Quantitative methods

Question 1

Multiple regression model assumptions

Answer 1

• linearity,
• homoskedasticity –> variance of residuals constant
• independence of errors, –> Residuals are not serially correlated
• normality, –> error term is normally distributed evaluated with QQ plot
• independence of independent variables.–> no linear relationships between independent variables

Question 2

MSR

Answer 2

MSR = RSS/k

Question 3

MSE

Answer 3

MSE = SSE/(n−k−1)

Question 4

SST

Answer 4

RSS+SSE

Question 5

R2

Answer 5

RSS/SST
oppure
(SST-SSE)/SST
oppure
(total variation – unexplained variation )/total variation

indica quanto l’indipendent variable puo spiegare

Question 6

Breusch pagan

Answer 6

n*R^2

Question 7

Adjusted R2

Answer 7

1-((n-1)/(n-k-1))*(1-R^2)

o measure of goodness of fit that adjusts for the number of independent variables
o adj R2<R2
o decreases when the added independent variable adds little value to regression model

Question 8

Cook’s D

Answer 8

If observation > √(k/n)–> influential point

Question 9

Odds
Prob given odds

Answer 9

Odds= e^coefficient
Prob with odds = odds/(1+odds)

Question 10

F statistic

Answer 10

((SSEr-SSEu)/q) / (SSEu/(n-k-1))
=MSR/MSE with K and N-K-1 df

H0 all coefficients are zero
reject H0 if F (test-statistic) > Fc (critical value)
to explain whether at least one coefficient is significant

Question 11

Conditional Heteroskedasticity

Answer 11

Residual variance is related to level of independent variables

• Coefficients consistent.
• St. errors underestimated
• Type I errors

DETECTION
* Breusch–Pagan chi-square test
* >5%  hetero
* <5%  no hetero

CORRECTIOn
robust or White-corrected standard errors

Question 12

Serial Correlation

Answer 12

Residuals are correlated with each other

• Coefficients consistent
• St errors underestimated
• Type I errors (positive correlation)

DETECTION
* Breusch–Godfrey (BG) F-test
* Durbin Watson (DW)
* DW<2–> pos. serial corre.

CORRECtION
Use robust or Newey–West corrected standard errors

Question 13

Multicollinearity

Answer 13

Two or more independent variables are highly correlated

• Coefficients are consistent (but unreliable).
• St errors are overestimated
• Type II errors

DETECTION
* Conflicting t and F-statistics
* variance inflation factors (VIF)
* VIF >5 o 10 problema

CORRECTION
* Drop 1 of the correl. variables
* use a different proxy for an included independent variable

Question 14

MISSPECIFICATIONS

Answer 14

Omission of important independent variable(s)–>May lead to serial correlation or heteroskedasticity in the residuals

Inappropriate transformation / variable form–> May lead to heteroskedasticity in the residuals

Inappropriate scaling–>May lead to heteroskedasticity in the residuals or multicollinearity

Data improperly pooled
Solve it by running regression for multiple periods–May lead to heteroskedasticity or serial correlation in the residuals

Question 15

prob with odds

Answer 15

P=(odds)/(1+odds)

Question 16

Autoregressive (AR) Model

Answer 16

• only 1 lag–>dependent variable is regressed against previous values of itself
• no distinction between the dependent and independent variables (i.e., x is the only variable).
• USE t-test to determine whether any of the correlations between residuals at any lag are statistically significant.
• if not covariance stationary To correct add one lag at a time–> first differencing
• Ex: pattern of currency using historical price
  add one lag at a time
• Chain rule forecasting

Question 17

Covariance Stationary

Answer 17

• Statistic significant = cov stationary
  o Constant and finite mean. E(xt) = E(xt-1) ATTENZIONE no growth rate della mean
  o Constant and finite variance.
  o Constant and finite covariance
• determine cov. StationaryDickey-Fuller test

Question 18

Mean Reversion

Answer 18

A time series is mean reverting if it tends towards its mean over tim
=b0/(1-b1)

Se b1 =1–> mean reverting è undefined perchè b0/0

Question 19

Unit Root = Random walk

Answer 19

• B1=1 devo first differencing I dati
  Undefined mean rev. level–>Not covariance stationary

Question 20

Random Walk

Answer 20

• random walk = value in one period is equal to the value in another period, plus a random error.
• Random walk without a drift: xt = xt−1 + εt b0=0 and b1=1
• Random walk with a drift (con b0): xt = b0 + xt−1 + εt b1=1

Question 21

Seasonality

Answer 21

• More than 1 lag
  o quarterly data = seasonal lag is 4;
  o monthly data = seasonal lag is 12.

Question 22

Root Mean Squared Error (RMSE)

Answer 22

to assess accuracy of autoregressive models.
* lower RMSE = better
* Out-of-sample forecasts

Question 23

• structural change

Answer 23

significant shift in the plotted data at a point in time that seems to divide the data into two distinct patterns

Question 24

• Cointegration:

Answer 24

two time series are economically linked (same macro variables) or follow the same trend and that relationship is not expected to change

Question 25

Autoregressive Conditional Heteroskedasticity (ARCH

Answer 25

variance of residuals in 1 time period is dependent on the variance of the residuals in another period.–> st. errors of the coefficients and the hypothesis tests are invalid.

Question 26

1. Penalized regression

Answer 26

Regression–> good when I have features reduces overfitting by imposing a penalty on—and reducing—the nonperforming features.

Question 27

2. Support vector machine

Answer 27

classification; separates the data into one of two possible classifiers based on a model-defined hyperplane.

Question 28

3. K-nearest neighbor

Answer 28

classification based on nearness to the observations in the training sample

Question 29

4. Classification and regression tree

Answer 29

. Classification of target variables
* when there are significant nonlinear relationships among variables.
* Binary classification (categorical data)
* Provides a visual explanation

Question 30

5. Ensemble learning

Answer 30

This combines predictions from multiple models, resulting in a lower average error rate.

Question 31

6. Random forest

Answer 31

This is a variant of the classification tree whereby a large number of classification trees are trained using data bagged from the same data set; solution for overfitting

Question 32

1. Dimension reduction=Principal components analysis

Answer 32

summarizes info into smaller set of uncorrelated factors called eigenvectors.

Question 33

2. K-means clustering.

Answer 33

split observations into k non-overlapping clusters; a centroid is associated with each cluster
* Hyperparameter = parameter set before analysis begins ex. 20 groups

Question 34

3. Hierarchical clustering

Answer 34

hierarchy of clusters without any predefined number of clusters

Question 35

• Neural networks

Answer 35

o input layer
o hidden layers (which process the input)
 The nodes in hidden layer =neurons–> summation operator (that calculates a weighted average) and an activation function (a nonlinear function).
o output layer.

o Good for speech recognition and natural language processing
o Good for modelling complex interactions among many features

Question 36

• Deep learning nets

Answer 36

many hidden layers (more than 20) useful for pattern, speech, and image recognition

Question 37

• Reinforcement learning

Answer 37

seek to learn from their own errors maximizing a defined reward.

Question 38

• precision (P)

Answer 38

true positives / (false positives + true positives)

Question 39

recall

Answer 39

= true positives / (true positives + false negatives)

Question 40

• accuracy

Answer 40

true positives + true negatives) / (all positives and negatives

Question 41

• F1 score

Answer 41

2 × P × R) / (P + R)

Question 42

Covariance stationarity

Answer 42

o Constant and finite mean. E(xt) = E(xt-1) ATTENZIONE no growth rate della mean
o Constant and finite variance.
o Constant and finite covariance with leading or lagged values

• determine cov. StationaryDickey-Fuller test

Question 43

Logistic model

Answer 43

dependent variable is binary

Question 44

steps in a data analysis project

Answer 44

1-conceptualization of the modeling task,
2- data collection,
3- data preparation and wrangling,
4- data exploration,
5- and model training.