The Strength of the Relationship Between Two Variables - Essay Example

Resource Environmental Economics Exercise Please enter and email address here: Answer the following questions Before you run the regression, what are your expectations regarding the signs of the coefficients for the following variables Also explain briefly why. (a) Expected sign of the coefficient for the CHARLES dummy variable is positive. Unless Charles River is extremely polluted, potential buyers are likely to perceive the house near the river more favourably than one that does not bound the river. Preference for river-side houses will cause readiness to pay price premium. Therefore, houses that bound the river should be, on average, priced higher than those that do not and the CHARLES variable coefficient should be positive. (b) I expect pupil-teacher ratio to have little influence on the housing prices. The PTRATIO coefficient would probably have the negative sign as low pupil-teacher ratio may be perceived as an indicator of better education in a particular town. Nevertheless, the opposite situation is also possible. (c) NOX variable reflects the pollution level in the area. The 'dirtier' in terms of pollution a town is, the less would the demand for housing in it be. Therefore, I expect nitric oxides concentration variable to be negatively correlated with price of houses and the corresponding coefficient to have negative sign. Now calculate the summary (descriptive) statistics and paste below. PRICE CRIME INDUS NOX ROOMS Mean 22.533 3.614 11.137 0.5547 6.2846 Std. Dev. 9.197 8.602 6.860 0.1159 0.7026 Median 21.200 0.254 9.690 0.5380 6.2080 Minimum 5.000 0.006 0.460 0.3850 3.5610 Maximum 50.000 88.976 27.740 0.8710 8.7800 Range 45.000 88.970 27.280 0.4860 5.2190 AGE DIS RAD TAX PTRATIO LSTAT Mean 68.57 3.795 9.549 408.24 18.456 12.653 Std. Dev. 28.15 2.106 8.707 168.54 2.165 7.141 Median 77.30 3.199 5.000 330.00 19.000 11.340 Minimum 2.90 1.130 1.000 187.00 12.600 1.730 Maximum 100.00 12.127 24.000 711.00 22.000 37.970 Range 97.10 10.997 23.000 524.00 9.400 36.240 For the CHARLES variable which takes only two possible values of 0 and 1 calculation and interpretation of mean and standard deviation would be misleading. Count and frequency describe this particular variable better. Out of 506 entries, 471 (93.08% of total) correspond to 0-value of CHARLES variable and houses that do not bound the river. CHARLES is equal to 1 for 35 entries (6.92% of total). (2) (a) Calculate the correlation between the independent variables and copy the results below. For correlations table see next page. Correlations Table: CRIME INDUS CHARLES NOX ROOMS AGE DIS RAD TAX PTRATIO INDUS 0.407 1.000 CHARLES -0.056 0.063 1.000 NOX 0.421 0.764 0.091 1.000 ROOMS -0.219 -0.392 0.091 -0.302 1.000 AGE 0.353 0.645 0.087 0.731 -0.240 1.000 DIS -0.380 -0.708 -0.099 -0.769 0.205 -0.748 1.000 RAD 0.626 0.595 -0.007 0.611 -0.210 0.456 -0.495 1.000 TAX 0.583 0.721 -0.036 0.668 -0.292 0.506 -0.534 0.910 1.000 PTRATIO 0.290 0.383 -0.122 0.189 -0.356 0.262 -0.232 0.465 0.461 1.000 LSTAT 0.456 0.604 -0.054 0.591 -0.614 0.602 -0.497 0.489 0.544 0.374 (b) Comment briefly on whether the calculated correlation of LSTAT with each of these variables makes logical sense or not, and why: CRIME, INDUS, CHARLES, NOX, ROOMS, and AGE. LSTAT variable demonstrates absence of correlation with CHARLES. Position of houses near a river and percentage of the lower status population in the area do not depend on each other. It is quite logical because the former is determined by the geographical factors while the latter is more of social nature. LSTAT is moderately correlated with PTRATIO, CRIME, RAD (positive correlation), and DIS (negative correlation). Based on the correlation coefficient it cannot be determined which variable influences which. For example, we can assume that high percentage of lower status of the population leads to increase in crime rates because of high poverty levels. However, it can also be argued that the relationship between variables goes in the opposite direction and higher status population tries to avoid housing in the areas with high crime rates. The regions with high percentage of low status population also tend to be more accessible to radial highways, located further from the main employment centres and have, on average, more pupils per one teacher. LSTAT demonstrates moderate to moderately strong correlation with TAX, AGE, NOX, INDUS (positive correlation), and ROOMS (negative correlation). The higher is the percentage of the low status population in the area, the higher are property taxes, higher ration of old buildings, less average number of rooms in the house, higher industrialization of the area and concentration of nitric oxides in the air. Two latter variables are intuitively perceived to be interdependent and the strong correlation between them (0.764) supports this hypothesis. (3) (a) Perform the regression and paste your results below. Summary Multiple R R-Square Adjusted R-Square Standard Error Of Estimate Durbin Watson 0.8533 0.7281 0.7221 4.848516072 1.0482 ANOVA Table Degrees of Freedom Sum of Squares Mean of Squares F-Ratio p-Value Explained 11 31103.29001 2827.571819 120.2807 < 0.0001 Unexplained 494 11613.0054 23.5081081 Coefficient Standard Error t-Value p-Value Lower Limit Upper Limit Constant 42.084 4.986 8.4404 < 0.0001 32.287 51.880 CRIME -0.112 0.033 -3.3633 0.0008 -0.177 -0.046 INDUS -0.009 0.062 -0.1440 0.8856 -0.132 0.114 CHARLES 2.885 0.879 3.2822 0.0011 1.158 4.612 NOX -19.237 3.889 -4.9462 < 0.0001 -26.878 -11.595 ROOMS 3.879 0.420 9.2471 < 0.0001 3.055 4.703 AGE -0.002 0.013 -0.1354 0.8924 -0.028 0.024 DIS -1.216 0.187 -6.5205 < 0.0001 -1.583 -0.850 RAD 0.265 0.067 3.9449 < 0.0001 0.133 0.397 TAX -0.010 0.004 -2.6211 0.0090 -0.017 -0.002 PTRATIO -1.077 0.127 -8.4794 < 0.0001 -1.326 -0.827 LSTAT -0.543 0.051 -10.6236 < 0.0001 -0.644 -0.443 Therefore, at 5% significance level the regression equation is as follows: PRICE = 42.084 - 0.112*CRIME + 2.885*CHARLES - 19.237*NOX + 3.879*ROOMS - 1.216*DIS + 0.265*RAD - 0.010*TAX - 1.077*PTRATIO - 0.543*LSTAT. (b) What is the R-squared for the model, and what does it imply R-squared is equal to 0.7281. It indicates that the used model explains 72.81% of the variation in housing prices. 27.19% of price variation is not determined by nitric oxides concentration, crime rate, property-tax rates, or any other independent variables used in the model. (4) (a) Comment on the coefficient signs, magnitude and statistical significance for the following variables: INDUS, CHARLES, NOX and LSTAT. INDUS coefficient is equal to -0.009 and, on average, additional 10% of non-retail business per town would decrease the price of the house by $90. However, the standard error is relatively very high and the range is from -0.132 up to 0.114. It means that while some houses cost less if proportion of non-retail business per acre of town is higher, for other the opposite is true. Extremely high p-value for INDUS variable (0.8856) indicates statistical insignificance of the variable for this model. Therefore, average price of the housing does not depend on proportion of non-retail business and INDUS variable is not included into the final regression equation. CHARLES coefficient equals to 2.885, lower and upper limits are correspondingly 1.158 and 4.612. If a house bounds the river, it adds additional $2,885 to its price. The variable is statistically significant (p-value of 0.0011). NOX is the most influential factor as demonstrated by its coefficient equal to -19.237. The variable is statistically significant (p-value Read More