CFA L2 Quant

Question 1

True or false: With financial instruments, we can typically use a one-factor linear regression model?

Answer 1

False, typically we need a multiple regression model.

Question 2

Multiple regression model

Answer 2

Regression models that allow to see the effects of multiple independent variables on one dependent variable.

Ex: Can the 10-year growth in the S&P 500 (dependent variable (Y)) be explained by the trailing dividend payout ratio of the index’s stocks (independent variable 1 (X1)) and the yield curve slope (independent variable 2 (X2))?

Question 3

What are the uses of multiple regression models?

Answer 3

• Identify relationships between variables.
• Forecast variables. (ex: forecast CFs or forecast probability of default)
• Test existing theories.

Question 4

Standard error

Answer 4

A statistical measure that shows how well the sample represents the population

Question 5

Residual (ε)

Answer 5

The difference between the observed Y value and the predicted Y value (ŷ).

ε = Y - ŷ
OR
Y - (b0 + b1x1 + b2x2 … + bnxn)

Question 6

P-value

Answer 6

The smallest level of significance for which the null hypothesis can be rejected.

• If the p-value is less than the significance level (α), the null hypothesis can be rejected and if it’s greater it is failed to be rejected.

Question 7

If the significance level is 5% and the p-value is .06, do we reject the null hypothesis?

Answer 7

No, we fail to reject the null hypothesis.

Question 8

Assumptions underlying a mutliple regression model:

Answer 8

• A linear relationship exists between the dependent and independent variables.
• The residuals are normally distributed.
• The variance of the error terms is constant.
• The residual of one observation ISN’T correlated w/ another.
• The independent variables ARE NOT random
• There is no linear relationship between any two or more independent variables.

Question 9

Q-Q plot

Answer 9

A plot used to compare a variable’s distribution to a normal distribution. The residual of the variable’s distribution should lie along a diagonal line if they follow a normal distribution.

Question 10

True or false: For a standard normal distribution, only 5% of the observations should be beyond -2 standard deviations of 0?

Answer 10

False, only 5% of the observations should be beyond -1.65 standard deviations.

Question 11

Analysis of variance (ANOVA)

Answer 11

A statistical test used to assess the difference between the means of more than two groups. At its core, ANOVA allows you to simultaneously compare arithmetic means across groups. You can determine whether the differences observed are due to random chance or if they reflect genuine, meaningful differences.

• A one-way ANOVA uses one independent variable.
• A two-way ANOVA uses two or more independent variables.

Question 12

Coefficient of determination (R^2)

Answer 12

The percentage of the total variation in the dependent variable explained by the independent variable(s).

R^2 = SSR/SST
OR
(SST - SSE) / SST

Ex: R^2 of 0.63 means that the model explains 63% of the variation in the dependent variable.

SSR= regression sum of squares. It’s the sum of the differences between the predicted value and the mean of the dependent variable.
RSS= regression sum of squares. It’s the total variation in the dependent variable explained by the independent variable.

Question 13

Adjusted R^2

Answer 13

Since R^2 almost always increases as more independent variables are added to the model, we must adjust it.

• If adding an additional independent variable causes the adjusted R^2 to decrease, it’s not worth adding that variable.

Question 14

Overfitting

Answer 14

When R^2 is high because there is a large # of indepedent variables, rather than a strong explanation.

Question 15

Akaike’s information criterion (AIC)

Answer 15

Looks at multiple regression models and determines which has the best forecast.

Calculation: (n * ln(SSE/n)) + 2(k+1)

• Lower values indicate a better model.
• Higher k values result in higher values of the criteria.

Question 16

Schwarz’s Bayesian information criteria (BIC)

Answer 16

Looks at multiple regression models and determines which has a better goodness of fit.

Calculation: (n * ln(SSE/n)) + (ln(n)*(k+1))

• Lower values indicate a better model.
• Higher k values result in higher values of the criteria.
• BIC imposes a higher penalty for overfitting than AIC.
• AIC and BIC are alternatives to R^2 and adjusted R^2 to determine the quality of the regression model.

Question 17

Nested models

Answer 17

Models that have a full model and an unrestricted model.

Question 18

Full model vs restricted model

Answer 18

Full model= A linear regression model that uses all k independent variables

Restricted model= A linear regression model that only uses some of the k independent variables

Question 19

Joint F-Test

Answer 19

Measures how well a set of independent variables, as a group, explains the variation in the dependent variable. Put simply, it tests overall model significance.

Calculation: [ (SSErestricted - SSEunrestricted) / Q ] / [ (SSEunrestricted) / (n - k - 1) ]
* Q = # of excluded variables in the restricted model.
* Decision rule: reject the null hypothesis if F-stat > F critical value.

Question 20

True or false: We could also use a t-test to evaluate the significance to see which variables are significant?

Answer 20

True, but the F-test provides a more meaningful evaluation since there is likely some amount of correlation among independent variables.

Question 21

True of false: The F-test will tell us if at least one of the slope coefficients in a multiple regression model is statistically different from 0?

Answer 21

TRUE

Question 22

True or false: When testing the hypothesis that all the regression coefficients are simultaneously equal to 0, the F-test is always a two tailed test?

Answer 22

False, when testing the hypothesis that all the regression coefficients are simultaneously equal to 0, the F-test is always a one tailed test.

Question 23

True or false: We can use the regression equation to make predictions about the dependent variable based on forecasted values of the independent variable?

Answer 23

True, we can make predictions.

Question 24

Predicting the dependent variable from forecasted values of the independent variable:

Answer 24

ŷ = predicted value of the intercept + (X1 * estimated slope coefficient for X1) + (X2 * estimated slope coefficient for X2)…

Question 25

Functional form misspecifications (A regression suffers from misspecification of the functional form when the functional form of the estimated regression model differs from the functional form of the population regression function):

Answer 25

• Omission of important independent variables: may lead to biased and inconsistent regression parameters OR serial correlation or heteroskedasticity in the residuals.
• Inappropriate variable form (ex: you may need to take the natural log of a variable): may lead to heteroskedasticity in the residuals. This can happen if there is no linear relationship between the independent & dependent variables.
• Inappropriate variable scaling (ex: common-size financial statements): May lead to heteroskedasticity in the residuals or multicollinearity.
• Data improperly pooled: May lead to heteroskedasticity or serial correlation in the residuals.

Question 26

Heteroskedasticity

Answer 26

When the variance of the residuals is not constant across all observations in the sample. This happens when there are subsamples that are more spread out than the rest of the sample.

Question 27

Unconditional heteroskedasticity

Answer 27

When the heteroskedasticity is not related to the # of independent variables meaning heteroskedasticity won’t increase/decrease as the amount of independent variables increase/decrease.

• Although it’s a violation of our assumptions, it is usually not a big problem.

Question 28

Conditional heteroskedasticity

Answer 28

Heteroskedasticity that is related to the # of independent variables. Creates significant problems for statistical interference if not corrected properly.

• Conditional heteroskedasticity DOES NOT affect the slope coefficients. It DOES affect the computed F-stat and and computed T-stat

Question 29

Effects of conditional heteroskedasticity

Answer 29

If the pattern of heteroskedasticity is low (most observations on the plot are low values): Standard errors (SEE) of the coefficients in a regression are affected by conditional heteroskedasticity and usually become unreliable estimates by being underestimated. This will lead to the T-stat being too large too often and thus rejecting the null too often, a.k.a type 1 error.

• For the F test (MSR/MSE), MSE is underestimated, and therefore the F-stat is often too large leading to the null is rejected too often, a.ka type 1 error.
• If the pattern of heteroskedasticity is high (most observations on the plot are high values): the same errors will happen but in the opposite direction.

Question 30

How to detect conditional heteroskedasticity

Answer 30

There are two methods of detection: examining scatter plots of the residuals and by using the Breusch-Pagan chi-square test.

Question 31

How to use scatterplots to detect heteroskedasticity?

Answer 31

Look at a scatterplot of the residuals vs the independent variables. If the variation is constant there is no heteroskedasticity. If it’s not constant, there is heteroskedasticity.

Question 32

Breusch-Pagan Chi-Square (BP) Test

Answer 32

A test used to detect heteroskedasticity. The BP test calls for the squared residuals (as the dependent variable) to be regressed on the original set of independent variables. If conditional heteroskedasticity is present, the independent variables will significantly contribute to the explanation of the variability in the squared residuals.

• We want a small R^2 when using a BP test.
• This is a one-tailed test because we are only concerned w/ large values.
• Use a chi-square dist. with k df

Question 33

How to correct heteroskedasticity?

Answer 33

We can use robust standard errors/white-corrected standard errors/heteroskedasticity-consistent standard errors

Question 34

Serial correlation/autocorrelation

Answer 34

When residuals are correlated with each other.

• Poses serious problems when using time series data.

Question 35

Positive serial correlation

Answer 35

When a positive residual in one time period increases the probability of observing a positive residual in the next time period.

• This type of correlation typically results in coefficient standard errors that are too small, causing T-stats or F-stats to be too large, which will lead to type 1 errors.

Question 36

Effect of serial correlation on model parameters

Answer 36

If the dependent variable’s reaction to the independent variable has a lag in a regression model, serial correlation causes the estimates of the slope coefficients to be inconsistent. If there is no lag, then the estimates of the slope coefficient will be consistent.

Question 37

How to detect serial correlation?

Answer 37

First, we can use a scatter plot. This will show very dramatic scenarios. We can also use a Durbin-Watston (DW) statistic or a Breusch-Godfrey (BG) test. The DW statistic is used to detect serial correlation at a single lag, whereas a BG test is used to detect serial correlation at multiple lags.

• The lower limit for the DW table is 15 observations.

Question 38

Breusch-Godfrey (BG) Test

Answer 38

The BG Test regresses the residuals against the original set of independent variables, plus one or more additional variables representing lagged residuals.

Calculation: ε = a1x1 + a2x2… + p1x1 + pnxn

• The null under the BG test is that there is no serial correlation (i.e p1=0).

Question 39

How to correct for serial correlation?

Answer 39

We can calculate robust standard errors/Newey-West corrected standard errors/heteroskedasticity-consistent standard errors

Question 40

Multicollinearity

Answer 40

When independent variables in a multiple regression are correlated w/ each other

• This inflates standard errors and lowers t-stats leading to the null failing to be rejected more often (type 2 error).
• Also causes the model’s coefficients to become unreliable.
• Multicollinearity has no effect on an F-stat

Question 41

Effect of multicollinearity on model parameters

Answer 41

Multicollinearity DOES NOT affect the consistency of slope coefficients. Multicollinearity DOES make those estimates imprecise and unpredictable.

Question 42

How to detect multicollinearity?

Answer 42

The most easily observable sign is when t-tests indicate none of the individual coefficients are significantly different than zero, but the F-test indicates that at least one of the coefficients is statistically significant and the R^2 is high. This means that none of the individual variables cause variation in the dependent variable but combined together they are highly correlated which washes out the individual effects. More formally we use a variance inflation factor (VIF) for each of the independent variables.

Question 43

Variance inflation factor (VIF)

Answer 43

Estimates how much of the variation in the dependent variable in a multiple regressions model is due to multicollinearity. We start by regressing one of the independent variables (making it a dependent variable) against the remaining independent variables.

VIF= 1 / (1 - Rj^2)
* VIF values >1 indicates that the variable is not highly correlated with other independent variables.
* VIF values >5 indicate further investigation.
* VIF values >10 indicate high correlation.

Rj^2 is the R^2 of J. J is the independent variable being regressed.

Question 44

How to correct multicollinearity?

Answer 44

The most common method to correct for multicollinearity is to omit one or more of the highly correlated independent variables. You can also use a proxy for one of the variables or increase the sample size.

Question 45

True or false: The coefficient on a variable in a multiple regression is the amount of return attributable to the variable?

Answer 45

TRUE

Question 46

True or false: Using actual instead of expected inflation will improve model specification?

Answer 46

False, using actual instead of expected inflation is likely to result in model misspecification.

Question 47

Outliers vs high-leverage points

Answer 47

Outliers: Extreme observations in the dependent (Y) variable

High-leverage points: Extreme observations in the independent (X) variable

Question 48

Leverage (in statistics)

Answer 48

This is a way of identifying extreme observations in the independent variable. A measure of the distance between the xth observation of the independent variable relative to its sample mean. Leverage values will be between 0 and 1. The closer to 1 the farther the distance. If a variable’s leverage is higher than three times the average ((3*(k+1))/n), it is considered potentially influential.

Question 49

Studentized residuals

Answer 49

An alternative way of identifying outliers than leverage. The studentized residual is the # of standard deviations the data point is from the regression line. For each data point, the residual ÷ standard division is its standardized residual. There are four main steps to this process:
1. Estimate the regression model using the original sample size and then delete one observation and re-estimate the regression. Perform this sequentially deleting a new observation each time.
2. Compare the actual Y values of the deleted observation to the predicted y-values. ei= Y-ŷ
3. The studentized residual is the residual in #2 ÷ standard deviation. t= ei / s
4. Compare the studentized residuals to critical values in a t-table using n-k-2 df. Points that fall in the rejection region are termed outliers and potentially influential.

Question 50

Influential data points

Answer 50

Extreme observations that, when excluded, cause a significant change to model coefficients.

Question 51

True or false: All outliers and high-leverage points are influential on the regression?

Answer 51

FALSE

Question 52

Cook’s Distance

Answer 52

A composite metric for evaluating if a high leverage and/or outlier is influential. Cook’s distance measures how much the estimated values of the regression change if certain high leverage points or outliers are deleted from the sample.

Calculation:
D= [ ei^2 / ((K+1) * MSE) ] * [ hi / (1-hx)^2 ]
* hi= leverage value for the xth observation
* ei= the residual for the ith observation

• Values > than √(k/n) indicate the observation is highly likely to be an influential data point.
• Generally, values > 1 indicate highly influential, whereas values > 0.5 indicate the need for further investigation.

Question 53

Dummy variables

Answer 53

Binary variables with only two options

• When assigning a numerical value, it can only be 0 and 1.
• Always use (n-1) dummy variables to avoid multicollinearity (i.e., 3 dummy variables for 4 quarters in a year).
• Ex: True/falseEx 2: On/off

Question 54

Dummy variables example:

Answer 54

EPS for four quarters:
EPS = 1.25 + 0.75Q1 - 0.20Q2 + 0.10Q3

Question 1: What this the predicted EPS for Q4?

Answer 1: EPS = 1.25 + 0.75(0) - 0.20(0) + 0.10(0) = 1.25
* omitted quarter shows as the intercept

Question 2: What is the predicted value for Q1?

Answer 2: EPS = 1.25 + 0.75(1) - 0.20(0) + 0.10(0) = 2.00

Question 3: What is the predicted EPS for Q1 of next year?

Answer 3: EPS = 1.25 + 0.75(1) - 0.20(0) + 0.10(0) = 2.00
* This simple model uses average EPS for any specific quarter over the past ten years as a forecast of EPS in its respective quarter of the following year.

Question 55

Logistic regression (logit) model

Answer 55

Estimates the probability of a DISCRETE binary variable occurring.

Calculation: ln(p/(1-p)) = b0 + b1x1 + b2x2 … + ε
* The intercept value is an estimate of log odds when the values of all independent variables is zero.
* The change in log odds when one of the independent variables change is dependent on the curvature of the function.
* Odds= e^y
* Probability = 1 / (1 + n(p/(1-p))) OR 1 / (1 + e^(-yhat))

• Logit models assume that residuals have a logistic distribution- similar to a normal distribution but with fatter tails.
• Logit models are nonlinear

Question 56

Likelihood ratio (LR) test

Answer 56

Similar to joint F-test but for logit models. Measures the goodness of fit of a logit model.

Calculation= -2 * (log likelihood restricted model - log likelihood unrestricted model).

• Recall, the restricted model has fewer independent variables.
• Always provides a negative value.
• Values closer to 0 indicate a better-fitting model.
• LR test is a chi-square distribution.

Question 57

Time-series data

Answer 57

A set of observations taken periodically (most often at equal intervals) at different points in time.

• A key feature of a time series is that new data can be added w/o affecting the existing data.
• Trends can be found by plotting these observations on a graph.

Question 58

Linear trend

Answer 58

1/2 broad types of trend models. A time-series trend that can be graphed using a straight line. The independent variable will be time. A downward sloping linear trend indicates a negative trend and vice versa for a positive trend.

Simplest form: Y= bo +b1(t) + b2(t) … + ε

Question 59

Log-linear trend model

Answer 59

1/2 broad types of trend models. This is used to model positive and negative exponential growth. Recall, exponential growth is some constant growth rate (positive or negative). Exponential growth will show a convex curve.

Simplest form: e^(b0 + b1(t))
* b1 is the constant rate of growth.
* Rather than trying to fit the nonlinear data with a linear (straight line) regression, we take the natural log of both sides and transform it into a linear trend line called the log-linear model. This increases the predictive ability of the model.

Form: ln(y) = ln(e^(b0 + b1(t)))

• Financial time series data is often modeled using log-linear trend models.

Question 60

How to determine if a linear or log-linear trend model should be used?

Answer 60

Plot the data. A linear trend model may be used if the data points are equally distributed above and below the regression line (ex: inflation data is usually modeled with a linear trend model). If, when plotted, the data plots with a curved shape, use a log-linear trend model (ex: financial data- stock indices and stock prices- are often modeled with log-linear trend models).

• If there is serial correlation, we will use an autoregressive model.

Question 61

True or false: For a time series model without serial correlation, the DW statistic should be approximately equal to 0?

Answer 61

False, for a time series model without serial correlation, the DW statistic should be approximately equal to 2. A DW that significantly differs from 2 suggests that the residuals are correlated.

Question 62

Autoregressive (AR) model

Answer 62

A time-series model that regresses the dependent variable against one or more lagged values of itself.

Ex: A regression of the sales of a firm against the sales of the firm in the previous month. In this model, past values are used to predict the current value of the variable.

Simplest form: Xt = bo + b1x_t-1 …. bpx_t-p + ε
* Xt= value of time series at time t
* X_t-1= value of time series at time t-1

• DW test stat cannot be used to test for serial correlation in AR model.

Question 63

Covariance stationary

Answer 63

An AR model is covariance stationary if:
* There is a constant and finite expected value: the expected value is constant over time.
* Constant and finite variance: the volatility around the time series’ mean is constant over time.
* The covariance between any two observations w/ equal distance apart will be equal.

Question 64

True or false: A nonstationary time series can still produce meaningful results sometimes?

Answer 64

False, we need stationary covariance. A nonstationary time series will produce meaningless results.

Question 65

True or false: We can use a DW or BG test to test for serial correlation in AR models?

Answer 65

False, we must use a t-test

• We can use a DW or BG test for a TREND model.

Question 66

T-stat for residual autocorrelations in AR model:

Answer 66

correlation of the error term with the kth lagged error term ÷ (1 ÷ √n)

Standard error = (1 ÷ √n)(n-2)
* dfn= # of observations.

• If data is monthly, check for 12 lags to see if there’s serial correlation. If quarterly, check for 4 lags.
• When there is statistically significant serial correlation in an AR model, it means that the model is incomplete. There’s still some pattern of data in the residuals that the model has failed to reveal.

Question 67

Mean reversion

Answer 67

When a time-series has a tendency to move towards its mean. In other words, the dependent variable has a tendency to decline when the current value is above the mean and rise when the current value is below the mean. If a time series is at its mean reverting level, the model predicts the next value of the time series will be the same as its current value.

Question 68

Mean reverting level calculation

Answer 68

Xt = b0 ÷ (1 - b1)

• The model will not be covariance stationary if b1 = 1
• If Xt > than the mean reverting level, the model predicts that x_t+1 will be lower than Xt and vice versa.
• All covariance stationary time series have a finite mean-reverting level.
• As forecasts become more distant, the value of the forecast will be closer to the mean reverting level.

Question 69

In-sample forecasts

Answer 69

Forecasts that are within the range of data used to estimate the model. This is where we compare how accurate our model is in forecasting the acutal data we used to develop the model.

Question 70

Out-of-sample forecasts

Answer 70

Forecasts that are made outside of the sample period. This is where we compare how accurate a model is in forecasting the y-variable value for a time period outside the period used to develop the model.

Question 71

Root mean squared error (RMSE)

Answer 71

Used to compare the accuracy of autoregressive models in forecasting out-of-sample values.

Ex: We have two AR models. To determine which model will more acurately forecast future values, we calculate the RMSE for the out-of-sample data.

• The model with the lower RMSE for the out-of-sample data will have lower forecast error and will be expected to have better predictive power in the future.

Question 72

True or false: Financial and economic time series inherently exhibit some form of instability or nonstationarity.

Answer 72

True. Since financial/economic conditions are dynamic, the coefficients in one period may be different from those in another period. Model with shorter estimated time periods are usually more stable for this reason. When selecting a time series sample, analysts should understand regulatory changes, changes to the economic environment, etc. If there have been large changes, the model may not be accurate.

Question 73

True or false: There is a trade-off between statistical reliability in the long run and statistical stability in the short run?

Answer 73

True. Statistical reliability= if you use a long time period, there is more statistical reliability.

Question 74

Random walk

Answer 74

When, in an AR model, the value of the dependent variable in one period is equal to the value of the series in the previous period plus a random error term.

Form: Xt = X_t-1 + ε
* b0 = 0
* b1 = 1

Question 75

Random walk with a drift/Unit root

Answer 75

The same concept as a random walk but the intercept term is not equal to zero. Thus, the time series model is expected to increase/decrease by the intercept term and the error term.

Form: Xt = b0 + X_t-1 + ε
* b1 = 1

Question 76

True or false: A random walk with or w/o a drift is NOT covariance stationary?

Answer 76

True, random walks will always have a unit root which makes them not covariance stationary.

Question 77

Why are unit roots problematic?

Answer 77

A unit root is when b1 = 1. If this occurs, then the mean reverting level (b0 ÷ (1 - b1)) is undefined.

Question 78

How to determine whether a time series is covariance stationary:

Answer 78

1. We can run an AR model and examine autocorrelations
2. Perform a Dickey-Fuller test

• We cannot use a T-test

Question 79

Dickey-Fuller Test

Answer 79

A test we use in an AR model to determine if there’s a unit root.

Calculation: Xt = b0 + b1X1 + ε ↠ Xt - X_t-1 = b0 * (b1 X_t-1) - X_t-1 + ε ↠ Xt - X_t-1 = b0 + (b1 -1) * X_t-1 + ε ↠Then, test whether the new coefficient (b-1) [(b-1) a.k.a G] = 0 using a t-test.

• The null hypothesis is that (b-1)= 0. If the null is failed to be rejected, the time series has a unit root and is nonstationary.

Question 80

True or false: The Dickey-Fuller test uses the standard T distribution to find the critical values?

Answer 80

False, it has its own distribution to calculate the critical values.

Question 81

First differencing

Answer 81

A procedure that transforms time series data w/ a random walk into a covariance stationary time series. The first differencing process involves subtracting the value of the time series (the dependent variable) in the immediately preceding period from the current value of the time series to define a new dependent variable, y.

Question 82

First differencing calculation

Answer 82

1. If the original time series has a unit root, then ε= Xt - X_t-1
2. Then we will create a new dependent variable: Yt = Xt - X_t-1 OR Yt = ε
3. Then, if we state it in the form of an AR model: Yt= B0 + B1*(Y_t-1) + εB0 = B1 = 0

Question 83

Seasonality

Answer 83

A characteristic of a time series in which the data experiences regular and predictable changes that recur every calendar year.

• If seasonality is present, we MUST adjust the AR model in order for it to be correctly specified.

Question 84

How to correct for seasonality?

Answer 84

We add an additional lag of the dependent variable to the original model as another independent variable. The lag will be X_t-4 in a quarterly model or X_t-12 in a monthly model.

Calculation: ln(Xt) = b0 + b1 * ln(X_t-1) + b2 * ln(X_t-4) + ε

Question 85

True or false: For a T-test with seasonality, the null hypothesis is that there is seasonality?

Answer 85

False. H0 = 0- no seasonality present; Ha ≠ 0- seasonality present.

Question 86

Autoregressive conditional heteroskedasticity (ARCH)

Answer 86

When the variance of the residuals in one period is dependent on the variance of the residuals in a previous period in an AR model. When ARCH exists, the standard errors of the coefficients and the hypothesis tests are invalid.

Question 87

ARCH model

Answer 87

A model used to test for ARCH.

• If a time-series model has been determined to contain ARCH errors, regression procedures that correct for heteroskedasticity, such as generalized least squares, must be used in order to develop a predictive model. Otherwise, the standard errors of the model’s coefficients will be incorrect, leading to invalid conclusions.

Question 88

How to predict future variance of errors in a time series model?

Answer 88

After we run an ARCH model, if we determine that a1 is significant (the time series has ARCH), future variance of errors can be predicted by using:

σ^2_t+1 = a0hat + a1hat * εt^2

• We cannot predict future variance if a1 is not significant.

Question 89

Multiple time series

Answer 89

When more than one time series is run at the same time.

Ex: Yt = b0 + b1 * Xt + εt ↠ Yt and Xt are two different time series.

• Either or both of these time series could be subject to nonstantionarity.

Question 90

How to test for nonstationarity in a multiple time series model?

Answer 90

Run separate DF tests for each time series.

• If either of the time series’ are nonstationary, the coefficients will be unreliable.

Question 91

Cointegration

Answer 91

When two time series are economically linked to the same macro variables or follow the same trend, and that relationship is not expected to change. If two time series are cointegrated, the error term from regressing one on the other is covariance stationary and the t-tests are reliable.

Question 92

Cointegration note

Answer 92

When working with two time series in a regression:
1. If neither time series has a unit root, then the regression can be used
2. If only one series has a unit root, the regression results will be invalid
3. If both time series have a unit root and are cointegrated, then the regression can be used
4. If both time series have a unit root but are not cointegrated, the regression results will be invalid.

Question 93

How to test whether two times series are cointegrated?

Answer 93

Regress one variable on the other:
Yt = b0 + b1 * Xt + ε
* Yt= value of time series ‘Y’ at time t.
* Xt= value of time series ‘X’ at time t.
Then, the residuals are tested for a unit root using the Dickey-Fuller test with critical t-values calculated by the Engle and Granger (DF-EG Test). If the DF test rejects the null hypothesis (Ho= no cointegration), then we conclude the error terms are covariance stationary and there is cointegration.

Question 94

Structural change

Answer 94

A significant shift in the plotted data at a point in time that essentially divides the data into two or more distinct patterns.

• If there is structural change present, you must run two different models- one incorporating data before the date and one after the date.

Question 95

Machine learning (ML)

Answer 95

Filters useful info from substantial amounts of data by learning from known examples to find a pattern in the data. Machine learning acts without human intervention.

Question 96

Target variable

Answer 96

The dependent variable.

• Target variables can be continuous, categorical, or ordinal.

Question 97

Feature

Answer 97

The independent variable

Question 98

Training data

Answer 98

The sample used to fit the model

Question 99

Hyperparameter

Answer 99

A model input specified by the researcher

Question 100

Supervised learning

Answer 100

1/3 types of ML. We teach the model, then with that knowledge have it predict future instances. Supervised learning uses labeled data- data where the target variable is defined. Supervised learning is used when the training data contains the ground truth= the target variable. Multiple regression is an example of supervised learning. Regression and classification are the two most common examples of supervised learning. If target variable is continuous then use a regression. If target variable is categorical or ordinal then use a classification model. Output of classification models looks to group observations.

Question 101

Unsupervised learning

Answer 101

1/3 types of ML. There is no labeled data, and instead the program seeks out patterns within the data.

Question 102

Deep learning networks (DLNs)

Answer 102

1/3 types of ML that is used for complex tasks such as imagine recognition, natural language processing, etc. Deep learning is based on neural networks. Deep learning is a self-teaching system. A type of NN with many hidden layers (at least two but often more than 20)

Question 103

Reinforced learning algorithms

Answer 103

Algorithms that have an agent seek for a reward given restraints. The RL does not rely on labeled data, but rather these programs learn from their own prediction errors.

Question 104

Neural networks

Answer 104

A group of ML algorithms applied to problems w/ significant nonlinearities.

Question 105

Generalization

Answer 105

The extent to which a ML program is able to make out-of-sample predictions.

Question 106

Overfitting for ML

Answer 106

When a large number of features (independent variables) are in the data set. Overfitting will decrease the accuracy of out-of-sample forecasts.

• The training sample will have a high R^2 and the test sample will have a low R^2.

Question 107

True of false: Under supervised learning, a training sample is used to train a ML algorithm and a separate test sample is used to evaluate the model’s ability to accurately predict new data?

Answer 107

TRUE

Question 108

How to measure the ability an ML program generalizes?

Answer 108

Create three overlapping data sets:
Training sample: In-sample data. Used to train the ML algorithm.
Validation sample: Out-of-sample data. Used to tune the training model.
Test sample: Out-of-sample data.

• A model that generalized well should have a high R^2 for in-sample and out-of-sample data.

Question 109

Bias errors

Answer 109

This is the in-sample error resulting from models with a poor fit.

• Occurs when there is underfitting.

Question 110

Variance error

Answer 110

This is the out-of-sample error resulting from overfitted models that do not generalize well. This is the extent to which the ML model’s results change in response to test and validation sample data.

• Associated with overfitting.
• Increases with model complexity.
• Nonlinear models tend to have high variance error.

Question 111

Base error

Answer 111

This is the out-of-sample error resulting from residual errors due to random noise. Just randomness in the data.

• Decreases with model complexity.
• Linear models tend to have high base error.

Question 112

Learning curve

Answer 112

Plots the accuracy rate in the test sample versus the size of the training sample. A ML model that generalizes well will show an improving accuracy rate as the sample size increases. The in-sample and out-of-sample error rates should converge toward the desired level as the sample size increases.

Question 113

In-sample accuracy rate calculation vs out-of-sample accuracy rate calculation vs base accuracy rate calculation

Answer 113

In-sample accuracy rate= 1 - bias error rate
Out-of-sample accuracy rate= 1 - variance error rate.
Base accuracy rate= 1 - base error rate.

Question 114

True or false: ML models with high bias error will not see the accuracy rates converge?

Answer 114

False, the accuracy rates will converge just far below the desired level.

Question 115

True or false: Models with high variance errors will see the accuracy rates of the in-sample data and out-of-sample data converge below the desired level?

Answer 115

False, only the in-sample accuracy rate will converge towards the desired level.

Question 116

How to minimize the effects of overfitting with an ML program?

Answer 116

Reduce complexity and cross validation.

Question 117

Cross validation

Answer 117

An estimate of out-of-sample error rates directly from the validation sample.

Question 118

Complexity reduction

Answer 118

A penalty imposed to exclude features that do not meaningfully contribute to out-of-sample prediction accuracy.

Question 119

Underfitting

Answer 119

When the ML algorithm fails to identify an actual relationship. This occurs when there is an oversimplified model.

• R^2 will be low for in-sample and out-of-sample data.
• High bias error
• Linear functions are susceptible to underfitting.

Question 120

K-fold cross validation

Answer 120

A method for alleviating the holdout sample problem: when the training set is reduced too much. This process eliminates sampling bias. There are four steps in this process:
1. Shuffle the data randomly.
2. Divide the data into k equal sub-samples.
3. K-1 samples will be training samples with the remaining sample a validation sample.
4. This process is then repeated k times. The average of the k validation errors is then taken as a reasonable estimate of the model’s out-of-sample error.

Question 121

Penalized regression models

Answer 121

Models that reduce the problem of overfitting by imposing a penalty based on the # of features in the model. The penalty increases w/ the # of features used. This will exclude features that do not meaningfully contribute to out-of-sample prediction accuracy. Penalized regression models seek to minimize the SSE.

• These models are used to forecast returns.

Question 122

Least absolute shrinkage and selection operator (LASSO)

Answer 122

This is a popular penalized regression model. LASSO attempts to minimize SSE and the sum of the absolute values of the slope coefficients of the regression. The penalty increases with number of features. There is a tradeoff in reducing SSE (increasing independent variables) and the penalty imposed. Investment analysts use LASSO to build parsimonious (few predictor variables) models.

Question 123

Regularization

Answer 123

A type of penalized regression. Forces the beta coefficients of nonperforming features towards zero. Regularization can be applied to non-linear models.

Question 124

Support Vector Machine (SVM)

Answer 124

A common supervised ML algorithm often used for textual, categorical data. The model assumes the data is linearly separable; An SVM is a linear classification algorithm. An SVM attempts to find the optimal hyperplane that separates two sets of data (classes) by the max amount using n features.

• Applications of SVM in investment management include classifying debt issuers into likely-to-default versus not-likely-to-default issuers, stocks-to-short versus not-to-short, and even classifying text (from news articles or company press releases) as positive or negative.

Question 125

Soft margin classification

Answer 125

Handles misclassified observations in the training data in an SVM.

Question 126

K-nearest neighbor (KNN)

Answer 126

A common supervised ML algorithm. New observations are classified by finding the new observation and its k-nearest piece of data in the current data set. This is used for categorical data.

• Investment applications of KNN include predicting bankruptcy, assigning a bond to a ratings class, predicting stock prices, and creating customized indices.

Question 127

Classification and regression trees (CART)

Answer 127

A common supervised ML algorithm. Classification trees are used when the target variable is categorical and typically when the target is binary. Regression trees are used when the target is continuous. Classification trees assign observations to one of two possible classifications at each node starting w/ the root node at the top, then moving to the decision nodes in the middle, and then the terminal nodes at the bottom.

• To avoid overfitting, regularization criteria such as maximum tree depth, maximum number of decision nodes, and so on are specified by the researcher. Alternatively, sections of tree with minimal explanatory power are pruned.
• Investment applications of CART include detecting fraudulent financial statements and selecting stocks and bonds.
• With Classification and Regression Trees (CART), one way that regularization can be implemented is via pruning which will reduce the size of the regression tree—sections that provide little explanatory power are pruned (i.e., removed).

Question 128

Ensemble learning

Answer 128

A common supervised ML algorithm that combines the predictions from multiple models rather than a single model. The different models cancel out noise and result in a lower average error rate. There are two types of ensemble methods: aggregation of heterogeneous learners and aggregations of homogeneous learners. Ensemble learning typically produces more stable and accurate results than single models. Aims to decrease variance (bagging), decrease bias (boosting), and improving predictions (stacking).

Question 129

Aggregation of heterogeneous learners

Answer 129

Different algorithms are combined together through a voting classifier and each algorithm gets a vote. The answer with the most votes is the model we go with.

Question 130

Aggregations of homogeneous learners.

Answer 130

The same algorithm is used but on different training data. The different training data used by the same model can be derived through bootstrap resampling (a.k.a bagging).

Question 131

Random Forest

Answer 131

A common supervised ML algorithm. This is a variation of a classification tree where a large # of classification trees are trained using bagged data from the same data set. A random subset of features is used in creating each tree, thus every tree is different. This process mitigates overfitting and reduce noise from errors. A drawback of Random Forests is that the transparency of CART is lost.

• Random forests can INCREASE the signal-to-noise ratio.
• Investment applications of random forest include factor-based asset allocation, and prediction models for the success of an IPO.

Question 132

Principal component analysis (PCA)

Answer 132

A common unsupervised ML algorithm. Problems w/ too much noise arise when there are excessive amts of features (high dimensionality). PCA seeks to reduce this excess noise by discarding the excess features. A PCA transforms the feature’s covariance matrix in order to reduce highly correlated features into a smaller # of uncorrelated features, called eigenvectors, which are linear combinations of the original feature. Each eigenvector has an eigenvalue: the proportion of total variance in the data set explained by the eigenvector. The end product is an algorithm with lower dimensionality, which makes the model easier to train and interpret.

• The process of reducing noise is called dimension reduction. Dimension reduction seeks to reduce this noise by discarding those attributes that contain little information.

Question 133

Scree plot

Answer 133

A plot that shows the proportion of total variance explained by each of the principal components.

Question 134

Clustering

Answer 134

A common unsupervised ML algorithm. Clustering is the process of grouping observations into categories based on similar attributes (a.k.a cohesion). The two most common types of clustering are: K-means clustering and hierarchical clustering.

Question 135

Cohesion

Answer 135

Grouping observations into categories based on the observations’ similarities.

Question 136

K-means clustering

Answer 136

1/2 main types of clustering that puts observations into k nonoverlapping clusters where k is a hyperparameter. Each cluster has a centroid (center of the cluster), and each new observation is assigned to a cluster based on its proximity to the centroid. As a new observation gets assigned to a cluster, its centroid is recalculated, which may result in reassignment of some observations, thus resulting in a new centroid and so forth until all observations are assigned and no new reassignment is made.

• One limitation of this type of algorithm is that the hyperparameter k is chosen before clustering starts, meaning that one has to have some idea about the nature of the data set.
• K-means clustering is used in investment management to classify thousands of securities based on patterns in high dimensional data.

Question 137

Hierarchical clustering

Answer 137

1/2 main types of clustering that builds a hierarchy of clusters without any predefined # of clusters.

Question 138

Agglomerative clustering/ Bottom-up clustering

Answer 138

1/2 types of hierarchical clusters. This starts with one observation as its own cluster and then adds other similar observations to that group, thus forming another nonoverlapping cluster. In the end, all observations are merged into a single cluster.

Question 139

Neural networks (NNs)

Answer 139

Made up of layers of neurons. The first layer is the input layer (node layer), which receives the input (the independent variables). The final layer is the output layer. In between exists hidden layers. Neurons of each layer are connected to neurons of the next layer through channels. There may be multiple hidden layers. The multiple layers allow the NN to model complex nonlinear functions. NNs are an adaptive system that computers use to learn from their mistakes and improve continuously. A group of ML algorithms applied to problems with significant nonlinearity.

Question 140

Divisive clusters/ top-down clustering

Answer 140

1/2 types of hierarchical clusters. The algorithm starts with one giant cluster, and then it partitions that cluster into smaller and smaller clusters. In the end, each cluster contains only one observation.

Question 141

Summation operator

Answer 141

Neurons comprise the summation operator which gathers the info from the neurons and assigns them a weighted average, then passes the info on to the activation function. The activation function then generates a value from the inputs. The value is then passed forward to other neurons in subsequent hidden layers- this process is called forward propagation.

Question 142

Backwards propagation

Answer 142

This is how the machine learns from its errors. When the weighted averages from the summation operators are adjusted as the algorithm learns from its errors.

Question 143

Steps in a supervised/ traditional ML model:

Answer 143

1. Conceptualization of the problem
2. Data collection
3. Data preparation and wrangling: cleaning the data set and preparing it for the model.
4. Data exploration: Feature selection and performing data analysis. Evaluating the data set and determining the most appropriate way to configure it for model training.
5. Model training: Determining which ML algorithm to use, using a training data set, and tuning the model.

Question 144

Steps in a unsupervised/ textual ML model:

Answer 144

1. Text problem formulation
2. Text curation: ensuring the quality of data, for example by adjusting for bad or missing data.
3. Text preparation and wrangling
4. Text exploration
5. Model training

Question 145

Big data

Answer 145

Immense amounts of data

Question 146

The 4 Vs of big data

Answer 146

• Volume= The amt of data
• Variety= The sources of data
• Velocity= The speed w/ which the data is created and collected
• Veracity= The quality of the data

• Big data often suffers from low veracity

Question 147

Data cleansing

Answer 147

Reducing errors in raw data. Common errors include:
* Missing values
* Invalid values
* Inaccurate values
* Non-uniform values
* Duplicate observations

• Removing HTML tags is part of the data cleansing step.

Question 148

Data wrangling

Answer 148

Prepping data for model use. This includes transforming and scaling. Data transformations include:
* Extraction
* Aggregation: consolidating two variables into one (using appropriate weighting)
* Filtration: removing irrelevant observations.
* Selection: removing features not needed for processing.
* Conversion of data of diverse types

Question 149

README Files

Answer 149

Contain info about how, what, and where the data is stored. Helps ensure validity.

Question 150

Application programming interface (API)

Answer 150

Data obtained from 3rd party sources

Question 151

Metadata

Answer 151

Data that describes other data by providing info about one or more aspects of the data. Essentially a summary.

Question 152

Winsorization

Answer 152

A way researchers exclude outliers. Instead of entirely excluding outliers, they substitute reasonable values in for them.

Question 153

Trimming

Answer 153

One way researchers exclude outliers. This type of means excludes a certain portion of the highest values and lowest values. For example, excludes lowest 1% and highest 1% of all values.

Question 154

Normalization

Answer 154

1/2 common types of scaling. Scales variable values between 0 and 1.

Calculation: (Xi - Xminimum) ÷ (Xmaximum - Xminimum)

• Sensitive to outliers.
• Use this when trying to understand where the variables lie within the data set.

Question 155

Cleansed text is normalized using these steps:

Answer 155

• Lowercasing: Ex: Dog ↠ dog
• Removal of stop words: super common unimportant words Ex: the, is, and, etc.
• Stemming: Take similar words and combine them into a single word. Ex: integrate ↠ integration ↠ integrating
• Lemmatization: Return the base of the word. Ex: saw ↠ see
• Bag-of-words (BOW): A bow is just the results of steps #1-#4. All the collected words or tokens are collected w/o regard to occurrence. If order doesn’t matter we can stop here.
• N-gram: If ordering is important, we can create a two-gram to look for two specific words that go together or three-gram that looks for three words that go together, and so on.
• Organizing the BOW and N-Gram into a document term matrix (DTM):

• Lemmatization, which takes places during the text wrangling/preprocessing process for unstructured data, is the process of converting inflected forms of a word into its morphological root (known as lemma). Lemmatization reduces the repetition of words occurring in various forms while maintaining the semantic structure of the text data, thereby aiding in training less complex ML models.

Question 156

Token (in text wrangling)

Answer 156

A word

Question 157

Black box approach to ML

Answer 157

ML models that give you a result without explaining how they get to their decision.

Question 158

Steps in data exploration

Answer 158

1. Exploratory data analysis (EDA)
2. Feature selection
3. Feature engineering

Question 159

Exploratory data analysis

Answer 159

Involves looking at data descriptors (stats, heat maps, word clouds, etc.) w/ the objective of understanding the data’s properties, finding patterns/relationships, and planning modeling in future steps.

Question 160

Feature selection

Answer 160

A process to select only the needed attributes of the data for ML model training

Question 161

True or false: In feature selection, we try to ONLY include the features that contribute to the model’s out-of-sample predictive power.

Answer 161

TRUE

Question 162

Feature extraction

Answer 162

When a feature is created from the data set.

Ex: Creating a value for age using date of birth data.

Question 163

Feature engineering (FE)

Answer 163

Involves optimizing and improving the selected features.

Question 164

One-hot encoding (OHE)

Answer 164

A type of feature engineering. The process is used to convert a categorical feature into a dummy variable.

Question 165

Techniques of feature selection:

Answer 165

• Term frequency= The # of times the token appears in the dataset
• Document frequency= The # of documents that a token appears in ÷ the # of documents.
• Chi-square Test= Ranks tokens by their usefulness to a certain class of info. Tokens with higher chi-square test-stat occur more frequently.
• Mutual information Test= A numerical value indicating the contribution of a token to a specific class. Tokens with less frequencies in a class compared to another class it will have a value close to 1, whereas if a token appears a lot in all classes it will have a value of 0.

Question 166

Techniques of feature engineering:

Answer 166

• Numbers= Tokens w/ standard lengths are converted into new tokens. Ex: 4 letter words converted into ‘#4’.
• N-Grams
• Name entity recognition (NER)= Assign tokens a NER tag based on their context. Ex: Europe-place ; Google-website.
• Parts of Speech= Assign tokens a POS tag based on their language structure. For example: Google- PPN (proper noun) ; 2000 - CDN (cardinal #).

Question 167

Procedures before model training:

Answer 167

The researcher must define the objective(s) of data analysis, identify useful data points, and conceptualize the model. Once a ML algorithm/method is selected, he should specify the hyperparameters.

Question 168

Common model fitting errors:

Answer 168

• Small training samples
• Low # of features in the model. This can lead to an underfitting problem because the model doesn’t have enough info to find patterns.

• Feature selection is important to mitigate underfitting and overfitting.
• Feature engineering can reduce underfitting.

Question 169

Three tasks of model training:

Answer 169

1. Method selection= choosing the right ML algorithm considering supervised/unsupervised learning, type of data, and size of data.
2. Performance evaluation
3. Tuning

Question 170

What type of ML algorithm do we use for text, numerical, and image data:

Answer 170

• Text= SVMs and Generalized linear models (GLMs)
• Numerical= Regression trees, CART methods, and classification methods.
• Image= Neural networks and deep learning networks.

Question 171

Techniques to measure model performance:

Answer 171

• Error analysis: Errors in classification problems can be false positives (type 1 errors) or false negatives (type 2 errors). We build confusion matrixes for type 1 and type 2 errors.
• Receiver operating characteristic (ROC)
• Root mean squared error (RMSE)

Question 172

Precision metric

Answer 172

A way to evaluate the fit of an ML algorithm. It’s the ratio of true positives (not false positives (type 1 errors)) to predicted positives. Use the precision metric when the cost of a type 1 error is large.

Calculation: True positives ÷ (True positives + false positives)

Question 173

Recall metric/ true positive rate

Answer 173

A way to evaluate the fit of an ML algorithm. It’s the ratio of true positives (not false positives (type 1 errors)) too all actual positives. Use when the cost of a type 2 error is large.

Calculation: True positives ÷ (True positives + false negatives)

Question 174

F1 score

Answer 174

A way to evaluate the fit of an ML algorithm. It’s the harmonic mean of precision and recall. The higher the better.

Calculation: (2 * precision * recall) ÷ (Precision + recall)

• More appropriate than the model accuracy metric when there are class imbalances.

Question 175

Receiver operating characteristic (ROC)

Answer 175

A curve that plots the tradeoff between false positives and true positives. The true positive rate (recall metric) is plotted on the y-axis, whereas the false positive rate is plotted on the x-axis. The area under the curve (AUC) is a value from 0 - 1. The closer the value is to 1 the higher the predictive accuracy of the model. AUCs = 0 mean it’s never right and 0.5 mean 50% of the time- just guessing. The higher convexity of the curve the higher its AUC.

Question 176

True or false: There is a tradeoff between bias error and variance error to where the model is overfitting and underfitting?

Answer 176

TRUE

Question 177

Fitting curve

Answer 177

A graph that plots error (in-sample error (training sample error) and out-of-sample error (cross-validation sample error) on the y-axis and model complexity on the x-axis. The graph shows two curves: a curve for training error and a curve for cross-validation prediction error.

Question 178

Ceiling analysis

Answer 178

An evaluation and tuning of each components in the model.

• Applied to complex models.

Question 179

What is the primary limitation of trend models?

Answer 179

The primary limitation of trend models is that they are not useful if the residuals exhibit serial correlation.

Question 180

True or false: The KNN is a parametric test?

Answer 180

False, it’s non-parametric: it makes no assumptions regrading the distribution of the data.

Question 181

What are LASSO models and regularization used for?

Answer 181

LASSO models are used to build parsimonious models and regularization is used for nonlinear models.

Question 182

What are SVMs used for?

Answer 182

Generates binary classifications, such as: classifying debt issuers into likely-to-default versus not-likely-to-default issuers, stocks-to-short versus not-to-short, and even classifying text (from news articles or company press releases) as positive or negative.

Question 183

What are KNNs used for?

Answer 183

Predicting bankruptcy, assigning a bond to a ratings class, predicting stock prices, and creating customized indices.

Question 184

What are CARTs used for?

Answer 184

Fraud detection in financial statements and selecting stocks/bonds.

Question 185

What are random forests used for?

Answer 185

Factor-based asset allocation and prediction models for the success of an IPO.

Question 186

True or false: NNs have an input layer node that consists of a summation operator and an activation function?

Answer 186

False, the hidden layer nodes (not the input layer nodes) each consist of a summation operator and an activation function; these nodes are where learning takes place.

Question 187

True or false: The coefficients on each dummy tells us about the difference in earnings per share between the respective quarter and the one left out?

Answer 187

TRUE

Question 188

True or false: The F-statistic enables us to make conclusions about how several independent variables affect a dependent variable?

Answer 188

False, it only allows us the reject the hypothesis that all regression coefficients are zero and accept the hypothesis that at least one isn’t.

Question 189

True or false: Serial correlation affects the consistency of regression coefficients?

Answer 189

FALSE

Question 190

Difference between RSS and R^2?

Answer 190

The RSS is just the absolute amount of explained variation, the R^2 is the (RSS/SST)- the absolute amount of variation as a proportion of total variation. It’s like saying NI is an absolute figure, whereas ROE is NI as a proportion of equity.

Question 191

Sum of squared errors (SSE)

Answer 191

The absolute amount of unexplained variation

Question 192

T-test

Answer 192

A statistical test to determine if there is a significant difference between the means of two groups and how they’re related. The t-stat tells us if we need to reject the null or not.

Question 193

True or false: The F-statistic does not enable us to conclude on both independent variables. It only allows us the reject the hypothesis that all regression coefficients are zero and accept the hypothesis that at least one isn’t?

Answer 193

True

Question 194

True or false: If it is determined that conditional heteroskedasticity is present in model one, both the regression coefficients and the standard errors will be biased?

Answer 194

False, regression coefficients will be unbiased but standard errors will be biased.

Question 195

True or false: In the presence of serial correlation, if the independent variable is a lagged value of the dependent variable, then regression coefficient estimates are invalid and coefficients’ standard errors are deflated, so t-statistics are inflated?

Answer 195

True

Question 196

Deep learning nets

Answer 196

Neural networks with many hidden layers—at least 3, but often more than 20 hidden layers—are known as deep learning nets.