Forecasting Inter Related Series
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Questions and Answers

What is the assumption underlying the derivation of interval forecasts in the direct multi-step ahead forecasting framework?

  • Stationarity of the time series
  • Linearity of the underlying model
  • Normality of forecast errors (correct)
  • Non-normality of forecast errors

    • What is the expression for the point forecast of x1t+hgiven the model parameters?

  • x1t+h|t = ϕ1 + ψ111 x1t + ψ121 x2t (correct)
  • x1t+h|t = ψ111 x1t + ψ121 x2t
  • x1t+h|t = ϕ1 - ψ111 x1t - ψ121 x2t
  • x1t+h|t = ϕ1 - ψ111 x1t + ψ121 x2t

    • What is the covariance between the forecast errors e1t+h and e2t+h?

  • Cov(e1t+h, e2t+h) ≠ 0 (correct)
  • Cov(e1t+h, e2t+h) = σ1h + σ2h
  • Cov(e1t+h, e2t+h) = σ1hσ2h
  • Cov(e1t+h, e2t+h) = 0

    • What is the expression for the forecast variance of x1t+h?

    <p>σ1t+h = E(e1t+h | Ωt) = σ1h</p> Signup and view all the answers

    What is the purpose of estimating the error covariance matrix in direct multi-step ahead forecasting?

    <p>To obtain interval forecasts</p> Signup and view all the answers

    What is the assumption underlying the expression for the forecast variance?

    <p>Normality of the forecast errors</p> Signup and view all the answers

    What is the role of the model parameters ψ111 and ψ121 in the direct multi-step ahead forecasting framework?

    <p>They are used to obtain point forecasts</p> Signup and view all the answers

    What is the advantage of using direct multi-step ahead forecasting compared to other forecasting methods?

    <p>It provides a more accurate estimate of the forecast variance</p> Signup and view all the answers

    What is a crucial assumption in forecasting with ARDL models?

    <p>That we have a complete knowledge of future realized values of x.</p> Signup and view all the answers

    How can we estimate forecast error variances in ARDL models?

    <p>By using historical data to estimate the error covariance matrix.</p> Signup and view all the answers

    What is a limitation of direct multi-step ahead forecasts in ARDL models?

    <p>They do not take into account the uncertainty of future realized values of x.</p> Signup and view all the answers

    How can we obtain forecasts in an ARDL model if we do not know future realized values of x?

    <p>By forecasting future realized values of x separately.</p> Signup and view all the answers

    What is a key difference between AR and ARDL models?

    <p>ARDL models incorporate an exogenous variable x.</p> Signup and view all the answers

    What is a characteristic of vector autoregression (VAR) models?

    <p>They model dynamic linkages between multiple economic variables.</p> Signup and view all the answers

    What is a common application of vector autoregression (VAR) models?

    <p>Analyzing the impact of monetary policy on economic variables.</p> Signup and view all the answers

    What is a key advantage of using vector autoregression (VAR) models?

    <p>They can model complex dynamic relationships between multiple variables.</p> Signup and view all the answers

    What is the formula for the point forecast of x1t+1 at time t?

    <p>x1t+1|t = α1 + π111 x1t + π121 x2t</p> Signup and view all the answers

    What is the formula for the forecast error of x2t+1 at time t?

    <p>e2t+1 = x2t+1 - x2t+1|t</p> Signup and view all the answers

    What is the formula for the forecast variance of x1t+1 at time t?

    <p>σ1t+1 = E(e1t+1|Ωt) = σ1</p> Signup and view all the answers

    What is the assumption required for obtaining interval forecasts?

    <p>Normality of forecast errors</p> Signup and view all the answers

    What is the formula for the point forecast of x1t+h at time t for a multi-step ahead forecast?

    <p>x1t+h|t = α1 + π111 x1t+h-1|t + π121 x2t+h-1|t</p> Signup and view all the answers

    What is the formula for the forecast error of x2t+h at time t for a multi-step ahead forecast?

    <p>e2t+h = π211 e1t+h-1 + π221 e2t+h-1 + ε2t+h</p> Signup and view all the answers

    What is the characteristic of the forecast variance for a multi-step ahead forecast?

    <p>It is a function of the error variances and covariances</p> Signup and view all the answers

    What is the method used to obtain interval forecasts?

    <p>Using the normal distribution</p> Signup and view all the answers

    What is a characteristic of economic variables?

    <p>They are often inter-related</p> Signup and view all the answers

    What type of model is suitable for modeling relationships between inter-related economic variables?

    <p>Autoregressive distributed lag model</p> Signup and view all the answers

    What criterion can be used to select the lag lengths in an ARDL model?

    <p>Akaike Information Criterion (AIC)</p> Signup and view all the answers

    What is the purpose of using an information criterion in an ARDL model?

    <p>To select the lag lengths</p> Signup and view all the answers

    What is an advantage of using an ARDL model over an autoregressive model?

    <p>It can account for both autoregressive and distributed lag components</p> Signup and view all the answers

    What is a potential issue with using an ARDL model?

    <p>It can be sensitive to the choice of lag lengths</p> Signup and view all the answers

    What is the goal of forecasting inter-related series?

    <p>To predict future values of the variables</p> Signup and view all the answers

    What is a type of forecast that can be generated using an ARDL model?

    <p>Multi-step ahead forecast</p> Signup and view all the answers

    Under what condition does X2 not Granger cause X1 in a bivariate VAR(p) model?

    <p>π121 = … = π12p = 0</p> Signup and view all the answers

    What is the Granger causality direction in the given example?

    <p>Unidirectional from crude oil prices to inflation</p> Signup and view all the answers

    What is the purpose of estimating the error covariance matrix in direct multi-step ahead forecasting?

    <p>To estimate the forecast variance</p> Signup and view all the answers

    What is the order of the VAR model used in the one-step ahead forecasting example?

    <p>VAR(1)</p> Signup and view all the answers

    What is the F statistic value when testing for Granger causality from inflation to crude oil prices?

    <p>1.98</p> Signup and view all the answers

    What is the advantage of using direct multi-step ahead forecasting compared to other forecasting methods?

    <p>It provides more accurate forecasts</p> Signup and view all the answers

    What is the purpose of computing the forecast variance in direct multi-step ahead forecasting?

    <p>To quantify the uncertainty of the forecasts</p> Signup and view all the answers

    What is the assumption underlying the expression for the forecast variance in direct multi-step ahead forecasting?

    <p>Normality of the error terms</p> Signup and view all the answers

    Study Notes

    Forecasting Oil Prices and Inflation

    • Direct multi-step ahead forecasts in a bivariate VAR model:
      • Estimated model: x1t = ϕ1 + ψ111 x1t−h + ψ121 x2t−h + υ1ht, x2t = ϕ2 + ψ211 x1t−h + ψ221 x2t−h + υ2ht
      • Forecast errors: e1t+h = υ1t+h, e2t+h = υ2t+h
      • Forecast variances: σ1h² = E(e1t+h |Ωt) = E(υ1t+h), σ2h² = E(e2t+h |Ωt) = E(υ2t+h)

    Direct Multi-Step Ahead Forecasts

    • Point forecasts: x1t+h|t = ϕ1 + ψ111 x1t + ψ121 x2t, x2t+h|t = ϕ2 + ψ211 x1t + ψ221 x2t
    • Forecast errors: e1t+h = υ1t+h, e2t+h = υ2t+h
    • Forecast variances: σ1h² = E(e1t+h |Ωt) = E(υ1t+h), σ2h² = E(e2t+h |Ωt) = E(υ2t+h)

    Forecasting with VAR Models: One-Step Ahead

    • Point forecasts: x1t+1|t = α1 + π111 x1t + π121 x2t, x2t+1|t = α2 + π211 x1t + π221 x2t
    • Forecast errors: e1t+1 = ε1t+1, e2t+1 = ε2t+1
    • Forecast variances: σ1² = E(e1t+1 |Ωt) = E(ε1t+1), σ2² = E(e2t+1 |Ωt) = E(ε2t+1)

    Forecasting with ARDL

    • Point forecasts: yt+1|t = α + β1 yt + β2 yt−1 + δ0 xt, yt+2|t = α + β1 yt+1|t + β2 yt + δ0 x^t+1 + δ1 xt + δ2 xt−1
    • Forecast errors: e1t+1 = ε1t+1, e2t+1 = ε2t+1
    • Forecast variances: σ1² = E(e1t+1 |Ωt) = E(ε1t+1), σ2² = E(e2t+1 |Ωt) = E(ε2t+1)

    Vector Autoregression

    • A system of equations modeling dynamic linkages between two (or more) economic variables
    • Forecasting inter-related series: economic variables are often inter-related, e.g., when incomes increase, people consume more

    Autoregressive Distributed Lag (ARDL)

    • Includes lagged values in the regression to model relationships between economic variables
    • Example: yt = α + β1 yt−1 + … + βp yt−p + δ0 xt + … + δq xt−q + εt

    A Relationship Between Oil Prices and Inflation

    • Example of an ARDL model: yt = α + β1 yt−1 + β2 yt−2 + δ0 xt + δ1 xt−1 + εt
    • Lag length selection using AIC and SIC: AIC suggests 4 lags, SIC suggests 2 lags

    Granger Causality

    • Testing in-sample Granger causality in a bivariate VAR(p) model
    • Example: Crude oil prices do Granger-cause 5-year expected inflation, but not vice versa

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    Forecasting Economics and Business PDF

    Description

    This quiz covers forecasting oil prices and inflation, including direct multi-step ahead forecasts with estimated models and error terms.

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