Glossary Flashcards
Random variable
A variable whose value depends on the outcome of a random experiment
Support (of a RV)
Set of possible values a RV can take
Probability mass/density function
Each point on the line represents the probability of getting that value
PMF = discrete, e.g. how many heads in 10 tosses of a coin
PDF = continuous, e.g. weight of females in California aged 18-25
fx(x) = P(X=x)
Cumulative distribution function
Integrate the PDF to get the CDF
This is cumulative, so the slope is always upward-sloping, and at each value of x, the corresponding value shows the probability of getting up to that point
Fx(x) = P(X≤x)
Variance
Distance from the mean
Statistic
A single measure of some attribute of a sample, calculated by applying a function to the set of data.
The function itself is independent of the sample’s distribution; that is, the function can be stated before realization of the data.
The mean of a population is not a statistic (no RV), but the mean of a sample is (sample variables chosen randomly)
Uniform distribution
Continuous RV between a and b - same probability of getting every point between a and b
Mean = (a+b)/2
Bernoulli distribution
Either take on the value of “p” or “1-p”
Support {0,1}
Population mean = p
Population variance = p(1-p)
Sample mean = X¯
Sample variance = n/n-1 X¯ ( 1 - X¯ )
Binomial distribution
Probability of something happening over a period of time - X is the number of successes in n independent Bernouilli trials
X~B(n,p)
E(X) = np
CLT: binomial is discrete, but as you do more trials, it starts to look continuous: as n→∞, B ~ N ( μ , σ2/n )
Normal distribution (skewness and kurtosis)
Averages more common than extremes, bell-shaped diagram
Skewness = measure of symmetry Kurtosis = fatness in tails
*z = X-μ/σ *or z = X-μ^/SE(μ^) with samples
Then use the tables: Φ(z) = P(X ≤ z)
Jensen’s inequality
If g(.) is concave, then E[g(X)] < g(E[X])
E.g. mean of the log < log of the mean
Marginal probability
Marginal probability = the probability of an event occurring, p(A)
The probability that A=1, regardless of the value of B
Joint probability
Joint probability = p(A and B), the probability of event A and event B occurring
So, the joint distribution is the set of probabilities of possible pairs of values
Conditional probability
p(A|B), the probability of event A occurring, given that event B occurs
Conditional mean
Mean of the conditional distribution → E[Y|X=x]
Law of iterated expectations
The mean of the conditional means is the mean → E[Y] = E[E[Y|X]]
E.g., suppose we are interested in average IQ generally, but we have measures of average IQ by gender. We could figure out the quantity of interest by weighting average IQ by the relative proportions of men and women
Covariance
Do X and Y vary together?
σ(X,Y) = E[(X - E[X])*(Y - E[X])] = E[XY] - E[X]E[Y]
Correlation
Measures only linear relationship - may be 0 is the relationship is perfect yet non-linear
ρX,Y = cov(X,Y) / σXσY
(σ are standard deviations)
E[aX + bY] =
Var[aX + bY] =
= aE[X] + bE[Y]
= a2Var[X] + b2Var[Y] + 2abCov(X,Y)
Random sample
When a random experiment is repeated n times, we obtain n independent identically distributed (IID) random variables
Drawing one person makes another no more likely, drawn from the same population
Sample mean
The mean of a subset of the population (and a random variable)
xbar = 1/n * Σxi
The larger the sample, the smaller the variance of the sample mean.
The sample mean is an unbiased estimator of the population mean (µ)
Law of large numbers
If Ynare IID, the sample mean converges in probability to the population mean as the sample size grows
As n → ∞, X¯ - μ → 0
Central limit theorem
If Yn are IID with mean µ and variance σ2 and n is large, then the sample mean (Ybar) is approximately normally distributed, with mean µ and variance σ2/n
Sample variance
s2 = 1/n-1 ∑(Xi - X¯)2
We have lost one degree of freedom: if you have n-1 numbers and the mean, you can work out the last number. The last number is not independent - we only have n-1 independent observations
The sample variance is an unbiased estimator of population variance - PROOF
Degrees of freedom
Depends on how many statistics have been found from the information. The mean is degree 1, variance 2, skewness 3…
Standard error
How far away an estimate is from the true value - property of expectations
SE(X¯) = s /√n
If, under a null hypothesis, two samples are drawn from the same distribution, they have the same sample means and the same sample variances - which can be pooled for the sample error.
SE = √[s2A / nA + s2B / nB] when pooling
SE = √[p(1-p)/n] for Bernoulli
SE = √[p1(1-p1)/n1 + p2(1-p2)/n2] when pooling Bernoulli
NB. Population: σ2→σ ; Sample: s2/n = Var(d^) → s/√n = SE(d^)
T-statistic
Due to the CLT, the sample mean is normally distributed when n is large
t = X¯ - μ / SE(X¯)
β^ is an estimator of the parameter β in a statistical model; β0 is a non-random, known constant; and se(β^) is the standard error of the estimator, β^.
Confidence intervals
“There is a 95% probability that the true XXX is between A and B”
CI = [ μ^ ± z • SE(μ^) ]
Hypothesis testing - what are the steps? [5]
1) State null and alternative hypotheses, decide if it’s a one- or two-tailed test
2) Suppose H0 is true: under H0, t = X¯-μ/se(X¯) ~ N(0,1), as it’s a large sample
3) Decision rule: reject H0 at (e.g. 5)% significance level if |t|>z (two-tailed), or if t>z (one-tailed)
4) Carry out test: plug in values
5) Decide whether or not to reject H0
* Significance levels: if the probability of getting μ<t></t>*
Type I error
When the null is rejected when it’s actually true, e.g. saying a drug works when it doesn’t
P(Type I error) = significance level, α = p-value, e.g. 6% of the time, the null will be accidentally rejected
Type II error
Accepting the null even though it is false - e.g. saying a drug doesn’t work when it does
P(Type II error) = β
Power of test = 1 - β
P value
Probability of obtaining a value at least as extreme as t under the null
P(|Z| ≥ t) = 2 - φt for a two-sided test (1 - φt for one-sided)
Probability of observing this t-statistic if the null were true
Smallest significance level to not reject the null
Estimators
Should be:
- Unbiased - E[µ^] = µ
- Consistent - converges in probability to true value with higher n
- Efficient - low variance
Real equivalised household income
Economist’s guide to welfare:
X = Y(P/C)
Y is nominal household income
P is cost of living index (inflation adjusted)
C represents changes in tastes (cost of achieving the same utility with different tastes - e.g. family type).
Equivalisation: C=1 (couple, no children), =0.67 (first adult), =0.33 (spouse), =0.33 (other adults and older kids), =0.2 (younger kids) - showing differing costs
Not an ideal measure - there are lots of other variables and externalities
Log normal distribution
X may not be normally distributed, but ln(X) might be
Kolmogorov Smirnov test
Compares sample CDF to hypothesised population CDF - looks for longest distance between them
Tests for whether they have the same distribution - e.g. is sample log normally distributed?
Power laws
Describe tails of some distributions - e.g. income, stock returns
These distributions have no mean or variance - they carry on indefinitely
What criteria should social welfare fuctions meet? [7]
- Monotonicity - more is weakly better than less
- Anonymity - blind to names
- Symmetry - swapping two people’s incomes has no effect on social welfare
- Dalton’s principle / quasiconcavity - inequality is bad, so bend towards origin
- Homogeneity of degree 1 - doube income, double welfare (not vital)
What are examples of inequality measures? [3]
- 90:10 ratio - only considers 2 data points, not overall inequality
- Gini - considers ranks, not absolute values
- Coefficient of variation - sensitive to top and bottom, so double everyone’s income and you double inequality
Atkinson index
SWF where you must explicitly decide on an inequality aversion parameter, ε
Problems with measuring poverty? [4]
Tough to measure - UK uses 60% of median income
Relative
Skewed incentives - e.g. taxing very poor to lift fairly poor aboce the line
Ignores distribution of poor’s income
Manski’s law of decreasing credibility
The credibility of inference decreases with the strength of the assumptions maintained
i.e. always make minimal assumptions for greater credibility in results
Sample linear regression
Calculated for a given sample - will be different for different samples, even if they are from the same population
Add hats to the population regression - sample estimates. Use sums, not expectations:
β^0 = E[Y] - β^1E[X]
Standard error of the regression
Estimator of the standard deviation of the regression error term u.
Cannot be calculated for population version (rely on unknown population figures)
The n-2 is adjustment for downward bias due to use of sample (-1 as normal, -1 for parameters from sample)
F-test: joint hypothesis testing
Cannot just do individual t-tests - chance of Type I error grows, but can instead use the rule of thumb F-stat:
- ur = unrestricted regression model - set tested parameters to zero (or whatever null says) in regression, then find R2
- q = number of restrictions
- k = number of regressors in unrestricted model
Autocorrelation
Correlation of a series with its own lagged values - first autocorrelation is corr(Yt,Yt-1), jth correlation is corr(Yt,Yt-j)
The jth sample autocorrelation is:
