Econometrics

model

assumed structure of data

estimator

a random variable & rule/method that uses sample data to produce estimates of unknown population parameters

types of econometric models for mean

• linear regression

• ARMA

• ADL

types of econometric models for volatility

GARCH

types of econometric models for return quantiles

quantile regression

What is a coefficient estimate?

A numerical approximation of a true parameter computed from the sample data via an estimation method.

regression analysis

describing/evaluating the linear relationship between the avg of the variable of interest (dependent) and other variables (independents)

linear regression model

explain expected value of y as a function of x

linear regression model equation

y = α + βx + u

alpha (α)

on average, the expected value of y when x=0

beta (β)

on average, how much a 1 unit change in x will change y

the error term (u)

represents factors other than the independent variables that affect y that the model cannot predict

• goal is to minimize u

correlation

measures the strength and direction of a linear relationship between 2 variables

• between -1 and 1

• Corr(x,y)=Cov(x,y)/√Var(x)Var(y)

Interepreting correlation

• -1= move in opposite directions

• 0 = no linear relationship

• 1= move in the same direction

covariance

measures the extent that 2 random variables change together

• cov(x,y)= E(xy) - E(x)E(y)

interpreting covariance

• positive: move together

• negative: move inversely

• if independent, cov(x,y)=0

regression vs correlation

• both: characterizations of linear dependence

• regression: one variable explains/predicts a characteristic of another

• correlation: how 2 variables move together, ignores cause and effect

arithmetic return in eviews

• used for single period calculations

• (x/x(-1))-1

log return in eviews

• used for multi period/compounding

• log(x/x(-1))

residual (û)

the difference between the actual y-value from the observed data and the corresponding fitted value on the regression line

• û = y - ŷ

residual analysis

how well a regression model explains data

interpreting residuals

• û = 0, perfect prediction

• û > 0, actual higher
  û < 0, actual lower

ideal residual characteristics

small, random, centered around 0

main estimation methods

• ordinary least squares (OLS)

• method of moments (MOM)

• max likelihood

which estimator should you use?

can be used for different model types (lin reg, arma, garch) but some fit better than others

unbiasedness

θ̂ is unbiased if its E(θ̂) is equal to the true parameter value E(θ̂) = θ

efficiency

θ̂ ₁ is more efficient than θ̂₂ if its variance is smaller

mean squared error

choose the smallest MSE estimator

consistency

estimator is consistent if as N increases, θ̂ converges to true value, discard inconsistent estimators

most important estimator quality

consistency

asymptotic distribution

distribution of estimator as N goes to infinity

• generally normal, centered around parameter true value

asymptotic efficiency

estimator variance as N goes to infinity

ordinary least squares (OLS)

an estimator that chooses α and β to minimize the sum of squared residuals (SSR)

OLS regression line

ŷ = a + bx

• a = y-intercept 

• b = slope

jarque bera test

goodness of fit test of whether sample data have skewness/kurtosis of a normal distribution

interpreting the jarque bera test

• 0= normal

• infinity= not normal

magnitude/volatility

the # of points the index rises or falls

• shown by the standard deviation

standard error

measures the precision of the sample statistic representing variability of stat’s sampling distribution (avg residual size, how much model’s prediction differs from actual value)

interpreting SE

• low = precise

• high = not precise

SE formula

√(SSR/n-k)

• simple linear regression: k = 2

hypothesis testing

decision rule using data sample and an estimator to discriminate between 2 hypothesis 

null hypothesis (Ho)

the assumed true statement

alternative hypothesis (Ha)

the opposite of the null

F-statistic

measure of overall model significance

• Ho = model isn’t statistically significant

• Ha = model is statistically significant

Interpreting f-test p-value

p-value < 0.01: reject Ho, at least one variable is statistically significant

• must reject model if p-value>1%

residual (ε hat) vs error (ε)

residual’s a number, error is a random variable

t-statistic

measure of an independent variable’s significance

• Ho = variable isn’t statistically significant

• Ha = variable is statistically significant

t-stat calculation

• ˆβ/S_^β

• ˆα/S_^α

gauss markov assumptions

assumptions where, if all are satisfied, the OLS is consistent, unbiased, efficient and BLUE

BLUE

best linear unbiased estimator

• smallest variance, efficient

• OLS estimators are linear

• the expected value equals the true value E(b)=β , E(a)=α

• the OLS estimator gives ^α and ^β

5 guass markov assumptions

1. errors have 0 mean

2. errors have constant, finite, variance over all i 

3. errors are independent across i 

4. errors aren’t correlated to explanatory variable

5. errors not normally distributed

homoskedacity

errors have constant, finite variances over all i

• rarely satisfied

which conditions are required for consistency

1 and 4

what does s represent

the standard error of the regression

biased estimator of variance formula (s²)

s² = SSR/n

unbiased estimator of variance formula (s²)

s² = SSR/n-2

determinants of standard errors

• u variance = higher variance, higher s

• sum of squared deviations of x from the mean= higher sum, lower s

• sample size = higher n, lower s

• sum of squared x values = higher sum, higher se(a)

distribution of ols estimators

• a ~ N(α, var(a))

• b ~ N(β, var(b))

estimated distributions of the OLS estimators

the distributions are no longer normal, they have t-distributions with n-x(# parameters estimated) degrees of freedom, called t-ratios or t-stats

• (a-α)/se(a) ~ t_n-2

• (b-β)/se(b) ~ t_n-2

computing t-stat formula

t = (b-β)/se(b)

• numerator: how close coefficient from data is to hypothesized value

• denominator: how precise slope estimate is

• t-stat is close to 0 if b is close to beta

• t-stat is large in magnitude if parameters are precisely estimated

multiple linear regression

variable y has multiple influences (x) on its avg behavior

matrices form of multiple linear regression

y = Xβ + u

f-test statistic formula

F = ((N-k)/m)((RRSS-URSS)/URSS)