Quantitative Methods

Question 1

Covariance Stationary

Answer 1

Mean and variance don’t change over time

Determined by:

1. Plotting data
2. Run an AR model and test correlations
3. Perform Dickey Fuller Test

Note: Most economic and financial time-series are not stationary

Question 2

T-Test Significance and Z-Test

90%, 95%, 99%

Answer 2

90%: 1.645

95%: 1.96

99%: 2.326

Question 3

When can we reject the null hypothesis

Answer 3

If the t-stat is too big

Question 4

What is covariance

Answer 4

How 2 variances move together

1. Very sensitive when only 2 variables
2. Can be negative infinite or positive infinite

Question 5

Increasing Adjusted R² means _______

Answer 5

The added variables are worth keeping

Question 6

T-Stat Formula

Answer 6

Coefficient / Standard Error

Question 7

Model Misspecification

Answer 7

Types:

1. Time-series: Serial correlation with a lagged variable, or forecasting the past
2. Functional: Omitting a variable or data pooled improperly

Question 8

Correlation Squared Purpose

Answer 8

Explains the variability

Question 9

Correlation

Answer 9

cov / (std of X * std of Y)

OR

(X - Xbar)(Y-Ybar) / √(X-Xbar)²(Y-Ybar)²

Question 10

MSE

Answer 10

SSE / n - k - 1

Question 11

ANOVA Table

Answer 11

Source DOF SOS Mean SOS
Regression(explained) K RSS MSR = RSS/K

Error (unexplained) n-k-1 SSE MSE = SSE/n-k-1

Total n-1 SST

Question 12

R² from ANOVA

Answer 12

Name: Coefficient of Determination

Formulas: RSS / SST
SST - SSE / SST
Correlation²

Purpose: This is the % of variability of Y explained by X’s

Analysis: The higher the better fit

Problem: Always increase as variables are added

Question 13

Random Walk

Answer 13

Unit root: coefficient = 1
This means the null (g = 0), cannot be rejected

Does not have a mean reverting level

Not stationary

Correct by first differencing

Question 14

MSR

Answer 14

RSS / k

Question 15

RMSE

Answer 15

Purpose: to compare the accuracy of AR models for our-of-sample

Formula: √Average squared error

Analysis: lower the better

Question 16

Multiple Regression Analysis Steps

Answer 16

1. Is there model mispecification
2. Is the t-test significant? If no, use another model
3. Is the F-Stat significant? If no, use another model
4. Check for Hetero, serial correlation, and multicollinearity

Question 17

When using Dummy Variables

Answer 17

They are either on of off

Always use n-1 or it will suffer from multcollinearity
Example: If using quarters per year, use 3

Question 18

Log-Linear Trend Model

Answer 18

Purpose: Used when there is exponential growth or there is serial correlation

Formula: y = e^b0 + b1(t)

Use the time/observations for t

Question 19

Sample Correlation Coefficient Other Formulas

Answer 19

Covariance / (Std X * Std Y)

OR

√R²

OR

Covariance / √X * √Y

Question 20

F-Stat

Answer 20

Use: To see if any X’s explain a significant portfolio of Y

Formula: MSR / MSE (only use when DOF is 1 or n-1)

Other forumla: (RSS/k) / (SSE / n-k-1)

Question 21

SST

Answer 21

RSS + SSE

Question 22

Sample Covariance Formula

Answer 22

(X-Xbar)(Y-Ybar) / n - 1

Question 23

Sample Covariance Steps

Answer 23

1. Create a table

Period 1 2 3 4
X-Xbar -.1 .2 .8 -0.9
Y-Ybar -.7 .5 1.1 -0.9 Sum
(X-Xbar)(Y-Ybar) .07 .10 .88 .86 1.86

Then take sum / n - 1

1.86 / 3 = 0.62

Question 24

Multicollinearity

Answer 24

Purpose: High correlation among X’s (Higher than 0.7)

Detect: T-test indicate no coefficients are different from 0
Correct: Drop a variable

Effects: F-Test is significant
R² is too high
All t-stats are below 2

Question 25

Sample Correlation Coefficient (R)

Answer 25

covariance / (sample √X) (sample √Y) (sample √X) = sum of (X-Xbar)² / n - 1 (sample √Y) = sum of (Y-Ybar)² / n - 1

Question 26

AR Models

Answer 26

Use previous values to get the next one. They build upon each other. Correct if autocorrelation of residuals not significant.

Question 27

Mean Reversion

Answer 27

b0 / (1 - b1)

Question 28

Adding additional variables are best evaluated by using....

Answer 28

Adjusted R²

Question 29

Seasonality

Answer 29

Purpose: Model will be misspecified unless the AR model incorporates the effects of seasonality Detect: statistically significant lagged error term Correct: Add an additional term (e.g. last year's quarter)

Question 30

Cointegration

Answer 30

Purpose: Two time-series are economically linked Correct: Regress one variable against the other with the Dickey Fuller Analysis: If null is rejected, they are covariance stationary

Question 31

What does the T-Test Mean

Answer 31

Gives more confidence Will be high if: correlation is high Sample is high

Question 32

Assumptions of Regression: Simple and Multiple

Answer 32

Simple 1. Linear relationship between X and Y 2. Expected value of error term = 0 3. Variance of error term is constant (Heteroskasticity) 4. Errors not serially correlated (Autocorrelation) 5. Error term normally distributed Multiple All the above plus: No exact linear relationship among X's (Multicolinearity)

Question 33

SEE

Answer 33

Name: Standard Error of Estimate (STANDARD DEVIATION) Purpose: Gauges the fit of the regression line. Smaller the better Formula 1: √MSE Formula 2: √ SSE / n - k - 1

Question 34

Covariance Formula

Answer 34

√R² * √X * √Y

Question 35

Confidence Interval

Answer 35

coefficient +/- (critical t value * standard error) Side note: Standard error is SEE

Question 36

When is the slope coefficient significant?

Answer 36

When zero is not included in the range

Question 37

Smallest Alpha to reject the null hypothesis? Under what value?

Answer 37

Answer: p-value Under: 0.5 If under .001 then ARCH exists

Question 38

In/Out of Sample Forecasts

Answer 38

In-sample: estimating data within the range provided Out-of-sample: Estimating outside the range Important b/c it proves whether the model describes the time-series

Question 39

T-Test Formula for Hypothesis

Answer 39

Estimate - Hypothesis / Standard Error (SEE)

Question 40

Limitations of Regression Analysis

Answer 40

1. Parameter Instability 2. Outliers may affect the estimated regression line 3. Spurious Correlation (appearance of a linear line)

Question 41

Degrees of Freedom

Answer 41

Simple Regression: n - 2 Multiple Regression n - k - 1

Question 42

Heteroskedasticity

Answer 42

Purpose: Data spread out on one tail Types: Unconditional and conditional (only conditional has issues) Effects: Non constant error variance F-Test is unreliable Biased T-Statistics Detect: Breusch-Pagan --> Takes errors and compares to X, If R² > 0 then it exists Correct: White-corrected Standard errors (AKA Robust)

Question 43

T-Test Steps

Answer 43

1. Use the formula to calculate the T-Test 2. Look up value of t-table 3. Reject if t > Tcritical or > -Tcritical

Question 44

Autocorrelation

Answer 44

AKA: serial correlation Purpose: correlation among error terms Detect: Durbin Watson (does not work with AR models) DW = 2 * (1 - correlation) If DW is close to 2: No serial correlation If DW is less than 2: Positively correlated If DW is greater than 2: negatively correlated Correct: Hansen

Question 45

Adjusted R² Formula

Answer 45

Purpose: Eliminates the impact of additional variables Formula: 1 - [(n-1 / n-k-1) * (1-R²)] Note: Will always be lower than R²

Question 46

Slope Coefficient and Intercept Term

Answer 46

Slope Coefficient: covariance / variance (or std²) Explain: How much coefficient will move for every 1% change Intercept Term: y - b1(x) Explain: when X is zero

Question 47

ARCH

Answer 47

Purpose: based on a regression of the squared residuals on their lagged values

Question 48

Effects of Model Misspecification

Answer 48

1. Coefficients are biased and inconsistent | 2. Lack of confidence in hypothesis