EC 827
Module 4
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
VAR Process: A Two Variable
Example
Two var iables: X t ; Z t
First Order VAR System:
X t  a 0  a1X t 1  b1Z t 1  e1t
Z t  c 0  c1X t 1  d1Z t 1  e 2 t
Special Case: b1  c1  0.0;
correlation of e1t , e 2 t  0.0
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
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?
 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 “overparameterized” and there is a loss of forecasting
efficiency
 AIC
and SIC again (MenuRATS VAR procudure
will compute their logs and print)
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 Indicators:
An Example
Two var iables: Xt ; Zt
First Order VAR System:
Xt  a 0  a1Xt 1  b1Zt 1  e1t
Zt  c0  c1Xt 1  d1Zt 1  e2t
Special Case:c1  0.0;
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
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
 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)
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
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
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 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
Differenced IP AR(1) Residuals
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
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
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.
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
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:
X t  a 0  a1X t 1  b1Z t 1  f1[gX t 1  hZ t 1 ]  e1t
Z t  c0  c1X t 1  d1Z t 1 f 2[gX t 1  hZ t 1 ]  e 2 t
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
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.
Predicted Change
Judging Forecasts I
10
5
Prediction-Realization Diagram
Turning Point
Errors
0
-5
-10
-10
Line of Perfect Forecast
-5
0
5
Actual Change
10
Judging Forecasts III
 Mean
Squared Error (MSE):
– obviously, smaller is better, again...
1 N
MSE   ( Pj  A j )2
N j1
Pj  Pr edicted Change
A j  Actual Change
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