Elevation, Aspect, Latitude, Land Use. y = β 0 + β 1X 1 + β 2X 2 + ε

Transcription

1 Lab 4 Graham Emde The purpose of this lab is to determine which factors have a relationship with average temperatures along the Front Range in The study analyzes average temperatures collected from 23 weather stations recorded temperatures in 1970 against elevation, aspect, latitude, and land use. The study runs two ordinary least squares (OLS) regressions, the first testing elevation, aspect, latitude, and land use against average temperature, and the second testing only elevation and land use against average temperature. The OLS tests show that elevation and land use are the only two variables that have a relationship with average temperature. Finally, the study includes a geographic weighted regression (GWR) to determine the relationships of elevation and land use to average temperature at each specific location. Although the Koenker statistic for elevation and land use shows that the relationships have stationarity, this study goes above and beyond by including a GWR of elevation and land use against average temperature as well. OLS Test #1 Results Elevation, Aspect, Latitude, Land Use y = β 0 + β 1X 1 + β 2X 2 + ε Average temperature = (Elevation) (Latitude) (Aspect) (Latitude) According to the summary table, elevation and land use each have a statistically significant relationship with average temperature for the time period. In the summary table, elevation and land use show p-values that are lower than the 0.05 level at which they were tested. The diagnostics table shows that the independent variables have a combined R-squared of 0.62, and the joint F- 1

2 statistic and Wald statistic were significant. The dependent variables did not pass the Koenker or the Jarque-Bera tests, which indicate that the results have a skewed distribution and do not show nonstationarity. Because the results failed these two tests, a GWR should not be considered. For this lab, we will continue the tests. I ran the Moran s I test on the residuals of the OLS to determine if there was spatial autocorrelation in the residuals. With a z-score of -0.74, the residuals seem to show randomness. OLS Test #2 OLS Results Elevation, Land Use 2

3 y = β 0 + β 1X 1 + β 2X 2 + ε Average temperature = (Elevation) (Latitude) According to second OLS test, elevation and land use still have statistically significant relationships with average temperature for the time period. Elevation and land use show p-values that are lower than the 0.05 level at which they were tested. The diagnostics table shows that the independent variables have a combined R-squared of 0.61, and the joint F-statistic and Wald statistic were significant. Like in the first OLS test, the dependent variables did not pass the Koenker or the Jarque-Bera tests, which indicate that the results have a skewed distribution and do not show nonstationarity. Because the results failed these two tests, a GWR should not be considered. For this lab, we will continue the tests. I ran the Moran s I test on the residuals of the second OLS test to determine if there was spatial autocorrelation in the residuals. With a z-score of -0.70, the residuals seem to show randomness. 3

4 GWR Test GWR Results Elevation, Land Use The R-squared value shows a significant relationship between the independent variables (elevation and land use) and the dependent variable (average temperature). Although these tests were successful, there are a number of reasons why the results should not be considered reliable. First, it is best not run regressions on data with less than 31 points. Our dataset had 23 unique points (the 23 weather stations), so our tests are not trustworthy. Secondly, before running a GWR, it is important that the preliminary OLS results pass each of the five tests (R-squared, F-statistic, Weld statistic, Koenker statistic, and Jarque-Bera statistic). For my first OLS, the results did not pass the Koenker or the Jarque-Bera tests. For this reason the second OLS test results cannot be trusted. For my second OLS, the results again failed the Koenker and Jarque-Bera tests. This shows that the data has stationarity, which means that a GWR is not necessary. OLS tests show global relationships, while GWR tests show local relationships. Since our OLS results failed the Koenker test for nonstationarity, we can determine that our results show global relationships and not local relationships. For this reason, running a GWR test is not advisable. Despite this, this study includes a GWR test, the results of which seem to show a relationship between the dependent variable and the explanatory variables. Because of the previously mentioned reasons for not trusting the results, the results of the GWR should also not be trusted. 4

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