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Introduction to Regression MSIT3000 Lecture 18 Objectives Learn key terms and uses of regression. Describe the assumptions needed for simple Ordinary Least Squares regression. Estimate the parameters for a simple linear probabilistic model. Text: 9.1, 9.2 & 9.3 MSIT3000 2 What is regression? To ‘regress’ one variable on another is to ‘fit’ a function. The simplest function to fit is: Y=A (Not very useful). The second simplest function to fit is: Y = A + Bx (Remarkably useful!) ‘Regression’ refers to finding values for A & B from values of X & Y. MSIT3000 3 Fitting a line to data: Example from p 455 5 4 3 Sales 2 1 0 0 1 2 3 4 5 6 Advertising MSIT3000 4 Data with “regression line” 'Fitted' Line y = 0.7x - 0.1 5 4 3 y 2 1 0 0 1 2 3 4 5 6 x y Predicted y Linear (Predicted y) MSIT3000 5 What is regression useful for? Marketing: advertising & sales models. Real estate: estimating the value of property and property attributes. Finance: Valuing assets. Modeling default risk. Establishing benchmarks. Accounting: Measuring financial performance – what is an appropriate benchmark? Organization Behavior: Relating performance to different kinds of pay or responsibilities. MSIT3000 6 Terminology Dependent variable. This is what you wish to model, explain and predict. In a sales-advertising model, you would want to predict sales based on how much you advertise. Independent variable (a.k.a. explanatory variable or predictor): This is the input to the model (advertising, in the sales advertising model). MSIT3000 7 Terminology Probabilistic vs deterministic models. Deterministic models have no room for ‘error’. I.e. if y = a + bx then that must be exactly true for all pairs of y and x. Probabilistic models recognize that there may be some ‘disturbance’ in our data. We therefore add noise to the model: y = a + bx + The noise term is denoted with and a.k.a. Disturbance Random error MSIT3000 8 Terminology Ordinary Least Squares regression: Ordinary refers to the deterministic part of the model being linear. We will expand on what “linear” means further when we get to multiple regression. ‘Least Squares’ refers to how we find the regression line. More on that shortly. MSIT3000 9 Where are we? We have a few terms and definitions. We have a set of problems in business that regression is useful for. We have found that it is possible to ‘fit’ a regression line by sight. The main problem with this method: it is subjective. This was terminology & motivation; now we will examine a method to find the regression line objectively. MSIT3000 10 Assumptions In order to fit a linear regression line, we need the following assumptions (cfr text): 1. Y = 0 + 1 x + (implied in text). 2. N(0,2) 3. i & j are independent if i j MSIT3000 11 Fitting an OLS The ‘fitted line’: ˆ ˆ Yhat = b0 + b1 x y 0 1 x ˆ We can find ‘errors’ [a.k.a. ‘prediction errors’ or ‘residuals’] for each pair of x & y: e = y- yhat How can we use the errors to find a “best” line through our data? MSIT3000 12 'Fitted' Line y = 0.7x - 0.1 5 4 3 y 2 1 0 0 1 2 3 4 5 6 x y Predicted y Linear (Predicted y) x Residual Plot 1 Residuals 0 0 1 2 3 4 5 6 -1 x Using the error terms In order to minimize the error in some meaningful way, we must first measure the overall error. How? we square each error to make sure each component of the overall error term is positive. then we sum all the squared error terms in order to get a measure for all of the data. finally we minimize that function; based on which ‘variables’? the parameter-estimates MSIT3000 14 Formulas When we minimize SSE using the parameter estimates, we find that: the slope 1hat = SS(xy)/SS(xx) the intercept 0hat = ybar - 1hat*xbar this is another way of saying that the OLS line passes through the pair of sample means, xbar and ybar. Where: SS ( x x)( y y ) x y x y i i xy i i i i n SS xx ( xi x) 2 x 2 x i 2 i n MSIT3000 15 Conclusion Objectives addressed: Terminology and some uses of regression. Assumptions needed OLS regression. Estimating the OLS parameters. Problem: Example on page 479. MSIT3000 16

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