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BUSI 650 – Business Analytics A Realtor in Massachusetts is analyzing the relationship between the sale price of a home (in $), it's the square footage (in ft2), the number of bedrooms, the number of bathrooms, and a colonial dummy variable (colonial equals 1 if a colonial-style home; 0 otherwise). The relater collects data of 36 sales in Massachusetts for the analysis. The attached file “Regression Assignment – Dataset 2” contains the data. 1. Estimate the linear regression model for the price. 2. Interpret the coefficients attached to area (square footage) and colonial. 3. Predict the price of 2100 ft2 colonial style home with two bedrooms and two bathrooms. Submit a PDF file for your answers as well as the excel sheet. Consumption DateConsumption Income Q1, 20002863431192 Q2, 20002883731438 Q3, 20002903931719 Q4, 20002922031742 Q1, 20012927631940 Q2, 20012928231836 Q3, 20012931332504 Q4, 20012968532020 Q1, 20022970932728 Q2, 20022979532821 Q3, 20022992732696 Q4, 20023001432773 Q1, 20033008432796 Q2, 20033035133197 Q3, 20033072233657 Q4, 20033088733713 Q1, 20043112733893 Q2, 20043126134152 Q3, 20043148134247 Q4, 20043172534589 Q1, 20053190034183 Q2, 20053217834380 Q3, 20053234534469 Q4, 20053238634666 Q1, 20063267735380 Q2, 20063277735354 Q3, 20063288435363 Q4, 20063313435734 Q1, 20073323535889 Q2, 20073327335878 Q3, 20073333535880 Q4, 20073329235819 Q1, 20083315035998 Q2, 20083313436677 Q3, 20083281435747 Q4, 20083234435892 Q1, 20093216735752 Q2, 20093195735932 Q3, 20093207635456 Q4, 20093199935328 Q1, 20103210635293 Q2, 20103230835686 Q3, 20103245135791 Q4, 20103271735967 Q1, 20113282636346 Q2, 20113283636228 Q3, 20113291336347 Q4, 20113295936299 Q1, 20123310436832 Q2, 20123310537054 Q3, 20123312836963 Q4, 20123315637860 Q1, 20133326336226 Q2, 20133328036380 Q3, 20133337036522 Q4, 20133358536528 Q1, 20143368836872 Q2, 20143394637289 Q3, 20143419037587 Q4, 20143450837907 Q1, 20153465938033 Q2, 20153485038336 Q3, 20153501538571 Q4, 20153514738785 Q1, 20163523638927 Q2, 20163555039148 Q3, 20163574339354 Q4, 20163598739254 WEEK 6: LINEAR REGRESSION RUSHDI ALSALEH 1 OUTLINE ¡ Basic Statistics ¡ Statistics using Excel 2 USE OF ANALYTICS BY FIELDS AND INDUSTRY ¡ Financial Institutions/Services ¡ Traditional FI’s ¡ Fintechs ¡ Payroll Accounting ¡ Social Media ¡ Energy ¡ Sports ¡ Construction ¡ Economics WHAT ARE YOU LOOKING FOR IN THE DATA? Patterns or Trends ¡ Cyclical: cycle occurs when the data exhibit rises and falls that are not of a fixed frequency ¡ Upward: increasing trend ¡ Downward: decreasing trend ¡ No pattern or trend: random fluctuations which do not appear to be very predictable, and no strong patterns that would help with developing a forecasting model. WHAT ARE YOU LOOKING FOR IN THE DATA? WHAT ARE YOU LOOKING FOR IN THE DATA? ¡ Outliers – Strange or a rare event/data ¡ Decision to include in data ¡ 1) If the occurrence is real, if yes split into two: a) with outlier b) without outlier ¡ 2) If the not real, exclude it STATISTICS ¡ Central Tendencies ¡ Average (Mean): A number expressing the central or typical value in a set of data. (can be effected significantly by the outliers). ¡ Median: The value separating the higher half from the lower half of a data sample (not effected significantly by the outliers). ¡ Mode: most repeated number or any qualitative data. CORRELATION ¡ Correlation (r) – explains co movement of the two variables that are interdependent. (x and y) ¡ Does: How two interdependent variables move. ¡ Doesn’t: If one is causing the change in another. ¡ Example: Inflation and Interest rate. Unemployment and Inflation. ¡ Perfect Positive (r=1) – Movement is in the same direction as well as proportion. ¡ Imperfect positive (r >0 - <1) –="" movement="" is="" in="" the="" same="" direction="" but="" in="" different="" proportion.="" ¡="" no="" correlation="" (r="0)" –="" there="" is="" no="" or="" little="" correlation.="" ¡="" perfect="" negative="" (r="-1)" –="" movement="" is="" in="" opposite="" direction="" but="" in="" the="" same="" proportion.="" ¡="" imperfect="" negative="" (r=""><0 -="">-1) Movement is in opposite direction but in the different proportion. CORRELATION ¡ Correlation (r) REGRESSION ¡ Regression is a linear relationship between two variables. One variable is independent, another is dependent. y is dependent on x. ¡ Regression measures changes in dependent variable (y) given the change in independent variable (x). Remember: Independent moves by itself. ¡ Independent: Explanatory variable. Dependent: Explained variable. ¡ Regression model: y = Alpha + Beta (x) ¡ Intercept: Alpha. What y equals to Alpha when x (or all independent variables) is (are) zero. ¡ Slope: Beta. ¡ Model Interpretation: How the independent variable explains the change in the dependent variable. ¡ Single Factor: Only one dependent and one independent variable. ¡ y = Alpha + Beta1 (x1) ¡ Multiple Factor: Only one dependent and more than one independent variable. ¡ y = Alpha + Beta1 (x1) + Beta2 (x2) HOW TO INTERPRET REGRESSION RESULTS? ¡ R Square: Fitness of the model. Combined % of change in the dependent variable explained by all the independent variables. Or. How much % change in the dependent variable is explained by all or one independent variable. ¡ Range: 0 to 1. What if the R Square is equal to 0.99? Spurious regression. ¡ Adjusted R Square: Is R Square adjusted for errors or noise in the data. Too much difference between R Sqr and Adj. R Sqr means more noise or error in the data. ¡ Intercept: When one more all or one the independent variables are equal to zero. Dependent variable = Intercept (Alpha). ¡ Slope: If the independent variable increases by 1 (unit of measurement), the dependent variable will increase or decrease by the slope (unit of measurement). Keeping all the other factors constant. ¡ t Stat: For Statistical significance. Refer to the table or compare it to 2. ¡ Multi-colinearity: When two independent variables are correlated (r > + or -0.75). HOW TO INTERPRET REGRESSION RESULTS? ¡ We cannot use ?! for model comparison when the competing models do not include the same number of predictor variables (but have the same response). ¡ ?! never decreases as we add more variables. ¡ May include variables with no economic or intuitive foundation. ¡ Adjusted ?! explicitly accounts for the sample size ? and the number of predictor variables ?. ¡ Imposes a penalty for any additional predictors. ¡ The higher the adjusted ?!, the better the model. ¡ When comparing models with the same response, the model with the higher adjusted ?! is preferred. EXAMPLES ¡ Correlation ¡ Inflation – Unemployment ¡ Regression ¡ In class example – Average prices in Vancouver (explained) –Variables explaining the change in explained variable (explanatory variable) ¡ Anova Table – Interpret – R² - Adjusted R² - Intercept – Coefficients – T – Statistics CAUTION! ¡ Correlation and Regression do not take into account the qualitative side of the data. ¡ So, high correlation between ice cream sales and crime rate means, people who eat more ice creams commit more crimes? ¡ Answer: ¡ Understand the model or correlation you run, better numbers don’t always mean the right results! Thank you! Questions? 15