The Predicted Value of a Dependent Variable

The Predicted Value of the Dependent Variable

The following steps are followed to predict the value of a dependent variable in a multiple regression model.

1. Determine the regression coefficient estimates.
2. Obtain the forecasted values of the independent variables.
3. Calculate the predicted value of the dependent variable using the equation:

$$\hat{Y_{i}}=\widehat{b_{0}}+\widehat{b_{1}}\hat{X_{1i}}+\widehat{b_{2}}\hat{X_{2i}}+…+\widehat{b_{k}}\hat{X_{ki}}$$

Where:

• \(\hat{Y}\) = the predicted value of the dependent variable.
• \(\widehat{b_{j}}\) = the estimated slope coefficient for the \(j^{th}\) independent variable.
• \(\hat{X_{ji}}\) = the forecasted value of the \(j^{th}\) independent variable, \(j = 1, 2, …, k\).

Example: Calculating the Predicted Value of a Dependent Variable

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

$$P=b_{0}+b_1INF+b_2IR+\epsilon_{t}$$

The following table gives the regression results:

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

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

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 rate of interest is 4%.

Solution

$$\hat{Y_{i}}=\widehat{b_{0}}+\widehat{b_{1}}\hat{X_{1i}}+\widehat{b_{2}}\hat{X_{2i}}+…+\widehat{b_{k}}\hat{X_{ki}}$$

$$\begin{align*}\hat{Y_{i}}&=81+(-276\times0.035)+(902\times0.04)\\&=$107.42\end{align*}$$

Question

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

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

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

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:

$$\text{ROC} = 8.653+0.0009S+0.0229DR+0.2996PM$$

$$\begin{align*}\text{ROC} &= 8.6531+ (0.0009\times 1000)+(0.0229\times 20) + (0.2996\times 20)\\&=16.03\%\end{align*}$$

Reading 2: Multiple Regression

LOS 2 (e) Calculate and interpret  a predicted value for the dependent variable, given an estimated regression model and assumed values for the independent variables.