Transcript y 1,T , y 2,T y 1,T+1 , y 2,T+1 y 1,T+1 , Y 2,T+1 y 1,T+2 - Ka

Forecasting with Regression Models
Ka-fu Wong
University of Hong Kong
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Linear regression models
Endogenous
variable
Exogenous
variables
Explanatory
variables
Rule, rather than exception: all variables are endogenous.
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Conditional forecasting
The h-step ahead forecast of y given some assumed h-stepahead value of xT+h.
Assumed h-step-ahead value
of the exogenous variables
Call it scenario analysis or contingency analysis – based on some assumed
h-step-ahead value of the exogenous variables.
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Uncertainty of Forecast

Specification uncertainty / error:
our models are only approximation (since no one knows the truth).
E.g., we adopt an AR(1) model but the truth is AR(2).
 Almost impossible to account for via a forecast interval.

Parameter uncertainty / sampling error:
parameters are estimated from a data sample. The estimate will always
be different from the truth. The difference is called sampling error.
 Can account for via a forecast interval if we do the calculation
carefully.

Innovation uncertainty:
errors that cannot be avoided even if we know the true model and true
parameter. This is, unavoidable.
 Often account for via a forecast interval using standard softwares.
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Quantifying the innovation and parameter
uncertainty
Consider the very simple case in which x has a zero mean:
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Density forecast that accounts for parameter
uncertainty
~
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Interval forecasts that do not acknowledge
parameter uncertainty
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Interval forecasts that do acknowledge
parameter uncertainty
The closer xT+h* is closer to its mean, the smaller is the prediction-error variance.
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Unconditional Forecasting Models
Forecast based on some other
models of x, say, by assuming x
to follow an AR(1).
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h-step-ahead forecast without modeling x explicitly
Based on unconditional forecasting models
 Standing at time T, with observations, (x1,y1), (x2,y2),…,(xT,yT)
 1-step-ahead:
 yt = b0 + b1 xt-1 + et
 yT+1 = b0 + b1 xT + et
 2-step-ahead:
 yt = b0 + b1 xt-2 + et
 yT+2 = b0 + b1 xT + et
 …
 h-step-ahead:
 yt = b0 + b1 xt-h + et
 yT+h = b0 + b1 xT + et
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h-step-ahead forecast without modeling x explicitly
Based on unconditional forecasting models
 Special cases:
 The model contains only time trends and seasonal components.
 Because these components are perfectly predictable.
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Distributed Lags
y depends on a distributed lags of past x’s
Parameters to be estimated: b0, d1,…,dNx
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Polynomial Distributed Lags
Parameters to be estimated:
b0, a, b, c
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Rational Distributed Lags
Example:
A(L) = a0 + a1L
B(L) = b0 + b1L
b0 yt + b1 yt-1 = a0 xt + a1 xt-1 + b0 et + b1 et-1
yt = [- b1 yt-1 + a0xt + a1xt-1 + b0 et + b1 et-1]/b0
yt = [- b1/b0] yt-1 + [a0/b0] xt + [a1/b0] xt-1 + et + [b1/b0] et-1
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Regression model with AR(1) disturbance
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ARMA(p,q) models equivalent to model with only a
constant regressor and ARMA(p,q) disturbances.
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Transfer function models
A transfer function is a mathematical representation of the relation between the
input and output of a system.
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Vector Autoregressions, VAR(p)
allows cross-variable dynamics
VAR(1) of two variables.
The variable
vector consists
of two elements.
Regressors consist of
the variable vector
lagged one period only.
The innovations
allowed to be
correlated.
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Estimation of Vector Autoregressions
Run OLS regressions equation by equation.
OLS estimation turns out to have very good statistical properties
when each equation has the same regressors, as in standard VARs.
Otherwise, a more complicated estimation procedure called
seemingly unrelated regression, which explicitly accounts for
correlation across equation disturbances, would be need to obtain
estimates with good statistical properties.
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The choice order
Estimation of Vector Autoregressions
Use AIC and SIC.
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Forecast
Estimation of Vector Autoregressions
Given the parameters, or parameter estimates
y1,T, y2,T
y1,T+1, Y2,T+1
y1,T+1, y2,T+1
y1,T+2, Y2,T+2
y1,T+2, y2,T+2
y1,T+3, Y2,T+3
y1,T+3, y2,T+3
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Predictive Causality
 Two principles
 Cause should occur before effect.
 A causal series should contain information useful for forecasting
that is not available in the other series.
 Predictive Causality in a VAR
y2 does not cause y1 if φ12 =0
In a bivariate VAR, noncausality in 1-step-ahead forecast will
imply noncausality in h-step-ahead forecast.
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Predictive Causality
 In VAR with higher dimension, noncausality in 1-step-ahead
forecast need not imply noncausality in h-step-ahead forecast.
 Example:
Variable i may 1-step-cause variable j
Variable j may 1-step-cause variable k
Variable i 2-step-causes variable k but does not 1-step-cause
variable k.
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Impulse response functions
All univariate ARMA(p,q) processes can be written as:
We can always normalize the innovations with a constant m:
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Impulse response functions
Impact of et on yt:
1 unit increase in et’ is equivalent to one standard deviation increase in et.
1 unit increase in et’ has b0’ impact on yt
1 standard deviation increase in et has b0s impact on yt, b1s impact on yt, etc.
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AR(1)
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VAR(1)
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Normalizing the VAR by the Cholesky factor
If y1 is ordered first,
Example: y1 = GDP, y2 = Price level
An innovation to GDP has effects on current GDP and price level.
An innovation to price level has effects only on current price level but not current GDP.
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Features of Cholesky decomposition
 The innovations of the transformed system are in standard
deviation units.
 The current innovations in the normalized representation have can
non-unit coefficients.
 The first equation has only one current innovation, e1,t. The
second equation has both current innovations.
 The normalization yields a zero covariance between the innovations.
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Normalizing the VAR by the Cholesky factor
If y2 is ordered first,
Example: y1 = GDP, y2 = Price level
An innovation to price level has effects on current GDP and price level.
An innovation to GDP has effects only on current GDP but not current price level.
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Impulse response functions
 With bivariate autoregression, we can compute four sets of
impulse-response functions:
 y1 innovations (e1,t) on y1
 y1 innovations (e1,t) on y2
 y2 innovations (e2,t) on y1
 y2 innovations (e2,t) on y2
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Variance decomposition
 How much of the h-step-ahead forecast error variance of variable i
is explained by innovations to variable j, for h=1,2,…. ?
 With bivariate autoregression, we can compute four sets of
variance decomposition:
 y1 innovations (e1,t) on y1
 y1 innovations (e1,t) on y2
 y2 innovations (e2,t) on y1
 y2 innovations (e2,t) on y2
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Example:
y1 = Housing starts, y2= Housing completions
(1968:01 – 1996:06)
group fig112 starts comps
freeze(Figure112) fig112.line(d)
Observation #1: Seasonal pattern.
Observation #2: Highly cyclical with business cycles.
Observation #3: Completions lag starts.
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Correlogram and Ljung-Box Statistics of housing
starts (1968:01 to 1991:12)
freeze(Table112) starts.correl(24)
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Correlogram and Ljung-Box Statistics of housing
starts (1968:01 to 1991:12)
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Correlogram and Ljung-Box Statistics of housing
completions (1968:01 to 1991:12)
freeze(Table113) comps.correl(24)
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Correlogram and Ljung-Box Statistics of housing
starts (1968:01 to 1991:12)
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Starts and completions, sample crosscorrelations
freeze(Figure115) fig112.cross(24) starts comps
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VAR regression by OLS (1)
equation Table114.ls starts c starts(-1) starts(-2) starts(-3) starts(-4) comps(-1) comps(-2) comps(-3) comps(-4)
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VAR regression by OLS (1)
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VAR regression by OLS (1)
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VAR regression by OLS (1)
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VAR regression by OLS (2)
equation Table116.ls comps c starts(-1) starts(-2) starts(-3) starts(-4) comps(-1) comps(-2) comps(-3) comps(-4)
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VAR regression by OLS (2)
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VAR regression by OLS (2)
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VAR regression by OLS (2)
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Predictive causality test
group tbl108 comps starts
freeze(Table118) tbl108.cause(4)
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Impulse response functions
(response to one standard-deviation innovations)
var fig1110.ls 1 4 starts comps
freeze(Figure1110) fig1110.impulse(36,m)
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Variance decomposition
freeze(Figure1111) fig1110.decomp(36,m)
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Starts:
History, 1968:01-1991:12
Forecast, 1992:01-1996:06
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Starts:
History, 1968:01-1991:12
Forecast, 1992:01-1996:06
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Completions:
History, 1968:01-1991:12
Forecast, 1992:01-1996:06
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Completions:
History, 1968:01-1991:12
Forecast, and Realization, 1992:01-1996:06
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End
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