MODULE 10

Question 1

simple linear regression / least squares regression

Answer 1

explain the variation in a dependent variable in terms of the variation in a single independent variable.
- y intercept and the slope coefficients and the error in the end (residual)
- residual = error
- the regression line minimizes the sum of squared residuals (e2)

Question 2

Regression line

Answer 2

one of many lines that can be drawn through the scatter plot. estimation of this line is the essence of lin regression. This line minimizes the sum of the squared differences between the predicted y and actual y. sometimes referred to as the ordinary last squares regression.

Question 3

sum of squared errors (SSE)

Answer 3

This is the UNEXPLAINED variation!! - var i dep not expl by ind. The model is not explaining this variation.

the difference between the point on the forecasted slope line and the actual value that exists at the point on the Y axis from that X value. We subtract these values and square them and sum them for all the observations.

Question 4

What are the assumptions of a linear regression line and draw them out

Answer 4

1. A linear relationship must exist between the dependent and independent variables.
2. The residual term’s variance remains constant across all observations (homoskedasticity).
3. The residual terms are independently distributed—meaning the residual of one observation does not correlate with another’s residual (in other words, the paired x and y observations are independent of each other).
   
   1. sometimes doesn’nt happen when there is seasonality in data (like people tend to shop way more during saturdays etc)
4. The residual term follows a normal distribution. THIS CAN BE RELAXED IF THE SAMPLE SIZES ARE LARGE
5. The dependent variable is uncorrelated with the residuals.

Question 5

SSR

Answer 5

• EXPLAINED variation
• subtract forecast value - mean value and square it and sum

Question 6

SST

Answer 6

SSE + SSR = SST

Question 7

Mean Squared Error

Answer 7

unexplained / n-k-1 = SSE / n-1 cuz we only work with 1 var here
- VARIANCE around forecast value
- square root to get SE of estimate (kinda like SD)

Question 8

What do you do with this

Answer 8

• calculate from ANOVA table - R^2 = correlation^2
• R squared = explained variation / total variation = SSR / SST - what % of movement in dep is explained by indep
• R squared = .0076 / (.0406 + .0076) = .1577
  
  • R2 high = good because things are more tight around our line
• Standard error of the estimate (SEE) = SQRT (MSE)
  
  • LOWER THE SEE THE BETTER THE MODEL FIT
  • R2 goes up, SEE down

Question 9

what is the test for significance

Answer 9

• F statistic - this is the global test
  
  • tests whether iten ind var explain variation in the dep (overall model significance)
    
    • H0 = all our coefficients are zero
    • Ha = at least one of our coefficients or slope is non zero
    • one tailed tests
    • Reject H0 = F statistic exceeds the critical value
    • df = 2
  • F stat calcu = MSR / MSE =

Question 10

How to tetst individual slope coefficients

Answer 10

• Regression coefficient t-Test
  
  • how many of the indep variables are significant actually?
  • if i fail to reject null = all are insignificant
  • reject null = at least 1 is significant - T TEST TO SEE WHICH ONE
• Null = b1 = 0
• alt = b1 ≠ 0
• df = n-2
• T stat = b1 - b1 hyp / SE
• Can also test int this way

Question 11

do the tests for ABC = .64, se = .26, n=36 sig = 5%

Answer 11

• t = .64 - 0 / .26 = 2.46
• critical stat = 2.03
• 2.46 > 2.03

Question 12

how do you calcualte excess return according ot a regression model

Answer 12

the difference between the actual return and the return predicted by the regression model.

Question 13

Answer 13

R stock = -2.3% + .64 (10%) = 4.1%

Question 14

why do we need to calculate the confidence interval

Answer 14

the int and the slope are sample values so they could give us errors. They’re not true population values
there are no error terms

Question 15

confidence interval information

Answer 15

• We need to have a conf interval to solve this!
  
  • need to use standard error of forecast
  • Critical t is TWO TAILED with n-2 df

Question 16

Answer 16

4.1% +- 2.03 (test stat) x 3.67(sf) = -3.4 , 11.6%

Question 17

types of lin reg relationships

Answer 17

• log-lin: taking natural log of y only
• lin-log: taking natural log of the x variable only
• log-log: taking natural log of both x and y variables

Question 18

what does ANOVA mean?

Answer 18

How good is x at explaining y

Question 19

how do you calculate the slope in a regression line

Answer 19

Cov(xy) / SD^2. OR covariance/variance

Question 20

What is the R squared value

Answer 20

it’s the level of variation in Y explained. The formula is SSR / SST. What percent of the variation in Y is explained

Question 21

What is the SEE?

Answer 21

it’s the volatility of the residual or Square root of the MSE. This is kinda like the SD

Question 22

what happens in a dataset when the sample size is really large

Answer 22

for linear regression, the Sf can be approximated with SEE for large samples because of this equation.

Question 23

when the relationship between x and y is not linear, we can’t fit a linear model. explain what to do and the different situations and outcomes

Answer 23

log - lin - means that we take ln(y) for plain x. This means that the relative change in Y, absolute change in X. Forecast Y is e(ln)

lin-log - means that we take normal Y and ln(x). Absolute change in Y, relative change X

log-log - means that we take ln(y) and ln(x). Relative change in Y, relative change X. Forecast Y is e(lny)

Question 24

Give the intuitive definition of SSE, SSR, SEE, R2, SE, F test, t test

Answer 24

SSE: How much variation the model failed to explain.
SSR: How much variation the model successfully explained.
SEE: Average size of the prediction error.
𝑅2 : Proportion of variation explained by the model.
SE: Precision of the slope estimate.
F-Test: Whether the overall model is significant.
t-Test: Whether individual predictors are significant.