Quant Methods

Residual

difference between the observed value Yi and the predicted value from the regression Y^i

P-value

Smallest significance level which we can reject Ho

if p-value < significance level, null hypothesis can be rejected

if p-value > significance level, null hypothesis cannot be rejected

R²; Coefficient of Determination

Explained Variation (RSS) / Total Variation (SST)

Total Variation - Unexplained Variation / Total

SST-SEE/ SST

Overfitting

relatively high R² may reflect the impact of a large set of independent variables, rather than how efficiently the set explains the dependent variable

In this case adjust R² for number fo independent variables

Adjusted R²

lower than R², adjsuted for the number of independent variables to address overfitting

does not indicate the quality of model fit or statistical significance of slope coefficients

Akaike’s Information Criterion (AIC)

Evaluates quality of model fit among competing models for the same dependent variable; used if goal is to have better forecast

Bayesian Information Criterion (BIC)

Evaluates quality of model fit among competing models for the same dependent variable; Used if goal is to have better goodness of fit; lower values mean better goodness of fit

Nested Models

"Full” or “unrestricted” model has higher # of independent variables while the other “restricted” model has only a subset of independent variables

F test is used to evaluate

Omission of Important Variables

model misspecification; biased and inconsistent parameters, may lead to serial correlation or heteroskedasticity

Variables are not Appropriate Form

Relationship between independent and dependent variable; may lead to heteroskedasticity in residuals

Inappropriate Variable Scaling

variables may need to be transformdel; may lead to heteroskedasticity or multicollinearity

Data Pooled Improperly

data from different structural regimes combined in the sample; may lead to serial correlation or heteroskedasticity

Heteroskedasticity

variance of the error term is nonconstant

Unconditional Heteroskedasticity

not related to independent variables; causes no major problems

Conditional Heteroskedasticity

related to independent variables; causes problems

as independent variables increase, variance increase

t-stat and F-stats are unreliable, so are std errors; coefficients are still consistent and unbiased

Breusch- Pagan Test

use to test for conditional heteroskedasticity; significant value is evidence of hs

test significance of resulting R²; Ho = no heteroskedasticity

Chi-square test; one-tailed test bc heteroskedacity is only a problem if stats are too high

White-corrected Standard errors

used to correct heteroskedasticity; recalculate t-stats using these robust errors

Serial Correlation

Residual terms are correlated

t-stats are unreliable; too high = positive correlation, too low = negative correlation

Durbin-Watson statistic

used to test for serial correlation; only detects one lag

Breush-Godfrey Test

used to test for serial correlation; tests for multiple lags

F-distribution, uses residuals from teh original regression as the dependent variable

if BGstat < Fstat, fail to reject Ho ; if BGstat > Fstat, reject Ho

Newsy-West corrected standard errors

can correct serial correlation by using using these robust std errors

Multicollinearity

two or more “X” variables are highly correlated with each other

inflates std errors, reduces t-stats (artificially small, so variables falsely look unimportant)

Signs: significant F-stat by all t-stats are insignificant, high VIF

Variance Inflation Factor (VIF)

each “X” variable is regressed against other remaining Xs; shows multicollinearity

VIF = 1 → no correlation

VIF > 5 → needs further investigation

VIF > 10 → serious multicollinearity

High-Leverage point

an observation with an extreme value of an “X” variable

leverage is a standardized measure of distance of independent variable observations j from the sample mean and between 0-1

if leverage is 3x the average then the observation is potentially influential

Studentized Residuals

measure of identifying an outlier; DOF = n-k-2

if it is greater than the critical value of the t-stat, then the observation is potentially influential

Dummy Variables

Binary, can only take values of 0 or 1

Dummy Variable Trap

always use d-1 dummy variables to avoid multicollinearity use 3 dummies for 4 quarters)

Logistic Regression Model

estimate probability of an event based on its logistic distribution; probability of event is first converted to odds p/1-p; the log of offs is then used as the dependent variable

Likelihood Ratio

similar to the joint F-test for nested models, this is used for logistic regressions

LR has chi-squared distribution w/ q DOF; q = omitted variables in restricted model

Linear Trend Models

regression w/ time as the independent variable; predicted change in y is b1

Log-Linear Trend Model

assumes the dependent financial variable exhibits exponential growth; slope coefficient b1 is the constant growth

Autoregressive (AR) Models

dependent variable is regressed on prior values of itself, no independent variable

Covariance Stationarity

time series must adhere to this use autoregressive model

constant and finite expected value, constant and finite variance, constant and finite covariance w/ leading or lagged values

Mean Reversion

the value of the dependent variable tends to fall when above its mean and rise when below its mean; can use for AR models

Regression Coefficient Instability

estimated regression coefficients change from period to period, exhibiting instability or nonstationarity

creates tradeoff between statistical reliability of long time series and stability of short time series

AKA: current underlying economic & market conditions are a primary concern when selecting a time series sample period

Random Walks

value in one period = value in previous period + random error

Unit Roots

characteristic of a time series that makes it non-stationary; b1 = 1

First Differencing

used to remove unit roots / nonstationarity; if unit root is present, Xt - (Xt-1) = Et

create new dependent variable, y, defined as the change in x

Seasonality

time series shows consistent seasonal patterns; incorporate the seasonal component in AR(1) [Xt-4 in quarterly or Xt-12 in monthly models]

Autoregressive Conditional Heteroskedasticity (ARCH1)

occurs when variance of error is conditional on variance of error in a previous period;

std. errors, t-stats, & conclusions are incorrect

Detect using BP test; Correct using white-corrected std errors

Cointegration

two time series are related to teh same macro variables or follow the same trend