Statistics Concepts

• Descriptive analysis summarizes data and establishes facts.
• Casual analysis understands how one variable (X) affects another (Y).
• Predictive analysis estimates how one variable (X) predicts another (Y).
• Confusing correlation with causation leads to many false claims.
• Descriptive statistics aims to reduce the number of numbers while retaining as much information as possible.
• Mean is the average value.
• Percentile is a value Q(p) such that a fraction (p) of observations are at most Q(p).
• Standard deviation indicates the average distance of observations from the mean.
• Variance measures how spread out the observations are.
• Coefficient of variance is a relative measure of variance used to compare across different datasets.

Lecture 2

• Quantiles are values Q(p) such that a fraction (p) of observations are at most Q(p).
• Percentile ratios indicate the width of a distribution.
• 90/10 ratio is calculated as Q(0.90) / Q(0.10).
• Probability random variable X takes a specific value x in a discrete density function. The probability of all x values is at most 1 and at least 0.
• Histograms display the distribution of observations.
• The width of histogram bars represents the range of observations.
• The height of histogram bars represents the frequency of observations in each bin.
• A normal distribution is symmetrically distributed.
• Right/positive skew has more values on the right and a long tail on the left.
• Left/negative skew has more values on the left and a long tail on the right.
• Skewedness implies that the mean and median are not close, while symmetry suggests they are close.
• Wide histogram bars indicate more spread-out data.
• Narrow histogram bars indicate data concentrated around the center value.
• The peak of a histogram indicates where most data points are concentrated.
• Density Function has a probability random variable X, that takes a specific value x within set A.
• Kernel density function is an estimate of the likelihood of values across a range, representing the local weighted average for each x variable.
• Bandwidth (h) in kernel density estimation determines how much data around x is used with larger bandwidths disregard more data, smaller bandwidths create more noise.
• Peak in kernel density estimation indicates the concentration of data.
• P(X > 40,000) = 1 - CDF(40,000) determines how many values are under 40,000.

Cumulative Density Function (CDF)

• Indicates the fraction of observations with values less than x.
• Z = (X - mean) / SD(X) for a standard normal distribution.
• Then create the cumulative density function by plotting points.

Sampling

• Population is the entire group to draw conclusions about.
• Sample is a specific group selected from the population and collected from, used to make inference of the population.
• Sampling bias happens when the sample does not represent the population.
• Random sampling removes bias by ensuring each object in the population has an equal chance of being selected.
• Sampling error is an exception of observations sampled by chance.
• As the sample size increases, the sample average tends to approach the population mean.
• Sample averages are distributed relatively symmetrically around the population mean.
• Law of large numbers states that the larger the sample size, the closer the sample average is to the population mean.
• Central limit theorem states that sample averages are distributed relatively symmetrically around the population mean.
• Joint distribution is used for small data sets of two variables, in a cross tabulation
• Rows represent values Y can take and Columns represent values X can take
• Crosstabulation can also be used to present the share.

Joint density

• Means the probability that random variable X takes value x and random variable Y takes value y.

Lecture 3

• Conditional expectation: expectation of random variable Y when another random variable X takes a value x.
• Marginal distribution of Y means the probability of Y not taking X into account.
• Marginal distribution of X means the probability of X not taking Y into account.
• Conditional distribution means the probability that Y takes y conditional on the fact that X takes x.
• Conditional expectation represents the population average of Y when X is fixed.
• Income distribution: the horizontal distance between different year CDFs represents the dollar change for each percentile per year, however, those may not be the same people as we are comparing percentiles with CDFs, not people.
• Scatter plots are used to see associations between variables.
• Covariance measures the direction of a relationship between two variables (positive, negative, or zero).
• Correlation measures the linear dependence between two variables and is bounded between -1 and 1.

Regression Model

• Y = βο + β₁X + €
• Y is the outcome or dependent variable.
• X is the regressor or independent variable.
• β₁ = Cov(X, Y) / Var(X)
• E is the error term, representing relevant unobserved factors.
• Setting the betas follows ordinary least squares, where values of βo and β1 that minimize the difference between the observed data and regression model prediction.
• The formula for correlation is closely related to Pearson correlation coefficient.
• Intergenerational mobility is the idea that everyone has equal opportunities.
• Children's position in life is measured by income distribution and parents position, and that is an incomplete but powerful tool to measure inequality.
• With the conditional expectation function, if Y is linear in X, the linear regression will recover the true relationship exactly and even if it not linear, the regression provides the best linear approximation.

Steps to setup

1. Create a scatter plot of data to Add some noise to make it more visible
2. Measure linear dependence to Find correlation and Estimate linear regression to Get the parameter estimates
3. Check if there are any helpful summary statistics
4. Compare sample average by x
5. Adjust regression to multivariate regression to see if it better fits the data to Fit sample better with higher order polynomials but be careful with overfitting.

Lecture 4

• R^2: measures variation in the dependent variable explained by the independent variable or fitted model with If R^2 = 1, the regression model perfectly predicts the data.
• Causal questions aim to compare counterfactual states of the world, specifically the impact of X on Y (how would Y change if we changed X, where Y is the outcome and X is the treatment?).
• Ceteris paribus means everything is the same except the treatment.
• Counterfactual asks what would have happened to the treated group in the absence of treatment.

1. Impact of...treatment
2. In comparison to...counterfactual
3. Impact On... outcome
4. Impact for... population

• Example: the impact of Jakarta's high-occupancy vehicle restriction (treatment) in comparison to unrestricted road travel(counterfactual) on drivers( population) travel time(outcome)
• Binary treatments: they either receive treatment or not by denoting the treatment status
• Is the treatment effect for an individual, where y is the outcome.
• Average treatment effect = ATE, that is expected the impact the treatment has for the population.
• External validity: how ATE or ATT applies to different populations and treatment may be heterogeneous, ATT may not be a good predictor of what would happen if expanded to larger population
• Structural models, they are quantitative models to construct alternative states of the world, or by:
• Control groups, which means compare treatment group to another otherwise similar control group where everything effects groups similarly except for the treatment.
• Control groups are used to approximate what would have happened in the absence of treatment.
• There could be Invalid control groups leads to selection bias.
• Randomization eliminates selection bias by ensuing T and C are on average statistically equivalent.
• Similar distributors of characteristics, where confounders are equally distrusted across T and C.

Lecture 5

• Columns 1 and 2 show the averages and Column 3 shows the difference in averages (The point estimate).
• Row 2 shows the standard errors.
• Point Estimate: the single best guess for the value of a parameter.
• Need to determine how likely we are to got that point estimate.
• If less than 0.05 or 5% then statistically significant.
• Simulating a test distribution, is Create a distributing of placebo treatments to see randomly occurring differences.
• Random splits can be large due to chance.
• P-value: probability of obtaining a result at least as extreme as the one observed under the null hypothesis (treatment is zero).
• P < 0.05: strong evidence against null hypothesis -> the effect is statistically significant.
• Experiments yield more precise evidence when outcome variable has less variation and when they are larger.
• T-statistic: how many standard errors the estimated coefficient is away from zero (the null hypothesis; treatment is 0).
• P > 0.05: strong evidence for the null hypothesis -> can't reject null hypothesis, the effect is not statistically significant.
• They represent the range consistent with the observed data and Values inside vary in their compatibility, for example the point estimates is most compatible.

Lecture 6

• Type 1 Error (false positive): claiming an effect when it doesn't exist.
• Type 2 Error (false negative): claiming there isnt an effect when it doesn't exist.
• Power: The probability of finding an effect when it exists.
• There is a trade-off between type 1 and type 2 errors.
• The likelihood of false positives does not vary with sample size; the probabililty of a false positive is set by the signicanece level.
• Small samples means there is more variability meaning that false positives show exaggerated effect, but in large samples the random fluctations balance out.
• Some problems are as follows:
• Can lead to bad policy choices, and garner media attention towards big effects over truth due to underpowered studies.
• Publication bias: academic journals are more likely to publish studies with significant results than insignificant
• File-drawer effect: researchers don't finish papers with statistically insignificant results as it wouldn't be published anyway.
• P-hacking: researchers only report only a speculation with p < 0.05.
• If too many tests are run there is an increased likelihood of false postives.

Approached to Human Mistakes

1. Pre-registration of randomized control trials (RCTs), meaning to Publicly document study design, hypothesis and analysis plan before collecting or analysing data.
2. Replication files to Must post code and data so other researchers can analyse the robustness of the results.
3. Running larger experiments, higher lower variability and more precise estimates.

• Power depends on:

1. True effect size
2. Sample size
3. Variability of the outcome variable
4. Statistical significance level.

• The moving to opportunity theory is:
• Housing vouchers to randomly selected low-income families in high-poverty areas, such as;
  • Experimental: vouchers with restriction (move to low-poverty neighbourhoods)
  • Section 8: unrestricted vouchers
  • Control: continue public housing with no vouchers.
• Found it had positive effects.
• Not everyone who was offered a voucher used it -> non-compliance, making hard to estimated true treatment effect.
• If there is self-selection bias such as those who take the treatment are different to the full treatment group.
• If a comparison was made between the treatment group and the entire control, the resulting estimate would be biased..

Questions cover descriptive statistics, data interpretation, distributions, and causal inference. Topics include standard deviation, coefficient of variation, percentile ratios, and the Central Limit Theorem. Focus on understanding statistical measures and their implications.