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Questions and Answers

What is the primary issue that arises from estimating betas in the first stage of testing the CAPM?

Reduction in sample size

Higher volatility in asset returns

Inaccurate prediction of returns

Measurement error leading to attenuation bias (correct)

What is a proposed solution to mitigate the effect of measurement error in betas when testing the CAPM?

Adjusting the model assumptions

Utilizing portfolio betas instead of individual betas (correct)

Employing robust regression techniques

Using individual stock returns instead of portfolio returns

How does measurement error in the explained variable affect parameter estimates in a regression model?

They become biased and inconsistent

They lead to higher statistical significance

They show no relationship with explanatory variables

They remain consistent and unbiased (correct)

What impact does measurement error in the explained variable have on standard errors?

They are enlarged relative to the situation without error Signup and view all the answers

What characteristic is essential for a statistically adequate empirical model?

Must satisfy the assumptions of the CLRM Signup and view all the answers

Which of the following is NOT mentioned as a motivation for including a disturbance term in a regression model?

To improve model parsimony Signup and view all the answers

What does attenuation bias in the context of CAPM testing refer to?

A weaker than expected relationship between beta and returns Signup and view all the answers

What is the desired property of a regression model regarding its theoretical interpretation?

It should align with theoretical foundations and be easily interpretable Signup and view all the answers

How many total ratings announcements were analyzed in the study between 1987 and 1994?

79 announcements Signup and view all the answers

What was the dependent variable used to measure market reaction in the study?

Changes in the relative spreads over the US T-bond Signup and view all the answers

Which of the following was NOT included as an explanatory variable in the study?

The market capitalization of the bond issuer Signup and view all the answers

Among the announcements analyzed, how many were actual ratings changes?

39 announcements Signup and view all the answers

What factor is likely to influence how the market reacts to a ratings announcement according to the study?

Whether the announcement was positive or negative Signup and view all the answers

What is ONE of the methods to deal with heteroscedasticity?

Transforming variables into logs. Signup and view all the answers

What effect does White's heteroscedasticity correction have on hypothesis testing?

Increases the standard errors of slope coefficients. Signup and view all the answers

Which of the following represents autocorrelated errors?

Residuals show a consistent pattern over time. Signup and view all the answers

In the context of lagged values, what does the symbol $ riangle y_t$ represent?

The difference between current and previous time period values. Signup and view all the answers

What does it imply when Cov(ui, uj) = 0 for i ≠ j in the CLRM?

The errors are independent. Signup and view all the answers

Which is NOT a method to address heteroscedasticity?

Random sampling. Signup and view all the answers

What is the primary goal of using robust standard errors?

To correct for heteroscedasticity and provide more reliable standard errors. Signup and view all the answers

When analyzing time series data, what does a lagged variable represent?

The value of the series from a previous time period. Signup and view all the answers

What is the primary purpose of Cantor and Packer's analysis?

To determine how ratings agencies arrive at their ratings. Signup and view all the answers

Which of the following is NOT listed as an explanatory variable in the study?

Government debt levels (%) Signup and view all the answers

In the model discussed, what transformation is applied to income and inflation data?

Taking logarithms Signup and view all the answers

Which of the following is an example of a dummy variable in the study?

Dummy for economic development Signup and view all the answers

Which credit rating corresponds to Moody's Aaa rating?

AAA Signup and view all the answers

What type of model is used for the analysis conducted by Cantor and Packer?

Ordinary least squares (OLS) model Signup and view all the answers

Which of the following describes an expected relationship mentioned in the model?

Greater fiscal balance leads to better credit ratings. Signup and view all the answers

What kind of balances are the determinants of sovereign ratings concerned with?

Both external and fiscal balances Signup and view all the answers

What is the primary focus of the Bera-Jarque test?

To test the normality of residuals in regression analysis. Signup and view all the answers

What does a leptokurtic distribution indicate in contrast to a normal distribution?

It has a higher peak and fatter tails. Signup and view all the answers

When testing for non-normality, which values are jointly tested in the Bera-Jarque statistic?

Coefficient of Skewness and Coefficient of Excess Kurtosis. Signup and view all the answers

What should often be done when evidence of non-normality is found?

Use dummy variables to adjust for outliers. Signup and view all the answers

What are coefficients of skewness and kurtosis defined in terms of?

Expected values of specific powers of residuals. Signup and view all the answers

What issue can often lead to a rejection of the normality assumption in a dataset?

One or two extremely high or low residuals. Signup and view all the answers

What does the Bera-Jarque test statistic formula include as part of its calculation?

The coefficients of skewness squared and functionality of sample size. Signup and view all the answers

What is a potential method to handle non-normal residuals without assuming normality?

Using a non-parametric method. Signup and view all the answers

What is the first difference of a variable yt?

yt - yt-1 Signup and view all the answers

What does a long-run static equilibrium solution imply about the variables yt and xt?

Both variables are at steady state and no longer changing. Signup and view all the answers

What is the first step in obtaining a long run static solution from a dynamic model?

Remove first difference terms. Signup and view all the answers

Which equation reflects the static solution derived from the dynamic model involving lagged terms?

β4yt-1 = - β1 - β3x2t-1 Signup and view all the answers

What issue arises when including lagged variables in a regression model to address autocorrelation?

It violates the assumption of non-stochastic RHS variables. Signup and view all the answers

What happens if there is still autocorrelation in the residuals after adding lagged variables?

The OLS estimators will not be consistent. Signup and view all the answers

In the context of dynamic models, what do the terms 'Δx2t' represent?

The change in the variable x2 from the previous time period. Signup and view all the answers

Which of the following is NOT a step in deriving the long run static equilibrium solution?

Introducing new independent variables. Signup and view all the answers

Study Notes

Classical Linear Regression Model Assumptions and Diagnostics

The Classical Linear Regression Model (CLRM) assumes specific characteristics of the disturbance terms.
These assumptions include:
- E(u) = 0: The expected value of the disturbance term is zero.
- Var(u₁) = σ² < ∞: The variance of the disturbance term is constant and finite.
- Cov (u₁, u₁) = 0, i ≠ j: Disturbance terms are uncorrelated with each other.
- The X matrix is non-stochastic or fixed in repeated samples: The independent variables are not random.
- u₁ ~ N(0, σ²): The disturbance terms follow a normal distribution with mean zero and constant variance.

Investigating Violations of CLRM Assumptions

Diagnostic tests are used to check if the CLRM assumptions are violated.
Potential violations can result in:
- Inaccurate coefficient estimates.
- Inaccurate standard errors.
- Incorrect distribution of test statistics.

Statistical Distributions for Diagnostic Tests

F-tests and χ²-tests (sometimes called LM tests) are employed for diagnostics.
F-tests compare restricted and unrestricted regression models.
χ²-tests, often called LM tests, have a single degree of freedom parameter.

Assumption 1: E(u) = 0

The mean of the disturbance terms is zero.
The mean of the residuals will always be zero if the model has an intercept (constant term).

Assumption 2: Var(u₁) = σ² < ∞

The variance of the errors (disturbances) is constant, a condition known as homoscedasticity.
If errors do not have a constant variance, they are heteroscedastic.
Heteroscedasticity implies that the variance of residuals varies across observations.

Detection of Heteroscedasticity: The GQ Test

The Goldfeld-Quandt (GQ) test detects heteroscedasticity by splitting the data into two sub-samples.
The null hypothesis is that the variances of the disturbances are equal across sub-samples.
The test statistic, GQ, is the ratio of the larger residual variance to the smaller.

Detection of Heteroscedasticity: The White Test

White's test is a general test for heteroscedasticity.
This test runs auxiliary regressions on the squared and cross-products of the explanatory variables.
The test statistic is related to the R² of the auxiliary regression.

Consequences of Using OLS in the Presence of Heteroscedasticity

OLS estimation still provides unbiased coefficient estimates but they are not efficient (BLUE property).
Using OLS in the presence of heteroscedasticity might yield inaccurate standard errors which could lead to incorrect inferences.

How do we Deal with Heteroscedasticity?

If the cause of heteroscedasticity is known—e.g. the variance of the errors is related to another variable—then Generalized Least Squares (GLS) can be used.
Other solutions include transforming variables (e.g. logs) or adopting White's corrected standard errors

Autocorrelation

Autocorrelation occurs when disturbance terms in a regression model are correlated across observations.
Autocorrelation can create bias in model estimates and make standard errors inaccurate.

Detection of Autocorrelation: Durbin-Watson Test

The Durbin-Watson (DW) test assesses first-order autocorrelation.
The null hypothesis of the test is no autocorrelation(p = 0).
The DW statistic is calculated from the residuals.
- Values close to 2 suggest little autocorrelation.
- Values away from 2 may indicate autocorrelation.

Another Test for Autocorrelation: Breusch-Godfrey Test

The Breusch-Godfrey test can detect multiple orders of autocorrelation.
The model includes lagged residuals as regressors to capture autocorrelation patterns.
The test statistic follows a χ² distribution.

Consequences of Ignoring Autocorrelation

Coefficient estimates derived using Ordinary Least Squares (OLS) can be inefficient (not Best Linear Unbiased Estimates) even with a large number of sample sizes.
R² values may be inflated if there is positive autocorrelation in the residuals.

"Remedies" for Autocorrelation

If the form of autocorrelation is known, Generalized Least Squares (GLS) procedures such as Cochrane-Orcutt can be used.

Dynamic Models

Static models assume the dependent variable depends only on current independent variables, while dynamic models consider the influence of previous values of both the dependent and independent variables.

Why Include Lags

Inertia, over-reactions, or overlapping moving averages are common reasons for including lags in macro-economic variables.
Lagged variables may indicate a significant relationship or omitted variables.

Models in First Difference

Transitioning to a first-difference model involves subtracting the previous period's values from the current values of variables.

The Long Run Static Equilibrium Solution

The long-run equilibrium value(s) of dependent variable is derived by simplifying the relationship.

Problems with Adding Lagged Regressors

Introducing lagged dependent variables might violate the assumption of exogeneity (non-stochasticity) of the regressors.

Multicollinearity

Multicollinearity occurs when there is a high linear correlation among the predictor variables in a multiple regression model.

Measuring Multicollinearity

Correlation matrix examination to determine how strongly the independent variables are correlated.

Solutions to the Problem of Multicollinearity

Traditional approaches (ridge regression, principal components) may create additional problems in the solution.
Strategies such as dropping one of the correlated variables, combining variables into a ratio, or increasing the sample size may solve multicollinearity problems.

Adopting the Wrong Functional Form

The RESET test diagnoses misspecification problems in the functional form of a regression model.
For example, adding polynomial terms of the fitted values to an auxiliary regression can help detect a mis-specified model.
Transforming variables into logs is one way to linearise non-linear structures.

Testing the Normality Assumption

The normality assumption is crucial for hypothesis testing's accuracy.
Bera and Jarque normality tests for errors (disturbances) look at skewness and kurtosis.
These tests look at whether the coefficients of skewness and kurtosis of the disturbance terms are jointly zero.
If the model violations in normality exist, dummy variables (e.g., for periods of particular economic factors) can be used to correct the misspecification.

What to do if Non-Normality?

Alternatives exist if normality testing for skewness and kurtosis produces non-zero coefficients.
Dummy variables may address particular outliers to correct the non-normal issue.

Omission of an Important Variable or Inclusion of an Irrelevant Variable

Omitting a relevant, correlated variable biases the estimates of other coefficients and the intercept.
Including irrelevant variables increases estimation complexity without improving accuracy.

Parameter Stability Tests

Parameter stability tests are used to identify whether the regression parameters (coefficients) remain constant throughout the sample period.

The Chow Test

The Chow test (analysis of variance test) divides the data into two sub-periods to compare the restricted and unrestricted models.
The test statistic follows an F-distribution and uses the ratio of the restricted (unrestricted) RSS from the models.
Test to see if the parameter values are stable between two periods using the estimated F-distribution statistical.

The Predictive Failure Test

The predictive failure test evaluates predictive performance.
The test statistic is obtained from RSS (restricted) and RSS1 (from a sub-sample).
The statistic follows an F-distribution to determine predictive validity.

Backwards vs. Forwards Predictive Failure

These models are used to determine the robustness of model coefficients depending if the model is estimated on a forward sub-sample or in backward direction.

Measurement Errors

Measurement error in the independent variables, but not in the dependent variable, will cause biased, inconsistent coefficient estimates towards zero.
Measurement error in the dependent variable does not cause the coefficient estimates to be inconsistent but will result in larger standard errors than expected in the model.

A Strategy for Building Econometric Models

An econometric model tries to provide a reasonably accurate estimation about variables’ relationship using the given data in the sample period.
The model should satisfy assumptions of the Classical Linear Regression Model (CLRM), be as parsimonious (simple) as possible, and be theoretically sound.

2 Approaches to Building Econometric Models

The specific-to-general approach starts with a simple model and progressively adds complexity, often overlooking detailed diagnostic checks.
The general-to-specific approach begins with a larger model to test the statistical and theoretical consistency of model coefficients.

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