Midterm

Question 1

Dependent Variable

Answer 1

Outcome we are interested in
Depends on the other variable
Goes on the y-axis

Question 2

Independent Variable

Answer 2

Intervention or treatment
The “cause”
Goes on the x-axis

Question 3

Confounder

Answer 3

Something related to both the IV and DV

Question 4

Comparability

Answer 4

in absence of treatment

Question 5

Treated Group

Answer 5

those who et some treatment of interest

Question 6

Control Group

Answer 6

Those who do not get the treatment of interest

Question 7

Observational Study

Answer 7

General term for research where you don’t get to randomize who get the treatment
Instead you just observe some relationship in the world

Question 8

Experimental Study & Randomized Control Trial (RCT)

Answer 8

common terms for research designs in which you do randomize who gets the treatment

Typically you can make causal claims from experimental studies

Question 9

Quasi-experimental research

Answer 9

research in which you have observational data, but you find ways to ensure that the treatment was effectively randomly distributed

Question 10

Internal Validity

Answer 10

Is the experiment well designed? Is it free from confounders or bias?

Question 11

External Validity

Answer 11

Is the finding generalizable to other populations, situations or cases? Does it apply outside of the context in which the finding was generated?

Question 12

Problems with experiments

Answer 12

Not everything can be randomized (democracy, gender)
Not everything should be randomized (wars, right to vote and medication for birth defects)
Ethical dilemma: randomized treatment means denying treatment to some but not randomizing means we don’t really know if its effective or not
Running experiment is expensive

Question 13

Yi

Answer 13

dependent variable, outcome variable, the thing we want to predict

Question 14

Xi

Answer 14

independent variable, the thing that predicts the DV

Question 15

Ei

Answer 15

error term
part of the DV or IV doesn’t explain
everything NOT in our model

Question 16

β1

Answer 16

slope coefficient
relationship between X & Y
Indicated how much change in Y is expected if X increases by 1 unit

Question 17

β0

Answer 17

constant
Value of Y when X is zero (intercept)
Indicates where the regression line crosses the Y-axis
Value of Y when X is 0

Question 18

Endogeneity

Answer 18

the IV is correlated with the error term
confounder: this means that there is another unmeasured variable (a confounder) that affects the IV which also affects the DV
We haven’t included this other confounding variable in our model

Question 19

To get our casual estimate

Answer 19

we need to create exogeneity

Question 20

Exogeneity

Answer 20

IV is unrelated to or uncorrelated with the error term

Accomplish this through random assignment

Question 21

Randomness

Answer 21

noise in the data
could go away with larger sample sizes
address some of these concerns with t-tests (p-values) and confidence intervals

Question 22

Randomization

Answer 22

using a coin toss to create treatment and control groups which creates exogeneity

Question 23

Population

Answer 23

the overall collection of individuals, beyond just the sample

Question 24

Sample

Answer 24

the collection of individuals on which statistical analyses are performed, and from which general trends for the population are inferred

Question 25

Individual

Answer 25

also called an “object” or “unit”

a single data point contributing to the sample

Question 26

Mean

Answer 26

average of a variable

X bar

Question 27

Minimum

Answer 27

lowest value of a variable (min)

Question 28

Maximum

Answer 28

the highest value of a variable (max)

Question 29

Sample size

Answer 29

the number of observations (N)

Question 30

Standard deviation

Answer 30

how widely dispersed the values of the variable are (population, sample or standard deviation)

Question 31

Probability Theory

Answer 31

Population distribution (Y)
—Estimand/Parameter
Then goes to sample (Y1, Y2,…Yn)
Then to Estimator/Statistic g(Y1, Y2,…Yn)
Finally to Estimate and back to population distribution

Question 32

Expectation

Answer 32

the best guess about what number will be drawn from the distribution

Question 33

Variance

Answer 33

how far the numbers you drew tend to be from the best guess

Question 34

E[X]

Answer 34

something that exists for any random variable

if you could draw repeatedly from a distribution and take an average, it would get closer and closer to the expectation

Question 35

Variance

Answer 35

measure of spread, or how fast you expect a random draw to be from the expectation

E[X-E[X]^2]

draw a bunch of numbers from the same distribution, squared the difference from each to the mean and averaged those

Question 36

E[X] & Var[X]

Answer 36

are properties of the distribution of X, not your data

Question 37

Normal Distribution

Answer 37

X~N (E [X], Var(X))

X ∼ N (µ, σ2), where µ is the expectation and σ 2 is the variance.

Question 38

Sample Mean

Answer 38

The Sample mean of X1, X2…XN is

X bar = 1/N  Xi = 1/N [X1 + X2 + … + XN]

This is an example of an estimator

Question 39

Law of Large Numbers

Answer 39

For Rv’s X1, X2…Xn the mean (x bar) gets closer and closer to E[X]

As N grows the estimator gets better

Question 40

Distribution of the mean

Properties

Answer 40

Property 1: the means distribution is centered around E[X}
 You only get a mean once, but you know it takes a value from a distribution centered on E[X]
 We don’t know E[X] (though we know X bar gets closer and closer as N grows)

Property 2: variance of the mean is V(X bar) which is the Var(X) divided by N
• So we can estimate Var(Xbar) =Varbar/N

Question 41

How is the mean distributed

Answer 41

o There is one more piece to understanding the distribution of the mean: we know its expectation and variance, but what about the shape?
o The shape must depend on the distribution of the underlying X, you would think
 Fortunately it does not
o Property 3: Central limit theorem (CLT): the distribution of the mean tends toward a normal distribution
 This is magical: regardless of how the original X is distributed, when you take the mean of multiple RVs drawn from the same distribution, it starts to look normal
 You do need N to be big enough for this to work, but that’s often not a problem
 We will discuss some rules for deciding if N is big enough and adjustments to use when it is not
 As N increases, the distribution starts to turn into a triangle shape, with the peak in the middle.
 The more N increases, the width of each box gets smaller
o Theoretical understanding of distribution
 Take the N=50 case
 We can estimate the center using X bar which is 0.5
 We can estimate the variance of X bar using (1/N) Varbar(X)
 We know the shape of the distribution of the mean is normal
 Using all this, we estimate that the mean should be distributed
• Xbar ~ N(µ = 0.5, σ2 = .08/N)

Question 42

Covariance

Answer 42

the mean value of the product of the deviations of two variables from their respective means

measure of the join variability of two random variables

IF the greater values of one variable mainly correspond with the grader values of the other variable, and the same holds for lesser values, the variables tend to show similar behavior, the covariance is positive

Positive association: when X is higher, we expect Y is usually higher
Negatively associated when X is higher, we expect Y is usually lower
Not associated: when X is higher it doesn’t tell us anything about Y

Problem with covariance: scale is not very natural

Question 43

Correlation

Answer 43

statistical technique that shows whether and how strongly pairs of variables are related

Correlation ranges between -1 & 1

A perfect positive relationship has Cor(X,Y) = 1
A perfect negative relationship has Cor(X,Y) = -1
Two perfectly unrelated variables have Cor (X,Y) = 0

Question 44

When we divide by the standard deviations

Answer 44

We are in effect standardizing the covariance and rescaling it from -1 to 1.

Question 45

Correlation only sees

Answer 45

linear relationships

Question 46

Regression logic

Answer 46

o How can we best figure out this relationship between X & Y
o Suppose we guess the slope and intercept. How do we assess whether it is a good guess for the relationship between X & Y?
o We use the sum of squared errors (the residuals added up and square) to see how well we are doing
o It turns out the best way to estimate this relationship is to choose our slope and intercept for X (slope and intercept) to minimize the value: the sum of squared errors.
o The residual is the distance between the line and any given point
o The SSE takes those residuals, squares them, and adds them up
o The regression shows us the best fitting line in terms of sum of squared errors

Question 47

Standard Error

Answer 47

o	If we took another sample of data from the same source and estimated βˆ.
o	SE (βˆ) estimates the standard deviation for that distribution of βˆ

Question 48

Compare modeled outcome to the simple mean

Answer 48

We can think of regression as a prediction machine that tells us our best guess of Y (growth) given our knowledge of X (yearsschool) for an observation

Question 49

Understanding the variance explained or R squared

Answer 49

Spread of points around the regression line should be smaller than the spread of points around the mean line
 If we average up the squared distances around the mean we get the variance of Y.
 If we average up the squared distances around the regression line we get mean squared error for the regression
 We were trying to minimize this form of prediction error in our choice of Beta.

Question 50

Causal Inference

Answer 50

o Just like difference in means reflected an association, correlations, covariance, and regression coefficients only reflect an observed relationship in the data
 The setup makes us focus on variation in IV as an explanation for DV
 But those countries that are higher or lower on IV are probably higher on other things
 These are things may be why DV differs not just from IV
 In terms of confounders

Question 51

What is bias

Answer 51

Unbiased estimate: on average, our estimate is equal to the true parameter

Biased estimate: our coefficient is systematically wrong, either too high or too low than the true parameter

Question 52

Omitted Variable Bias

Answer 52

o OLS does not necessarily create unbiased estimates
o Omitted variable bias: this a specific form of endogeneity
 X is correlated with something else that influences Y; the error term is correlated with Y
 This is often the reason why, if you change the model specification, your estimates change. We say, our model is not robust to alternative specifications
 Therefore our model is missing a key confounder, and we haven’t estimated a causal relationship
 Theoretically, you could include this confounder in your model (multi-variable regression) but this is often hard to do: maybe you don’t know what the confounder is, or you don’t have data on it.

Question 53

Homoscedasticity

Answer 53

when the random variable, X, HAS the same variance for all observations of X. This isn’t a problem

Question 54

Heteroscedasticity

Answer 54

when the random variable, X, DOES NOT have the same variance for all observations of X. This is a fixable problem
o Remember this only affects our standard errors, not our slope estimate. Therefore it doesn’t cause bias
o So what the solution? We use slightly different estimator for our standard error calculations. Intuitively, we estimate the variance in our standard errors.

Question 55

Outliers

Answer 55

o An observation that is extremely difference from the rest of the observations in the sample. “One of these things is not like the others
o Reminder, what does an outlier do to our estimate of the mean? It drags it toward the outlier. The mean is sensitive to outliers
o Intuitively, then what would outliers do to our regression estimates? It would also drag our estimate of the slope towards the outlier.

Question 56

Two sample tests:

Answer 56

o How can we compare one sample to another sample to ask: “Are these samples from the same population or different populations?
 We call this two sample tests, comparing the samples
• We will try to understand how likely we are to observe some difference if there really is not a difference
• That will require thinking about a null distribution, describing how “weird” or result is if there really is no difference (our null hypothesis)

Question 57

Difference in means

Answer 57

o The fundamental quantity of interest today will be the difference in means between two groups on some outcome
 Difference in mean income across two states
 Difference in voter turnout among two groups of people (Students vs employed)
 Difference in probability of war in two groups of countries (autocracies vs democracies)
 Difference in proportion of heads after tossing one coin N1 time and another coin N2 times.

Question 58

One sided vs. Two sided tests

Answer 58

Two sided: is one group different, either bigger or smaller than one?

One sided: is one group bigger (smaller) than the other

Question 59

Hypothesis

Answer 59

o A good null: H0: VoteR = VoteD
o Alternative: VoteR ≠ VoteD
o We find that:
 Among Republicans, 20/25 report they will vote. (VoteR=0.8)
 Among Democrats, 22/35 report they will vote (VoteD=0.63)
 Thus VoteR – VoteD = 0.17
 What is the key thing we need to assess how “weird” this result is?

Question 60

What is a p-value?

Answer 60

o A p-value is the probability of observing a difference in mean or a coefficient as big as what we observed, if the null were true.
o First, you need to decide is this a one-sided or two-sided test?
o Then you need to set your critical value- how different would I want these two distributions to be before I conclude they probably weren’t from the same distribution?
o Based on this critical value, you can say whether it seems like these two groups are significantly different

Question 61

For two sample test with a difference in means

Answer 61

 Null Hypothesis: difference in means between two groups is zero
 Alternative: the two samples are probably from different groups
 Critical value: how certain do I want to be that they’re really different? Often 1.96
 P-value: how often you would get a result at least as extreme as you got under the null. Usually we set the cut-off for significance level to 0.05-1 out of 20 cases

Question 62

Hypothesis testing with regression

Answer 62

o What if we wanted to know if the relationship we found between X and Y was real, versus we saw it just by chance?
o It could be that the sample we draw shows a relationship between X & Y but if we had a slightly different sample we wouldn’t see any relationship
o We need to set decision criteria: how different from zero do we want the relationship to be before we decide we’ve really uncovered a real relationship. We call this the critical value.

Question 63

Hypothesis testing with regression

Answer 63

o Null: β1 = 0- there is not relationship between X and Y
o Alternative: β1 ≠ 0- there is a relationship between X and Y
o How would we set our cut off for making sure we don’t make mistakes? Pick our critical value

Question 64

The standard normal distribution

Answer 64

o Up to 1.64 SD gets 95% confidence interval
o -1.64 to 1.64 SDs gets central 90% confidence interval
o Up to 1.96 SDs gets 97.5% confidence interval
o -1.96 to 1.96 SDs gets central 95% confidence interval

Question 65

Hypothesis Testing

Answer 65

o State null hypothesis
o Determine if hypothesis test is one sided or two sided
o State alternative hypothesis
o Run Regression
o Test statistic: take coefficient and standardize it
 Divide coefficient by standard error
 We do this so we can apply coefficient to normal distribution
 This will let us know how likely it would be that we got this coefficient just by chance (if the RA scrambled our data on accident)
o Decide on a critical value (typically 1.96 for two sided test)
o State whether we can reject the null in favor of the alternative
 If critical value < |test statistic|: we reject the null
 If critical value > |test statistic|: we fail to reject the null

Question 66

Fearon and Laitin Dataset

Answer 66

o Research question: does ethnic fractionalization explain how long civil wars last?
 Dependent variable? Year of civil war
 Independent variable? Ethnic fractionalization
 Null hypothesis?
• There is no association between ethnic fractionalization and years of civil war
 One or two sided? Lets go with two-sided to be safe
 Alternative hypothesis?
• There is an association (either positive or negative) between ethnic fractionalization and years of a civil war
 Lets get test statistic
• Step 1: get β1 (coefficient for ethnic fractionalization)
• Step 2: Standardize β1
o Estimate/std. error
o For year of civil war: Estimate for the intercept (44.571), Std. error (2.079). t value (21.439), p-value (2e^-16)
o For ethfrac: Estimate (-9.830), std error (4.205), t-value (-2.338) , p-value (0.0207)
o For standardizing β1: (-9.830/4.205) = -2.338
 Which is your test statistic for ethfrac!
• Lets decide on a critical value: 1.96
• |test statistic| = 2.338
• Critical value = 1.96
• Therefore, critical value < |test statistic|
o In other words, we are very unlikely to see such a large estimate with this data due to chance. How unlikely? Such an estimate will come up only about 2% of the time
o We reject the null hypothesis
 Why do we use p-values?
• Intuitive: its just the probability of your coefficient being due to chance
• Easy short-hand: run the regression, if your p-value is below 0.05 reject the null. If not, you fail to reject the null
o Confidence intervals
 Given we have just one sample, and we know there is variability from just having one sample rather than the full population of data, how confident should we be in our results—our point estimate?
 What factors would make you more confident in your estimates? Larger sample size, smaller variance in data
 What factors would make you less confident in your estimates? Smaller sample size, large variance in your data
o Calculating confidence intervals
 As long as you have a relatively big sample size then you can use the following basic formula
 Notice this formula is also dependent on choosing your critical value