Ka-fu Wong University of Hong Kong

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Ka-fu WongUniversity of Hong Kong

Forecasting with Regression Models

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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-step-ahead 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 explicitlyBased on unconditional forecasting models

Standing at time T, with observations, (x1,y1), (x2,y2),…,(xT,yT)

1-step-ahead: yt = 0 + 1 xt-1 + t

yT+1 = 0 + 1 xT + t

2-step-ahead: yt = 0 + 1 xt-2 + t

yT+2 = 0 + 1 xT + t

… h-step-ahead:

yt = 0 + 1 xt-h + t

yT+h = 0 + 1 xT + t

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h-step-ahead forecast without modeling x explicitlyBased 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:0, 1,…,Nx

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Polynomial Distributed Lags

Parameters to be estimated:0, a, b, c

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Rational Distributed Lags

Example: A(L) = a0 + a1LB(L) = b0 + b1L

b0 yt + b1 yt-1 = a0 xt + a1 xt-1 + b0 t + b1 t-1

yt = [- b1 yt-1 + a0xt + a1xt-1 + b0 t + b1 t-1]/b0

yt = [- b1/b0] yt-1 + [a0/b0] xt + [a1/b0] xt-1 + t + [b1/b0] t-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

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

Given the parameters, or parameter estimates

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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 jVariable j may 1-step-cause variable kVariable 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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1 unit increase in t’ is equivalent to one standard deviation increase in t.

1 unit increase in t’ has b0’ impact on yt

1 standard deviation increase in t has b0impact on yt, b1 impact on yt, etc.

Impact of t on yt:

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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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With bivariate autoregression, we can compute four sets of impulse-response functions: y1 innovations (1,t) on y1

y1 innovations (1,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 (1,t) on y1




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Example:y1 = Housing starts, y2= Housing completions (1968:01 – 1996:06)

Observation #1: Seasonal pattern.

Observation #2: Highly cyclical with business cycles.

Observation #3: Completions lag starts.

group fig112 starts compsfreeze(Figure112) fig112.line(d)

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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 completions (1968:01 to 1991:12)

freeze(Table113) comps.correl(24)

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Starts and completions, sample cross-correlations

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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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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Predictive causality test

group tbl108 comps startsfreeze(Table118) tbl108.cause(4)

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Impulse response functions(response to one standard-deviation innovations)

var fig1110.ls 1 4 starts compsfreeze(Figure1110) fig1110.impulse(36,m)

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Variance decomposition

freeze(Figure1111) fig1110.decomp(36,m)

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Starts: History, 1968:01-1991:12Forecast, 1992:01-1996:06

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Starts: History, 1968:01-1991:12Forecast, 1992:01-1996:06

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Completions: History, 1968:01-1991:12Forecast, 1992:01-1996:06

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Completions: History, 1968:01-1991:12Forecast, and Realization, 1992:01-1996:06

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Ka-fu Wong University of Hong Kong