Explanatory variables

What Are Explanatory Variables?

Explanatory variables are a cornerstone in statistical modeling, representing the factors or inputs that are believed to influence or account for changes in a specific outcome. In the realm of finance and economics, these variables are crucial for understanding relationships between different data points and for making predictions. They are often used in regression analysis to model how one or more independent variables explain the variation in a dependent variable. Properly identifying and utilizing explanatory variables is essential for robust data analysis and informed decision-making.

History and Origin

The concept underlying explanatory variables, particularly within the framework of regression, traces its origins to the late 19th century. Sir Francis Galton, a polymath and cousin of Charles Darwin, first introduced the term "regression" in his studies on heredity. Galton observed that extreme characteristics, such as the height of parents, tended to "regress" or move towards the average in their offspring. He published his observations in the 1886 paper "Regression towards mediocrity in hereditary stature." This pioneering work laid the conceptual groundwork for what would become linear regression analysis, a method for quantifying the relationship between variables. Karl Pearson, a contemporary of Galton, later advanced the mathematical framework for correlation and regression, developing the product-moment correlation coefficient and expanding on the techniques of multiple regression.¹⁵, ¹⁶, ¹⁷ The insights from these early studies established the foundation for using explanatory variables to understand and predict phenomena across various scientific and economic disciplines.

Key Takeaways

• Explanatory variables are inputs in a statistical model used to predict or explain changes in an outcome.
• They are also known as independent variables or predictor variables in regression analysis.
• Understanding these variables is fundamental for building effective forecasting models and conducting empirical research in finance.
• Identifying appropriate explanatory variables and validating their relationships is critical to avoid misleading conclusions.
• The relationship between explanatory variables and a dependent variable can be linear or non-linear.

Formula and Calculation

In a simple linear regression analysis, the relationship between a single dependent variable and a single explanatory variable is often represented by the following formula:

Where:

• ( Y ) represents the dependent variable (the outcome being explained or predicted).
• ( X ) represents the explanatory variable (the input believed to influence Y).
• ( \beta_0 ) is the y-intercept, representing the expected value of Y when X is 0.
• ( \beta_1 ) is the regression coefficient, indicating the change in Y for a one-unit change in X.
• ( \epsilon ) is the error term, representing the unexplained variation in Y.

In multiple linear regression, the model expands to include several explanatory variables:

Here, ( X_1, X_2, \dots, X_n ) are the multiple explanatory variables, and ( \beta_1, \beta_2, \dots, \beta_n ) are their respective coefficients. The goal of fitting such a statistical model is to estimate the beta coefficients that best describe the relationship between the explanatory variables and the dependent variable.

Interpreting the Explanatory Variables

Interpreting explanatory variables involves understanding their impact on the dependent variable within a given model. The coefficient associated with each explanatory variable ((\beta)) indicates the magnitude and direction of its effect. For instance, a positive coefficient suggests that as the explanatory variable increases, the dependent variable tends to increase, assuming all other explanatory variables remain constant. Conversely, a negative coefficient implies an inverse relationship.

The statistical significance of an explanatory variable, often assessed through hypothesis testing and p-values, determines the likelihood that the observed relationship is not due to random chance. A variable might be considered statistically significant if its p-value is below a predetermined threshold (e.g., 0.05). However, statistical significance does not automatically imply practical or economic significance. Analysts must consider the real-world implications of the coefficient's magnitude and the overall context of the data analysis to draw meaningful conclusions.

Hypothetical Example

Consider a financial analyst attempting to predict a company's stock price (the dependent variable) based on its quarterly earnings per share (EPS) and the prevailing interest rate. Here, EPS and interest rate would be the explanatory variables.

Scenario: A tech company, "InnovateCo," is set to release its quarterly earnings.

• Dependent Variable: InnovateCo's Stock Price (Y)
• Explanatory Variables:
  • Quarterly Earnings Per Share (EPS, (X_1))
  • Federal Funds Rate (Interest Rate, (X_2))

Hypothetical Model: Based on historical time series data, the analyst develops a simple linear model:
Stock Price = ( \beta_0 ) + (( \beta_1 ) * EPS) + (( \beta_2 ) * Interest Rate) + ( \epsilon )

Let's assume the estimated model is:
Stock Price = $50 + ($15 * EPS) - ($200 * Interest Rate)

Walkthrough:

1. Current Data: InnovateCo's last reported EPS was $3.00. The current Federal Funds Rate is 5% (0.05).

2. Prediction: Using the model, the predicted stock price would be:
   Stock Price = $50 + ($15 * 3.00) - ($200 * 0.05)
   Stock Price = $50 + $45 - $10
   Stock Price = $85

3. Interpretation:

   • The coefficient of $15 for EPS suggests that, holding the interest rate constant, every $1 increase in EPS is associated with a $15 increase in InnovateCo's stock price.
   • The coefficient of -$200 for the Interest Rate suggests that, holding EPS constant, every 1% (0.01) increase in the Federal Funds Rate is associated with a $2 decrease in InnovateCo's stock price (since ( $200 \times 0.01 = $2 )).
     This example demonstrates how explanatory variables are used to quantify relationships and make predictions in financial markets.

Practical Applications

Explanatory variables are fundamental to virtually all quantitative financial analysis and econometrics. Their practical applications span diverse areas:

• Investment Analysis: Analysts use explanatory variables like economic indicators (GDP growth, inflation, interest rates), industry-specific metrics, and company financial ratios to build models that predict asset prices, corporate earnings, or credit risk. This is crucial for portfolio management and security selection.
• Economic Forecasting: Central banks and financial institutions extensively employ complex statistical models with numerous explanatory variables to forecast key economic indicators such as inflation, unemployment rates, and GDP growth. For example, the Federal Reserve Board utilizes models like FRB/US for forecasting and policy analysis, which incorporates various economic explanatory variables.¹⁴
• Risk Management: In risk management, explanatory variables are used to model and predict financial risks, such as market risk, credit risk, or operational risk. For instance, models might use volatility, interest rate changes, or credit ratings as explanatory variables to estimate potential losses. Academic journals frequently publish research on these applications; recent studies in the Journal of Risk and Financial Management exemplify the use of macroeconomic factors as explanatory variables in predicting non-performing loans and assessing financial stability.¹³
• Policy Making: Governments and regulatory bodies use models with explanatory variables to assess the potential impact of new policies on the economy or specific sectors. Understanding these relationships helps in designing effective regulations and economic stimuli.

Limitations and Criticisms

Despite their widespread utility, explanatory variables and the models that employ them have several limitations and are subject to criticism. A primary concern is the distinction between correlation and causation. A strong statistical relationship between an explanatory variable and a dependent variable does not inherently mean one causes the other. There might be a "common response variable" or "confounding factor" that influences both, leading to a spurious correlation. For instance, ice cream sales and drowning incidents may correlate, but both are likely driven by a third variable: warm weather.¹²

Other significant limitations include:

• Model Specification Errors: Choosing the wrong explanatory variables or an inappropriate functional form (e.g., assuming a linear relationship when it's non-linear) can lead to biased or inconsistent estimates.¹⁰, ¹¹
• Multicollinearity: When two or more explanatory variables in a model are highly correlated with each other, it can make it difficult to determine the unique contribution of each variable and may lead to unstable coefficient estimates.⁸, ⁹
• Outliers and Influential Observations: Extreme data points can disproportionately influence the estimated relationships, distorting the model's accuracy.⁶, ⁷
• Assumptions Violations: Regression models rely on specific assumptions (e.g., linearity, independence of errors, homoscedasticity, normality of residuals). Violations of these assumptions can lead to unreliable results and invalid inferences.⁴, ⁵
• Data Quality and Availability: The accuracy and completeness of historical data can significantly impact the reliability of models built using explanatory variables. Incomplete or biased data can lead to skewed results.³
• Overfitting: Including too many explanatory variables, particularly those that do not genuinely explain the underlying phenomenon, can lead to a model that performs well on historical data but poorly on new, unseen data.²

The paper "Spurious Relations" by L. Scholtens, from Ethics in Econometrics, highlights that spurious relationships often arise when a model is not well-specified, potentially leading to inappropriate conclusions in econometric studies.¹ Analysts must exercise caution and apply rigorous testing to ensure their models are robust and their conclusions are meaningful.

Explanatory Variables vs. Confounding Variables

While explanatory variables are the independent factors explicitly included in a statistical model to explain a dependent variable, confounding variables are external factors that can influence both the explanatory variables and the dependent variable, creating an apparent, but not true, causal relationship.

The key difference lies in their intentional inclusion and their effect on the validity of observed relationships. Researchers aim to identify and control for confounding variables to ensure that the observed effects of explanatory variables are genuine.

FAQs

What is the primary purpose of using explanatory variables?

The primary purpose of using explanatory variables is to understand, explain, and predict the behavior of a dependent variable. They help quantify the relationships between different factors in a statistical model.

Can an explanatory variable also be a dependent variable?

In different models, a variable can certainly switch roles. For example, GDP growth might be an explanatory variable when predicting stock market returns, but it becomes a dependent variable when an economist attempts to explain its changes based on interest rates or government spending.

How many explanatory variables should I use in a model?

The optimal number of explanatory variables depends on the complexity of the phenomenon being modeled, the availability of relevant data, and the need to avoid issues like overfitting. Including too many variables without strong theoretical justification can lead to less reliable models, while too few might result in an under-specified model that misses key relationships.

What is the difference between an explanatory variable and a predictor variable?

The terms "explanatory variable" and "predictor variable" are often used interchangeably, especially in regression analysis. Both refer to the independent variables in a model that are used to forecast or explain changes in an outcome.

What happens if an important explanatory variable is omitted from a model?

Omitting an important explanatory variable can lead to "omitted variable bias." This occurs when the excluded variable is correlated with both the dependent variable and one or more of the included explanatory variables. This bias can distort the estimated coefficients of the included variables, leading to incorrect inferences and less accurate forecasting.