Time Series Analysis 2
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Questions and Answers

In forecasting models, what is the purpose of ARMA and GARCH models?

  • To provide forecasts based solely on own historical observations (correct)
  • To include multiple equation models for forecasting purposes
  • To construct forecasting models based on gut feeling
  • To test theories and investigate dynamics in detail

    • What is the forecasting function for an AR(1) process, $y_t = a_0 + a_1y_{t-1} + \epsilon_t$?

  • $y_{t+1} = a_0 + a_1y_{t-1} + \epsilon_t$
  • $y_{t+1} = a_0 + a_1y_{t-1} + \epsilon_{t+1}$
  • $y_{t+1} = a_0 + a_1y_t + \epsilon_t$
  • $y_{t+1} = a_0 + a_1y_t + \epsilon_{t+1}$ (correct)

    • What type of models provide the most accurate (out-of-sample) forecasts even if R2 is relatively low?

  • Models with more estimated parameters
  • Parsimonious models (correct)
  • Complex models with high R2
  • Simple models with low R2

    • In forecasting, what does the true DGP stand for?

    <p>True Data Generating Process</p> Signup and view all the answers

    What is the main purpose of including other variables with predictive power in forecasting models?

    <p>To improve the accuracy of forecasts</p> Signup and view all the answers

    What is the key characteristic of coefficients in forecasting models?

    <p>They don’t need to be causal</p> Signup and view all the answers

    What is the focus of pure ARMA and GARCH models in forecasting?

    <p>To provide forecasts based solely on own historical observations</p> Signup and view all the answers

    What type of models are often used to get better forecasts for asset returns and other economic variables based on historical data?

    <p>ARMA and GARCH models</p> Signup and view all the answers

    In ARMA process forecasting, what happens to prediction accuracy as the forecast horizon increases?

    <p>Prediction accuracy decreases</p> Signup and view all the answers

    What is a property of a good prediction model in time series analysis?

    <p>Mean of prediction errors equal to 0 and small prediction error variance</p> Signup and view all the answers

    How are longer-horizon forecasts computed in ARMA process?

    <p>By iterating forward using forecast functions</p> Signup and view all the answers

    What is used to evaluate forecast accuracy in time series analysis?

    <p>Mean squared prediction error (MSPE)</p> Signup and view all the answers

    What does estimating models based on observed data introduce into future forecasts?

    <p>Coefficient uncertainty</p> Signup and view all the answers

    What can be computed for different forecast horizons in ARMA models?

    <p>Prediction errors</p> Signup and view all the answers

    What is involved in comparing out-of-sample forecasts in time series analysis?

    <p>Computing prediction errors and analyzing forecast accuracy using various criteria</p> Signup and view all the answers

    What can be computed for any ARMA(p,q) model using the iteration technique?

    <p>Forecasts</p> Signup and view all the answers

    What can be used to conduct statistical significance testing of MSPE differences in time series analysis?

    <p>Granger-Newbold and Diebold-Mariano tests</p> Signup and view all the answers

    What are dynamic forecasts in time series analysis?

    <p>Shorter horizon predictions used to compute longer horizon forecasts</p> Signup and view all the answers

    What are unbiased predictions for future values in ARMA models?

    <p>Forecasts</p> Signup and view all the answers

    What decreases as the forecast horizon increases in stationary ARMA models?

    <p>Prediction accuracy</p> Signup and view all the answers

    What is the purpose of using a rolling window in forecasting models?

    <p>To estimate a model with a fixed in-sample period, with start and end dates successively increasing by one observation</p> Signup and view all the answers

    What is the key characteristic of coefficients in forecasting models when using a rolling window?

    <p>They are updated for each new prediction, catering for possible structural changes</p> Signup and view all the answers

    What does the GARCH(p,q) model allow for in forecasting?

    <p>Forecasting the volatility</p> Signup and view all the answers

    What is involved in forecast evaluation in EViews II?

    <p>Calculating various statistics such as RMSE, Mean Absolute Error, Mean Absolute Percent Error</p> Signup and view all the answers

    What can be computed for different forecast horizons in ARMA models?

    <p>Pseudo out-of-sample forecast errors</p> Signup and view all the answers

    What does choosing the 'Static' option for forecasting result in, based on the empirical example using OMXH Small Cap weekly returns?

    <p>Unbiased forecasts with slightly more volatile actual returns than forecasted values</p> Signup and view all the answers

    What is the focus of pure ARMA and GARCH models in forecasting?

    <p>Forecasting the mean and volatility</p> Signup and view all the answers

    What decreases as the forecast horizon increases in stationary ARMA models?

    <p>Prediction accuracy</p> Signup and view all the answers

    What is the main purpose of estimating models based on observed data in forecasting?

    <p>To capture historical patterns and relationships</p> Signup and view all the answers

    What happens to prediction accuracy as the forecast horizon increases in ARMA process forecasting?

    <p>It decreases</p> Signup and view all the answers

    What can be used to conduct statistical significance testing of MSPE differences in time series analysis?

    <p>The F-test</p> Signup and view all the answers

    What is a key characteristic of a non-stationary time series?

    <p>Permanent components in mean and variance</p> Signup and view all the answers

    In the context of time series analysis, what does stationarity refer to?

    <p>Covariance stationarity (&quot;weak&quot; stationarity)</p> Signup and view all the answers

    What is the distinguishing feature of the variance in non-stationary time series?

    <p>No finite value when t approaches infinity</p> Signup and view all the answers

    What happens to the sample autocorrelations in non-stationary time series?

    <p>They die out very slowly</p> Signup and view all the answers

    What is the null hypothesis in the Dickey-Fuller (DF) test for unit root?

    <p>The parameter is equal to 1</p> Signup and view all the answers

    What does the presence of a unit root in a time series indicate?

    <p>A stochastic trend and non-stationarity</p> Signup and view all the answers

    What is the key purpose of the Augmented Dickey-Fuller (ADF) test?

    <p>Address complications such as autocorrelation in the residual and include lags of the dependent variable in the model</p> Signup and view all the answers

    What type of time series may show a cointegrating relationship?

    <p>Non-stationary time series</p> Signup and view all the answers

    What is the impact of the lag length chosen for the ADF test?

    <p>The test result</p> Signup and view all the answers

    What is the purpose of verifying whether time series are stationary or non-stationary before conducting time series analysis?

    <p>To ensure the validity of standard inference and testing procedures</p> Signup and view all the answers

    What is the main consequence of non-stationary time series in regression?

    <p>Spurious regression</p> Signup and view all the answers

    Which variables are usually stationary?

    <p>Differenced variables, such as GDP growth and stock market returns</p> Signup and view all the answers

    What is commonly used for lag length selection in the ADF test?

    <p>SBC, AIC, and MAIC</p> Signup and view all the answers

    What can occur due to spurious correlation in non-stationary time series?

    <p>Spurious regression</p> Signup and view all the answers

    What should be included in unit root tests based on the nature of the time series?

    <p>Deterministic variables, such as a constant or trend</p> Signup and view all the answers

    What can lead to a long-run equilibrium relationship among variables?

    <p>Non-stationary time series</p> Signup and view all the answers

    What is a potential issue with the ADF test?

    <p>It has relatively low power and accepts the null hypothesis of unit root too often</p> Signup and view all the answers

    What is a complication associated with the Phillips-Perron (PP) test?

    <p>It suffers from similar low power as the ADF test</p> Signup and view all the answers

    What does the KPSS test reject the null hypothesis for?

    <p>U.S. default risk premium, indicating it is a non-stationary series</p> Signup and view all the answers

    What is recommended for univariate analysis of trend stationary series?

    <p>Removing deterministic trend or including it as an explanatory variable</p> Signup and view all the answers

    What is recommended for difference stationary series?

    <p>Taking the difference</p> Signup and view all the answers

    What may be involved in multivariate analysis if there are multiple I(1) series?

    <p>Testing for cointegration</p> Signup and view all the answers

    What type of p-values are used for accurate p-values in unit root tests according to MacKinnon (1996)?

    <p>One-sided p-values</p> Signup and view all the answers

    What happens if there are structural changes/breaks in time series?

    <p>It can complicate unit root tests and lead to erroneous conclusions</p> Signup and view all the answers

    What does the ADF test conclude if the test value is less than -3.13 at 10% significance level?

    <p>Series is stationary around a deterministic trend</p> Signup and view all the answers

    What is a potential issue with the ADF test for relatively small number of observations?

    <p>It may not work well</p> Signup and view all the answers

    What is a general recommendation for multivariate analysis in the presence of multiple I(1) series?

    <p>Testing for cointegration</p> Signup and view all the answers

    In the context of cointegration, what did Engle and Granger show in 1987?

    <p>Variables with a unit root may have a linear combination that is stationary</p> Signup and view all the answers

    What is the implication of having both yt and zt as non-stationary, I(1), and cointegrated?

    <p>It is meaningful to estimate the model (even with OLS)</p> Signup and view all the answers

    In the regression model yt = a0 + a1zt + et, what does it mean if et is not I(0)?

    <p>It is not meaningful to estimate such model</p> Signup and view all the answers

    What is the key characteristic of cointegration?

    <p>Variables with a unit root may have a linear combination that is stationary</p> Signup and view all the answers

    What is the implication of having both yt and zt as integrated of different orders (e.g., one being I(0) and the other I(1)) with a non-I(0) residual?

    <p>It is not meaningful to estimate such model</p> Signup and view all the answers

    What did Engle and Granger receive the economics Nobel prize for in 2003?

    <p>Showing that variables with a unit root may have a linear combination that is stationary</p> Signup and view all the answers

    What is the significance of having both yt and zt as non-cointegrated non-stationary I(1)* variables?

    <p>The model cannot be estimated with OLS, as et is non-stationary</p> Signup and view all the answers

    What is the main implication of having both yt and zt as stationary, making OLS a suitable estimator?

    <p>OLS is a suitable estimator</p> Signup and view all the answers

    What is a key limitation of the Engle-Granger method?

    <p>It can estimate only up to one cointegrating relationship between the variables</p> Signup and view all the answers

    What is a distinguishing feature of the Johansen Maximum Likelihood (ML) method?

    <p>It can estimate and test for the presence of multiple cointegrating vectors</p> Signup and view all the answers

    What is the purpose of the Trace Test in the context of cointegration testing?

    <p>It tests the number of cointegrating vectors against a specific alternative</p> Signup and view all the answers

    What is a potential issue associated with cointegration testing?

    <p>Lag length selection</p> Signup and view all the answers

    What is the number of cointegrating vectors in a system with n stochastic variables according to Johansen ML method?

    <p>At most n-1</p> Signup and view all the answers

    What is the focus of the Hansen instability test in the context of cointegration testing?

    <p>Testing for structural breaks in the cointegrating relationship</p> Signup and view all the answers

    What is the main advantage of the Johansen ML method over the Engle-Granger method?

    <p>It can estimate and test for the presence of multiple cointegrating vectors</p> Signup and view all the answers

    What is recommended for lag length selection in cointegration testing?

    <p>Info criteria</p> Signup and view all the answers

    What is used to test for residual stationarity in cointegration analysis?

    <p>KPSS-test</p> Signup and view all the answers

    Which test is recommended for testing residual stationarity in cointegration analysis?

    <p>KPSS-test</p> Signup and view all the answers

    What should not be included in the tested equation for residual stationarity in cointegration analysis?

    <p>Deterministic variables</p> Signup and view all the answers

    What does rejection of the null hypothesis in cointegration analysis indicate?

    <p>et is stationary and yt and zt are cointegrated</p> Signup and view all the answers

    Who provides the critical values for the Engle-Granger test?

    <p>Engle and Yoo</p> Signup and view all the answers

    What is recommended for cointegrating regression estimation?

    <p>Fully-Modified OLS (FMOLS)</p> Signup and view all the answers

    What does the FMOLS estimator aim to remove in cointegrating regression estimation?

    <p>Bias</p> Signup and view all the answers

    When is the error correction model used?

    <p>When variables are cointegrated</p> Signup and view all the answers

    What is the equilibrium error in the error-correction model?

    <p>The residual from the estimated regression</p> Signup and view all the answers

    What do error-correction coefficients indicate in time series analysis?

    <p>The adjustment speeds of variables towards the equilibrium relation</p> Signup and view all the answers

    What does the VECM enable in time series analysis?

    <p>Forecasting</p> Signup and view all the answers

    What is the purpose of the Engle-Granger method in time series analysis?

    <p>To test for cointegration by estimating a regression involving testing for the order of integration of each variable</p> Signup and view all the answers

    What does a cointegrating vector represent in time series modelling?

    <p>The long-term equilibrium relationship between variables</p> Signup and view all the answers

    What does the equilibrium error in cointegration measure?

    <p>Deviations from the long-term equilibrium relationship</p> Signup and view all the answers

    What does the OLS regression yield in the context of cointegration?

    <p>Super-consistent estimators of the cointegrating parameters if the model is stationary</p> Signup and view all the answers

    What is the focus of the Engle-Granger method in testing for cointegration?

    <p>Testing for the order of integration of each variable</p> Signup and view all the answers

    What implications does cointegration have in time series modelling?

    <p>Implications for predictability, long-term dynamics, and optimal portfolio allocation</p> Signup and view all the answers

    What is a potential application of cointegration in time series modelling?

    <p>Testing theories such as PPP</p> Signup and view all the answers

    What are examples of potential cointegration mentioned in the text?

    <p>Spot and futures prices for a commodity, exchange rates, equity prices and dividends, interest rates, and housing prices</p> Signup and view all the answers

    What does the OLS regression yield if the model is stationary in cointegration?

    <p>Super-consistent estimators of the cointegrating parameters</p> Signup and view all the answers

    What is the key characteristic of the cointegrating vector in time series modelling?

    <p>Represents the long-term equilibrium relationship between variables</p> Signup and view all the answers

    What does the equilibrium error measure in cointegration?

    <p>Deviations from the long-term equilibrium relationship</p> Signup and view all the answers

    What is the key characteristic of a vector autoregressive (VAR) model?

    <p>It includes at least two stochastic variables that are assumed to interact with each other.</p> Signup and view all the answers

    What is the structural form of the simpliest VAR model?

    <p>A system of two or more equations with the same explanatory variables in each estimated equation.</p> Signup and view all the answers

    What is the significance of Christopher A. Sims in the context of VAR modeling?

    <p>He was awarded the Nobel Prize for Economics in 2011, highlighting the importance of VAR modeling in economics.</p> Signup and view all the answers

    What does a VAR model illustrate about the stochastic variables it includes?

    <p>They are assumed to interact or have prediction power with respect to each other.</p> Signup and view all the answers

    What do accumulated impulse response functions (IRFs) show?

    <p>The sum of responses of a given variable to a given shock over time</p> Signup and view all the answers

    What is the purpose of the accumulated response of a variable to a shock?

    <p>Show the impact on the levels of the variables over time</p> Signup and view all the answers

    What should the accumulated IRFs converge to over the long run?

    <p>Some value (does not need to be zero)</p> Signup and view all the answers

    What does the accumulated response of a variable to a shock enable the interpretation of?

    <p>The nature of the shocks</p> Signup and view all the answers

    What test can be used to select the lag length in Vector Autoregression (VAR) Models?

    <p>Multivariate versions of AIC and SBC</p> Signup and view all the answers

    What does the Likelihood Ratio (LR) test assume about the test statistic distribution?

    <p>Chi-squared distribution</p> Signup and view all the answers

    What does the Null hypothesis for the LR test in lag length selection state?

    <p>The last w lags in the model are together insignificant and can be removed</p> Signup and view all the answers

    What is a potential issue with using the F-test on separate equations for lag selection?

    <p>It may not be suitable for testing whether dropping specific lags is suitable for every equation</p> Signup and view all the answers

    What must be the same when comparing VAR models with different lag lengths?

    <p>Effective sample size</p> Signup and view all the answers

    What does the LR test adjust for when estimating models?

    <p>Lost observations from the effective sample</p> Signup and view all the answers

    What distribution does the LR test assume for the errors?

    <p>Normal distribution</p> Signup and view all the answers

    What can be used as alternative test criteria for lag length selection in VAR models?

    <p>Multivariate versions of AIC and SBC</p> Signup and view all the answers

    What can be used to test for residual autocorrelation in VAR models?

    <p>Portmanteau test</p> Signup and view all the answers

    What can be used to test for residual normality in VAR models?

    <p>Jarque-Bera test</p> Signup and view all the answers

    What does the LR test depend on for giving conclusions?

    <p>Testing approach</p> Signup and view all the answers

    What can be used to evaluate forecast accuracy in EViews for VAR models?

    <p>Forecast evaluation II</p> Signup and view all the answers

    In a reduced form VAR model with a lag length of 1, how are the equations estimated?

    <p>The lagged variables are included and equations are estimated separately using ordinary least squares (OLS)</p> Signup and view all the answers

    What is a key advantage of VAR models?

    <p>They allow for detailed economic dynamics analysis</p> Signup and view all the answers

    What is a potential limitation of VAR models?

    <p>They are atheoretical</p> Signup and view all the answers

    In lag length selection for monthly data, what is a suitable starting point?

    <p>12 lags</p> Signup and view all the answers

    What constitutes the reduced form residuals in a VAR model?

    <p>Innovations from the white noise processes</p> Signup and view all the answers

    What is crucial to avoid in VAR model lag length selection?

    <p>Model misspecification or wasted degrees of freedom</p> Signup and view all the answers

    What is the primary characteristic of the error terms in a reduced form VAR model?

    <p>They are white noise processes</p> Signup and view all the answers

    What is the key consideration in the selection of variables for a VAR model?

    <p>Theoretical relevance</p> Signup and view all the answers

    What is the role of feedback in a reduced form VAR model?

    <p>It introduces endogeneity in the system</p> Signup and view all the answers

    What is the appropriate lag length selection procedure in VAR models?

    <p>Starting with the longest plausible lag length and removing insignificant lags</p> Signup and view all the answers

    What is the primary requirement for the error terms in a reduced form VAR model?

    <p>They must be independent of each other</p> Signup and view all the answers

    What is the primary advantage of the reduced form VAR model over the structural form?

    <p>It resolves the simultaneous equation bias</p> Signup and view all the answers

    What is a key method to identify a VAR model by imposing a recursive ordering among the variables?

    <p>Choleski decomposition</p> Signup and view all the answers

    What can be computed for impulse response functions to show the uncertainty of the estimates?

    <p>Confidence intervals</p> Signup and view all the answers

    In what way can theoretical impulse response functions show asymmetry?

    <p>They decay slower for positive shocks</p> Signup and view all the answers

    What can be used to obtain empirical examples of impulse response functions and simulate confidence bands with repetitions?

    <p>Choleski decomposition</p> Signup and view all the answers

    What is the expected long-term behavior of impulse response functions with stationary variables?

    <p>They should converge to zero</p> Signup and view all the answers

    What is a way to compare the impulse response functions obtained by different orderings if there are significant correlations between error terms?

    <p>Choleski decomposition</p> Signup and view all the answers

    What can be used to understand the effects of shocks on the variables in time series analysis?

    <p>Impulse response functions</p> Signup and view all the answers

    What is a potential approach to identify the VAR model and compute impulse response functions other than Choleski decomposition?

    <p>Structural VAR (SVAR) model</p> Signup and view all the answers

    What is a way to obtain empirical examples of impulse response functions and simulate confidence bands with repetitions?

    <p>Choleski decomposition</p> Signup and view all the answers

    What is a method to identify a VAR model by imposing a recursive ordering among the variables based on economic theory or previous empirical examinations?

    <p>Choleski decomposition</p> Signup and view all the answers

    What is a way to identify the VAR model by imposing a recursive ordering among the variables based on the magnitude of correlation coefficients between error terms?

    <p>Choleski decomposition</p> Signup and view all the answers

    What is a method to identify the VAR model by imposing a recursive ordering among the variables based on previous empirical examinations?

    <p>Choleski decomposition</p> Signup and view all the answers

    What is the aim of the LR test in VAR models?

    <p>To test for the significance of seasonal dummies</p> Signup and view all the answers

    In a two-variable 1st order VAR, how many parameters are there?

    <p>10 parameters</p> Signup and view all the answers

    What is the purpose of imposing the restriction b21= 0 in VAR models?

    <p>To estimate the reduced form</p> Signup and view all the answers

    What is the focus of Granger causality in VAR analysis?

    <p>Investigating predictive power between variables</p> Signup and view all the answers

    What is the purpose of innovation accounting in VAR analysis?

    <p>To investigate dynamic interrelationships among endogenous variables</p> Signup and view all the answers

    What is the implication of imposing more restrictions in a VAR model?

    <p>The system becomes over-identified</p> Signup and view all the answers

    What do impulse response functions in VAR models show?

    <p>The reactions of each variable to a given shock in the structural form</p> Signup and view all the answers

    What is the aim of block-exogeneity tests in EViews for VAR models?

    <p>To investigate whether a variable is exogenous to the system</p> Signup and view all the answers

    What is the purpose of including seasonal dummies in VAR models?

    <p>To reduce the need for lags</p> Signup and view all the answers

    What is the significance of diagnostic checks in VAR models?

    <p>To address significant autocorrelation</p> Signup and view all the answers

    What does Granger causality imply in VAR analysis?

    <p>Predictive relationships</p> Signup and view all the answers

    What is the purpose of identification in VAR models?

    <p>To estimate the structural parameters</p> Signup and view all the answers

    Study Notes

    Forecasting in Time Series Analysis

    • Forecasting in ARMA process involves predicting future values based on past observations and forecast errors
    • Longer-horizon forecasts can be computed by iterating forward using forecast functions
    • Prediction accuracy decreases as the forecast horizon increases in stationary ARMA models
    • Prediction errors for ARMA models are not perfectly accurate and can be computed for different forecast horizons
    • Forecasts are unbiased predictions for future values in ARMA models, and confidence bands can be computed for the predictions
    • Forecasts can be computed for any ARMA(p,q) model using the iteration technique
    • Estimating models based on observed data introduces coefficient uncertainty into future forecasts
    • Comparing out-of-sample forecasts involves computing prediction errors and analyzing forecast accuracy using various criteria
    • Dynamic forecasts use shorter horizon predictions to compute longer horizon forecasts
    • Properties of a good prediction model include a mean of prediction errors equal to 0 and a small prediction error variance
    • Mean squared prediction error (MSPE) is a key criterion for evaluating forecast accuracy in time series analysis
    • Testing statistical significance of MSPE differences can be conducted using Granger-Newbold and Diebold-Mariano tests

    Forecasting with GARCH Model and Forecast Evaluation in EViews II

    • A rolling window involves using a fixed in-sample period to estimate a model, with the start and end dates successively increasing by one observation.
    • The forecasts are based on a model estimated with the last 100 observations, even if there are more historical observations.
    • Model coefficients are updated for each new prediction, which caters better for possible structural changes in the coefficients.
    • GARCH(p,q) model allows for forecasting the volatility.
    • After estimating the model in EViews II, one can choose between dynamic and static forecasting methods, each with its own characteristics and uses.
    • Forecast evaluation in EViews II involves calculating various statistics such as Root Mean Squared Error (RMSE), Mean Absolute Error, Mean Absolute Percent Error, Theil Inequality Coefficient, Bias Proportion, Variance Proportion, and others.
    • It is possible to compute the pseudo out-of-sample forecast errors and check their mean and standard deviation.
    • An empirical example using OMXH Small Cap weekly returns shows the use of non-normal residuals and QML estimation, with coefficients estimates remaining significant.
    • Choosing the "Static" option for forecasting in the empirical example results in unbiased forecasts with slightly more volatile actual returns than forecasted values.
    • Choosing the "Dynamic" option for forecasting in the empirical example shows forecasts for return quickly converging upon the long-term unconditional mean.
    • The text compares various models for one-step ahead forecasts and discusses the Theil U2 coefficient and RMSE as criteria for selecting the best model.
    • The content provides detailed information on forecasting with GARCH models and evaluating forecasts using EViews II, including practical examples and statistical measures.

    Unit Root Tests and Stationarity in Time Series Analysis

    • Augmented Dickey-Fuller (ADF) test with 500 observations at 10% significance level concludes series is stationary around a deterministic trend if test value < -3.13
    • Simulations show ADF test has relatively low power and accepts the null hypothesis of unit root too often
    • ADF test may not work well with relatively small number of observations
    • Phillips-Perron (PP) test suffers from similar complications as ADF test
    • Structural changes/breaks in time series can complicate unit root tests and may lead to erroneous conclusions
    • Empirical example using Eviews for unit root tests on Neste share total return index and U.S. default risk premium
    • MacKinnon (1996) one-sided p-values are used for accurate p-values in unit root tests
    • DF-GLS and KPSS are alternative unit root tests to consider
    • KPSS test rejects the null of stationarity for U.S. default risk premium, indicating it is a non-stationary series
    • Recommendations for univariate analysis include removing deterministic trend or including it as an explanatory variable for trend stationary series
    • For difference stationary series, taking the difference is recommended
    • Multivariate analysis may involve testing for cointegration if there are multiple I(1) series

    Cointegration in Time Series Modelling

    • Cointegration refers to a long-run relationship between variables that cannot wander away from each other over time
    • It has implications for predictability, long-term dynamics, optimal portfolio allocation, and can be used to test theories such as PPP
    • Examples of potential cointegration include spot and futures prices for a commodity, exchange rates, equity prices and dividends, interest rates, and housing prices
    • A cointegrating vector represents the long-term equilibrium relationship between variables, and the equilibrium error measures deviations from this relationship
    • The Engle-Granger method is used to test for cointegration, involving testing for the order of integration of each variable and estimating a regression
    • The OLS regression yields "super-consistent" estimators of the cointegrating parameters if the model is stationary, indicating cointegration between the variables

    Understanding VAR Models for Time Series Analysis

    • Seasonal dummies reduce the need for lags in VAR models and are preferred by information criteria, leading to the inclusion of three seasonal dummies in the model.
    • The significance of dummies can be tested using the LR test, which is not readily available in EViews.
    • Different criteria suggest different lag lengths, and diagnostic checks may require the inclusion of more lags to address significant autocorrelation.
    • Granger causality does not imply actual causality; it is about predictive relationships and can indicate lead-lag relationships between variables.
    • Granger causality is tested using standard F-tests and block-exogeneity tests in EViews, with the aim of investigating predictive power between variables.
    • Identification is not necessary for forecasting purposes, but for VAR models, the reduced form equations are estimated due to the feedback inherent in the VAR process.
    • In a two-variable 1st order VAR, there are 10 parameters, and to identify the structural model, at least one restriction needs to be imposed.
    • If exactly one parameter of the structural form is restricted, the system is just identified, while more restrictions lead to an over-identified system.
    • Impose the restriction b21= 0 to estimate the reduced form and compute the structural parameters based on the estimated reduced form model.
    • Innovation accounting in VAR analysis involves investigating dynamic interrelationships among endogenous variables and computing impulse response functions and variance decomposition to understand the effect of shocks on variable values.
    • Impulse response functions, computed in practice by expressing the VAR model as a vector moving average (VMA) process, show the reactions of each variable to a given shock in the structural form.
    • The formal derivations of impulse responses and variance decomposition are presented in Enders, providing a more in-depth understanding of the analysis.

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    Test your knowledge of time series analysis with this quiz covering forecasting in ARMA process, GARCH model, and forecast evaluation in EViews II. Explore topics such as prediction accuracy, forecast errors, model estimation, out-of-sample forecasts, forecast evaluation statistics, dynamic and static forecasting methods, and practical examples with non-normal residuals and QML estimation.

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