Forecasting Multiple Variables: VAR Systems, Predictive Causality, and Leading Indicators, Slides of Economics

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Forecasting Multiple Variables from

their own Histories

Multiple Variable AR Systems

 Vector Autoregression (VAR) » equations for several variables where each variable depends not only on its own history, but also the history of all the other variables. » multiple variable extension of an AR model

 In principle could specify multiple variable MA or multiple variable ARMA models » in practice such models are difficult to specify » typically low order VARs are adequate to approximate MA or ARMA processes

Predictive Causality

(Diebold p.303)

 Says that information in the history of one variable can be used to improve upon the forecasts of a second variable compared to just forecasting from the history of the second variable.

 Z has predictive causality for X if b 1 not equal to zero. X has predictive causality for Z if c 1 not equal to zero

VAR Processes: Higher Order

Systems

 Only one lag on X and Z appears in each of the above equations.

• systems with one lag are referred to as first order systems.
• higher order systems involve more than one lag in at least one of the variables

 Not necessary to limit the size of the forecasting problem to only two variables

• data limitations preclude systems with very large number of variables (say > 10)

How Long for the Lags? II

 One strategy:

• start with a longer lag
• check that autocorrelations of residuals are small (i.e. you’re not wasting information).
• shorten the lag and re-estimate » do AIC and/or SIC increase or decrease? (smaller is better!) » check on the stability of the estimated coefficients as the lag length is shortened

How Long for the Lags? III

 Generally are not going to need a lot of lags for seasonally adjusted data

• 3-4 lags for quarterly observations
• 5-7 lags for monthly observations

 For non-seasonally adjusted data, be careful about autocorrelations at seasonal frequencies

• may need short continuous lag, then another lag at seasonal frequency.

Leading Indicator

 When c 1 = 0.0 then the history of the X variable does not influence the future values of the Z variable (no predictive causality of X for Z)

 As long as b 1 not equal to zero, the history of Z has predictive value for future outcomes of X

• under these conditions we say that Z is a leading indicator of X.
• good or bad leading indicator depends on the size of b 1 and the variance of e1t

Testing for Leading Indicators

 Question of interest is whether all the coefficient on lagged values on one variable are zero in the regression in which some other variable is the dependent variable?

 F test can be used to examine the hypothesis that multiple regression coefficients are jointly equal to zero.

Leading Indicators

 Commerce (or Conference Board) Composite Leading Indicator viewed as a forecasting device for future expansions or recessions.

• autocorrelations are not 1.0; not a perfect forecasting device by any means
• frequently generates false signals of recessions
• accuracy somewhat improved (though not perfect by any means) by looking at average behavior over several months.

Industrial Production

Autocorrelations

Log of Industrial Production

-1.00 1 3 5 7 9 11 13 15 17 19

-0.

-0.

-0.

0.

0.

0.

0.

1.

Log Difference IP AR(1) Model

Dependent Variable DQIP - Estimation by Least Squares Monthly Data From 47:02 To 94: R Bar **2 0. Standard Error of Estimate 0. Durbin-Watson Statistic 2.

Variable Coeff Std Error T-Stat

---

1. Constant 0.0018 0.0004 4.
2. DQIP{1} 0.40 0.0385 10.

Differenced IP AR(1) Residuals

• -1.00 Log Difference IP AR(1)Residuals
• -0.
• -0.
• -0. - 0. - 0. - 0. - 0. - 1.

Forecasting from VAR Models

 One period ahead forecasts:

• multiply coefficients of model by most recently observed values of time series and add the terms up = forecast of next period value (tXt+1)

 Multiple period ahead forecasts:

• most recent data values are not available
• for (^) tXt+2 use predicted values, (^) tXt+1 for Xt-1, Xt for Xt-2, etc.
• for (^) tXt+3 use predicted values, (^) tXt+2 for Xt-1, tXt+1 for Xt-2, Xt for Xt-3, etc. docsity.com

Cointegration

 Suppose that you have several variables that are generated by unit root processes

• random walks with or without drift

 Suppose that such variables are “tied together” - there are linear combinations of the variables that are stationary

• such variables are said to be cointegrated