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 = Ï•1 + ψ111 x1t + ψ121 x2t (correct)
  • x1t+h|t = ψ111 x1t + ψ121 x2t
  • x1t+h|t = Ï•1 - ψ111 x1t - ψ121 x2t
  • x1t+h|t = Ï•1 - 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 (D)</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 (D)</p> Signup and view all the answers

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

<p>Normality of the forecast errors (C)</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 (B)</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 (B)</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. (A)</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. (D)</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. (C)</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. (D)</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. (A)</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. (C)</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. (C)</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. (D)</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 (C)</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 (B)</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 (D)</p> Signup and view all the answers

What is the assumption required for obtaining interval forecasts?

<p>Normality of forecast errors (C)</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 (D)</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 (A)</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 (B)</p> Signup and view all the answers

What is the method used to obtain interval forecasts?

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

What is a characteristic of economic variables?

<p>They are often inter-related (C)</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 (A)</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) (B)</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 (D)</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 (C)</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 (A)</p> Signup and view all the answers

What is the goal of forecasting inter-related series?

<p>To predict future values of the variables (D)</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 (A)</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 (D)</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 (D)</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 (A)</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) (C)</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 (A)</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 (A)</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 (B)</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 (C)</p> Signup and view all the answers

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Study Notes

Forecasting Oil Prices and Inflation

  • Direct multi-step ahead forecasts in a bivariate VAR model:
    • Estimated model: x1t = Ï•1 + ψ111 x1t−h + ψ121 x2t−h + Ï…1ht, x2t = Ï•2 + ψ211 x1t−h + ψ221 x2t−h + Ï…2ht
    • Forecast errors: e1t+h = Ï…1t+h, e2t+h = Ï…2t+h
    • Forecast variances: σ1h² = E(e1t+h |Ωt) = E(Ï…1t+h), σ2h² = E(e2t+h |Ωt) = E(Ï…2t+h)

Direct Multi-Step Ahead Forecasts

  • Point forecasts: x1t+h|t = Ï•1 + ψ111 x1t + ψ121 x2t, x2t+h|t = Ï•2 + ψ211 x1t + ψ221 x2t
  • Forecast errors: e1t+h = Ï…1t+h, e2t+h = Ï…2t+h
  • Forecast variances: σ1h² = E(e1t+h |Ωt) = E(Ï…1t+h), σ2h² = E(e2t+h |Ωt) = E(Ï…2t+h)

Forecasting with VAR Models: One-Step Ahead

  • Point forecasts: x1t+1|t = α1 + Ï€111 x1t + Ï€121 x2t, x2t+1|t = α2 + Ï€211 x1t + Ï€221 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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