what univariate models tell us about multivariate macroeconomic models · what univariate models...

What univariate models tell us about multivariatemacroeconomic models

James Mitchell (Warwick), Donald Robertson (Cambridge) andStephen Wright (Birkbeck)

NBP Workshop on Forecasting 2016 (Warsaw, Poland)

21 November 2016

Mitchell, Robertson & Wright (NBP Workshop on Forecasting 2016 (Warsaw, Poland))Univariate forecasting 21 November 2016 1 / 25

Puzzle 1: The Predictive Puzzle

Multivariate macro models often fail to outpredict univariate models

Evidence from Nelson (1972, AER) through to Stock & Watson(2015)...

AR(MA) is a tough benchmark to beat

for inflation, exchange rates, stock prices, consumption...it’s challenging to find predictor variables that persistently predict


Puzzle 2: The Order Puzzle

The univariate models that compete with multivariate models inpredictive terms tend to be of low order

But the multivariate models commonly have large numbers of statevariables

which should imply high-order ARMA reduced forms (see Wallis, 1977Econometrica; Cubbada et al., 2009 Journal of Econometrics)


Our Aims

The paper offers an explanation for both Puzzles - in terms ofpopulation properties

we work backwards, observing just the univariate history of yt , datafor which are assumed to be generated by an (unknown) multivariatemacroeconomic model and show that

univariate properties can tightly constrain the properties of the (truebut unknown) multivariate macro model


Our key results - The predictive puzzle

1 We show how much better we could forecast yt if we conditioned onthe true state variables of the underlying multivariate model, ratherthan just its history, y t

The one-step-ahead predictive R2 must lie between bounds, R2min andR2max, that are strictly within [0, 1]

bounds derive from the ARMA parameters based on y t

results generalise to time-varying parameter models

the gap between R2min and R2max can be narrow for some yt processes

searching for reliable predictors is then a thankless task


Our key results - The order puzzle

2. If yt is ARMA(p, q) in population then (absent tight restrictions) the# unique predictors from the macro model, r = q

In finite samples, p and q may never be knowableSo to explain the order puzzleeither, the true multivariate model does have many state variables, butit’s close to satisfying parameter restrictions that imply (near)cancellation of AR and MA roots in the reduced form. Why?or, there are only a few distinct eigenvalues driving the macroeconomy- as in factor models


The true multivariate model

Let yt be generated by a multivariate macroeconomic model in then-vector of states zt

zt = Azt−1 +Bstyt = Czt−1 +Dst

Under (weak, and standard) Assumptions, ABCD implies a truepredictive regression for the first element of yt , where r ≤ n due to(possible) linear dependence between the states

yt = β′xt−1 + utxt(r×1)

= Mxt−1 + vt

where M =diag(µ1, ..., µr ) contains the distinct eigenvalues of A


Assumptions

A1 The autoregressive matrix A of the state variables zt can bediagonalised as A = T−1M∗T where M∗ is an n× n diagonal matrix, withfirst r diagonal elements M∗ii = µi , i = 1, ..., r , r ≤ n being the distincteigenvalues of AA2 |µi | < 1, i = 1, ..., rA3 st is an s × 1 vector of mutually orthogonal Gaussian IID processeswith E (sts′t ) = IsA4 DD′ has non-zero diagonal elements.


Lemma(The Macroeconomist’s ARMA) The true predictive regression and theprocess for the associated true predictor vector xt−1 together imply that ythas a unique fundamental ARMA(r , r) representation with parametersλ = (λ1, ...,λr ) and θ = (θ1, ..., θr )

λ (L) yt = θ (L) εt

where λ (L) ≡ ∏ri=1 (1− λiL) ≡ det (I −ML) and θ (L) ≡

∏ri=1 (1− θiL), |θi | ≤ 1, ∀i


The macroeconomist’s ARMA - a simple illustration

Consider an ABCD model with a single state variable zt and a 2× 1vector of structural shocks. This implies the predictive system

yt = βxt−1 + utxt = µxt−1 + vt

with A = µ, C = β, vt = Bst , ut = DstThis system is common in predictive return regressions

with yt some measure of returns or excess returns and xt somestationary valuation criterion


The macroeconomist’s ARMA(1,1) - a simple illustration

By substitution

(1− µL) yt = βvt−1 + (1− µL) ut

i.e. yt admits a fundamental ARMA(1,1) representation

(1− µL) yt = (1− ψL) εt

with |ψ| < 1, where ψ = ψ (A,B,C,D) is the solution to

−ψ

1+ ψ2=

µσ2u − βσuv

(1+ µ2) σ2u + β2σ2v + 2µβσuv

ψ−1 is also a solution

but this results in a nonfundamental representation, with a differentIID shock process, which cannot be recovered from the history y t (werevert to this representation in due course)


What the ARMA tells us about the true predictiveregression

We now work backwards, and ask: if we only observed the history ofyt , what would its univariate properties (as captured by λ and θ) tellus about the underlying multivariate model?

They do not, in general, tell us about the structural shocks stst ∝ εt only when n = 1 and the ARMA is fundamental

But the ARMA is informative about some properties of theunderlying multivariate model

1 The predictive power of the multivariate model, expressed in terms ofits one-step-ahead predictive R2, must lie between bounds definedw.r.t. the ARMA parameters


Bounds for the predictive R-sq

Under A1-A4, the predictive R2 (A,B,C,D) = 1− σ2u/σ2y of theregression derived from the underlying ABCD macro model

0 < R2min ≤ R2 ≤ R2max < 1

where R2min is the predictive R2 from the ARMA representation and so is a

function of λ and θ alone, and

R2max = R2min +

(1− R2min

)(1−

r

∏i=1

θ2i )

so R2max is also a function of λ and θ alone


The fundamental ARMA(1,1)

(1− λL) yt = (1− θL) εt

λ and θ are less than unity in modulus

εt are then derived from the history of yt

εt =

(1− λL1− θL

)yt =

∞

∑i=0

θi (yt−i − λyt−1−i )

andR2min = R

2F = 1− σ2ε /σ2y


The non-fundamental ARMA(1,1)

yt =

(1− θ−1L1− λL

)ηt

Now the (structural) shocks cannnot be derived from the history of yt .Instead

ηt =

(1− λL

1− θ−1L

)yt = −θF

(1− λL1− θF

)yt

= −∞

∑i=1

θi [yt+i − λyt+i−1]

where F = L−1

ηt is a linear combination of current and future values of yt


Bounds for the predictive R-sq: ARMA(1,1)

R2min = R2F =(θ − λ)2

1− λ2 + (θ − λ)2

Rmax = R2min +(1− R2min

) (1− θ2

)=

(1− λθ)2

1− λ2 + (θ − λ)2

If θ is close to λ, yt is close to white-noise, and R2min is close to 0

If θ is close to 0, R2max is close to 1

Iff θ and λ are both suffi ciently close to 0 (yt and xt are white-noise)the bound opens up to [0, 1]

As |θ| → 1, both R2min and R2max →

1−sgn(θ)λ2 : i.e. the R2 bound

collapses to a single point


Intuition

Lower bound: the predictions from the fundamental ARMA conditiononly on the history of yt

they cannot be worsened by increasing the information set to includethe true predictor

Upper bound: ηt has minimum innovation variance

ηt are a linear combination of the history and the future of yt

While we can improve our prediction of yt by using the true statevariable, there is a limit to this improvement

and this limit is determined by the univariate properties of yt


What the ARMA tells us about the dimensions of the truemultivariate model

Proposition(ARMA order and the ABCD Model) Let yt admit a minimalARMA(p, q) representation in population, with

q = r −#{θi = 0} −#{θi = λi 6= 0}p = r −#{λi = 0} −#{θi = λi 6= 0}

Under A1 to A3, the data yt must have been generated by a multivariatemodel in which, in the absence of exact restrictions over the parameters(A,B,C,D), A has r = q distinct eigenvalues. Thus there must be a truepredictive system for yt with r = q predictors


What the ARMA tells us about the dimensions of the truemultivariate model

CorollaryThe predictor vector xt in the true predictive regression can only haver > q elements if the parameters of the structural model (A,B,C,D)satisfy r − q exact restrictions that each ensure θi (A,B,C,D) = 0 forsome i, or θj (A,B,C,D) = λk (A,B,C,D), for some j and k .


Stock & Watson’s (2007) model of US inflation

Find little evidence that it’s possible to out-forecast a univariatemodel, particularly in recent data

Their preferred model lets the variances of the permanent andtransitory innovations vary over time: a UC-SV model

Their UC-SV is equivalent to a time-varying IMA(1,1) model ofinflation, yt = ∆πt

yt = (1− θtL) εt

where λ = 0

θt thus determines time-varying R2 predictive bounds

λ = 0 tells us that any good predictor (in composite) must be close toIID


Time-varying MA(1) parameter - median, 16.5% and83.5% quantiles of the posterior distribution


Time-varying R-sq bounds: posterior medians


Time-varying 16.5% and 83.5% quantiles for R-sq max -R-sq min


Implications of Stock & Watson’s (2007) model of inflation

The changing univariate properties of inflation noted by SW dictatethat even the true multivariate model would now struggle toout-predict a univariate model

The IMA(1,1) model implies that, in the absence of cancellation,there is a single IID predictor

No wonder macroeconomists struggle to find output gap measures thatforecast inflationMore promise in looking for “news” type predictorsWhile higher-dimensional multivariate models (with cancellation) canescape the R2 bounds, why should this be a feature of themacroeconomy? Smets and Wouters’(2007) DSGE model does notescape the bounds, as its predictions for inflation are too persistent


Conclusions

What we know about the univariate time-series properties of yt tellsus a lot about the properties and performance of the true (butunknown) multivariate predictive model

Helps us explain why it’s often hard to out-forecast a univariate model

And why researchers often struggle to find reliable predictors; e.g. whythe output gap struggles to forecast inflationMultivariate models with news variables and/or with few distincteigenvalues may fare better?


what univariate models tell us about multivariate macroeconomic models · what univariate models...

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