Forecasting Multiple Variables Using Vector Autoregression (VAR) Models

1. EC 827 Module 4 Forecasting Multiple Variables from their own Histories

2. 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

3. VAR Process: A Two Variable Example

4. 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 b1 not equal to zero. X has predictive causality for Z if c1 not equal to zero

5. 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)

6. How Long for the Lags? • Long enough to get rid of any significant autocorrelation in the residuals of each equation (otherwise there is information to improve on the forecasts • Not so long that the model is “over-parameterized” and there is a loss of forecasting efficiency • AIC and SIC again (MenuRATS VAR procudure will compute their logs and print)

7. 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

8. 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.

9. Leading Indicators: An Example

10. Leading Indicator • When c1 = 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 b1 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 b1 and the variance of e1t

11. 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.

12. Leading Indicators • Until recently the U.S Department of Commerce published a leading indicator series • Recently “privatized” (or “outsourced”) to Conference Board (NY research operation) • not a single variable, but a weighted sum of 11 variables (a composite leading indicator) • alleged systematic autocorrelation between this composite variable and real output (real GDP)

13. 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.

14. Log of Industrial Production 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 1 3 5 7 9 11 13 15 17 19 Industrial Production Autocorrelations

15. Log Differences of Industrial Production 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 1 3 5 7 9 11 13 15 17 19 Log Change in IP Autocorrelations

16. Log Difference IP AR(1) Model Dependent Variable DQIP - Estimation by Least Squares Monthly Data From 47:02 To 94:12 R Bar **2 0.15 Standard Error of Estimate 0.00996 Durbin-Watson Statistic 2.07 Variable Coeff Std Error T-Stat ******************************************* 1. Constant 0.0018 0.0004 4.12 2. DQIP{1} 0.40 0.0385 10.27

17. Log Difference IP AR(1)Residuals 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 1 3 5 7 9 11 13 15 17 19 Differenced IP AR(1) Residuals

18. IP-Composite Leading Indicator Model Dependent Variable DQIP - Estimation by Least Squares Monthly Data From 47:02 To 94:12 R Bar **2 0.27 Standard Error of Estimate 0.0092941152 Durbin-Watson Statistic 1.967783 Variable Coeff Std Error T-Stat ******************************************** 1. Constant 0.0017 0.0004 4.19 2. DIND{1} 0.19 0.09 2.08 3. DIND{2} 0.47 0.09 5.18 4. DIND{3} 0.07 0.09 0.79 5. DIND{4} 0.25 0.09 2.92 6. DQIP{1} 0.20 0.04 4.66

19. 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.

20. 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

21. Cointegration and Vector Error Correction Models (VECM) • Variables that are cointegrated can be represented by a special kind of VAR - A Vector Error Correction Model • Two Variable VECM:

22. Cointegration and VECM’s • gXt-1 + hZt-1 is called the cointegrating vector (the linear combination of X and Z that is stationary) • f1 and f2 are called the error correction coefficients • if f1 and f2 are both equal to zero, then the VECM is just an ordinary VAR in first differences of X and Z

23. Cointegration and VECM’s • Advantage of VECM specification • VAR in differences ignores the information that the levels of the variables cannot wander aimlessly, but are tied together in the long run. • may be able to improve forecasts over intermediate to long-run over just VAR in differences.

24. Judging Forecasts I Prediction-Realization Diagram 10 Turning Point Errors 5 Predicted Change 0 -5 Line of Perfect Forecast -10 -10 -5 0 5 10 Actual Change

25. Judging Forecasts III • Mean Squared Error (MSE): • obviously, smaller is better, again...

26. Judging Forecasts IV • Theil Inequality Proportions (add to 1.0) • UM = Bias Proportion • large values are bad; indicates systematic differences in actual and average changes • US = Variance Proportion • large values indicate unequal variances of actual and predicted changes • UC = Covariance Proportion • zero = perfect correlation between actual and predicted changes