Best test for unconditional heteroskedasticity is:

Best test for conditional heteroskedasticity is:

Which is more serious than the other?

No tests for unconditional.

Breusch-Pagan for conditional

Conditional considered more serious than unconditional

The Durbin-Watson test is for serial correlation. The Breusch-Pagan test is for conditional heteroskedasticity; it tests to see if the size of the independent variables influences the size of the residuals. Although tests for unconditional heteroskedasticity exist, they are not part of the CFA curriculum, and unconditional heteroskedasticity is generally considered less serious than conditional heteroskedasticity. 

Multicollinearity is indicated

=RFR + S₁RFP₁ + S₂RFP₂ + ...

Given a three-factor arbitrage pricing theory APT model, what is the expected return on the Freedom Fund? 

The factor risk premiums to factors 1, 2, and 3 are 10%, 7% and 6%, respectively. 

The Freedom Fund has sensitivities to the factors 1, 2, and 3 of 1.0, 2.0 and 0.0, respectively. 

The risk-free rate is 6.0%. 

Your answer: A was incorrect. The correct answer was C) 30.0%. 

The expected return on the Freedom Fund is 6% + (10.0%)(1.0) + (7.0%)(2.0) + (6.0%)(0.0) = 30.0%.

=(# of categories) - 1

Spurious correlation occurs when the analysis erroneously indicates a relationship between two variables when none exists. 

F = MSR / MSE

The F-statistic is equal to the ratio of the mean squared regression to the mean squared error.

MSR = RSS / k

MSE = SSE / (n-k-1)

Random walk time series (formula)

For a random walk, the long-run mean is (answer and formula)

x_t = x_t-1 + e_t

Undefined. The slope coefficient is one, b₁=1, and that is what makes the long-run mean undefined:

mean = b₀/(1-b₁)

b₀ / (1 − b₁)

Relationship between two variables when there is a standard error of estimate (SEE) that is high relative to total variability

Weak. The SEE is the standard deviation of the error terms in the regression, and is an indicator of the strength of the relationship between the dependent and independent variables. The SEE will be low if the relationship is strong and conversely will be high if the relationship is weak.

Assumptions of linear regression:

(1) Expected value/mean of the residuals is ______

(2) residuals are _____ distributed

(3) ______ variance

(1) zero

(2) independently

(3) constant

An indication of multicollinearity is when:

(1) if two ind. variables only--high corr. between the two

(2) high R² and significant F-stat but not significant coefficients on ind. vars.

The R² is the ratio of the:

The Durbin-Watson test is used to:

detect serial correlation

The Breusch-Pagan test is used to:

detect heteroskedasticity

Estimation of first differences:

Sales_t = b₀ + b₁ Sales _t-1+ ε_t 

What is the specification of the model if first differences are used?

(Sales_t - Sales_t-1)= b₀ + b₁ (Sales _t-1 - Sales _t-2) + ε_t. 

A time series x that is a random walk with a drift is best described as:

x_t = b₀ + b₁x_{t − 1} + ε_t

The standard error of the estimate (SEE) in a regression is the (def) (2 formulas) (smaller/larger is better fit?)

residuals of the regression. It is the standard deviation of the residuals.

SEE = √[SSE/(n-2)]

SEE = sqrt(MSE)

Smaller is better fit.

When is there economic significance to a strategy?

Market inefficiency is avaliable when:

When there are abnormal returns AND they can cover transaction costs

When excess returns are available after covering transaction costs

A simple linear regression equation had a coefficient of determination (R²) of 0.8. What is the correlation coefficient between the dependent and independent variables and what is the covariance between the two variables if the variance of the independent variable is 4 and the variance of the dependent variable is 9?

The correlation coefficient is the square root of the R², r = 0.89. 

To calculate the covariance multiply the correlation coefficient by the product of the standard deviations of the two variables: 

COV = 0.89 × √4 × √9 = 5.34

The root mean squared error (RMSE) criterion is used to (formula + interpret values):

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

RMSE = sqrt(average squared error)

The model with the smallest RMSE is the preferred model.

The RMSE for Model 1 is √10.429 = 3.23, while the RMSE for Model 2 is √11.642 = 3.41. Since Model 1 has the lowest RMSE, that is the one Zox should conclude is the most accurate. 

The coefficient of determination / R² (definition + formula)

the percentage of total variation in the dependent variable explained by the independent variable

R² = (RSS / SST)

A probit model is:

a qualitative dependant variable which is based on a normal distribution

A logit model is:

a qualitative dependant variable which is based on the logistic distribution

A discriminant model: 

When utilizing a proxy for one or more independent variables in a multiple regression model, what error is likely to occur?

Model misspecification

By using a proxy for an independent variable in a multiple regression analysis, there is some degree of error in the measurement of the variable.

The mean regression sum of squares (MSR) formula:

RSS/k

The regression sum of squares divided by the number of independent variables

RSS = SST - SSE

The residual sum of squares is the difference between the total sum of squares and the regression sum of squares

|t| > critical t

-or-

p-value < alpha

The standard error of estimate (SEE)

SEE = √(SSE / (n-2))

Positive serial correlation is the condition where a positive regression error in one time period increases the likelihood of having a positive regression error in the next time period. The residual terms are correlated with one another, leading to coefficient error terms that are too small.

The Durbin-Watson statistic tests for serial correlation. For large samples, the Durbin-Watson statistic is approximately equal to:

2 * (1 - r)

r = sample correlation between the regressions residuals from one period and the residuals from the previous period

If r < (lower) DW value on chart, REJECT H₀ of no corr.

For 90 observations, use 90 df for DW

t =

[r*(sqrt(n-2)) / (sqrt(1-r²))]

df = n - 2

David Wellington, CFA, has estimated the following log-linear trend model: LN(x_t) = b₀ + b₁t + ε_t. Using six years of quarterly observations, 2001:I to 2006:IV, Wellington gets the following estimated equation: LN(x_t) = 1.4 + 0.02t. The first out-of-sample forecast of x_t for 2007:I is closest to:

The first out-of-sample forecast of x_t for 2007:I is closest to:

Your answer: A was incorrect. The correct answer was B) 6.69. 

Wellington’s out-of-sample forecast of LN(x_t) is 1.9 = 1.4 + 0.02 × 25, and e^1.9 = 6.69.

Dickey-Fuller Test tests for (+ formula):

nonstationarity; which uses a modified t-statistic

The Dickey-Fuller test estimates the equation:

(x_t – x_t-1) = b₀ + (b₁ - 1) * x_t-1 + e_t and tests if H₀: (b₁ – 1) = 0.

Using a modified t-test, if it is found that (b₁–1) is not significantly different from zero, then it is concluded that b₁ must be equal to 1.0 and the series has a unit root.

The Hansen method adjusts for problems associated with both serial correlation and heteroskedasticity

Used if DW tests finds positive autocorrelation

Serial correlation always impacts the statistical inference about:

the parameters

If one of the independent variables is a lagged value of the dependent variable, ______ will cause an inaccurate parameter estimate.

There are too many dummy variables specified, so the equation will suffer from:

Simple Linear Regression:

Correlation (formula)

Estimated slope coefficient (formula)

Estimated intercept (formula)

Confidence interval for predicted Y-value

Correlation:

r_xy = covxy / [s_xs_y]

Estimated slope coefficient:

ESC: cov_xy / s_x²

Estimated intercept:

EI: b(hat)₀ = Y(bar) - b(hat)₁X(bar)

Confidence interval for predicted Y-value:

Y(hat) +/- [t_c*SE(forecast)]

Multiple Regression:

Test statistical significance of b; H₀=?; #df?; reject if?

Confidence interval

H₀: b=0

t = b(hat) / s_b(hat) with n-k-1 df

Reject if |t|>t_c or p-value < (alpha)

Confidence Interval:

b(hat)_j +/- (t_c*s_b(hat))

SST = RSS + SSE

(Regression sum of squares + sum of squared errors)

Measures the total variation in the dependent variable. Not the same as variance.

Non-constant error variance

Detect with Breusch-Pagan test

Correct with White-corrected standard errors

Correlation among error terms; error terms are not independent

Detect with Durbin-Watson test

Positive autoc. if DW<d₁

Correct by adjusting standard errors using Hansen method

High correlation among Xs (independent variables)

Detect if F-test significant + t-tests insignificant

Correct by dropping X variables

Linear trend model

Log-linear trend model

y_t = b₀ + b₁t + e_t

ln(y_t) = b₀ + b₁t + e_t 

mean and variance don't change over time

To determine:

(1) plot data

(2) run an AR model and test correlations

(3) perform Dickey Fuller test

coefficient on lagged dependent variable = 1. Series with UR is not covariance stationary.

First differencing will often eliminate the unit root

Indicated by statistically significant lagged err. term

Correct by adding lagged term

Autoregressive Conditional Heteroskedasticity (ARCH):

-detected by estimating

-Variance of ARCH series

e(hat)_t² = a₀ + a₁e(hat)²_t-1 + (mu)_t

Variance:

se(hat)²_t+1 = a(hat)₀ + a(hat)₁*error(hat)_t²

cov_xy = Sigma[[X_i - X(mean)]*[Y_i-Y(mean)]]

/

(n-1)

The coefficient on each dummy tells us:

the difference in (whatever) between the respective quarter and the one left out

H₀: r = 0

If DW statistic > upper boundary, FAIL TO REJECT

(no serial corr.)

If DW statistic < lower boundary, REJECT

(there is serial corr.)

If lower bound<DW<upper bound, INCONCLUSIVE