EXAM 1 Flashcards
Econometrics
The science of using statistics and economic theory to acquire economic data
Causality
An action is said to cause an outcome if the outcome is the direct
result, or consequence, of that action
Controlled Experience
- control group
- treatment group
-random assignment
In a controlled experiment, the control group doesn’t receive treatment, while treatment group does. The assignment to each group is random.
Casual Effect
Effect on an outcome of a given action/treatment
Experimental Data
experiment designed to evaluate a treatment
Observational Data-
Observes actual behavior outside an experimental setting
- treatments are not assigned
Cross sectional data
data on different entities for a single period of time
Time series data
data for single entity at multiple times period
Panel data
data for multiple entities, which each entity is observed at 2+ periods
Probability Theory
basic language of uncertainty + forms the basis for statistical inference
Outcomes
mutually exclusive potential results of a random process
Sample Space
set of all possible outcomes
Event
Subset of the sample space; set that contains more than one outcome
Probability of Outcome
the proportion of the time that the outcome occurs in the long run
Probability of Event
the sum of the probabilities of the outcomes in the event
Random Variable
numerical summary of a random outcome
Probability Distribution
The probability distribution of a discrete random variable is the list of all possible values of the variable and the probability that each value will occur.
Cumulative Probability Distribution
the probability that the random variable is less than
or equal to a particular value.
Bernoulli Distribution
p and 1-p
Probability Density Function (continuous)
probability that the random
variable falls between those two points
CDF for continuous
the probability that the random variable is less than or equal
to a particular value
Expected Value of Discrete RV
- long run average value of the random variable over many repeated trials
- weighted average of the
possible outcomes of that random variable where the weights are the
outcomes’ probabilities.
Expected Value of Continuous RV
an uncountable infinite many
possible values
mean+SD
measure the center of the
distribution and its spread
Skewness
measures the lack of symmetry
- symmetric, skewness =0
- long right tail = + skewness
-long left tail = - skewness
Kurtosis
how thick or heavy the tails of distributions are
- the greater the kurtosis, the more likely the outliers
- a normal distributed RV is 3
Standard Deviation + Variance
measures of dispersion of distribution
- the variance is an expected value of the square of the deviation of Y from its mean
Moments of Distribution
- Mean of Y; E[Y] is first moment
- E[Y^2] is the second moment
- E[Y^r] is the rth moment
- variance is function of 1st and 2nd moment
-skewness is the function of 1st-3rd moment
- kurtosis is the 1st-4th moment
Joint Probability Distribution
probability that the random variables simultaneously take on certain x and y values
Marginal Probability Distribution of Y
Adding up all the probabilities possible for which Y takes on a specific value
Conditional Distribution
The distribution of Y conditional on X taking a specific value
P( X|Y)= P(X,Y)/P(Y)
Law of Iterated Expectation
weighted average of the P(Y|X), weighted by the distribution of X
- the expected value of Y is equal to the expectation of the
conditional expectation of Y given X
E [Y ] = E [E [Y |X ]]
LIE
- computed using the conditional distribution of Y given X , and the outer expectation is computed
using the marginal distribution of X . - implies that if the conditional mean of Y given X is zero, then the mean of Y is zero.
- applies to expectations that are conditioned on
multiple random variables
Independence
2 RV are independent if knowing the value of one variable provides no info about the others
Covariance
The extent the two variables move together
- if X & Y are independent, covariance is zero
Correlation
How much one variable depends on the other variable
- X&Y are uncorrelated if corr(x,y)=0
-1<corr<1
funfact
If the conditional mean of Y does not depend on X , then Y and X are
uncorrelated
Standard normal distribution
convenient representation for RV that is normally distributed
Multivariate Normal Distribution
FOUR IMPORTANT PROPERTIES
Represents the joint distribution of two (bivariate normal) or more
MULTIVARIATE NORMAL DISTRIBUTION- PROPERTY 1
If X and Y have bivariate normal distribution with covariance σXY , then for
constants a and b
MULTIVARIATE NORMAL DISTRIBUTION- PROPERTY 2
If a set of variables has a multivariate normal distribution, then the marginal
distribution of each of the variables is normal.
MULTIVARIATE NORMAL DISTRIBUTION- PROPERTY 3
If variables with multivariate normal distribution have covariances that
equal zero, then the random variables are independent.
- if X
and Y have a bivariate normal distribution with σXY = 0, then X and Y
are independent.
- uncorrelation implies independence (not true in
general)
MULTIVARIATE NORMAL DISTRIBUTION- PROPERTY 4
If X and Y have a bivariate normal distribution, then the conditional
expectation of Y given X is linear in X that is E [Y |X = x] = a + bx, where a and b are constant.
Joint normality implies linearity of conditional
expectations, but linearity of conditional expectation does not imply joint
normality
Chi Squared Distribution
sum of M square independent standard normal RV
- et Z1,
Z2 and Z3, be independent standard normal random variables. Then,
Z1 + Z2+Z3 has a Chi-squared distribution with 3 degrees of freedom
Student T Distribution
ratio of a standard normal random variable, divided by the square root of an independently distributed chi-squared random variable with M degrees of freedom divided by M
F Distribution
with M and N degrees of freedom, denoted by FM,N is
defined to be the distribution of the ratio of a chi-squared random variable
with M degrees of freedom, divided by M, to an independent chi-squared
distribution with N degrees of freedom, divided by N
Random Sampling
Random sampling procedures n objects are selected random from a population
Y1, …, Yn, where
Y1 is the first observation, Y2 is the second observation, and so forth.
Each of these Yi ’s are random variables.
Identically distributed
each Yi has the same marginal distribution
Sample Average
= 1/n (Y1+..+Yn)
E[Y] = μY
Var [Y] = σ Y = σ^2/n .
When the distribution of Y is not normal
the exact distribution of the sample mean is typically
complicated and depends on the distribution of Y
Large Sample Approximation
Large sample approach uses approximations to sampling distribution that rely on on n>30
Asymptotic Distribution
approximation becomes exact, n-> infinity
Law of Large #
sample size is large, the average be very close to mean with very high probability
Central Limit Theorem
when sample size is large, the sampling distribution of standardized sample average is approximately normal
Asymptotic Theory
While exact sampling distributions are complicated and depend on the
distribution of Y , asymptotic distributions are simple.
Convergence in Probability
converges in probability to μY (or equivalently ̄Y
is consistent for μY if the probability that the the sample average ̄Y is in
the range μY − c to μY − c becomes arbitrarily close to 1 as n increases
for any constant c > 0
Statistics
science of using data to learn about the world around us
Estimator
function of a sample of data to be drawn from population
- is a RV because it’s a function of random sample observations
Estimate
numerical value of the estimator when it’s actually computed using data from specific sample
- not random sample
ESTIMATION PROPERTY 1: Unbiasedness
We say ˆμY is an unbiased estimator of μY if E [ˆμY ] = μY .
- bias of the estimator is E [ˆμY ] − μY
- if we compute the value of the estimator for different samples, on average, we get the right number
ESTIMATION PROPERTY 2: Consistency
Let ˆμY be an estimator of μY . We say ˆμY is a consistent estimator of μY
if ˆμY converges in probability to μY
- when the sample size is large, the uncertainty about the value of μY arising from random variation in the sample is very small.
Convergence in Probability
A sequence of random variables { Xn } converges in probability to X if for
all > 0
lim
x→∞ Pr (|Xn − X | > ) = 0.
ESTIMATION PROPERTY 3: Efficiency
Let ˆμY and ̃μY be unbiased estimators of μY .
We say that ˆμY is more
efficient than ̃μY if V [ˆμY ] < V [ ̃μY ]. In other words, an estimator is more
efficient than other if it has a tighter sampling distribution.
SBU Econometrics Fall 2021 8 / 43
Properties of the average Y
- The sample mean Y is an unbiased and consistent estimator of μY
- The sample mean Y is the best linear unbiased estimator (BLUE), where “best” stands for more efficient here.
- The sample mean Y is also the least squares estimator of μY .
P Value
probability of drawing a statistic at least as unfavorable to the null hypothesis as the value actually computed with your data,
assuming that the null hypothesis is true. One often “rejects the null
hypothesis” when the p-value is less than the significance level α
Significance level
The significance level of a test is a pre-specified probability of incorrectly rejecting the null, when the null is true.
- probability of type 1 error
Critical Value
the value of the test statistic for which the test just rejects the null hypothesis at the chosen significance level
Sample Variance
the square root of the sample
variance. The sample variance is an unbiased and consistent estimator of the population variance.
Standard Error
an estimator of the standard deviation and is denoted by SE
n known, variance unknown
p value= (- (Yaverage-mean)/SE))
Type 1 Error
Rejecting null when it’s true
Type 2 Error
Accepting null when it’s false
Rejection Region
the set of values of test statistics for which the null hypothesis is rejected
Acceptance region
the set of values of test statistics for which null hypothesis is not rejected
Size of Test
probability of type 1 error
Power of test
probability of rejecting the Ho when alternative is true
Confidence Interval
an interval that contains the true value of mean in 95% of repeated samples
When to use t statistics
when sample size is rlly small
The sample covariance is a consistent estimator of the population covariance.
The sample correlation lies between −1 and 1
The sample correlation coefficient measures the strength of the linear
association between X and Y in the sample of n observations.
