) = Price of the US Dollar index.

\(INF\) = Annual inflation rate.

\(IR\) = Annual real rate of interest.

The regression equation is interpreted as follows:

The intercept term of 81 implies that the price of USDX is $81 when both the inflation rate and real interest rate are 0.

A 1% increase in the inflation rate leads to a $276 decrease in the price of USDX, keeping real interest rates constant. On the other hand, a 1% increase in the real interest rate leads to a $902 increase in the price of USDX, keeping the inflation rate constant.

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 debt ratio.

$$ \textbf{SUMMARY OUTPUT} \\ \begin{array}{c|c|c|c|c} & \textbf{Coefficients} & \textbf{Standard} & \textbf{t Stat} & \textbf{P-value} \\ & & \textbf{Error} & & \\ \hline \text{Intercept} & 8.6531 & 0.9174 & 9.4323 & 0.0000 \\ \hline \text{Sales} & 0.0009 & 0.0005 & 1.7644 & 0.0922 \\ \hline \text{Debt ratio} & 0.0229 & 0.0165 & 1.3880 & 0.1797 \\ \hline \text{Profit Margin (%)} & 0.2996 & 0.0564 & 5.3146 & 0.0000 \end{array} $$

Which independent variable(s) is (are) most likely statistically and significantly different from zero at the 5% significance level, assuming the sample size is 25?

1. Profit margin.
2. Sales and profit margin.
3. Sales and debt ratio.

Solution

The correct answer is A.

An independent variable is statistically significant if its p-value is less than the significance level, in this case, 5% or 0.05. Therefore, only the profit margin is statistically and significantly different from zero at the 5% significance level.

B and C are incorrect. At a 5% significance level, only the profit margin is statistically significantly different from zero.