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Ordinary least squares regression

What Is Ordinary Least Squares Regression?

Ordinary least squares (OLS) regression is a fundamental statistical method used within statistical analysis to estimate the relationship between a dependent variable and one or more independent variables. At its core, OLS aims to find the "best-fit" line (or hyperplane in higher dimensions) that minimizes the sum of the squared differences between the observed values and the values predicted by the model58, 59. This method is a cornerstone of regression analysis, providing a powerful tool for understanding and modeling relationships in data. The "least squares" aspect refers directly to its objective: minimizing the sum of the squared residuals, which are the vertical distances between each data point and the regression line56, 57.

History and Origin

The concept behind the method of least squares, which underpins ordinary least squares regression, emerged in the late 18th and early 19th centuries, driven by challenges in astronomy and geodesy, such as determining planetary orbits from imperfect observations. While the French mathematician Adrien-Marie Legendre publicly published the method in 1805 in his "Nouvelles méthodes pour la détermination des orbites des comètes," Carl Friedrich Gauss claimed to have used it as early as 1795. G54, 55auss's later, more sophisticated exposition in 1809 provided a robust theoretical framework, linking the method to probability theory. Stephen Stigler, a prominent historian of statistics, argues that Gauss's work fundamentally shaped the future of science, business, and society by providing a rigorous foundation for OLS.. T53he term "regression" itself was later introduced by Sir Francis Galton in the late 1800s, observing the "regression to the mean" phenomenon in his studies of heredity, which then became associated with the least squares method of prediction.

51, 52## Key Takeaways

  • Ordinary least squares (OLS) regression is a widely used statistical technique for modeling linear relationships between variables.
  • Its primary objective is to minimize the sum of the squared differences between observed and predicted values, defining a "best-fit" line.
  • OLS is foundational in econometrics, financial modeling, and various scientific fields for prediction and understanding variable relationships.
  • The method assumes linearity, independence of errors, homoscedasticity, and normality of residuals for optimal and unbiased estimates.
  • While powerful, OLS is sensitive to outliers and issues like multicollinearity, which can impact the reliability of its results.

Formula and Calculation

The goal of ordinary least squares (OLS) regression is to find the coefficients that define the line of best fit. For a simple linear regression with one independent variable, the model can be represented as:

Yi=β0+β1Xi+ϵiY_i = \beta_0 + \beta_1 X_i + \epsilon_i

Where:

  • (Y_i) is the observed value of the dependent variable for the (i)-th observation.
  • (X_i) is the observed value of the independent variable for the (i)-th observation.
  • (\beta_0) is the intercept (the expected value of (Y) when (X) is zero).
  • (\beta_1) is the slope (the change in (Y) for a one-unit change in (X)).
  • (\epsilon_i) is the error term, representing the difference between the observed and predicted values ((Y_i - \hat{Y}_i)).

The OLS method estimates (\beta_0) and (\beta_1) by minimizing the sum of the squared errors (SSE), also known as the sum of squared residuals:

min⁡∑i=1n(Yi−(β0+β1Xi))2\min \sum_{i=1}^{n} (Y_i - (\beta_0 + \beta_1 X_i))^2

The formulas for the estimated slope ((\hat{\beta}_1)) and intercept ((\hat{\beta}_0)) in simple linear regression are:

β^1=∑i=1n(Xi−Xˉ)(Yi−Yˉ)∑i=1n(Xi−Xˉ)2\hat{\beta}_1 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i=1}^{n} (X_i - \bar{X})^2}

β^0=Yˉ−β^1Xˉ\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X}

Where (\bar{X}) and (\bar{Y}) are the means of the independent and dependent variables, respectively. F50or multiple independent variables, the calculation involves matrix algebra, but the underlying principle of minimizing the sum of squared residuals remains the same.

Interpreting the Ordinary Least Squares Regression

Interpreting the results of an ordinary least squares (OLS) regression involves examining several key statistics to understand the relationship between variables and assess the model performance. The primary outputs include coefficients, R-squared, and p-values.

48, 49The coefficients ((\beta) values) indicate the estimated change in the dependent variable for a one-unit change in the corresponding independent variable, assuming all other independent variables are held constant. F47or example, if a coefficient for "years of experience" is 500, it suggests that for every additional year of experience, the dependent variable (e.g., salary) is expected to increase by $500. The intercept ((\beta_0)) represents the predicted value of the dependent variable when all independent variables are zero.

46R-squared ((R^2)) is a statistic that measures the proportion of the variance in the dependent variable that can be explained by the independent variables in the OLS model. A45 higher (R^2) value (closer to 1) indicates that the model explains a larger proportion of the variability and generally provides a better fit to the data points. H43, 44owever, a high (R^2) alone does not guarantee a good model; it's also crucial to consider the statistical significance of the individual coefficients, often indicated by p-values. A41, 42 p-value less than a predetermined significance level (e.g., 0.05) suggests that the relationship between the independent and dependent variable is statistically significant.

40## Hypothetical Example

Consider a hypothetical financial analyst who wants to understand if advertising expenditure influences quarterly sales for a retail company. The analyst collects data for the past 20 quarters, noting the advertising spend (in thousands of dollars) and the corresponding total sales (in millions of dollars).

  • Dependent Variable (Y): Quarterly Sales (in millions of dollars)
  • Independent Variable (X): Advertising Expenditure (in thousands of dollars)

The analyst performs an ordinary least squares regression on this data. After running the OLS model, the results yield the following equation:

(\text{Sales} = 0.5 + 0.15 \times \text{Advertising Spend})

In this equation:

  • The intercept (0.5) suggests that if there were zero advertising expenditures, the predicted quarterly sales would be $0.5 million.
  • The coefficient for Advertising Spend (0.15) indicates that for every additional $1,000 spent on advertising, the predicted quarterly sales increase by $0.15 million (or $150,000).

If the company plans to spend an additional $100,000 on advertising next quarter, the analyst can use this OLS model to predict the impact on sales:
(\text{Predicted Sales} = 0.5 + 0.15 \times (100) = 0.5 + 15 = 15.5) million dollars.

This simple example illustrates how ordinary least squares provides a quantifiable relationship, allowing for predictions and informing decisions based on historical data points.

Practical Applications

Ordinary least squares (OLS) regression is widely applied across various domains, particularly in finance and economics, due to its simplicity and interpretability. In financial modeling, OLS is frequently used to:

  • Predict Asset Prices: Analysts might use OLS to model the relationship between a stock's price and various factors such as earnings per share, interest rates, or market indices. T38, 39his can help in portfolio management and investment strategy development.
  • Risk Management: OLS can be employed to estimate beta in the Capital Asset Pricing Model (CAPM), which measures a security's systematic risk relative to the overall market.
  • Economic Forecasting: Economists use OLS to forecast macroeconomic indicators like Gross Domestic Product (GDP) growth, inflation rates, or unemployment, by analyzing their relationships with other economic variables. F37or instance, it can be used to model the relationship between GDP growth and unemployment, as highlighted in Okun's law. The Federal Reserve System, for example, often uses various regression models to analyze economic data and inform monetary policy decisions.
  • Time Series Analysis: While more advanced methods exist for complex time series analysis, OLS can be used to identify linear trends in financial data, such as commodity prices or exchange rates.
  • Corporate Finance: Companies might use OLS to understand the drivers of revenue or expenses, for example, by regressing sales on marketing spend or production costs on output volume.

Limitations and Criticisms

Despite its widespread use and foundational role, ordinary least squares (OLS) regression has several important limitations and criticisms that can affect the reliability and validity of its results. Understanding these drawbacks is crucial for appropriate application and interpretation.

One significant limitation is OLS's sensitivity to outliers in the data. O35, 36utliers are extreme observations that can disproportionately influence the regression line, potentially leading to biased coefficient estimates and misleading conclusions. B34ecause OLS minimizes the sum of squared residuals, large errors from outliers are heavily penalized, pulling the line towards them.

33Another critical aspect involves the assumptions underlying OLS. While OLS provides the "Best Linear Unbiased Estimator" (BLUE) under certain conditions (known as Gauss-Markov assumptions), violations of these assumptions can compromise the validity of the model. K32ey assumptions include:

  • Linearity: The relationship between the independent and dependent variables is assumed to be linear. If the true relationship is non-linear, OLS may provide a poor fit.
    *30, 31 Independence of Errors: Residuals are assumed to be independent of each other. Issues like autocorrelation, common in time series analysis, violate this assumption.
    *29 Homoscedasticity: The variance of the error term is assumed to be constant across all levels of the independent variables. 28Heteroscedasticity, where the variance of errors is not constant, can lead to inefficient coefficient estimates and unreliable standard errors.
    *27 Normality of Errors: While OLS estimators are unbiased even if errors are not normally distributed, this assumption is important for accurate hypothesis testing and confidence interval construction, especially with small sample sizes.

26Furthermore, multicollinearity, where independent variables are highly correlated with each other, can make OLS estimates unstable and difficult to interpret. T24, 25his makes it challenging to ascertain the individual impact of each independent variable. W23hen these assumptions are violated, the OLS model may yield erroneous results. F22or instance, a study published in Medicine & Science in Sports & Exercise highlighted that using OLS to analyze repeated measures data can produce "substantially inflated probabilities of Type I errors" when the data's variance/covariance structure is not compound symmetric, suggesting that alternative methods like generalized least squares (GLS) may be more appropriate. R21esearchers should always conduct diagnostic checks, such as examining residual plots or running tests for multicollinearity, to ensure the assumptions of OLS are met.

20## Ordinary Least Squares Regression vs. Multiple Linear Regression

The terms ordinary least squares (OLS) regression and multiple linear regression are often used interchangeably or cause confusion, but they represent different, albeit related, concepts.

Multiple linear regression (MLR) is a type of linear regression that involves two or more independent variables to predict a single dependent variable. I18, 19t is a model that describes a linear relationship. For example, predicting house prices based on square footage, number of bedrooms, and location would involve multiple linear regression.

Ordinary least squares (OLS) regression, on the other hand, is an estimation method used to determine the coefficients of a linear regression model, whether it's simple (one independent variable) or multiple (two or more independent variables). O15, 16, 17LS is the technique employed to find the "best-fit" line by minimizing the sum of the squared differences between observed and predicted values. T14herefore, multiple linear regression can be estimated using the OLS method. O12, 13LS refers to the specific optimization technique, while multiple linear regression describes the structure of the model itself.

FAQs

What is the main goal of Ordinary Least Squares (OLS) regression?

The main goal of ordinary least squares regression is to find the line or hyperplane that best fits a set of data points by minimizing the sum of the squared differences between the actual observed values and the values predicted by the model. T10, 11his helps in identifying and quantifying the linear relationship between variables.

What are residuals in OLS regression?

Residuals in OLS regression are the differences between the observed values of the dependent variable and the values predicted by the regression model. T9he OLS method specifically seeks to minimize the sum of these residuals, squared, to determine the optimal fit. A8nalyzing residuals is a crucial part of checking the validity of an OLS model.

Can OLS regression be used for prediction?

Yes, ordinary least squares regression is commonly used for prediction, allowing analysts to forecast future outcomes based on the established linear relationship between variables. O6, 7nce the OLS model's coefficients are determined, they can be used with new independent variable values to predict the corresponding dependent variable. However, predictions outside the range of the original data should be made with caution.

What are some common assumptions of OLS regression?

Key assumptions for valid ordinary least squares regression include linearity in the relationship between variables, independence of errors, constant variance of errors (homoscedasticity), and normality of the error distribution. V4, 5iolation of these assumptions can lead to unreliable or inefficient estimates.

3### Is OLS the same as linear regression?
OLS is a widely used method for estimating the parameters of a linear regression model. W2hile often used interchangeably, linear regression refers to the statistical model that assumes a linear relationship, and OLS is the particular optimization technique used to find the best parameters for that linear model. A1 linear regression model can be estimated using methods other than OLS, though OLS is the most common.