Quant

Front 1

Sample Covariance

Back 1

Statistical measure of the degree to which two variables move together

[Sum of (Xi - X mean)(Yi - Y mean)]/(n-1)

Front 2

Sample Correlation Coefficient

Back 2

Measure of the strength of the linear relationship (correlation) between two variables. Ranges from -1 to 1. Measured by (r)

(Covariance of X and Y)/[(Sample Standard Deviation of X)(Sample Standard Deviation of Y)

Front 3

Limitations on Correlation Analysis

Back 3

1. Outliers can greatly influence

2. Nonlinear relationships are not measured.

3. Spurious correlations that have no logical backing can occur.

Front 4

Testing Statistical Significance of Correlation Coefficient

Back 4

t = (r*sqrt(n-2)) / sqrt (1-r^2)

Front 5

General Form of Simple Linear Regression

Back 5

Dependent Variable = Intercept + Slope Coefficient * Independent Variable + Residual Error

Front 6

Independent Variable vs. Dependent Variable for Linear Regression

Back 6

Independent (X): Variable used to explain the variation in the dependent variable (Y)

Front 7

Assumptions Made With Simple Linear Regression

Back 7

1. There is a linear relationship between dependent and independent variable.

2. Independent variable is uncorrelated with residuals

3. The expected value of the residual term is zero

4. The variance of the residual term is constant for all variables

5. The residual term is independently and normally distributed

Front 8

Sum of Squared Errors

(SSE)

Back 8

The sum of the squared vertical distances between the estimated and actual Y-Values

Front 9

Slope Coefficient

Back 9

b1 The change in Y (Dependent variable) for a one unit change in X. Beta of a stock is the slope coefficient.

Front 10

Degrees of Freedom in Linear Regression

Back 10

(n - k - 1)

k: the number of independent variable used in a linear regression.

Front 11

Test of Statistical Significance

Back 11

Test to see if correlation does not equal 0.

Front 12

Conducting Hypothesis Tests for Linear Regressions

Back 12

If the test does not yield 0 when calculating

slope coefficient (b1) +/- ( t * standard error of coefficient)

then do not reject the hypothesis.

Front 13

Standard Error of Estimate (SEE)

Back 13

Measures the variability of the actual Y-values relative to estimated Y-Values from a regression equation, smaller error, the better the line "fits". The standard deviation of of the regression error terms.

for simple linear regression:

SEE = sqrt ((SSE)/(n-2))

Front 14

Coefficient of Determintion

Back 14

Proportion of the total variation of the dependent variable explained by the independent variable. An R^2 of .63 indicates that 63% of the variance of the variation in the dependent variable is explained by the independent variable.

R^2 = RSS/SST

Front 15

Regression Sum of Squares

Back 15

RSS: Measures the variation in the dependent variable explained by the independent variable. Sum of the squared distances between the predicted Y-Values and the mean of Y

Front 16

Total Sum of Squares

Back 16

Measures the total variation in the dependent variable. SST is equal to the sum of the squared differences between the actual Y-values and the mean of Y:

Sum of (Yi - Mean Y)^2

SST = Regression sum of squares + Sum of Squared Errors

Not the same as variance, which is SST/n-1

Front 17

ANOVA Tables

Back 17

Front 18

Mean Regression Sum of Squares

Back 18

MSR: Regression sum of squares/k

Front 19

Mean Squared Error

Back 19

Sum of Squared Errors/(n - k - 1)

Front 20

Calculating Coefficient of Determination (R^2)

Back 20

R^2 = (Total Variation (SST) - Unexplained Variation (SSE))/ Total Variation (SST)

Front 21

Simple Linear Regression (R^2) Calculation

Back 21

covXY ^ 2

Front 22

F-Statistic Calculation

Back 22

MSR/MSE

Front 23

Limitations of Linear Regression Testing

Back 23

1. Parameter instability (Relationships change over time)

2. Public knowledge of relationships eliminate usefulness to traders

3. Assumptions can be violated.

Front 24

Coefficient of Variation Next to Dependent Variable

Back 24

What percent increase/decrease in a one unit change in the independent variable.

Front 25

Alpha in Regression Lines

Back 25

The intercept is a measure of alpha

Front 26

F-Statistic

Back 26

Always one-tailed. Assesses how well the set of independent variables, as a group, explains the variation in the dependent variable.

MSR/MSE

Front 27

Standard Error of Forecast

Back 27

Value to be used for confidence testing.

Front 28

t-Statistic For Multiple Linear Regression Formula

Back 28

(Estimated Slope Coefficient - Hypothesized Slop e Coefficient)/ Standard Error

If you are testing to make sure null does not equal 0, then hypothesized slope is 0 for this calculation. Reject if absolute value of t-stat is is greater than the absolute value of the critical stat.

Front 29

p-Values for Multiple Linear Regression

Back 29

The probability to the right of the computed value (the confidence interval). Reject the null if the p-value is less than the significance value. Compare p-values to alpha. Will reflect whether it was a 1 or two tail test.

p-values default to a two tailed test. To test significance in a 1 tail, divide p value by 2.

Front 30

High p-values in Multiple Linear Regression

Back 30

If p-value is above significance level, then it is not significantly significant

Front 31

Critical t-Stat Depends On:

Back 31

1. Significance (alpha) = 1 - confidence level

2. Degrees of freedom: n - k - 1

3. Confidence internals are always two tailed

Front 32

ANOVA Tables: Sum of Squares

Back 32

RSS (explained variance) + SSE (unexplained variance) = SST (total variance)

Front 33

F-Statistics in Multiple Linear Regressions

Back 33

Tests the ratio of the average explained variation over the average unexplained variation. Tests whether any of the independent variables explain variation in the dependent variable. One tailed test.

Reject null if F-statistic exceeds critical value.

Front 34

Determining Degrees of Freedom for F-Statistics

Back 34

Numerator: explained variation per independent variable (RSS/k or MSR)

Denominator: (SSE/(n - k - 1) Amount of unexplained variation.

Front 35

F-Statistic Testing

Back 35

MSR/MSE = F-statistic

High F statistic means that there is a lot of average explained variation for every unit of unexplained variation.

Front 36

Adjusted R^2

Back 36

Unadjusted R^2 will always go up as more variables are added, adjusted R^2 corrects this issue. The model with higher adjusted R^2 is always the preference over one with higher unadjusted R^2

Adjusted R^2 = 1 - [(n-1)/(n - k - 1) * (1 - r^2)]

Front 37

R^2

Back 37

Explained Variance (RSS)/ Total Variance (SST)

Always goes up as new variables are added.

Front 38

Dummy Variables

Back 38

Can only take the values of 0 or 1. An example is calendar studies.

Front 39

Dummy Variable Trap

Back 39

Always use n-1 dummy variables to avoid multicollinearity. ex. 3 dummies for 4 quarters in a year, or 11 dummies for 12 months in the year.

Need at least one as a base.

Intercept is equal to the omitted dummy.

Front 40

Heteroskedasticity Definition/Assumption Violation

Back 40

Violates assumption that error term has constant variance

Heteroskedasticity means that the error term variance is non-constant.

Front 41

Serial Correlation Definition/Assumption Violation

Back 41

Violates assumption error terms are not correlated with each other

Each term tends to move with the previous term. Positive Autocorrelation means they move in the same direction (more common than negative autocorrelation)

Front 42

Multicollinearity Definition/Assumption Violation

Back 42

Violates assumption that their is no exact linear relationship among "X" variables.

Two or more X variables are highly correlated with each other.

Front 43

Effects of Heteroskedasticity

Back 43

Estimated value of slope is unimpacted, but the standard error of the slope tend to be too small, causing t-stat to rise, reject null (type I error)

Front 44

Effects of Serial Correlation

Back 44

Slope is unaffected, standard errors are too low, t-statistic is inflated, increasing the chances of type I errors.

Front 45

Effects of Multicollinearity

Back 45

Inflates standard errors, reducing t-stats, and increases your chance of type II errors. Variables appear falsely unimportant.

Front 46

How to Detect Heteroskedasticity

Back 46

Either look at a scatter plot or conduct a Breush-Pagan test. Regress squared residuals against X. Compute R^2, hoping the value is low since you want errors to 0. Test null that R^2 of this second regression is 0. Compare against critical chi-squared variable.

Front 47

How to Detect Serial Correlation

Back 47

Use a scatter plot

Use the Durbin-Watson Statistic (roughly 2(1 - r)

Front 48

How to Detect Multicollinearity

Back 48

F-stat says the model is valid and R^2 is high, yet all t-stats are insignificant.

If correlation between two X values is high, then there is likely multicollinearity. Only applies to regressions with 2 independent variables.

Front 49

How to Correct Heteroskedasticity

Back 49

1. Use robust standard errors (white corrected standard errors). This inflates standard errors, lowers t-stats, and makes conclusions more accurate.

2. Use generalized least squares (change the model)

Front 50

How to Correct Serial Correlation

Back 50

Use the Hansen method, inflates standard errors and allows for recalculation of t-statistics. Chance of type I error declines. Hansen method can solve conditional heteroskedacity as well.

Front 51

How to Correct Multicollinearity

Back 51

Omit one or more X variables.

Front 52

Conditional vs. Unconditional Heteroskedasticity

Back 52

Unconditional: Variance of the error is not constant, but it is not dependent on independent variable. Does not cause major problems.

Conditional: Variance is not constant and is related to the value of the independent variable. ex. the higher the value of X, the higher the variance of errors. Does cause issues.

Front 53

End Points of Durbin-Watson Model

Back 53

0: perfect negative correlation

2: no correlation

4: perfect positive correlations.

If Durbin watson computed is less than the Durbin lower, then there is positive serial correlation.

Front 54

Functional Form Misspecifications

Back 54

1. Important variables are omitted

2. Variables are not transformed properly

3. Data may be pooled improperly

Front 55

Time-Series Misspecifications

Back 55

1. X is lagged Y with serial correlation present

2. Forecasting the past

3. Measurement error.

Front 56

Probit Model

Back 56

Estimate the probability of default given values of X based on the normal distribution

Front 57

Logit Models

Back 57

Estimate probability of default based on logistic distribution

Front 58

Discriminant Model

Back 58

Produce a score or ranking used to classify into categories (credit score)

Front 59

Standard Error of Estimate Calculation

Back 59

sqrt ( SSE/(1 - n - k))

Front 60

Log-Linear Trend Model Formula

Back 60

ln y(t) = b0 + b1(t) + error

Appropriate for exponential data.

Front 61

Covariance Stationary

Back 61

Requirement for using Autoregressive Models.

Requires:

1. Constant and finite expected value

2. Constant and finite variance

3. Constant and finite covariance with leading or lagged variables

Front 62

Autoregressive Models

Back 62

The dependent variable is regressed on prior values of itself. Requires Covariance Stationarity.

Front 63

Testing for Serial Correlation in AR Models

Back 63

Use t-statistic. If t-stat is above 2, the model is not correctly specified. Add a lag.

Front 64

Autocorrelation Tables

Back 64

Autocorrelation column shows correlation between x at lag one and x at lag t - 1.

Standard Error (SE): 1/sqrt(n)

Front 65

Testing if AR time-series Model is Correctly Specified

Back 65

1. Estimate the AR model being evaluated using linear regression.

2. Calculate the autocorrelations's of the model's residuals

3. Test whether autocorrelations are significant.

Front 66

Mean Reverting Level for AR(1)

Back 66

b0/ (1 - b1)

A time series is mean reverting if the dependent variable falls to a mean. If b1 = 1, then you are dividing by zero and essentially saying:

x(t) = b0 + 1*x(t-1) + error

Since error is impossible to estimate, this would be useless.

Forecast values will always revert to MRL

Front 67

Root Mean Squared Error (RMSE)

Back 67

Variable used to compare accuracy of AR models in forecasting out-of-sample values. Use lowest RMSE model when predicting out of sample values.

Square root of MSE = RMSE = SEE = Standard deviation of the error term.

Front 68

Instability of Time-Series Models

Back 68

Estimated coefficients can change over time. Larger sample sizes are more dependable, but also contain data that may no longer be relevant.

Front 69

Random Walk

Back 69

Value in time series data that is equal to the value of another period + error. Does not have mean reverting level and is not stationary.

Front 70

Unit Roots

Back 70

Occur when the coefficient of a lagged dependent variable is equal to 1. This will not be covariance stationary. Data must be differenced before using in time series model.

Unit roots can be tested for using Dickey-Fuller Test

Front 71

Seasonality

Back 71

Tested by calculating the autocorrelations of the error terms. Add seasonal coefficient to counteract.

Front 72

Dickey-Fuller Test

Back 72

Tests to see if there is a unit root (b1 = 0). If the null is rejected (b1 does not = 0) the series does not have a unit root.

Front 73

Cointegration

Back 73

Two variables are linked to the same same trend and that relationship is not expected to change. Model can still be good under this circumstance, ex. a stock and the overall market are correlated.

Front 74

One Period Ahead vs. Two Period Ahead Forecasts

Back 74

One: x(t+1) = b0 + b1x(t)

Two: x(t+2) = b0 + b1x(t+1)

Front 75

Standard Error of Autocorrelations Calculation

Back 75

1/ sqrt(number of observations)

Front 76

Steps for Running a Simulation

Back 76

1. Determine Problematic Variables (uncertain inputs that influence the value of an investment)

2. Define probability distributions for these variables

3. Check for correlation among these variables

4. Run the simulation

Front 77

How To Treat Correlation Across Variables in A Simulation

Back 77

1. Allow only one variable that has the greatest influence to vary

2. Build the rules of correlation into the model

Front 78

Three Ways to Define the Probability Distributions for a Simulation's Variables

Back 78

1. Use historical data

2. Use cross sectional data (data from a similar firm)

3. Pick a distribution and use your discretion to estimate the parameters (ad-hoc)

Front 79

Advantages of Using Simulations in Decision Making

Back 79

1. Input accuracy can be greater, since they are manually input

2. Expected value is a distribution, not just a point estimate.

Front 80

Constraints on Simulations

Back 80

1. Book Value Constraints (regulatory reasons)

2. Earnings and cash flow constraints

3. Market value contraints

Front 81

Issues in Simulations

Back 81

1. Garbage in, garbage out

2. Real data may not follow normal distributions

3. Non-stationarity (what worked yesterday, may not work tomorrow)

4. Dynamic correlations

Front 82

Scenario Analysis

Back 82

Defines best case, worst case, and base case. Sum of probabilities <1, ignores all other possibilities. Better suited for discrete outcomes.

Front 83

Decision Trees

Back 83

Best suited for sequential and discrete risks. Correlated risks are difficult to model in Decision trees. Best with historical data since you need to estimate probabilities at each node.

Front 84

Simulations

Back 84

Considers all possible outcomes. Better suited for continuous risks. Allow for explicitly modelling correlations of input variables.

Front 85

Simulations vs. Scenario Analysis vs. Decision Trees Chart.

Back 85

a