CFA Level 2 - Quantitative Methods

n-grams

A technique that compares runs of letters in two values to get a match score from 0 to 100 percent

Deep Learning Algorithms

Algorithms such as neural networks and reinforced learning learn from their own prediction errors and are used for complex tasks such as image recognition and natural language processing.

Log-linear regression

Assumes financial independent variable grows at constant rate

How do you test for Conditional Heteroskedascity?

Breusch Pagan / Chi Squared Test

Ensemble Learning

Combines predictions from models

How to correct overfitting

Complexity Reduction & Cross Validation, penalties are introduced

neural networks

Comprise an input layer, hidden layers (which process the input), and an output layer. The nodes in the hidden layer are called neurons, which comprise a summation operator (that calculates a weighted average) and an activation function (a nonlinear function).

three conditions of covariance stationarity

Constant and finite mean. Constant and finite variance. Constant and finite covariance with leading or lagged values.

What probabilistic models do not account for correlated variables

Decision Trees

Qualitative dependent variables

Dummy variables used as dependent variables rather than as independent variables (Zero or One)

How to test for Serial Correlation

Durbin Watson Test

Durbin-Watson Test Decision Rule

If DW stat < dL then reject H0 >> Positive Serial Correlation If dL < DW < dU then test is inconclusive If DW is > dU then do not reject

Residual Sum of Squares (SSR)

In multiple regression analysis, the sum of the squared OLS residuals across all observations (RSS/K)

Supervised Learning

Inputs and outputs are identified for the computer, and the algorithm uses this labeled training data to model relationships.

Overfitting & Issues

Large number of variables, but low out of sample accuracy/r2

Can the Durbin Watson Test be used with AR Models?

NO

Underfit Model

Underfitting describes a machine learning model that is not complex enough to describe the data it is meant to analyze. An underfit model treats true parameters as noise and fails to identify the actual patterns and relationships

What regression do you use when rate of change is constant?

Whenever the rate of change is constant over time, the appropriate model is a log-linear trend model. The other two choices are a linear trend model and an autoregressive model.

How to correct conditional heteroskedascity

White-Corrected Standard errors

correcting for structural shift

estimate the models before and after the change

When to use classification and regression tree

for non-linear data

Standard error of estimate

gives a measure of the standard distance between the predicted Y values on the regression line and the actual Y values in the data...if the relationship between the dependent and independent variables is very strong, the SEE will be low.

F-test (ANOVA)

has two distinct degrees of freedom, one associated with the numerator (k, the number of independent variables) and one associated with the denominator (n − k − 1). The critical value is taken from an F-table. The decision rule for the F-test is reject H0 if F > Fcritical. Remember that this is always a one-tailed test.

what does adjusted r squared measure

he adjusted R2 provides a measure of the goodness of fit that adjusts for the number of independent variables included in the model

K-Nearest Neighbor

his is used to classify an observation based on nearness to the observations in the training sample.

When is linear trend model used?

if the data points seem to be equally distributed above and below the line and the mean is constant.

presence of autoregressive conditional heteroskedasticity (ARCH)

indicates that the variance of the error terms is not constant. This is a violation of the regression assumptions upon which time series models are based.

Random Forest

large number of classification trees

AR Model Formula

xt = b0 + b1xt − 1 + b2xt − 2 + ... + bpxt − p + εt

Log Linear Regression Formula

yt = e^ [b0 + b1(t)] ln(yt) = ln(eb0 + b1(t)) ⇒ ln(yt) = b0 + b1(t)

Sum of Squared Errors (SSE)

-Measures the unexplained variation in the dependent variable that is explained by the independent variable -The sum of squared vertical distances between estimated and actual Y-values -Sum of squared residuals (SSE)

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

Regression Assumptions

1) Linearity 2) independence of errors 3) homoscedasticity 4) normality of error distribution The assumptions include a normally distributed residual with a constant variance and a mean of zero.

Conditional Heteroskedasticity: 1. What is it? 2. Effect? 3. Detection? 4. Correction?

1. Residual variance related to level of independent variables. 2. Standard errors are unreliable, but the slope coefficients are consistent and unbiased. 3. BP Chi squared test 4. White-Corrected Std. Errors or Hansen method

Serial correlation (autocorrelation) 1. What is it? 2. Effect? 3. Detection? 4. Correction?

1. Residuals are correlated. 2.Type I errors (for positive correlation) but the slope coefficients are consistent and unbiased. 3. Durbin Watson Test 4. Hansen adjusted standard errors

Sum of Squares Total (SST)

= RSS + SSE

Chain Rule of Forecasting

A forecasting process in which the next period's value as predicted by the forecasting equation is substituted into the right-hand side of the equation to give a predicted value two periods ahead. ˆxt+1=ˆb0+ˆb1xt

Autoregressive model

A regression model in which the independent variables are previous values of the time series.

Multicollinearity

A situation in which several independent variables are highly correlated with each other. This characteristic can result in difficulty in estimating separate or independent regression coefficients for the correlated variables. Too many Type II errors and causes coefficient estimates to be unreliable and standard errors to be biased.

F statistic

A statistical test to determine the relationship between the variances of two or more samples. (MSR/MSE)

ANOVA table

A table used to summarize the analysis of variance computations and results. It contains columns showing the source of variation, the sum of squares, the degrees of freedom, the mean square, the F value(s), and the p-value(s). 1. Regression (RSS) / df=k 2. Error (SSE) / df= n-k-1 3. Total (SST) / df = n-1

data curation

Activities to preserve/maintain the quality of data

correction for seasonality

Add a seasonal Lag - Plot data. Check seasonal residuals (autocorrelations) for significance. If residuals are significant, add the appropriate lag (e.g., for monthly data, add the 12th lag of the time series).

Machine Learning Summary

Find the pattern, apply the pattern

Correcting for Linear Trend

First Differencing

Likely candidates for linear trend model

Growth & GDP

Cointegration

Occurs when two time series are moving with a common pattern due to a connection between the two time series

How to Correct Multicollinearity

Omit one or more of the correlated independent variables

Six common misspecifications of the regression model

Omitting a variable. Transforming variable. Incorrectly pooling data. Using a lagged dependent variable as an independent variable. Forecasting the past. Measuring independent variables with error.

How to test for covariance stationarity

Plot the data to see if the mean and variance remain constant. Perform the Dickey-Fuller test (which is a test for a unit root, or if b1 − 1 is equal to zero or b1=1)

positive serial correlation

Positive serial correlation is the condition where a positive regression error in one time period increases the likelihood of having a positive regression error in the next time period. The residual terms are correlated with one another, leading to coefficient error terms that are too small.

Machine Learning Metrics

Precision (P) = true positives / (false positives + true positives) recall (R) = true positives / (true positives + false negatives) accuracy = (true positives + true negatives) / (all positives and negatives) F1 score = (2 × P × R) / (P + R)

k-means clustering

Process of organizing observations into one of k groups based on a measure of similarity.

RMSE (root mean squared error)

Root mean squared error (RMSE) is used to assess the predictive accuracy of autoregressive models (FOR OUT OF SAMPLE DATA) For example, you could compare the results of an AR(1) and an AR(2) model. The RMSE is the square root of the average (or mean) squared error. The model with the lower RMSE is better.

Data Normalization

Scales variables between 0 and 1

When to use a log-linear trend model

Seasonality, data series that exhibits a trend or for which the residuals are correlated or predictable or the mean is non-constant, exponential growth

Structural Change (Coefficient Instability)

Shorter time periods have higher coefficient stability....structural change is indicated by a significant shift in the plotted data at a point in time that seems to divide the data into two distinct patterns.

t statistic formula

Single Variables: The numerator measures the actual difference between the sample data (M) and the population hypothesis (μ). The estimated standard error in the denominator measures how much difference is reasonable to expect between a sample mean and the population mean.

Standard error of Estimate Formula

Square Root (SSresidual/df) or Square root (MSE)

Support Vector Machine

Supervised learning classification tool that seeks a dividing hyperplane for any number of dimensions can be used for regression or classification

Dickey Fuller-Engle Granger (DF-EG)

Test to cointegration. Dickey-Fuller test (which is a test for a unit root, or if b1 − 1 is equal to zero). Null Hypothesis of a unit root

Tokenization

Text is considered to be a collection of tokens, where a token is equivalent to a word. Tokenization is the process of splitting a given text into separate tokens. Bag-of-words (BOW) is a collection of a distinct set of tokens from all the texts in a sample dataset. Stemming is the process of converting inflected word forms into a base word.

Unsupervised Machine Learning

The computer is not given labeled data; rather, it is provided unlabeled data that the algorithm uses to determine the structure of the data

Effects of Model Misspecification

The effects of the model misspecification on the regression results are basically the same for all the misspecifications: regression coefficients are biased and inconsistent, which means we can't have any confidence in our hypothesis tests of the coefficients or in the predictions of the model.

hierarchal clustering

This builds a hierarchy of clusters without any predefined number of clusters.

First Differencing

This is used to correct AR when it has a unit root. You subtract the previous value from the current value to define NEW DEPENDENT variable. Here you graph the CHANGE in the values which is essential the error term. (Yt = = xt − xt−1 ⇒ yt = εt) b0=b1=0 & Yt = Et

Unit Root

To test: 1. Examine autocorrelations 2. Use Dicky Fuller Test If b1 is close or equal to 1 then it is similar to Random Walk which means it is NON STATIONARY.

How to correct for serial correlation

Use Hansen method to adjust Standard Errors

Root Mean Squared Error

[Sum (Predicted - Actual)^2 / n] ^ 0.5 Lower RMSE equals higher predictive power

Principal Component Analysis (PCA)

a dimension-reduction tool that can be used to reduce a large set of variables to a small set that still contains most of the information in the large set

low latency

a network that doesn't have delay

Deep Learning Nets

are neural networks with many hidden layers (more than 20) useful for pattern, speech, and image recognition.

Mean Reversion Formula

b0 / (1-b1)

Data standardization

centers variables at mean of zero and std. dev of 1

Autoregressive Conditional Heteroskedasticity (ARCH)

describes the condition where the variance of the residuals in one time period within a time series is dependent on the variance of the residuals in another period. When this condition exists, the standard errors of the regression coefficients in AR models and the hypothesis tests of these coefficients are invalid.

Confidence Interval

predicted Y value ± (critical t-value)(standard error of forecast)

bag of words model

procedure then collects all the tokens in a document

Confidence Interval of regression coefficient

regression coefficient ± (critical t-value)(standard error of regression coefficient) If zero is contained in the confidence interval constructed for a coefficient at a desired significance level, we conclude that the slope is not statistically different from zero.

Correcting for Exponential trend

take natural log and first difference

Mean Squared Error (MSE)

the average of the squared differences between the forecasted and observed values (SSE / n - k - 1 )

What is the mean reverting level when a unit root is present?

the mean reverting level is undefined (b1 = 1), so the series is not covariance stationary.

R squared

the proportion of the total variation in a dependent variable explained by an independent variable (coefficient of determination) (RSS/SST)

If a time series has a random walk then >>

unit root means that the least squares regression procedure that we have been using to estimate an AR(1) model cannot be used without transforming the data first. A time series with a unit root will follow a random walk process. Since a time series that follows a random walk is not covariance stationary, modeling such a time series in an AR model can lead to incorrect statistical conclusions, and decisions made on the basis of these conclusions may be wrong. Unit roots are most likely to occur in time series that trend over time or have a seasonal element.

How to test for serial correlation in an AR Model

use a t-test to see if correlations at any lag are statistically significant