R E G R E S S I O N D I S C O N T I N U I T Y I PMAP 8521: Program Evaluation for Public Service November 4, 2019 Fill out your reading report on iCollege!
P L A N F O R T O D A Y Jumps and cutoffs Measuring the size of the discontinuity Main RDD concerns RDD with R
J U M P S & C U TO F F S
Think of five social programs that use eligibility cutoffs to determine who can access the program How is eligibility measured? What’s the cutoff? Federal/state/local governments; school districts; nonprofits; etc.
K E Y T E R M S Running/forcing variable Index or measure that determines eligibility Cutoff/cutpoint/threshold Number that formally assigns access to program Outcome The thing you want to see the causal effect on
M A I N I N T U I T I O N People right before and right after the cutoff are essentially the same This mimics the idea of treatment and control groups
Treatment group Control group Causal effect Cutoff
M E A S U R I N G T H E S I Z E O F T H E D I S C O N T I N U I T Y
The size of the discontinuity depends on how you draw the trend lines on each side of the cutoff There’s no one right way to draw lines! Parametric Nonparametric Bandwidths Kernels
<latexit sha1_base64="5SUf5zqAKKliBbeqXXuPfFCcMOY=">AAAB8XicbVBNSwMxEJ31s9avqkcvwSIIQtmtgl6EohePFewHtkvJptk2NMkuSVZclv4LLx4U8eq/8ea/MW33oK0PBh7vzTAzL4g508Z1v52l5ZXVtfXCRnFza3tnt7S339RRoghtkIhHqh1gTTmTtGGY4bQdK4pFwGkrGN1M/NYjVZpF8t6kMfUFHkgWMoKNlR5SdIXEEzpFQa9UdivuFGiReDkpQ456r/TV7UckEVQawrHWHc+NjZ9hZRjhdFzsJprGmIzwgHYslVhQ7WfTi8fo2Cp9FEbKljRoqv6eyLDQOhWB7RTYDPW8NxH/8zqJCS/9jMk4MVSS2aIw4chEaPI+6jNFieGpJZgoZm9FZIgVJsaGVLQhePMvL5JmteKdVap35+XadR5HAQ7hCE7AgwuowS3UoQEEJDzDK7w52nlx3p2PWeuSk88cwB84nz+RYY+K</latexit> <latexit sha1_base64="8eZ/az91uVfG6S71lzMSeHDHUwM=">ACEHicbVDLSsNAFL2pr1pfUZduBosoCWJgm6EohuXFewD2hAm02k7dPJgZiKG0E9w46+4caGIW5fu/BunbQRtPXDh3HPuZeYeP+ZMKsv6MgoLi0vLK8XV0tr6xuaWub3TkFEiCK2TiEei5WNJOQtpXTHFaSsWFAc+p01/eDX2m3dUSBaFtyqNqRvgfsh6jGClJc8TNEF6vhUYc9Cxzmz0b2un87RneOZatiTYDmiZ2TMuSoeZnpxuRJKChIhxL2batWLkZFoRTkelTiJpjMkQ92lb0xAHVLrZ5KAROtBKF/UioStUaKL+3shwIGUa+HoywGogZ72x+J/XTlTv3M1YGCeKhmT6UC/hSEVonA7qMkGJ4qkmAim/4rIAtMlM6wpEOwZ0+eJw2nYp9UnJvTcvUyj6MIe7APR2DGVThGmpQBwIP8AQv8Go8Gs/Gm/E+HS0Y+c4u/IHx8Q3JMZnq</latexit> P A R A M E T R I C L I N E S Formulas with parameters y = β 0 + β 1 x 1 + β 2 x 2 y = mx + b
<latexit sha1_base64="UcRWUsyhJyM/Kyi/JWdQzvRY5c0=">ACAnicbZDLSsNAFIYnXmu9RV2Jm8EiCEJqAboejGZQV7gTaEyfSkHTqZhJmJGEpx46u4caGIW5/CnW/jtM1CW38Y+PjPOZw5f5BwprTjfFsLi0vLK6uFteL6xubWtr2z21BxKinUacxj2QqIAs4E1DXTHFqJBIFHJrB4Hpcb96DVCwWdzpLwItIT7CQUaKN5dv7Gb7EnQA08R18kpOLH3zXt0tO2ZkIz4ObQwnlqvn2V6cb0zQCoSknSrVdJ9HekEjNKIdRsZMqSAgdkB60DQoSgfKGkxNG+Mg4XRzG0jyh8cT9PTEkVJZFJjOiOi+mq2Nzf9q7VSHF96QiSTVIOh0UZhyrGM8zgN3mQSqeWaAUMnMXzHtE0moNqkVTQju7Mnz0KiU3dNy5fasVL3K4yigA3SIjpGLzlEV3aAaqiOKHtEzekVv1pP1Yr1bH9PWBSuf2UN/ZH3+AP/slUE=</latexit> y = β 0 + β 1 x 1
<latexit sha1_base64="5SUf5zqAKKliBbeqXXuPfFCcMOY=">AAAB8XicbVBNSwMxEJ31s9avqkcvwSIIQtmtgl6EohePFewHtkvJptk2NMkuSVZclv4LLx4U8eq/8ea/MW33oK0PBh7vzTAzL4g508Z1v52l5ZXVtfXCRnFza3tnt7S339RRoghtkIhHqh1gTTmTtGGY4bQdK4pFwGkrGN1M/NYjVZpF8t6kMfUFHkgWMoKNlR5SdIXEEzpFQa9UdivuFGiReDkpQ456r/TV7UckEVQawrHWHc+NjZ9hZRjhdFzsJprGmIzwgHYslVhQ7WfTi8fo2Cp9FEbKljRoqv6eyLDQOhWB7RTYDPW8NxH/8zqJCS/9jMk4MVSS2aIw4chEaPI+6jNFieGpJZgoZm9FZIgVJsaGVLQhePMvL5JmteKdVap35+XadR5HAQ7hCE7AgwuowS3UoQEEJDzDK7w52nlx3p2PWeuSk88cwB84nz+RYY+K</latexit> <latexit sha1_base64="8eZ/az91uVfG6S71lzMSeHDHUwM=">ACEHicbVDLSsNAFL2pr1pfUZduBosoCWJgm6EohuXFewD2hAm02k7dPJgZiKG0E9w46+4caGIW5fu/BunbQRtPXDh3HPuZeYeP+ZMKsv6MgoLi0vLK8XV0tr6xuaWub3TkFEiCK2TiEei5WNJOQtpXTHFaSsWFAc+p01/eDX2m3dUSBaFtyqNqRvgfsh6jGClJc8TNEF6vhUYc9Cxzmz0b2un87RneOZatiTYDmiZ2TMuSoeZnpxuRJKChIhxL2batWLkZFoRTkelTiJpjMkQ92lb0xAHVLrZ5KAROtBKF/UioStUaKL+3shwIGUa+HoywGogZ72x+J/XTlTv3M1YGCeKhmT6UC/hSEVonA7qMkGJ4qkmAim/4rIAtMlM6wpEOwZ0+eJw2nYp9UnJvTcvUyj6MIe7APR2DGVThGmpQBwIP8AQv8Go8Gs/Gm/E+HS0Y+c4u/IHx8Q3JMZnq</latexit> <latexit sha1_base64="06f6zKjKig8XkZ2WeVJGs7FS9z8=">ACEnicbVDLSsNAFJ34rPUVdelmsAiKUJIo6EYounFZwT6gjWEynbRDJ5MwMxFD6De48VfcuFDErSt3/o2TNoK2Hrhw7jn3MnOPHzMqlWV9GXPzC4tLy6WV8ura+samubXdlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/vMz91h0Rkb8RqUxcUPU5zSgGCkteZhCs9h1ycKeRY8KpgN73X9dE7e3TqeWbGq1hwltgFqYACdc/87PYinISEK8yQlB3bipWbIaEoZmRU7iaSxAgPUZ90NOUoJNLNxieN4L5WejCIhC6u4Fj9vZGhUMo09PVkiNRATnu5+J/XSVRw5maUx4kiHE8eChIGVQTzfGCPCoIVSzVBWFD9V4gHSCsdIplHYI9fIsaTpV+7jqXJ9UahdFHCWwC/bAbDBKaiBK1AHDYDBA3gCL+DVeDSejTfjfTI6ZxQ7O+APjI9vBHGajQ=</latexit> P A R A M E T R I C L I N E S Formulas with parameters y = β 0 + β 1 x 1 + β 2 x 2 y = mx + b Not just for straight lines! Make curvy with exponents y = β 0 + β 1 x 1 + β 2 x 2 1
<latexit sha1_base64="UcRWUsyhJyM/Kyi/JWdQzvRY5c0=">ACAnicbZDLSsNAFIYnXmu9RV2Jm8EiCEJqAboejGZQV7gTaEyfSkHTqZhJmJGEpx46u4caGIW5/CnW/jtM1CW38Y+PjPOZw5f5BwprTjfFsLi0vLK6uFteL6xubWtr2z21BxKinUacxj2QqIAs4E1DXTHFqJBIFHJrB4Hpcb96DVCwWdzpLwItIT7CQUaKN5dv7Gb7EnQA08R18kpOLH3zXt0tO2ZkIz4ObQwnlqvn2V6cb0zQCoSknSrVdJ9HekEjNKIdRsZMqSAgdkB60DQoSgfKGkxNG+Mg4XRzG0jyh8cT9PTEkVJZFJjOiOi+mq2Nzf9q7VSHF96QiSTVIOh0UZhyrGM8zgN3mQSqeWaAUMnMXzHtE0moNqkVTQju7Mnz0KiU3dNy5fasVL3K4yigA3SIjpGLzlEV3aAaqiOKHtEzekVv1pP1Yr1bH9PWBSuf2UN/ZH3+AP/slUE=</latexit> y = β 0 + β 1 x 1
<latexit sha1_base64="06f6zKjKig8XkZ2WeVJGs7FS9z8=">ACEnicbVDLSsNAFJ34rPUVdelmsAiKUJIo6EYounFZwT6gjWEynbRDJ5MwMxFD6De48VfcuFDErSt3/o2TNoK2Hrhw7jn3MnOPHzMqlWV9GXPzC4tLy6WV8ura+samubXdlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/vMz91h0Rkb8RqUxcUPU5zSgGCkteZhCs9h1ycKeRY8KpgN73X9dE7e3TqeWbGq1hwltgFqYACdc/87PYinISEK8yQlB3bipWbIaEoZmRU7iaSxAgPUZ90NOUoJNLNxieN4L5WejCIhC6u4Fj9vZGhUMo09PVkiNRATnu5+J/XSVRw5maUx4kiHE8eChIGVQTzfGCPCoIVSzVBWFD9V4gHSCsdIplHYI9fIsaTpV+7jqXJ9UahdFHCWwC/bAbDBKaiBK1AHDYDBA3gCL+DVeDSejTfjfTI6ZxQ7O+APjI9vBHGajQ=</latexit> y = β 0 + β 1 x 1 + β 2 x 2 1
<latexit sha1_base64="ndiwNQlURvYkUS4lA/F3iS+7x9Q=">ACInicbVDLSsNAFJ34rPUVdelmsAiCUJUBdC0Y3LCvYBbQyT6aQdOnkwMxFD6Le48VfcuFDUleDHOEmjaOuBC+ecey8z97gRo0Iaxoc2N7+wuLRcWimvrq1vbOpb2y0RxhyTJg5ZyDsuEoTRgDQlYx0Ik6Q7zLSdkcXWb9S7igYXAtk4jYPhoE1KMYSWU5+mkCz2DPJRI5BjwsmAnvVH0rK1M31o+u5brm6BWjauSAs8QsSAUaDj6W68f4tgngcQMCdE1jUjaKeKSYkbG5V4sSITwCA1IV9EA+UTYaX7iGO4rpw+9kKsKJMzd3xsp8oVIfFdN+kgOxXQvM/rdWPpndgpDaJYkgBPHvJiBmUIs7xgn3KCJUsUQZhT9VeIh4gjLFWqZRWCOX3yLGlZVbNWta6OKvXzIo4S2AV74ACY4BjUwSVogCbA4B48gmfwoj1oT9qr9j4ZndOKnR3wB9rnF1i1n9s=</latexit> y = β 0 + β 1 x 1 + β 2 x 2 1 + β 3 x 3 1
<latexit sha1_base64="9Lt8zBRTz0oJIOCIkOazb1/Iz94=">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</latexit> M E A S U R I N G T H E G A P y = β 0 + β 1 Running variable (centered) + β 2 Indicator for treatment running_var_ ID outcome running_var treatment centered 1 90.0 64 -6 FALSE 2 85.7 70 0 TRUE 3 85.8 73 3 TRUE 4 85.7 60 -10 FALSE 5 84.4 71 1 TRUE
N O N P A R A M E T R I C L I N E S Lines without parameters
Loess
Loess Y = mx + b
Loess Y = mx + b Y = mx + nx 2 + b
B A N D W I D T H S All you really care about is the area right around the cutoff Observations far away from cutoff don’t really matter Bandwidth = window around cutoff where you focus your analysis
Loess Y = mx + b Y = mx + b
K E R N E L S You care the most about observations right by the cutoff, so give them extra weight Kernel = method for assigning importance to values by distance to cutoff
M A I N R D D C O N C E R N S
G R E E D Y M E T H O D You need lots of data, since you’re throwing lots of it away
Bandwidth = $20,000
Bandwidth = $10,000
L A T E V S . A T E You’re only measuring the ATE for people in the bandwidth Local Average Treatment Effect (LATE)
N O N C O M P L I A N C E People on the margin of the discontinuity might end up in/out of the program The ACA, Medicaid, and 138% of the poverty line
Sharp discontinuity
Fuzzy discontinuity Address with instrumental variables (next week!)
R D D W I T H R
1: Is assignment to treatment rule-based? If not, stop! 2: Is design fuzzy or sharp? Either is fine; sharp is easier. 3: Is there a discontinuity in running variable at cutpoint? Hopefully not. 4: Is there a discontinuity in outcome variable at cutpoint in running variable? Hopefully. 5: How big is the gap? Measure parametrically and nonparametrically.
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