Regression Analysis By Example
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Methods of regression analysis are clearly demonstrated, and examples containing the types of irregularities commonly encountered in the real world are provided. Each example isolates one or two techniques and features detailed discussions, the required assumptions, and the evaluated success of each technique. Additionally, methods described throughout the book can be carried out with most of the currently available statistical software packages, such as the software package R.
Regression analysis is a set of statistical methods used for the estimation of relationships between a dependent variable and one or more independent variables. It can be utilized to assess the strength of the relationship between variables and for modeling the future relationship between them.
Regression analysis includes several variations, such as linear, multiple linear, and nonlinear. The most common models are simple linear and multiple linear. Nonlinear regression analysis is commonly used for more complicated data sets in which the dependent and independent variables show a nonlinear relationship.
Multiple linear regression analysis is essentially similar to the simple linear model, with the exception that multiple independent variables are used in the model. The mathematical representation of multiple linear regression is:
Multiple linear regression follows the same conditions as the simple linear model. However, since there are several independent variables in multiple linear analysis, there is another mandatory condition for the model:
Regression analysis comes with several applications in finance. For example, the statistical method is fundamental to the Capital Asset Pricing Model (CAPM). Essentially, the CAPM equation is a model that determines the relationship between the expected return of an asset and the market risk premium.
When forecasting financial statements for a company, it may be useful to do a multiple regression analysis to determine how changes in certain assumptions or drivers of the business will impact revenue or expenses in the future. For example, there may be a very high correlation between the number of salespeople employed by a company, the number of stores they operate, and the revenue the business generates.
Python and R are both powerful coding languages that have become popular for all types of financial modeling, including regression. These techniques form a core part of data science and machine learning where models are trained to detect these relationships in data.
Imagine this: you are provided with a whole lot of different data and are asked to predict next year's sales numbers for your company. You have discovered dozens, perhaps even hundreds, of factors that can possibly affect the numbers. But how do you know which ones are really important Run regression analysis in Excel. It will give you an answer to this and many more questions: Which factors matter and which can be ignored How closely are these factors related to each other And how certain can you be about the predictions
Regression analysis helps you understand how the dependent variable changes when one of the independent variables varies and allows to mathematically determine which of those variables really has an impact.
Technically, a regression analysis model is based on the sum of squares, which is a mathematical way to find the dispersion of data points. The goal of a model is to get the smallest possible sum of squares and draw a line that comes closest to the data.
In statistics, they differentiate between a simple and multiple linear regression. Simple linear regression models the relationship between a dependent variable and one independent variables using a linear function. If you use two or more explanatory variables to predict the dependent variable, you deal with multiple linear regression. If the dependent variable is modeled as a non-linear function because the data relationships do not follow a straight line, use nonlinear regression instead. The focus of this tutorial will be on a simple linear regression.
As an example, let's take sales numbers for umbrellas for the last 24 months and find out the average monthly rainfall for the same period. Plot this information on a chart, and the regression line will demonstrate the relationship between the independent variable (rainfall) and dependent variable (umbrella sales):Linear regression equationMathematically, a linear regression is defined by this equation:
The linear regression equation always has an error term because, in real life, predictors are never perfectly precise. However, some programs, including Excel, do the error term calculation behind the scenes. So, in Excel, you do linear regression using the least squares method and seek coefficients a and b such that:
Below you will find the detailed instructions on using each method.How to do linear regression in Excel with Analysis ToolPakThis example shows how to run regression in Excel by using a special tool included with the Analysis ToolPak add-in.
This will add the Data Analysis tools to the Data tab of your Excel ribbon.Run regression analysisIn this example, we are going to do a simple linear regression in Excel. What we have is a list of average monthly rainfall for the last 24 months in column B, which is our independent variable (predictor), and the number of umbrellas sold in column C, which is the dependent variable. Of course, there are many other factors that can affect sales, but for now we focus only on these two variables:With Analysis Toolpak added enabled, carry out these steps to perform regression analysis in Excel:
As you have just seen, running regression in Excel is easy because all calculations are preformed automatically. The interpretation of the results is a bit trickier because you need to know what is behind each number. Below you will find a breakdown of 4 major parts of the regression analysis output.
R Square. It is the Coefficient of Determination, which is used as an indicator of the goodness of fit. It shows how many points fall on the regression line. The R2 value is calculated from the total sum of squares, more precisely, it is the sum of the squared deviations of the original data from the mean.
In our example, R2 is 0.91 (rounded to 2 digits), which is fairy good. It means that 91% of our values fit the regression analysis model. In other words, 91% of the dependent variables (y-values) are explained by the independent variables (x-values). Generally, R Squared of 95% or more is considered a good fit.
Standard Error. It is another goodness-of-fit measure that shows the precision of your regression analysis - the smaller the number, the more certain you can be about your regression equation. While R2 represents the percentage of the dependent variables variance that is explained by the model, Standard Error is an absolute measure that shows the average distance that the data points fall from the regression line.
Observations. It is simply the number of observations in your model.Regression analysis output: ANOVAThe second part of the output is Analysis of Variance (ANOVA):Basically, it splits the sum of squares into individual components that give information about the levels of variability within your regression model:
The ANOVA part is rarely used for a simple linear regression analysis in Excel, but you should definitely have a close look at the last component. The Significance F value gives an idea of how reliable (statistically significant) your results are. If Significance F is less than 0.05 (5%), your model is OK. If it is greater than 0.05, you'd probably better choose another independent variable.Regression analysis output: coefficientsThis section provides specific information about the components of your analysis:The most useful component in this section is Coefficients. It enables you to build a linear regression equation in Excel:
In a similar manner, you can find out how many umbrellas are going to be sold with any other monthly rainfall (x variable) you specify.Regression analysis output: residualsIf you compare the estimated and actual number of sold umbrellas corresponding to the monthly rainfall of 82 mm, you will see that these numbers are slightly different:
Why's the difference Because independent variables are never perfect predictors of the dependent variables. And the residuals can help you understand how far away the actual values are from the predicted values:For the first data point (rainfall of 82 mm), the residual is approximately -2.8. So, we add this number to the predicted value, and get the actual value: 17.8 - 2.8 = 15.How to make a linear regression graph in ExcelIf you need to quickly visualize the relationship between the two variables, draw a linear regression chart. That's very easy! Here's how:
And this is how our improved regression graph looks like:Important note! In the regression graph, the independent variable should always be on the X axis and the dependent variable on the Y axis. If your graph is plotted in the reverse order, swap the columns in your worksheet, and then draw the chart anew. If you are not allowed to rearrange the source data, then you can switch the X and Y axes directly in a chart.
The LINEST function uses the least squares regression method to calculate a straight line that best explains the relationship between your variables and returns an array describing that line. You can find the detailed explanation of the function's syntax in this tutorial. For now, let's just make a formula for our sample dataset:
The following screenshot shows all these Excel regression formulas in action:Tip. If you'd like to get additional statistics for your regression analysis, use the LINEST function with the stats parameter set to TRUE as shown in this example.
That's how you do linear regression in Excel. That said, please keep in mind that Microsoft Excel is not a statistical program. If you need to perform regression analysis at the professional level, you may want to use targeted software such as XLSTAT, RegressIt, etc.To have a closer look at our linear regression formulas and other techniques discussed in this tutorial, you are welcome to download our sample workbook below. Thank you for reading! 59ce067264