ECON 493 PS 6

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University of British Columbia *

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493

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Statistics

Date

Feb 20, 2024

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pdf

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Uploaded by pragnya030202

Answer 1 1. The regression analysis for attendance yields a coefficient of 0.0278363, with a t-statistic value of 4.01. The t-statistic exceeds the conventional significance cutoff of 1.96, indicating significant statistical relevance. The magnitude of this effect is considerable, translating to an increase of 2.78 percentage points in the standardized final exam score for each additional class attended. 2. The primary issue with the uncontrolled regression from part (1) when assessing the causal impact of attendance on exam scores is the potential for omitted variable bias. Factors like a student's GPA / personal ambition could also affect final exam outcomes. If these variables are not accounted for, it could skew the observed effect of attendance on exam scores due to selection bias. To mitigate this problem, it is necessary to adjust for all known variables that are thought to influence exam performance. 3. Adding termgpa to the regression: a) Adding termgpa as a control variable partially addresses the issue of omitted variable bias and advances the quest for establishing a causal relationship between class attendance and exam performance. Despite a significant change in the attendance coefficient upon controlling for termgpa, the consideration of additional variables is imperative for a more accurate analysis. b) The increased standard error in the attendance coefficient upon including termgpa indicates that this control variable is more predictive of the propensity to attend than of the final exam scores themselves. It is also critical to evaluate that termgpa may be influenced by attendance, suggesting its potential ineffectiveness as a control due to the possibility of introducing selection bias, thereby complicating the causal interpretation of the attendance effect. c) The stated attendance coefficient of -0.0351458 suggests that an increase in attendance by one unit is associated with an approximate decrease of 3.5 percentage points in standardized final exam scores when termgpa is held steady. Adding priGPA to the regression: a) Factoring in priGPA as a control variable mitigates some of the omitted variable bias, moving us toward isolating the causal impact of attending lectures. The substantial shift in the attendance coefficient upon controlling for priGPA indicates that other factors still need to be considered for a thorough analysis. The observation that the standard error for attendance increased suggests that priGPA more accurately forecasts attendance patterns than stndfnl. b) Probably not. While the standard error for the coefficient on attend increased, unlike termgpa, priGPA is not a consequence of attendance, which qualifies it as an effective control in this regression model. c) With an attendance coefficient of -0.0017479, the data suggest a minor decrease of about 0.17 percentage points in standardized final exam scores per unit

increase in attendance, with priGPA held constant. Yet, the insignificance of this relationship is highlighted by a low t-statistic of -0.23 and a high p-value of 0.815, casting doubt on the existence of any substantive link between attendance and exam performance. Adding ACT to the regression: a) By incorporating the ACT score as a control variable, we address some aspects of the omitted variable bias and edge closer to determining the causal effect of attending lectures on exam scores. However, recognizing that other variables also merit consideration is crucial for a robust model. The altered attendance coefficient after adjusting for the ACT score signifies that there's more to the equation. b) It's critical to consider that ACT scores may not be an effect of lecture attendance, which contrasts with earlier assumptions about termgpa. Including ACT as a control could introduce selection bias, thus obstructing a clear causal inference from the attendance coefficient. c) With the attendance coefficient at 0.0375987, there's an indicated average increase of 3.76 percentage points in final exam scores for each additional unit of lecture attendance, keeping ACT scores constant. This relationship implies that despite potential biases, there may be a positive association between lecture attendance and exam performance. 4. Upon reviewing the data in part (3), priGPA, or the cumulative GPA before the current term, emerges as the preferred control variable for exploring the causal influence of lecture attendance on exam scores. The selection of priGPA is strategic because it is not influenced by current lecture attendance, unlike termgpa and ACT scores, which may both be affected by the treatment variable, attendance. Using termgpa or ACT as controls could potentially lead to selection bias, as they might be consequences of attending lectures. By contrast, priGPA is established prior to and therefore independent of the current term's lecture attendance, rendering it a stable and reliable control for regression analysis aimed at identifying causal relationships. The attendance coefficient, marked at -0.0017479 with a t-statistic of -0.23 and a p-value of 0.815, fails to meet the criteria for statistical significance. The t-statistic's absolute value is below the critical value of 1.96, and the p-value exceeds the threshold of 0.05, leading to the conclusion that the attendance coefficient is statistically inconsequential. Consequently, the null hypothesis, which states there is no difference from zero for the attendance coefficient, cannot be rejected based on the t-statistic, p-value, and the 95% confidence interval provided. Thus, it is not possible to assert that class attendance has a tangible impact on the standardized final exam scores. 5. Adding hwrte to the regression: In this regression analysis, the completing homework variable is considered as a potential factor that could unveil the causal relationship between lecture attendance and student performance. The coefficient for completing homework stands at 0.0031609 with a t-statistic of 1.17, which is below the significance threshold of 1.96, indicating its

statistical insignificance. Furthermore, the coefficient for attendance shows a reduction from -0.0017479 to -0.0008707 upon including homework in the model. With a corresponding t-statistic of -0.93 and a p-value of 0.353 for attendance, these values fall short of the significance benchmarks. As such, both the t-statistic and the p-value do not support rejecting the null hypothesis that the attendance coefficient differs from zero. Therefore, it can be concluded that there is no statistically significant evidence to suggest that lecture attendance influences standardized final exam scores, even after accounting for the variable of completing homework. 6. Adding skipped to the regression in (4): The variable 'skipped' is excluded from the regression due to collinearity, which arises when two or more predictor variables in the model are highly interrelated. This interdependence complicates the model's ability to isolate the specific impact of each predictor. The rationale for the removal of 'skipped' is that it is inherently related to 'attend'—not attending class inherently means the class was skipped. Therefore, 'attend' and 'skipped' are so closely linked that they do not provide distinct information for the purposes of regression analysis, thus justifying the omission of 'skipped' to avoid redundancy and to clarify the model's predictive capability. Answer 2 1. The coefficients for 'black' and 'Hispanic' in the regression model quantify the average height differences between black or Hispanic individuals and those who are not—typically white or other races in the sample. In this analysis, the coefficient for 'black' is -0.20 , suggesting that black males are, on average, 0.20 inches shorter than their non-black counterparts. Conversely, the coefficient of 0.04 for black females, implying they are, on average, 0.04 inches taller than non-black females. A similar approach is taken when interpreting the 'Hispanic' coefficient, which compares the average height of Hispanic individuals to non-Hispanic ones within the specified gender groups. 2. To test if there's a statistically significant difference in the Hispanic variable's coefficient between males and females, an interaction term between gender (coded as 0 for males and 1 for females) and the Hispanic variable can be introduced in the regression model. The null hypothesis (H0) posited would assert that the Hispanic coefficient is identical for both genders. In contrast, the alternative hypothesis (H1) would contend that the Hispanic coefficient varies between males and females. The regression model would include these interaction terms to observe any significant differences in the coefficient, allowing for a nuanced analysis of the variable's impact by gender. Heighti = α + β1Blacki + β2Hispanici + γASVAB + β4Dummyi + β5Hispanici*Dummyi

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3. Running separate regressions for males and females gives distinct coefficients for the constant term, the 'Hispanic' variable, and 'cognitive ability'. From the male-specific regression we have an intercept α(constant) of 68.24, a 'Hispanic' coefficient (β1) of -2.20, and a 'cognitive ability' coefficient (γ) of 0.046. The female-specific regression provides an intercept of 62.27, a 'Hispanic' coefficient of -1.76, and a 'cognitive ability' coefficient of 0.038. The coefficient for 'white' (β2) is not provided for either regression, which implies that it is either omitted or subsumed within the constant term as the reference category against which the 'Hispanic' variable's effect is measured. Do-file: . use "/Users/pragnyasanghvi/Downloads/attend.dta" . regress stndfnl attend, robust Linear regression Number of obs = 674 F(1, 672) = 16.08 Prob > F = 0.0001 R-squared = 0.0219 Root MSE = .98242 ------------------------------------------------------------------------------ | Robust stndfnl | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- attend | .0278363 .0069417 4.01 0.000 .0142063 .0414662 _cons | -.702364 .1816686 -3.87 0.000 -1.05907 -.3456577 ------------------------------------------------------------------------------ . regress stndfnl attend termgpa, robust Linear regression Number of obs = 674 F(2, 671) = 112.49 Prob > F = 0.0000 R-squared = 0.2968 Root MSE = .83362 ------------------------------------------------------------------------------ | Robust stndfnl | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- attend | -.0351458 .0075528 -4.65 0.000 -.0499757 -.0203159 termgpa | .8518788 .0593675 14.35 0.000 .7353104 .9684471

_cons | -1.273935 .1709605 -7.45 0.000 -1.609617 -.9382536 ------------------------------------------------------------------------------ . regress stndfnl attend priGPA, robust Linear regression Number of obs = 674 F(2, 671) = 48.04 Prob > F = 0.0000 R-squared = 0.1377 Root MSE = .92311 ------------------------------------------------------------------------------ | Robust stndfnl | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- attend | -.0017479 .0074481 -0.23 0.815 -.0163722 .0128764 priGPA | .6843534 .0775414 8.83 0.000 .5321006 .8366063 _cons | -1.698644 .2101562 -8.08 0.000 -2.111287 -1.286001 ------------------------------------------------------------------------------ . regress stndfnl attend ACT, robust Linear regression Number of obs = 674 F(2, 671) = 68.73 Prob > F = 0.0000 R-squared = 0.1702 Root MSE = .90557 ------------------------------------------------------------------------------ | Robust stndfnl | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- attend | .0375987 .0064706 5.81 0.000 .0248938 .0503037 ACT | .1107491 .0101418 10.92 0.000 .0908356 .1306626 _cons | -3.448301 .3042148 -11.34 0.000 -4.045628 -2.850973 ------------------------------------------------------------------------------ . regress stndfnl attend priGPA, robust Linear regression Number of obs = 674 F(2, 671) = 48.04 Prob > F = 0.0000 R-squared = 0.1377 Root MSE = .92311

------------------------------------------------------------------------------ | Robust stndfnl | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- attend | -.0017479 .0074481 -0.23 0.815 -.0163722 .0128764 priGPA | .6843534 .0775414 8.83 0.000 .5321006 .8366063 _cons | -1.698644 .2101562 -8.08 0.000 -2.111287 -1.286001 ------------------------------------------------------------------------------ . regress stndfnl attend priGPA hwrte, robust Linear regression Number of obs = 674 F(3, 670) = 33.20 Prob > F = 0.0000 R-squared = 0.1400 Root MSE = .92257 ------------------------------------------------------------------------------ | Robust stndfnl | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- attend | -.008707 .0093685 -0.93 0.353 -.0271021 .0096881 priGPA | .6782941 .0781394 8.68 0.000 .5248666 .8317216 hwrte | .0031609 .002702 1.17 0.242 -.0021446 .0084663 _cons | -1.777928 .2193038 -8.11 0.000 -2.208533 -1.347322 ------------------------------------------------------------------------------ . regress stndfnl attend priGPA skipped, robust note: skipped omitted because of collinearity. Linear regression Number of obs = 674 F(2, 671) = 48.04 Prob > F = 0.0000 R-squared = 0.1377 Root MSE = .92311 ------------------------------------------------------------------------------ | Robust stndfnl | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- attend | -.0017479 .0074481 -0.23 0.815 -.0163722 .0128764 priGPA | .6843534 .0775414 8.83 0.000 .5321006 .8366063 skipped | 0 (omitted)

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_cons | -1.698644 .2101562 -8.08 0.000 -2.111287 -1.286001 ------------------------------------------------------------------------------ .

ECON 493 PS 6

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