Model Design
We used a Quantile Regression model to explore the relationship between the dependent variable 'rate_log' and independent variables represented by principal components (pca0 through pca44). Quantile Regression allows us to examine the impact of these independent variables at different quantiles of the dependent variable's distribution, providing a more comprehensive understanding of the relationships.
Methodology
We employed Quantile Regression using the Least Squares method on a dataset with 586,095 observations. The model's performance was evaluated using the pseudo R-squared value.
In conclusion, this Quantile Regression model demonstrates various relationships between the rate_log and the principal components used as independent variables. Many of these principal components have a statistically significant impact on rate_log, while a few do not show a significant relationship.
Results Summary
Pseudo R-squared: Our model explains approximately 16.76% of the variability in the dependent variable 'rate_log' as indicated by the pseudo R-squared value of 0.1676.
Coefficients: The coefficients (coef) represent the change in rate_log for a one-unit increase in the corresponding principal component while holding all other principal components constant. The model includes 45 principal components as independent variables. Some coefficients have a positive relationship with rate_log, while others have a negative relationship. This suggests that different principal components have varying effects on the dependent variable, rate_log.
Statistical Significance: Statistical significance is assessed using the p-values (P>|t|) for each principal component. Most of the principal components show a statistically significant relationship with the rate_log, as their p-values are less than 0.05. However, there are a few exceptions: pca7 (p=0.215) and pca24 (p=0.732), which do not exhibit a statistically significant relationship with the dependent variable, rate_log.
Coefficients Table
QuantReg Regression Results
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Dep. Variable: rate_log Pseudo R-squared: 0.1676
Model: QuantReg Bandwidth: 0.06131
Method: Least Squares Sparsity: 3.650
Date: Wed, 19 Apr 2023 No. Observations: 586095
Time: 15:42:07 Df Residuals: 586049
Df Model: 45
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coef std err t P>|t| [0.025 0.975]
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const 7.0413 0.002 3410.821 0.000 7.037 7.045
pca0 0.0254 0.001 39.592 0.000 0.024 0.027
pca1 -0.0122 0.001 -13.963 0.000 -0.014 -0.011
pca2 -0.1010 0.001 -104.836 0.000 -0.103 -0.099
pca3 0.0379 0.001 32.552 0.000 0.036 0.040
pca4 0.0459 0.001 36.850 0.000 0.043 0.048
pca5 -0.0040 0.001 -2.995 0.003 -0.007 -0.001
pca6 0.0323 0.001 24.266 0.000 0.030 0.035
pca7 -0.0017 0.001 -1.241 0.215 -0.004 0.001
pca8 0.1054 0.001 70.355 0.000 0.102 0.108
pca9 -0.0107 0.001 -7.426 0.000 -0.014 -0.008
pca10 0.0657 0.002 42.974 0.000 0.063 0.069
pca11 -0.0995 0.002 -64.663 0.000 -0.103 -0.096
pca12 0.0780 0.002 49.256 0.000 0.075 0.081
pca13 -0.0730 0.002 -44.422 0.000 -0.076 -0.070
pca14 -0.0976 0.002 -57.408 0.000 -0.101 -0.094
pca15 0.0418 0.002 23.280 0.000 0.038 0.045
pca16 -0.0808 0.002 -46.741 0.000 -0.084 -0.077
pca17 -0.0612 0.002 -34.441 0.000 -0.065 -0.058
pca18 0.0738 0.002 42.169 0.000 0.070 0.077
pca19 -0.1477 0.002 -84.887 0.000 -0.151 -0.144
pca20 0.0287 0.002 16.043 0.000 0.025 0.032
pca21 0.0910 0.002 52.688 0.000 0.088 0.094
pca22 -0.0616 0.002 -34.849 0.000 -0.065 -0.058
pca23 -0.0663 0.002 -35.591 0.000 -0.070 -0.063
pca24 -0.0006 0.002 -0.342 0.732 -0.004 0.003
pca25 -0.1712 0.002 -88.744 0.000 -0.175 -0.167
pca26 0.0786 0.002 43.153 0.000 0.075 0.082
pca27 -0.0537 0.002 -29.857 0.000 -0.057 -0.050
pca28 0.1190 0.002 61.396 0.000 0.115 0.123
pca29 0.0460 0.002 23.522 0.000 0.042 0.050
pca30 0.1588 0.002 86.822 0.000 0.155 0.162
pca31 0.1207 0.002 62.591 0.000 0.117 0.125
pca32 0.2390 0.002 123.489 0.000 0.235 0.243
pca33 0.0216 0.002 11.081 0.000 0.018 0.025
pca34 0.1053 0.002 54.100 0.000 0.101 0.109
pca35 0.2268 0.002 120.584 0.000 0.223 0.230
pca36 -0.1766 0.002 -92.504 0.000 -0.180 -0.173
pca37 0.0791 0.002 40.369 0.000 0.075 0.083
pca38 -0.1473 0.002 -73.149 0.000 -0.151 -0.143
pca39 -0.0901 0.002 -46.953 0.000 -0.094 -0.086
pca40 -0.2922 0.002 -150.323 0.000 -0.296 -0.288
pca41 -0.0763 0.002 -41.556 0.000 -0.080 -0.073
pca42 -0.2067 0.002 -102.645 0.000 -0.211 -0.203
pca43 -0.0766 0.002 -37.745 0.000 -0.081 -0.073
pca44 -0.1364 0.002 -73.916 0.000 -0.140 -0.133
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