Suppose you have the daily stock price data of a company for the last one year. You want to find out what pattern does the data follow over time. Using this, you might want to predict future stock prices. Some of the questions that might arise are:

• Can the data available to us be modelled using some equation??
• Can we forecast future trend of the stock’s price using historical price data??
• How accurate the predicted prices would be??
• How accurate the predicted prices would be??

All the above concerns can be answered using a technique called “Regression Analysis”. This is a very important topic covered in Quants of FRM-I.

Regression analysis in brief:-

Consider the following table carefully!!
Following are the marks obtained in a test:
If I study for five more hours, will it actually increase my marks??? If the same question lingers over your mind, then the answer to this is hidden in Predictive Modeling using Linear Regression. Predictive Modeling basically involves statistical analysis of past behaviour to simulate future results.
Uses of Linear Regression
1) It is widely used for prediction and forecasting
2) Or explaining the impact of changes in an independent variable on the dependent variable.
For instance in the above example, we can find out the impact of studying for more no of hours (independent variable) on the total marks obtained (dependent variable).
How do you get started?
1) What do you require or what do you need to know –We want to know whether increasing the no of hours of study will really help in increasing total marks obtained.

2) What do you know? – We have a small sample dataset as depicted in the table above.

3) What do you wish to know? - We need to know population regression function ie how are marks obtained and the no of hours of study related to each other.

4) What is predictable? – We need to predict the population regression function from sample regression function as shown below:
5) What predicts? – In this session, you will learn how to use “Ordinary Least Squares method” to predict the population function. We will also see how to detect errors (TSS, RSS etc) in regression.

In daily life, we generally encounter problems in which the output of our actions is influenced by more than one factors. In other words, we move from the simple linear regression model with one predictor to the multiple linear regression model with two or more predictors.
MULTIPLE LINEAR REGRESSION
For better understanding, let’s have a look at a real life example where multiple linear regression model is used.
Mini Case Example:
Does an individual’s brain size and body size (height, weight etc) predictive of his/her intelligence??

Now to answer this research question, we first need to identify the predictor and response variables.
• Response variable (y): Performance IQ Score (PIQ) of an individual
• Potential Predictor Variable (x1): Brain Size obtained from MRI scan provided by doctor
• Potential Predictor Variable (x2): Height in inches
• Potential Predictor Variable (x3): Weight in kgs

A common way of investigating the relationships among all of the variables is by way of a "scatter plot matrix." which contains a scatter plot of each pair of variables arranged in an orderly arrangement. Here's how a scatter plot matrix looks like for our brain and body size case study:

Image Courtesy: onlinecourses.science.psu.edu

Below we can see the multiple linear regression model with three quantitative predictors (brain size, height and weight) :
Where independent error terms εi follow a normal distribution with mean 0 and equal variance σ 2.

The multiple regression model formulated above will try to answer the below question concerns:
• Which predictors (if any) -brain size, height, or weight - explain some of the variation in intelligence scores? Or conduct hypothesis tests for testing whether the slope parameters are 0.
• What is the effect of brain size on PIQ, after taking into account height and weight? Or calculate and interpret a confidence interval for the PIQ slope parameter.
• What is the PIQ of an individual with a given brain size, height, and weight? Or Calculate and interpret a prediction interval for the response.

However, one point to be noted is that whether the individual predictor variables (brain size, height, weight etc) are correlated with each other or not?

• If yes, then it will create a problem in running linear regression, also referred to as Multicollinearity. It is not a mistake in model specification but happens due to the nature of data at hand.
• A simple test for detecting multicollinearity is to conduct artificial regressions between each independent variable (as the “dependent” variable) and the remaining independent variables
• High R2, highly significant F-test, but few or no statistically significant t tests are a symptom of the presence of multicollinearity in the model.
• Manual variable selection - By excluding those independent variables which appear to causing problem.
• If possible, one should obtain more data for the sample considered. This is the preferred solution. More data can produce more precise parameter estimates (with lower standard errors).

EduPristine has many more theories and examples to understand the concept in details. Write to us help@edupristine.com to know further!

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