Time Series Analysis and Tests Quiz

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Questions and Answers

What does a positive autocorrelation in the residuals indicate?

The residuals are independent of each other.

There is a constant variance in the residuals.

The residuals show a predictable pattern that can be utilized. (correct)

The residuals are randomly distributed over time.

Which value represents the change in $y$ (Δyt) for 1990M02?

-4.0

2.3

4.0 (correct)

-1.7

In the context of time series, what does yt-1 represent?

The value of y one time period before t. (correct)

The maximum value of y in the dataset.

The current value of y at time t.

The average of all previous y values.

Which condition is violated if there is autocorrelation present in the residuals?

The independence of observations. Signup and view all the answers

Which is the correct interpretation of the change in yt from 1989M10 to 1989M11?

The value of y decreased significantly. Signup and view all the answers

What is the null hypothesis in the context of the Chow Test for this example?

The parameters of the regression model remain constant over time. Signup and view all the answers

How is the test statistic calculated in the Chow Test?

It compares the RSS of the whole sample with the sum of both sub-sample RSS. Signup and view all the answers

What does a test statistic value greater than the critical value from the F-distribution signify?

The null hypothesis of stability is rejected. Signup and view all the answers

In the Chow Test example, what is the value of $T$ for the whole sample period from 1981 to 1992?

$144$ Signup and view all the answers

What is the role of $k$ in the formula for the test statistic in the Chow Test?

It is the number of regressors in the unrestricted regression. Signup and view all the answers

What is the null hypothesis in the Goldfeld-Quandt test for heteroscedasticity?

The variances of the disturbances are equal. Signup and view all the answers

What are the null hypotheses in the Breusch-Godfrey Test for autocorrelation?

All ρ coefficients are equal to 0 Signup and view all the answers

What does the test statistic from the Breusch-Godfrey Test resemble?

Chi-squared distribution Signup and view all the answers

Which of the following describes the calculation of the GQ test statistic?

The ratio of the larger residual variance to the smaller residual variance. Signup and view all the answers

What is a consequence of ignoring autocorrelation in regression analysis?

Standard errors may be underestimated Signup and view all the answers

What is the distribution of the GQ test statistic under the null hypothesis of homoscedasticity?

F distribution. Signup and view all the answers

Which aspect of the Goldfeld-Quandt test is potentially problematic?

The choice of where to split the sample. Signup and view all the answers

Which method can be used to correct for the known form of autocorrelation?

Generalized Least Squares Signup and view all the answers

What does it mean for an estimator to be referred to as BLUE?

It has the smallest variance among all linear estimators. Signup and view all the answers

In White’s test for heteroscedasticity, what is done after obtaining the residuals?

Run an auxiliary regression. Signup and view all the answers

Which of the following is a consequence of using OLS in the presence of heteroscedasticity?

Standard errors may be inappropriate. Signup and view all the answers

What characterizes perfect multicollinearity in regression models?

At least one explanatory variable is a linear combination of others Signup and view all the answers

How is the chi-squared statistic related to White’s test for heteroscedasticity?

It is related to R-squared from the auxiliary regression. Signup and view all the answers

How can we remove heteroscedasticity if its cause is known?

By using generalized least squares (GLS). Signup and view all the answers

What typically happens to R² when near multicollinearity is present?

R² is likely to be inflated Signup and view all the answers

What assumption makes White’s test preferable for detecting heteroscedasticity?

It makes few assumptions about the form of heteroscedasticity. Signup and view all the answers

What is a common problem that arises from high standard errors of individual coefficients due to multicollinearity?

It complicates determining the importance of predictors Signup and view all the answers

What happens to the estimated standard errors if heteroscedasticity is present?

They could be either too large or too small. Signup and view all the answers

What does the term 'heteroscedasticity' refer to in regression analysis?

The varying variances of residuals. Signup and view all the answers

What is the relationship between the error variance and another variable in the context of heteroscedasticity?

Error variance can be directly related to specific variables. Signup and view all the answers

What should be considered when applying a GLS procedure to correct autocorrelation?

Assumptions about the form of the autocorrelation Signup and view all the answers

What will be true about the disturbances in a regression equation after applying GLS?

They will be homoscedastic. Signup and view all the answers

Which assumption is NOT necessary for OLS to be considered BLUE?

Large sample size. Signup and view all the answers

What characteristic of OLS allows it to provide unbiased coefficient estimates?

It uses a linear function of the data. Signup and view all the answers

What is a potential issue when the regression becomes very sensitive to small changes in specification?

Wide confidence intervals for parameters Signup and view all the answers

Which method can be used to measure the extent of multicollinearity among variables?

Variance Inflation Factor Signup and view all the answers

Which of the following is NOT a suggested solution for multicollinearity problems?

Increase the sample size with unrelated data Signup and view all the answers

What does the Ramsey’s RESET test help to identify?

Mis-specification of functional form Signup and view all the answers

If the RESET test statistic exceeds the critical value of a chi-squared distribution, what should be concluded?

The functional form likely requires a change Signup and view all the answers

Which of the following is true about high correlation between y and predictor variables?

It does not imply multicollinearity Signup and view all the answers

What is one common approach to alleviate multicollinearity issues?

Collect a longer run of relevant data Signup and view all the answers

What higher-order terms does the RESET test incorporate for testing mis-specification?

Quadratic and cubic terms of fitted values Signup and view all the answers

Study Notes

Classical Linear Regression Model Assumptions and Diagnostics

Classical linear regression model assumes certain characteristics of the error term.
The error term's expected value is zero (E(εt) = 0).
The variance of the error term is constant (Var(εt) = σ²).
Error terms are uncorrelated (cov(εi, εj) = 0 for i ≠ j).
The X matrix is non-stochastic or fixed in repeated samples.
Error terms follow a normal distribution (εt ~ N(0, σ²)).

Violation of Classical Linear Regression Model Assumptions

Violations of these assumptions can lead to incorrect coefficient estimates, inaccurate standard errors, and inappropriate test statistics.
Multiple violations of these assumptions are possible.

Investigating Violations of CLRM Assumptions

Studying how to test for violations
Identifying causes of violations
Defining the consequences of violations.

Assumption 1: E(εt) = 0

The mean of the error terms should be zero.
Residuals are used to test this assumption as the error terms cannot be observed directly.
The mean of residuals will be zero when there is a constant (intercept) term in the regression model.

Assumption 2: Var(εt) = σ²

Constant variance of error terms, known as homoscedasticity.
Non-constant variance is heteroscedasticity.
Detection methods like graphical or formal tests (e.g. Goldfeld-Quandt test and White's test) can be used.

Detection of Heteroscedasticity: The Goldfeld-Quandt Test

Splits the dataset into two sub-samples.
Calculates residual variances on each sub-sample.
The ratio of the variances forms the test statistic.
This statistic is F(T₁-k, T₂-k) distributed under the null hypothesis of equal variances.
Choice of split point is arbitrary, affecting the test.

Detection of Heteroscedasticity: The White's Test

Assumes a general form for heteroscedasticity.
Builds an auxiliary regression using squared and cross-products of predictor variables.
The test statistic is the multiplied R-squared from the auxiliary regression with the sample size (TR²).
This is a x²(m) distribution, where m is the number of regressors in the auxiliary regression.

Consequences of Using OLS in the Presence of Heteroscedasticity

OLS estimation provides unbiased coefficient estimates but is no longer Best Linear Unbiased Estimator (BLUE).
Standard errors calculated via the traditional OLS formula are unreliable.
Inference made from conclusions based on OLS is potentially incorrect.
R-squared might be inflated for positively correlated residuals.

Remedies for Heteroscedasticity

If the form of heteroscedasticity is known, generalized least squares (GLS) can correct.
Dividing by the variable influencing variances can result in a homoscedastic model.

Autocorrelation

Models assume errors are uncorrelated (Cov(ε₁, εj) = 0, for i ≠ j).
Autocorrelation (serial correlation) occurs when errors at one period (εt) are correlated with error terms of a past period (εt-1).
Visual inspection of the residuals plot can indicate autocorrelation.
Positive autocorrelation indicates cyclical patterns.
Negative indicates alternating patterns.

Detecting Autocorrelation: The Durbin-Watson Test

Tests for first-order autocorrelation.
The test statistic (DW) ranges from 0 to 4.
Values near 2 suggest little evidence of autocorrelation.
Critical values (d_l and d_u) allow for rejection of the null hypothesis (no autocorrelation).
Intermediate values of DW cannot assist in either rejecting or not rejecting the null hypothesis.

Another Test for Autocorrelation: The Breusch-Godfrey Test

More general test for rth-order autocorrelation.
The test runs a regression with residuals and lagged residuals.
The test statistic (derived from the regression's R²) follows a x²(r) distribution.
Values exceeding the critical value lead to rejection of the null hypothesis of no autocorrelation.

Consequences of Ignoring Autocorrelation

Unbiased coefficient estimates may be inefficient (i.e., not BLUE even in large samples).
Incorrect standard errors can lead to invalid inferences.
R-squared values might be inflated (especially for positive autocorrelation).

Remedies for Autocorrelation

If the form is known, GLS (like Cochrane-Orcutt) can account for it.
However, these procedures rely on assumptions of the correlation's form.
In some cases, modelling adjustments may be needed rather than simply correcting for autocorrelation.

Multicollinearity

Explanatory variables are highly correlated.
Perfect multicollinearity prohibits estimating all coefficients.
Near multicollinearity results in high R-squared but unreliable coefficient standard errors and potentially erroneous inferences.

Measuring Multicollinearity

Method 1: Correlation matrix (looking at correlations between variables).
Method 2: Variance Inflation Factor (VIF).
High correlation between a dependent and independent variable is not considered a violation of the model.

Solutions to the Problem of Multicollinearity

Traditional methods (ridge regression, principal components) can worsen issues.
If model validity is otherwise sound, multicollinearity might be ignored.
Transformations of highly correlated variables, collecting more data, or shifting to a higher frequency can potentially resolve issues.

Adopting the Wrong Functional Form

Linearity is not always the appropriate form (can be multiplicative).
Ramsey's RESET test is a general test to assess model specification.
Adding higher-order terms of fitted values to the model can be a way to detect misspecifications.
Test statistic following a x²(p-1) distribution is created from the regression's R squared.

Testing the Normality Assumption

Normality is inherent to many hypothesis testing procedures.
Bera-Jarque test tests for departures from normality in residuals (by testing whether skewness and excess kurtosis are jointly zero).
Skewness and excess kurtosis are standardized third and fourth moments of a distribution.

What if Non-Normality is Detected?

Method that doesn't assume normality may be applied.
One solution involves dummy variables to account for extreme residuals.

Omission of an Important Variable or Inclusion of an Irrelevant Variable

Omission of a relevant covariate or inclusion of an irrelevant variable leads to biased and inconsistent estimates.
Relevant model considerations include that any omitted relevant covariate must not be correlated with any other included variable.

Parameter Stability Tests

Tests whether model parameters remain constant across different data periods.
Splitting data into sub-samples and comparing the residual sum of squares (RSS) across models is used as a test.
Chow test is an analysis of variance test that examines stability across sub-sample regressions.

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Description

Test your understanding of key concepts in time series analysis, including autocorrelation, Chow Test, and heteroscedasticity. This quiz covers definitions, interpretations, and statistical tests relevant to the field. Perfect for students looking to solidify their knowledge in econometrics.