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Homework answers / question archive / Multiple Regression: Home Prices 1 Multiple Regression: Predicting Home Selling Prices California State University, Sacramento Professor Sudhir Thakur Multiple Regression: Home Prices 2 Introduction This paper presents the statistical analysis made using multiple regression in evaluating how home prices or value of the houses are influenced by the different set of independent variables such as the house floor area (square feet), its lot area (square feet), the number of bedrooms and bathrooms, age of house (years), median income, and thereby defining their relationships
Multiple Regression: Home Prices 1 Multiple Regression: Predicting Home Selling Prices California State University, Sacramento Professor Sudhir Thakur Multiple Regression: Home Prices 2 Introduction This paper presents the statistical analysis made using multiple regression in evaluating how home prices or value of the houses are influenced by the different set of independent variables such as the house floor area (square feet), its lot area (square feet), the number of bedrooms and bathrooms, age of house (years), median income, and thereby defining their relationships. House selling price can then be predicted based on the outcome of the fitted models. The motivation behind this analysis is to find out the best set of independent variables that trigger the increase in home prices in the US, and so interestingly, we can perform a similar statistical analysis model to determine home prices in other places such as in Canada, France, UK, and Australia, where housing demands are high, and from there a study comparison can be created. Literature Review Although there are several studies and reviews made on home prices, this paper intends to answer a theory about the relationship between the home prices and several independent variables on the selected 150 home purchase transactions compiled in the California State University Sacramento (CSUS) data files. In the multiple regression analysis, the theory supported the historical data that the size of a house’s lot area and its floor area plus the number of bedrooms can determine the house’s selling price and its value in the market. These predictor variables particularly the size of lot area seem to be a reasonable expectation as the home built with bigger land area command higher sales price than a home with less land as highlighted by Pardoe (2006), in his article, Modeling Home Prices Using Realtor Data, Journal of Statistics Education v16n2. However, a few other determinants and variables such as the addition of extra bathrooms and fireplaces may cause some inconsistencies with the first theory. Pardoe (2006), in the same article, Modeling Home Prices Using Realtor Data, indicated that both bath and bed have resulted to large P-values and contradicted the theory that home prices should increase with the number of bedroom and bathrooms. Another study by Alexander Gustafsson and Sebastian Wogenius (2014) in their article Modelling Apartment Prices with the Multiple Regression Model, the study result suggested that an apartment is worth more if it has a fireplace. Problem Statement The purpose of the study is to look at the variable relationships between the home prices (response variable) and its size, house structure and special house features (explanatory variables) and if indeed these determinants can influence the increase (or decrease) of house prices that may be available in the market. If the variable model proved that there is no relationship or significance to the null hypothesis→ H?: ?? = 0, then H? can be rejected. The alternate hypothesis (H?) →H?: ß? ≠ 0, state that there are relationships, and therefore H? can be accepted. Methodology It is intended to use multiple regression. Firstly, the model result from the gathered sets of data by CSUS for the 150 home purchase transaction records is analyzed. This is then followed by a hypothesis testing as stated in the problem statement. The data files 2|Page Multiple Regression: Home Prices 3 in Excel is then uploaded into Statgraphics to perform multiple regression through the following steps; Simple model: Relate→Multiple Factors→Multiple Regression; Input home price as the dependent variable; input sq feet, bathrooms, lot size, median income and age of the house as independent variables; then analyze the results. Subsequently running the next model by multiple regression; Modified model: Relate→Multiple Factors→Multiple Regression; Input home price as the dependent variable; input sq feet, bathrooms, lot size, median income and age of the house as independent variables, and by this time add on the dummy variable, fireplace dummy (1=fireplace; 0=no fireplace) to the list of independent variables; then analyze the results. Analysis A. Simplified Model: Multiple Regression - Price ($1000) Dependent variable: Price ($1000) Independent variables: Sq Feet (000) Bathrooms Lot Size (000 sq ft) Median Income ($M) Age (Years) Number of observations: 150 Parameter CONSTANT Sq Feet (000) Bathrooms Lot Size (000 sq ft) Median Income ($M) Age (Years) Standard Estimate Error 169.932 31.9288 14.2834 5.66821 0.208986 T Statistic PValue 47.2679 3.59509 0.0004 6.29299 5.07371 0.0000 10.9884 1.29986 0.1957 2.27734 2.48896 0.0140 0.0546441 3.8245 0.0002 -9.05927 4.39011 Analysis of Variance Source Sum of Squares Model 1.24483E6 Residual 824243. Total 2.06907E6 (Corr.) Df Mean Square 5 248966. 144 5723.91 149 -2.06356 0.0409 F-Ratio PValue 43.50 0.0000 R-squared = 60.1636 percent R-squared (adjusted for d.f.) = 58.7804 percent 3|Page Multiple Regression: Home Prices 4 Standard Error of Est. = 75.6565 Mean absolute error = 58.7232 Durbin-Watson statistic = 2.00079 (P=0.5019) Lag 1 residual autocorrelation = -0.00856 Resulted linear equation of the fitted model: Price ($1000) = 169.932 + 31.9288*Sq Feet (000) + 14.2834*Bathrooms + 5.66821*Lot Size (000 sq ft) + 0.208986*Median Income ($M) - 9.05927*Age (Years) Summary Table A: Independent Variable Intercept ß-slope Interpretation P-value Interpretation 169.932 If all the independent variables equal to zero (0), the home price is estimated to be 169.932 0.0004 P-value is less than α(0.05), thus the null hypothesis is rejected, at 95% confidence level Sq Feet (000) 31.9288 For every 1 sq ft increase in floor area, there is an estimated increase in home prices by this factor. 0.0000 P-value is less than α(0.05), thus the null hypothesis is rejected, at 95% confidence level Bathrooms 14.2834 For every 1 bathroom added in the house, there is an estimated increase in home prices by this factor 0.1957 P-value is more than α(0.05), thus the null hypothesis is accepted, at 95% confidence level Lot Size (000 sq ft) 5.66821 For every 1 sq ft increase in lot area, 0.0140 P-value is less than α(0.05), thus there is an estimated increase in home the null hypothesis is rejected, at prices by this factor. 95% confidence level Median Income ($M) 0.208986 For every 1 unit increase in median 0.0002 P-value is less than α(0.05), thus income, there is an estimated increase the null hypothesis is rejected, at in home prices by this factor. 95% confidence level Age (Years) -9.05927 For every 1 year increase in house's age, there is an estimated decrease in home prices by this factor. 0.0409 P-value is less than α(0.05), thus the null hypothesis is rejected, at 95% confidence level B. Modified Model: Multiple Regression - Price ($1000) 4|Page R² Interpretation 60.1636 R² suggests that 60.1636% of the variation in home price (dependent variable) is influenced by the variability of the independent variables. Multiple Regression: Home Prices 5 Dependent variable: Price ($1000) Independent variables: Sq Feet (000) Bathrooms Lot Size (000 sq ft) Median Income ($M) Age (Years) Fireplace Dummy (1= fireplace, 0 Number of observations: 150 Standard Estimate Error T Statistic PValue 172.254 47.2449 3.64597 0.0004 34.6691 6.70024 5.1743 0.0000 12.8547 11.0402 1.16435 0.2462 5.88851 2.28192 2.58051 0.0109 0.203276 0.0547845 3.71046 0.0003 -9.31529 4.38954 -2.12216 0.0355 -16.6229 14.0947 -1.17937 0.2402 Parameter CONSTANT Sq Feet (000) Bathrooms Lot Size (000 sq ft) Median Income ($M) Age (Years) Fireplace Dummy (1= fireplace, 0 Analysis of Variance Source Sum of Squares Model 1.25277E6 Residual 816303. Total 2.06907E6 (Corr.) Df Mean Square 6 208795. 143 5708.41 149 R-squared = 60.5474 percent R-squared (adjusted for d.f.) = 58.892 percent Standard Error of Est. = 75.554 Mean absolute error = 59.0027 Durbin-Watson statistic = 2.00276 (P=0.5067) Lag 1 residual autocorrelation = -0.00867774 5|Page F-Ratio PValue 36.58 0.0000 Multiple Regression: Home Prices 6 Resulted linear equation of the fitted model: Price ($1000) = 172.254 + 34.6691*Sq Feet (000) + 12.8547*Bathrooms + 5.88851*Lot Size (000 sq ft) + 0.203276*Median Income ($M) - 9.31529*Age (Years) 16.6229*Fireplace Dummy (1= fireplace, 0 Summary Table B: Independent Variable Intercept ß-slope Interpretation P-value Interpretation 172.254 If all the independent variables equal to zero (0), the home price is estimated to be 172.254 0.0004 P-value is less than α(0.05), thus the null hypothesis is rejected, at 95% confidence level Sq Feet (000) 34.6691 For every 1 sq ft increase in floor area, there is an estimated increase in home prices by this factor. 0.0000 P-value is less than α(0.05), thus the null hypothesis is rejected, at 95% confidence level Bathrooms 12.8547 For every 1 bathroom added in the house, there is an estimated increase in home prices by this factor 0.2462 P-value is more than α(0.05), thus the null hypothesis is accepted, at 95% confidence level Lot Size (000 sq ft) 5.88851 For every 1 sq ft increase in lot area, there is an estimated increase in home prices by this factor. 0.0109 P-value is less than α(0.05), thus the null hypothesis is rejected, at 95% confidence level Median Income ($M) 0.203276 For every 1 unit increase in median income, there is an estimated increase in home prices by this factor. 0.0003 P-value is less than α(0.05), thus the null hypothesis is rejected, at 95% confidence level Age (Years) -9.31529 For every 1 year increase in house's age, there is an estimated decrease in home prices by this factor. 0.0355 P-value is less than α(0.05), thus the null hypothesis is rejected, at 95% confidence level Fireplace Dummy (1= fireplace, 0=no fireplace) -16.6229 For every 1 fireplace added in the house, there is an estimated decrease in home prices by this factor. 0.2402 P-value is more than α(0.05), thus the null hypothesis is accepted, at 95% confidence level R² Interpretation 60.5474 R² suggests that 60.5474% of the variation in home price (dependent variable) is influenced by the variability of the independent variables. Conclusion Based on the summarized information in Tables A and B, it can be concluded that there is sufficient evidence showing the relationship between home prices and the floor 6|Page Multiple Regression: Home Prices 7 area of the house in square feet, the lot size, number of bedrooms, the median income and the age of the house, at 95% confidence level. In the simplified model in Table A, the interesting finding was that the bathrooms parameter has resulted in a P-value that is greater than 0.05 (α). Also, in the modified model, the addition of a fireplace dummy in Table B has resulted in a decrease in house price as well as a P-value that is greater than 0.05 (α), so in this case, we cannot reject the null hypothesis. It can then be concluded that, based on the study from the given data sets, the addition of an extra bathroom and a fireplace in the house, didn’t trigger a rise in the house price, as normally expected. Therefore it is recommended that a further study and analysis is conducted in order to find out the best set of independent variables and combinations that can trigger an increase in home prices. References Pardoe, Iain (2006). Modeling Home Prices Using Realtor Data, Journal of Statistics Education v16n2. Retrieved from http://jse.amstat.org/v16n2/datasets.pardoe.pdf Alexander Gustafsson and Sebastian Wogenius (2014). Modelling Apartment Prices with the Multiple Linear Regression Model. Retrieved from https://www.divaportal.org/smash/get/diva2:725045/FULLTEXT01.pdf Library, California State University Sacramento (2019) CSUS data files Economic http://www.economagic.com/ Appendix 7|Page Multiple Regression: Home Prices 8 S. No. Price ($1000) Sq Feet (000) Bathrooms Lot Size (000 sq ft) Median Income ($M) Age (Years) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 531 489 256 447 459 452 381 301 183 228 388 380 472 233 826 400 120 275 141 523 172 321 259 198 317 288 284 281 6.4 3.2 3.4 5.7 5.9 4.1 6.3 2.4 2.9 2.8 3.7 4.9 5.9 3.1 12.8 5.5 4.1 4.5 2.4 9.6 2.3 3.5 3.3 2.2 6.1 4 5.4 4.6 4 2 2 3 3 3 4 2 2 2 2 3 3 2 7 3 3 2 2 4 1 1 2 2 4 3 2 3 16 13 5 13 8 8 9 8 9 7 9 9 10 4 17 10 7 7 8 11 1 9 4 5 6 3 6 9 288 77 42 314 82 57 42 179 75 68 238 32 535 32 483 294 98 212 68 425 56 152 36 108 58 133 161 74 10 10 10 10 10 10 8 8 8 10 8 8 10 8 10 10 10 10 12 10 10 8 11 12 8 12 8 9 8|Page Fireplace Dummy (1= fireplace, 0= no fireplace) 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 0 1 Multiple Regression: Home Prices 9 S. No. Price ($1000) Sq Feet (000) Bathrooms Lot Size (000 sq ft) Median Income ($M) Age (Years) 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 243 111 265 214 121 381 258 209 294 270 202 485 344 356 274 397 203 298 408 262 502 465 396 201 452 365 158 397 192 298 309 427 345 237 369 202 354 186 346 358 556 143 432 378 207 332 382 2.6 2.2 4.4 2.1 2.3 5.1 3.3 3.7 4.7 2.8 2.3 5.2 4.9 2.4 2.5 4.1 2.1 3 6.5 3.1 4.6 5.8 5.2 3.5 4.4 3.4 4.3 2.7 3.5 5.7 3.6 6.2 4 2.1 2.7 3.2 3.9 4.2 3.2 2.9 7.1 2.3 3.8 3.6 3.1 4.4 4.1 2 1 3 2 1 2 2 1 3 2 2 2 3 2 2 2 2 2 4 2 2 3 2 1 3 2 2 2 2 3 2 2 3 2 2 2 2 2 1 2 2 2 2 2 2 3 3 7 3 11 6 9 5 5 8 6 8 6 9 10 9 6 8 8 7 13 8 7 7 10 9 6 8 5 2 2 11 10 8 11 5 5 3 7 8 5 5 14 6 8 7 3 1 8 36 133 63 84 76 310 145 99 259 62 163 134 453 153 39 50 98 338 71 80 146 61 219 110 401 131 36 57 97 85 200 320 147 41 170 334 53 102 268 58 479 43 43 110 39 312 414 10 11 11 12 9 8 9 9 8 9 8 10 8 8 9 8 12 12 10 12 10 10 8 10 10 8 12 8 10 9 8 10 8 7 8 9 8 12 8 8 10 11 10 8 11 8 8 9|Page Fireplace Dummy (1= fireplace, 0= no fireplace) 0 1 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 0 Multiple Regression: Home Prices 10 S. No. Price ($1000) Sq Feet (000) Bathrooms Lot Size (000 sq ft) Median Income ($M) Age (Years) 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 254 324 337 265 371 265 285 247 358 279 416 282 423 467 335 316 136 319 309 353 277 347 216 347 200 346 351 392 156 192 477 208 371 258 172 452 260 197 650 409 174 243 155 380 474 395 286 3.2 4.6 3.7 2.9 6.1 2.2 4.6 3.6 2.3 3.3 6.5 2.2 3.9 4 3.2 5.2 2.9 3.6 2.9 2.5 2.2 3.1 4.7 4 3.7 3.4 3.3 8.3 2.6 2.8 7.3 2.2 2.9 2.4 3.8 5 2.3 3.3 10 4.9 2.7 2.9 3.3 4.6 4.3 4.7 2.5 1 2 2 1 4 1 3 2 2 1 4 2 2 2 2 1 2 2 1 2 2 2 2 3 1 2 2 3 2 1 4 2 2 2 2 3 2 2 4 2 2 2 2 3 1 3 2 9 8 4 1 9 6 7 4 11 6 14 6 7 9 7 13 3 8 11 9 7 4 9 8 7 14 8 9 6 7 10 4 9 10 7 10 6 9 21 8 8 7 14 6 9 16 8 53 57 263 173 121 44 224 254 93 80 268 64 195 145 387 112 59 184 128 59 232 214 43 37 31 102 118 54 79 63 165 86 141 136 60 363 67 41 104 101 86 253 38 191 154 233 35 11 8 8 12 8 13 12 12 8 11 10 12 10 10 8 8 8 8 8 8 12 8 11 8 11 8 8 8 8 11 10 10 8 8 11 10 12 8 10 10 9 9 9 8 10 8 9 10 | P a g e Fireplace Dummy (1= fireplace, 0= no fireplace) 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 0 Multiple Regression: Home Prices 11 S. No. Price ($1000) Sq Feet (000) Bathrooms Lot Size (000 sq ft) Median Income ($M) Age (Years) 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 232 522 235 377 196 286 432 285 392 510 77 281 329 420 250 438 213 429 364 332 315 679 345 149 345 484 350 239 3 6.3 2.1 4.4 3.2 3.5 4.2 2.8 2.9 5.6 2.9 2.7 4.9 2.5 3.5 4.8 2.8 4.1 4.7 2.5 3.2 9.1 5.2 2.7 5.3 7.9 4.5 4 2 3 2 3 2 2 3 2 2 3 2 1 3 2 2 3 2 3 2 2 2 4 3 2 2 2 3 2 8 13 9 6 12 6 4 10 7 8 2 9 7 7 9 9 3 3 7 6 6 13 6 12 9 13 7 6 69 35 74 188 115 43 184 222 92 475 55 84 35 98 194 214 35 469 188 226 177 608 336 71 66 668 139 289 9 10 10 8 9 8 10 9 8 10 12 8 8 10 12 10 8 10 8 8 8 10 8 10 8 10 8 8 11 | P a g e Fireplace Dummy (1= fireplace, 0= no fireplace) 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 1 California State University, College of Business Administration DS 101 Data analysis for Managers, Spring 2021 Project-2 (Extra Credit!) Due Date: May 20, 2021 Maximum Points: 50 Motivation: In the second project, you will run a multiple regression model where you will explain a phenomenon (one dependent variable) with a number of explanatory variables (at least four). I will provide you with free data or you can get your own data. Yu will run the model in JMP software and develop an analytical report where you provide interpretation of the regression output. Implementation: Each student will work on the project individually. It is the responsibility of the student to get the data (or use fee data), export data in a spreadsheet/JMP, run a multiple regression analysis and follow the instructions below to analyze and write a project report in not more than five-six pages double space or three pages single space (excluding tables, graphs, and maps). You will use 11-12 font size letters. The reports should be professionally prepared. You would submit a word-processed and grammatically error free document. Please proof read the document before submitting it. Failure to do so will mean points will be deducted. Attach data at the end pf the report. Points will not be deducted if you exceed the suggested page length. Regression Model: You will run the following multiple regression model: Y ? ? 0 ? ?1 X 1 ? ? 2 X 2 ? ... ? ? k X k ? ? Look at the free data folder and see if there is dataset that interests you. Pick a dataset and think if you can explain a problem/phenomenon utilizing one of the datasets. Import the data in JMP save the file and run the multiple regression (Analyze? Fit model? Insert dependent variable?explanatory variables? Run. Copy paste/print output in word and follow the steps below for interpreting the regression output. I. Cover Page ? Title of Report/Name II. Introduction [3 points] ? What motivates you to do this study? ? Include a few lines about the importance of this research. If doing housing market analysis than you may say California, /US economy went through a recession and is currently facing a pandemic. This has affected the real estate market and so this research examines the determinants of housing market. III. Literature Review [5 points] ? Review any two references 1|Page ? ? ? What is the theory that you are testing? What is the expected relationship between the dependent variable and independent variables Example: car prices decreases as mileage increases, or age of the car increases. You would expect used car price to decrease as mileage and age increases. Thus, there is a negative relation between explanatory variables and dependent variables. What does literature suggest? What are the findings of other studies? IV. Problem Statement [5 points] ? What is your question? ? Propose a hypothesis. What is it that you are testing? ? What are your null and alternate hypotheses statements? V. Methodology [7 points] ? How will you implement the project? What steps will you use to get the results? ? What are your data sources? ? What variables have you selected? Provide a definition of the variable (attach units, $, kms, pounds etc.) ? You will use Four explanatory variables and thirty time periods or data points such as cities, states, counties ? You will run a multiple regression model-? Must include dummy variables or lagged variables (either one of both or both from same category) ? Four explanatory variables ? You will run two models ? Simple Model and Modified ? Simple with 1 dependent and four independent variables ? Modified- one dependent and four independent with one of them as a dummy variable or lag variable VI. Analysis: Prepare two tables, summarize your results and provide INTERPRETATION [20 points] ? Report results regression output ? R square, standard error of estimate/RMSE ? Tests of Significance (5%) and coefficient interpretation of each variable and P value interpretation ? What is the relationship between the dependent variables and independent variables? ? Has the analysis contributed to a better understanding of the determinants of the phenomenon? ? Compute the confidence intervals for the response variable in relation to independent variables and provide interpretation 2|Page VII. Further Analysis [5 points] ? Examine and comment on any one the following that applies 1. Residual plot and outliers 2. Multicollinearity problem 3. Autocorrelation VIII. Conclusions [5 points] ? Report your findings succinctly in a few lines. References ? Provide a list of references used in writing this report.
