CFA Level II Quick Sheet - Quantitative Methods
F1 Score
(2 x P x R)/(P + R)
T - Test
(B^ - B1) Divided By Standard Error DF = N-2
Standard Error of Estimate
(SSE)^1/2
Accuracy
(True Positives +True Negatives) / (True Positives + False Negatives)
Normalization
(X1 - Xmin)/ (XMax -XMin)
Breusch-Pagan test
A test for conditional heteroskedasticity in the error term of a regression. Tests if R^2 is equal to zero which would show that the error is not explained. One Tailed Test
Deep Learning Algorithms
Algorithm such as neural networks and reinforced learning learn from their own prediction errors are used for complex tasks such as image recognition and natural language processing
Supervised Learning
Algorithm uses labeled training data to model relationship
Unsupervised Machine Learning
Algorithm uses unlabeled data to determine the structure of the data
Serial Correlation
Autocorrelation divided by the standard error. Standard Error 1/(N)^1/2 Tested with a T - Test to show variable with significance and misspecification
Mean Reverting Level
B0 divided by (1-B1)
Out of Sample Error
Bias Error + Variance Error + Base Error
Regressing a Random Walk
Both Variables are Covariance Stationary --> Yes One Variable is Covariance Stationary --> No Neither are covariance stationary --> Check for Cointegration
Categories of Supervised Learning
Classification --> Categorical and Ordinal Regression --> Continous
Data Sets for Supervised Learning
Classification and Prediction
Unit Root
Coefficients are equal to one so the series is not covariances non stationary.
Hansen method
Corrects Autocorrelation by adjusting standard errors upward.
White-Corrected Standard Errors
Corrects Heteroskedasticity by inflating the standard errors.
Autocorrelation
Correlation among error terms. Each error term seems to be in the direction of the previous error term. Standard Error is too small.
F - Test - Multicollinearity
Detects Multicollinearity and helps arrive at a variable to omit.
Random Forest
Does not reduce signal to noise ration
Support Machine Vector
Linear Classifier that seeks an optimal hyperplane which separates into two data sets by the maximum margin
Test Significance of Regression Formula (F-Test)
MSR / MSE With K and N-K-1 Degrees of Freedom
F Test Formula
MSR/MSE RSS/k Divided by SSE/K-N-1
Covariance Stationary
Mean and Variance do not change over time. There is a mean reverting level with a constant expected value, covariance and correlation. For Autoregression models to work they must be covariance stationary.
Ensemble learning provides
More accurate and stable predictions
If Regression Shows no significant T-Tests but F-Test is significant than it results in
Multicollinearity
Dummy Variables Used
N-1 The Omitted variable becomes the Intercept
Cointegration
Two Time series are related to the same macroeconomic event which always the ability to test for Beta
Multicollinearity
Two or more "X" Variables are correlated with each other while in reality they are supposed to be independent. Standard Error is too low - Null is tougher to reject and increases chance of type II error.
Neural Networks
Work Well in the Prescence of non-linear and complex interaction amongst variables
Standard X1
X1 - Mean/ Standard Deviation
Confidence Interval for Predicted Y-Value
Y^ + and - (Two Tailed Critical Value at n-2 degrees of freedom) x (Standard Error of Forecast)
A Significant A1 in the ARCH Model allows
for the estimation of the variance of the error term
Big Data is defined as
high volume, velocity and variety which usually leads to low latency
The Coefficient of Determination shows
how much of the independent variable is explained by the variance in the dependent variables
The Presence of Conditional heteroskedasticity of residuals would lead to
invalid standard errors and statistical tests
Autoregression Models use first differencing to
make adjustments for the data to become stationary
A 2 or higher Durbin Watson Score shows
The possibility of negative serial correlation
Dickey Fuller Test
-Test for covariance stationary condition. -Subtract a one-period lagged variable from each side of an autoregressive model, then test to see if the new coefficient is different from 0. -If not different than 0, coefficient must be a unit root
Assumptions of Linear Regression
1. X and Y share linear relationship 2. Variance of the error term (residual) is constant. 3. Residual term is independently distributed and normally distributed.
An Autoregression Model regresses
A Dependent variable against one or more lagged values
Durbin Watson Test
A test to determine whether first-order Autocorrelation is present. Durbin Watson Statistic = 2(1-Correlation) If DWS is lower than 1.34 (positive autocorrelation) or higher than 2.66 (Negative autocorrelation) then there is evidence of auto correlation.
Seasonality
A time series shows consistent seasonal patterns. Shown by Statistically significant lagged error term through a T-Test. Corrected by adding lagged terms
Model Misspecification
Effects Hypothesis Testing and leads to bias coefficients of tests. - Function Form - Incorrect Pooling of Data - Variables should be transformed - Using lagged dependent variables as independent variables - Forecasting the Past - Measuring independent variables with Error
Data explanation
Encompasses Data, Future Selection and Future Engineering
Forecasting the Past
Financial Statements are published with a lag
Random Walk
Happens with B1 = 1 Xt +Xt-1 + E
Autoregression Conditional Heteroskedasticity Model (ARCH)
Heteroskedasticity occurs when variance is non-constant and conditional on the independent variable. The Error Term today correlates to the Error Term Yesterday Et^2 = A0 +A1Et +E
Heteroskedasticity
Non Constant Error Variance Type II Error --> As the independent variable increases the residual variance expands. Stand Errors are too low.
Durbin Watson Test Parameters
Null Hypothesis is that no autocorrection exists Reject Null if DWS is lower than 1.34 or higher than 2.66
F-Test Degrees of Freedom
Numerator --> Independent Variables Used Denominator --> Sample Size - Independent Variables - 1
Measurement Error
Occurs when you are testing a proxy variable (i.e. corporate governance).
Least absolute shrinkage and selection operator (LASSO)
Penalty Regression where the term compromises summing the absolute values of the regression coefficient
Incorrect Pooling of Data
Pooling data from a government that had a recent change in regime.
Sum of Squares Total Formula
RSS + SSE
Mean Squared Regression Formula (MSR)
RSS / k
R^2
Regression Sum of Squares (RSS) divided by Total Sum of Squares
Effects of Model Misspecification
Regression coefficients are often biased and/or inconsistent, which means we can't have any confidence in our hypothesis tests of the coefficients or in the predictions of the model.
Dimension Reduction Seeks to
Remove Noise and the excessive number of features from a data set
First Differencing
Removes the Unit Root from the Data Set
Mean Squared Error Formula (MSE)
SSE/N-K-1
Test of Fitness
Smaller SEE is better fit Higher R^2 is better Fit
Sum of Squared Error Formula (SSE)
Sum of (Y1-Y^)^2
Residual Sum of Squares Formula (RSS)
Sum of (Y^- YMean)^2
Autoregression
The Dependent Variable is regressed on prior values them self To Model sales in the futures we lost at sales in the past. Specified correct if autocorrelation of residuals are not significant
The F-Test Indicates
The Joint Significance of the independent variables
Underfit Model
Treats Parameters as noise
Precision
True Positives / (False Positives + True Positives)
Recall
True Positives/ (True Positives + False Negatives)
Root Mean Square Error
Used when the target variable is continuous. Sum (Predicted - Actual)^2 Divided by N
Log-Linear Model
Used when you do not want a negative outcomes where B1 is the constant rate of growth. LNY = B0 +B1t or y= E^B0t +B1t
Function Form Misspecification
Variables are omitted. If omitted variables are correlated with each other than the residual will be bias.
Variance of ARCH
Variance = A0 + Ai*Et Tested through a T-STAT
Linear Trend Model
Yt = B0 +B1t+ Error
White Standard Errors in heteroskedastic are higher than
bias errors leading to lower computer t-test and less frequent rejection of the null (Type I Error)
In K-fold cross validation technique the data is
randomly separated into K-1 equal sets than then one set become validation sample and the rest are training samples
The Standard Error of Estimate is the difference of
the Estimated values to the actual values of the dependent variables
