The Use of Multiple Regression for Forecasting

Multiple regression uses the same process employed in simple regression for predicting the dependent variable’s value. It, however, does so with more items summed up, as shown below:

$$ \widehat{Y_f}={\hat{b}}_0+{\hat{b}}_1X_{1f}+{\hat{b}}_2X_{2f}+\ldots+{\hat{b}}_kX_{kf}={\hat{b}}_0+\sum_{j=1}^{k}{{\hat{b}}_jX_{jf}} $$

Where:

\({\hat{Y}}_f\) = Predicted (forecasted) value of the dependent variable.

\({\hat{b}}_jX_f\) = This value is the estimated slope of the coefficient multiplied by the assumed value of the variable.

\({\hat{b}}_0\) = Estimated intercept coefficient.

A note on predicting with multiple regression models.

• Any prediction of the dependent variable must also include all five independent variables, even if they are statistically insignificant. This, for example, applies if all five independent variables are used to estimate a regression model. To estimate slope coefficients, correlations between these variables were taken into account.
• The intercept term must also be included in any prediction of the dependent variable.

Example: Calculating the Predicted Value of a Dependent Variable

Consider the following regression equation of the price of USDX on inflation and real interest rates.

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

The following table gives the regression results:

$$ \begin{array}{l|c} \text{Multiple R} & 0.8264 \\ \hline \text{R Square} & 0.6830 \\ \hline \text{Adjusted R Square} & 0.5924 \\ \hline \text{Standard Error} & 5.3537 \\ \hline \text{Observations} & 10 \end{array} $$

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

Use the estimated regression equation above to calculate the predicted price of the US dollar index (USDX), assuming the inflation rate is 3.5% and the real interest rate is 4%.

Solution

$$ \begin{align*} \widehat{Y_i} &= \widehat{b_0}+ \widehat{b_1}\widehat{X_{1i}}+ \widehat{b_2}\widehat{X_{2i}}+\ldots+ \widehat{b_k}\widehat{X_{ki}} \\  & =81+(-276\times0.035)+(902\times0.04)=$107.42 \end{align*} $$

Question

Consider the following multiple regression results of the return on capital (ROC) on performance measures (profit margin (%), sales, and debt ratio).

$$ \begin{array}{l|c}   \text{Multiple R} & 0.7906 \\ \hline \text{R Square} & 0.6251 \\ \hline \text{Adjusted R Square} & 0.5715 \\ \hline \text{Standard Error} & 1.1963 \\ \hline \text{Observations} & 25 \end{array} $$

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

Given that sales = 1000, debt ratio = 20, and profit margin = 20%, the predicted value of the return on capital (ROC) according to the regression model is closest to:

1. 7.38%.
2. 8.29%.
3. 16.03%.

Solution

The correct answer is C.

The regression equation is expressed as:

$$ \begin{align*} ROC & =8.653+0.0009S+0.0229DR+0.2996PM \\ ROC & =8.6531+\left(0.0009\times1000\right)+\left(0.0229\times20\right)+\left(0.2996\times20\right) \\ & =16.03\% \end{align*} $$

A and B are incorrect. From the resulting calculation, the correct answer is 16.03%.