Regression analysis

Examples of Regression analysis in the following topics:

• Regression Analysis for Forecast Improvement

  • Regression Analysis is a causal / econometric forecasting method that is widely used for prediction and forecasting improvement.
  • Regression Analysis is a causal / econometric forecasting method.
  • A large body of techniques for carrying out regression analysis has been developed.
  • The performance of regression analysis methods in practice depends on the form of the data generating process and how it relates to the regression approach being used.
  • Regression analysis shows the relationship between a dependent variable and one or more independent variables.

• Predictions and Probabilistic Models

  • In statistics, regression analysis is a statistical technique for estimating the relationships among variables.
  • In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function, which can be described by a probability distribution.
  • Regression analysis is widely used for prediction and forecasting.
  • In restricted circumstances, regression analysis can be used to infer causal relationships between the independent and dependent variables.
  • Explain how to estimate the relationship among variables using regression analysis

• Slope and Intercept

  • In the regression line equation the constant $m$ is the slope of the line and $b$ is the $y$-intercept.
  • Regression analysis is the process of building a model of the relationship between variables in the form of mathematical equations.
  • A simple example is the equation for the regression line which follows:
  • The case of one explanatory variable is called simple linear regression.
  • For more than one explanatory variable, it is called multiple linear regression.

• Analyzing Data and Drawing Conclusions

  • Quantitative data can be analyzed in a variety of ways, regression analysis being among the most popular .
  • More specifically, regression analysis helps one understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed.
  • A large body of techniques for carrying out regression analysis has been developed.
  • In practice, the performance of regression analysis methods depends on the form of the data generating process and how it relates to the regression approach being used.
  • Since the true form of the data-generating process is generally not known, regression analysis often depends to some extent on making assumptions about this process.

• The Equation of a Line

  • In statistics, linear regression can be used to fit a predictive model to an observed data set of $y$ and $x$ values.
  • In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable.
  • Simple linear regression fits a straight line through the set of $n$ points in such a way that makes the sum of squared residuals of the model (that is, vertical distances between the points of the data set and the fitted line) as small as possible.
  • Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.
  • If the goal is prediction, or forecasting, linear regression can be used to fit a predictive model to an observed data set of $y$ and $X$ values.

• Qualitative Variable Models

  • In statistics, particularly in regression analysis, a dummy variable (also known as a categorical variable, or qualitative variable) is one that takes the value 0 or 1 to indicate the absence or presence of some categorical effect that may be expected to shift the outcome.
  • Dummy variables are "proxy" variables, or numeric stand-ins for qualitative facts in a regression model.
  • In regression analysis, the dependent variables may be influenced not only by quantitative variables (income, output, prices, etc.), but also by qualitative variables (gender, religion, geographic region, etc.).
  • Analysis of variance (ANOVA) models are a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups).
  • Break down the method of inserting a dummy variable into a regression analysis in order to compensate for the effects of a qualitative variable.

• Polynomial Regression

  • For this reason, polynomial regression is considered to be a special case of multiple linear regression.
  • In the 20th century, polynomial regression played an important role in the development of regression analysis, with a greater emphasis on issues of design and inference.
  • Although polynomial regression is technically a special case of multiple linear regression, the interpretation of a fitted polynomial regression model requires a somewhat different perspective.
  • Polynomial regression is one example of regression analysis using basis functions to model a functional relationship between two quantities.
  • This is similar to the goal of non-parametric regression, which aims to capture non-linear regression relationships.

• Least-Squares Regression

  • The criteria for determining the least squares regression line is that the sum of the squared errors is made as small as possible.
  • The process of fitting the best- fit line is called linear regression.
  • Therefore, this best fit line is called the least squares regression line.
  • Ordinary Least Squares (OLS) regression (or simply "regression") is a useful tool for examining the relationship between two or more interval/ratio variables assuming there is a linear relationship between said variables.
  • If the relationship is not linear, OLS regression may not be the ideal tool for the analysis, or modifications to the variables/analysis may be required.

• Checking the Model and Assumptions

  • There are a number of assumptions that must be made when using multiple regression models.
  • When working with multiple regression models, a number of assumptions must be made.
  • These assumptions are similar to those of standard linear regression models.
  • If the data does not appear as linear, but rather in a curve, it may be necessary to transform the data or use a different method of analysis.
  • Error will not be evenly distributed across the regression line.

• Introduction to multiple regression

  • The principles of simple linear regression lay the foundation for more sophisticated regression methods used in a wide range of challenging settings.
  • In Chapter 8, we explore multiple regression, which introduces the possibility of more than one predictor, and logistic regression, a technique for predicting categorical outcomes with two possible categories.
  • Multiple regression extends simple two-variable regression to the case that still has one response but many predictors (denoted x1 , x2 , x3 , ...).
  • Multiple regression will help us answer these and other questions.
  • Multiple regression also allows for categorical variables with many levels, though we do not have any such variables in this analysis, and we save these details for a second or third course.